INTERFACES AND BOUNDARIESTHE SOLID-SOLID INTERFACE IN THERMAL PHONON RADIATION

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INTERFACES AND BOUNDARIESTHE

SOLID-SOLID INTERFACE IN THERMAL PHONON RADIATION

O. Weis

To cite this version:

O. Weis. INTERFACES AND BOUNDARIESTHE SOLID-SOLID INTERFACE IN THER- MAL PHONON RADIATION. Journal de Physique Colloques, 1972, 33 (C4), pp.C4-49-C4-56.

�10.1051/jphyscol:1972411�. �jpa-00215088�

IN TERFA CES AND BOUNDARIES

THE SOLID-SOLID INTERFACE IN THERMAL PHONON RADIATION

Institute fiir Angewandte Physik, Universitat Heidelberg, Germany

Abstract. - The problem of thermal phonon transmission across a solid-solid interface between two dissimilar materials is discussed especially in connexion with the generation of THz-phonon beams by thermal phonon radiators.

The thermal phonon radiators are current heated thin metal films deposited on cooled dielectric monocrystals which serve as transmission media.

Measurements of the radiation temperature as function of the transmitted phonon power are reported and show that the acoustic-mismatch model describes the thermal phonon transmission quite well in a frequency range where dispersion effects are still not important. In order to ensure that the formulas of acoustic-mismatch between two isotropic solids are applicable without too large error, these experiments have been performed by using crystals

-

like sapphire and especially diamond - showing only a small mechanical anisotropy.

To avoid such limitations in future, an extension is given for the general case of an interface between an isotropic radiator and an anisotropic substrate. No large differencies are to be expect- ed in most crystals for the emissivities in comparison to the values calculated from the isotropic approximation, since one has to integrate over the half space. By far more important seems to be this extension of theory for the evaluation of the radiant characteristics, respectively for the eva- luation of the time dependent phonon power falling onto phonon detectors or other devices.

1. Introduction. - The aim of this paper is twofold.

First I want to discuss the thermal phonon transmis- sion through a solid-solid interface between two diffe- rent materials as it happens if a metal film deposited on a dielectric crystal is heated and emits thermal pho- nons into the substrate. There is now experimental evidence justifying the use of the acoustic-mismatch model over a wide frequency range. Second I would like to point out that thermal phonon radiation is a simple and powerful method to generate phonon pulses with phonon frequencies adjustable up to the acoustic or even optical cutoff frequencies of the radiator material, i. e. up to some 10''

...

¹⁰¹³ Hz. This last aspect opens for direct phonon pulse experiments a wide frequency range inaccessible to any other present method.

The heat transfer by phonons across an interface between two dissimilar, isotropic solids at low tempe- ratures was treated first by W. A. Little [I] in 1959 under the assumption that thermal phonons are reflected, transmitted and transfered into other polari- zations at the interface according to the laws of conti- nuum acoustics, respectively to the theory of elasticity.

Meanwhile many thermal-conductivity measurements have been performed with combinations of a large variety of materials to determine or to verify the thermal boundary resistance predicted by this ^(< acous- tic-mismatch model

>>.

But no convicting results were achieved by this experimentally not quite easy method, which requires a reliable extrapolation of the tempera-

ture gradient along both materials to get the tempera- tures a t the two sides of the interface and requires of course an excellent microscopic contact between such dissimilar bulk materials which has to withstand fur- thermore large temperature changes. A review article including this subject was recently published by Cheeke [2]. A direct excitation of phonons in the dielectric substrate by electrons of the metal film across the contact has also been considered. But this electronic contribution seems to be not important in the experiments to be described and will be neglected in the following. The interest in phonon transmission across a solid-solid interface was stimulated especially in 1964 by the heat pulse experiments of von Gutfeld and Nethercot 131. By observing the temperature decay of light pulse heated metal films evaporated onto several crystals, von Gutfeld, Nethercot and Arm- strong [4] deduced data giving no decision wether the acoustic-mismatch model is valid or the model of perfect match, i. e. a model which assumes a unhinder- ed transmission of phonons across the contact, or any other model.

I n connexion with the experimental investigation of very hot thermal phonon radiators, Herth and Weis [5]

introduced an electric pulse reflection method to determine the dependence of radiation temperature on emitted phonon radiation power and gave results for thin films of gold, copper, nickel and lead on sapphire in good accordance with the acoustic-mismatch model.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972411

Measurements by Kappus and Weis [6] using the same metals on diamond show the same behaviour. The acoustic mismatch-model is also supported by many measurements reported by Cheeke 171 in this issue, using the electric pulse reflection method too. Whereas these measurements were performed with radiator temperatures above 10 OK, recently Wigmore [8]

investigated the thermal emission from a constantan radiator deposited on a MgO-crystal by exploiting the paramagnetic ions Fe2' as a tunable phonon scatterer in the region below 6 OK and found full agreement with predictions of the acoustic mismatch model.

2. Crystal substrates approximated as isotropic solid.

- In order to justify the use of formulas derived for the phonon transmission across the interface between two isotropic solids, strictly speaking, only crystals should be used for an experimental comparison, which have a small anisotropy in phase velocities. To get an impression of this kind of approximation, we have to look at the angular dependence of the phase velocities in crystals. In the long wavelength region, where dis- persion can be neglected, the wave vector q of phonons of polarization a is connected to the phonon frequency o(q, a) by the relation

en 0)

q =

~o(en) ' (2

-

1) where c,(en) denotes the phase velocity of phonons of polarization a and wave normal en ⁼ q/l q

I.

From (2. l), it is obvious that all constant frequency surfaces, respectively energy surfaces in q-space have the same shape as the reciprocal phase velocity surface (slowness surface) which can be constructed taking for each direction en the reciprocal phase velocity l/c,(en) as radius vector to the surface. Figure 1 shows these surfaces for diamond, sapphire and quartz. The elastic constants and mass densities used in numerical compu- tation of the phase velocities are listed in table I.

According to figure 1 an approximation of the energy surfaces by spheres is quite good for diamond and less for sapphire, but seems bad for quartz. Especially for the calculation of the directional phonon emission the last materials demand an extension of the theory to

FIG. I. -Normalized energy surfaces of long wavelength phonons in diamond, sapphire and quartz, characterized by contour lines of constant absolute value of the wave vector and hence of phase velocity too. The minimum distance to the origo of q-space is taken as reference and denoted by 1 .O. Phonons described by a certain surface point propagate in direction of the surface normal at this point. In each case, only one half of the

energy surface is shown.

include also anisotropic substrates, whereas the evapo- rated metal phonon radiators are isotropic from a macroscopic point of view and will therefore further be treated as an isotropic material.

3. Phonon emission spectra.

-

In calculating the phonon transmission the metal film is regarded as a half space. This seems admissible if the free phonon path length is short in comparison to the film thickness, or if diffuse phonon reflections occur at the free surface of the metal film. Further we assume that there exists a thermal equilibrium distribution for all phonons in the radiator material [I], corresponding t o , a certain temperature TI. The phonon state density will be described by a modified Debye model which assumes that transverse and longitudinal phonons habe the same maximum wave vector ( q,,

I,

i. e. in medium (1) the Brillouin zone of each acoustic phonon polarization is replaced by a sphere with radius

I

q,,,

I

and hence

Data for diamond, sapphire and quartz used in numerical calculations. The elastic constants are taken from the work of McSkimin and Bond [9] for diamond, Bernstein [lo] for sapphire and Farnell [ l l ] for quartz. The last column show the phase velocities used in the isotropic approximation.

Mass Phase velocities

density Elastic constants (1012 dyn/cm2) (km/s)

(g/cm3> C 1 l C12 C 4 4 C 3 3 C 1 3 C14 (A ^Ct

- -

-

-

-

- - - -

Diamond.

. .

3.512 10.76 1.25 5.76 17.50 12.80

Sapphire

. . .

3.986 4.902 1.654 1.454 4.902 1.130 - 0.232 11.10 6.04 a-quartz.

. . .

2.650 0.869 4 0.069 6 0.576 2 1.068 0 0.156 0 0.174 3

THE SOLID-SOLID INTERFACE I N THERMAL PHONON RADIATION C4-51

each polarization branch has its own cutoff frequency can be given (analogous to the treatment of Weis [12]) (ou)max. Any possible inelastic phonon effects are for the spectral phonon power p,(o, TI, To) of polari- neglected at the interface. zation z being emitted across the contact area A into

Under these assumptions the following expression the crystal (0) being at the temperature To :

I o

^else.

where z means the polarization of the quasilongitudinal phonons (z = L) or of the quasitransverse phonons

( x = TI,

T,)

emitted into the anisotropic substrate (0).

In the isotropic radiator (1) designate the polariza- tion of the longitudinal phonons (z = D) or of the transverse phonons having a polarization in the plane of incidence (z = S,) or perpendicular (z = S2).

ejlO)(w) is the spectral (half space) emissivity for pho- nons of polarization z, defined as the ratio

ejiO)(o) = P,(u, TI, To = )0'

(3 .2)

Pz (07 = 0°) Iperfect match

and hence gives directly the spectral reduction factor for the emission of phonons of polarization z into the whole substrate (0) in comparison to an undisturbed transmission. This quantity reflects in an integral manner the phonon reflection, refraction and polariza- tion change at the interface. As already mentioned in the long wavelength range the appropriate model to calculate emissivities is the acoustic-mismatch model.

This gives emissivities independent of frequency.

Certainly, at phonon frequencies where dispersion becomes important the acoustic mismatch-calculation will cease to be valid. Since there exists today no other model at these very high frequencies (it should be a lattice dynamic model) it is convenient to take the frequency independent emissivities of the acoustic- mismatch model over the whole frequency range in calculating the emitted phonon power - at least for the purpose of comparison with experimental results.

Figure 2 gives the spectrum of phonons emitted from a constantan radiator at different radiation tempera- tures Tl into a substrate (0) held at absolute zero temperature under perfect matching conditions (e!lO'(w) = 1). In order to come nearer to reality, these spectra must still be multiplied by the unknown frequency dependent emissivities ejlO'(o).

4. Emissivities and radiant characteristics assuming acoustic-mismatch between isotropic radiator and aniso- tropic substrate. - We consider now in more detail the acoustic-mismatch in order to derive expressions suitable for numerical computation of the half space emissivities into anisotropic media and to get some insight in the determination of radiant characteristics of thermal phonon radiators.

+Frequency Hz -+ Frequency Hz

FIG. 2. -Thermal phonon emission spectra of a constantan radiator under perfect matching conditions, valid for longitu- dinal phonons. Phonon dispersion is neglected on the left side and put equal to the dispersion of a linear chain on the right.

Constantan temperature as indicated, substrate at 0 OK.

At the interface, the continuity of particle dispIace- ment and of stress is required by the theory of elasticity.

As a consequence of these boundary conditions an incoming plane wave of polarization o gives rise to reflected and transmitted waves of the same and of the other polarizations z having all the same wave vector component parallel to the interface. This corresponds for the transmitted waves to the law of refraction

sin 8;') _- sin 0:O'

(1)

c!o)(e.))

co

and the condition

&') = (1)

CPU 9 (4.1)

valid for each z = L, T1 or T , together with o = D, S, or S,. Here the wave normals are denoted for the incoming wave by

eil) = (cos q12l). sin 8L1), sin

qy).

sin 8L1), cos 8:))

( 4 . 2 ~ )

and for the outgoing waves of polarization z by eiO) = (COS q!'). sin 8!O), sin q!') .sin 0!O), cos 8:'))

.

Since these wave normals are in a unique relation to each other by (4. I), we will choose either the angles of eil) or these of eiO', if this is more convenient to desi- gnate the values of the phonon transmission probabi- lities of incoming phonons of polarization a and transmitted with z into the substrate :

(0) - (10) (1)

t:f0)(e;l))

=

tSo)(e, ) = to, (0, , qg')

- ^{(10) (0)}

= tar (or ~ 1 ~ ' )

.

(4.3) These transmission probabilities must be computed numerically for each wave normal from expressions for the power transmission factor of plane elastic waves

falling onto such an interface. For details see the books of Federov [13] and of Musgrave [14]. In the following it will be assumed that these transmission probabilities are already determined.

The spectral energy flux of phonons of polarization (a = D, S, or S,) coming from the solid angle sin ~ ~ ' ) . d f ? ~ " . d ~ ~ ~ ' and striking the contact area A is given by

= sin 8:'). d8y).dqi1). i,(w, TI). A.cos 8,") (4.4) where i,(w, TI) means the isotropic spectral energy flux per solid angle (= spectral radiant intensity) in the phonon radiator material. The integration over the half space gives the whole spectral radiation power pJ1'(o, Ti) of polarization o falling onto the contact

area

P C ) ( ~ , T,) =

jni2

^{/ 2 n} dOJ1).dg)!'). iil)(o, TI). A.sin ^B~').COS 0;') e y ) = o ^{q(')- a - 0}

being equal to the transmitted spectral radiation power under perfect matching conditions and a substrate temperature To = 0 OK. According to (4.5) and (3. I ) the spectral radiant intensities of different polarizations are related in medium (1) to each other in the following way

Under acoustic-mismatch conditions only the fraction t2f0'(0:', qy') of the phonon power (4.4) will be transmitted into a phonon beam with polarization z filling the solid angle sin 8?'.d01°).dqj0) in medium (0) and hence gives rise to a spectral radiant intensity with a wave normal eLO)

sin 0:'). do:'). dqL1) igO)(o, TI, B!'), qjO)). A = i:"(o, T,). A.cos

05".

t ~ : O ' ( 8 ~ , q"') sin

dq:O)

.

This can be rewritten using the relations (4.1) and (4.6) :

'1' 2 sin g?' dO(l)

T

8'0)

ZuT

,, ^{, ,}

^{qiO)) =} i!"(w, T,). t::~)(e:~),

,a)] . .

^--

^.

^L

^.

^{cos 8L1'}

C, sin dl0) dB!')

Summing over all three polarizations o = D, S, and S, we get the spectral radiant intensity including all phonons of polarization z emitted into the substrate (0) and having a wave normal e?) :

( 1 ) 2 sin 0:' d0'"

= 1 ,)

(

~::O)(BI'),

- .

²

.

^{cos 8:'} ( 4 . 7 ~ )

CU sin 81'' dB!')

or using the refraction law (4.1)

(1) 2 1

jlY)(o, TI, @jO), g)(0)) = i!')(~, TI)

(%) (

^{cos 0:')}

^-

^{sin 9p) -}

. dx ^{] . Z}

I%O)(O:~), qlO) )

.

(4.76)

C, CT (0) T

The expression (4.7a) is especially suitable for the determination of the half space emissivities and the expression (4.7b) for the phonon radiant characteristics.

THE SOLID-SOLID INTERFACE I N THERMAL PHONON RADIATION C4-53

Combining (3.2), (4.5) and ( 4 . 7 4 we finally get the formula for the half space emissivities of phonons of polarization z (z = L, T I , T2) valid for an anisotropic medium (0)

iJO)(o, TI, 0!O), cp!')) .sin O!O).dO!O).dp!O). A P, ^{( ~} TI, To ⁷ = 0)

ey

^{= -}

-

P r (07 ⁼ 0) [perfect match i:l)(o, TI).n.A

Integration in (4.8) must be done numerically since the transmission probabilities can not be presented in a closed form. Simplifications occur, if the substrate (0) is isotropic or can be approximated in this way, since then no p:')-angular dependence exists (for details see [12]). The remaining expressions are then in accor- dance with the treatment of Little [5], whereby the I'-values are just the half of the ernissivities. In table 2 values of the emissivities are given for several

direction and that moreover there may exist more than one propagation velocity (group velocity) in a given propagation direction for the same phonon polariza- tion (this latter property can easily be seen from the energy surface of TI and T2 phonons in quartz of figure 1, remembering that each phonon with a q-vector ending at the energy surface at a certain point has a direction of group velocitygiven by the normal of the surface in this point). Contrary, for a given wave

Values of parameters used in calculating the phonon emissivities of several metals on diamond and sapphire.

Sound velocities and mass densities of metals are taken from Bergmann [15]

Constan-

Cu Ni Au Pb In Sn Bi Ag A1 tan

- - - - - - - -

- -

Mass density y ( ' ) g/cm3 8.93 8.81 19.3 11.34 7.28 7.30 9.80 10.50 2.70 8.80 Phase velocity of longi-

j

^c;i)

tudinal phonons km/s 4.70 5.63 3.24 2.16 2.51 3.32 2.18 3.60 6.26 5.24

Phase velocities of

}

^=;I) _2.26

transverse phonons Phonon emissivities

into diamond from 0.068 3 0.095 0 0.026 1 0.012 5 0.016 9 0.028 9 0.010 9 0.042 2 0.094 9 0.082 3 a c o u s t i c - m i s m a t c h

model 0.027 8 0.048 6 0.007 59 0.002 04 0.005 94 0.012 4 0.004 24 0.013 3 0.034 3 0.036 2 Phonon emissivities

into sapphire from ^{0.174 0.246 0.078 7 0.033 7 0.043 4} 0.080 9 0.034 5 0.099 8 0.257 0.216 a c o u s t i c - m i s m a t c h

model ^{0.130 0.219 0.035 6 0.011 0 0.0404} 0.070 2 0.029 7 0.064 3 0.216 0.176

metal radiators on sapphire and diamond calculated in the isotropic approximation using data given in table I . Thermal radiant characteristics were discussed within the isotropic approximation by Weis [12], but in the case of an anisotropic substrate the problem becomes by far more complicated. This is due to the fact that in anisotropic media the wave normal ehO' of phonons deviate in general from their propagation

normal ey' there is only one phase velocity for each polarization.

In calculating

ti,"'

the essential quantities are the phase velocities and mass densities at both sides of the interface. The group velocities are not involved at this stage. It seems convenient to introduce a directional weighting factor g!O'(O!O), q110') for the q-space, defined by

1 ^{(1) 7.}

-

--. (5) (

^cos

^ojO)

^{- sin 0:') ----} ₍₀₎ ¹ ₍₀₎

^.

^{dc(O) -i}

] 2

'::O)(s:O), p:")

.

n.e!lo) ^{C ,} (en ) dB!') ^@ (4.9)

Integration of (4.9) over the half space gives unity, calculation of this directional weighting factor can hence this normalized factor represents the distribution again only be done for each wave normal e?) by a of the emitted phonons over all q-space directions. The digital computer. By transforming the phonon energy

C4-54 0. WETS

to be found in solid angles sin ~ : ~ ' . d O ~ ~ ' . d q , in Tf one succeeds to measure at the same time the

q-space : temperature of the radiator and the whole emitted

i:O'(w, TI, O?', q y ) ) . sin

OlO).

dolo). dq!" = phonon power one gets a n experimental decision which model is to prefer. As Herth and Weis [5] showed, this

(0) g(0) )

= gr (

,

^{r o )} .P!')(co, T ~ , T' = 0) x is possible if instead of constantan radiators metals are

used showing a temperature dependent resistance x sin 8:". d~!". d d O ) (Fig. 4). In this case, the momentary temperature can into the corresponding angles in real space around the Allen-Brodley

reststor

direction of group velocity, the radiant characteristic results. This transformation is again computer work.

For further details of this last step compare the publi-

cations of Taylor, Maris and Elbaum [16], [17], [18]. Phonon rodlotor (metot fllrn n = n i r l l

5. Experimental results. - The integration over the

expression (3.1) of the spectral power p,(o, T I , To) Calibrated

can be performed if the emissivities are known or if they are assumed to be independent of fre- quency as in the models of perfect match and of acoustic-mismatch. As long as the Debye tempe- ratures

@jl)

= A(co!~)),,,/~, are large in comparison to Tl and To, the integration can be estended to infinity.

Including all three polarizations these yields the follow- ing Stephan-Boltzmann law

In figure 3 this theoretical dependence between radiator temperature and radiation power per area is

Power denslly + i ~ l m m ' l

FIG. 3. - Radiation temperature of a constantan radiator as

RG. 4. - Schematic representation of the electric pulse reflec- tion method suitable for the determination of the radiator

temperature as function of the emitted phonon power.

be determined from measuring the momentary resis- tance. But this last quantity follows, according to simple formulas of transmission line theory, from the ratio of the reflected to the direct pulse amplitude and the known characteristic impedance of the transmis- sion line. This method is limited to radiation tempera- ture above the temperature at which the residual resis- tance of the thin films become the dominant part.

As can be seen from figure 5, this experimental

LOO l ° K 1

K:

rn 8 0

L.

2 60

;

^LO

- 5

g ²⁰

.-

-

0_

73 0 10

"

⁸

6

4

. - -

lo0 10' 1 02 l o 3 loL

Power d e n s h y 1 ~ l r n r n ~ l

function of the emitted phonon power per area according to the

model of perfect match and acoustic-mismatch. Substrate FIG. 5. - Radiation temperature of gold phonon radiators on temperature is 4.2 OK. Solid curves : dispersion neglected, diamond and on sapphire according to [61 and [5]. The theoreti- cal curves have the same meaning as in figure 3. The deviation dashed curves : dispersion law of a linear chain.

at low radiation temperatures is due to the gold/helium contact.

plotted for a constantan radiator, assuming perfect method gives a good agreement with the predictions of match, acoustic-mismatch to sapphire and acoustic the acoustic-mismatch model in the case of gold mismatch to diamond within the isotropic approxima- radiator on sapphire and on diamond. The deviation tion. At higher temperatures expressions are used below 25

...

30 OK are obviously due to the direct instead of (5.1) explained in detail in [12]. contact of the metal film to the liquid helium, which

THE SOLID-SOLID INTERFACE IN THERMAL PHONON RADIATION (24-55

was neglected in the theoretical treatment, but existed in these experiments. Measurements with lead as radiator material given in figure 6 reveal this, since by

COO [OK1

200

ll0O 80 60

2, LO m E

2 20

C .-

-

0

g 10 n 8

6 L

1 o0 10' l o 2 lo3 t o 4 Power d e n s ~ t y ---c [w/rnrn2 I

FIG. 6. - Lead phonon radiator on diamond in and without contact to liquid helium of 4,2 O K 161.

avoiding a direct helium contact this tenlperature drop does not appear. Measurements with lead as radiator are especially cumbersome due to contamination of the surface which will give rise to a better acoustic match- ing. Performing these measurements on diamond it was not possible in our experimental set up to clean the surface as good as we wanted. So we d o not know with certainty whether the small deviations from the acoustic-mismatch model in figure 6 and also in figure 7 are due to intermediate impurity layers or to other effects or even to an incorrectness in the assumed data of phase velocities and mass density of the metal films. Nevertheless the agreement with the acoustic- mismatch calculation is so obvious that the perfect match model should be abandoned in any case. I would like to emphasize, that the material combination Icad/diamond shows an extremely high acoustic- mismatch according to the large difference in phase velocities and mass densities envolved. As seen from figures 5, 6 and 7 the influence of a direct contact between phonon radiator and liquid helium is by no means negligible as usually assumed and requires a detailed experimental and theoretical investigation to get an insight Into this heat transfer.

Using the experimental method of figure 4, Cheeke

LOO [OK1

200

80

GI 60 2 LO

P

GI a

i 2 0

..-

C

.- 0

%

.- 10

::

8 6 4

l o 0 lo1 l o 2 1 03 10' Power density

-

[ ~ / r n r n ~ l

FIG. 7. -Copper phonon radiator on diamond in contact with liquid helium of 4,2 O K [6].

has studied a large variety of other material combina- tions and found good agreement with the acoustic- mismatch model (compare his contribution in this volume).

Further support is given as already mentioned by measurements of Wigmore [8] using the paramagnetic system MgO : Fe2+ as tunable phonon scatterer. His results are plotted in figure 8. Summarizing recent

l°K1

t

^'O

2 e 10

:

⁸

-

6 E L .-

-

0

.- D

2 2

10- 10

Power d e n s ~ t y

-

[ ~ l r n r n ~ l

FIG. 8. - Constantan phonon radiator on MgO. Radiation temperatures observed by Wigmore [8].

experimental results we may conclude that the acoustic- mismatch model for the description of phonon trans- mission across an interface between two dissimilar media is now well supported by many experiments.

References

[I] LITTLE (W. A.), Can. J. Phys., 1959, 37, 334. [5] HERTH (P.), WEIS (O.), 2. Angew. Phys., 1970,29,101.

[21 CHEEKE ^(J. D. N-), ^{J .} Physique, Paris, 1970, 31, [6] KAPPUS (W.), WEIS (O.), to be published.

Suppl. C 3, 129.

[7] CHEEKE (J. D. N.), in this issue.

[3] VON GUTFELD (R. J.), NETHERCOT (A. H.), Phys.

Rev. Lett., 1964, 12, 641. [8] WIGMORE (J. K.), Phys. Rev., 1972, B 5 , 700.

[4] VON GUTFELD (R. J.), NETHERCOT (A. H.), ARM- I91 MCSKIMIN (H. J.), BOND (W. L.), Phys. Rev., 1957,

STROXG (.I. A.), Phys. Rev., 1966, 142, 436. 105, 116.

[lo] BERNSTEIN (R. T.), J . Appl. Phys., 1963, 34, 169.

[ l l ] FARNELL (G. W.), Can. J. Phys., 1961, 39, 65.

[12] WEIS (O.), 2. Angew. Phys., 1969, 26, 325.

[I31 FEDEROV (F. I.), Theory of Elastic Waves in Crystals, New York : Plenum 1968.

[14] MUSGRAVE (M. J. P.), Crystal Acoustics, London : Holden-Day Inc. Publ. 1970.

H. BUDD.

-

Without discussing the details of the single temperature model, which N. Perrin and I did this morning, it is obvious that as one goes to lower power levels this approximation becomes increasingly poor. This is perhaps relevant to the lack of agreement between theory and experiment in this range.

0. WEIS.

-

From our experimental results of figure 6, showing the difference in temperature of a lead film being in contact with liquid helium and being tiot in contact, I conclude that the observed drop is due to heat transfer to the liquid helium.

Since the helium bath is 4.2 OK whereas the metal film is above 25 OK, obviously, this energy transfer is of gas-kinetic type and not carried by phonons.

W. EISENMENGER.

-

Have the measurements of Wigmore at low energies been made on contact to liquid helium.

0. WEIS. - Yes. Wigmore reports in his paper that during these measurements the whole specimen assembly was immersed directly in liquid helium pumped to below its

A

point. It is very surprising for me that under these conditions a full agreement was observed with the curve of acoustic mismatch.

From the results of your group, reported yesterday, we know that energy transfer to He I1 is by far not negligible. I would like to present an example too, showing the influence of the adjacent helium on the shape of the emitted phonon pulses. The phonon signals of the following diagram were observed recently in our group by R. Batzner.

A constantan radiator deposited on sapphire was in contact with liquid helium at a temperature of 3.72 OK, the working temperature of the Sn-bolometer. At low input powers the observed longitudinal and the trans- verse phonon pulses are nearly rectangular as expected.

Increasing the input power slowly, there is a strong distortion to be seen, starting at the end of each pulse

1151 BERGMANN (L.), Ultraschall, Table 85, Stuttgart :

Hirzel Verlag, 1954.

[I 61 TAYLOR (B.), MARIS (H. J.), ELBAUM (C.), Phys. Rev.

Lett., 1969, 23, 416.

1171 TAYLOR (B.), MARIS (H. J.), ELBAUM (C.), Phys. Rev., 1971, B 3, 1462.

[IS] MARIS (H. J.), J . ACOUS~. SOC. Am., 1971, 50, 812.

JSSION

Sn-Bolometer

1 7

Time

-

and growing until the familiar pulse shape is formed.

If one takes a longer pulse the onset of the triangle is the same, but also no flat roof is visible showing that under such conditions no stationary point can be achieved. Since the pulse shape is the same for the longitudinal as well as for the transverse pulse at the same power level, we concludethat the constantan radia- tor is the source of this pulse shapening. Obviously, at the beginning there exists a normal contact between metal and liquid helium and hence a phonon transfer across this interface. But this contact is destroyed by building up a gas layer which is growing thicker and thicker and hence isolating better and better the radiator from the liquid helium. If one repeats these experiments with the constantan radiator and the bolometer in helium gas no shoulder can be seen, whereas the other distortions become stronger.

Tn both cases, after the end of Joule heating the hot gas film gives back part of its energy to the constantan film and extends in this way the duration of phonon radiation.

A. ZYLBERSZTEJN.

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I would like to point out that B. Pannetier and I report essentially the same obser- vations (this conference), with exactly the same inter- pretation.