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arXiv:physics/0001001v1 [physics.flu-dyn] 1 Jan 2000Under consideration for publication in J. Fluid Mech. 1
Capillary-gravity wave transport over
spatially random drift
By GUILLAUME BAL∗and TOM CHOU†
∗Department of Mathematics, University of Chicago, Chicago , IL 60637
†Department of Mathematics, Stanford University, Stanford , CA 94305
(Received 2 February 2008)
We derive transport equations for the propagation of water w ave action in the pres-
ence of a static, spatially random surface drift. Using the W igner distribution W(x,k,t)
to represent the envelope of the wave amplitude at position xcontained in waves with
wavevector k, we describe surface wave transport over static flows consis ting of two length
scales; one varying smoothly on the wavelength scale, the ot her varying on a scale com-
parable to the wavelength. The spatially rapidly varying bu t weak surface flows augment
the characteristic equations with scattering terms that ar e explicit functions of the cor-
relations of the random surface currents. These scattering terms depend parametrically
on the magnitudes and directions of the smoothly varying dri ft and are shown to give
rise to a Doppler coupled scattering mechanism. The Doppler interaction in the presence
of slowly varying drift modifies the scattering processes an d provides a mechanism for
coupling long wavelengths with short wavelengths. Conserv ation of wave action (CWA),
typically derived for slowly varying drift, is extended to s ystems with rapidly varying
flow. At yet larger propagation distances, we derive from the transport equations, an
equation for wave energy diffusion. The associated diffusion constant is also expressed
in terms of the surface flow correlations. Our results provid e a formal set of equations
to analyse transport of surface wave action, intensity, ene rgy, and wave scattering as a
function of the slowly varying drifts and the correlation fu nctions of the random, highly
oscillatory surface flows.
1. Introduction
Water wave dynamics are altered by interactions with spatia lly varying surface flows.
The surface flows modify the free surface boundary condition s that determine the dis-
persion for propagating water waves. The effect of smoothly v arying (compared to the
wavelength) currents have been analysed using ray theory (P eregrine (1976), Jonsson
(1990)) and the principle of conservation of wave action (CW A) (cf. Longuet-Higgins
& Stewart (1961), Mei (1979), White (1999), Whitham (1974) a nd references within).
These studies have largely focussed on long ocean gravity wa ves propagating over even
larger scale spatially varying drifts. Water waves can also scatter from regions of un-
derlying vorticity regions smaller than the wavelength Fab rikant & Raevsky (1994) and
Cerda & Lund (1993). Boundary conditions that vary on capill ary length scales, as well
as wave interactions with structures comparable to or small er than the wavelength can
also lead to wave scattering (Chou, Lucas & Stone (1995), Gou , Messiter & Schultz
(1993)), attenuation (Chou & Nelson (1994), Lee et al. (1993)), and Bragg reflections
(Chou (1998), Naciri & Mei (1988)). Nonetheless, water wave propagation over random2 Bal & Chou
static underlying currents that vary on both large and small length scales, and their
interactions, have received relatively less attention.
In this paper, we will only consider static irrotational cur rents, but derive the transport
equations for surface waves in the presence of underlying flo ws that vary on bothlong
and short (on the order of the wavelength) length scales. Rat her than computing wave
scattering from specific static flow configurations (Gerber ( 1993), Trulson & Mei (1993),
Fabrikant & Raevsky (1994)), we take a statistical approach by considering ensemble
averages over realisations of the static randomness. Differ ent statistical approaches have
been applied to wave propagation over a random depth (Elter & Molyneux (1972)), third
sound localization in superfluid Helium films (Kleinert (199 0)), and wave diffusion in the
presence of turbulent flows (Howe (1973), Rayevskiy (1983), Fannjiang & Ryzhik (1999)).
In the next section we derive the linearised capillary-grav ity wave equations to low-
est order in the irrotational surface flow. The fluid mechanic al boundary conditions are
reduced to two partial differential equations that couple th e surface height to velocity
potential at the free surface. We treat only the “high freque ncy” limit (Ryzhik, Papani-
colaou, & Keller (1996)) where wavelengths are much smaller than wave propagation dis-
tances under consideration. In Section 3, we introduce the W igner distribution W(x,k,t)
which represents the wave energy density and allows us to tre at surface currents that
vary simultaneously on two separated length scales. The dyn amical equations developed
in section 2 are then written in terms of an evolution equatio n forW. Upon expanding
Win powers of wavelength/propagation distance, we obtain tr ansport equations.
In Section 4, we present our main mathematical result, equat ion (4.1), an equation
describing the transport of surface wave action. Appendix A gives details of some of
the derivation. The transport equation includes advection by the slowly varying drift,
plus scattering terms that are functions of the correlation s of the rapidly varying drift,
representing water wave scattering. Upon simultaneously t reating both smoothly varying
and rapidly varying flows using a two-scale expansion, we find that scattering from rapidly
varying flows depends parametrically on the smoothly varyin g flows. In the Results and
Discussion, we discuss the regimes of validity, consider sp ecific forms for the correlation
functions, and detail the conditions for doppler coupling. CWA is extended to include
rapidly varying drift provided that the correlations of the drift satisfy certain constraints.
We also physically motivate the reason for considering two s cales for the underlying
drift. In the limit of still larger propagation distances, a fter multiple wave scattering,
wave propagation leaves the transport regime and becomes di ffusive (Sheng (1995)). A
diffusion equation for water wave energy is also given, with a n outline of its derivation
given in Appendix B.
2. Surface wave equations
Assume an underlying flow V(x,z)≡(U1(x,z),U2(x,z),Uz(x,z))≡(U(x,z),Uz(x,z)),
where the 1,2 components denote the two-dimensional in-pla ne directions. This static
flow may be generated by external, time independent sources s uch as wind or inter-
nal flows beneath the water surface. The surface deformation due to V(x,z) is denoted
¯η(x) where x≡(x,y) is the two-dimensional in-plane position vector. An addit ional
variation in height due to the velocity v(x,z) associated with surface waves is denoted
η(x,t). When all flows are irrotational, we can define their associa ted velocity poten-
tialsV(x,z)≡(∇x+ˆz∂z)Φ(x,z) andv(x,z,t)≡(∇x+ˆz∂z)ϕ(x,z,t). Incompressibility
requires
∆ϕ(x,z,t) +∂2
zϕ(x,z,t) = ∆Φ( x,z) +∂2
zΦ(x,z) = 0, (2.1)Water wave transport 3
+ε−y
|y|~1/k
x1x2
O(1)X
-+y
O(L)x
x
Figure 1. The relevant scales in water wave transport. Initially, the system size, observation
point, and length scale of the slowly varying drift is O(L), with surface wave wavelength and
scale of the random surface current of O(1). Upon rescaling, the system size becomes O(1), while
the wavelength and random flow variations are O(ε).
where ∆ = ∇2
xis the two-dimensional Laplacian. The kinematic condition (Whitham
(1974)) applied at z= ¯η(x) +η(x,t)≡ζ(x,t) is
∂tη(x,t) +U(x,ζ)· ∇xζ(x,t) =Uz(x,z=ζ) +∂zϕ(x,z=ζ,t). (2.2)
Upon expanding (2.2) to linear order in ηandϕabout the static free surface, the right
hand side becomes
Uz(x,ζ) +∂zϕ(x,ζ,t) =Uz(x,¯η) +η(x,t)∂zUz(x,¯η) +∂zϕ(x,¯η,t) +O(η2).(2.3)
At the static surface ¯ η,U(x,¯η)·∇x¯η(x) =Uz(x,¯η). Now assume that the underlying flow
is weak enough such that Uz(x,z≈0) and ¯ηare both small. A rigid surface approximation
is appropriate for small Froude numbers U2/c2
φ∼ |∇x¯η|2∼Uz(x,0)/|U(x,0)| ≪1 (cφ
is the surface wave phase velocity) when the free surface bou ndary conditions can be
approximately evaluated at z= 0 (Fabrikant & Raevsky (1994)). Although we have
assumedUz(x,z≈0) =∂zΦ(x,z≈0)≈0 and a vanishing static surface deformation
¯η(x)≈0,∇x·U(x,0) =−∂zUz(x,0)∝ne}ationslash= 0.
Combining the above approximations with the dynamic bounda ry conditions (derived
from balance of normal surface stresses at z= 0 (Whitham (1974))), we have the pair of
coupled equations
∂tη(x,t) +∇x·(U(x,z= 0)η(x,t)) = lim
z→0−∂zϕ(x,z,t)
lim
z→0−[ρ∂tϕ(x,z,t) +ρU(x,z)· ∇xϕ(x,z,t)] =σ∆η(x,t)−ρgη(x,t)(2.4)
whereσandgare the air-water surface tension and gravitational accele ration, respec-
tively. Although it is straightforward to expand to higher o rders in ¯η(x) andη(x,t), or to
include underlying vorticity, we will limit our study to equ ations (2.4) in order to make
the development of the transport equations more transparen t.
The typical system size, or distance of wave propagation sho wn in Fig. 1 is of O(L) with4 Bal & Chou
L≫1. Wavelengths however, are of O(1). To implement our high frequency (Ryzhik,
Papanicolaou, & Keller (1996)) asymptotic analyses, we res cale the system such that all
distances are measured in units of L≡ε−1. We eventually take the limit ε→0 as an
approximation for small, finite ε. Surface velocities, potentials, and height displacement s
are now functions of the new variables x→x/ε,z→z/εandt→t/ε. We shall further
nondimensionalise all distances in terms of the capillary l engthℓc=/radicalbig
σ/gρ. Time,
velocity potentials, and velocities are dimensionalised i n units of/radicalbig
ℓc/g,/radicalbig
gℓ3c, and√gℓc
respectively, e.g.for water,U= 1 corresponds to a surface drift velocity of ∼16.3cm/s.
SinceUz(x,z≈0)≈0, we define the flow at the surface by
U(x,z= 0)≡U(x) +√εδU(x/ε). (2.5)
In these rescaled coordinates, U(x) denotes surface flows varying on length scales of
O(1) much greater than a typical wavelength, while δU(x/ε) varies over lengths of O(ε)
comparable to a typical wavelength. The amplitude of the slowly varying flow U(x) is
O(ε0), while that of the rapidly varying flow δU(x/ε), is assumed to be of O(√ε). A more
detailed discussion of the physical motivation for conside ring the√εscaling is deferred
to the Results and Discussion. After rescaling, the boundar y conditions (2.4) evaluated
atz= 0 become
∂tη(x,t) +∇x·/bracketleftbig/parenleftbig
U(x) +√εδU(x/ε)/parenrightbig
η(x)/bracketrightbig
= lim
z→0−∂zϕ(x,0)
∂tϕ(x,t) +U(x)· ∇xϕ(x,t) +√εδU(x/ε)· ∇xϕ(x,t) =ε∆η(x,t)−ε−1η(x,t).
(2.6)
Although drift that varies slowly along one wavelength can b e treated with characteristics
and WKB theory, random flows varying on the wavelength scale r equire a statistical
approach. Without loss of generality, we choose δUto have zero mean and an isotropic
two-point correlation function ∝an}b∇acketle{tδUi(x)δUj(x′)∝an}b∇acket∇i}ht ≡Rij(|x−x′|), where (i,j) = (1,2) and
∝an}b∇acketle{t...∝an}b∇acket∇i}htdenotes an ensemble average over realisations of δU(x).
We now define the spatial Fourier decompositions for the dyna mical wave variables
ϕ(x,−h/lessorequalslantz/lessorequalslantζ,t) =/integraldisplay
qϕ(q,t)e−iq·xcoshq(h+z)
coshqh, η(x,t) =/integraldisplay
qη(q,t)e−iq·x,
(2.7)
the static surface flows
U(x) =/integraldisplay
qU(q)e−iq·x, δU/parenleftigx
ε/parenrightig
=/integraldisplay
qδU(q)e−iq·x/ε, (2.8)
and the correlations
Rij(x) =/integraldisplay
qRij(q)e−iq·x, (2.9)
where q= (q1,q2) is an in-plane two dimensional wavevector, q≡ |q|=/radicalbig
q2
1+q2
2, and/integraltext
q≡(2π)−2/integraltext
dq1dq2. The Fourier integrals for ηexclude q= 0 due to the incompress-
ibility constraint/integraltext
xη(x,t) = 0, while the q= 0 mode for ϕgives an irrelevant constant
shift to the velocity potential. Note that ϕin (2.7) manifestly satisfies (2.1). SubstitutingWater wave transport 5
(2.8) into the boundary conditions (2.4), we obtain,
∂tη(k,t)−i/integraldisplay
qη(k−q)U(q)·k−i√ε/integraldisplay
qη(k−q/ε)δU(q)·k=ϕ(k,t)ktanhεkh
∂tϕ(k,t)−i/integraldisplay
qU(q)·(k−q)ϕ(k−q)−i√ε/integraldisplay
qδU(q)·(k−q/ε)ϕ(k−q/ε)
=−(εk2+ε−1)η(k).
(2.10)
where theδU(q) are correlated according to
∝an}b∇acketle{tδUi(p)δUj(q)∝an}b∇acket∇i}ht=Rij(|p|)δ(p+q). (2.11)
Since the correlation Rij(x) is symmetric in i↔j, and depends only upon the magnitude
|x|,Rij(|p−q|) is real.
In the case where δU= 0 and U(x)≡U0is strictly uniform, equations (2.10) can
be simplified by assuming a e−iωtdependence for all dynamical variables. Uniform drift
yields the familiar capillary-gravity wave dispersion rel ation
ω(k) =/radicalbig
(k3+k)tanhkh+U0·k≡Ω(k) +U0·k. (2.12)
However, for what follows, we wish to derive transport equat ions for surface waves (action,
energy, intensity) in the presence of a spatially varying dr ift containing two length scales:
U=U(x) +√εδU(x/ε).
3. The Wigner distribution and asymptotic analyses
The intensity of the dynamical wave variables can be represe nted by the product of two
Green functions evaluated at positions x±εy/2. The difference in their evaluation points,
εy, resolves the waves of wavevector |k| ∼2π/(εy). Elter & Molyneux (1972) used this
representation to study shallow water wave propagation ove r a random bottom. However,
for the arbitrary depth surface wave problem, where the Gree n function is not simple,
and where two length scales are treated, it is convenient to u se the Fourier representation
of the Wigner distribution (Wigner (1932), G´ erard et al. (1 997), Ryzhik, Papanicolaou,
& Keller (1996)).
Defineψ= (ψ1,ψ2)≡(η(x),ϕ(x,z= 0)) and the Wigner distribution:
Wij(x,k,t)≡(2π)−2/integraldisplay
eik·yψi/parenleftig
x−εy
2,t/parenrightig
ψ∗
j/parenleftig
x+εy
2,t/parenrightig
dy (3.1)
where xis a central field point from which we consider two neighbouri ng points x±εy
2,
and their intervening wave field. Fourier transforming the xvariable using the definition
(2.7) we find,
Wij(p,k,t) = (2πε)−2ψi/parenleftbiggp
2−k
ε,t/parenrightbigg
ψ∗
j/parenleftbigg
−p
2−k
ε,t/parenrightbigg
. (3.2)
The total wave energy, comprising gravitational, kinetic, and surface tension contribu-6 Bal & Chou
tions is
E=1
2/integraldisplay
x/bracketleftbig
|∇xη(x)|2+|η(x)|2/bracketrightbig
+1
2/integraldisplay
x/integraldisplay0
−hdz|U(x,z) +ˆ zUz(x,z) +v(x,z)|2
−1
2/integraldisplay
x/integraldisplay0
−hdz|U(x,z) +ˆ zUz(x,z)|2
=1
2/integraldisplay
k(k2+ 1)|η(k)|2+ktanhkh|ϕ(k,z= 0)|2.(3.3)
The energy above has been expanded to an order in η(x,t) andϕ(x,z,t) consistent with
the approximations used to derive (2.4). In arriving at the l ast equality in (3.3), we have
integrated by parts, used the Fourier decompositions (2.7) and imposed an impenetrable
bottom condition at z=−h. The wave energy density carried by wavevector kis (G´ erard
et al. (1997))
E(k,t) =1
2Tr [A(k)W(k,t)], (3.4)
whereA11(k) =k2+1,A22(k) =ktanhkh,A 12=A21= 0. Thus, the Wigner distribution
epitomises the local surface wave energy density.
In the presence of slowly varying drift, we identify W(x,k,t) as the localWigner
distribution at position xrepresenting waves of wavevector k. The time evolution of its
Fourier transform W(p,k,t), can be derived by considering time evolution of the vector
fieldψimplied by the boundary conditions (2.4):
˙ψj(k,t) +iLjℓ(k)ψℓ(k,t) =i/integraldisplay
qU(q)·(k−qδj2)ψj(k−q,t)
+i√ε/integraltext
qδU(q)·(k−qδj2/ε)ψj(k−q/ε,t),(3.5)
where the operator L(k) is defined by
L(k) =
0i|k|tanhε|k|h
−i(εk2+ε−1) 0
. (3.6)
We have redefined the physical wavenumber to be k/εso thatk∼O(1). Upon using
(3.5) and the definition (3.2), (see Appendix A)
˙Wij(p,k,t) =iWiℓ(p,k,t)L†
ℓj/parenleftbiggk
ε+p
2/parenrightbigg
−iLiℓ/parenleftbiggk
ε−p
2/parenrightbigg
Wℓj(p,k,t)
+i/integraldisplay
qU(q)·/parenleftbigg
−k
ε+p
2−qδi2/parenrightbigg
Wij(p−q,k+εq/2,t)
−i/integraldisplay
qU(q)·/parenleftbigg
−k
ε−p
2+qδj2/parenrightbigg
Wij(p−q,k−εq/2,t)
+i√ε/integraldisplay
qδU(q)·/parenleftbigg
−k
ε+p
2−q
εδi2/parenrightbigg
Wij(p−q/ε,k+q/2,t)
−i√ε/integraldisplay
qδU(q)·/parenleftbigg
−k
ε−p
2+q
εδj2/parenrightbigg
Wij(p−q/ε,k−q/2,t),(3.7)
where only the index ℓ= 1,2 has been summed over. If we now assume that W(x,k,t)
can be expanded in functions that vary independently at the t wo relevant length scales,Water wave transport 7
functions of the field p(dual to x) can be replaced by functions of a slow variation in p
and a fast oscillation ξ/ε;p→p+ξ/ε.
This amounts to the Fourier equivalent of a two-scale expans ion in which xis replaced
byxandy=x/ε(Ryzhik, Papanicolaou, & Keller (1996)). The two new indepe ndent
wavevectors pandξare both of O(1). Expanding the Wigner distribution in powers of√εand using p→p+ξ/ε,
W(p,k,t)→W0(p,ξ,k,t) +√εW1/2(p,ξ,k,t) +εW1(p,ξ,k,t) +O(ε3/2),(3.8)
we expand each quantity appearing in (3.7) in powers of√εand equate like powers.
Upon expanding the off-diagonal operator L(−k/ε+p/2) =ε−1L0(k)+L1(k,p)+O(ε),
where
L0(k) =
0iktanhkh
−i(k2+ 1) 0
,L1(k,p)≡
0ip·kf(k)
ip·k 0
(3.9)
and
f(k)≡ −hk+ sinhkhcoshkh
2kcosh2kh. (3.10)
3.1.Orderε−1terms
The terms of O(ε−1) in (3.7) are
W0(p,ξ,k,t)L†
0(k+)−L0(k−)W0(p,ξ,k,t) = 0,k±≡k±ξ
2(3.11)
To solve (3.11), we use the eigenvalues and normalised eigen vectors for L0and its complex
adjoint L†
0,
τΩ(k)−iγ,bτ=
iτ/radicalbig
α(k)/2
1/radicalbig
2α(k)
;τΩ(k) +iγ,cτ=
iτ/radicalbig
2α(k)
/radicalbig
α(k)/2
,(3.12)
whereα(k)≡Ω(k)
k2+ 1,τ=±1, andiγ→0 is a small imaginary term. A W0(p,ξ,k,t)
that manifestly satisfies (3.12) can be constructed by expan ding in the basis of 2 ×2
matrices composed from the eigenvectors:
W0(p,ξ,k,t) =/summationdisplay
τ,τ′=±aττ′(p,k,t)bτ(k−)b†
τ′(k+).(3.13)
Right[left] multiplying (3.11) (using (3.13)) by the eigen vectors of the adjoint prob-
lem,cτ(k−)/bracketleftbig
c†
τ(k+)/bracketrightbig
, we find that a+−=a−+= 0, anda−−(x,k,t)≡a−(x,k,t) =
a++(x,−k,t)≡a+(x,−k,t). Furthermore, a+,a−∝ne}ationslash= 0 only ifξ= 0. Thus W0has the
form
W0(p,ξ,k,t) =W0(p,k,t)δ(ξ). (3.14)
From the definition of W0, we see that the (1,1) component of W0is the local envelop
of the ensemble averaged wave intensity |η(x,k,t)|2≃a+(x,k,t)α(k). Similarly, from the
energy (Eq. (3.4)), we see immediately that the local ensemb le averaged energy density
∝an}b∇acketle{tE(x,k,t)∝an}b∇acket∇i}ht=A11(k)α(k)∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht+A22(k)∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht
= Ω(k)∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht.(3.15)8 Bal & Chou
Therefore, since the starting dynamical equations are line ar, we can identify ∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}htas
the ensemble averaged local wave action associated with wav es of wavevector k(Henyey
et al. (1988)). The wave action ∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht, rather than the energy density ∝an}b∇acketle{tE(x,k,t)∝an}b∇acket∇i}ht
is the conserved quantity (Longuet-Higgins & Stewart (1961 ), Mei (1979), Whitham
(1974)).
The physical origin of γarises from causality, but can also be explicitly derived fr om
considerations of an infinitesimally small viscous dissipa tion (Chou, Lucas & Stone
(1995)). Although we have assumed γ→0, for our model to be valid, the viscosity
need only be small enough such that surface waves are not atte nuated before they have
a chance to multiply scatter and enter the transport or diffus ion regimes under consid-
eration. This constraint can be quantified by noting that in t he frequency domain, wave
dissipation is given by γ= 2νk2(Landau (1985)) where νis the kinematic viscosity and
cg(k)≡ |∇kΩ(k)| (3.16)
is the group velocity. The corresponding decay length k−1
d∼cg(k)/(νk2) must be greater
than the relevant wave propagation distance. Therefore, we require
ε2cg(k/ε)
νk2≫(1,ε−1) (3.17)
for (transport, diffusion) theories to be valid. The inequal ity (3.17) gives an upper bound
for the viscosity
νk2≪(εcg(k/ε),ε2cg(k/ε)) (3.18)
which is most easily satisfied in the shallow water wave regim e for transport. Otherwise
we must at least require ν <o(√ε). The upper bounds for ν(and hence γ) given above
provide one criterion for the validity of transport theory.
3.2.Orderε−1/2terms
Collecting terms in (3.7) of order ε−1/2, we obtain
W1/2(p,ξ,k,t)L†
0(k+)−L0(k−)W1/2(p,ξ,k,t) +/integraldisplay
qU(q)·ξW1/2(p−q,ξ−q,k,t)
−/integraldisplay
qδU(q)·k−W0(p,ξ−q,k+q/2,t) +/integraldisplay
qδU(q)·k+W0(p,ξ−q,k−q/2,t)
−/integraldisplay
qδU(q)·q[W0(p,ξ−q,k+q/2,t)S+SW0(p,ξ−q,k−q/2,t)] = 0
(3.19)
where S=/bracketleftbigg
0 0
0 1/bracketrightbigg
.
Similarly decomposing W1/2in the basis matrices composed of bτ(k−)b†
τ′(k+) (as in
3.13), substituting W0(p,0,k,t)δ(ξ) from (3.13 into (3.19), and inverse Fourier trans-
forming in the slow variable p, we obtain
W1/2(x,k,ξ,t) =/summationdisplay
τ,τ′=±δU(ξ)·Γτ,τ′(x,ξ,k,t)bτ(k−)b†
τ′(k+)
τ′Ω(k+)−τΩ(k−) +U(x)·ξ+ 2iγ, (3.20)Water wave transport 9
where
Γτ,τ′(x,ξ,k,t)≡k−aτ′(x,k+,t)c†
τ(k−)bτ′(k+)−k+aτ(x,k−,t)b†
τ(k−)cτ′(k+)
+ξ
2/summationdisplay
µ=±/bracketleftbig
aµ(x,k+,t)c†
τ(k−)bµ(k+) +aµ(x,k−,t)b†
µ(k−)cτ′(k+)/bracketrightbig
.
(3.21)
3.3.Orderε0terms
The terms of order ε0in (3.7) read
˙W0(p,k,t) =iW0(p,k,t)L†
1(−p)−iL1(p)W0(p,k,t)−i/integraldisplay
qk·U(q)q· ∇kW0(p−q,ξ,k,t)
+i/integraldisplay
qU(q)·pW0(p−q,ξ,k,t)−i/integraldisplay
qU(q)·q[SW0(p−q,ξ,k,t) +W0(p−q,ξ,k,t)S]
+i/integraldisplay
qδU(q)·k+W1/2(p,ξ−q,k−q/2,t)−i/integraldisplay
qδU(q)·k−W1/2(p,ξ−q,k+q/2,t)
−/integraldisplay
qδU(q)·q/bracketleftbig
SW1/2(p,ξ−q,k+q/2,t) +W1/2(p,ξ−q,k−q/2,t)S/bracketrightbig
+iW1L†
0−iL0W1+/integraldisplay
qU(q)·ξW1(p−q,ξ,k,t).
(3.22)
To obtain an equation for the statistical ensemble average ∝an}b∇acketle{ta+(x,k,t)∝an}b∇acket∇i}ht, we multiply
(3.22) by c†
+(k) on the left and by c+(k) on the right and substitute W1/2from equation
(3.20). We obtain a closed equation for a(x,k,t)≡ ∝an}b∇acketle{ta+(x,k,t)∝an}b∇acket∇i}ht(we henceforth suppress
the∝an}b∇acketle{t...∝an}b∇acket∇i}htnotation for a(x,k,t) andE(x,k,t)) by truncating terms containing W1. Clearly,
from (3.12), c†
+(k)(iW1L†
0−iL0W1)c+(k) = 0. Furthermore, we assume ∝an}b∇acketle{tξW1(p−
q,ξ,k,t)∝an}b∇acket∇i}ht ≈0 which follows from ergodicity of dynamical systems, and ha s been used in
the propagation of waves in random media (see Ryzhik, Papani colaou, & Keller (1996),
Bal et al. (1999)). The transport equations resulting from t his truncation are rigorously
justified in the scalar case (Spohn (1977), Erd¨ os & Yau (1998 )).
4. The surface wave transport equation
The main mathematical result of this paper, an evolution equ ation for the ensemble
averaged wave action a(x,k,t) follows from equation (3.22) above (cf. Appendix A) and
reads,
˙a(x,k,t) +∇kω(x,k)· ∇xa(x,k,t)− ∇xω(x,k)· ∇ka(x,k,t)
=−Σ(k)a(x,k,t) +/integraldisplay
qσ(q,k)a(x,q,t),(4.1)
where
ω(x,k) =/radicalbig
(k3+k)tanhkh+U(x)·k≡Ω(k) +U(x)·k. (4.2)
The left hand side in (4.1) corresponds to wave action propag ation in the absence of
random fluctuations. It is equivalent to the equations obtai ned by the ray theory, or
a WKB expansion (see section 5.1). The two terms on the right h and side of (4.1)
represent refraction, or “scattering” of wave action out of and into waves with wavevector10 Bal & Chou
krespectively. In deriving (4.1) we have inverse Fourier tra nsformed back to the slow field
point variable x, and used the relation ( α(k)−f(k)α−1(k))k≡ ∇kΩ(k). To obtain (4.1),
we assumed Rij(q)qi=Rij(q)qj= 0, which would always be valid for divergence-free
flows in two dimensions. Although the perturbation δUis not divergence-free in general,
∇·δU(x,z= 0) = −∂zδUz(x,0)∝ne}ationslash= 0, using symmetry considerations, we will show in
section 5.2 that Rij(q)qi=Rij(q)qj= 0.
Explicitly, the scattering rates are
Σ(k)≡2π/integraldisplay
qqiRij(q−k)kj/summationdisplay
τ=±b†
+(k)cτ(q)b†
τ(q)c+(k)δ(τω(x,τq)−ω(x,k))
σ(q,k)≡2π/summationdisplay
τ=±τqiRij(τq−k)kj|b†
τ(τq)c+(k)|2δ(τω(x,q)−ω(x,k))
(4.3)
where
b†
+(k)cτ(q)b†
τ(q)c+(k) =(τα(k) +α(q))(τα(q) +α(k))
4α(k)α(q)
|b†
τ(k)cτ′(q)|2=(τα(q) +α(k))2
4α(k)α(q).(4.4)
Physically, Σ( k) is a decay rate arising from scattering of action out of wave vector k.
The kernel σ(q,k) represents scattering of action from wavevector qintoaction with
wavevector k. Note that the slowly varying drift U(x) enters parametrically in the scat-
tering viaω(x,k) in theδ−function supports. The arguments ω(x,k) in theδ−functions
mean that we can consider the transport of waves of each fixed f requencyω0≡ω(x,k)
independently.
The typical distance travelled by a wave before it is signific antly redirected is defined
by the mean free path
ℓmfp=cg(k)
Σ(k)∼O(1). (4.5)
The mean free path described here carries a different interpr etation from that considered
in weakly nonlinear, or multiple scattering theories (Zakh arov, L’vov & Falkovich (1992))
where one treats a low density of scatterers. Rather than str ong, rare scatterings over
every distance ℓmfp∼O(1), we have considered constant, but weak interaction with
an extended, random flow field. Although here, each scatterin g isO(ε) and weak, over
a distance of O(1), approximately ε−1interactions arise, ultimately producing ℓmfp∼
O(1).
5. Results and Discussion
We have derived transport equations for water wave propagat ion interacting with
static, random surface flows containing two explicit length scales. We have further as-
sumed that the amplitude of δUscales asεβwithβ= 1/2: The random flows are
correspondingly weakened as the high frequency limit is tak en. Since scattering strength
is proportional to the power spectrum of the random flows and i s quadratic in δU, the
mean free path can be estimated heuristically by ℓmfp∼cg(k)/Σ(k)ε1−2β. Forβ >1/2,
the scattering is too weak and the mean free path diverges. In this limit, waves are nearly
freely propagating and can be described by the slowly varyin g flows alone, or WKB the-
ory. Ifβ <1/2,ℓmfp→0 and the scattering becomes so frequent that over a propagat ion
distance of O(1), the large number of scatterings lead to diffusive (cf. Se ction 5.4) be-Water wave transport 11
haviour (Sheng (1995)). Therefore, only random flows that ha ve the scaling β= 1/2
contribute to the wave transport regime.
We also note that β > 0 precludes any wave localisation phenomena. In a two-
dimensional random environment, the localisation length o ver which wave diffusion is
inhibited is approximately (Sheng (1995))
ℓloc∼ℓmfpexp/parenleftig
ε−1kℓmfp/parenrightig
∼ε1−2βexp/parenleftbig
ε−2β/parenrightbig
. (5.1)
As long as the random potential is scaled weaker ( β >0),ℓloc→ ∞, and strong lo-
calisation will not take hold. In the following subsections , we systematically discuss the
salient features of water wave transport contained in Eq. (4 .1) and derive wave diffusion
for propagation distances /greaterorsimilarO(1).
5.1.Slowly varying drift: U(x)∝ne}ationslash= 0,δU= 0
First consider the case where surface flows vary only on scale s much larger than the
longest wavelength 2 π/kconsidered, i.e.,δU= 0. The left-hand side in (4.1) represents
wave action transport over slowly varying drift and may desc ribe short wavelength modes
propagating over flows generated by underlying long ocean wa ves.
We first demonstrate that the nonscattering terms of the tran sport equation (4.1) is
equivalent to the results obtained by ray theory (WKB expans ion) and conservation of
wave action (CWA) (Longuet-Higgins & Stewart (1961), Mei (1 979), Peregrine (1976),
White (1999), Whitham (1974)). Assume the WKB expansion (Ke ller (1958), Bender &
Orszag (1978))
ηε=Aη(x,t)eiS(x,t)/εandϕε=Aϕ(x,t)eiS(x,t)/ε, (5.2)
with smoothly varying AηandAϕ. Upon using the above ansatz in (3.1) and setting
ε→0, we have a(x,k,t) =|A|2(x,t)δ(k− ∇xS(x,t)) where |A|2= 2α(k)|Aϕ|2=
2α−1(k)|Aη|2. Substitution of this expression for a(x,k,t) into (4.1), we obtain the fol-
lowing possible equations for S(x,t) and|A|2(x,t)
∂tS+ω(x,∇xS) = 0, (5.3)
∂t|A|2(x,t) +∇x·/parenleftig
|A|2∇kω(x,∇xS)/parenrightig
= 0. (5.4)
The first equation is the eikonal equation, while the second e quation is the wave action
amplitude equation. Recalling that |Aη|2=α(k)|A|2/2, we obtain the following transport
equation for the height amplitude:
∂t/parenleftig|Aη|2
α(∇xS)/parenrightig
+∇x·/parenleftig|Aη|2
α(∇xS)∇kω(x,∇xS)/parenrightig
= 0. (5.5)
Equation (5.5) is the same as Eq. (8) of White (1999), except t hat his ¯Ω is replaced here
withαdue to our inclusion of surface tension.
Wave action conservation can be understood by noting that
d
dta(X(t),K(t),t) = 0, (5.6)
where the characteristics ( X(t),K(t)) satisfy the Hamilton equations
dX(t)
dt=∇kω(X(t),K(t)), anddK(t)
dt=−∇xω(X(t),K(t)). (5.7)
The solutions to the ordinary differential equations (5.7) a re the characteristic curves
used to solve (5.3) and (5.4) (Courant & Hilbert (1962)).12 Bal & Chou
5.2.Correlation functions and conservation laws
We now consider the case where δU∝ne}ationslash= 0. The scattering rates defined by (4.3) depend
upon the precise form of the random flow correlation Rij. There are actually six additional
terms in (4.3) in the calculation of σand Σ, which vanish because
2/summationdisplay
j=1Rij(q)qj= 0 for i= 1,2. (5.8)
We prove relation (5.8) provided that δUz(k,kz) andδUz(k,−kz) have the same proba-
bility distribution. Thus,
∝an}b∇acketle{tδUi(p,pz)δUz(k,kz)∝an}b∇acket∇i}ht=∝an}b∇acketle{tδUi(p,pz)δUz(k,−kz)∝an}b∇acket∇i}ht (5.9)
This symmetry condition is reasonable, and is compatible wi th the divergence-free con-
dition forδUin three dimensions. We show that Hypothesis (5.9) implies ( 5.8) by first
using incompressibility/summationtext2
j=1δUj(k,kz)kj+δUz(k,kz)kz= 0:
2/summationdisplay
j=1δ(p+k)Rij(k)kj=2/summationdisplay
j=1∝an}b∇acketle{tδUi(p,0)δUj(k,0)kj∝an}b∇acket∇i}ht
=2/summationdisplay
j=1∝an}b∇acketle{tδUi(p,0)/integraldisplay
kzδUj(k,kz)kj∝an}b∇acket∇i}ht
=−/integraldisplay∞
−∞∝an}b∇acketle{tδUi(p,0)δUz(k,kz)∝an}b∇acket∇i}htkzdkz
= 0,
where the last equality follows from (5.9). Thus, (5.8) is ve rified, and (4.3) derived.
The formRij(|q|)qi=Rij(|q|)qj= 0, requires the correlation function to be transverse:
Rij(|q|) =R(q)/bracketleftbigg
δij−qiqj
q2/bracketrightbigg
, (5.10)
whereR(q) is a scalar function of q. The correlation kernels in the scattering integrals
can now be written as
qiRij(|τq−k|)kj=R(|τq−k|)/bracketleftbigg
q·k−q·(τq−k)k·(τq−k)
|τq−k|2/bracketrightbigg
=τR(|τq−k|)
|τq−k|2q2k2sin2θ(5.11)
whereθdenotes the angle between qandk. The scattering must also satisfy the support
of theδ-functions; for U(x) = 0 only |q|=|k|satisfy the the δ−function constraints. In
the presence of slowly varying drift, the evolution of a(x,|k| ∝ne}ationslash=|q|) can “doppler” couple
toa(x,q,t).
It is straightforward to show from the explicit expressions (4.3) that
Σ(k) =/integraldisplay
qσ(k,q). (5.12)
This relation indicates that the scattering operator on the right hand side of (4.1) is
conservative: Integrating (4.1) over the whole phase space yields
d
dt/integraldisplay
x/integraldisplay
ka(x,k,t) = 0. (5.13)Water wave transport 13
Figure 2. (a). Contour plot of ω(q). Each grayscale corresponds to a different constant value
ofω(q) =ω(k)≡ω0. (b). The band of qthat satisfies 0 .625< ω0<0.6625. Wavevectors qand
kthat lie in this band can couple a(x,k, t) toa(x,q, t) via wave scattering.
Equation (5.13) is the generalization of CWA to include scat tering of action from rapidly
varying random flows δU(x/ε). Although a(x,k,t) is conserved, the total water wave
energy E(x,k,t) = Ω( k)a(x,k,t) will not be conserved. For example, if U(x) is small
enough such that the δ−function in the σ(q,k) integral is triggered only when τ= +1,
d
dtE=d
dt/integraldisplay
x/integraldisplay
k[ω(x,k)−k·U(x)]a(x,k,t) =−d
dt/integraldisplay
x/integraldisplay
kk·U(x)a(x,k,t)∝ne}ationslash= 0.(5.14)
This nonconservation results from the energy that must be su pplied in order to sustain
the stationary underlying flow. For small U(x), the quantity ω(x,k)a(x,k,t) is conserved.
In that case, the evolution of ω(x,k)a(x,k,t) obeys an equation identical to (4.1). When
there is doppler coupling with τ=−1, an additional term arises and ω(x,k)a(x,k,t) is
no longer conserved under scattering.
5.3.Doppler coupled scattering
In addition to the correlation functions, the wave action sc attering terms involving Σ( k)
and integrals over σ(q,k) depend also on the support of the δ−function. Consider action
contained in water waves of fixed wavevector k. When U(x) = 0, only τ= +1 terms
contribute to the the integration over qas long as |q|=|k|. In this case, we can define
the angle q·k=k2cosθand reduce the cross-sections to single angular integrals o ver
qiRij(|q−k|)qj=R/parenleftbigg/vextendsingle/vextendsingle2ksinθ
2/vextendsingle/vextendsingle/parenrightbiggk2
4sin2θ
sin2θ
2, τ= +1. (5.15)
In this case ( U(x) = 0), assuming R(|q|) is monotonically decreasing, the most important
contribution to the scattering occurs when qandkare collinear.
When U(x)∝ne}ationslash= 0, andτ= +1, the sets of qwhich satisfy Ω( q) +U(x)·q= Ω(k) +
U(x)·k≡ω0trace out closed ellipse-like curves and are shown in the con tour plots
ofω(q) in Figure 2(a). The parameters used are U(x)·k1=−0.5k1andh=∞(the
−k1,−q1directions are defined by the direction of U(x)).
Each grayscale corresponds to a curve defined by fixed ω(k) =ω0. All wavevectors q
in each contour contribute to the integration in the express ions for Σ( k) andω(q,k).
Thus, slowly varying drift can induce an indirect doppler co upling between waves with14 Bal & Chou
different wavenumbers, with the most drastic coupling occur ring at the two far ends of
a particular oval curve. For example, in Figure 2(b), the dar k band denotes qsuch that
ω(q) =ω0when 0.625< ω0<0.6625. The wavevectors q≈(−0.3,0) and q≈(0.8,0)
are two of many that contribute to the scattering terms. Ther efore, the evolution of
a(x,k≈(−0.3,0),t) also depends on a(x,q≈(0.8,0),t) via the second term on the
right side of (4.1).
Provided U(x) is sufficiently large, the τ=−1 terms can also contribute to scatter-
ing. The dissipative scattering rate Σ( k)a(x,k,t) will change quantitatively since addi-
tional q’s will contribute to Σ( k). However, this decay process depends only on kand
is not coupled to a(x,|q| ∝ne}ationslash=|k|,t). Wavevectors qthat satisfy the δ−function in the
σ(q,k)a(x,q,t) term will, as when τ= +1, lead to indirect doppler coupling. This oc-
curs whenω(q) =−ω0and, as we shall see, allows doppler coupling of waves with mo re
widely varying wavelengths than compared to the τ= +1 case. Observe that if τ=−1
terms arise, the drift frame energy a(x,k,t)ω(x,k) is no longer conserved. Figure 3(a)
plotsω(q1,q2= 0) forU(x) = 1<√
2,U(x) =√
2, andU(x) = 1.6>√
2. Sinceω0
andω(q) are identical functions, −ω0can take on values below the upper dotted line
(ω0/lessorsimilar0.22 forU= 1.6). Therefore, coupling for τ=−1 andq2= 0 occurs for values of
−ω0between the dotted lines. Note that depending upon the value ofω0, coupling can
occur at two or four different points q= (q1,0). Figure 3(b) shows a contour plot of |ω(q)|
as a function of ( q1,q2). A level set lying between the dotted lines in ( a) will slice out two
bands; one band corresponds to all values of kthat couple to qlying in the associated
second band. The two bands determined by the interval 0 .414<−ω0<0.468 are shown
in Fig. 3(c). For any klying in the inner band of Fig. 3(c), all qlying in the outer band
will contribute to doppler coupling for τ=−1, and vice versa . As−ω0is increased, the
inner(outer) band decreases(increases) in size, with the c entral band vanishing when −ω0
approaches the upper dotted line in ( a) where the τ=−1 coupling evaporates. If −ω0
is decreased, the two bands merge, then disappear as −ω0reaches the lower limit. Fig.
3(d) is an expanded view of the two bands for small 0 .0756<−ω0<0.1368. Note that
a small island of qorkappears for very small wavevectors. The water wave scatteri ng
represented by σ(q,k) can therefore couple very long wavelength modes with very s hort
wavelength modes (the two larger bands to the right in Fig. 3( d)). However, the strength
of this coupling is still determined by the magnitude of qiRij(|q−k|)kj, which may be
small for |q−k|large.
The depth dependence of doppler coupling will be relevant wh enhq,hk /lessorsimilar1 whereq
andkare the magnitudes of the wavevectors of two doppler-couple d waves. For τ= +1,
finite depth reduces the ellipticity of the coupling bands, r esulting in weaker doppler
effects. Since the water wave phase velocity decreases with h, a finite depth will also
reduce the critical U(x) required for τ=−1 doppler coupling. For small U(x), it is clear
that theδ−functions associated with the τ=−1 terms inσ(q,k) are first triggered when
theqandkare antiparallel, U·k=−k|U|,U·q= +q|U|.
Figure 4(a) shows the phase velocity for various depths h. In order for τ=−1 to
contribute to scattering, U/greaterorequalslantcφ(k;h). ForU≈1.6, this condition holds in the h=∞case
for 0.5/lessorsimilark/lessorsimilar2 (the dashed region of cφ(k,∞)). Recall that our starting equations (2.4)
are valid only in the small Froude number limit. However, for water waves propagating
over infinite depth, τ=−1 coupling requires U >U min= min k{cφ(k)}, withcφ(kmin)≃
22cm/s. Therefore, in such “supersonic” cases, where τ=−1 is relevant, our treatment
is accurate only at wavevectors k∗such thatU≪cφ(k∗;h),e.g.,the thick solid portion
ofcφ(k;∞). ForU/greaterorsimilarUmin, theτ=−1 term can couple wavevectors q≈0≪kmin
withk≈2−3≫kmin. The rich τ=−1 doppler coupling displayed in Figures 3 isWater wave transport 15
Figure 3. Conditions for doppler coupling when τ=−1. (a). Plot of ω(q1, q2= 0;h=∞) for
U= 1,U=√
2, and U= 1.6. Only for U >√
2 does ω(q1, q2= 0;h=∞)<0. (b). Contour
plot of |ω(q)|. Each grayscale corresponds to a different constant value of ω(q) =ω(k)≡ −ω0.
(c). The bands of qsatisfying 0 .414<−ω0<0.468. (d). An expanded view of the coupling
bands for 0 .0756<−ω0<0.1368. Note that wavenumbers of very small modulus can couple
with wavenumbers of significantly larger modulus.
particular to water waves with a dispersion relation ω(q) that behaves as q3/2,U·q, or
q1/2depending on the wavelength. Doppler coupling in water wave propagation is very
different from that arising in acoustic wave propagation in a n incompressible, randomly
flowing fluid (Howe (1973), Fannjiang & Ryzhik (1999), Vedant ham & Hunter (1997))
whereω(q) =cs|q|. An additional doppler coupling analogous to the τ=−1 coupling
for water waves arises only for supersonic random flows when U(x)/greaterorequalslantcs, independent of
q. In such instances, compressibility effects must also be con sidered.
Figure 4(b) plots the minimum drift velocity Umin(h) whereτ=−1 doppler coupling
first occurs at any wavevector. The wavevector at which coupl ing first occurs is also
shown by the dashed curve. For shallow water, h≪√
3,Umin(h)∝√
hand very long
wavelengths couple first (small k(Umin)). For depths h >√
3 (∼3cm for water), the
minimum drift required quickly increases to U∗(∞) =√
2, while the initial coupling
occurs at increasing wavevectors until at infinite depth, wh ere the first wavevector to
doppler couple approaches k→1 (in water, this corresponds to wavelengths of ∼6.3cm).
The conditions for τ=−1 doppler coupling outlined in Figures 2 and 3 apply to both16 Bal & Chou
0 1 2 3 4 5
k0123 phase velocity cφ(k;h)
h=∞
h=2
h=1
h=0.5
0 1 2 3 4
h00.511.5Umin for τ=−1 coupling
k(Umin)
Umin1.021/2
(a).(b).
Figure 4. U > c φ(k) is required for τ=−1 coupling. (a). The phase velocity cφ(k) for various
depths h. The velocity shown by the solid horizontal line U≈1.6> cφ(k;h=∞) for 0 .5/lessorsimilark/lessorsimilar2.
(b). The minimum Umin(h) required for existence of τ=−1 coupling at any wavevector k, and
the wavevector k(Umin) at which this first happens.
Σ(k) andσ(q,k), with the proviso that qandkare parallel for Σ( k) and antiparallel
forσ(q,k). However, even when U <U minsuch that only τ= +1 applies, the set of q
corresponding to a constant value of ω(k) =ω0, traces out a noncircular curve. There is
doppler coupling between wavenumbers q∝ne}ationslash=kas long asU∝ne}ationslash= 0.
5.4.Surface wave diffusion
We now consider the radiative transfer equation (4.1) over p ropagation distances long
compared to the mean free path ℓmfp. Imposing an additional rescaling and measuring all
distances in terms of the mean free path, we introduce anothe r scalingǫ−1, proportional
to the number of mean free paths travelled. Since β= 1/2, transport of wave action
prevails when O(ε)<|x| ∼O(1), while diffusion holds when O(ǫ−1)∼ |x|<ℓloc.
Since waves of each frequency satisfy (4.1) independently, we consider the diffusion of
waves of constant frequency ω0. To derive the diffusion equation, we assume for simplicity
thatUis constant and small such that ω0+ω(x,q)∝ne}ationslash= 0 (theτ=−1 terms are never
triggered by the δ−functions). Expanding all quantities in the transport equa tion (4.1)
in powers of ǫ, we find
˙a0+¯U· ∇xa0− ∇x·D· ∇xa0= 0. (5.16)
The derivation of this equation is given in Appendix B. The di ffusion tensor Dis given
in (B 12) and is a function of the power spectrum Rij. The effective drift ¯Uis given by
(B 7):
¯U=/integraltext
k∇kω(k)δ(k·U+ Ω(k)−ω0)/integraltext
kδ(k·U+ Ω(k)−ω0). (5.17)
Up to a change of basis, we can assume that U=Ue1, whereU >0. Then the set of
points k·U+ Ω(k)−ω0= 0 is symmetric with respect to the x1−axis and ¯Uis parallel
toU. Also notice that the total energy given in (3.15) is asympto tically conserved in theWater wave transport 17
diffusive regime. Indeed, the total energy variations are gi ven by (5.14). Assuming that
all water waves have frequency ω0, we have in the diffusive regime
d
dtE=−d
dt/integraldisplay
x/integraldisplay
kk·U(x)a(x,k,t)
≈ −/parenleftig/integraldisplay
x˙a0(x,t)/parenrightig/integraldisplay
kk·Uδ(ω0−ω(k)),
sinceUis constant. Recasting the diffusion equation as ˙ a0=−∇x·(¯Ua0+D· ∇xa0),
we deduce that/integraldisplay
x˙a0(x,t) = 0,
which conserves the total energy E.
Now consider the simplified case U≡0,h=∞and Ω ∞(k) =√
k3+k. Since U=
¯U= 0, (5.15) holds and we have for all k,
/integraldisplay
qqiRij(|q−k|)qjq=0. (5.18)
We deduce that the corrector χin (B 8) is given by
χ(k) =−∇kΩ∞(k)
Σ(k)=−|∇kΩ∞(k)|
Σ(k)ˆk=−cg
Σ(k)ˆk,
where k=kˆk. The isotropic diffusion tensor Dis thus given by
D=1
Σ(k)Vω0/integraldisplay
q|∇qΩ∞(q)|2ˆqˆqTδ(Ω∞(q)−ω0) =c2
g(k)
2Σ(k)I, (5.19)
whereIis the 2 ×2 identity matrix. Thus, the diffusion equation for a0(x,t) assumes the
standard form (Sheng (1995))
˙a0−c2
g(k)
2Σ(k)∆a0= 0. (5.20)
6. Summary and Conclusions
In this paper, we have used the Wigner distribution to derive the transport equations
for water wave propagation over a spatially random drift com posed of a slowly varying
partU(x), and a rapidly varying part√εδU(x/ε). The slowly varying part determines
the characteristics on which the waves propagate. We recove r the standard result obtained
from WKB theory: conservation of wave action. Provided Rij(q)qj= 0, we extend CWA
to include wave scattering from correlations Rijof the rapidly varying random flow.
Evolution equations for the nonconserved wave intensity an d energy density can be read-
ily obtained from (4.1). Moreover, conservation of drift fr ame energy a(x,k,t)ω(x,k)
requires small U <U minand absence of τ=−1 contributions to scattering.
Explicit expressions for the scattering rates Σ( k) andσ(q,k) are given in Eqs. (4.3).
For fixedω(k), we find the set of qsuch that the δ−functions in (4.3) are supported.
This set of qindicates the wavevectors of the background surface flow tha t can me-
diate doppler coupling of the water waves. Although widely v arying wavenumbers can
doppler couple, supported by the δ−function constraints, particularly for τ=−1, the
correlation Rij(|q−k|) also decreases for large |q−k|. For long times, multiple weak scat-
tering nonetheless exchanges action among disparate waven umbers within the transport18 Bal & Chou
regime. Our collective results, including water wave actio n diffusion, provide a model for
describing linear ocean wave propagation over random flows o f different length scales.
The scattering terms in (4.1) also provide a means to correla te sea surface wave spectra
to statistics Rijof finer scale random flows.
Although many situations arise where the underlying flow is r otational (White (1999)),
the irrotational approximation used simplifies the treatme nt and allows a relatively simple
derivation of the transport and diffusive regimes of water wa ve propagation. The recent
extension by White (1999) of CWA to include rotational flows a lso suggests that an
explicit consideration of velocity and pressure can be used to generalise the present
study to include rotational random flows. Other feasible ext ensions include the analysis
of a time varying random flow, as well as separating the underl ying flows into static and
wave dynamic components.
The authors thank A. Balk, M. Moscoso, G. Papanicolaou, L. Ry zhik, and I. Smol-
yarenko for helpful comments and discussion. GB was support ed by AFOSR grant 49620-
98-1-0211 and NSF grant DMS-9709320. TC was supported by NSF grant DMS-9804780.
Appendix A. Derivation of the transport equation
Some of the steps in the derivation of (4.1) are outlined here . By taking the time
derivative of Wijin (3.2) and using the definition (3.5) for ˙ψ, we obtain
(2πε)2˙Wij(p,k,t) = (2πε)2iWiℓ(p,k)L∗
ℓj/parenleftbiggk
ε−p
2/parenrightbigg
−(2πε)2iLiℓ/parenleftbiggk
ε+p
2/parenrightbigg
Wℓj(p,k)
+i/integraldisplay
qU(q)·/parenleftbigg
−k
ε+p
2−qδi2/parenrightbigg
ψi/parenleftbigg
−k
ε+p
2−q/parenrightbigg
ψ∗
j/parenleftbigg
−k
ε−p
2/parenrightbigg
−i/integraldisplay
qU∗(q)·/parenleftbigg
−k
ε−p
2−qδj2/parenrightbigg
ψi/parenleftbigg
−k
ε+p
2/parenrightbigg
ψ∗
j/parenleftbigg
−k
ε−p
2−q/parenrightbigg
+i√ε/integraldisplay
qδU(q)·/parenleftbigg
−k
ε+p
2−q
εδi2/parenrightbigg
ψi/parenleftbigg
−k
ε+p
2−q
ε/parenrightbigg
ψ∗
j/parenleftbigg
−k
ε−p
2/parenrightbigg
−i√ε/integraldisplay
qδU∗(q)·/parenleftbigg
−k
ε−p
2−q
εδj2/parenrightbigg
ψi/parenleftbigg
−k
ε+p
2/parenrightbigg
ψ∗
j/parenleftbigg
−k
ε−p
2−q
ε/parenrightbigg
(A 1)
To rewrite the above expression as a function of Wijonly, we relabel appropriately,
e.g.,
−k
ε−p
2=−k′
ε−p′
2
−k
ε+p
2−q=−k′
ε+p′
2(A 2)
for the third term on the right hand side of (A1). Similarly re labelling for all relevant
terms yields the integral equation (3.7).
TheO(ε−1/2) terms of (3.7) determine W1/2. Decomposing
W1/2(p,ξ,k)≡/summationdisplay
τ,τ′=±a(1/2)
τ,τ′(p,ξ,k)bτ(k−)b†
τ′(k+) (A 3)
and substituting into (3.19) we find the coefficients a(1/2)
τ,τ′, where in this case a(1/2)
+−,a(1/2)
−+∝ne}ationslash=Water wave transport 19
0. Due to the nonlocal nature of the third term on the right of ( 3.19), we must first inverse
Fourier transform the slow wavevector variable back to x.
To extract the O(ε0) terms from (3.7) we need to expand Lto orderε0, theL1term.
Similarly, the terms W(p−q,ξ,k±εq/2) must be expanded:
W(p−q,ξ,k±εq/2) =W(p−q,ξ,k)±ε
2q· ∇kW(p−q,ξ,k) +O(ε2).(A 4)
Theεq·∇kW(p−q,ξ,k) terms combine with the −ε−1U(q)·k−+ε−1U(q)·k+terms
from the third and fourth terms in (3.7) to give the third term on the right of Eq. (3.22).
TheδU-dependent, order ε0terms (the sixth, seventh, and eighth terms on the right side
of (3.22)) come from collecting
±√εδU(q)·/parenleftbigg
−k
ε±ξ
2ε/parenrightbigg√εW1/2(p,ξ−q,k±q/2) (A 5)
from the last two terms in (3.7). The ensemble averaged time e volution of the Wigner
amplitudeaσ(x,k) can be succinctly written in the form:
˙a+(x,k,t)− ∇xω(x,k)· ∇ka+(x,k,t) +∇kω(x,k)· ∇xa+(x,k,t)
= Σ+,µ(k)aµ(x,k,t) +/integraldisplay
qσ+,µ(q,k)aµ(x,k,t).
(A 6)
Using the form for W0found from (3.11) in (3.19) to find W1/2, we substitute into
(3.22) to find (4.1), the transport equation for one of the dia gonal intensities of the Wigner
distribution. We have explicitly used eigenbasis orthonor mality b†
τ(k)·cτ′(k) =δτ,τ′and
the fact that a−(x,k,t) =a+(x,−k,t).
Appendix B. Derivation of the diffusion equation
The derivation of diffusion of water wave action is outlined b elow and follows the
established mathematical treatment of Larsen & Keller (197 4) and Dautray & Lions
(1993). For simplicity we assume that the flow Uis constant and small enough so that
for a considered range of frequencies, the relation ω(q) +ω(k) = 0 is never satisfied
for any kandq∝ne}ationslash=0. The diffusion approximation is valid after long times and la rge
distances of propagation X(see Fig. 1) such that the wave has multiply scattered and its
dynamics are determined by a random walk. We therefore resca le time and space as
˜t=t
ǫ2, ˜x=x
ǫ. (B 1)
The small parameter ǫin this further rescaling represents the transport mean fre e path
ℓmfpand not the wavelength as in the initial rescaling used to der ive the transport
equation. We drop the tilde symbol for convenience and rewri te the transport equation
in the new variables:
˙aǫ(x,k,t)+1
ǫ∇kω(k)·∇xaǫ(x.k,t) =1
ǫ2/integraldisplay
qQ(q,k)(aǫ(x,q,t)−aǫ(x,k,t))δ(ω(q)−ω(k)),
(B 2)
with obvious notation for Q(q,k). Since the frequency is fixed, the equation is posed
forksatisfyingω(k) =ω0. The transport equation assumes the form (B 2) because the
scattering operator is conservative. Since U∝ne}ationslash= 0, wave action is transported by the flow,
and diffusion takes place on top of advection. Therefore, we i ntroduce the main drift ¯U,20 Bal & Chou
which will be computed explicitly later, and define the drift -free unknown ˜ aǫ(x,k,t) as
˜aǫ(x,k,t) =aǫ(x+¯U
ǫt,k,t). (B 3)
It is easy to check that ˜ aǫsatisfies the same transport equation as aǫwhere the drift
term∇kωhas been replaced by ∇kω−¯U.
We now derive the limit of ˜ aǫasǫ→0. Consider the classical asymptotic expansion
˜aǫ= ˜a0+ǫ˜a1+ǫ2˜a2+.... (B 4)
Upon substitution into (B 2) and equating like powers of ǫ, we obtain at order ǫ−2, for
fixed frequency ω0,
/integraldisplay
qQ(q,k)(˜a0(x,q,t)−˜a0(x,k,t))δ(ω(q)−ω0) = 0. (B 5)
It follows from the Krein-Rutman theory (Dautray & Lions (19 93)) that ˜a0is independent
ofq. At orderǫ−1, we obtain
(∇kω(k)−¯U)· ∇x˜a0=/integraldisplay
qQ(q,k)(˜a1(x,q,t)−˜a1(x,k,t))δ(ω(q)−ω0). (B 6)
The compatibility condition for this equation to admit a sol ution requires both sides to
vanish upon integration over δ(ω(k)−ω0)dk. Therefore, ¯Usatisfies
¯U=1
Vω0/integraldisplay
k∇kω(k)δ(ω(k)−ω0), where Vω0=/integraldisplay
kδ(ω(k)−ω0).(B 7)
Once the constraint is satisfied, we deduce from Krein-Rutma n theory the existence of a
vector-valued mean zero corrector χsolving
(∇kω(k)−¯U) =/integraldisplay
qQ(q,k)(χ(q)−χ(k))δ(ω(q)−ω0)≡ Lχ. (B 8)
There is no general analytic expression for χ, which must in practice be solved numer-
ically. This is typical of problems where the domain of integ ration in qdoes not have
enough symmetries (cf. Allaire & Bal (1999), Bal (1999)). We now have
˜a1(x,k,t) =χ(k)· ∇x˜a0(x,t). (B 9)
It remains to consider O(ǫ0) in the asymptotic expansion. This yields
˙˜a0+ (∇kω(k)−¯U)· ∇x˜a1=L(˜a2). (B 10)
The compatibility condition, obtained by integrating both sides over k, yields the wave
action diffusion equation
˙˜a0− ∇x·D· ∇x˜a0= 0, (B 11)
where the diffusion tensor is given by
D=−1
Vω0/integraldisplay
k(∇kω(k)−¯U)χT(k) =−1
Vω0/integraldisplay
kL(χ)χT(k). (B 12)
The second form shows that Dis positive definite since Lis a nonpositive operator. The
formal asymptotic expansion can be justified rigorously usi ng the techniques in Dautray
& Lions (1993). As ǫ→0, we obtain that the error between ˜ aǫand ˜a0is at most of
orderǫ. Therefore, we have that aǫconverges to a0satisfying the following drift-diffusion
equation
˙a0+¯U
ǫ· ∇xa0− ∇x·D· ∇xa0= 0, (B 13)Water wave transport 21
with suitable initial conditions. Equation (B 13) is the coo rdinate-scaled version of (5.16).
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arXiv:physics/0001002v1 [physics.gen-ph] 3 Jan 2000We can know more in double-slit experiment
Gao Shan
Institute of Quantum Mechanics
11-10, NO.10 Building, YueTan XiJie DongLi, XiCheng Distri ct
Beijing 100045, P.R.China
E-mail: gaoshan.iqm@263.net
We show that we can know more than the orthodox view does, as on e example, we make a new
analysis about double-slit experiment, and demonstrate th at we can measure the objective state
of the particles passing through the two slits while not dest roying the interference pattern, the
measurement method is to use protective measurement.
Double-slit experiment has been widely discussed, and near ly all textbooks about quantum mechanics demonstrated
the weirdness of quantum world using it as one example, as Fey nman said, it contains all mysteries of quantum
mechanics, but have we disclosed these mysteries and unders tood the weirdness in double-slit experiment? as we
think, the answer is definitely No.
When discussing double-slit experiment, the most notoriou s question is which slit the particle passes through in
each experiment, it is just this problem that touches our sor e spots in understanding quantum mechanics, according
to the widely-accepted orthodox view, this question is actu ally meaningless, let’s see how it gets this bizarre answer, it
assumes that only an measurement can give an answer to the abo ve question, then detectors need to be put near both
slits to measure which slit the particle passes through, but when this is done the interference pattern will disappear,
thus the orthodox view asserts that the above question is mea ningless since we can not measure which slit the particle
passes through while not destroying the interference patte rn.
In fact, the above question is indeed meaningless, and at it h appens the orthodox answer is right, but its reason
is by no means right, the genuine reason is that if the particl e passes through only one slit in each experiment, the
interference pattern will not be formed at all∗, thus it is obviously wrong to ask which slit the particle pas ses through
in each experiment, it does not pass through a single slit at a ll!
On the other hand, we can still ask the following meaningful q uestion, namely how the particle passes through the
two slits to form the interference pattern? now as to this que stion, the deadly flaw of the orthodox view is clearly
unveiled, what is its answer? as we know, its answer will be th ere does not exist any objective motion picture of the
particle, the question is still meaningless, but how can it g et this conclusion? it can’t! and no one can.
Since we have known that the particle does not pass through a s ingle slit in each experiment, the direct position
measurement near both slits is obviously useless for finding the objective motion state of the particle passing through
the two slits, and it will also destroy the objective motion s tate of the particle, then the operational basis of the
orthodox view disappears, it also ruins, thus the orthodox d emonstrations can’t compel us to reject the objective
motion picture of the particle†, it only requires that the motion picture of classical conti nuous motion should be
rejected, this is undoubtedly right, since the motion of mic roscopic particle will be not classical continuous motion a t
all, it will be one kind of completely different motion.
Once the objective motion picture of the particle can’t be es sentially rejected, we can first have a look at it using
the logical microscope, since the particle does not pass thr ough a single slit in each experiment, it must pass through
both slits during passing through the two slits, it has no oth er choices! this kind of bizarre motion is not impossible
since it will take a period of time for the particle to pass thr ough the slits, no matter how short this time interval is,
so far as it is not zero, the particle can pass through both sli ts during this finite time interval, what it must do is just
discontinuously move, nobody can prevent it from moving in s uch a way! in fact, as we have demonstrated [4], this
is just the natural motion of particle.
∗Here we assume the only existence of particle, thus Bohm’s hi dden-variable theory [3] is not considered.
†Why we can’t detect which slit the particle passes through wh en not destroying the interference pattern is not because th ere
does not exist any objective motion picture of the particle, but because the particle does not pass through a single slit a t all.
1On the other hand, in order to find and confirm the objective mot ion picture of the particle passing through the
two slits, which will be very different from classical contin uous motion, we still need a new kind of measurement,
which will be very different from the position measurement, f ortunately it has been found several year ago [1,2], its
name is protective measurement, since we know the state of th e particle beforehand in double-slit experiment, we
can protectively measure the objective motion state of the p article when it passes through the two slits, while the
state of the particle will not be destroyed after such protec tive measurement, and the interference pattern will not
be destroyed either, thus by use of this kind of measurement w e can find the objective motion picture of the particle
passing through the two slits while not destroying the inter ference pattern, and the measurement results will reveal
that the particle indeed passes through both slits as we see u sing the logical microscope.
Now, the above analysis has strictly demonstrated that we ca n know more than the orthodox view does in double-
slit experiment, namely we know that the particle passes thr ough both slits to form the interference pattern, while
the orthodox view never knows this.
[1] Y.Aharonov, L.Vaidman, Phys.Lett.A 178, 38 (1993)
[2] Y.Aharonov, J.Anandan, and L.Vaidman, Phys.Rev.A. 47, 4616 (1993)
[3] D.Bohm, Phys.Rev. 85,166-193. (1952)
[4] Gao Shan, quant-ph/.
2 |
arXiv:physics/0001003v1 [physics.chem-ph] 2 Jan 2000Tentative statistical interpretation of
non-equilibrium entropy
Lajos Di´ osi∗
Research Institute for Particle and Nuclear Physics
H-1525 Budapest 114, POB 49, Hungary
January 1, 2000
Abstract
We suggest a certain statistical interpretation for the ent ropy pro-
duced in driven thermodynamic processes. The exponential f unction
of the irreversible entropy re-weights the probability of t he standard
Ornstein-Uhlenbeck-type thermodynamic fluctuations.
∗E-mail: diosi@rmki.kfki.hu
0In 1910 Einstein [1], paraphrasing Boltzmann’s lapidary fo rmula S=
logW, expressed the probability distribution of thermodynamic variables x
through the entropy function S(x):
W(x)∼eS(x). (1)
This equation describes thermodynamic fluctuations in Gaus sian approxima-
tion properly. Going beyond the stationary features, the ti me-dependence of
fluctuations xtcan be characterized by a certain probability functional W[x]
over complete paths {xt;t∈(−∞,∞)}. I suggest that, in driven thermody-
namic processes, this probability is related to the irrever sible entropy Sirr[x].
Symbolically, we can write the following relationship:
W[x]∼WOU[x−¯x]eSirr[x], (2)
where ¯ xtis the ‘driving’ value of parameter xtandWOU[z] turns out to
correspond to fluctuations ztof Ornstein-Uhlenbeck type. This relationship
offers Sirra certain statistical interpretation, somehow resembling Einstein’s
suggestion (1) for the equilibrium entropy S(x). In this short note, Ein-
stein’s approach to the thermodynamic fluctuations is outli ned and standard
equations of time-dependent fluctuations are invoked from i rreversible ther-
modynamics. Then I give a precise form to the relationship (2 ) for driven
thermodynamic processes.
The equilibrium conditions for isolated composite thermod ynamic sys-
tems derive from the maximum entropy principle:
S(x) =max , (3)
where S(x) is the total entropy of the system in function of certain fre e
thermodynamic parameters x[2]. If the function S(x) is maximum at x= ¯x
then ¯xis the equilibrium state. For example, xmay be the temperature
Tof a small (yet macroscopic) subsystem in the large isolated system of
temperature T= ¯x. Then, the function S(x) must be the total entropy of the
isolated system, depending on the variation of the subsyste m’s temperature
around its equilibrium value. The equilibrium value ¯ x[as well as S(x) itself]
may vary with the deliberate alteration of the initial condi tions. Surely,
in our example the temperature Tof the whole isolated system can always
be controlled at will. For later convenience, especially in treating driven
1thermodynamic processes, we may prefer the explicit detail ed notation S(x|¯x)
forS(x). Though S(¯x)−S(x) might qualify the lack of equilibrium, nearby
values x≈¯xhave no interpretation in phenomenological thermodynamic s.
They only have it in the broader context of statistical physi cs. In finite
thermodynamic systems there are fluctuations around the equ ilibrium state
¯xand their probability follows Eq. (1):
W(x|¯x)dx=NeS(x|¯x)−S(¯x|¯x)dx . (4)
Assume, for simplicity, that there is a single free variable x. The Taylor
expansion of the entropy function yields Gaussian fluctuati ons:
W(x|¯x) =1√
2πσ2exp/parenleftBig
−1
2σ2(x−¯x)2/parenrightBig
, (5)
where
1
σ2=−S′′(¯x)≡ −∂2S(x|¯x)
∂x2/vextendsingle/vextendsingle/vextendsingle
x=¯x. (6)
In our concrete example σ2=T2/Cwhere Cis the specific heat of the
subsystem.
We are going to regard the time-dependence of the parameter xtfluctu-
ating around ¯ x, according to the standard irreversible thermodynamics [2 ].
The time-dependent fluctuation zt≡xt−¯xis an Ornstein-Uhlenbeck (OU)
stochastic process [3] of zero mean /an}bracketle{tzt/an}bracketri}ht ≡0 and of correlation
/an}bracketle{tztzt′/an}bracketri}htOU=σ2e−λ|t−t′|. (7)
The relaxation rate λof fluctuations is related to the corresponding Onsager
kinetic constant γbyλ=γ/σ2. It can be shown that the probability dis-
tribution of xt=zt+ ¯xat any fixed time tis the Gaussian distribution (5)
as it must be. For the probability of the complete fluctuation pathzt, the
zero mean and correlation (7) are equivalent with the follow ing functional
measure:
WOU[z]Dz= exp/parenleftBig
−1
2γ/integraldisplay
(˙z2
t+λ2z2
t)dt/parenrightBig
Dz , (8)
where a possible constant of normalization has been absorbe d into the func-
tional measure Dz.
In order to construct and justify a relationship like (2) one needs to
proceed to driven thermodynamic processes. In fact, we assu me that we
2are varying the parameter ¯ xwith small but finite velocity. Formally, the
parameter ¯ xbecomes time-dependent. For simplicity’s sake we assume th at
the coefficients σ, γdo not depend on ¯ xor, at least, that we can ignore
their variation throughout the driven range of ¯ xt. We define the irreversible
entropy production during the driven process as follows:
Sirr[x|¯x] =1
σ2/integraldisplay
(¯xt−xt)dxt. (9)
In our concrete example dSirr= (C/T2)(T−T)dT≈dQ(T−1−T−1) which
is indeed the entropy produced randomly by the heat transfer dQfrom the
surrounding to the subsystem. By partial integration, Eq. ( 9) leads to an
alternative form:
Sirr[x|¯x] =1
σ2/integraldisplay
(¯xt−xt)d¯xt+1
σ2(x−∞−¯x−∞)2−1
σ2(x∞−¯x∞)2.(10)
In relevant driven processes the entropy production is macr oscopic, i.e.,
Sirr≫1 inkB-units, hence it is dominated by the integral term above.
I exploit this fact to replace expression (9) by
Sirr[x|¯x] =1
σ2/integraldisplay
(¯xt−xt)d¯xt (11)
which vanishes for constant ¯ x[4]. In the sense of the guess (2), I suggest the
following form for the probability distribution of the driv en path:
W[x|¯x] =N[¯x]WOU[x−¯x]eSirr[x|¯x]. (12)
The non-trivial normalizing pre-factor is a consequence of ¯x’s time-depen-
dence and will be derived below. Since the above distributio n is a Gaussian
functional and Sirr[x|¯x] is a linear functional (11) of x, we can easily calculate
the expectation value of the irreversible entropy:
Sirr[¯x]≡ /an}bracketle{tSirr[x|¯x]/an}bracketri}ht=1
2σ2/integraldisplay /integraldisplay
˙¯xt˙¯xt′e−λ|t−t′|dtdt′. (13)
In case of moderate accelerations ¨¯x≪λ˙¯x, this expression reduces to the
standard irreversible entropy γ−1/integraltext˙¯x2
tdtof the phenomenological theory of
driven processes [6]. Coming back to the normalizing factor in Eq. (12), we
3can relate it to the mean entropy production (13): N[¯x] =exp(−Sirr[¯x]).
Hence, the ultimate form of Eq. (12) will be:
W[x|¯x] =WOU[x−¯x]eSirr[x|¯x]−Sirr[¯x]. (14)
This result gives the precise meaning to our symbolic relati onship (2). If
the entropy production Sirrwere negligible then the thermodynamic fluctu-
ations xt−¯xtwould follow the OU statistics (7) like in case of a steady sta te
¯xt=const. Even in slow irreversibly driven processes Sirrmay grow essen-
tial and exp[Sirr] will re-weight the probability of OU fluctuations. The true
stochastic expectation value of an arbitrary functional F[x] can be expressed
by the OU expectation values of the re-weighted functional:
/an}bracketle{tF[x]/an}bracketri}ht=/angbracketleftBig
F[x]eSirr[x|¯x]−Sirr[¯x]/angbracketrightBig
OU. (15)
I can verify the plausibility of Eq. (14) for the special case of small ac-
celerations. Let us insert Eqs. (8,11) and also Eq. (13) whil e ignore ¨¯xin
comparison with λ˙¯x. We obtain:
W[x|¯x] =WOU[x−¯x+λ−1˙¯x]. (16)
Obviously, the fluctuations of the driven system are governe d by the OU
process zt(7) in the equilibrium case when ˙¯x≡0. In driven process, when
˙¯x/ne}ationslash= 0, there is only a simple change: The OU fluctuations happen a round
the retarded value ¯ xt−τ˙¯x≈¯xt−τof the driven parameter. The lag τis
equal to the thermodynamic relaxation time 1 /λ. Consequently, the driven
random path takes the following form:
xt= ¯xt−τ+zt, (17)
where ztis the equilibrium OU process (7). This result implies, in pa rticular,
the equation /an}bracketle{txt/an}bracketri}ht= ¯xt−τwhich is just the retardation effect well-known in the
thermodynamic theory of slightly irreversible driven proc esses. For example,
in case of an irreversible heating process the subsystem’s a verage temperature
will always be retarded by τ˙Twith respect to the controlling temperature
T[6].
Finally, let us summarize the basic features of Einstein’s f ormula (1) and
of the present proposal (2). They characterize via pretty co mpact formu-
lae the lack of equilibrium in steady and driven states, resp ectively. They
4do it in terms of thermodynamic entropies while they refer to a statistical
context lying outside both reversible and irreversible the rmodynamics. Both
formulae are only valid in the lowest non-trivial order and t heir correctness in
higher orders is questionable [7]. Contrary to their limite d validity, they can
no doubt give an insight into the role of thermodynamic entro py in statistical
fluctuations around both equilibrium or non-equilibrium st ates.
Acknowledgments. I thank Bjarne Andresen, Karl Heinz Hoffmann, At-
tila R´ acz, and Stan Sieniutycz for useful remarks regardin g the problem in
general. This work enjoyed the support of the EC Inco-Copern icus program
Carnet 2 .
References
[1] A. Einstein, Ann.Phys.(Leipzig) 33, 1275 (1910).
[2] L.D. Landau and E.M. Lifshitz, Statistical Physics (Clarendon, Oxford,
1982).
[3] G.E.Uhlenbeck and L.S.Ornstein, Phys.Rev. 36, 823 (1930).
[4] I am puzzled by the fact that one could retain the original definition (9)
forSirrif one used the well-known Onsager–Machlup functional [5]:
WOM[z]Dz= exp/parenleftBig
−1
2γ/integraldisplay
( ˙zt+λzt)2dt/parenrightBig
Dz ,
instead of (8). With this replacement all results would rema in, including
the central ones (14,15). Unfortunately the OM functional, unlike the
OU functional, can not be considered a resonable equilibriu m distribu-
tion since, e.g., it orders the same probability to the class of unbounded
fluctuations {zt+cexp(−λt);c∈(−∞,∞)}.
[5] L. Onsager and S. Machlup, Phys.Rev. 91, 1505 (1953); 91, 1512 (1953);
R.L. Stratonovitch, Sel.Transl.Math.Stat.Prob. 10, 273 (1971); R. Gra-
ham, Z.Phys. B26, 281 (1976).
[6] L.Di´ osi, Katalin Kulacsy, B.Luk´ acs and A.R´ acz, J.Ch em.Phys. 105,
11220 (1996).
5[7] Einstein’s ansatz fails obviously beyond the Gaussian a pproximation.
Our present proposal is first of all limited to small velociti es˙¯x. In fact,
the fluctuations of the thermodynamic parameters are govern ed by the
phenomenological Langevin equation (see, e.g., in [2]):
˙xt=−λ(xt−¯x) +/radicalBig
2γ wt
which can be generalized for time-dependent ¯ xt. To lowest order in ˙¯x
the result (17) comes out. In higher orders the Langevin equa tion gives
different results from the present proposal.
6 |
arXiv:physics/0001004v1 [physics.acc-ph] 3 Jan 2000SINGLE–PASS LASER POLARIZATION OF
ULTRARELATIVISTIC POSITRONS.
Alexander Potylitsyn
Tomsk Polytechnic University,
pr. Lenina 2A, Tomsk, 634050, Russia
e-mail: pap@phtd.tpu.edu.ru
Abstract
The new method for producing of the polarized relativistic p ositrons is suggested.
A beam of unpolarized positrons accelerated up to a few GeV ca n be polarized
during a head-on collision with an intense circularly polar ized lazer wave. After a
multiple Compton backscattering process the initial posit rons may lose a substantial
part of its energy and, as consequence, may acquire the signi ficant longitudinal
polarization. The simple formulas for the final positron ene rgy and polarization
degree depended on the laser flash parameters have been obtai ned. The comparison
of efficiences for the suggested technique and known ones is ca rried out. Some
advantages of the new method were shown.
The experiments with polarized electron-positron beams in future linear colliders will
furnish a means for studying a number of intriguing physical problems [1]. While the
problem of generation and acceleration of longitudinally p olarized electron beams seems
to be solved [2], the approach for production of polarized po sitron beams with the required
parameters has not been finally defined yet. In [3-7] methods w ere proposed for the
generation of longitudinally polarized positrons during e+e−– pair production by circularly
polarized photons with the energy ω∼101MeV, which are, in their turn, generated by
either passing electrons with the energy ∼102GeV through a helical undulator [3],
or through Compton backscattering of circularly polarized laser photons on a beam of
electrons with the energy ∼5 GeV [4,5], or through bremsstrahlung of longitudinally
polarized ∼50 MeV electrons [6,7]. To achieve the needed intensity of a p ositron source
(Ne+,pol∼1010particles/bunch) it is suggested to use an undulator of the l ength L >100
m [8], or to increase the laser power [9], or to use a high-curr ent accelerator of polarized
electrons [10].
The present paper considers an alternative way to approach t his problem.
A beam of unpolarized positrons from a conventional source b eing cooled in a damping
ring and preliminary accelerated to an energy E 0can be polarized during a head-on-
collision with a high-intensity circularly polarized lase r wave.
It is well known that during Compton backscattering of circu larly polarized laser pho-
tons on unpolarized positrons (electrons) with the energy E 0∼100 GeV the scattered
photon takes up to 90% of the initial positron energy while th e recoil positron aquires
∼100% longitudinal polarization [11,12]. At E 0≤10 GeV, however, the positron loses
too little of its energy during single Compton backscatteri ng (a few percent), and the lon-
gitudinal polarization of the recoil positron is, therefor e, of the same order of magnitude.
Current advances of laser physics make it possible to obtain parameters of laser flash such
1that the positron successively interacts with N ≫1 identical circularly polarized photons.
It is apparent that in this case the positron can lose a substa ntial fraction of its energy
(comparable with E 0). To evaluate the resulting polarization of the recoil posi tron, let us
consider multiple Compton backscattering in greater detai l.
Let us carry out calculations in a positron rest frame (PRF) a nd in a laboratory frame
(LF). Following [13], let us write the Compton scattering cr oss section in PRF where
spin correlations of three particles will be viewed– initia l photon, and initial and recoil
positrons (upon summation over the scattered photon polari zations):
dσ
dΩ= 2r2
0/parenleftBigk
k0/parenrightBig2/braceleftBig
Φ0+ Φ2(Pc,/vectorξ0) + Φ 2(Pc,/vectorξ) + Φ 2(/vectorξ0,/vectorξ) + Φ 3(Pc,/vectorξ0,/vectorξ)/bracerightBig
(1)
Here r 0is the electron classical radius; k 0, k are the initial and scattered photon energy;
Pc=±1 is the circular polarization of the initial photon; and /vectorξ0,/vectorξare the spin vectors
of the initial and final positrons. Functions Φ 0,Φ2,Φ3were obtained in paper [13].
In (1) and further in the paper use is made of the system of unit s ¯ h = m e= c = 1,
unless otherwise indicated.
Since the scattered photons are not detected, the cross sect ion (1) has to be integrated
over the photon outgoing angles. Due to azymuthal symmetry, it will depend on the
average longitudinal polarization components ξ0l, ξlsolely. On this basis we will keep
only these components which remain the same in LF.
For positrons with γ0≤104(γ0is the Lorentz-factor of the initial positron), the laser
photon energy in PRF ( ω0∼1 eV in LF ) will satisfy the relation
k0= 2γ0ω0≪1 (2)
Using (2) let us write the expression for the scattered photo n energy in PRF:
k =k0
1 + k 0(1−cosθ)≈k0[1−k0(1−cosθ)] (3)
Hereθis the polar angle of the scattered photon in PRF.
Leaving the terms not higher than k2
0, let us write in explicit form the Φ ifunctions
derived in [13] for electrons :
Φ0=1
8/bracketleftBig
1 + cos2θ+ k2
0(1−cosθ2)/bracketrightBig
,
Φ2(Pc, ξ0l) =−1
8Pcξ0lk0cosθ , (4)
Φ2(Pc, ξl) =−1
8Pcξl(1−cosθ)/bracketleftBig
2k0cosθ−k2
0(cosθ−cos2θ+ sin2θ)/bracketrightBig
,
Φ2(ξ0l, ξl) =1
8ξ0lξl[1 + cos2θ−k2
0cosθsin2θ],
Φ3(Pc, ξ0l, ξl) = 0.
2Upon routine integration we obtain:
σ=πr2
0
2/braceleftBig8
3(1−2k0) +4
3Pcξ0lk0(1−2k0) +ξl/bracketleftBig8
3ξ0l(1−2k0) +4
3Pck0/bracketrightBig/bracerightBig
(5)
It is obvious that in averaging with respect to the initial pa rticles spin and taking the
summation with respect to two spin states of the recoil posit ron, instead of (5) we get
Klein-Nishina’s cross section for k 0≪1 [11]:
σ=8
3πr2
0(1−2k0) (6)
From (5) follows that longitudinal polarization of a recoil positron (electron) is deter-
mined by both its initial polarization and the circular pola rization of a photon (later the
longitudinal polarization indices lwill be omitted):
ξ=ξ0∓k0
2Pc
1∓k0
2Pcξ0(7)
The upper (lower) sing refers to a positron (electron).
If the initial positron is unpolarized ( ξ0= 0), then upon a single interaction with a
laser photon the recoil positron becames polarized :
|ξ(1)|=|−k0
2Pc| ≪1. (8)
In order to consider the next scattering act, let us calculat e the average longitudinal
momentum <k/bardbl>along the initial photon direction and the average energy <k>of
the scattered photon in PRF using the same approximation as b efore:
<k/bardbl>=/integraltextk cosθ/parenleftBigk
k0/parenrightBig2Φ0dΩ
/integraltext/parenleftBigk
k0/parenrightBig2Φ0dΩ=6
5k2
0, (9)
<k>=/integraltextk/parenleftBig
k
k0/parenrightBig2Φ0dΩ
/integraltext/parenleftBigk
k0/parenrightBig2Φ0dΩ= k0(1−k0).
Thus, upon the first event of interaction, the photon in LF aqu ires, on average, the energy
< ωsc>=γ0(<k>−β0<k/bardbl>)≈γ0<k>=γ0k0. (10)
In (10) β0= 1−γ−2
0
2is the velocity of PRF with respect to LF.
3It is apparent that the recoil positron loses its energy (10) and hence
γ(1)=γ0−< ωsc>=γ0(1−k0) =γ0(1−2γ0ω0). (11)
In PRF before the second interaction the initial photon, in v iew of (11), will have a lower
energy
k(1)= 2γ(1)ω0= 2γ0ω0(1−2γ0ω0) = k 0(1−k0), (12)
and the recoil positron will have a polarization:
ξ(2)=ξ(1)−k(1)
2Pc
1−k(1)
2Pcξ(1). (13)
Substituting its value from (8) for ξ(1), we obtain:
ξ(2)=−Pck0
2+k(1)
2
1−Pck0
2k(1)
2,|ξ(2)|>|ξ(1)| (14)
It follows from (14) that as a result of multiple Compton back scattering the longitudinal
polarization of positrons builds up, while their energy dec reases in LF (so-called laser
cooling, see [14, 15]).
Let us write expressions relating the polarization and ener gy for two subsequent acts
of scattering:
γ(i+1)=γ(i)(1−2ω0γ(i)), (15)
ξ(i+1)=ξ(i)−γ(i)ω0Pc
1−γ(i)ω0PCξ(i). (16)
From these we can obtain the equations for the finite differenc es:
∆γ(i)=γ(i+1)−γ(i)= 2ω0γ2
(i), (17)
∆ξ(i)=ξ(i+1)−ξ(i)≈ −ω0Pcγ(i)(1−ξ2
(i)). (18)
When N ≫1, instead of (17) and (18) we can arrive at differential equat ions, whose
solution with proper initial conditions will yield
γ(N)=γ0
1 + 2γ0ω0N(19)
ξ(N)=γ0ω0N
1 +γ0ω0N. (20)
When deriving the above relation, there was taken the left ci rcular polarization P c=−1
for the sake of simplicity.
4Equations (19) and (20) describe the positron characterist ics after Ncollisions with
circularly polarized laser photons. The number of collisio ns N is controlled by the lumi-
nosity of the process L:
N =Nscat
N+e= N 0L = N 08
3πr2
0
2π(σ2
e++σ2
ph). (21)
In (21) N 0= A/ω0is the number of photons per laser flash, A is the laser energy, and
σph, σe+, are the laser focus and positron bunch radii. We can expect t hat after cooling in
the damping ring σe+≪σph. In this case, substituting (21) into (19) and (20), we obtai n
the following simple formulas for positron’s characterist ics:
γ(N)=γ0
1 + 2µ, (22)
ξ(N)=µ
1 +µ, (23)
which depend on the dimensionless parameter µsolely
µ=γ0ω0N =4
3A
mc2γ0/parenleftBigr0
σph/parenrightBig2. (24)
It follows from (24) that the µparameter depends linearly on the laser flash energy and
the initial positron energy, but it is inversely proportion al to the laser focus area and does
not depend on the interaction time (duration flash). Having w ritten (22) as:
γ0
γ(N)= 1 + 2 µ , (25)
we will compare the result with the estimate by V. Telnov [15] obtained in a classical
approximation. Substituting into (24) the estimate used in [15]σ2
ph=λ0le
8π(λ0is the
laser photon wavelength and l eis the positron bunch length), we get :
γ0
γ(N)= 1 +64
3A
mc2γ0πr2
0
λ0le. (26)
The resulting expression is closed to a similar one in [15] bu t the second term in (26)
is by a factor of πsmaller. This dicrepancy is connected with rough calculati on of the
luminosity (constant area of the laser focus) used in (21).
By way of illustration let us consider an example (see [15]):
γ0= 104,A = 5 J , λ0= 500 nm ,le= 0.2 mm, σ2
ph=λ0le/8π . (27)
In this case µ= 1.6 and, therefore, γ(N)≈0.3γ0;ξl≈60%.
5Thus, when a positron bunch interacts with a laser flash of the given parameters, all the
positrons acquire longitudinal polarization of about 60%. The change in the polarization
sign is obtained by inverting the sign of the circular polari zation of laser radiation.
It should be noted that with a proper selection of the sign of c ircular polarization,
the process of laser cooling would give rise to a longitudina l polarization increase of the
electrons rather than to depolarization of electrons beam ( as in the case of unpolarized
laser radiation considered in [15]).
Note that, generally speaking, the laser parameters (27) co rrespond to the so-called
”strong” field, when the contribution from non-linear Compt on scattering [4] would be
considerably high.
Non-linear processes, i.e., simultaneous scattering of a f ew laser photons on the moving
positron, are characterized by an increase in the effective p ositron mass in PRF, which,
in its turn, leads to a decrease in the Lorentz-factor and the energy transferred to the
positron through scattering. It is to be expected that the µparameter (24) for a fixed
value of the laser flash energy A will be sufficiently lower for a non-linear case as compared
to the linear one, and hence a lower attainable polarization (23).
In order to reach a linear mode of the Compton scattering proc ess, one has to stretch
the laser flash (the length of its interaction with the positr on bunch) (see, for instance,
[16]).
In conclusion, let us estimate the energy A +,polnecessary to obtain one polarized
positron with the energy E +>101MeV and the longitudinal polarization ξl>0,5 i.e.,
the parameters acceptable for consequent acceleration.
i) According to the estimates [8] an electron with the energy E−∼200 GeV passing
through a helical undulator of the length L ∼150 m can generate a number of circu-
larly polarized photons needed to obtain one polarized posi tron to be later accelerated
(conversion efficiency η= N e+,pol/Ne−≈1). Hence, A +,pol∼E−/η= 200 GeV.
ii) The author of the paper [9] considered a scheme for produc tion of ∼Ne+= 109
polarized positrons when the laser radiation of the total en ergy A Σ∼20 J is scattered on
an electron bunch with E −= 5 GeV and N e−= 1010e−/ bunch. Thus
A+,pol≈Ne−E−+ AΣ
Ne+≈170 GeV .
iii) In [6] the author estimated the conversion efficiency for longitudinally polarized
electrons with the energy E −= 50 MeV:
η≈10−3.
Therefore A +,pol∼E−/η= 50 GeV.
iv) For the method suggested in the present paper, evaluatio n of A +,polcan be made
for parameters of the unpolarized positron source used in SL AC [17].
The conversion efficiency for the electron energy E −= 33 GeV equals:
η≈1.
6Therefore, for a bunch with N e+= 1010and the positron energy E 0= 5 GeV interacting
with the laser flash (A = 5 J) we have:
A+,pol=E−
η+ E0+A
Ne+= 33 GeV + 5 GeV + 3 GeV ∼40 GeV .
Thus, the scheme proposed here seems to be most energy effecti ve.
The author is grateful to V. Telnov and J. Clendenin for stimu lating discussions.
References
1. P.M. Zerwas. Preprint DESY 94-001, 1994.
2. J.E. Clendenin, R. Alley, J. Frish, T. Kotseroglou, G.Mul hollan, D.Schultz, H.Tang,
J.Turner and A.D. Yeremian. The SLAC Polarized Electron Sou rce. AIP Conf.
Proceedings, N. 421, pp. 250-259, 1997.
3. V.E. Balakin, A.A.Mikhailichenko. Preprint INP 79-85, N ovosibirsk, 1979.
4. Yung Su Tsai. Phys.Rev.D, v.48 (1993), pp.96-115.
5. T. Okugi, Y. Kurihara, M. Chiba, A. Endo, R. Hamatsu, T. Hir ose, T. Kumita, T.
Omori, Y. Takeuchi, M. Yoshioka. Jap.J.Appl.Phys. v. 35(19 96), pp. 3677-3680.
6. A.P. Potylitsyn. Nucl. Instr. and Meth. v. A398 (1997), pp .395-398.
7. E.G. Bessonov, A.A. Mikhailichenko. Proc. of V European P article Accelerator
Conference, 1996, pp. 1516-1518.
8. K. Flottmann. Preprint DESY 93-161, 1993.
9. T. Omori. KEK- Proceedings 99-12, 1999, pp. 161-179.
10. T. Kotseroglou, V. Bharadwaj, J.E. Clendenin, S. Ecklun d, J. Frisch, P. Krejcik,
A.V. Kulikov, J. Liu, T. Maruyama, K.K. Millage, G. Mulholla n, W.R. Nelson,
D.C. Schultz, J.C. Sheppard, J. Turner, K. Van Bibber, K. Flo ttmann, Y. Namito.
Particle Accelerator Conference (PAC’99). Proceedings (t o be published).
11. H. Tolhoek. Rev. of Mod. Phys. v. 28 (1956), pp. 277-298.
12. G.I.Kotkin, S.I. Polityko, V.G. Serbo. Physics of Atomi c Nuclei, v. 59 (1996), pp.
2229-2234.
13. F.W.Lipps, H.A.Tolhoek. Physika, v. 20 (1954), pp. 395- 405.
14. P.Sprangle, E.Esarey. Phys. Fluids. v.B4 (1992), pp. 22 41-2248.
15. V. Telnov. Phys. Rev. Lett. v.78 (1997), pp.4757-4760.
716. I.V. Pogorelsky, I. Ben-Zvi, T. Hirose. BNL Report No.65 907, October, 1998.
17. S. Ecklund. SLAC-R-502 (1997), pp. 63-98.
8 |
arXiv:physics/0001005v1 [physics.class-ph] 3 Jan 2000An asymptotic form of the reciprocity theorem
with applications in x-ray scattering
Ariel Caticha
Department of Physics, University at Albany-SUNY,
Albany, NY 12222, USA.
ariel@cnsvax.albany.edu
Abstract
The emission of electromagnetic waves from a source within o r near
a non-trivial medium (with or without boundaries, crystall ine or amor-
phous, with inhomogeneities, absorption and so on) is somet imes studied
using the reciprocity principle. This is a variation of the m ethod of Green’s
functions. If one is only interested in the asymptotic radia tion fields the
generality of these methods may actually be a shortcoming: o btaining
expressions valid for the uninteresting near fields is not ju st a wasted ef-
fort but may be prohibitively difficult. In this work we obtain a modified
form the reciprocity principle which gives the asymptotic r adiation field
directly. The method may be used to obtain the radiation from a pre-
scribed source, and also to study scattering problems. To il lustrate the
power of the method we study a few pedagogical examples and th en, as a
more challenging application we tackle two related problem s. We calcu-
late the specular reflection of x rays by a rough surface and by a smoothly
graded surface taking polarization effects into account. In conventional
treatments of reflection x rays are treated as scalar waves, p olarization
effects are neglected. This is a good approximation at grazin g incidence
but becomes increasingly questionable for soft x rays and UV at higher
incidence angles.
PACs: 61.10.Dp, 61.10.Kw, 03.50.De
1 Introduction
The principle of reciprocity can be traced to Helmholtz in th e field of acoustics.
It states that everything else being equal the amplitude of a wave at a point
Adue to a source at point Bis equal to the amplitude at Bdue to a source
atA. With its extension to electromagnetic waves by Lorentz [1] and later to
quantum mechanical amplitudes [2], the applicability to al l sorts of fields was
made manifest. Nowadays the principle is regarded as a symme try of Green’s
functions when the source point and the field point are revers ed. This symmetry
is actually quite general. As shown in [3] the conditions of t ime-reversal invari-
1ance and hermiticity of the Hamiltonian are sufficient to guar antee reciprocity,
but they are not necessary; in fact, reciprocity holds even i n the presence of
complex absorbing potentials. In the case of electromagnet ic waves the only
requirement is that the material medium be linear and descri bed by symmetric
permittivity and permeability tensors [4][5]. This exclud es plasmas and ferrite
media in the presence of magnetic fields.
In the field of x-ray optics the principle was used by von Laue [ 6] to explain
the diffraction patterns generated by sources within the cry stal, the so-called
Kossel lines [7]. More recently there has been a widespread r ecognition that
these interference patterns contain information not just a bout intensities but
also about phases and can be thought of as holographic record s from which real
space images of the location of the internal sources can be re constructed. Thus,
under the modern name of ‘x-ray holography’ there has been a c onsiderable
revival of interest in this subject [8].
However, powerful as it is, the usual formulation of the reci procity principle
suffers from a rather serious drawback: it refers to the excha nge of source and
fieldpoints . As a consequence, a careful application of the principle re quires one
to consider the emission of spherical waves which in crystal line media or even
in the mere presence of plane boundaries, can be surprisingl y difficult (recall
e.g.studying the radiation by an antenna in the vicinity of the co nducting
surface of the Earth [9] or of layered media [10]). Furthermo re, one is typically
interested in the asymptotic radiation fields so the relevan t exchange should
involve a source point herewith a field point at infinity .
These technical difficulties have not deterred the users of re ciprocity from
using the principle to make valuable predictions, but a high price has been paid.
The required asymptotic limits are usually taken verbally and no accounts are
given of where and how spherical waves are replaced by plane w aves. Such
sleights of hand, because skillfully performed, have not le ad to wrong results,
but intensities are predicted only up to undetermined propo rtionality factors
and this excludes applications to classes of problems where absolute intensities
are needed. Moreover one is left with the uneasy feeling that the validity of
the predictions is justified mostly on the purely pragmatic g rounds that for the
problem at hand they seem to work which, again, limits applic ations to problems
that are already familiar.
The main goal of this paper (section 2) is to obtain a modified f orm of the
reciprocity theorem that gives the asymptotic radiation fie lds directly and that
accommodates plane waves and both point and extended source s in a natu-
ral way. Remarkably the resulting expressions, which inclu de all the relevant
proportionality factors and yield absolute, not just relat ive intensities, are very
simple.
For many problems the Asymptotic Reciprocity Theorem (ART) obtained
here represents an improvement not only over the usual form o f the reciprocity
theorem but also over the method of Green’s functions. Compu ting the Green’s
function requires solving a boundary value problems for sph erical waves in the
presence of plane boundaries and/or periodic media; this ma y well be an in-
tractably difficult problem. Furthermore, a considerable eff ort is wasted by first
2obtaining both near fields and far fields and then discarding t he uninteresting
near fields. The ART is a shortcut that discards the near fields before, rather
than after they are computed.
To illustrate the power of the method we consider several app lications. The
first three (section 3) are brief pedagogical examples of inc reasing complexity.
First the ART is used to calculate the fields radiated by an arb itrary prescribed
source in vacuum; next as an application to scattering probl ems we reproduce
the kinematical theory of diffraction by crystals. The third example, the radi-
ation by a current located near a plane dielectric boundary, is straightforward
when the ART is used but not if other methods are used. One must emphasize
that what is new in these examples are not the results, but the method; the first
two are standard textbook material, a special case of the thi rd is treated in [9].
As a more involved application of the ART, in section 4 we comb ine ideas from
the three previous examples to study two other related scatt ering problems, the
specular reflection of polarized x rays by a rough surface and by a continuously
graded surface.
The technique of the grazing-incidence reflection of x-rays has received con-
siderable attention [11]-[17] from both the theoretical an d the experimental sides
as a means to obtain structural information about surfaces. The effect of sur-
face roughness on the reflection is taken into account by mult iplying the Fresnel
reflectivity of an ideal sharp and planar surface by a “static Debye-Waller” fac-
tor. The problem is to calculate this corrective factor. The calculation has
been carried out in several different approximations. The Ra yleigh or Born ap-
proximation [11] is satisfactory for rough surfaces with lo ng lateral correlation
lengths but for x-rays the situations of interest generally involve short lateral
correlation lengths. Here other approximations such as the distorted-wave Born
approximation [13][14] and the Nevot-Croce approximation [15] are used. For
variations and interpolations between these two methods se e [16], and for a gen-
eralization to surfaces with non-Gaussian roughness and to graded interfaces of
arbitrary profile see [17]. In these treatments ([14] is an ex ception) the x rays
are treated as scalar waves. One expects this approximation to hold at graz-
ing incidence but at higher incidence angles ( e.g., for soft x rays) its validity
becomes increasingly questionable. Using a modified first Bo rn approximation
Dietrich and Haase [14] took the vector character of the x ray s into account but
they point out that the validity of their approximation is no t in general easy to
assess and they restrict themselves to studying special int erface profiles.
In section 4 we study this problem using a different approxima tion; we use
the ART to develop approximations of the Nevot-Croce type [1 7]. There is, of
course, a trivial polarization dependence that is already d escribed by Fresnel
formulas for the reflectivity of the ideal flat step surface. T he question we
address here is whether the “static Debye-Waller” factor sh ows any additional
dependence on polarization. The final result is remarkably s imple: the “static
Debye-Waller” factor for the specularly reflected vector wa ves is the same for
both polarizations and coincides with that for scalar waves . Finally, some brief
concluding remarks are collected in section 5.
3Figure 1: (a) In the usual form of the reciprocity theorem the surface Sencloses
the medium, and all sources. (b) For the asymptotic form of th e reciprocity
theorem the connecting field /vectorEcis a radiation field, its source lies outside the
surface S=S++S′.
2 The reciprocity theorem and its asymptotic
form
We wish to calculate the asymptotic radiation fields /vectorEand/vectorHgenerated by a
prescribed current /vectorJ(t,/vector r) located near or within a linear medium,
Di=εijEjand Bi=µijHj.
We will assume that the tensors εij(/vector r) and µij(/vector r) are symmetric, but otherwise
the situation remains quite general, the medium may have an i rregular shape,
or be inhomogeneous, crystalline or amorphous, absorbing, dispersive, etc.
As in the usual deduction of the reciprocity theorem (see e.g., [4]), we con-
sider a second set of fields /vectorEcand/vectorHc, which we will call the “connecting fields”,
generated by a source /vectorJc(see fig.1a). For simplicity we will also assume that all
fields and sources are monochromatic /vectorE=/vectorE(/vector r)e−iωt,/vectorJ=/vectorJ(/vector r)e−iωt, etc. For
linear media this is not a restriction.
From Maxwell’s equations
∇ ×/vectorE=iK/vectorB and ∇ ×/vectorH=−iK/vectorD+4π
c/vectorJ, (1)
where K≡ω/c, one easily obtains the following identity
∇ ·/parenleftBig
/vectorE×/vectorHc−/vectorEc×/vectorH/parenrightBig
=4π
c/parenleftBig
/vectorEc·/vectorJ−/vectorE·/vectorJc/parenrightBig
,
4which, on integrating over a large volume Vbounded by the surface S, can be
rewritten as
/integraldisplay
S/parenleftBig
/vectorE×/vectorHc−/vectorEc×/vectorH/parenrightBig
·d/vector s=4π
c/integraldisplay
V/parenleftBig
/vectorEc·/vectorJ−/vectorE·/vectorJc/parenrightBig
dv. (2)
This expression simplifies if one deals with point sources. F or example, consider
oscillating point dipoles /vector poe−iωtand/vector pce−iωt, located at /vector roand/vector rcrespectively.
The current density /vectorJis given by /vectorJ=−iω/vector poδ(/vector r−/vector ro)e−iωtand/vectorJcis given by
an analogous expression. Further simplification is achieve d if one assumes that
the surface Sis so remote that the surface integral is negligibly small [1 8], then
/vectorEc(/vector ro)·/vector po=/vectorE(/vector rc)·/vector pc. (3)
This is the usual form of the reciprocity theorem; it says tha t if we know /vectorEcat
the location of /vector powe can calculate /vectorEat the location of /vector pc. This elegant result
takes us a long way toward a final answer for /vectorE, but the remaining problem of
calculating /vectorEc, that is, the calculation of how the spherical wave generate d by
/vector pcis scattered by the medium, can still be too difficult.
A more useful version of the theorem can be obtained once one r ealizes
that the connecting field is merely a tool that codifies inform ation about the
influence of the non-trivial medium. Above, the field /vectorEchas been introduced
by first specifying a source /vectorJc, but clearly this is an unnecessary additional
complication. In fact, since the most convenient /vectorJcis that which results in
the simplest /vectorEcit is best to focus attention directly on the field rather than
its source. Thus we move /vectorJcoutside the surface S, to infinity (see fig.1b) so
that throughout the volume Vthe connecting field /vectorEcis a pure radiation field.
Furthermore, let the surface Sitself be so distant that on Sitself both /vectorEand
/vectorEcarevacuum radiation fields. Then
/integraldisplay
S/parenleftBig
/vectorE×(∇ ×/vectorEc)−/vectorEc×(∇ ×/vectorE)/parenrightBig
·d/vector s=4πiK
c/integraldisplay
V/vectorEc·/vectorJ dv. (4)
At this point it is not yet clear that this form of the reciproc ity theorem is
simpler than eq. (3) but one remarkable feature can already b e seen: eq. (4)
relates the field /vectorEat a distant surface Sto its source /vectorJwithin a nontrivial
medium without having to calculate /vectorEin the vicinity of /vectorJ. The “connection”
between the distant radiation field /vectorEand its source /vectorJis achieved through the
much simpler (i.e., hopefully calculable) “connecting” fie ld/vectorEc.
To bring eq. (4) into a form that is manifestly simpler than (3 ) the surface
Sis chosen as a cube with edges of length L→ ∞. In fig.1b the upper face,
defined by a constant zcoordinate, z=z+, has been singled out as S+, the
remaining seven faces are denoted S′.
On the upper face S+we write the field /vectorEas a superposition of outgoing
plane waves of wave vector /vectorksatisfying /vectork·/vectork=ω2/c2=K2andkz>0,
/vectorE(/vector r) =/integraldisplay
kz>0d3k
(2π)32πδ(k−K)/vectorE(/vectork)ei/vectork·/vector r, (5)
5where, in a self-explanatory notation, /vectork·/vector r=/vectork⊥·/vector r⊥+kzz+. It is here, by the
very act of writing /vectorEin this form, that the asymptotic limit of discarding near
fields is being taken. For z << z +additional terms describing the near fields
should be included.
The integral over dkzis most easily done using
δ(k−K) =K
kz/bracketleftbigg
δ/parenleftbigg
kz−/radicalBig
K2−k2
⊥/parenrightbigg
−δ/parenleftbigg
kz+/radicalBig
K2−k2
⊥/parenrightbigg/bracketrightbigg
.(6)
The result is
/vectorE(/vector r) =/integraldisplayd2k⊥
(2π)2K
kz/vectorE(/vectork)ei/vectork·/vector r, (7)
where kz= +/radicalbig
K2−k2
⊥.
The choice of the connecting field /vectorEcis dictated purely by convenience. A
particularly good choice for /vectorEcis the superposition of an incoming plane wave
of unit amplitude (we use ˆ to denote vectors of unit length) a nd wave vector
/vectorkc, with /vectorkc·/vectorkc=K2andkcz<0, plus all the waves scattered by the medium,
/vectorEc(/vector r) = ˆecei/vectorkc·/vector r+/vectorE′
c(/vector r). (8)
OnS+, the scattered field /vectorE′
cis a superposition of outgoing plane waves and is
given also in a form analogous to eq. (7),
/vectorE′
c(/vector r) =/integraldisplayd2k′
⊥
(2π)2K
k′
z/vectorE′
c(/vectork′)ei/vectork′·/vector r, (9)
with/vectork′·/vector r=/vectork′
⊥·/vector r⊥+k′
zz+andk′
z= +/radicalBig
K2−k′2
⊥.
Now we are ready to calculate the surface integral on the left hand side of
eq. (4). Substituting (8) into (4) the integral over S+separates into two terms,
one due to the incoming plane wave ˆ ecei/vectorkc·/vector r, and the other due to the scattered
waves /vectorE′
c(/vector r). The first term is
I1=/integraldisplay
S+dxdy ˆez·/parenleftBig
/vectorE×(i/vectorkc׈ec)−ˆec×(∇ ×/vectorE)/parenrightBig
ei/vectorkc·/vector r, (10)
and substituting (7) its evaluation is straightforward. Th e integral over dxdy
yields (2 π)2δ(/vectork⊥+/vectorkc⊥). Since k2=k2
c=K2,kcz<0 and kz>0 this implies
that the plane waves superposed in (7) yield a vanishing cont ribution except
when /vectork=−/vectorkc. Thus,
I1=−2iKˆec·/vectorE(−/vectorkc) (11)
The contribution of the scattered waves /vectorE′
c(/vector r),
I2=/integraldisplay
S+dxdy ˆez·/parenleftBig
/vectorE×(∇ ×/vectorE′
c)−/vectorE′
c×(∇ ×/vectorE)/parenrightBig
, (12)
6is calculated in a similar way. Substitute (7) and (9) and int egrate over dxdy
to obtain a delta function. This eliminates all Fourier comp onents except those
with/vectork⊥=−/vectork′
⊥. Since k2=k′2=K2, and both kz, k′
z>0 this implies kz=
k′
z>0. Thus,
I2=/integraldisplayd2k′
⊥
(2π)2K2
k′2
zˆez·/bracketleftBig
/vectorE(/vectork)×/parenleftBig
i/vectork′×/vectorE′
c(/vectork′)/parenrightBig
−/vectorE′
c(/vectork′)×/parenleftBig
i/vectork×/vectorE(/vectork)/parenrightBig/bracketrightBig
,(13)
where /vectork=−/vectork′+ 2k′
zˆez. Further manipulation using /vectork′·/vectorE′
c(/vectork′) =/vectork·/vectorE(/vectork) = 0
gives ˆez·[···] = 0, so that
I2= 0. (14)
According to eq. (11) and (14) the only contributions to the s urface integral
over the distant plane S+come from products of outgoing with incoming waves.
Products of two outgoing waves yield vanishing contributio ns. This result ap-
plies also to the remaining seven faces of the cube S. Since on each of these
faces there are only outgoing waves we find that the integral o verS′makes no
contribution to the left hand side of (4). Incidentally, thi s argument completes
our previously unfinished deduction of the usual form of the r eciprocity theo-
rem, eq. (3): if both sources /vectorJand/vectorJcare internal to the surface Sthe surface
integral in eq. (2) vanishes because it only involves produc ts of outgoing waves.
Substituting (11) into (4) leads us to the main result of this paper, the
asymptotic reciprocity theorem,
/hatwidee·/vectorE(/vectork) =−2π
c/integraldisplay
V/vectorEc·/vectorJ dv. (15)
In words:
The field /vectorE(/vectork)radiated in a direction /vectorkwith a certain polarization /hatwideeis
−2π/ctimes the “component” of the source /vectorJ(/vector r)“along” a connecting field
/vectorEc(/vector r)with incoming wave vector /vectorkc=−/vectorkand polarization /hatwideec=/hatwidee.
Typically one is interested in the intensity radiated into a solid angle dΩ;
since the amplitude /vectorE(/vectork) that appears in (7) and (15) is not quite the Fourier
transform of /vectorE(/vector r) it may be useful to derive an explicit expression for dW/d Ω.
The total power radiated through the plane S+is given by the flux of the time-
averaged Poynting vector,c
8πRe [/vectorE×/vectorB∗],
W=/integraldisplay
d2x⊥c
8πRe [/vectorE×/vectorB∗]·ˆez=/integraldisplay
dΩdW
dΩ(16)
Using (7) and d2k⊥=k⊥dk⊥dφ=KkzdΩ (where φis the usual azimuthal angle
about the zaxis) we get
W=c
8π/integraldisplayd2k⊥
(2π)2K
kz/vectorE(/vectork)·/vectorE∗(/vectork), (17)
so that
dW
dΩ=c
8π/parenleftbiggK
2π/parenrightbigg2
/vectorE(/vectork)·/vectorE∗(/vectork). (18)
In the next section we offer a few illustrative examples of the ART in action.
73 Some simple examples
The ART, eq. (15), holds for an arbitrary linear medium. In pa rticular it holds if
the medium is vacuum. Our first trivial example is the radiati on by a prescribed
current in vacuum. Next, to show that the ART can be used to stu dy scattering
problems we deal with another equally trivial example, the k inematical theory
of diffraction by crystals. The third example, the radiation by currents located
near a dielectric boundary, is also straightforward. What i s remarkable here is
the ease with which the results are obtained compared to conv entional methods
[9][10].
3.1 Radiation in vacuum
In this case the connecting field is just an incoming plane wav e,/vectorEc(/vector r) = ˆecei/vectorkc·/vector r.
The ART, eq. (15), gives the radiated field with polarization ˆe= ˆecas
/hatwidee·/vectorE(/vectork) =−2π
c/integraldisplay
V/vectorJ(/vector r)·ˆe e−i/vectork·/vector rdv=−2π
cˆe·/vectorJ(/vectork), (19)
so that
/vectorE(/vectork) =2π
cˆk×(ˆk×/vectorJ(/vectork)). (20)
The radiated power, eq. (18), is
dW
dΩ=K2
8πc/vextendsingle/vextendsingle/vextendsingleˆk×(ˆk×/vectorJ(/vectork))/vextendsingle/vextendsingle/vextendsingle2
, (21)
as expected. (For radiation by a point dipole just substitut e/vectorJ(/vectork) =−icK/vector p.)
3.2 Bragg diffraction
Consider a crystal described by its dielectric susceptibil ityχ(/vector r) [19] which for
x rays is quite small (typically about 10−5or less). An incident plane wave
/vectorEoei/vectorko·/vector rinduces a current
/vectorJ(/vector r) =−iω/vectorP(/vector r) =−iω
4πχ(/vector r)/vectorEoei/vectorko·/vector r, (22)
which radiates. The connecting field needed to calculate thi s radiation is a
simple incoming plane wave, /vectorEc(/vector r) = ˆecei/vectorkc·/vector r, and the ART, eq. (15), gives the
radiated field as
/vectorE(/vectork) =−iω
2cˆk×(ˆk×/vectorEo)χ(/vectork−/vectorko). (23)
The scattered field is proportional to the Fourier transform of the susceptibility
of the medium; for a periodic medium this is Bragg diffraction .
83.3 Radiation in the vicinity of a reflecting surface
Consider a current /vectorJin(/vector r) located within a uniform medium with dielectric sus-
ceptibility χ0occupying the region z <0 (see fig. 2). To calculate the radiation
in the direction /vectorkwith polarization ˆ ewe choose as connecting field an incom-
ing plane wave with wave vector /vectorkc=−/vectorkand unit amplitude ˆ ec= ˆeplus the
corresponding reflected and transmitted waves,
/vectorEc(/vector r) =/braceleftBigg
ˆecei/vectorkc·/vector r+/vector εcrei/vectorkcr·/vector r,forz >0
/vector εctei/vectorkct·/vector r, forz <0(24)
The various wave vectors are given by
/vectorkc=−Kcosθˆex−qˆez=−/vectork , (25)
/vectorkcr=−Kcosθˆex+qˆez, (26)
/vectorkct=−Kcosθˆex−¯qˆez, (27)
where K=ω/c, and the normal components qand ¯qare given by
q=Ksinθand ¯ q=K/parenleftbig
sin2θ+χ0/parenrightbig1/2. (28)
The amplitudes /vector εcrand/vector εctof the reflected and transmitted waves are given by
the Fresnel expressions
/vector εcr=rsˆecrwhere rs=q−¯q
q+ ¯qˆecr·ˆect
ˆec·ˆect, (29)
and
/vector εct=tsˆectwhere ts=2q
q+ ¯q1
ˆec·ˆect. (30)
(ˆecrand ˆectare unit vectors describing the polarization of the specula r reflected
and transmitted waves.)
Then for a source /vectorJin(/vector r) located within the medium, the ART, eq.(15), gives
the radiated field as
/hatwidee·/vectorE(/vectork) =−2π
c/integraldisplay
z<0/vectorJin(/vector r)·/vector εctei/vectorkct·/vector rdv . (31)
On the other hand, had the source /vectorJout(/vector r) been located outside the dielectric
medium ( z >0) the corresponding radiated field would be
/hatwidee·/vectorE(/vectork) =−2π
c/integraldisplay
z>0/vectorJout(/vector r)·/parenleftBig
ˆecei/vectorkc·/vector r+/vector εcrei/vectorkcr·/vector r/parenrightBig
dv . (32)
For an oscillating dipole on the zaxis, /vectorJ(/vector r) =−iω/vector pδ(/vector r−zpˆez), eq.(31) and
(32) give
/hatwidee·/vectorE(/vectork) =/braceleftbigg
2πiK/parenleftbig
/vector p·ˆee−iqzp+/vector p·ˆecrrseiqzp/parenrightbig
ifzp>0
2πiK /vector p ·ˆecttse−i¯qzp ifzp<0.(33)
9Figure 2: The connecting field for radiation in the presence o f a reflecting
medium includes reflected and transmitted waves. Here the so urce/vectorJinis shown
within the medium ( z <0).
The power radiated with polarization ˆ e, eq.(18), is
dW
dΩ=/bracketleftBigc
8πK4(ˆe·/vector p)2/bracketrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 +ˆecr·/vector p
ˆe·/vector prse2iqzp/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
forzp>0, (34)
and
dW
dΩ=/bracketleftBigc
8πK4(ˆe·/vector p)2/bracketrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆect·/vector p
ˆe·/vector ptse−i¯qzp/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
forzp<0. (35)
In these two expressions we can recognize the first factor (in square brackets)
as the power radiated by a dipole in vacuum. The second factor accounts for
the presence of the dielectric medium.
4 Specular reflection of polarized x rays
In this section ideas from the three previous examples are co mbined to study
two similar and considerably more involved scattering prob lems, the specular
reflection of polarized x rays by a rough surface and by graded interfaces. We
show that within approximations of the Nevot-Croce type gra ding and roughness
affect the specular reflectivity in a manner that is independe nt of the polarization
of the incident radiation.
104.1 Reflection by rough surfaces
The dielectric susceptibility χ(/vector r) that describes the rough surface from which
we wish to scatter x rays is given by
χ(x, y, z ) =/braceleftbigg
0 for z > ζ(x, y)
χ0forz < ζ(x, y)(36)
where the height ζ(x, y), is a Gaussian random variable with zero mean, /angb∇acketleftζ/angb∇acket∇ight= 0,
and variance/angbracketleftbig
ζ2/angbracketrightbig
=σ2(see fig. 3).
To apply the ART it is convenient to rewrite χ(/vector r) as
χ(/vector r) =χs(/vector r) +δχ(/vector r), (37)
where χs(/vector r) represents a medium with an ideally flat surface at z0,
χs(/vector r) =/braceleftbigg0 for z > z 0
χ0forz < z 0(38)
andδχ(/vector r) represents the roughness.
Let ˆe0ei/vectork0·/vector rbe the incident field. The total scattered field /vector ε(/vector r) includes the
wave/vector εs(/vector r) specularly reflected by the step χs(/vector r) plus waves δ/vector ε(/vector r) scattered by
δχ(/vector r)
/vector ε(/vector r) =/vector εs(/vector r) +δ/vector ε(/vector r). (39)
The first term on the right is
/vector εs(/vector r) = ˆerrse−2iqz0ei/vectorkr·/vector r, (40)
where
rs=q−¯q
q+ ¯qˆer·ˆet
ˆe0·ˆet(41)
(ˆerand ˆetare unit vectors describing the polarization of the specula r reflected
and transmitted waves). The second contribution in eq.(39) , the field δ/vector ε(/vector r)
includes a specular component plus diffusely scattered and e vanescent waves,
δ/vector ε(/vector r) =δεrˆerei/vectorkr·/vector r+δ/vector εd(/vector r). (42)
Using eq.(7) this may be written as
δ/vector ε(/vector r) =/integraldisplayd2k′
⊥
(2π)2K
k′
zδ/vector ε(/vectork′)ei/vectork′·/vector r, (43)
where
δ/vector ε(/vectork′) =k′
z
Kδεrˆer(2π)2δ(/vectork′
⊥−/vectork0⊥) +δ/vector εd(/vectork′). (44)
To calculate δ/vector ε(/vector r) we can proceed exactly as in the previous section (3.3): δ/vector ε(/vector r)
is the field radiated by a current δ/vectorJ(/vector r) in the presence of the medium χs(/vector r).
The current
δ/vectorJ(/vector r) =−iω
4πδχ(/vector r)/vectorE(/vector r), (45)
11Figure 3: The problem of scattering by a rough surface can be t ackled using the
ART by adding a fictitious overlayer δχ.
originates in the polarization of the roughness δχ(/vector r) by the total electric field
/vectorE(/vector r) due to the incident and all scattered waves, including thos e generated by
the roughness itself. Thus, the challenge here is that the fie ld/vectorE(/vector r) is itself
unknown; an approximation for it must be obtained as part of o ur solution.
We can exploit the arbitrariness in the separation of χ(/vector r) into χs(/vector r) plus
δχ(/vector r) to suggest a self-consistent approximation for /vectorE. Suppose we choose z0
positive and considerably larger than the roughness σ(see fig. 3). Then δχ(/vector r)
represents a fictitious overlayer that extends well into the vacuum; the sign of
δχ(/vector r) is opposite to that of χs(/vector r) and in the vicinity of z0they completely cancel
out. The field δ/vector ε(/vectork) in a direction /vectorkwith polarization ˆ eis given by eq.(31)
/hatwidee·δ/vector ε(/vectork) =iK
2/integraldisplay
dv δχ(/vector r)/vectorE(/vector r)·/vector εctei/vectorkct·/vector r, (46)
where the connecting field is precisely as in eqs.(24)-(30) e xcept for phase shifts
due to the reflecting surface being at z0,
/vector εcr=e−2iqz0rsˆecrand /vector εct=ei(¯q−q)z0tsˆect. (47)
The reason behind the somewhat surprising choice for z0will now become
clear: slightly above z0, in vacuum, the exact field is
/vectorE(/vector r) = ˆe0ei/vectork0·/vector r+/vector ε(/vector r) = ˆe0ei/vectork0·/vector r+/vector εs(/vector r) +δ/vector ε(/vector r), (48)
but slightly below z0and, in fact, over all of the extension occupied by δχ(/vector r),
we are also in vacuum ( δχ(/vector r) and χs(/vector r) cancel each other) and therefore /vectorE(/vector r) is
12given by the same expression (48). The last term δ/vector ε(/vector r), given by (42), includes
some weak diffusely scattered and evanescent waves δ/vector εd(/vector r). Our approximation
consists of neglecting them. Therefore,
/vectorE(/vector r)≈ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r, (49)
where the specular reflections by χs(/vector r) and δχ(/vector r) have been combined into the
single, and still unknown, reflection coefficient r,
r=rse−2iqz0+δεr. (50)
Substituting into eq.(46) yields
/hatwidee·δ/vector ε(/vectork) =−iKχ0
2/integraldisplay
dxdy/integraldisplayz0
ζ(x,y)dz/bracketleftBig
ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r/bracketrightBig
·/vector εctei/vectorkct·/vector r.(51)
From now on we focus our attention on the specularly reflected component; let
ˆe= ˆec= ˆer, ˆecr= ˆe0,/vectork=/vectorkr=−/vectorkc. Substituting eq.(44) into the left hand
side (l.h.s.), using (2 π)2δ(k⊥−k0⊥) = (2 π)2δ(0) =/integraltext
dxdy we get
l.h.s.=q
Kδεr(2π)2δ(0) =q
K/parenleftbig
r−rse−2iqz0/parenrightbig/integraldisplay
dxdy. (52)
This shows that the unknown reflection coefficient rwe want to calculate appears
in both the left and the right hand sides of (51), as part of the radiated field and
also as part of the field that induces the source; eq.(51) perm its a self-consistent
calculation of r.
The integral over dzin the right hand side of eq.(51) is elementary and the
remaining integral over dxdy is performed using the identity
/integraltext
dxdy e−iQζ(x,y)
/integraltext
dxdy=/angb∇acketlefte−iQζ/angb∇acket∇ight=e−Q2σ2/2, (53)
where ζis a Gaussian random variable with zero mean, /angb∇acketleftζ/angb∇acket∇ight= 0, and variance/angbracketleftbig
ζ2/angbracketrightbig
=σ2. The right hand side ( r.h.s.) of eq.(51) becomes
r.h.s. =Kχ0
2/parenleftbigg/integraldisplay
dxdy/parenrightbigg /braceleftbiggˆe0·/vector εct
q+ ¯q/bracketleftBig
e−i(q+¯q)z0−e−(q+¯q)2σ2/2/bracketrightBig
−rˆer·/vector εct
q−¯q/bracketleftBig
ei(q−¯q)z0−e−(q−¯q)2σ2/2/bracketrightBig/bracerightbigg
, (54)
which can be further rewritten by substituting /vector εctas given by eq.(47), and using
¯q2−q2=K2χ0, and
ˆe0·ˆect
ˆe·ˆect=ˆecr·ˆect
ˆe·ˆect=ˆer·ˆet
ˆe0·ˆetandˆer·ˆect
ˆe·ˆect= 1. (55)
13Finally, equating eq.(52) to (54) yields a self-consistent approximation to r,
r=q−¯q
q+ ¯qˆer·ˆet
ˆe0·ˆete−2q¯qσ2=rse−2q¯qσ2. (56)
This coincides exactly with the Nevot-Croce result for the p olarization ˆ e0= ˆet=
ˆerfor which the ratio ˆ e0·ˆet/ˆer·ˆetis unity, and provides the correct generalization
to all polarizations. According to this approximation the s pecular reflection
coefficient rhas no polarization dependence beyond that already implici t in the
reflection coefficient rsfor the ideal flat step surface; the “static Debye-Waller”
factor exp(−2q¯qσ2) is polarization independent.
Notice that any possible dependence on the arbitrary choice ofz0has can-
celled out.
4.2 Reflection by smoothly graded surfaces
The problem of scattering by a smoothly graded interface is s imilar and some-
what simpler. Here the susceptibility χ(z) depends only on the normal coor-
dinate zand not on the transverse coordinates xandy. This implies that the
tangential component of momentum is conserved in the scatte ring; there are no
diffuse waves, there is only specular scattering.
As before, it is convenient to separate χ(z) into
χ(z) =χs(z) +δχ(z), (57)
where χs(z) represents an ideally flat surface at z0,
χs(z) =/braceleftbigg
0 for z > z 0
χ0forz < z 0(58)
andδχ(z) is an overlayer (see fig.4) describing the smooth transitio n from bulk
to vacuum.
Let ˆe0ei/vectork0·/vector rbe the incident field. The total scattered field /vector ε(/vector r), eq.(39),
/vector ε(/vector r) =/vector εs(/vector r) +δ/vector ε(/vector r). (59)
includes the wave /vector εs(/vector r) reflected by the step χs(z), eq.(40), plus waves δ/vector ε(/vector r)
scattered by the overlayer δχ(z). While diffusely scattered waves are not present
inδ/vector ε(/vector r), faint evanescent waves could be; these are weak near field e ffects and
we neglect them. Thus
δ/vector ε(/vector r) =δεrˆerei/vectorkr·/vector r, (60)
and the Fourier expansion, eq.(43), and transform δ/vector ε(/vectork), eq.(44), remain other-
wise unchanged. Once again, δ/vector ε(/vector r) is radiated by a current
δ/vectorJ(/vector r) =−iω
4πδχ(z)/vectorE(/vector r), (61)
where the field /vectorE(/vector r) includes the incident and the unknown reflected waves;
/vectorE(/vector r) must be self-consistently obtained as part of the solution . Then the ART,
14Figure 4: The problem of reflection by a graded surface can be t ackled using
the ART by adding a fictitious overlayer δχ. The hatched region shows the
transition region from δχ= 0 to δχ=−χ0.
in the form of eq.(31), gives the field δ/vector ε(/vectork) in a direction /vectorkwith polarization ˆ e
as
/hatwidee·δ/vector ε(/vectork) =iK
2/integraldisplay
dv δχ(z)/vectorE(/vector r)·/vector εctei/vectorkct·/vector r, (62)
with the same connecting field given back in eq.(47).
The approximation we use for /vectorE(/vector r) is the same as in last section. The
arbitrariness of z0can be exploited by choosing it large enough that the overlay er
extends well into the vacuum. Near z0the overlayer and the sharp step χs(/vector r)
cancel each other out; slightly above z0, in vacuum, the field is
/vectorE(/vector r) = ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r, (63)
where ris the unknown reflection coefficient we want to calculate,
r=rse−2iqz0+δεr. (64)
Slightly below z0and over most of the extension occupied by δχ(z) we are also
in vacuum (provided the bulk to vacuum transition is not too g radual) and we
approximate /vectorE(/vector r) by the same expression, eq.(63). Substituting into eq.(62 )
yields an equation for r,
q
K/parenleftbig
r−rse−2iqz0/parenrightbig
(2π)2δ(k⊥−k0⊥) =
15=iK
2/integraldisplayz0
−∞dz δχ(z)/integraldisplay
dxdy/bracketleftBig
ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r/bracketrightBig
·/vector εctei/vectorkct·/vector r.(65)
The integral over dxdy yields a delta function, (2 π)2δ(k⊥−k0⊥), and we can
substitute ˆ e= ˆec= ˆer, ˆecr= ˆe0,/vectork=/vectorkr=−/vectorkc. The integral over zis
conveniently expressed as
/integraldisplayz0
−∞dz δχ(z)e−iQz=χ0
iQ[e−iQz0+χ′(Q)
χ0], (66)
where χ′(Q) is the Fourier transform of dχ(z)/dz,
χ′(Q) =/integraldisplay+∞
−∞dzdχ(z)
dze−iQz(67)
Eq.(66) is proved by integrating the left hand side by parts, using δχ(z0)≈ −χ0,
anddδχ(z)/dz=dχ(z)/dz. Using ¯ q2−q2=K2χ0and the identities in eq.(55)
the final result is
r=rsχ′(¯q+q)
χ′(¯q−q). (68)
Notice that any possible dependence on the arbitrary choice ofz0has cancelled
out.
This coincides exactly with the scalar wave result [17] and p rovides the
correct generalization to all polarizations. Within these approximations the
specular reflection coefficient rhas no polarization dependence beyond that
already implicit in the reflection coefficient rsfor the ideal flat step surface; the
“static Debye-Waller” factor is polarization independent .
To conclude we mention some illustrative examples:
(a) The error-function profile
χ(z) =χ0√
2πσ2/integraldisplayz
−∞dxexp−/parenleftbiggx2
2σ2/parenrightbigg
, (69)
gives
χ′(¯q+q)
χ′(¯q−q)=e−2q¯qσ2, (70)
the same factor obtained in the previous section for a Gaussi an rough surface.
This is as expected, the error function is the averaged profil e for the Gaussian
rough surface.
(b) The Epstein (or Fermi distribution) profile [17]
χ(z) =χ0
1 +e−z/σ, (71)
gives
χ′(¯q+q)
χ′(¯q−q)=¯q+q
¯q−qsinh[πσ(q−¯q)]
sinh[πσ(q+ ¯q)]. (72)
16(c) The triangular profile
χ(z) =
χ0 forz <−σ/2
χ0(1−2z/σ) for |z|< σ/2
0 for z > σ/ 2, (73)
gives
χ′(¯q+q)
χ′(¯q−q)=q−¯q
q+ ¯qsin[(q+ ¯q)σ/2]
sin[(q−¯q)σ/2]. (74)
The reliability of these approximations was studied in [17] in the case of
scalar waves. There is no reason to expect any difference from the conclusions
reached there: the “static Debye-Waller” in eq.(68) provid es a remarkably good
approximation for the intensities reflected by interfaces o f arbitrary grading
profile even for transition regions that are quite wide ( σas large as several
nanometers). The phase of the reflected waves is however more sensitive; eq.(68)
provides a good approximation for more abrupt transitions ( σof the order of 1
nmor less).
5 Conclusion
The main result of this work, eq.(15), is an asymptotic form o f the reciprocity
theorem which can be used as the basis for a practical method f or calcula-
tions. The theorem states that the field radiated in the prese nce of a nontriv-
ial medium, in a certain direction and with a given polarizat ion, is a suitable
‘component’ of the radiating source. This ‘component’ is to be extracted by
introducing an auxiliary ‘connecting’ field which contains the necessary infor-
mation about the medium. The practical advantage of the meth od lies in the
simplifications achieved by systematically avoiding unnec essary calculations; it
thereby allows one to tackle problems of increasing complex ity.
In forthcoming papers we will further explore the applicati on of the ART to
the study of the dynamical diffraction of radiation generate d by sources within
a crystal, the so-called Kossel lines. Even this well explor ed topic has not been
exhausted. Of particular interest are situations where the Bragg angle lies close
toπ/2 and the Kossel cones degenerate into single beams [20], and situations
where the source location is revealed by the oscillatory ‘Pe ndell¨ osung’ structure
of the diffraction pattern [21]. Other applications will inc lude a new approach
to thermal diffuse scattering under conditions of dynamical diffraction [22].
Acknowledgments . I am indebted to P. Zambianchi and E. Sutter for
valuable discussions.
References
[1] H. A. Lorentz, Proc. Amsterdam Acad., 8, 401 (1905).
[2] See e.g., J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics
(Dover, 1979); E. Merzbacher, Quantum Mechanics (Wiley, 1998).
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5, 435 (1964).
[4] L. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Contin-
uous Media (Butterworth-Heinemann, 1984).
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[6] M. v. Laue, Ann. Physik 23, 705 (1935).
[7] W. Kossel, V. Loeck and H. Voges, Zeit. f. Physik, 94, 139 (1935).
[8] M.Tegze and G. Feigel, Europhys. Lett. 16, 41 (1991), Nature (London)
380, 49 (1996); T. Gog et al., Phys. Rev. B 51, 6761 (1995), Phys. Rev.
Lett.76, 3132 (1996); P. M. Len et al., Phys. Rev. B 55, R3323 (1997).
[9] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York 1941).
[10] W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand
Reinhold, New York 1990); J. R. Wait, Electromagnetic Waves in Stratified
Media (IEEE-Oxford, New York 1996).
[11] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves
from Rough Surfaces (Artech House, Norwood, MA 1987); J. Lekner, The-
ory of Reflection of Electromagnetic and Particle Waves (Martinez Nijof,
Dordrecht, Holland).
[12] D. G. Stearns, J. Appl. Phys. 65, 491 (1989) and 71, 4286 (1992); A. V.
Vinogradov et al., Sov. Phys. JETP 62, 1225 (1985) and 67, 1631 (1988);
W. Weber and B. Lengeler, Phys. Rev. B 46, 7953 (1992); J. C. Kimball
and D. Bittel, J. Appl. Phys. 74, 887 (1993).
[13] S. K. Sinha, et al., Phys. Rev. B 38, 2297 (1988); V. Holy et al., Phys.
Rev.B 47, 15896 (1993).
[14] S. Dietrich and A. Haase, Phys. Rep. 260, 1-138 (1995).
[15] L. Nevot and P. Croce, Revue Phys. Appl. 15, 761 (1980).
[16] R. Pynn, Phys. Rev. B 45, 602 (1992); D. K. G. de Boer, Phys. Rev. B
49, 5817 (1994).
[17] A. Caticha, Phys. Rev. B 52, 9214 (1995).
[18] This is a nontrivial statement; its justification is giv en in the paragraph
following eq. 14.
[19] To avoid factors of 4 πelsewhere it is usual in x-ray optics to define the
susceptibility by 4 πP=χE.
[20] P. Zambianchi and A. Caticha, “Dynamical diffraction of x rays generated
within the crystal: the case of θB≈π/2.”
18[21] E. Sutter and A. Caticha, “Dynamical diffraction of x ray s generated within
the crystal: the Laue case.”
[22] P. Zambianchi and A. Caticha, “Thermal diffuse dynamica l diffraction of
x rays.”
19JcEcEc
E E
J JS'S
ε(r) ε(r)
(b) (a)S+kcr
S+
kctkc z
θ x
Jin |
arXiv:physics/0001006v1 [physics.bio-ph] 4 Jan 2000Temporal correlations and neural spike train entropy
Simon R. Schultz1and Stefano Panzeri2
1Howard Hughes Medical Institute and Center for Neural Scien ce, New York University,
4 Washington Place, New York, NY 10003, U.S.A.
2Department of Psychology, Ridley Building, University of N ewcastle upon Tyne,
Newcastle upon Tyne, NE1 7RU, U.K.
Sampling considerations limit the experimental condition s under which information theoretic
analyses of neurophysiological data yield reliable result s. We develop a procedure for computing the
full temporal entropy and information of ensembles of neura l spike trains, which performs reliably for
extremely limited samples of data. This approach also yield s insight upon the role of correlations
between spikes in temporal coding mechanisms. The method is applied to recordings from the
monkey visual cortex, yielding 1.5 and 0.5 bits per spike for simple and complex cells respectively.
PACS numbers: 87.19.Nn,87.19.La,89.70.+c,07.05.Kf
Cells in the central nervous system communicate by
means of stereotypical electrical pulses called action po-
tentials, or spikes [1]. The information content of neural
spike trains is fully described by the sequence of times of
spike emission. In principle, the pattern of spike times
provides a large capacity for conveying information be-
yond that due to the code commonly assumed by phys-
iologists, the number of spikes fired [2]. Reliable quan-
tification of this spike timing information is made diffi-
cult by undersampling problems that can increase expo-
nentially with the precision of spike observation. While
advances have been made in experimental preparations
where extensive sampling may be undertaken [3–6], our
understanding of the temporal information properties of
nerve cells from less accessible preparations such as the
mammalian cerebral cortex is limited.
This Letter presents an analytical expression which al-
lows the ensemble spike train entropy to be computed
from limited data samples, and relates the entropy and
information to the instantaneous probability of spike oc-
currence and the temporal correlations between spikes.
This is achieved by power series expansion of the entropy
in the time window of observation [7], keeping terms of
up to second order, and subtraction of an analytical es-
timate of the bias due to finite sampling. Comparison
is made with other procedures such as the ‘brute force’
approach [4,9]; the analytical expression is found to give
substantially better performance for data sizes of the or-
der typically obtained from mammalian neurophysiology
experiments, as well as providing insight into potential
coding mechanisms.
Consider a time period of duration T, associated with
a dynamic or static sensory stimulus, during which the
activity of Ccells is observed. The neuronal population
response to the stimulus is described by the collection
of spike arrival times {ta
i},ta
ibeing the time of the i-th
spike emitted by the a-th neuron. The spike time is ob-
served with finite precision ∆ t, and this bin width is used
to digitise the spike train. The total entropy of the spike
train ensemble is
H({ta
i}) =−/summationdisplay
ta
iP({ta
i})log2P({ta
i}), (1)where the sum over ta
iis over all possible spike times
within Tand over all possible total spike counts from
the population of cells. This entropy quantifies the to-
tal variability of the spike train. Each different stimu-
lus history (time course of characteristics within T) is
denoted as s. The noise entropy, which quantifies the
variability to repeated presentations of the same stimu-
lus, is Hnoise=/an}bracketle{tH({ta
i}|s)/an}bracketri}hts, where the angular brackets
indicate the average over different stimuli,/summationtext
s∈SP(s).
The mutual information that the responses convey about
which stimulus history invoked the spike train is the dif-
ference between these two quantities.
These entropies may be expanded as a Taylor series in
the time window of measurement,
H=THt+T2
2Htt+O(T3). (2)
This becomes essentially an expansion in the total num-
ber of spikes emitted; the only responses which contribute
to order kare those with up to kspikes emitted in total.
The conditional firing probabilities can be written
P(ta
i|tb
j;s)≡ra(ta
i;s) ∆t/bracketleftbig
1 +γab(ta
i, tb
j;s)/bracketrightbig
+O(∆t2),(3)
and are assumed to scale proportionally to ∆ tin the
short timescale limit. Only the first order conditional
firing probability affects the entropy to second order in
the time window. In the above, ra(t;s) is the time-
dependent instantaneous firing rate and is measurable
from the data. The bar indicates the average over mul-
tiple trials in which the same stimulus history was pre-
sented. The scaled correlation function γabis measured
as [8,10]:
γab(ta
i, tb
j;s) =ra(ta
i;s)rb(tb
j;s)
ra(ta
i;s)rb(tb
j;s)−1, a/ne}ationslash=borta
i/ne}ationslash=tb
j
γaa(ta
i, ta
i;s) =−1. (4)
Denoting the no spikes event as 0and the joint occur-
rence of a spike from cell aat time ta
1and a spike from cell
bat time tb
2asta
1tb
2, the conditional response probabilities
are:P(0|s) = 1 −C/summationdisplay
a=1/summationdisplay
ta
1ra(ta
1;s)∆t+1
2/summationdisplay
ab/summationdisplay
ta
1/summationdisplay
tb
2ra(ta
1;s)rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig
∆t2
P(ta
1|s) =ra(ta
1;s)∆t−ra(ta
1;s)C/summationdisplay
b=1/summationdisplay
tb
2rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig
∆t2a= 1,· · ·, C
P(ta
1tb
2|s) =1
2ra(ta
1;s)rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig
∆t2a= 1,· · ·, C, b = 1,· · ·, C. (5)
where/summationtext
abindicates the sum over both aandbfrom 1 to C. The unconditional response probabilities are simply
p({ta
i}) =/an}bracketle{tp({ta
i}|s)/an}bracketri}hts. Inserting p({ta
i}) into Eq. 1 and keeping only terms up to and including O(T) yields for the
first order total entropy
THt=1
ln 2/summationdisplay
a/summationdisplay
ta
1/an}bracketle{tra(ta
1;s)∆t/an}bracketri}hts−/summationdisplay
a/summationdisplay
ta
1/an}bracketle{tra(ta
1;s)∆t/an}bracketri}htslog2/an}bracketle{tra(ta
1;s)∆t/an}bracketri}hts. (6)
Similarly, inserting p({ta
i}|s) yields an expression for the first order noise entropy THnoise
twhich is identical, except
that there is a single stimulus average /an}bracketle{t·/an}bracketri}htsaround the entire second term. Continuing the expansion, th e additional
terms up to second order are:
T2
2Htt=1
2 ln2/summationdisplay
ab/summationdisplay
ta
1/summationdisplay
tb
2/braceleftbig/angbracketleftbig
ra(ta
1;s)rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig/angbracketrightbig
s− /an}bracketle{tra(ta
1;s)/an}bracketri}hts/angbracketleftbig
rb(tb
2;s)/angbracketrightbig
s/bracerightbig
∆t2
+/summationdisplay
ab/summationdisplay
ta
t/summationdisplay
tb
2/angbracketleftbig
ra(ta
1;s)rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig
∆t2/angbracketrightbig
slog2/an}bracketle{tra(ta
1;s)/an}bracketri}hts/radicalBig/angbracketleftbig
ra(ta
1;s)rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig/angbracketrightbig
s(7)
T2
2Hnoise
tt=1
2 ln2/summationdisplay
ab/summationdisplay
ta
1/summationdisplay
tb
2/angbracketleftbig
ra(ta
1;s)rb(tb
2;s)γab(ta
1, tb
2;s)/angbracketrightbig
s∆t2
+/summationdisplay
ab/summationdisplay
ta
t/summationdisplay
tb
2/angbracketleftBigg
ra(ta
1;s)rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig
∆t2log2ra(ta
1;s)/radicalBig
ra(ta
1;s)rb(tb
2;s)/bracketleftbig
1 +γab(ta
1, tb
2;s)/bracketrightbig/angbracketrightBigg
s.(8)
It is easily verified that the difference between the total
and noise entropies gives the expression for the mutual
information detailed in [10].
It has recently been found that correlations, even if
independent of the stimulus identity, can increase the
information present in a neural population [11,8]. This
effect is governed by the similarity of tuning of the cells
across the stimuli, and applies both to correlations be-
tween neurons and between different spikes emitted by
the same neuron [10]. The equations derived above ex-
plain how this is realised in terms of entropy. The sec-
ond order total entropy can be rewritten in a form which
shows that it depends only upon the grand mean firing
rates across stimuli, and upon the correlation coefficient
of the whole spike train, Γ( ·) (defined across alltrials
rather than for fixed stimulus as with Eqn. 4):
T2
2Htt=T2
2 ln 2/summationdisplay
ab/summationdisplay
ta
1/summationdisplay
tb
2/an}bracketle{tra(ta
1;s)/an}bracketri}hts/angbracketleftbig
rb(tb
2;s)/angbracketrightbig
s(9)
×/braceleftbig
Γab(ta
i, tb
j)−[1 + Γ ab(ta
i, tb
j)] ln[1 + Γ ab(ta
i, tb
j)]/bracerightbig
.
It follows that the second order entropy is maximal when
Γ(·) = 0, and non-zero correlations in the spike trains
(indicating statistical dependence) always decrease thetotal response entropy. However, statistical dependence
in the full spike train recorded across all stimuli does not
necessarily imply neuronal interaction [12]. If the signal
correlation
νab(ta
i, tb
j) =<ra(ta
i;s)rb(tb
j;s)>s
<ra(ta
i;s)>s<rb(tb
j;s)>s−1 (10)
is negative, then positive γ(s)’s reduce the overall sta-
tistical dependency and thus increase the entropy of the
whole spike train. The entropy increase is maximum for
theγvalue which leads to exactly Γ = 0. The effect
is opposite when signal correlation is positive. In com-
parison, the noise or conditional entropy is always de-
creased by γ/ne}ationslash= 0 - at fixed stimulus the only statistical
dependencies in the spike train are those measured by
γ. The increase/decrease of the population information
depending upon the signs of the signal and noise corre-
lation is thus not due to a change in the behaviour of
the noise entropy, but to the increase or decrease of the
total entropy. Statistical dependence always decreases
entropy, but neuronal (or spike time) interaction may in-
crease entropy itself, by eliminating or reducing the sta-
tistical dependencies introduced by other covariations. I tis intriguing to speculate a specific role for synaptic in-
teractions in compensating for the statistical dependency
introduced by necessary covariations such as firing rate
and slow systemic covariations.
The rate and correlation functions must be estimated
from a limited number of experimental trials, which leads
to a bias in each of the entropy components. This bias
can be estimated by the method derived in [13]:
Hbias=−R
2Nln 2Hnoise
bias=−1
2Nln 2/summationdisplay
s∈SRs(11)
where Ris the number of relevant (non-zero) response
bins. For the first order entropy, it is the number of non-
zero bins of the dynamic rate function; for the second
order entropy, it is the number of relevant bins in the
space of pairs of spike firing times. For the noise en-
tropy, the conditional response space is used, and for the
frequency entropy, the response space is that of the full
temporal words.
There is some subtlety as to how the number of ‘rele-
vant’ response bins should be counted. If zero occupancy
count is observed, it is ambiguous whether that indicates
true zero occupancy probability or local undersampling.
Naive counting of the bins based on raw occupancy prob-
abilities results in underestimation of the bias (in fact
providing a lower bound upon it) and thus underesti-
mation of the entropy. An alternative strategy is to use
Bayes’ theorem to reestimate the expectation value of the
number of occupied bins, as described in [13].
These procedures were compared by estimating the
entropy of a time-dependent simulated Poisson process
for different data sizes. The series estimator using the
Bayesian bias estimate is the only one which gives accept-
able performance in the range of 10-20 trials per stimulus,
although there is little to choose between it and the naive
series estimator above about 50 trials. The uncorrected
frequency (‘brute force’) estimator is inadequate.
Also shown in Fig. 1 is the Ma lower bound upon the
entropy [14], which performs comparatively poorly. The
Ma bound has been proposed as a useful bound which
is relatively insensitive to sampling problems [6]. The
Ma bound is tight only when the probability distribution
of words at fixed spike count is close to uniform; this is
not the case in general. To understand the behaviour of
the Ma bound for short time windows, we calculated se-
ries terms. The Ma entropy already differs from the true
entropy at first order:
THMa
t=1
ln 2/summationdisplay
a/summationdisplay
ta
1/an}bracketle{tra(ta
1;s)∆t/an}bracketri}hts
−/summationdisplay
a;ta
1/an}bracketle{tra(ta
1;s)∆t/an}bracketri}htslog2/summationtext
a;ta
1/an}bracketle{tra(ta
1;s)/an}bracketri}ht2
s∆t
/summationtext
a;ta
1/an}bracketle{tra(ta
1;s)/an}bracketri}hts(12)
The first order approximation coincides with the true en-
tropy rate if and only if there are no variations of rate
across time and cells. If there were higher frequency ratevariations, or more cells with different response profiles,
the Ma bound would be worse than depicted.
10010110210310400.511.522.5
Trials per stimulusEntropy (bits)<
<
total entropy (Bayes counting)
noise entropy (Bayes counting)
total entropy (naive counting)
frequency entropy (naive counting)
frequency entropy (uncorrected)
Ma bound 1001011021031040.10.20.30.40.50.6 coding efficiency
FIG. 1. Estimates of the entropy of an inhomogeneous
Poisson process with mean rate r(t) = 50 sin(2 π50t)
spikes/sec. in response to one stimulus, and zero spikes/se c.
for a second equiprobable stimulus. 30 ms time windows of
data are used for all curves, with a bin width of 3 ms. The ar-
rowhead at the upper right corner indicates the true entropy ,
calculated analytically. Inset : the effect of entropy and in-
formation bias on the estimated coding efficiency. Compared
are the bias-corrected series entropy (our best estimate; s olid
line) and the uncorrected frequency entropy (dot-dashed li ne).
The mutual information suffers less from sampling
problems than do the individual entropies, since to some
extent the biases of the total and noise entropies elim-
inate. This is particularly true for a small number of
stimuli; the effect will diminish as the number of stim-
uli increases, and the bias behaviour of the information
can be expected to become worse (see Eqn. 11). A re-
lated quantity often used to characterise neural coding,
the coding efficiency [15,5] (defined as the mutual infor-
mation divided by the total entropy), does not have this
elimination property, and in fact compounds the effects
of both total and noise entropy biases. This is shown
in the inset of Fig. 1, which shows the coding efficiency
versus the number of trials of data per stimulus for both
the bias-corrected series estimator (solid) and the raw
frequency (‘brute-force’ ) estimator (dot-dashed). One
might be cautioned against use of the brute-force ap-
proach for calculating the information efficiency.
To demonstrate their applicability, we applied these
techniques to data recorded from the primary visual cor-
tex (V1) of anaesthetised macaque monkeys [17]. Theone empirical assumption made in this analysis (Eq. 3 -
that the probability of observing a spike at time tigiven
that one has been observed at time tjscales with ∆ t) was
examined by computing the average conditional spiking
probability as the bin width is decreased (i.e. spikes are
observed with higher precision). This assumption might
be expected to break down if there were spikes synchro-
nised with near-infinite precision. For all cells examined,
the assumption was valid, as is shown in the inset of
Fig. 2.
2 3 4 5 6 7 8 9 1000.511.522.533.544.5 bits in 30 ms
Precision ∆t (ms)Entropies
Informationcomplex cells
simple cells
1 1010−310−210−1100
Precision ∆ t (ms)Probability
00.10.20.3
Coding efficiencyEfficiencies
FIG. 2. Temporal entropy and information in spike trains
recorded from 7 simple and 14 complex cells in the monkey
primary visual cortex. For clarity population mean and stan -
dard error are shown. Black symbols at the far right indicate
complex cell spike count entropy (diamond) and information
(circle). White symbols similarly for simple cells. Coding ef-
ficiencies are also shown (lines without error bars; right ax es).
Inset : The conditional spiking probability scales with ∆ tas
the binwidth becomes small. Each symbol type represents a
different cell (black, a complex cell and white, a simple cell ).
Fig. 2 shows entropy estimates for two classes of V1
cells. The entropy of the spiking process continued to
rise as the observation precision was increased, up to a
resolution of 2 ms. For 30 ms time windows and 2 ms
bin width, the information rate for the complex cells was
9±1 (s.e.m.) bits/sec., or 0.5 bits per spike. For the
simple cells it was 11 ±2 bits/sec. or 1.5 bits per spike.
The coding efficiencies of up to 19 and 31% (maximal in
the spike count limit) for this type of stimulation were
substantially below the >50% efficiencies that have been
reported for insect visual neurons [5,6,16].
As neuroscience enters a quantitative phase of develop-
ment, information theoretic techniques are being found
useful for the analysis of data from physiological experi-
ments. Sampling considerations have however prevented
their application to many interesting experimental prepa-rations. The methods developed here broaden the scope
of the study of neuronal information properties consider-
ably. In particular, they make possible the reliable anal-
ysis of recordings from both the anaesthetised and awake
mammalian cerebral cortex.
SRS is supported by the HHMI, and SP by the Well-
come Trust.
[1] N. Wedenskii, Bull. de l’Acad. de St. Petersbourg
XXVIII , 290 (1883); E. D. Adrian, J. Physiol. (Lond.)
61, 49 (1926).
[2] D. MacKay and W. S. McCulloch, Bull. Math. Biophys.
14, 127 (1952).
[3] F. Theunissen et al., J. Neurophys. 75, 1345, 1996.
[4] R. R. de Ruyter van Steveninck et al., Science 275, 1805
(1997);
[5] F. Rieke, D. Warland, R. de Ruyter van Steveninck, and
W. Bialek, Spikes: exploring the neural code (MIT Press,
Cambridge, MA, USA, 1997).
[6] S. Strong et al., Physical Review Letters 80, 197 (1998).
[7] A number of previous studies have reported first order
expansions of the information: W. E. Skaggs, B. L. Mc-
Naughton, K. Gothard, and E. Markus, in Advances in
Neural Information Processing Systems , eds. S. Hanson,
J. Cowan, and C. Giles (Morgan Kaufmann, San Mateo,
1993), Vol. 5, pp. 1030–1037; S. Panzeri et al.,Network
7, 365 (1996).; N. Brenner et al.physics/9902067 . Sec-
ond order series expansion of the spike count information
from an ensemble of cells was performed in [8]. An alter-
native cluster expansion method has also been used by
M. de Weese, Network 7, 325 (1996).
[8] S. Panzeri et al.Proc. R. Soc. Lond. B 266, 1001 (1999).
[9] G. T. Buracas et al., Neuron 20, 959 (1998).
[10] S. Panzeri and S. R. Schultz, physics/9908027 .
[11] L.F. Abbott and P. Dayan, Neur. Comp. 11, 91-101
(1999); M. W. Oram et al., Trends in Neurosci. 21, 259-
265 (1998).
[12] C. F. Brody, Neur. Comp. 11, 1536 (1999).
[13] S. Panzeri and A. Treves, Network 7, 87 (1996).
[14] S.-K. Ma, J. Stat. Phys. 26, 221 (1981).
[15] F. Rieke, D. Warland and W. Bialek, Europhys. Lett. 22,
151 (1993).
[16] A. Dimitrov and J. P. Miller, Neurocomputing, in press.
[17] The data used was from procedures to extract the ori-
entation tuning of V1 cells. The stimuli were sinusoidal
gratings of 16 different orientations placed over the re-
ceptive field of the cell. Each cycle of the moving grating
was considered to be an experimental trial; only cells with
≥96 cycles available were selected from the database, in
order to study timing precisions as high as 2 ms. J. R.
Cavanaugh, W. Bair and J. A. Movshon, Society for Neu-
roscience Abstracts, 24, 1875 (1998); J. R. Cavanaugh,
W. Bair and J. A. Movshon, Society for Neuroscience Ab-
stracts, 25, 1048 (1999). See J. A. Movshon and W. T.
Newsome, J. Neurosci. 16, 7733, 1996 for experimental
methods from this laboratory. We thank J. Cavanaugh,
W. Bair and J.A. Movshon for making their data avail-
able to us. |
arXiv:physics/0001007v1 [physics.ins-det] 4 Jan 2000The preliminary results of fast neutron flux measurements
in the DULB laboratory at Baksan
J.N.Abdurashitova, V.N.Gavrina, A.V.Kalikhova, A.A.Klimenkoa,b, S.B.Osetrova,b,
A.A.Shikhina, A.A.Smolnikova,b, S.I.Vasilieva,b, V.E.Yantza, O.S.Zaborskayaa.
aInstitute for Nuclear Research, 117312 Moscow, Russia
bJoint Institute for Nuclear Research, 141980 Dubna, Russia
(4 Jan 2000)
One of the main sources of a background in underground physic s experiments (such as the investi-
gation of solar neutrino flux, neutrino oscillations, neutr inoless double beta decay, and the search for
annual and daily Cold Dark Matter particle flux modulation) a re fast neutrons originating from the
surrounding rocks. The measurements of fast neutron flux in t he new DULB Laboratory situated
at a depth of 4900 m w.e. in the Baksan Neutrino Observatory ha ve been performed. The relative
neutron shielding properties of several commonly availabl e natural materials were investigated too.
The preliminary results obtained with a high-sensitive fas t neutron spectrometer at the level of
sensitivity of about 10−7neutron cm−2s−1are presented and discussed.
PACS numbers: 06.90.+v, 29.30.Hs
INTRODUCTION
It is well known that one of the main sources of a back-
ground in underground physics experiments (such as the
investigation of solar neutrino flux, neutrino oscillation s,
neutrinoless double beta decay, and the search for annual
and daily Cold Dark Matter particle flux modulation)
are fast neutrons originating from the surrounding rocks.
The sources of the fast neutrons are ( α, n) reactions on
the light elements contained in the rock (C, O, F, Na, Mg,
Al, Si). Neutrons from spontaneous fission of238U take
an additional contribution in a total fast neutron flux
of about 15-20%. Several research groups have investi-
gated the neutron background at different underground
laboratories [1–3]. Some of them used6Li-dopped liquid
scintillator technique [3], and others used in addition a
Pulse Shape Discrimination technique [1].
The measurements of fast neutron flux in the Deep Un-
derground Low Background Laboratory of Baksan Neu-
trino Observatory (DULB BNO) have been performed
with using of a special, high-sensitive fast neutron spec-
trometer [4]. This laboratory is located under Mt.
Andyrchy (Northen Caucasus Mountains, Russia) in a
tunnel that penetrates 4.5 km into the mountain, at a
depth of 4900 meters of water equivalent.
The results of such measurements lead to a conclu-
sion that a neutron background places a severe limitation
on the sensitivity of current and planned experiments.
Owing this fact, the development of new cost-effective,
high-strength radiation shielding against neutrons be-
comes a very important task for modern non-accelerator
physics experiments. For such purposes the relative neu-
tron shielding properties of several commonly available
natural materials were investigated too. Specially, these
materials are planned for use in the construction of large-
volume underground facilities which will be covered with
suitable shielding materials and are situated in the DULB
Laboratory at Baksan.NEUTRON DETECTOR
The spectrometer was constructed to measure
low background neutron fluxes at the level up to
10−7cm−2s−1in the presence of intensive gamma-ray
background.
The detector consists of 30 l liquid organic scintilla-
tor viewed by photomultipliers with 19 neutron coun-
ters (3Heproportional counters) uniformly distributed
through the scintillator volume (see [4] for detail). The
spectrometer schematic view and the principle of opera-
tion are shown in Fig. 1.
Fast neutrons with En>1 MeV entering the liquid
scintillator (LS) are moderated down to thermal energy,
producing a LS signal. Then they diffuse through the de-
tector volume to be captured in3Hecounters or on pro-
tons in the scintillator. The LS signal starts the recording
system. After triggering the system waits a signal from
any of the helium counters for a specific time. This time
window corresponds to the delay time between correlated
events in the scintillator and in the helium counters. This
is one of specific features of the detector. The signal from
the LS is ’marked’ as a coincident with a neutron capture
in the3Hecounters only in the case if a single counter
is triggered during the waiting period. An amplitude of
the ’marked’ LS signal corresponds to an initial neutron
energy. This method allows us to suppress the natural
γ-ray background considerably.
The described event discriminating procedure allows
us to measure extremely low neutron fluxes at the level
up to 10−7cm−2s−1reliably even if the LS counting rate
is as large as several hundred per second. The dead time
of the detector is equal to the delay time (variable value,
but generally about 120 µs) plus about 400 µs, which is
needed to analyze a LS event whether it corresponds to
neutron or not. The detection efficiency depends in a
complicated manner on the response function of the de-
tector. As a rough estimation, we use the value of the
efficiency, which is equal to 0 .04±0.02 in the energy range
1FIG. 1. The neutron spectrometer schematic view and a
principle of operation. (1) is a PMT, (2) is a liquid scintill ator
and (3) is a3Hecounter.
from 1 to 15 MeV. This is based on preliminary measure-
ments performed with a Pu-Be source. Owing this fact,
an absolute values of the neutron fluxes can be estimated
with an uncertainty of 50% on the basis of available cal-
ibration data. The delay time is a specific feature of the
detector and depends on the detector design. The acqui-
sition system allows us to measure the delay time for the
neutron events directly. Such measurements were carried
out using a Pu−Besource with a time window selected
to be equal to 300 µs. A typical delay time distribution
is shown in Fig. 2 A fitting procedure leads to a time
constant of T1/2∼55µs. According to this result it is
sufficient to select the time window to be equal 120 µsfor
an actual measurement.
MEASUREMENTS
A. The geometry
It has been mentioned that we have no yet precise
information about the detection efficiency, that is why
one can calculate absolute value of neutron fluxes with
only 50% certainty. However, it is possible to measure
the relative neutron absorption abilities of various shiel ds
FIG. 2. Delay time distribution for Pu-Be neutron source
TABLE I. Concentration of U, Th and K in the rock sam-
ples.
Sample238U, g/g232Th, g/g40K, g/g
Quartzite (1 .1±0.1) 10−7(4.3±0.1) 10−7(1.9±0.03) 10−7
Serpentine (2 .2±0.5) 10−8(2.0±0.9) 10−8<1.2 10−8
Surrounding (1 .6±0.3) 10−6(4.0±0.1) 10−6(1.6±0.1) 10−6
mine rock
with high precision. This information will be very use-
ful for development of new low background experiments
and searching for cost-effective neutron absoption shields .
Such measurements were carried out in the DULB BNO
with using the described neutron spectrometer. This new
laboratory, consisted of 8 separate counting facilities, i s
located under Mt. Andyrchy in a tunnel, which pen-
etrates 4.3 kminto the mountain, at a depth of 4900
m w.e.
Quartzite and serpentine were selected as materials to
be tested because of comparatively low concentrations of
uranium- and thorium-bearing compounds contained in
these rocks. For instant, the measured concentrations
of uranium and thorium for rock serpentine are about
10−8g/g in comparison with 10−6g/g for the surround-
ing rock. As for potassium (40K) contained in serpen-
tine, it has been found less than 10−8g/g in comparison
with 10−6g/g for the surrounding rock. Measurements of
gamma-activity of different rock samples have been per-
formed with using a well-type NaI gamma spectrometer
with level of sensitivity of about 10−9g/g, operated in one
of the underground low counting facilities at BNO [5].
The measured Th, U, and K concentrations in different
rock samples are given in Table I.
Four series of measurements were performed with the
neutron spectrometer surrounded by different radiation
2FIG. 3. Schematic view of cross-section of the DULB ex-
perimental site and the detector inside the neutron shield. All
sizes in millimeters.
shields. In the first series the spectrometer was sur-
rounded with a lead shield 4 cmthickness (to reduce the
natural gamma-ray counting rate), and measurements of
the natural neutron background radition field existing in
the open experimental site were performed. In the second
and third series the spectrometer was surrounded with
shields of quartzite and serpentine, respectively. The
rock shields consisted of broken pieces of various sizes,
ranging from 1 cmto 15 cm, with an effective shield
thickness of 35 cmin all directions. The mean relaxation
length of fast neutrons in these shields is about 15 cm(25
g/cm2for quartzite and 21 g/cm2for serpentine). In the
fourth series we measured the internal background of the
detector using a neutron-absorbing shield consisted of 40-
cm thick section of polyethylene containing an admixture
of boron and water about 30 cmthick. Schematic view of
one of the investigated neutron shield and cross-section
of the DULB experimental site are shown in Fig.3.
B. Calibration
A60Co γ-source has been used to calibrate the LS-
channel. The energy of the middle of the Compton edge
was assumed to be equal to 1 MeV in the electron en-
ergy scale, which corresponds to ∼3MeV in the neutron
energy scale (see Fig.4a). A Pu-Be source was used to
calibrate the NC-channel of3Hecounters. The spectrum
produced by the Pu-Be source in the3Hecounters has
a specific shape due to a wall effect which distorts the
counter event spectrum (see Fig. 4b). In spite of this
distortion, the range of energies observed for true neu-
tron events is less narrow compared to the broad back-
FIG. 4. Calibration spectra. (a) liquid scintillator irra-
diated with60Co; (b)3He counters irradiated with Pu-Be
source.
ground spectrum produced by internal alphas. Using of
the only events from the neutron window coincident with
LS signals makes it possible to suppress the internal back-
ground of the detector.
C. Conditions of measurements
Main conditions for all series of measurements, such as
measuring times, LS- and NC-counting rates are given in
the Table II.
The typical exposure time for each series was a few
weeks. The γ-ray background in the open experimen-
tal site is high enough that leads to γ-counting rate in
the LS-channel of about 700 s−1. Due to this fact, fol-
lowing values of dead time were determined for different
series: 12% of the total exposure time for measurements
with the lead shield, 4 .3% for quartzite series, 2 .7% for
serpentine series, and 1 .5% for measurements with the
polyethylene/water shield. To calculate the true neutron
counting rates a proper dead time correction has been
performed.
3TABLE II. Conditions of measurements.
Value No shield Quartzite Serpentine Water+
(5 cm lead) Polyethylene
The measuring time, h 400 290 950 605
Dead Time, % 12 4.3 2.7 1.5
Total LS-rate, s−1202 83 62 21
Total NC-rate, h−1123±0.6 103 ±0.6 92 ±0.3 95 ±0.4
NC-rate in neutron window, h−158±0.4 37 ±0.4 27 ±0.2 27 ±0.2
Random coincidences rate, h−11.41±0.005 0.38 ±0.002 0.19 ±0.001 0.07 ±0.001
Rnneutron counting rate, h−129.6±0.5 9.6 ±0.5 -0.2 ±0.3 —
DATA TREATMENT AND RESULTS
Contamination of222Rngas inside the experimental
site can make a considerable contribution (up to 20%)
to the background γ-counting rate, which can influence
results of the performed measurements because222Rn
activity can vary significantly for a period of a measure-
ment.
To suppress the count rate variation effect we used a
special procedure for treatment of experimental data. It
consists of the following steps.
Two types of data files are stored as a result of a mea-
surement. One of them contains the information about
neutron energy losses ( LS-signal amplitudes),3Hecoun-
ters signal amplitudes, and delay time for each ’neutron’
candidate event. Data accumulation was stopped every
half-hour and overall numbers of NC-counts, LS-counts,
LS-counts above 1 MeV, and elapsed time were saved in
a file. Total background γ- spectra for every half-hour
run were measured simultaneously and saved in a sec-
ond file to make it possible to take into account a time
variation of the background γ-counting rate.
We consider three contributions into the experimen-
tally measured counting rate Rmeas: the random coinci-
dence rate Rrnd, the internal detector background count-
ing rate Rbkg, and the ’neutron’ counting rate Rn, so that
Rn=Rmeas−Rrnd−Rbkg (1)
We have made obvious assumption that the total back-
ground γ-spectrum and the random coincidence spec-
trum have the same shapes. To obtain random coin-
cidence spectrum for further subtraction procedure the
total background γ-spectrum has been normalized with
a factor corresponding to the calculated random coinci-
dence rate. The maximal evaluation for the random co-
incidence rate, if the LS- and NC- events are absolutely
independent, can be calculated by the following way:
Rrand=rγrw
n∆t, (2)
where rγis the γ-rate, rw
nis the3He-counters count-
ing rate in the determined neutron energy window, ∆ t
is the time window. In the case of the performed mea-
surements ( RLS≫RHe), this evaluation is very close
to the real counting rate of random coincidences. Due
to a variation in time of the222Rnactivity, the current
value of ri
γdepends on time too. Owing to this fact,we applied the described subtraction procedure to each
half-hour run with corresponding current value of Ri
rnd,
and then summarized resulting neutron spectra in a total
serial spectrum. The accumulated LS-spectra of all coin-
cidented events ( Rmeas) and the recalculated spectra of
random coincidence ( Rrnd) for the no-shield, quartzite,
and serpentine series are presented in Fig. 5.
An internal detector background spectrum Rbkghas
been accumulated inside the neutron-absorbing shield
consisting of polyethylene and water. Obtained count-
ing rate of the internal background correlated (neutron-
type, but non-neutron) events was measured as 27 counts
per hour, which in terms of a neutron flux corresponds
to (8.1±0.5) 10−7s−1cm−2. The residual LS-spectra
(Rmeas−Rrnd) in comparison with the internal back-
ground LS-spectrum ( Rbkg) are presented in Fig. 6.
Performing the total subtraction procedure in accord-
ing with the equation (1) we obtain values of the neutron
counting rate Rnfor the no-shield, quartzite, and serpen-
tine series. Taking into account the detection efficiency
uncertainty ( ε= 0.04±0.02) the obtained values of fast
neutron fluxes (above 700 keVof neutron energy) are
presented here in a following way:
a(3.5±1.1) 10−7s−1cm−2for the no-shield measure-
ment,
a(2.9±1.1) 10−7s−1cm−2for quartzite shield,
a(0.6±0.7) 10−7s−1cm−2for serpentine shield,
were a = ( ε+ ∆ε)/ε.
One can see that the resulting neutron flux measured
when the serpentine shield was in place were found to
be at about the minimum level of sensitivity of the spec-
trometer. It means that a neutron background inside the
serpentine shield is consisted with a neuron flux less than
0.7 10−7s−1cm−2. It indicates that serpentine is indeed
clear from uranium and thorium, and is, therefore, the
most likely candidate for use as a cost-effective neutron
shield component material for large-scale low background
experiments.
A delay time distribution analysis was performed to
understand the origin of a high level of the internal de-
tector background.
DELAY TIME DISTRIBUTIONS
Decays of Bi and Po radioactive isotopes, such as
4FIG. 5. The accumulated LS-spectra of all coincidented
events ( Rmeas, solid) and the recalculated spectra of random
coincidences ( Rrnd, dashed)
214Bi(e,˜ν)164µs→214Po(α)→..., (3)
which can take place in the helium counter walls, have
been considered as main possible sources of the signifi-
cant internal background. To imitate an actual neutron
event beta decay of214Bican fire the liquid scintillator,
followed by a delayed capture α- signal from Po decay
in helium counters. The delay time distribution of the
neutron-type coincidented events obtained for the series
in the water shield is shown in Fig. 7. Fitting procedure
leads to the time constant T1/2= 164 µs.
It means that, as it was supposed, the origin of the
internal background of our detector is mostly due to con-
tamination of214Biin the3He-counter walls. The delay
time distributions for other series of measurements are
shown in Fig. 8.
The following fitting function was used to analize these
distributions ( tis expressed in µs):
A+Ne−t ln2/55+Be−t ln2/164, (4)
where A is a constant, N is an amplitude corresponding
to neutrons and B corresponds to internal background.
FIG. 6. Residual LS-spectra ( Rmeas-Rrnd, solid) in com-
parison with the internal background LS-spectrum ( Rbkg,
dashed).
The ratio N/B, which was obtained in this manner, de-
creases from measurements in the lead shield to the mea-
surements in the serpentine shield.
CONCLUSIONS
The main results of the measurements can be summa-
rized as follows.
(I). The preliminary results obtained from the fast
neutron spectrum accumulated in the open experimen-
tal site of the DULB Laboratory at Baksan is consisted
with a neutron flux (for neutrons with energy above 700
keV) estimated as values from 5 .3×10−7cm−2s−1to
1.8×10−7cm−2s−1depending on the present uncertainty
in determination of the detection efficiency.
(II). The neutron spectrometer sensitivity in a shielded
experimental site is estimated as 0 .5×10−7cm−2s−1for
a measuring time of about 1000 h.
(III). It is shown that the main source of the detec-
tion sensitivity limitation, rather then random coinci-
dences, is the internal background of the spectrometer,
5FIG. 7. Delay time distribution for the coincident events
measured in the water shield.
which is mostly due to the presence of α-particle emitters
(214Bi−214Podecays) in the3He-counters walls.
(IV). The achieved neutron background inside the ser-
pentine shield is consisted with a neutron flux less than
0.7 10−7s−1cm−2. It indicates that serpentine is one of
the more likely candidate for use as a cost-effective neu-
tron shield component material for large-scale low back-
ground experiments.
We have obtained the presented results using the sim-
ple event discrimination procedure and did not use pulse
shape discrimination yet. Nevertheless, it takes us a pos-
sibility to measure extremely low neutron fluxes up to
10−7cm−2s−1even when external γ-counting rate is more
than 200 s−1.
ACKNOWLEDGMENTS
We are grateful to I.I.Pyanzin for the management in
proving of reserves and quarrying of the domestic ultra
basic rock samples. We thank P.S.Wildenhain for care-
ful reading of this article and his critical remarks. We
acknowledge the support of the Russian Foundation of
Basic Research. This research was made possible in part
by the grants of RFBR No. 98–02 16962 and No. 98-02-
17973.
FIG. 8. Delay time distributions for the coincident events
measured in the series with no shield, quartzite and serpen-
tine.
[1] V. Chazal, B. Chambon, M. De Jesus et al. Astroparticle
Physics, v.9, n.2 (1998) pp. 163-172
[2] F. Arneodo, F. Cavanna, S. Parlatti et al. INFN/AE-
97/52 (1997)
[3] R.Alexan et al. NIM A274 203(1989).
[4] J.N. Abdurashitov, V.N. Gavrin, G.D. Efimov, A.V. Ka-
likhov, A.A. Shikhin and V.E. Yants, “Instrum. and Exp.
Tech.” , Vol. 40, No 6, 1997, pp. 741–752.
[5] A.A.Klimenko, A.A.Pomansky, A.A.Smolnikov, NIM B17
445(1986).
6 |
arXiv:physics/0001008v1 [physics.atom-ph] 5 Jan 2000A semi–classical over–barrier model for charge exchange be tween highly charged ions
and one–optical electron atoms
Fabio Sattin∗
Consorzio RFX, Corso Stati Uniti 4, 35127 Padova, ITALY
Absolute total cross sections for electron capture between slow, highly charged ions and alkali
targets have been recently measured. It is found that these c ross sections follow a scaling law
with the projectile charge which is different from the one pre viously proposed basing on a classical
over–barrier model (OBM) and verified using rare gases and mo lecules as targets. In this paper
we develop a ”semi–classical” (i.e. including some quantal features) OBM attempting to recover
experimental results. The method is then applied to ion–hyd rogen collisions and compared with
the result of a sophisticated quantum-mechanical calculat ion. In both cases the present method is
found to underestimate by a factor two the correct result but , where comparison can be made, it is
superior to other OBMs. A qualitative explanation for the di screpancies is also given.
PACS numbers: 34.70+e, 34.10.+x
I. INTRODUCTION
The electron capture processes in collisions of slow, highl y charged ions with neutral atoms and
molecules are of great importance not only in basic atomic ph ysics but also in applied fields such as
fusion plasmas and astrophysics.
In the past years a number of measurements have been carried o n the collisions between highly
charged ions and rare gases [1] or molecules [2], in which one or several electrons were transferred
from the neutral target to a charged projectile:
A+q+B→A(q−j)++Bj+. (1)
Their results-together with those from a number of other lab oratories-yielded a curve which can
be fitted within a single scaling law (a linear relationship) when plotting cross section σversus
projectile charge q: it is almost independent of the projectile species and of th e impact velocity
v(at least in the low–speed range v <1 au). When one extends experiments to different target
species, the same linear relation holds between σandq/I2
t, with Itthe ionization potential of the
target [3,4].
It is found that this scaling law could to be predicted, in the limit of very high projectile charge, by
a modification of an extended classical over–barrier model ( ECBM), allowing for multiple electron
capture, proposed by Niehaus [5]. Quite recently a confirmat ion of this scaling has come from a
sophisticated quantum–mechanical calculation [6].
Similar experiments were carried on more recently for colli sions between ions and alkali atoms
[7]. The results show that the linear trend is roughly satisfi ed, but the slope of the straight line is
grossly overestimated by the ECBM: in Fig. 1 we show some data points (stars with error bars)
together with the analytical curve from the ECBM (dashed cur ve) which, for one–electron atoms,
is written [3,4]
σ= 2.6×103q/I2
t[10−20m2] (2)
(Itin eV). It should be noticed that experimental data are inste ad well fitted by the results of a
Classical Trajectory MonteCarlo (CTMC) code [7].
The ECBM of ref. [3] works in a simplified one-dimensional geo metry where the only physically
meaningful spatial dimension is along the internuclear axi s. It does not take into account the fact
that the electrons move in a three-dimensional space. This m eans that only a fraction of the electrons
actually can fulfil the conditions dictated by the model. For rare gases and molecules, which have a
large number of active electrons, this can be not a trouble (i .e., there are nearly always one or more
∗E-mail: sattin@igi.pd.cnr.it
1electrons which can participate to the collision). For alka li atoms with only one active electron, on
the other hand, an overestimate of the capture probability b y OBM’s could be foreseen.
With present–days supercomputers there are relatively few difficulties in computing cross sec-
tions from numerical integration of the time-dependent Sch r¨ odinger equation (e.g. refer to ref. [6]).
Notwithstanding this, simple models are still valuable sin ce they allow to get analytical estimates
which are easy to adapt to particular cases, and give physica l insight on the features of the problem.
For this reason new models are being still developed [8,9].
In this paper we present a modified OBM which allows to get a bet ter agreement with the experi-
mental data of ref. [7].
II. THE MODEL
We start from the same approach as Ostrovsky [8] (see also [10 ]): berthe electron vector relative
to the neutral atom ( T) andRthe internuclear vector between Tand the projectile P(see Fig.
2 for a picture of the geometry: it is an adaptation of Figure 1 from ref. [8]). Let us consider the
plane containing the electron, PandT, and use cylindrical polar coordinates ( ρ,z, φ ) to describe the
position of the electron within this plane. We can choose the angle φ= 0 and the zdirection along
the internuclear axis. We will assume that the target atom ca n be described as an hydrogenlike
atom, which is not a bad approximation when dealing with alka li atoms.
The total energy of the electron is
E=p2
2+U=p2
2−Zt/radicalbig
ρ2+z2−Zp/radicalbig
ρ2+ (R−z)2. (3)
ZpandZtare the charge of the projectile and the effective charge of th e target seen by the electron,
respectively, and we are using atomic units.
We can also approximate Eas
E(R) =−En−Zp
R. (4)
Enis given by the quantum–mechanical value: En=Z2
t/(2n2). This expression is asimptotically
correct as R→ ∞.
On the plane (e, P,T) we can draw a section of the equipotential surface
U(z, ρ,R ) =En−Zp
R. (5)
This represents the limit of the region classically allowed to the electron. When R→ ∞ this region is
decomposed into two disconnected circles centered around e ach of the two nuclei. Initial conditions
determine which of the two regions actually the electron liv es in.
AsRdiminishes there can be eventually a time where the two regio ns become connected. It is easy
to solve eq. (5) for Rby imposing that ρm= 0 and that there must be an unique solution for z
with 0 < z < R :
Rm=Zt+/radicalbig
ZtZp
En. (6)
In the spirit of OBMs it is the opening of the equipotential cu rve between PandTwhich leads
to a leakage of electrons from one nucleus to another, and the refore to charge exchange. Along the
internuclear axis the potential Uhas a maximum at
z=z0=R√Zt/radicalbig
Zp+√
Zt. (7)
Whether the electron crosses this potential barrier depend s upon its initial conditions. These are
chosen from a statistical ensemble, which we will leave unsp ecified for the moment. Let NΩbe the
fraction of trajectories which lead to electron loss at the t imetandW(t) the probability for the
electron to be still bound to the target, always at time t. The fraction of losses in the interval t, t+dt
is given by
dW(t) =−NΩdt
TemW(t), (8)
2withTemthe period of the electron motion along its orbit. A simple in tegration yields the leakage
probability
Pl= 1−exp/parenleftbigg
−1
Tem/integraldisplay+∞
−∞NΩdt/parenrightbigg
. (9)
In order to actually integrate Eq. (9) we need to know the coll ision trajectory; an unperturbed
straight line with bimpact parameter is assumed:
R=/radicalbig
b2+ (vt)2. (10)
At this point it is necessary to give an explicit expression f orNΩ. The electron is supposed to be in
the ground state ( n= 1, l=m= 0). Tembecomes therefore [11]
Tem= 2π/Z3
t. (11)
Ref. [8] adopts a geometrical reasoning: the classical elec tron trajectories, with zero angular momen-
tum, are ellipses squeezed onto the target nucleus. The only trajectories which are allowed to escape
are those whose aphelia are directed towards the opening wit hin the angle ±θm. The integration
over this angle yields an analytical expression for NΩ(Eq. 17 of ref. [8]). In Fig. 1 we show the
results obtained using Ostrovsky’s model ( dotted curve–eq ns. 8,17 of ref. [8])1. Notice that from
direct inspection of the analytical formula, one sees that t he scaling law is not exactly satisfied, at
least at small values of the parameter q/I2
t, and this is clearly visible in the plot. The result is
almost equal to the scaling (2).
The present approach is based on the electron position instead than on electron direction . The
recipe used here is (I) to neglect the dependence from the ang le: all electrons have the same proba-
bility of escaping, regardless of their initial phase. Inst ead, (II) the lost electrons are precisely those
which, when picked up from the statistical ensemble, are fou nd farther from nucleus Tthan the
distance z0:
NΩ=/integraldisplay∞
z0f(r)dr , (12)
withf(r) the electron distribution function.
There is not a unique choice for f(r): the (phase-space) microcanonical distribution
˜f(r,p)∝δ/parenleftbigg
En+p2
2−Zt
r/parenrightbigg
(13)
(δis the Dirac delta) has been often used in literature since th e works [12] as it is known that,
when integrated over spatial coordinates, it reproduces th e correct quantum–mechanical momentum
distribution function for the case of the electron in the gro und state [13] (more recently the same
formalism has been extended to Rydberg atoms [14]). After in tegration over momentum variables
one gets instead a spatial distribution function [15]
fmc(r) =Zt(2Zt)3/2
πr2/radicalbigg
1
r−Zt
2, r < 2/Zt (14)
and zero elsewhere (The lowerscript ”mc” is to emphasize tha t it is obtained from the microcanonical
distribution). However, this choice was found to give poor r esults. It could be expected on the basis
of the fact that (14) does not extend beyond r= 2/Ztand misses therefore all the large impact–
parameter collisions. In the spirit of the present approach , it should be instead important to have an
accurate representation of the spatial distribution. We us e therefore for f(r) the quantum mechanical
formula for an electron in the ground state:
f1s(r) = 4Z3
tr2exp(−2Ztr) (15)
which, when substituted in (12), gives
1Beware of a small difference in notation between the present p aper and [8]: here we use an effective charge for the target,
Zt=√2En, while [8] uses an effective quantum number nt= 1/√2Enwith the effective charge of the target set to 1.
3NΩ=/bracketleftbig
1 + 2r0Zt+ 2(r0Zt)2/bracketrightbig
exp(−2r0Zt). (16)
Since the choice for f(r) does not derive from any classical consideration, we call t his method a
“semi–classical” OBM.
Notice that, in principle, one could go further and compute f(r) from a very accurate wavefunction,
fruit of quantum mechanical computations (see [16]), but th is is beyond the purpose of the present
paper (it could be worthy mentioning a number of other attemp ts of building stationary distributions
f(r), mainly in connections with CTMC studies, see [17–19]).
Thef(r) of Eq. (15) does not reproduce the correct momentum distrib ution, nor the correct energy
distribution (which could be obtained only by using eq. (13) . However, it is shown in [15] that this
choice gives an energy distribution for the electrons, f(E), peaked around the correct value En, and
< E > =En, where < . . . > is the average over f(E).
Some important remarks are to be done here. First of all, a que stion to be answered is: why use an
unperturbed distribution, when the correct one should be se nsitively modified by the approaching of
the projectile. The answer is, obviously, that this choice a llows to perform calculations analitically.
We are doing here a sort of classical counterpart of a quantum –mechanical Born calculation: there,
too, the matrix elements are computed as scalar products ove r unperturbed states, regardless of
any perturbation induced by the projectile. In the followin g, however, some considerations about
possible improvements over this simple approximation will be done.
A second question regards the meaning of the factor dt/T emin eq. (8): in Ostrovsky’s paper
this is the fraction of electrons which enter the loss zone du ring the time interval dtand is valid
under the hypothesis of a uniform distribution of initial ph ases of the electrons. In our case this
this assumption ceases to be valid: electrons actually spen d different fractions of their time at
different radial distances from T, depending on their energy. We will do a (hopefully not too se vere)
assumption by assuming that, on the average, the expression (8) still holds.
III. RESULTS
A. Iodine - Cesium
This study has been prompted by the ion-atom experiments of [ 7]: first of all, therefore, we apply
the above model to the process of electron capture
Iq++ Cs→I(q−1)++ Cs+(17)
withq= 6÷30. Impact energy is 1 .5×qkeV [7]. The ionization potential of Cesium is It= 3.9
eV. Solid line in Fig. 1 is the result of the present model: the ratio between the OBM result and
the true one is passed from a factor near three to something le ss than 50% (which means that, in
absolute value, the disagreement is reduced).
We will attempt here a possible explanation of this underest imate: we have used in Eq. (15)
an unperturbed wavefunction, while the electron distribut ion function must be modified by the
approaching of the projectile. Since the electron is attrac ted by P, the distribution function is likely
to be more weighted at larger r’s: in a quantum mechanical treatment, the perturbation of t he
electron due to the projectile would translate in an excitat ion to higher states. We will find in the
following subsection some results in favour of this hypothe sis.
B. Bare ions - Hydrogen
As second test, we have computed cross section for captures
H + O8+→H++ O7+(18)
and
H + He2+→H++ He+(19)
and compared it with similar calculations done using the mol ecular approach by Harel et al[20].
The results are summarized in fig. 3. There is a sharp discrepa ncy in the behaviour for v→0,
where the present model predicts an increasing cross sectio n. At very low speed it is the concept
4itself of atomic distribution function which becomes quest ionable, and molecular aspects become
important. Besides, quantum effects such as the discretenes s of the energy levels also play a major
role and are completely missed by this approach. In the highe r velocity part, the present model
underestimates the more accurate value by a factor 2 for proc ess (18), but the error is much less,
just 25 %, for process (19). These two ions have been chosen ad hoc : they correspond to values of
the ratio Zt/Zp= 1/8 and 1/2 respectively. In the (I, Cs) test this ratio ranged f rom≈1/12 to
≈1/60 depending upon the projectile charge. This means that in t he former case the perturbation
of the projectile on the electron distribution function is c omparable to the (I, Cs) case, while in the
latter it is much less. We expect the electron distribution f unction to be more and more perturbed
asZt/Zp→0.
IV. SUMMARY AND CONCLUSIONS
We have developed in this paper a very simple OBM for charge ex change. It exploits some features
of the quantum mechanical version of the problem, thus differ ing from similar models which are solely
classical. The agreement with experiment is better than pre vious calculations where a comparison
could be made. It is far from excellent, but reasons for the (p artial) failure have been suggested.
As it stands, the model is well suited for one-optical-elect ron atoms (since it uses hydrogen–like
wavefunctions), therefore we do expect that other classica l OBM’s can still work better in the
many-electrons targets studied in previous experiments.
Some improvements are likely to be added to the present model : a possible line of investigation
could be coupling the present method with a very simplified ca lculation of the evolution of the
wavefunction, using quantum mechanics. From this one shoul d not compute the fas coming from
a single state, but as a linear combination including also ex cited wavefunctions (the relative weights
in the combination should be given by the quantum mechanical calculation). Work in this direction
is currently underway.
ACKNOWLEDGMENTS
It is a pleasure to thank the staff at National Institute for Fu sion Science (Nagoya), and in
particular Prof. H. Tawara and Dr. K. Hosaka for providing th e data of ref. [7] and for useful
discussions about the subject. The referees through their s uggestions and criticism have made the
manuscript readable.
5[1] Hiroyuki A et al1997 Fus Eng Design 34-35 785
[2] Hosaka K et al1997 Phys Scr T73273
[3] Kimura M et al1995 J Phys B: At Mol Opt Phys 28L643
[4] Hosaka K et al1997 Fus Eng Design 34-35 781
[5] Niehaus A 1986 J Phys B: At Mol Phys 192925
[6] Nagano R, Yabana K, Tazawa T and Abe Y 1999 J Phys B: At Mol Op t Phys 32L65
[7] Hosaka K et al Electron capture cross sections of low energy highly ch arged ions in collisions with alkali atoms , poster
presented at the International Seminar on Atomic Processes in Plasmas held in Toki (Japan, 1999). To be published as a
report of the National Institute for Fusion Science (NIFS)
[8] Ostrovsky V N 1995 J Phys B: At Mol Opt Phys 283901
[9] Ivanovski G, Janev R K, and Solov’ev E A 1995 J Phys B: At Mol Opt Phys 284799
[10] Ryufuku H, Sasaki K and Watanabe T 1980 Phys Rev A 21745
[11] Landau L D and Lifshitz E M 1977 Quantum Mechanics (Oxford, Pergamon) Eq. (48.5)
[12] Abrines R and Percival I C 1966 Proc Phys Soc 88861
[13] Sattin F and Bolzonella T 1998 Phys Scr 5853
[14] Samengo I 1998 Phys Rev A 582767
[15] Cohen J S 1985 J Phys B: At Mol Phys 181759
[16] Ema J et al1999 At Data Nucl Data Tables 7257
[17] Eichenauer D, Gr¨ un N and Scheid W 1981 J Phys B: At Mol Phy s143929
[18] Hardie D J W and Olson R E 1983 J Phys B: At Mol Phys 161983
[19] Montemajor V J and Schiwietz G 1989 J Phys B: At Mol Opt Phy s222555
[20] Harel C, Jouin H and Pons B 1998 At Data Nucl Data Tables 68279
6FIGURE CAPTIONS
FIG. 1. Comparison between experimental data and predictio n from models for electron capture cross section of process
(17). Stars, experiment with 20% error bar; dashed line, sca ling law from Niehaus (Eq. 2); dotted line, Ostrovsky’s scal ing
law; solid line, scaling law from present model. σis in 10−20m2,Itin eV.
FIG. 2. Geometry of the scattering. PandTare the projectile and target nucleus respectively. The env eloping curve shows
a section of the equipotential surface U=E, i.e., it is the border of the region classically accessible to the electron. Ris the
internuclear distance. The parameter ρmis the radius of the opening which joins the potential wells, θmthe opening angle
fromT;z0is the position of the potential’s saddle point.
7FIG. 3. Capture cross section versus impact velocity. Upper, H–O8+collisions; lower, H–He2+collisions. Diamonds, data
from ref. 20; solid line, present model.
8 |
"arXiv:physics/0001009v1 [physics.atom-ph] 5 Jan 2000An Exact Approach to the Oscillator Radiation(...TRUNCATED) |
"arXiv:physics/0001010v1 [physics.atom-ph] 5 Jan 2000Spectrum of atomic radiation at sudden\npertu(...TRUNCATED) |
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