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Heat waves in rough-walled dielectric cylinders at low temperatures
Brown,
C. R.
Physica
1967
35
114-118
HEAT WAVES IN ROUGH-WALLED DIELECTRIC CYLINDERS AT LOW TEMPERATURES by C. R. BROWN Department
of Physics,
University
of British
Columbia,
Vancouver,
Canada
Synopsis The
modified
heat
equation
cussion
of heat
pulse
transmission
which phonons for the speed
are subject
used by Brown,
to diffuse
and attenuation
in sapphire boundary
of high
frequency
Chung
and Matthews
and helium, scattering heat
is derived
only.
waves
1) in a disfor a model
Expressions under
these
in
are given conditions.
1. Intro&&on. Several authors, including Howling, Mendoza and Zimmermans), Englmana) and Vernottea), have pointed out the necessity of modifying the Fourier equation for heat conduction,
in order to obtain a finite limit to the speed of propagation of high frequency heat waves. In a recent paper, Brown, Chung and Matthewsl) used V e rno t t e’s4) phenomenological modified heat equation
(1) (in which T is a relaxation time), in a discussion of the propagation of heat pulses in A1203 crystals at temperatures of 5.O”K and 3.8”K. At these temperatures, boundary scattering was the dominant scattering ‘process. Elsewhere, Vernotte’s heat equation has been employed as the basis of a discussion of the possibility of observing second sound in dielectric solids5). An attempt is made in this paper to derive a modified heat equation for a situation approximating that discussed by Brown et alii. 2. The modified heat eqztation. Consider a dielectric single crystal cut in the form of a cylinder having a surface which is rough in the sense that all incident phonons are diffusely reflected. For simplicity, the phonon spectrum is assumed to be isotropic and dispersionless. It is assumed that the temperature is sufficiently low and the crystal sufficiently pure, that the flow of -
114 -
HEAT
WAVES
phonons is completely mire),
IN ROUGH-WALLED
controlled
DIELECTRIC
by boundary
115
CYLINDERS
scattering.
Following
it is assumed that, in the process of diffuse reflection, as a “black” surface, absorbing all incident phonons
Casi-
the boundary and radiating
acts phonons according to a distribution which is symmetric in the wave vector Q, and which defines the boundary temperature. In the following discussion, the symmetric distribution at any boundary point rB is given a time dependence that will produce zero net radial energy flux at the boundary at all times, and is taken to have cylindrical symmetry in the space coordinates. This distribution is described in terms of the set of phonon state occupation numbers {PZO(Q,r& t)}. Let {?z(q, r, t)} be the distribution at a point r inside the crystal and let {na(~, r, t)} be identical with the distribution on the boundary at its intersection with the cross section containing the point r. It is assumed that each occupation number n(q, r, t) may be split up as ti(q, r, t) = IZO(Q,r, t) + ~zr(q, r, t). ~1, which is assumed to be small compared with IZO(Q,r, t), is therefore related to the net phonon flux in the asymmetric distribution {PZ(~, r, t)}. The wavelength of the heat waves will be assumed to be somewhat longer than the crystal diameter. Since there is no scattering inside the specimen, all phonons of wave vector q at a point r at time t, left the same point rB on the boundary at time t - (Ir - rBl/uO). Hence
where vector
(r - rB) is parallel q. Making a Taylor
to (%60/q)q, the velocity of a phonon of wave expansion of no(q, r, t) about r, one has
fi(q, rJ 4 = "0(q, r,t) + mkz,
c
IzO(%
rpt) +
(rB
r, 4
- r) - vnO(q,r,
t) -
u.- r’ -‘s (q, r, t). jrB
If 6 is defined as the angle between (r - rB) and the 2 (cylinder) axis in a system of cylindrical coordinates p, z, 4; in, 2~2,+B, then, dropping the arguments of rto(q, z, t), one has
The symmetric part of 1~1 (i.e. the second term on the right) gives a frequency dependent contribution to the energy density at internal points r. This leads to time dependent radial heat fluxes and temperature gradients inside the specimen. It is this symmetric part of ni which gives the wavelike correction to the Fourier heat equation. The fact that nr is not necessarily zero at the boundary implies a temperature discontinuity between the boundary and the adjacent phonon gas. This result, while perhaps disturbing at first sight, has a well established analogue in the physics of rarified molecular gases 7). Knudsen 7) defined an “accommodation coef-
116
C. R. BROWN
ficient”
which is a measure
of the extent
to which gas molecules
leaving
a
wall behave as if emitted from a mass of gas at the wall temperature. Thus an accommodation coefficient of unity has been assumed in this model. Briefly, the result for a rarified molecular gas adjacent to a wall with an accommodation coefficient of unity is that a temperature discontinuity AT c L aT/an exists, where L is the mean free path and aTIan is the temperature gradient normal to the walls). In the absence of scattering inside the crystal, the Boltzmann equation for phonons of wave vector q reduces to
On substituting rc(q, r, t) '= n0 + nl in eq. (3), using eq. (2) to express in terms of no and carrying out the differentiation, one arrives at
ano
It-B-
at+
a2fio ano ~ = 3240 cos e __ + z40cos2f3 IrB at2 a.2
r(
240
nl
ako r/ F.
With appropriate values of rB and 8, this equation holds for all values of q. Two averages, X and Z are now defined. X(Z’, p’, I#‘, q) is the average of x over all phonons with wave vectors of magnitude q arriving at z’, p’, 4’. Z(z, q) is the average of X over the disc Z’ = z, 0 < p < pB. Thus cos 8 and hence cos 0 are zero. If No(q, z, t) dq is defined as the number of phonons per unit length of cylinder at z, t with wave numbers in the range q, q + dq, given by the distribution {no(q, z, t)} th en, using the above definitions, aN0 at+---
(10 -
4)
a2N0
No(q, z, t) and the corresponding the wave equation.
energy
a2Eo -_at2
a2No rj) a22.
= u0 Cm2 e(lrg -
at2
density
E&,
t) therefore
COS2 6 IrB -
-& -
(
p I0 -
rj
rl
satisfy
a2Eo azz’
If the wall temperature has the form To + Tl(z, t), where IT11 < To then T1 also satisfies eq. (4), which has the form of Vernotte’s modified heat equation, eq. (1). For an infinite cylinder of radius pB = R, 2n
m
COS2 6 Irg -
rl =
f&
s --oo
dZB
d& .&[R s [R2 + p2 -
p cos #'El
2pR cos ‘#B + z;]”
0
and [R -
____
--co
d’B 0
[R2 + p2 -
p Cm +B] 2pR cos +B + Z;]
HEAT
WAVES
IN ROUGH-WALLED
since the solid angle dQ subtended
DIELECTRIC
CYLINDERS
117
at a point z = 0, 4 = 0, p by the area
dS = R d& dzg on the curved surface is
RR -
dQ=
[Rs + p2 -
P ~0s
9~1
dzBd+B
2pR cos 4~ + &];l .
The integrals 1
jr - rBI = ~ dSlr-rBI, ?CR2
~r-r~~cos~0=~~dScos20~ra--r~
disc
disc
may be evaluated by the method given by Kennards) for similar integrals arising in the transport theory of rarified gases. The results are 4R
IrB - r/ = -, Thus the equation ature becomes
satisfied a2T1
at2
3
toss 0 IrB -
rl = F.
by the time dependent 3zQ aT1 +_.__---_~4R at
part of the wall temper-
24; a2T1 a22 ’
2
(5)
which is to be compared with the equation used by Brown et al. As remarked above, the total energy density per unit length includes a contribution Ei(z, t) from the various ni(q, r, t). Using eq. (2) and integrating as before, one obtains
El(z, t) = -
4R __ 3~
aEO __
at
(6
t).
Thus, for sinusoidal variations, the time dependent part of the energy density will lag the time dependent part of the wall temperature by a phase angle tani( where T = 4R/3~a. 3. Properties of the modified heat equation. If a solution of the form TI(z, t) = Tl exp[i(Rz - cot)] is assumed for eq. (5), then, putting K = = j3 + iu and solving for /?zand &‘, one finds a2 = p
1/[2Gi~~( 1 + (1 + (COT)-2)*)]
= f
[l + (1 + bd-2Yl
where T = 4R/3ue as before. Thus in the low frequency limit I/31= \a/= (~/zti~)* and in the high frequency limit /? = 2Q0/ue, a = I/(~*uoT). Therefore at high frequencies a limiting speed of propagation ua/2t and a constant attenuation coefficient
118 a =
HEAT
WAVES
IN ROUGH-WALLED
3/2*4R are predicted.
DIELECTRIC
At low frequencies
CYLINDERS
eq. (5) reduces to the diffusion
equation a2T1
aT1 ---_DD at
a22
with D = (2ztaR/3). 4. Conclusions. somewhat greater
The assumption that the heat waves have a wavelength than the crystal diameter may now be made more spe-
cific. In view of the result Ir - t-B/ = 4R/3, the above treatment is expected to begin to fail for wavelengths such that (A/4) c 4R/3. For a wavelength of 5R, a speed of roughly 0.88(ua/2&) is predicted, whereas a (phase) velocity of 1.67~0 is obtained from the Fourier heat equation. At a wavelength of 2R eqs. (4) and (5) predict a speed of 0.99(ua/2*). In a pulse experiment, one would expect to see a small signal travelling at ua, due to direct phonon transmission down a finite cylinder, followed by a pulse with its leading edge travelling at roughly uo/2*, but with this speed modified slightly by small amounts of specular reflection and a few “normal collisions”. It should perhaps be emphasized that the factor 24 reflects the geometry of the specimen, whereas the factor 31 in the expression for the speed of second sound in solids5)s) has a statistical origin. Acknowledgements. I wish to thank Dr. P. W. Matthews for helpful discussions and for making work available prior to publication. I am indebted to Professor R. E. Burgess for his constructive criticism. This work was supported by the National Research Council of Canada. Received
20-10-66
REFERENCES
1) 4
Brown,
3)
Englman,
R., Proc.
4) 5) 6)
Vernotte,
D., Compt.
Howling,
J, B., Chung,
I). Y. and Matthews,
D. H., Mendoza, Phys.
P. W., Phys.
E. and Zimmermann, Sot.
Rend.
72 (1958)
246
(1958)
Letters 21 (1966) 241.
J. E., Proc.
roy.
Sot. 11229
(1955)
391. 3154.
Chester, M., Phys. Rev. 131 (1963) 2013. Casimir, H. B. G., Physica 5 (1938) 495. M., Ann. Physik 34 (1911) 593. 7) Knudsen, E. H., Kinetic Theory of Gases (McGraw Hill, New York, 8) Kennard, R. A. and Krumhaast J. A., Phys. Rev. 148 (1966) 766. 9) Guyer,
1938) Chap. 8.
86.
C. R.
Physica
1967
35
114-118
HEAT WAVES IN ROUGH-WALLED DIELECTRIC CYLINDERS AT LOW TEMPERATURES by C. R. BROWN Department
of Physics,
University
of British
Columbia,
Vancouver,
Canada
Synopsis The
modified
heat
equation
cussion
of heat
pulse
transmission
which phonons for the speed
are subject
used by Brown,
to diffuse
and attenuation
in sapphire boundary
of high
frequency
Chung
and Matthews
and helium, scattering heat
is derived
only.
waves
1) in a disfor a model
Expressions under
these
in
are given conditions.
1. Intro&&on. Several authors, including Howling, Mendoza and Zimmermans), Englmana) and Vernottea), have pointed out the necessity of modifying the Fourier equation for heat conduction,
in order to obtain a finite limit to the speed of propagation of high frequency heat waves. In a recent paper, Brown, Chung and Matthewsl) used V e rno t t e’s4) phenomenological modified heat equation
(1) (in which T is a relaxation time), in a discussion of the propagation of heat pulses in A1203 crystals at temperatures of 5.O”K and 3.8”K. At these temperatures, boundary scattering was the dominant scattering ‘process. Elsewhere, Vernotte’s heat equation has been employed as the basis of a discussion of the possibility of observing second sound in dielectric solids5). An attempt is made in this paper to derive a modified heat equation for a situation approximating that discussed by Brown et alii. 2. The modified heat eqztation. Consider a dielectric single crystal cut in the form of a cylinder having a surface which is rough in the sense that all incident phonons are diffusely reflected. For simplicity, the phonon spectrum is assumed to be isotropic and dispersionless. It is assumed that the temperature is sufficiently low and the crystal sufficiently pure, that the flow of -
114 -
HEAT
WAVES
phonons is completely mire),
IN ROUGH-WALLED
controlled
DIELECTRIC
by boundary
115
CYLINDERS
scattering.
Following
it is assumed that, in the process of diffuse reflection, as a “black” surface, absorbing all incident phonons
Casi-
the boundary and radiating
acts phonons according to a distribution which is symmetric in the wave vector Q, and which defines the boundary temperature. In the following discussion, the symmetric distribution at any boundary point rB is given a time dependence that will produce zero net radial energy flux at the boundary at all times, and is taken to have cylindrical symmetry in the space coordinates. This distribution is described in terms of the set of phonon state occupation numbers {PZO(Q,r& t)}. Let {?z(q, r, t)} be the distribution at a point r inside the crystal and let {na(~, r, t)} be identical with the distribution on the boundary at its intersection with the cross section containing the point r. It is assumed that each occupation number n(q, r, t) may be split up as ti(q, r, t) = IZO(Q,r, t) + ~zr(q, r, t). ~1, which is assumed to be small compared with IZO(Q,r, t), is therefore related to the net phonon flux in the asymmetric distribution {PZ(~, r, t)}. The wavelength of the heat waves will be assumed to be somewhat longer than the crystal diameter. Since there is no scattering inside the specimen, all phonons of wave vector q at a point r at time t, left the same point rB on the boundary at time t - (Ir - rBl/uO). Hence
where vector
(r - rB) is parallel q. Making a Taylor
to (%60/q)q, the velocity of a phonon of wave expansion of no(q, r, t) about r, one has
fi(q, rJ 4 = "0(q, r,t) + mkz,
c
IzO(%
rpt) +
(rB
r, 4
- r) - vnO(q,r,
t) -
u.- r’ -‘s (q, r, t). jrB
If 6 is defined as the angle between (r - rB) and the 2 (cylinder) axis in a system of cylindrical coordinates p, z, 4; in, 2~2,+B, then, dropping the arguments of rto(q, z, t), one has
The symmetric part of 1~1 (i.e. the second term on the right) gives a frequency dependent contribution to the energy density at internal points r. This leads to time dependent radial heat fluxes and temperature gradients inside the specimen. It is this symmetric part of ni which gives the wavelike correction to the Fourier heat equation. The fact that nr is not necessarily zero at the boundary implies a temperature discontinuity between the boundary and the adjacent phonon gas. This result, while perhaps disturbing at first sight, has a well established analogue in the physics of rarified molecular gases 7). Knudsen 7) defined an “accommodation coef-
116
C. R. BROWN
ficient”
which is a measure
of the extent
to which gas molecules
leaving
a
wall behave as if emitted from a mass of gas at the wall temperature. Thus an accommodation coefficient of unity has been assumed in this model. Briefly, the result for a rarified molecular gas adjacent to a wall with an accommodation coefficient of unity is that a temperature discontinuity AT c L aT/an exists, where L is the mean free path and aTIan is the temperature gradient normal to the walls). In the absence of scattering inside the crystal, the Boltzmann equation for phonons of wave vector q reduces to
On substituting rc(q, r, t) '= n0 + nl in eq. (3), using eq. (2) to express in terms of no and carrying out the differentiation, one arrives at
ano
It-B-
at+
a2fio ano ~ = 3240 cos e __ + z40cos2f3 IrB at2 a.2
r(
240
nl
ako r/ F.
With appropriate values of rB and 8, this equation holds for all values of q. Two averages, X and Z are now defined. X(Z’, p’, I#‘, q) is the average of x over all phonons with wave vectors of magnitude q arriving at z’, p’, 4’. Z(z, q) is the average of X over the disc Z’ = z, 0 < p < pB. Thus cos 8 and hence cos 0 are zero. If No(q, z, t) dq is defined as the number of phonons per unit length of cylinder at z, t with wave numbers in the range q, q + dq, given by the distribution {no(q, z, t)} th en, using the above definitions, aN0 at+---
(10 -
4)
a2N0
No(q, z, t) and the corresponding the wave equation.
energy
a2Eo -_at2
a2No rj) a22.
= u0 Cm2 e(lrg -
at2
density
E&,
t) therefore
COS2 6 IrB -
-& -
(
p I0 -
rj
rl
satisfy
a2Eo azz’
If the wall temperature has the form To + Tl(z, t), where IT11 < To then T1 also satisfies eq. (4), which has the form of Vernotte’s modified heat equation, eq. (1). For an infinite cylinder of radius pB = R, 2n
m
COS2 6 Irg -
rl =
f&
s --oo
dZB
d& .&[R s [R2 + p2 -
p cos #'El
2pR cos ‘#B + z;]”
0
and [R -
____
--co
d’B 0
[R2 + p2 -
p Cm +B] 2pR cos +B + Z;]
HEAT
WAVES
IN ROUGH-WALLED
since the solid angle dQ subtended
DIELECTRIC
CYLINDERS
117
at a point z = 0, 4 = 0, p by the area
dS = R d& dzg on the curved surface is
RR -
dQ=
[Rs + p2 -
P ~0s
9~1
dzBd+B
2pR cos 4~ + &];l .
The integrals 1
jr - rBI = ~ dSlr-rBI, ?CR2
~r-r~~cos~0=~~dScos20~ra--r~
disc
disc
may be evaluated by the method given by Kennards) for similar integrals arising in the transport theory of rarified gases. The results are 4R
IrB - r/ = -, Thus the equation ature becomes
satisfied a2T1
at2
3
toss 0 IrB -
rl = F.
by the time dependent 3zQ aT1 +_.__---_~4R at
part of the wall temper-
24; a2T1 a22 ’
2
(5)
which is to be compared with the equation used by Brown et al. As remarked above, the total energy density per unit length includes a contribution Ei(z, t) from the various ni(q, r, t). Using eq. (2) and integrating as before, one obtains
El(z, t) = -
4R __ 3~
aEO __
at
(6
t).
Thus, for sinusoidal variations, the time dependent part of the energy density will lag the time dependent part of the wall temperature by a phase angle tani( where T = 4R/3~a. 3. Properties of the modified heat equation. If a solution of the form TI(z, t) = Tl exp[i(Rz - cot)] is assumed for eq. (5), then, putting K = = j3 + iu and solving for /?zand &‘, one finds a2 = p
1/[2Gi~~( 1 + (1 + (COT)-2)*)]
= f
[l + (1 + bd-2Yl
where T = 4R/3ue as before. Thus in the low frequency limit I/31= \a/= (~/zti~)* and in the high frequency limit /? = 2Q0/ue, a = I/(~*uoT). Therefore at high frequencies a limiting speed of propagation ua/2t and a constant attenuation coefficient
118 a =
HEAT
WAVES
IN ROUGH-WALLED
3/2*4R are predicted.
DIELECTRIC
At low frequencies
CYLINDERS
eq. (5) reduces to the diffusion
equation a2T1
aT1 ---_DD at
a22
with D = (2ztaR/3). 4. Conclusions. somewhat greater
The assumption that the heat waves have a wavelength than the crystal diameter may now be made more spe-
cific. In view of the result Ir - t-B/ = 4R/3, the above treatment is expected to begin to fail for wavelengths such that (A/4) c 4R/3. For a wavelength of 5R, a speed of roughly 0.88(ua/2&) is predicted, whereas a (phase) velocity of 1.67~0 is obtained from the Fourier heat equation. At a wavelength of 2R eqs. (4) and (5) predict a speed of 0.99(ua/2*). In a pulse experiment, one would expect to see a small signal travelling at ua, due to direct phonon transmission down a finite cylinder, followed by a pulse with its leading edge travelling at roughly uo/2*, but with this speed modified slightly by small amounts of specular reflection and a few “normal collisions”. It should perhaps be emphasized that the factor 24 reflects the geometry of the specimen, whereas the factor 31 in the expression for the speed of second sound in solids5)s) has a statistical origin. Acknowledgements. I wish to thank Dr. P. W. Matthews for helpful discussions and for making work available prior to publication. I am indebted to Professor R. E. Burgess for his constructive criticism. This work was supported by the National Research Council of Canada. Received
20-10-66
REFERENCES
1) 4
Brown,
3)
Englman,
R., Proc.
4) 5) 6)
Vernotte,
D., Compt.
Howling,
J, B., Chung,
I). Y. and Matthews,
D. H., Mendoza, Phys.
P. W., Phys.
E. and Zimmermann, Sot.
Rend.
72 (1958)
246
(1958)
Letters 21 (1966) 241.
J. E., Proc.
roy.
Sot. 11229
(1955)
391. 3154.
Chester, M., Phys. Rev. 131 (1963) 2013. Casimir, H. B. G., Physica 5 (1938) 495. M., Ann. Physik 34 (1911) 593. 7) Knudsen, E. H., Kinetic Theory of Gases (McGraw Hill, New York, 8) Kennard, R. A. and Krumhaast J. A., Phys. Rev. 148 (1966) 766. 9) Guyer,
1938) Chap. 8.
86.