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Casimir effect in dielectrics
ANNALS
OF PHYSICS
115,
1-23 (1978)
Casimir JULIAN
SCHWINGER,
Effect in Dielectrics*
LESTER L. DERAAD,
JR., AND KIMBALL
Department of Physics, University of California,
Los
Angeles,
A. MILTON
California
90024
Received December 5, 1977
We reconsider the Casimir (van der Waals) forces between dielectrics with plane, parallel surfaces for arbitrary temperature, using the methods of source theory. The general results of Lifshitz are confirmed, and are shown to imply the correct forces on metal surfaces. The same phenomena give rise to contributions to the surface tension and the latent heat of an idealized liquid, contributions which, unfortunately, are not well defined since they depend upon a momentum cutoff. However, with a reasonable value for this cutoff, qualitative agreement with the experimentally observed surface tension and latent heat of liquid helium at absolute zero is obtained.
1. INTRODUCTION
One of the least intuitive consequences of quantum electrodynamics is the force between uncharged conducting surfaces, the so-called Casimir force [l]. However, another manifestation of this force is known to have a molecular origin. That is, the “Casimir” forces between bodies having different dielectric constants [2, 31 can be interpreted, in the limit of tenuous media, to arise from the retarded (I/R’) and shortrange (l/P) van der Waals potentials between the molecules that make up the bodies. The Casimir effect for plane dielectric surfaces was first worked out by Lifshitz over two decades ago [3]. His results, for zero temperature, have been confirmed in a beautiful experiment employing acoustical interferometry to accurately measure the thickness of a thin helium film [4]. (The measurements of forces between solid bodies are less conclusive [5].) However, questions remain in connection with his finite temperature results [6]. For the Casimir force between metal plates, obtained by setting the dielectric constant equal to infinity in his general formula, he found a temperature dependence which disagrees with that found in other calculations of this effect [7]. This, together with various theoretical objections, has raised doubts concerning the validity of his results for dielectrics. In this paper we will lay those doubts to rest. We begin by showing that Lifshitz’ general formula for the force, including the temperature dependence, is correct, and that the errors arise only in the limits taken to recover the conductor case. We will accomplish
this confirmation
methods for computing
by adopting
far superior
and more physically
transparent
the force (Sect. 2). We will consider carefully the limiting
* Work supported in part by the National Science Foundation 1
and the Alfred P. Sloan Foundation. OOO3-4916/78/1151-OOO1S05.00/0
All
Copyright 0 1978 by Academic Press. Inc. rights of reproduction in any form reserved.
2
SCHWINGER, DERAAD, AND MILTON
conductor case, and the limit of rarified media, from which the van der Waals force may be deduced (Sect. 3). Our major new efforts concern this quantum electrodynamical contribution to the surface tension, and the latent heat, of a liquid. The former requires a calculation of the change in the energy of the systemwhen the surface is slightly deformed (Sect. 4), while the latter only requires knowledge of the force betweena molecule and a dielectric layer (Sect. 3). The calculation of both of thesequantities necessitatesthe introduction of a cutoff in transverse momenta, expressing the fact that important contributions arise from short distances, where our continuum picture breaks down. Nevertheless, with a reasonable value for this cutoff, we obtain qualitative agreement with the measured values of the surface tension and latent heat of liquid helium at 0°K (Sect. 5). 2. CASIMIR EFFECT BETWEEN DIELECTRICS The Casimir effect is usually thought of as arising from zero-point fluctuations in the vacuum [8]. This posesa challenge for source theory, where the vaccum is regarded as truly a state with all physical properties equal to zero. A successfulresponseto this challenge has been supplied by one of us (J.S.) elsewhere[9], where the Casimir force between perfectly conducting parallel plates was calculated; however, there the significant simplification was made that the virtual “photons” exchanged be spinless. Although the electrodynamic results are simply a factor of 2 larger, corresponding to the two polarization states of the photon, a thoroughly consistent calculation requires considerably more effort. We will make that effort here for the more general circumstance of parallel interfaces between dielectric media (see Fig. 1). We will calculate the force on the surface at z = 0 in two ways? (A) by considering the change
z qo Z =-a
FIG. I. surfaces.
Two semi-infinite
dielectric slabs, separated by a third dielectric with plane, parallel
1 We use Heaviside-Lorentz units, as well as set (except in final numerical results) fi = c = 1. Other notation includes (dx) = (dr) dr = d.udydzdr, and, in the Appendix, a metric with diagonal elements (-1, 1, 1, 1).
CASIMIR
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DIELECTRICS
in the energy of the system when the separation between the surfaces is changed by an infinitesimal amount 6a, and (B) by calculating the stress tensor component, T,, , measuring the flux of momentum flowing across the interface. (We will consider a third variant of this method in a subsequent publication, in which we calculate the total Casimir energy of an uncharged, perfectly conducting. spherical shell.) A. Method of Energy Variation We begin with an appropriate action expression for the macroscopic fields produced by an external polarization source P: W =
* (d..u)[P . (-A J
-
V+) +
-
electromagnetic
04) - H . (V x A) + ;H? -
+&I, (2.1)
where E is the dielectric constant of the medium. The principle of stationary action now implies the following Maxwell equations, arising from appropriate field variations, 6H: H = V x A,
(2.2a)
6E: E = -A
(2.2b)
- V+,
6A: V x H = & + P,
(2.2c)
84: V . (EE + P) = 0.
(2.2d)
The resulting wave equation for E is -vx(VxE)-&=@.
(2.3)
It is possible to rewrite Win a form making reference only to the sources by noting that the linear relation between source and field implies E(X) = 1 (CIx’) ?(.Y, s’) . P(Y),
(2.4)
where ‘i is a Green’s dyadic. Then, we may integrate the response of the action to a source variation I&W=
I’(&)GP.E
to obtain for the numerical value of W W = 4 = 3
i s
(dx) P(x) (dx)(dx’)
E(X) P(x) . Y(X, s’) . P(X’).
4
SCHWINGER,
DERAAD,
AND
MILTON
Since the action (2.1) is stationary under field variations, infinitesimal change in the dielectric constant, 86, is
its variation
6, W = j (dx) 8~ 4E’.
(2.6)
Such a variation acts as a source of polarization; that is, by comparing iteration of the source term in (2. l), or symbolically [lo], i
s
under an
(2.6) with the
8~ &E2 t) eiw = . . . + $ [i j E . P12,
we identify the effective product of polarization iP(X) P(X’)ief* = l&
sources so induced: 6(X - X’).
(2.7)
By inserting these effective sources into (2.9, we deduce the change in the action when the dielectric constant is varied slightly, SW = - ; j (d.u) SE(X) rJJx,
x),
(2.8)
where we sum over repeated indices. Our formal discussion to this point has neglected dispersion. The latter is included, for zero temperature, by introducing a Fourier transform in time, (2.9) so that the change in the energy implied by (2.8) is (2.10) Thz temperature-dependent that r, is periodic in time2
Green’s FB(t -
dyadic, ?, , is Fourier-analyzed
t’) = &(t -
t’ & i/3),
(2.11)
/3 = l/kT, so that for T > 0 we have a Fourier transform, Yo(x, x’) = + 2 See, for example,
Ref.
[lo,
Vol.
I, pp.
(2.12)
series in imaginary =$ e5”(t+(r, n m 15Off.l.
by noting
time rather than a Fourier
r’, <,),
(2.13)
CASIMIR
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3
where (2.14)
5, = 2743
represents an imaginary frequency. In the following we will deal with finite temperatures and zero temperature on the same footing by making a complex rotation in the latter case, [ + real. (2.15) w + i<, Both cases can therefore be handled in a uniform manner. Henceforth, for ease in writing, we will assume T = 0 until the final stage. Our problem nowis to find the Green’s function occurring in (2.10). From the defining relation for r, (2.4), and the differential equation for E, (2.3), we obtain the Green’s function equation -v
x (V x F) + u%;: = -4
TS(r - r’).
(2.16)
It is convenient to obtain first-order equations, by defining V
(2.17)
Xi;=iuZ
so that (2.16) is equivalent to -V
X
Z -
iwe?
=
iu
i+S(r - r’).
(2.18)
For the situation we are considering here, illustrated in Fig. 1, the dielectric constant varies only in the z direction, so we may introduce a transverse spatial Fourier transform,, (2.19) We can choose a particular coordinate system in which k, points along the +x axis. If we take the x and z components of (2.17) and (2.18) (k = 1k, 1,f is the unit vector in the x direction, etc., and we use a vector notation for the second tensor index),
(2.20)
” rz = - 5 6(z- z’) - $ 9,)
6
SCHWINGER,
DERAAD,
AND
and we substitute these into they components for ry and ap,,
[ [
-- ii f $ + T - w2] av = -jw$
z
‘(’
(2.21a)
T “I
- kw % 6(z - z’).
(2.21b)
is [c’ = ~(.a’)] ry = wygE(z, au = -s
where the “transverse
electric”
%
z’),
(2.22)
gH(z, z’) + iw ; and “transverse
cj2
[ [ respectively.
of (2.17) and (2. IS), we obtain equations
w% ] rv = coy S(z - z’),
-$+k”-
The implication
MILTON
&
gH(z, z’),
magnetic”
Green’s
(2.23) functions
1
a,--o + k2 -
w2e g”(z, z’) = 8(z - z’),
al;! k2 - - - - + - - w2 az E az E I
The remaining components
gfyz,
z')
zz.z
S(z
-
z')
satisfy (2.24a) (2.24b)
of? can now be calculated from (2.20): (2.25a) (2.25b) (2.25~) (2.25d)
.k rzz= -‘2-pH.
%
(2.25e)
Ehen the interface is displaced, the change in the energy, (2.10), involves the trace 0f
r:
6E -=A
i f@ -1 2 s 2T (242
dz Se(z) r&z,
z, k, , w).
In evaluating this trace, we take the limit z’ ---f z, so we consequently functions appearing in (2.25), thereby obtaining
(2.26) omit the 6-
(2.27)
CASIMIR
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DIELECTRICS
For a variation in the position of the interface between media 2 and 3, the change in the dielectric constant is (2.28)
SE(Z) = -S6a(E, - Es) S(z).
We now interpret the limiting process in (2.27) as one in which z and z’ approach the interface from opposite sides. Therefore the force per unit area on the interface between media 2 and 3 is
where we have recognized gE, gH, and
that the continuity
of E, , E, , and EE; implies that of
12 1 6 E 2; E’ Pz r gH. Finally we must solve (2.24) for the Green’s functions gE and gH. We introduce the quantity K2 = k2 - w26, (2.30) which is positive because w3 is negative for T + 0, and for T = 0, may be made negative by a complex rotation. Consequently the electric Green’s function has the simple form in region 2 (z, Z’ > 0)
where the reflection coefficient is given by 1 -t r =
2KP
K. i _
+ K3
The magnetic Green’s function,
4K2K3 K32
-
K22
K3
+
Kl
K3
+
K2
K3
-
KI
K3
-
K.?
e*‘+ -
11-l.
(2.31b)
g H, has the same form but with the replacement K
-
K/C
=
K’,
(2.31~)
except in exponentials. Evaluating these Green’s functions just outside the interface, we find for the force on the surface per unit area
8
SCHWINGER,
DERAAD,
AND
MILTON
where the first square bracket comes from the E polarization, and the second from the H polarization. The first term in each bracket, which does not make reference to the separation a between the surfaces, is seen to be a change in the volume energy of the system. These terms correspond to the electromagnetic energy required to replace medium 2 by medium 3 in the displacement volume.s The remaining terms give the Casimir force. We rewrite the latter by making the change of frequency integration contour, (2.15): f;;iimir
=
T
‘d
+
K;
K;
f
K;.
Kj
-
K;
K$
-
K;
K3
+
K~
K3
+
K.,
K3
-
K1
K3
-
K2
e2ti,a
_
1]-1
-1
p3n
_
1
1
K2
= k2 + Q.
From this, we obtain the finite temperature expression immediately
(2.33)
by the substitutions (2.34)
the prime being a reminder to count the y1 = 0 term with half weight. These results agree with those of Lifschitz et al. [3]. To make this agreement explicit, we could adopt their variablep, K3
=
P&;/2,
=
E3
kP = j2c3(pz -
l),
(2.35)
so that K1,2
“2g~1,2/~3 + p2 -
1)1’2 = $“I!&
2)
(2.36)
which, however, is not well defined for n = 0. For this reason we prefer to obtain the finite temperature version of (2.33) by simply making the substitution (2.34). Thus, the general temperature dependence found by Lifshitz et al. [3] is confirmed. Experimental confirmation is, as yet, lacking, although the T = 0 result (2.33), has been checked experimentally for the case of thin helium films on crystalline substrates [41.
Before proceeding with applications of this result, we will supply another derivation of this Casimir force, which employs the stress tensor of the electromagnetic field. B. Stress Tensor Method4 The force per unit area on the interface is given in terms of the flux of the momentum incident on the surface, that is, by the zz component of the stress tensor, T,, = &[HL2 - Hz2 + e(EL2 - Ez2)]. 3 Since this term in the energy is already phenomenologically appropriate contact term. 4 Another stress tensor approach is given in Ref. [l 1I.
(2.27)
described, it must be canceled by an
CASIMIR
IN
EFFECT
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DIELECTRICS
What are the electric and magnetic fields present in the dielectric ? By comparing the second..order iteration of the source terms, Q[i 1 E . PI”, with (2.Q we see that the effective product of electric fields is i&(r)
&@‘I
leff
=
rdr,
r’,
~1,
(2.38)
and, from the Maxwell equation VxE=--Iti,
(2.39)
we have also the effective magnetic field product
(The same results may also be obtained from a Green’s function referring to potentials. See the Appendix.) We next introduce the transverse spatial transform (2.19). By means of (2.25), all the r’s in a given region can be written in terms of I’,, and I’Y,: (2.41a) (2.41b) (2.41~) Thus, if we ignore 6 functions, the stress tensor (2.37) can be written in the following simple form for T = 0,
(2.42) where the limit x’ + x is implicit. In region 2, where the Green’s functions have the form of (2.31a), the computed value of 7;, is constant, independent of the separation y2) zz
=
--i
(dk,) f
da,
(2,rr)Z 27-r K2 *
(2.43)
This term would arise from the formalism even if the dielectric 2 extended over all space. No physical meaning can be ascribed to such a term, however, since no stress
10
SCHWINGER, DERAAD, AND MILTON
can arise from a homogeneous dielectric .5 The Green’s function r,, r&z-,
2’) = 2
cash K3(E<
[Kg
jm
U)
+-
K1
Sitlh
K3(Z<
-..
in region 3 is a)]
3
:;
-qr f
[~3
cash K+> -
sinh QZ;]
K2
w” yK3+ K1)(K3 +..Ka),@-z’l+o) (K3 -
K1)(K3 -
K2) e-4--z’i+q,
(2.44)
where in the second form we have omitted terms annihilated by the differential operator in (2.42). The factor in the denominator is d
=
3[(K3
+
K,)(K,
+
K2)
6”’
-
(K3
-
K1)(K3
-
K2)
EpK3’].
(2.45)
The expression for r,, is obtained from (2.44) and (2.45) by making the substitution, except in the exponents, K
--f
(2.46)
-d/K.
When we substitute these Green’s functions into (2.42), we obtain, as before, a constant term in the force, which is the same as that of (2.43), with the replacement K2
+
K3
.
Subtracting these two constant stresseswe obtain (2.47) which is physically unobservable. Note that this term is the sameasthat found in (2.32), which was omitted there becauseit corresponds to a volume energy. The remaining terms combine to give just the Casimir force, (2.33).
3. APPLICATIONS Having derived the general expression for the Casimir force due to two parallel dielectric interfaces, we now consider a few special cases. First (A), specializing to conducting surfaces, we will reexamine the temperature dependence as well as the effects of finite conductivity (imperfect conductors). Second (B), we will infer the van der Waals force between molecules by considering the limit of tenuous dielectrics. Finally (C), by a method similar to that employed in Section 2A, we will calculate the energy of interaction between a polarizable molecule and a planar dielectric.
Assuchit canbecanceledby a contactterm.
5
CASIMIR
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A. Conductors Formally, the situation of a perfect conductor can be obtained by letting the dielectric constant tend to infinity, since this corresponds to an infinite conductivity. However, special care is required for the static situation, w = 0. In order to correctly enforce the electrostatic boundary conditions, we adopt the prescription that we take the limit E -+ co before we set w = 0. Thus the Casimir force, for T = 0, between two perfectly conducting plates separated by vacuum (cX = 1) is obtained from (2.33) by setting K~ = K~ = co and K;
=
K:
=
0:
(3.1) with K2 = k2 + [“. If we adopt polar coordinates k =
K COS
(3.2)
according to 8,
c
=
K
sin 8,
dk d[ =
K
dK d0,
(3.3)
we obtain for the force per unit area, (3. l), J’ = - &
jn12 de
COS
0
ti jrn dy & 0
(3.4)
7r2 1 z -_240 a4 ’
the well-known Casimir force between metal plates at T = 0 [l]. For T #: 0, we must apply the above-mentioned limiting procedure to the temperature-depemlent version of (2.33) thus obtaining (es = 1) (3.5) where K2 := k2 + (~Tvz/~)“. Notice that if we had simply let .zl,? + 00, the first denominatfor structure in (2.33) for II = 0 would not contribute, which, among other consequences, would imply an incorrect T + 0 limit. Defining y
=
2KU,
(3.6)
and t = 4rra/& we have, for arbitrary
(3.7)
temperature, (3.8)
12
SCHWINGER,
DERAAD,
AND
MILTON
There are two simple limits to consider here. The high-temperature limit, I > 1, is particularly easily obtained from our general formula, for then the n = 0 term is the only one that is not exponentially small. Including the first of these exponential corrections (from n = 1) we find for large T, fT e - +
l(3) - +
(1 + t + i fz) f+,
p < 45-a.
(3.9)
This result coincides with that first found by Sauer and Mehra [7]. Our form, (3.8), is especially suited to obtain the high-temperature limit, in contradistinction to the forms obtained by Sauer and Mehra [7] and Brown and Maclay [ll]. On the other hand, for small values of t, the form (3.8) is not practically useful6 as it stands sincefT is not analytic as t -+ 0. Instead, we can re-sum the series by means of the Poisson sum formula [12]. This states that if C(CY)= &
1-1 b(x) e-i”x dx,
(3.10)
f b(n) = 2%. f c(2?rn). n=-m 12=--m
(3.11)
the following identity holds,
In order to apply this formula to rewrite (3.Q we define 0)
= I,St Y2 dY $J n
(3.12)
3
implying for the Fourier transform for E # 0, c(a) = + lorn dx cos ax j-m y2 dy -& st 1 d2 n__~_ _ coth rrz - - 1 n-a dz2 [ 2 2z
---
1 %-ci
4,rr3
e-7;“’
et2f;;;f’t)
m
dq
-
$1.
l
Here we have interchanged the order of integration so eU--1
IIz=alt (3.13)
and used the fact that
1 sin zy = 2 coth rz - - . 2 22
(3.14)
The evaluation of c(0) is easily accomplished either directly, or by expanding coth nz in the above, yielding c(0) = 7?/15t. (3.15) OIn particular, result.
note that the Euler-Maclaurin
sum formula applied to (3.8) gives an incorrect
CASIMIR
EFFECT
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DIELECTRICS
13
We therefor’e find an alternative form for the sum in (3.8), (3.16) which, apart from a factor, expresses the general temperature dependence of the Casimir force. From this result, it is very easy to obtain the low-temperature limit, (3.17) again in agreement with Sauer and Mehra [7]. Therefore, we conclude that Lifshitz’ formula for the temperature dependence of the Casimir force between dielectric surfaces is correct; the errors in his papers [3] lie in the manner of extracting the case of metals fr’om his general formula. For an “imperfect” metal, we must employ the correct, frequency-dependent, dielectric constant. For frequencies in the infrared and higher, an adequate expression for ‘the dielectric constant is E(W) = 1 - WD2/6J2, where we see the appearance of the “plasma”
(3.18)
frequency,
cog2 = 4rre2N/m,
(3.19)
where N is the number density of free electrons in the conductor. For lower frequencies, the usual connection with the static conductivity, u, is C(W) -+ 477b/w.
(3.20)
If we assume identical, imperfect conducting walls separated by vacuum, the main frequency range contributing to the Casimir force at zero temperature, (2.33), must satisfy 5 - l/a. (3.21) Therefore, for separations that are not too large (a 5 1 pm), the relevant dielectric constant is given by (3.18). For a separation of the order of microns, a2wD2 is large but finite; ‘we wish here to calculate the first-order deviations from the infinite dielectric constant results found above, (3.4). Again using the change of variable (3.3), the Casimir force per unit area at T = 0 for an imperfect conductor is, from (2.33), f N - &
I/“:”
de
cos
e iom 3
0
1 Jw’2 de am, o
- -
= - &i
cos
[l - &
e(l + sin2 e) Jam(,‘:eId$2/
& ($gy
(3.22)
in agreement with Hargreaves [6], who pointed out an error in Lifshitz’ evaluation [3].
14
SCHWINGER,
DERAAD,
AND
MILTON
B. Van der Waals Force Tf we assumethat dielectrics 1 and 2 are of extremely low density, •r,~ N 1, and we assumevacuum in region 3, then the force between the dielectrics can be interpreted asthe resultant of the forces between individual molecules, the so-calledvan der Waals forces [2]. With the definitions, E1,2 --1
f a,,, ,
h.2
<
(3.23)
1,
the force per unit area at T = 0, (2.33), can be rewritten, upon introducing the variable P, (2.35), as f=-m
1
x * [I + (2p2 - 1)2] j5 d[c3 8182ep2Q’a. s1 $ 0
(3.24)
If the separation is large, only small values of c contribute, in which range a,,, are essentially constant. For this case,the Casimir force is
f--f&&,
(3.25)
for a large compared to wavelengths for which 6, and 6, have significant variation. If the separation is small, both p and 5 can become large so 6,,? can no longer be considered constants. The leading contribution in this caseis
where it has been assumedthat the significant values of 5 are those for which ca < 1. The force between the dielectrics can be viewed as arising from an interaction between the molecules composing the dielectrics. If the energy of interaction between two molecules a distance r apart is assumedto have the form V = -B/t+,
(3.27)
then the force per unit area between two parallel dielectric media separated by a separations, distance a will be
f =
-N,N,
29~B
(y - 2)& - 3) a:-,
’
where N,,, are the number densities of molecules in the dielectrics. For large separations, comparison with (3.25) shows that V = -2301,ci,/4m7,
(3.29)
where the polarizibilities, q2 , are defined by ~1.2
= 1 + ~~N,,GL,,
(3.30)
CASIMIR
EFFECT
IN
at zero frequency. For small separations, comparison 3 16+V
v=--p-
15
DIELECTRICS
1 m d&E, NJ-v’, so
with (3.26) shows that l)(Q - 1).
These results, (3.29) and (3.31), are in agreement with those found by Lifshitz et al. [3] from the Casimir force. Naturally, they correspond to the van der Waals forces found by Clasimir and Polder [2]. C. Interact,ion between a Molecule and a Dielectric Plate
As a final application of these ideas, we will calculate the energy of interaction between a molecule of polarizability a(~) and a dielectric slab. This energy is given by (2.10) with (3.32)
&(r, w) = 47ra(w) 6(r - R),
which expresses the change in the dielectric constant when a molecule is inserted in the vacuum. at R. We will suppose that the dielectric slab occupies the region of space z < 0 with vacuum above it. The appropriate Green’s functions here, referring to a single interface, are trivially obtained from those discussed in Section 2. In region 2, g”: has the form of (2.31a) with the reflection coefficient r given by (3.33)
1 + r = 2K/(K + ‘Q),
which is obtained by taking the limits a + co, Ed = I, E~ = Ed. The energy is then CR, = 4 47dw)
rm(z,
z, k, , ~1,
(3.34)
where, from (2.27)
a az ~azf I gH IzIez c
rkk
=
co2gE +
= $
[
k2 +- L
+ -&
[ w2 z
(3.35)
+ (2k2 - w2) 2:: y z: ] e-2KZ,
with K2
=
k2
-
&?
q2
=
k2 - E+J~.
(3.36)
The necessary contact term here is easily deduced from the physical requirement that the energy of interaction go to zero as the separation gets large, z -+ co, which effectively removes the w”/K term in r,, . Therefore, the interaction energy between a molecule and a dielectric slab is 1 -s” dc 4m(LJ lrn dk2 + E = - 16n2 ,, 0 x
-52"~"1 [
K
+ (2kz + i2) :‘: 1
5 z: ] e-2KZ.
(3.37)
16
SCHWINGER, DERAAD, AND MILTON
One possible application of this result refers to the attraction perfectly conducting plate (Q ---f a~). It is then easy to show that E = -3ci/8nz4,
of a molecule by a (3.38)
a result’ first calculated by Casimir and Polder [2]. A second, particularly interesting possibility occurs when the molecule is of the same type as those composing the dielectric slab. When the common dielectric constant is close to unity, the energy of interaction, to lowest order in E - 1, is
where we have expressed the polarizability of the molecule in terms of the dielectric constant according to (3.30). Lifshitz et al. [3] have considered the limiting behavior for large separations (small 0, where E can be considered to be constant. An application of (3.39) to liquid helium will be considered in Section 5.
4. SURFACE TENSION
The same phenomena that give rise to forces between dielectrics should also be an important source of surface tension in a liquid. We recall that the surface tension may be defined by the change in energy of the liquid due to a deformation of its surface, in which the total volume of the liquid remains unaltered. We here wish to calculate the Casimir contribution to this change in energy. Accordingly, we consider a single interface, at z = 0, between two dielectrics 4)
= e1 ,
z > 0,
E2>
z < 0.
(4.1)
We then imagine an infinitesimal change in the location of the interface, which depends upon the transverse coordinates, 6z(x, v), in such a way that the volume of either dielectric does not alter,
s
dx dy 6z(x, y) = 0.
(4.2)
We can parametrize this alteration of the interface by means of a scale factor h, so that the dielectric constant may be written, in terms of the unit step function, 7, as &j(r)
= l 2 + (Ed - Ed)7j(z - h 6z(x, y)).
(4.3)
7It mightbepossible to measure theeffectdirectly,by using,for example,a variationof Fairbank’s experiment,Ref. 1131.(T. Erber, private communication.)
CASIMIR
The first- and second-order
EFFECT
IN
17
DIELECTRICS
derivatives of &) with respect to X are
2 dA) = Sz(x, y)(+ ah
- EJ S(z -
x Sz),
(4.4)
a2 E(A)= -(,Q - E,)[SZ(S, y)]" S'(z - x Sz). 6h2 The change in energy due to a variation ;; E(A) = ; 1 (dr) g
(c2 -
where we have noted that the alteration Clearly, at h = 0, (4.6) vanishes,
because of (4.2). The second-order
X
[
(43
in A is given by (2. IO), which for this case reads El) Sz(x, y) S(z - h Sz) I$:,)(& r, w),
(4.6)
of the interface changes the Green’s function.
change in the energy, at h = 0, is
-Sz S’(z) Tkk(r, r, w) + 6(z) $ l$(r,
r, w)/
1.
(4.8)
A=0
This change in the energy is directly related to the surface tension. Before we can evaluate (4.Q we must calculate (a/ax) I-‘;;’ jAzo. The Green’s function still satisfies the differential equation (2.16). Under a change in A, 6X, the change in the Green’s function, 6I’, satisfies -v Since ?is
x (V x s’;, + w% ST: = --co2 ss
the Green’s function
for this differential
operator,
Si!(r, r”) = J (dr’) i!(r, r’) . &(r’)
(4.9) the solution to (4.9) is
. F(r), f),
(4.10)
which entails
Because we will ultimately have in mind a liquid helium-vacuum interface for which cIV2N 1, we will here pick out the leading behavior in the difference in the dielectric constants. The first term in (4.8) involves (a/az) I’,, , which can be evaluated by using the explicit Green’s functions for a single interface. (For example, see (3.33).)
18
SCHWINGER,
In the approximation rewrite the result as
DERAAD,
of dielectric constants
n (dk,)
4 J m
= -Cc2 -
e
AND
MILTON
close to unity, it is straightforward
iky(r--r’)q&
zt
k 2
4 WZ’,
I,
2, k, , 4,
to
(4.12)
with r’ ---f r. This is closely related to (a/?%) r,, (Eq. (4.1 I)), occurring in the second term of (4.8), so that the two terms there fit together to yield (d,S = dxdy) dS dS’ g
(eZ -
cl)” 6z(x, y)[6z(x’,
y’) -
Sz(x, y)]
x @!fd 0 eickek’jL‘(‘-“)LI’kz(z, 0, k 12 co) T&O, z, k; , co)3 (4 .13) (27f)2 (2,rr)2 with z -+ 0. With the effective replacement Sz(x’, y’) -
Sz(x, y) -
; [(r’ - r)L . VI2 Sz
(4.14) -Jvv 2
a b szaa ak, ak;’
a, b = 1, 2,
the second-order change in the energy may be rewritten as
a
x b r,,(z - 0, 0, k, , w) ak r,,(O, z - 0, k, , u). (4.15) b ak, Being careful about setting z = 0, and writing the dyadic “r in a coordinate independent way, we find, using (2.25). (4.16) An integration by parts and the substitution w -+ i[ finally yields -$
E(A)
1 j- dS ; [V, 62(x, y)]” Ioa d&c, - d2 16rr2
= -
n=o
x irn dk2 (2 -
&
-
-&).
(4.17)
CASIMIR
EFFECT
IN
Since this change in the energy is related to a deformation
_22 /+v 8x"
19
DIELECTRICS
of the surface,
= Z's j d&s= z j ds; [V, Sz(s,y)]2,
(4.18)
A=0
the surface tension, Z, is 1 ,r z ~ jm dc(c, 16~~ o
- &).
(4.19)
As it stands, the surface tension is infinite.s What is required is a cutoff in k2, that is, in transverse momentum. The implication is that short-distance behavior makes a significant contribution to the surface tension. The continuum approach applied here is only accurate in the large-distance regime. Consequently, we see that to calculate the van der Waals contribution to the surface tension convincingly, we would have to graft on a detailed microscopic theory of short-range behavior. Such a theory, at present, does not exist. Nevertheless, in the concluding section, we will employ a crude cutoff to demonstrate that the correct order of magnitude of the surface tension in liquid helium is reasonably well reproduced.
5.
CONCLUSIONS
Jn the above we have verified Lifshitz’ formula [3] for the Casimir force between parallel dielectric interfaces, including the temperature dependence. Where the Russian calculations went wrong [6] was in their specialization to metal plates. A careful reading of their papers shows that, through relatively trivial errors, they obtained incorrect results for the Casimir force, for perfect conductors at finite temperatures, and for imperfect conductors at zero temperature. Beyond this clarification, we have investigated the Casimir contribution to the surface tension of a liquid at T = 0 as given by (4.19). As noted there, the transverse momentum integral requires a cutoff, k, = fiw,jc, implying that short-distance phenomena. play an important role. However, we can estimate the continuum contribution by taking for w, a frequency corresponding to atomic separations. For liquid He a rough model for the dielectric constant is provided by [4] E-
l +
0.05 1 + c2/wo2
(5.la)
1016set-l.
(5.lb)
with w. ‘v 3.5 Substituting
this into (4.19), and retaining
8 It is interesting to note that the contribution
X
only the leading dependence in the cutoff, due to the E polarization
is finite.
20
SCHWINGER,
we find for the Casimir contribution zero
DERAAD,
AND
MILTON
to the surface tension of liquid helium at absolute
ef 1 x 10-36[W,(sec-1)]2 erg/cm2.
(5.2)
Taking w e = 10ls set-‘, which corresponds
to a reasonable interatomic z Casimir
E
distance, we obtain
1 w/Cm2,
which is to be compared with the experimental
(5.3)
(5.4)
value [14]
z ew = 0.37 erg/cm2,
(5.5)
a factor of 3 smaller. That this qualitative agreement is not fortuitous may be seen by considering the Casimir model for the latent heat. That is provided by (3.39), which expressesthe energy of interaction between a molecule and a dielectric slab, E(z), where z is the separation between the molecule and the surface. The work required to separate a molecule from the surface and remove it to infinity is, if this is the only force involved, --E(O). The latent heat, q, is the energy required to vaporize a gram of substance, so 4 = --E@Wf). (54 Here N is the number density and p is the massdensity, which for liquid helium is p = 0.15 g/cm”. Then using (5.1) for the dielectric constant, we find for the leading behavior of the Casimir contribution to the latent heat
z
7 x 10-4i[w,(sec-1)]3 erg/g,
(5.7)
where w, is the transverse momentum cutoff. For the samerepresentative value of w, , (5.3), we find qCasimir
F
7 X 10' erg/g,
(5.8)
while the experimental value is two times larger [ 151 4exp = 15 x 10’ erg/g.
(5.9)
We seethat qualitative agreement with the experimental values for both the surface tension and the latent heat is achieved with the samevalue of the transverse momentum cutoff, (5.3). We are evidently lacking a quantitative theory, which would require
21
CASIMIR EFFECT IN DIELECTRICS
detailed microscopic considerations. However, we can fairly conclude that the Casimir effect, a manifestation of van der Waals forces, is responsible for a significant part of these phenomena.
APPENDIX:
GREEN'S FUNCTION
FOR POTENTIAL
Rather than introduce a Green’s function for the electric field strength, as in Section 2, it is possible to employ one for the vector potential. Since this method exhibits sorne interesting features of gauge covariance, and makes closer contact with the approach of Lifshitz et al. [3], we will discuss it briefly here. If A, is the vector potential, the corresponding Green’s function is defined by A.(x) = j (lid) D&x, x’) P(x’) + a&x),
(AlI
where J,, is the current density. To find the differential equation satisfied by D,, we note that in any region where E is constant, Maxwell’s equations can be written as L”“A
Y
=
JI* >
WI
where
LOi
=
=-c--j, a
LiO
(A31
at
Lij = S;&~“) + vivj ) where 3 = --E g
+ v2.
(A41
Since J, is not an arbitrary vector field, but is conserved, a,J“
= 0,
645)
we conclude that the equation for the Green’s function is L””
D&Y,
x’) = (g””
-
F)
6(x -
x’).
We solve this equation by making a specific choice of gauge, which is similar to the
22
SCHWINGER, DERAAD, AND MILTON
radiation gauge, in which Ai (and not A,) are the dynamically important That is, we can easily show that a solution to (A6) is achieved if D,, = D<, = Doi = 0,
(A74
and (-8 Once Dij is determined, the analog of (2.38),
variables.
aij + VJ,) Djk(x, x’) = &, 6(x - x’).
(A7b)
the effective product of vector potentials is determined by b&(x) Aj(x
= D&x, x’).
648)
The corresponding electric and magnetic field products are given by i.&(x) Ej(x’)leff = $ 4 iHi
Hj(x’)]
D&c, x’),
(A94
eft-= l itm~j~nVzV;Dmn(x, x’).
Wb)
For the geometry of Fig. 1, the Fourier decomposition (dk,)
ik1.br’)l
Dij(X, x’> = 1 mc
e
“w I
e-iw~t-t’)g,f(z
2rr
z ,
z~>
is appropriate. As before, we can choose k, to point along the x axis. In this case the only nonzero off-diagonal Green’s functions are g,, , g,, . The boundary conditions on gij at the surfaces z = 0, --a follow from the continuity of H, and of E, , E, , and EE,: E, , J& , (Hz) : g,, , g,, , g,, continuous,
(Al la)
H, : g g,, continuous,
(Allb)
Hv : 2 gxi - ikgzt continuous,
(Al lc)
EE, : Egsi continuous.
(Al Id)
We can now easily solve the differential equations (A7b) for the gij . The result is W”g~j
= ri* ,
(A13
where ? is the Green’s function discussed in Section 2, given, for example, in the intermediate region 2 by (2.44), (2.46), and (2.41). Consequently, the electric and magnetic field products are the same as before, given by (2.38) and (2.40). ACKNOWLEDGMENT We would like to thank Professor S. Putterman for arousing our interest in this problem, and for many stimulating discussions.
CASIMIR
EFFECT
IN
DIELECTRICS
23
REFERENCES 1. H. B. G. CASIMIR, Proc. Kon. Ned. Akad. Wetensch. 51 (1948), 193-795. 2. H. B. G. CASIMIR AND D. POLDER, Phys. Rev. 73 (1948), 360-372. 3. E. M. LIFSHITZ, Zh. Eksp. Teor. Fiz. 29 (1955), 94110. [English transl.: Soviet Phys. JETP 2 (1956), 73.-83.1; I. D. DZYALOSHINSKII, E. M. LIFSHITZ, AND L. P. PITAEVSKII, Usp. Fiz. Nauk 73 (1961), 381-422. [English transl.: Souiet Phys. Usp. 4 (1961), 153-176.1; L. D. LANDAU AND E. M. LIF~HITZ, “Electrodynamics of Continuous Media,” pp. 368-376, Pergamon, Oxford, 1960. 4. E. S. SABISKY AND C. H. ANDERSON, Phys. Rev. A 7 (1973), 79&806. 5. B. V. DEFXAGIN AND I. I. ABRIKOSOVA, Zh. Eksp. Theor. Fiz. 30 (1956), 993-1006; 31 (1956), 3-13. [English transl.: Sooiet Phys. JETP 3 (1957), 819-829; 4 (1957), 2-101; A. KITCHENER AND A. P. PROISSER, Proc. Roy. Sot. (London) A 242 (1957), 403409; W. BLACK, J. G. V. DE JONGH, J. TH. G. OVERBECK, AND M. J. SPARNAAY, Trans. Faraday Sot. 56 (1960), 1597-1608; A. VAN SILFHOUT, Proc. Kon. Ned. Akad. Wetensch. B 69 (1966), 501-541; R. H. S. WINTERTON, Contemp. Phys. 11 (1970), 559-574; J. N. ISRAELACHIVILI AND TABOR, Proc. R. Sot. Lond. A 331 (1972), 19-38. 6. C. M. HARGREAVES, Proc. Kon. Ned. Akad. Wetensch. B 68 (1965), 231-236. 7. F. SAUER, dissertation, Gottingen, 1962, unpublished; J. MEHRA, Physica 37 (1967), 145-152. 8. T. H. BOY ER, Ann. Phys. (N. Y.) 56 (1970), 474-503. 9. J. SCHWINGER, Lett. Math. Phys. 1 (1975), 4347. 10. J. SCHWINGER, “Particles, Sources, and Fields,” Vols. I, II, Addison-Wesley, Reading, Mass., 1970, 1973. 11. L. S. BROWN AND G. J. MACLAY, Phys. Rev. 184 (1969), 1272-1279. 12. M. FIERZ, Heh. Phys. Acta 33 (1960), 855-858. 13. G. S. LAR.uE, W. M. FAIRBANK, AND A. F. HEBARD, Phys. Rev. Lett. 38 (1977), 1011-1014. 14. J. M. MAGERLEIN AND T. M. SANDERS, JR., Phys. Rev. Lett. 36 (1976), 258-261; K. R. ATKINS AND Y. NARAHARA, Phys. Rev. 138 (1965), A437-A441. 15. W. E. KEI-LER, “Helium-3 and Helium-4,” pp. 112-117, Plenum, New York, 1969.
OF PHYSICS
115,
1-23 (1978)
Casimir JULIAN
SCHWINGER,
Effect in Dielectrics*
LESTER L. DERAAD,
JR., AND KIMBALL
Department of Physics, University of California,
Los
Angeles,
A. MILTON
California
90024
Received December 5, 1977
We reconsider the Casimir (van der Waals) forces between dielectrics with plane, parallel surfaces for arbitrary temperature, using the methods of source theory. The general results of Lifshitz are confirmed, and are shown to imply the correct forces on metal surfaces. The same phenomena give rise to contributions to the surface tension and the latent heat of an idealized liquid, contributions which, unfortunately, are not well defined since they depend upon a momentum cutoff. However, with a reasonable value for this cutoff, qualitative agreement with the experimentally observed surface tension and latent heat of liquid helium at absolute zero is obtained.
1. INTRODUCTION
One of the least intuitive consequences of quantum electrodynamics is the force between uncharged conducting surfaces, the so-called Casimir force [l]. However, another manifestation of this force is known to have a molecular origin. That is, the “Casimir” forces between bodies having different dielectric constants [2, 31 can be interpreted, in the limit of tenuous media, to arise from the retarded (I/R’) and shortrange (l/P) van der Waals potentials between the molecules that make up the bodies. The Casimir effect for plane dielectric surfaces was first worked out by Lifshitz over two decades ago [3]. His results, for zero temperature, have been confirmed in a beautiful experiment employing acoustical interferometry to accurately measure the thickness of a thin helium film [4]. (The measurements of forces between solid bodies are less conclusive [5].) However, questions remain in connection with his finite temperature results [6]. For the Casimir force between metal plates, obtained by setting the dielectric constant equal to infinity in his general formula, he found a temperature dependence which disagrees with that found in other calculations of this effect [7]. This, together with various theoretical objections, has raised doubts concerning the validity of his results for dielectrics. In this paper we will lay those doubts to rest. We begin by showing that Lifshitz’ general formula for the force, including the temperature dependence, is correct, and that the errors arise only in the limits taken to recover the conductor case. We will accomplish
this confirmation
methods for computing
by adopting
far superior
and more physically
transparent
the force (Sect. 2). We will consider carefully the limiting
* Work supported in part by the National Science Foundation 1
and the Alfred P. Sloan Foundation. OOO3-4916/78/1151-OOO1S05.00/0
All
Copyright 0 1978 by Academic Press. Inc. rights of reproduction in any form reserved.
2
SCHWINGER, DERAAD, AND MILTON
conductor case, and the limit of rarified media, from which the van der Waals force may be deduced (Sect. 3). Our major new efforts concern this quantum electrodynamical contribution to the surface tension, and the latent heat, of a liquid. The former requires a calculation of the change in the energy of the systemwhen the surface is slightly deformed (Sect. 4), while the latter only requires knowledge of the force betweena molecule and a dielectric layer (Sect. 3). The calculation of both of thesequantities necessitatesthe introduction of a cutoff in transverse momenta, expressing the fact that important contributions arise from short distances, where our continuum picture breaks down. Nevertheless, with a reasonable value for this cutoff, we obtain qualitative agreement with the measured values of the surface tension and latent heat of liquid helium at 0°K (Sect. 5). 2. CASIMIR EFFECT BETWEEN DIELECTRICS The Casimir effect is usually thought of as arising from zero-point fluctuations in the vacuum [8]. This posesa challenge for source theory, where the vaccum is regarded as truly a state with all physical properties equal to zero. A successfulresponseto this challenge has been supplied by one of us (J.S.) elsewhere[9], where the Casimir force between perfectly conducting parallel plates was calculated; however, there the significant simplification was made that the virtual “photons” exchanged be spinless. Although the electrodynamic results are simply a factor of 2 larger, corresponding to the two polarization states of the photon, a thoroughly consistent calculation requires considerably more effort. We will make that effort here for the more general circumstance of parallel interfaces between dielectric media (see Fig. 1). We will calculate the force on the surface at z = 0 in two ways? (A) by considering the change
z qo Z =-a
FIG. I. surfaces.
Two semi-infinite
dielectric slabs, separated by a third dielectric with plane, parallel
1 We use Heaviside-Lorentz units, as well as set (except in final numerical results) fi = c = 1. Other notation includes (dx) = (dr) dr = d.udydzdr, and, in the Appendix, a metric with diagonal elements (-1, 1, 1, 1).
CASIMIR
EFFECT
IN
3
DIELECTRICS
in the energy of the system when the separation between the surfaces is changed by an infinitesimal amount 6a, and (B) by calculating the stress tensor component, T,, , measuring the flux of momentum flowing across the interface. (We will consider a third variant of this method in a subsequent publication, in which we calculate the total Casimir energy of an uncharged, perfectly conducting. spherical shell.) A. Method of Energy Variation We begin with an appropriate action expression for the macroscopic fields produced by an external polarization source P: W =
* (d..u)[P . (-A J
-
V+) +
-
electromagnetic
04) - H . (V x A) + ;H? -
+&I, (2.1)
where E is the dielectric constant of the medium. The principle of stationary action now implies the following Maxwell equations, arising from appropriate field variations, 6H: H = V x A,
(2.2a)
6E: E = -A
(2.2b)
- V+,
6A: V x H = & + P,
(2.2c)
84: V . (EE + P) = 0.
(2.2d)
The resulting wave equation for E is -vx(VxE)-&=@.
(2.3)
It is possible to rewrite Win a form making reference only to the sources by noting that the linear relation between source and field implies E(X) = 1 (CIx’) ?(.Y, s’) . P(Y),
(2.4)
where ‘i is a Green’s dyadic. Then, we may integrate the response of the action to a source variation I&W=
I’(&)GP.E
to obtain for the numerical value of W W = 4 = 3
i s
(dx) P(x) (dx)(dx’)
E(X) P(x) . Y(X, s’) . P(X’).
4
SCHWINGER,
DERAAD,
AND
MILTON
Since the action (2.1) is stationary under field variations, infinitesimal change in the dielectric constant, 86, is
its variation
6, W = j (dx) 8~ 4E’.
(2.6)
Such a variation acts as a source of polarization; that is, by comparing iteration of the source term in (2. l), or symbolically [lo], i
s
under an
(2.6) with the
8~ &E2 t) eiw = . . . + $ [i j E . P12,
we identify the effective product of polarization iP(X) P(X’)ief* = l&
sources so induced: 6(X - X’).
(2.7)
By inserting these effective sources into (2.9, we deduce the change in the action when the dielectric constant is varied slightly, SW = - ; j (d.u) SE(X) rJJx,
x),
(2.8)
where we sum over repeated indices. Our formal discussion to this point has neglected dispersion. The latter is included, for zero temperature, by introducing a Fourier transform in time, (2.9) so that the change in the energy implied by (2.8) is (2.10) Thz temperature-dependent that r, is periodic in time2
Green’s FB(t -
dyadic, ?, , is Fourier-analyzed
t’) = &(t -
t’ & i/3),
(2.11)
/3 = l/kT, so that for T > 0 we have a Fourier transform, Yo(x, x’) = + 2 See, for example,
Ref.
[lo,
Vol.
I, pp.
(2.12)
series in imaginary =$ e5”(t+(r, n m 15Off.l.
by noting
time rather than a Fourier
r’, <,),
(2.13)
CASIMIR
EFFECT
IN
DIELECTRICS
3
where (2.14)
5, = 2743
represents an imaginary frequency. In the following we will deal with finite temperatures and zero temperature on the same footing by making a complex rotation in the latter case, [ + real. (2.15) w + i<, Both cases can therefore be handled in a uniform manner. Henceforth, for ease in writing, we will assume T = 0 until the final stage. Our problem nowis to find the Green’s function occurring in (2.10). From the defining relation for r, (2.4), and the differential equation for E, (2.3), we obtain the Green’s function equation -v
x (V x F) + u%;: = -4
TS(r - r’).
(2.16)
It is convenient to obtain first-order equations, by defining V
(2.17)
Xi;=iuZ
so that (2.16) is equivalent to -V
X
Z -
iwe?
=
iu
i+S(r - r’).
(2.18)
For the situation we are considering here, illustrated in Fig. 1, the dielectric constant varies only in the z direction, so we may introduce a transverse spatial Fourier transform,, (2.19) We can choose a particular coordinate system in which k, points along the +x axis. If we take the x and z components of (2.17) and (2.18) (k = 1k, 1,f is the unit vector in the x direction, etc., and we use a vector notation for the second tensor index),
(2.20)
” rz = - 5 6(z- z’) - $ 9,)
6
SCHWINGER,
DERAAD,
AND
and we substitute these into they components for ry and ap,,
[ [
-- ii f $ + T - w2] av = -jw$
z
‘(’
(2.21a)
T “I
- kw % 6(z - z’).
(2.21b)
is [c’ = ~(.a’)] ry = wygE(z, au = -s
where the “transverse
electric”
%
z’),
(2.22)
gH(z, z’) + iw ; and “transverse
cj2
[ [ respectively.
of (2.17) and (2. IS), we obtain equations
w% ] rv = coy S(z - z’),
-$+k”-
The implication
MILTON
&
gH(z, z’),
magnetic”
Green’s
(2.23) functions
1
a,--o + k2 -
w2e g”(z, z’) = 8(z - z’),
al;! k2 - - - - + - - w2 az E az E I
The remaining components
gfyz,
z')
zz.z
S(z
-
z')
satisfy (2.24a) (2.24b)
of? can now be calculated from (2.20): (2.25a) (2.25b) (2.25~) (2.25d)
.k rzz= -‘2-pH.
%
(2.25e)
Ehen the interface is displaced, the change in the energy, (2.10), involves the trace 0f
r:
6E -=A
i f@ -1 2 s 2T (242
dz Se(z) r&z,
z, k, , w).
In evaluating this trace, we take the limit z’ ---f z, so we consequently functions appearing in (2.25), thereby obtaining
(2.26) omit the 6-
(2.27)
CASIMIR
EFFECT
IN
I
DIELECTRICS
For a variation in the position of the interface between media 2 and 3, the change in the dielectric constant is (2.28)
SE(Z) = -S6a(E, - Es) S(z).
We now interpret the limiting process in (2.27) as one in which z and z’ approach the interface from opposite sides. Therefore the force per unit area on the interface between media 2 and 3 is
where we have recognized gE, gH, and
that the continuity
of E, , E, , and EE; implies that of
12 1 6 E 2; E’ Pz r gH. Finally we must solve (2.24) for the Green’s functions gE and gH. We introduce the quantity K2 = k2 - w26, (2.30) which is positive because w3 is negative for T + 0, and for T = 0, may be made negative by a complex rotation. Consequently the electric Green’s function has the simple form in region 2 (z, Z’ > 0)
where the reflection coefficient is given by 1 -t r =
2KP
K. i _
+ K3
The magnetic Green’s function,
4K2K3 K32
-
K22
K3
+
Kl
K3
+
K2
K3
-
KI
K3
-
K.?
e*‘+ -
11-l.
(2.31b)
g H, has the same form but with the replacement K
-
K/C
=
K’,
(2.31~)
except in exponentials. Evaluating these Green’s functions just outside the interface, we find for the force on the surface per unit area
8
SCHWINGER,
DERAAD,
AND
MILTON
where the first square bracket comes from the E polarization, and the second from the H polarization. The first term in each bracket, which does not make reference to the separation a between the surfaces, is seen to be a change in the volume energy of the system. These terms correspond to the electromagnetic energy required to replace medium 2 by medium 3 in the displacement volume.s The remaining terms give the Casimir force. We rewrite the latter by making the change of frequency integration contour, (2.15): f;;iimir
=
T
‘d
+
K;
K;
f
K;.
Kj
-
K;
K$
-
K;
K3
+
K~
K3
+
K.,
K3
-
K1
K3
-
K2
e2ti,a
_
1]-1
-1
p3n
_
1
1
K2
= k2 + Q.
From this, we obtain the finite temperature expression immediately
(2.33)
by the substitutions (2.34)
the prime being a reminder to count the y1 = 0 term with half weight. These results agree with those of Lifschitz et al. [3]. To make this agreement explicit, we could adopt their variablep, K3
=
P&;/2,
=
E3
kP = j2c3(pz -
l),
(2.35)
so that K1,2
“2g~1,2/~3 + p2 -
1)1’2 = $“I!&
2)
(2.36)
which, however, is not well defined for n = 0. For this reason we prefer to obtain the finite temperature version of (2.33) by simply making the substitution (2.34). Thus, the general temperature dependence found by Lifshitz et al. [3] is confirmed. Experimental confirmation is, as yet, lacking, although the T = 0 result (2.33), has been checked experimentally for the case of thin helium films on crystalline substrates [41.
Before proceeding with applications of this result, we will supply another derivation of this Casimir force, which employs the stress tensor of the electromagnetic field. B. Stress Tensor Method4 The force per unit area on the interface is given in terms of the flux of the momentum incident on the surface, that is, by the zz component of the stress tensor, T,, = &[HL2 - Hz2 + e(EL2 - Ez2)]. 3 Since this term in the energy is already phenomenologically appropriate contact term. 4 Another stress tensor approach is given in Ref. [l 1I.
(2.27)
described, it must be canceled by an
CASIMIR
IN
EFFECT
9
DIELECTRICS
What are the electric and magnetic fields present in the dielectric ? By comparing the second..order iteration of the source terms, Q[i 1 E . PI”, with (2.Q we see that the effective product of electric fields is i&(r)
&@‘I
leff
=
rdr,
r’,
~1,
(2.38)
and, from the Maxwell equation VxE=--Iti,
(2.39)
we have also the effective magnetic field product
(The same results may also be obtained from a Green’s function referring to potentials. See the Appendix.) We next introduce the transverse spatial transform (2.19). By means of (2.25), all the r’s in a given region can be written in terms of I’,, and I’Y,: (2.41a) (2.41b) (2.41~) Thus, if we ignore 6 functions, the stress tensor (2.37) can be written in the following simple form for T = 0,
(2.42) where the limit x’ + x is implicit. In region 2, where the Green’s functions have the form of (2.31a), the computed value of 7;, is constant, independent of the separation y2) zz
=
--i
(dk,) f
da,
(2,rr)Z 27-r K2 *
(2.43)
This term would arise from the formalism even if the dielectric 2 extended over all space. No physical meaning can be ascribed to such a term, however, since no stress
10
SCHWINGER, DERAAD, AND MILTON
can arise from a homogeneous dielectric .5 The Green’s function r,, r&z-,
2’) = 2
cash K3(E<
[Kg
jm
U)
+-
K1
Sitlh
K3(Z<
-..
in region 3 is a)]
3
:;
-qr f
[~3
cash K+> -
sinh QZ;]
K2
w” yK3+ K1)(K3 +..Ka),@-z’l+o) (K3 -
K1)(K3 -
K2) e-4--z’i+q,
(2.44)
where in the second form we have omitted terms annihilated by the differential operator in (2.42). The factor in the denominator is d
=
3[(K3
+
K,)(K,
+
K2)
6”’
-
(K3
-
K1)(K3
-
K2)
EpK3’].
(2.45)
The expression for r,, is obtained from (2.44) and (2.45) by making the substitution, except in the exponents, K
--f
(2.46)
-d/K.
When we substitute these Green’s functions into (2.42), we obtain, as before, a constant term in the force, which is the same as that of (2.43), with the replacement K2
+
K3
.
Subtracting these two constant stresseswe obtain (2.47) which is physically unobservable. Note that this term is the sameasthat found in (2.32), which was omitted there becauseit corresponds to a volume energy. The remaining terms combine to give just the Casimir force, (2.33).
3. APPLICATIONS Having derived the general expression for the Casimir force due to two parallel dielectric interfaces, we now consider a few special cases. First (A), specializing to conducting surfaces, we will reexamine the temperature dependence as well as the effects of finite conductivity (imperfect conductors). Second (B), we will infer the van der Waals force between molecules by considering the limit of tenuous dielectrics. Finally (C), by a method similar to that employed in Section 2A, we will calculate the energy of interaction between a polarizable molecule and a planar dielectric.
Assuchit canbecanceledby a contactterm.
5
CASIMIR
EFFECT
IN
11
DIELECTRICS
A. Conductors Formally, the situation of a perfect conductor can be obtained by letting the dielectric constant tend to infinity, since this corresponds to an infinite conductivity. However, special care is required for the static situation, w = 0. In order to correctly enforce the electrostatic boundary conditions, we adopt the prescription that we take the limit E -+ co before we set w = 0. Thus the Casimir force, for T = 0, between two perfectly conducting plates separated by vacuum (cX = 1) is obtained from (2.33) by setting K~ = K~ = co and K;
=
K:
=
0:
(3.1) with K2 = k2 + [“. If we adopt polar coordinates k =
K COS
(3.2)
according to 8,
c
=
K
sin 8,
dk d[ =
K
dK d0,
(3.3)
we obtain for the force per unit area, (3. l), J’ = - &
jn12 de
COS
0
ti jrn dy & 0
(3.4)
7r2 1 z -_240 a4 ’
the well-known Casimir force between metal plates at T = 0 [l]. For T #: 0, we must apply the above-mentioned limiting procedure to the temperature-depemlent version of (2.33) thus obtaining (es = 1) (3.5) where K2 := k2 + (~Tvz/~)“. Notice that if we had simply let .zl,? + 00, the first denominatfor structure in (2.33) for II = 0 would not contribute, which, among other consequences, would imply an incorrect T + 0 limit. Defining y
=
2KU,
(3.6)
and t = 4rra/& we have, for arbitrary
(3.7)
temperature, (3.8)
12
SCHWINGER,
DERAAD,
AND
MILTON
There are two simple limits to consider here. The high-temperature limit, I > 1, is particularly easily obtained from our general formula, for then the n = 0 term is the only one that is not exponentially small. Including the first of these exponential corrections (from n = 1) we find for large T, fT e - +
l(3) - +
(1 + t + i fz) f+,
p < 45-a.
(3.9)
This result coincides with that first found by Sauer and Mehra [7]. Our form, (3.8), is especially suited to obtain the high-temperature limit, in contradistinction to the forms obtained by Sauer and Mehra [7] and Brown and Maclay [ll]. On the other hand, for small values of t, the form (3.8) is not practically useful6 as it stands sincefT is not analytic as t -+ 0. Instead, we can re-sum the series by means of the Poisson sum formula [12]. This states that if C(CY)= &
1-1 b(x) e-i”x dx,
(3.10)
f b(n) = 2%. f c(2?rn). n=-m 12=--m
(3.11)
the following identity holds,
In order to apply this formula to rewrite (3.Q we define 0)
= I,St Y2 dY $J n
(3.12)
3
implying for the Fourier transform for E # 0, c(a) = + lorn dx cos ax j-m y2 dy -& st 1 d2 n__~_ _ coth rrz - - 1 n-a dz2 [ 2 2z
---
1 %-ci
4,rr3
e-7;“’
et2f;;;f’t)
m
dq
-
$1.
l
Here we have interchanged the order of integration so eU--1
IIz=alt (3.13)
and used the fact that
1 sin zy = 2 coth rz - - . 2 22
(3.14)
The evaluation of c(0) is easily accomplished either directly, or by expanding coth nz in the above, yielding c(0) = 7?/15t. (3.15) OIn particular, result.
note that the Euler-Maclaurin
sum formula applied to (3.8) gives an incorrect
CASIMIR
EFFECT
IN
DIELECTRICS
13
We therefor’e find an alternative form for the sum in (3.8), (3.16) which, apart from a factor, expresses the general temperature dependence of the Casimir force. From this result, it is very easy to obtain the low-temperature limit, (3.17) again in agreement with Sauer and Mehra [7]. Therefore, we conclude that Lifshitz’ formula for the temperature dependence of the Casimir force between dielectric surfaces is correct; the errors in his papers [3] lie in the manner of extracting the case of metals fr’om his general formula. For an “imperfect” metal, we must employ the correct, frequency-dependent, dielectric constant. For frequencies in the infrared and higher, an adequate expression for ‘the dielectric constant is E(W) = 1 - WD2/6J2, where we see the appearance of the “plasma”
(3.18)
frequency,
cog2 = 4rre2N/m,
(3.19)
where N is the number density of free electrons in the conductor. For lower frequencies, the usual connection with the static conductivity, u, is C(W) -+ 477b/w.
(3.20)
If we assume identical, imperfect conducting walls separated by vacuum, the main frequency range contributing to the Casimir force at zero temperature, (2.33), must satisfy 5 - l/a. (3.21) Therefore, for separations that are not too large (a 5 1 pm), the relevant dielectric constant is given by (3.18). For a separation of the order of microns, a2wD2 is large but finite; ‘we wish here to calculate the first-order deviations from the infinite dielectric constant results found above, (3.4). Again using the change of variable (3.3), the Casimir force per unit area at T = 0 for an imperfect conductor is, from (2.33), f N - &
I/“:”
de
cos
e iom 3
0
1 Jw’2 de am, o
- -
= - &i
cos
[l - &
e(l + sin2 e) Jam(,‘:eId$2/
& ($gy
(3.22)
in agreement with Hargreaves [6], who pointed out an error in Lifshitz’ evaluation [3].
14
SCHWINGER,
DERAAD,
AND
MILTON
B. Van der Waals Force Tf we assumethat dielectrics 1 and 2 are of extremely low density, •r,~ N 1, and we assumevacuum in region 3, then the force between the dielectrics can be interpreted asthe resultant of the forces between individual molecules, the so-calledvan der Waals forces [2]. With the definitions, E1,2 --1
f a,,, ,
h.2
<
(3.23)
1,
the force per unit area at T = 0, (2.33), can be rewritten, upon introducing the variable P, (2.35), as f=-m
1
x * [I + (2p2 - 1)2] j5 d[c3 8182ep2Q’a. s1 $ 0
(3.24)
If the separation is large, only small values of c contribute, in which range a,,, are essentially constant. For this case,the Casimir force is
f--f&&,
(3.25)
for a large compared to wavelengths for which 6, and 6, have significant variation. If the separation is small, both p and 5 can become large so 6,,? can no longer be considered constants. The leading contribution in this caseis
where it has been assumedthat the significant values of 5 are those for which ca < 1. The force between the dielectrics can be viewed as arising from an interaction between the molecules composing the dielectrics. If the energy of interaction between two molecules a distance r apart is assumedto have the form V = -B/t+,
(3.27)
then the force per unit area between two parallel dielectric media separated by a separations, distance a will be
f =
-N,N,
29~B
(y - 2)& - 3) a:-,
’
where N,,, are the number densities of molecules in the dielectrics. For large separations, comparison with (3.25) shows that V = -2301,ci,/4m7,
(3.29)
where the polarizibilities, q2 , are defined by ~1.2
= 1 + ~~N,,GL,,
(3.30)
CASIMIR
EFFECT
IN
at zero frequency. For small separations, comparison 3 16+V
v=--p-
15
DIELECTRICS
1 m d&E, NJ-v’, so
with (3.26) shows that l)(Q - 1).
These results, (3.29) and (3.31), are in agreement with those found by Lifshitz et al. [3] from the Casimir force. Naturally, they correspond to the van der Waals forces found by Clasimir and Polder [2]. C. Interact,ion between a Molecule and a Dielectric Plate
As a final application of these ideas, we will calculate the energy of interaction between a molecule of polarizability a(~) and a dielectric slab. This energy is given by (2.10) with (3.32)
&(r, w) = 47ra(w) 6(r - R),
which expresses the change in the dielectric constant when a molecule is inserted in the vacuum. at R. We will suppose that the dielectric slab occupies the region of space z < 0 with vacuum above it. The appropriate Green’s functions here, referring to a single interface, are trivially obtained from those discussed in Section 2. In region 2, g”: has the form of (2.31a) with the reflection coefficient r given by (3.33)
1 + r = 2K/(K + ‘Q),
which is obtained by taking the limits a + co, Ed = I, E~ = Ed. The energy is then CR, = 4 47dw)
rm(z,
z, k, , ~1,
(3.34)
where, from (2.27)
a az ~azf I gH IzIez c
rkk
=
co2gE +
= $
[
k2 +- L
+ -&
[ w2 z
(3.35)
+ (2k2 - w2) 2:: y z: ] e-2KZ,
with K2
=
k2
-
&?
q2
=
k2 - E+J~.
(3.36)
The necessary contact term here is easily deduced from the physical requirement that the energy of interaction go to zero as the separation gets large, z -+ co, which effectively removes the w”/K term in r,, . Therefore, the interaction energy between a molecule and a dielectric slab is 1 -s” dc 4m(LJ lrn dk2 + E = - 16n2 ,, 0 x
-52"~"1 [
K
+ (2kz + i2) :‘: 1
5 z: ] e-2KZ.
(3.37)
16
SCHWINGER, DERAAD, AND MILTON
One possible application of this result refers to the attraction perfectly conducting plate (Q ---f a~). It is then easy to show that E = -3ci/8nz4,
of a molecule by a (3.38)
a result’ first calculated by Casimir and Polder [2]. A second, particularly interesting possibility occurs when the molecule is of the same type as those composing the dielectric slab. When the common dielectric constant is close to unity, the energy of interaction, to lowest order in E - 1, is
where we have expressed the polarizability of the molecule in terms of the dielectric constant according to (3.30). Lifshitz et al. [3] have considered the limiting behavior for large separations (small 0, where E can be considered to be constant. An application of (3.39) to liquid helium will be considered in Section 5.
4. SURFACE TENSION
The same phenomena that give rise to forces between dielectrics should also be an important source of surface tension in a liquid. We recall that the surface tension may be defined by the change in energy of the liquid due to a deformation of its surface, in which the total volume of the liquid remains unaltered. We here wish to calculate the Casimir contribution to this change in energy. Accordingly, we consider a single interface, at z = 0, between two dielectrics 4)
= e1 ,
z > 0,
E2>
z < 0.
(4.1)
We then imagine an infinitesimal change in the location of the interface, which depends upon the transverse coordinates, 6z(x, v), in such a way that the volume of either dielectric does not alter,
s
dx dy 6z(x, y) = 0.
(4.2)
We can parametrize this alteration of the interface by means of a scale factor h, so that the dielectric constant may be written, in terms of the unit step function, 7, as &j(r)
= l 2 + (Ed - Ed)7j(z - h 6z(x, y)).
(4.3)
7It mightbepossible to measure theeffectdirectly,by using,for example,a variationof Fairbank’s experiment,Ref. 1131.(T. Erber, private communication.)
CASIMIR
The first- and second-order
EFFECT
IN
17
DIELECTRICS
derivatives of &) with respect to X are
2 dA) = Sz(x, y)(+ ah
- EJ S(z -
x Sz),
(4.4)
a2 E(A)= -(,Q - E,)[SZ(S, y)]" S'(z - x Sz). 6h2 The change in energy due to a variation ;; E(A) = ; 1 (dr) g
(c2 -
where we have noted that the alteration Clearly, at h = 0, (4.6) vanishes,
because of (4.2). The second-order
X
[
(43
in A is given by (2. IO), which for this case reads El) Sz(x, y) S(z - h Sz) I$:,)(& r, w),
(4.6)
of the interface changes the Green’s function.
change in the energy, at h = 0, is
-Sz S’(z) Tkk(r, r, w) + 6(z) $ l$(r,
r, w)/
1.
(4.8)
A=0
This change in the energy is directly related to the surface tension. Before we can evaluate (4.Q we must calculate (a/ax) I-‘;;’ jAzo. The Green’s function still satisfies the differential equation (2.16). Under a change in A, 6X, the change in the Green’s function, 6I’, satisfies -v Since ?is
x (V x s’;, + w% ST: = --co2 ss
the Green’s function
for this differential
operator,
Si!(r, r”) = J (dr’) i!(r, r’) . &(r’)
(4.9) the solution to (4.9) is
. F(r), f),
(4.10)
which entails
Because we will ultimately have in mind a liquid helium-vacuum interface for which cIV2N 1, we will here pick out the leading behavior in the difference in the dielectric constants. The first term in (4.8) involves (a/az) I’,, , which can be evaluated by using the explicit Green’s functions for a single interface. (For example, see (3.33).)
18
SCHWINGER,
In the approximation rewrite the result as
DERAAD,
of dielectric constants
n (dk,)
4 J m
= -Cc2 -
e
AND
MILTON
close to unity, it is straightforward
iky(r--r’)q&
zt
k 2
4 WZ’,
I,
2, k, , 4,
to
(4.12)
with r’ ---f r. This is closely related to (a/?%) r,, (Eq. (4.1 I)), occurring in the second term of (4.8), so that the two terms there fit together to yield (d,S = dxdy) dS dS’ g
(eZ -
cl)” 6z(x, y)[6z(x’,
y’) -
Sz(x, y)]
x @!fd 0 eickek’jL‘(‘-“)LI’kz(z, 0, k 12 co) T&O, z, k; , co)3 (4 .13) (27f)2 (2,rr)2 with z -+ 0. With the effective replacement Sz(x’, y’) -
Sz(x, y) -
; [(r’ - r)L . VI2 Sz
(4.14) -Jvv 2
a b szaa ak, ak;’
a, b = 1, 2,
the second-order change in the energy may be rewritten as
a
x b r,,(z - 0, 0, k, , w) ak r,,(O, z - 0, k, , u). (4.15) b ak, Being careful about setting z = 0, and writing the dyadic “r in a coordinate independent way, we find, using (2.25). (4.16) An integration by parts and the substitution w -+ i[ finally yields -$
E(A)
1 j- dS ; [V, 62(x, y)]” Ioa d&c, - d2 16rr2
= -
n=o
x irn dk2 (2 -
&
-
-&).
(4.17)
CASIMIR
EFFECT
IN
Since this change in the energy is related to a deformation
_22 /+v 8x"
19
DIELECTRICS
of the surface,
= Z's j d&s= z j ds; [V, Sz(s,y)]2,
(4.18)
A=0
the surface tension, Z, is 1 ,r z ~ jm dc(c, 16~~ o
- &).
(4.19)
As it stands, the surface tension is infinite.s What is required is a cutoff in k2, that is, in transverse momentum. The implication is that short-distance behavior makes a significant contribution to the surface tension. The continuum approach applied here is only accurate in the large-distance regime. Consequently, we see that to calculate the van der Waals contribution to the surface tension convincingly, we would have to graft on a detailed microscopic theory of short-range behavior. Such a theory, at present, does not exist. Nevertheless, in the concluding section, we will employ a crude cutoff to demonstrate that the correct order of magnitude of the surface tension in liquid helium is reasonably well reproduced.
5.
CONCLUSIONS
Jn the above we have verified Lifshitz’ formula [3] for the Casimir force between parallel dielectric interfaces, including the temperature dependence. Where the Russian calculations went wrong [6] was in their specialization to metal plates. A careful reading of their papers shows that, through relatively trivial errors, they obtained incorrect results for the Casimir force, for perfect conductors at finite temperatures, and for imperfect conductors at zero temperature. Beyond this clarification, we have investigated the Casimir contribution to the surface tension of a liquid at T = 0 as given by (4.19). As noted there, the transverse momentum integral requires a cutoff, k, = fiw,jc, implying that short-distance phenomena. play an important role. However, we can estimate the continuum contribution by taking for w, a frequency corresponding to atomic separations. For liquid He a rough model for the dielectric constant is provided by [4] E-
l +
0.05 1 + c2/wo2
(5.la)
1016set-l.
(5.lb)
with w. ‘v 3.5 Substituting
this into (4.19), and retaining
8 It is interesting to note that the contribution
X
only the leading dependence in the cutoff, due to the E polarization
is finite.
20
SCHWINGER,
we find for the Casimir contribution zero
DERAAD,
AND
MILTON
to the surface tension of liquid helium at absolute
ef 1 x 10-36[W,(sec-1)]2 erg/cm2.
(5.2)
Taking w e = 10ls set-‘, which corresponds
to a reasonable interatomic z Casimir
E
distance, we obtain
1 w/Cm2,
which is to be compared with the experimental
(5.3)
(5.4)
value [14]
z ew = 0.37 erg/cm2,
(5.5)
a factor of 3 smaller. That this qualitative agreement is not fortuitous may be seen by considering the Casimir model for the latent heat. That is provided by (3.39), which expressesthe energy of interaction between a molecule and a dielectric slab, E(z), where z is the separation between the molecule and the surface. The work required to separate a molecule from the surface and remove it to infinity is, if this is the only force involved, --E(O). The latent heat, q, is the energy required to vaporize a gram of substance, so 4 = --E@Wf). (54 Here N is the number density and p is the massdensity, which for liquid helium is p = 0.15 g/cm”. Then using (5.1) for the dielectric constant, we find for the leading behavior of the Casimir contribution to the latent heat
z
7 x 10-4i[w,(sec-1)]3 erg/g,
(5.7)
where w, is the transverse momentum cutoff. For the samerepresentative value of w, , (5.3), we find qCasimir
F
7 X 10' erg/g,
(5.8)
while the experimental value is two times larger [ 151 4exp = 15 x 10’ erg/g.
(5.9)
We seethat qualitative agreement with the experimental values for both the surface tension and the latent heat is achieved with the samevalue of the transverse momentum cutoff, (5.3). We are evidently lacking a quantitative theory, which would require
21
CASIMIR EFFECT IN DIELECTRICS
detailed microscopic considerations. However, we can fairly conclude that the Casimir effect, a manifestation of van der Waals forces, is responsible for a significant part of these phenomena.
APPENDIX:
GREEN'S FUNCTION
FOR POTENTIAL
Rather than introduce a Green’s function for the electric field strength, as in Section 2, it is possible to employ one for the vector potential. Since this method exhibits sorne interesting features of gauge covariance, and makes closer contact with the approach of Lifshitz et al. [3], we will discuss it briefly here. If A, is the vector potential, the corresponding Green’s function is defined by A.(x) = j (lid) D&x, x’) P(x’) + a&x),
(AlI
where J,, is the current density. To find the differential equation satisfied by D,, we note that in any region where E is constant, Maxwell’s equations can be written as L”“A
Y
=
JI* >
WI
where
LOi
=
=-c--j, a
LiO
(A31
at
Lij = S;&~“) + vivj ) where 3 = --E g
+ v2.
(A41
Since J, is not an arbitrary vector field, but is conserved, a,J“
= 0,
645)
we conclude that the equation for the Green’s function is L””
D&Y,
x’) = (g””
-
F)
6(x -
x’).
We solve this equation by making a specific choice of gauge, which is similar to the
22
SCHWINGER, DERAAD, AND MILTON
radiation gauge, in which Ai (and not A,) are the dynamically important That is, we can easily show that a solution to (A6) is achieved if D,, = D<, = Doi = 0,
(A74
and (-8 Once Dij is determined, the analog of (2.38),
variables.
aij + VJ,) Djk(x, x’) = &, 6(x - x’).
(A7b)
the effective product of vector potentials is determined by b&(x) Aj(x
= D&x, x’).
648)
The corresponding electric and magnetic field products are given by i.&(x) Ej(x’)leff = $ 4 iHi
Hj(x’)]
D&c, x’),
(A94
eft-= l itm~j~nVzV;Dmn(x, x’).
Wb)
For the geometry of Fig. 1, the Fourier decomposition (dk,)
ik1.br’)l
Dij(X, x’> = 1 mc
e
“w I
e-iw~t-t’)g,f(z
2rr
z ,
z~>
is appropriate. As before, we can choose k, to point along the x axis. In this case the only nonzero off-diagonal Green’s functions are g,, , g,, . The boundary conditions on gij at the surfaces z = 0, --a follow from the continuity of H, and of E, , E, , and EE,: E, , J& , (Hz) : g,, , g,, , g,, continuous,
(Al la)
H, : g g,, continuous,
(Allb)
Hv : 2 gxi - ikgzt continuous,
(Al lc)
EE, : Egsi continuous.
(Al Id)
We can now easily solve the differential equations (A7b) for the gij . The result is W”g~j
= ri* ,
(A13
where ? is the Green’s function discussed in Section 2, given, for example, in the intermediate region 2 by (2.44), (2.46), and (2.41). Consequently, the electric and magnetic field products are the same as before, given by (2.38) and (2.40). ACKNOWLEDGMENT We would like to thank Professor S. Putterman for arousing our interest in this problem, and for many stimulating discussions.
CASIMIR
EFFECT
IN
DIELECTRICS
23
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