HAL Id: jpa-00215079
https://hal.archives-ouvertes.fr/jpa-00215079
Submitted on 1 Jan 1972
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
RADIATION OF PHONONS BY METALLIC FILMS
H. Maris
To cite this version:
H. Maris. RADIATION OF PHONONS BY METALLIC FILMS. Journal de Physique Colloques, 1972, 33 (C4), pp.C4-3-C4-10. �10.1051/jphyscol:1972402�. �jpa-00215079�
JOURNAL DE PHYSIQUE Colloque C4, suppliment au no 10, Octobre 1972, page C4-3
RADIATION OF PHONONS BY RaETALLIC FILMS
H. J. MARIS (*)
Physics Department, Brown University, Providence, Rhode Island, U. S. A.
RBsumB. - On consid&re la generation de phonons par une couche mktallique mince, chauffh par un courant Qectrique, en contact avec un cristal diklectrique maintenu a basse temperature.
Le taux de g6nbration des phonons par l'interaction electrons-phonons est obtenu. On montre que dans le cas des m6taux contenant une concentration suffisante d'impuretk, la g6neration des phonons due la diffusion inelastique des Blectrons par les impuretes peut constituer un processus important.
Abstract. - The generation of phonons by a heated metallic film on a dielectric crystal substrate at low temperatures is considered. The film is assumed to be heated by an electric current. The rate of generation of phonons by the electron-phonon interaction is derived. It is shown that in metals containing a large number of impurities an important mechanism of phonon generation may be the inelastic scattering of electrons at the impurities.
1 . Introduction. - In a typical heat pulse experi- ment [l] a metallic film evaporated onto the surface of a crystal is heated by an electric current. The film radiates phonons which, after passing through the crystal, are detected by a thin-film superconducting bolometer. Although many experiments have been performed using heat pulses, the precise mechanism by which phonons are generated is still uncertain. The simplest approximation is obtained by ignoring the details of the generation mechanism and assuming that the film radiates as a (( phonon black-body >>. A somewhat more sophisticated approach is the (( acous- tic-mismatch model )). In this it is assumed that the phonons incident on the film-substrate interface from the film side are described by a Planck distribution.
The distribution of phonons actually entering the crystal differs from the distribution incident on the boundary because
(a) some phonons are reflected at the interface due to the difference between the acoustic impedances of the film and the crystal ;
(b) the phonons passing through the interface are refracted because of the different velocities in the source and detector.
These corrections to the phonon radiation arising fi-om reflection and refraction can be calculated in a fairly straightforward way [2]-[4].
In this paper we investigate the mechanism by which the phonons are initially produced in the film and discuss the validity of the acoustic-mismatch model. For simplicity we restrict attention to non- superconducting metals and assume the temperature of the film is much less than the Debye temperature.
(*) Work supported in part by the National Science Founda- tion and the Advanced Research Projects Agency.
We find that in some situations of experimental inte- rest the (( ordinary >) electron-phonon interaction is too weak to provide the observed coupling between the electrons and phonons. We next consider the production of phonons by inelastic collisions of elec- trons with impurity atoms. I% appears that in dirty metals such as constantan this may be the dominant mechanism for the production of phonons.
2. General considerations regarding heat-pulse generation. - We may conveniently divide the gene- ration process into the following stages :
(a) The electrons in the metal film. These are acce- lerated by, and receive energy from, the electric field applied to the film.
(b) The phonons in the film. These phonons are produced in some as yet unspecified way by the hot electron gas.
(c) The radiated phonons in the substrate crystal.
These are phonons which have passed through the film-substrate interface.
Let us begin by assuming that the electron gas (a) is described by a Fermi-Dirac distribution function with a well-defined temperature T,. In general, the rate at which phonons of a particular type are radiated by a hot electron gas will depend in a very complicated way on phonon wave-vector and polarization. Thus, one does not expect a priori that the rate at which phonons are produced in the film will necessarily correspond to a Planck distribution. However, if the probability of a phonon passing through the interface is sufficiently small, the distribution of phonons in the film will build up within a fairly short time to a Planck distribution corresponding to the same tem- perature T, as that of the electron gas. The phonons
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972402
MARIS
Power s u p p l i e d by e l e c t r i c f i e l d
R a d i a t i o n o f phonons by unknown p r o c e s s
Phonon g a s i n f i l m
(b)
Escape t h r o u g h
f i l m - s u b s t r a t e i n t e r f a c e
I
Phonon g a s i n s u b s t r a t eI
I
FIG. 1. - Energy-flow diagram representing the generation of heat pulses.
incident on the interface will then have a Planck distribution and the acoustic-mismatch model will be valid.
Now we must consider what constitutes a suffi- ciently small transmission probability. Suppose the film has area A and thickness d. Consider an initial situation where the electron gas is at temperature Te and there are no phonons present in the film. We define a time z equal to how long it would take the phonon system to come to statistical equilibrium with the electrons if no phonons could escape from the film. Let the number of phonons per unit volume of the film in this case be Neq and let the average phonon velocity be c,,. Suppose now we assume that there is a probability p of an incident phonon passing through the interface. It is straightforward to show that the number of phonons incident on the interface per unit time is
Hence, the number escaping is
The rate at which phonons can be produced inside the film is of the order of
Thus, one expects the distribution to be only slightly disturbed by the escapes provided that
where A,,, = zc,,. This length has the physical signi- ficance of being the average distance a phonon travels before interacting with the electron gas.
Eq. (1) provides a condition on p but it is not the only condition. Consider when the film is sufficiently thick that
Eq. (1) implies that under these circumstances the interface may be perfectly transmitting and the phonons inside the film will still be described by a Planck distribution. A more careful examination shows that for p = 1 this is true only on the average in the sense that a Planck distribution exists every- where in the film except in a thin layer of thickness of the order of A ; , , adjacent to the substrate [ 5 ] . However, it is precisely the phonons from this thin layer which are incident on the interface with the substrate. T o ensure that these phonons in the boundary layer are characterized by a Planck distribution we require the extra condition
3. Electron-phonon interaction. - Consider now the rate at which phonons are produced by the electron- phonon interaction. Let the electron gas have a distri- bution function
where ck is the energy of an electron with wave vector k
and E F is the Fermi energy. We have neglected the
small drift velocity of the electron gas corresponding to the electric current in the film. Let the phonon dis- tribution function be nqj where q is the wave vector and j the polarization. The electron-phonon interaction matrix element has been discussed in great detail by Ziman [6]. At temperatures much less than the Debye temperature, the phonon wavelengths are much greater than the lattice spacing. Hence we need only consider phonon wave vectors q such that
where k , is the Fermi wavenumber. In this limit a number of different approaches give the same result for the electron-phonon matrix element [6]. For nearly- free electrons the matrix element describing the scat-
RADIATION OF PHONONS BY METALLIC FILMS C4-5
tering of an electron from state k to k' with the emis- sion of a phonon of wave vector
where p is the density, V the volume, and eqj and oqj are respectively the polarization vector and frequency of the phonon qj. The matrix element for the reverse process in which an electron in state k' absorbs the phonon q j and goes to state k is
< k, (nqj - 1) I H' I k', nqj > =
For an elastically isotropic solid these results simplify because for longitudinal waves
and for transverse waves
Thus, no transverse phonons are produced by this mechanism, at least for nearly free electrons and isotropic elasticity. For longitudinal phonons we can calculate the rate of change of nqj with time using the matrix elements given above, together with time- dependent perturbation theory. The calculation is straightforward and the result is
ntj is thus the number of phonons of type qj when the phonon gas is in thermal equilibrium a t the same temperature as the electrons. The characteristic time 7;; that phonons of type qj will take to come to equilibrium due to the electron-phonon interaction is thus
The characteristic distance of interaction between the electron gas and phonons of type qj is thus
For transverse phonons 28; and A;; are infinite. To get an idea of the magnitudes of these quantities for longitudinal phonons we express the frequency in temperature units using
Then we may write
where .c? is the interaction time for a longitudinal phonon of energy 1 OK and A;' is the corresponding interaction length. We have
where c, is the longitudinal velocity of sound, m is the A, ep = 6 PC: h
electron mass, n is the number of electrons per unit nk, nmv, ' (11) volume, vF is the Fermi velocity, and
In table I we show ztP, and A? for a number of metals.
0 1
nqj = The values of n, VF, p , C, were taken from Kittel [7].
exp[hoqj/kR Te] - 1 ' In all cases c, has been calculated for a longitudinal
Parameters used in calculating the rate of generation of phonons due to the electron-phonon interaction n
Metal cm-3
- -
Copper ... 8.5 x lo2'
Silver. ... 5.9
Gold ... 5.9
Aluminium. ... 18.1
Indium ... 11.5
Lead ... 13.2
Constantan ... 7.0
0,
cm. s-' - 1.57 x 10' 1.39 1.39 2.02 1.74 1.82 1.47
C4 -6 H. J. MARIS wave propagating in the [001] direction. For constan-
tan we use the same values of p and c, as Weis 131.
The number of atoms per cm3 in constantan is 8.7 x 10''. We have arrived at a value for n by assum- ing 0.8 free electrons per atom [8].
It can be seen from table I that for phonons of energy a few OK the length Aqj is considerably greater than the thickness of films typically used in heat pulse experiments (500 or 1 000 A). Consider, for example, a constantan or gold film at 3 OK. A phonon of energy 3 k, Twill then have an interaction length of 42 000 A
and hence for a 1 000 A film condition (1) becomes
The calculations of Little [2] and Weis [3] show that, although for some substratesp may be as small as 0.01, in generalp is greater than 0.1 and hence condition (12) is not satisfied. Moreover, the problem becomes more serious for phonons of energy less than 3 k, T, and for lower film temperatures. For aluminium, indium, and lead the prospects for the acoustic-mismatch model are much better. The interaction lengths are almost an order of magnitude shorter than those for constantan and The condition corresponding to (12) is then the same as condition (2), i. e.
in most cases this is not a serious error-usually leading to an inaccuracy of no more than a factor of 2 for copper, silver, gold, aluminium and indium. For lead the available evidence [lo], [ l l ] indicates that the free-electron model overestimates the electron-phonon interaction by a factor of 3. It should be noted, howe- ver, that indium and lead have Debye temperatures of about 100 OK. Hence, even for phonons of energy as low as 20 or 30 OK, there may be significant contri- butions from Umklapp processes. We discuss the vali- dity of the free-electron model for constantan in section 5.
4. Inelastic scattering of electrons at defects. - We now investigate the possible importance of pho- nons generated when electrons strike impurities in a metal. The inelastic scattering of electrons at impu- rities is a controversial subject. Koshino [12]-[14]
considered an impurity atom with equilibrium posi- tion in the lattice x, and an additional small displace- ment u, due to the vibration of the lattice. If the poten- tial field of the impurity is V(X), where X is a position vector relative to the instantaneous position of the impurity, then the actual potential seen by an elec- tron at position x is
There is still a problem with phonons of energy 1 OK or less. At this energy for aluminium, indium or lead the condition is again that p must be less than 0.1
A more careful treatment of the electron-phonon interaction does not seem to change the situation in any important way. If we relax the assumption of elastic isotropy, At: will be finite for transverse phonons. However, there is no reason to believe that even under the most favorable conditions A; would be significantly less than the value predicted by eq. (7).
Another possible defect in our calculation is that we have assumed implicitly that the phonons involved have sufficiently high frequencies to satisfy the condi- tions
q6 > 1 (13)
- ., . ,
where 6 is the skin-depth at the frequency of the phonon and A , is the electron mean free path [9]-[lo].
These conditions may or may not be satisfied in typical heat pulse experiments. If condition (13) is violated, there is a finite interaction for transverse phonons even assuming elastic isotropy and nearly free elec-
Since u, fluctuates in time, this appears to an electron as a time-dependent potential and gives an addi- tional interaction between the electrons and the lat- tice vibrations. Koshino calculated the additional scattering of electrons by this time-dependent poten- tial using the first Born approximation. His primary interest was in deviations from Matthiessen's rule and his result suggested that this inelastic scattering was an important effect. Taylor [15], [16] criticized Koshino's calculations and showed that there were higher order terms in the perturbation series that nearly cancelled out the first order term considered by Koshino. Taylor shows that the inelastic scattering of electrons at impurities may be treated by a method similar to that used in discussing the Mossbauer effect [17], [18]. We now describe Taylor's method and use it to calculate the radiation by an electron gas in the presence of scattering impurities.
Consider first a fictitious situation in which the impurity is rigidly fixed in space. Let the scattering matrix be FQc, k'). Then the probability per unit time that an electron initially in state k will be scat- tered to a final state k' is
trons. However, the interaction is still at most only of 2 .n
the same order of magnitude as that given by eq. (7). P(k, kt) = F I F(k, k') 12ff(1 - f:) 6(ek - eke) .
If condition (14) is not satisfied, the interaction
between phonons and electrons .is made weaker and (15)
thus this is of no help. Another possible source of The factor f is the probability that there is initially error in our calculation is the use of the nearly free an electron in the state k, and (1 - f:,) is the proba- electron approximation. This has been discussed in bility that state k' will be vacant and. able to receive great detail in reference [lo]. The conclusion is that an electron. We now allow the impurity to move.
RADIATION OF PHONONS BY METALLIC FILMS C4-7 If the displacement of the impurity is u,, the scatter-
ing matrix is changed to
F(k, k t , ul) = eiK"" F(k, k t ) (16) where the momentum transfer is
K = k - k ' .
Assume that initially the phonons in the film are described by some distribution function nqj. Let the initial vibrational state of the film be denoted by
1 i > and let the final state after the electron has collided with the impurity be If >. Then the proba- bility of an electron being scattered from k to k' is
For electrons having energies of the order of 8,
This is equivalent to the statement that the atomic displacements are a small fraction of the interatomic spacing. Thus :
F(k, k', u3 z F(k, k') [ I + i K . u , + ...I . (18) The displacement u, may be expressed in terms of phonon annihilation and creation operators as
,tj "
I =
[
N M w q e,j[a(qj) - ai (- d)] eiqVx1 -(19) Combining eq. (17)-(19), we find that the probability of an electron being scattered from k to k' and the lattice remaining in state I i > is
to lowest non-vanishing order in I K. u, I. This cor- responds to the (( recoilless-fraction )) in the Moss- bauer effect [17], [IS].
The probability of scattering from k to k t with the emission of a phonon of type qj is
n ( F(k, k') l2
~ : ( k , k', q j ) = - 81 2
I K.eqj I PV wqj
'The corresponding probability for absorption of a phonon is
Pii(k k', q j ) = - 71. I F(k, k') j2 I K . e q j l2f:(l - f,9) PV
The net rate at which phonons of type qj are produced is
This can be simplified to
72 2
= -(nqj - n t j ) Z 1 t ' ( k , k t ) l2 1 I . e q j I
--
at ~ K O ~ ~kk'
The important terms in the summation are those for which k and kt are very close to the Fermi surface.
Hence, we assume that F(k, k t ) may be well approxi- mated by a function F(6,,,) which only depends on 6,,., the angle between k and k'. Then it is straight- forward to show that
8% - -
- - vm3 nu, F:,,
(nqj - n t j )
B t nph4 (24)
where
1
F:. = f I F(B,,,) I2 ( 1 - cos B,,.) d cos 8,,. .
- 1
Assume now that there are ni impurities per unit volume and that these impurities scatter indepen- dently. Then the characteristic time z&j to come to equilibrium is
To make numerical estimates the safest procedure is to relate ni F:~ to the electrical conductivity o. Using standard results [19], one can show that
where r e , is the electron mean free time appearing in the relation
where o is the electrical conductivity. Combining eq. (24)-(27) gives the very simple result
For the interaction length we find
where cqj is the velocity of phonon qj.
In Table I1 we give values of x i j and ' 4 i j for some typical alloys. These values are independent of phonon energy, and zdj is the same for longitudinal and trans-
Parameters used in calculating the rate of generation of phonons by inelastic scattering of electrons at impurities
0 zi /lij (longitudinal) 4,. (transverse)
Alloy ohm-' .cm-I s
- - - A A -
90 % Cu-10 % Au . . . 0.18 x lo6 0.9 x lo-g 38 700 26 000 94 % In-6 % Sn . . . 0.46 1 .O 26 500 10 500
. . .
Constantan 0.02 0.14 7 300 3 700
verse phonons. Comparing tables I and I1 we see that the inelastic scattering dominates over the elec- tron-phonon interaction at temperatures u p to 1.5 OK in indium-tin, 5 OK in copper-gold and 50°K in constantan.
Taylor's approach has been criticized by Kle- mens [20], and by Damon, Mathur and Klemens [21].
They argue that the displacement u, of the impurity atom relative to a fixed frame of reference is not relevant. Instead they propose that eq. (16) should contain u, (re]), the displacement of the impurity relative to its neighbors, instead of u,. The phonons we are concerned with have wavelengths much greater than the lattice spacing. Hence neighboring atoms will move approximately in phase and thus u, (rel) will be much smaller that u,. The effect of this is roughly to multiply Taylor's results for z i j and nij by a factor of the order of
If this argument is indeed correct, the interaction times and lengths given in table I1 would be in creas- ed by several orders of magnitude and inelastic scattering would be of no importance in the present context.
If we accept Taylor's basic approach, there is still a question about the validity of the results for very long wavelength phonons. We have assumed in deri- ving eq. (25) that the impurities scatter independently and that their contributions to the rate of phonon production can therefore be added. However, an electron makes successive collisions with impurities which are typically separated by a distance of the order of the electron mean free path A,. If we are considering phonons of wavelength A, these impurities will be moving in phase if
This is equivalent to the condition
It would seem that under these conditions the rate
at which phonons are produced should be reduced, presumably by a factor of the order of qA,, or even (qAJ2. For constantan A,, as estimated from the electrical resistance, is 15 A, whereas q for a 10 OK energy transverse phonon is 5 x lo6 cm-'. Thus qA, is 0.75. This argument would suggest that for phonons with energies below about 10 OK the interac- tion time and length should increase as the energy decreases.
5. Discussion. - The most careful studies of pho- non radiation from thin films have been performed by Herth and Weis [22] and Wigmore [23]. Herth and Weis investigated films of copper, nickel, gold and lead on sapphire substrates. In all cases they found agreement with the acoustic-mismatch model.
This is consistent with our predictions for the tempe- rature range they investigated. In the case of gold, for example, they made measurements down to 30OK. At this temperature a typical phonon energy is 100 OK. The electron-phonon interaction length is, from table I, about 3 900 '4, and the film thickness 155 A. Hence, from eq. (1) the acoustic-mismatch model should be valid provided
For gold on sapphire p is 0.08 for longitudinal phonons and 0.04 for transverse phonons. Hence, this condi- tion is satisfied. For copper and nickel the inequality is only just satisfied at the lowest temperatures studied by Herth and Weis. Lead was investigated to 10 OK and easily satisfies the inequality down t o this temperature. Note that these conclusions are based only on the strength of the ordinary electron- phonon interaction. Any contribution from inelastic scattering at impurities would increase the validity of the acoustic-mismatch model.
Wigmore [23] has made a detailed study of cons- tantan films on magnesium oxide substrates. He inves- tigated film temperatures down to 3 OK and found good agreement with the acoustic-mismatch model.
The film thickness in these experiments was 680 A.
The transmission probabilities into MgO are 0.46 for longitudinal phonons and 0.16 for transverse pho- nons [23]. Then, according to our eq. (I), the acoustic-
RADIATION O F PHONONS BY METALLIC FILMS C4-9
mismatch theory should be valid provided that A,,, is less than
4 d
This is 6 000 A for longitudinal phonons and 17 000 A
for transverse phonons. At 3 OK the typical phonon energy is about 10 OK and the free-electron estimate of A:: is 38 000 A. If we use Taylor's theory, ntj
is 7 300 for transverse phonons. Since q A , is of the order of unity, these estimates would not be changed drastically by the correction for the phase relations between the motions of impurities. If we accept the modification due to Damon e t al. [21], however, the contribution from inelastic scattering is negligible. The only way we could then justify the acoustic-mismatch model would be to assume that for constantan the free-electron model underestimates the electron-phonon interaction by at least a factor of 5. An analysis by B. Taylor [24] of the lattice thermal conductivity of constantan indicates that the average phonon mean free path at 10 OK is 5 000 A
and is approximately inversely proportional to tem- perature below this temperature. However, this does not determine unambiguously whether the mean free path is determined by the electron-phonon interaction or by inelastic scattering.
Finally, we note that experiments on very then films at low temperatures would be of great interest.
A film a t 0.3 OK would be expected t o radiate phonons with energies around 1 OK. I n a typical solid these phonons have wavelengths of the order of 2 000 A.
I n this case it is possible for the wavelength to be considerably greater than the film thickness. Under these conditions it clearly does not make much sense t o talk of phonons propagating through the film and being reflected or transmitted at the interface. There are to date no theoretical calculations for this regime.
It is possible that some form of resonance generation occurs for phonons whose wavelengths are harmoni- cally related to the film thickness. For films with low impurity content, a significant energy transfer due to inelastic-scattering of electrons at the film-substrate interface may be observable.
References [I] VON GUTFELD (J.), Physical Acoustics, edited by
W. P. iMason (Academic Press, New York, 1968), Vol. V, p. 233.
[2] LITTLE (W. A.), Can. J. Phys., 1959, 37, 334.
[3] WEIS (O.), Z . Angew. Phys., 1969, 26, 325.
[4] TAYLOR (B.), MARIS (H. J.) and ELBAUM (C.), Phys.
Rev., 1971, 3, 1462.
[5] This point is discussed in more detail by BUDD (H.) and VANN~MENUS I(J.), P h y ~ . Rev. Left., 1971, 26, 1637.
161 ZIMAN (J.), Electrons and Phonons (Oxford Univer- sity Press, London, 1960), chapter V.
[7] KITTEL (C.), Introduction to Solid State Physics (Wiley, New York, 1971).
[8] For want of a better value we have simply used the average of copper (1) and nickel (0.6).
[9] For a discussion, see PIPPARD (A. B.), LOW Tempe- rature Physics, edited by C. de Witt, B. Dreyfus and P. G. de Gennes (Gordon and Breach, New York, 1962), p. 1.
[lo] RAYNE (J. A.) and JONES (C. K.), Physical Acoustics, edited by W. P. Mason and R. N. Thurston (Academic Press, New York, 1970), Vol. VII, p. 149.
[Ill FAIE (W. A.), Phys. Rev., 1968, 172, 402.
[12] KOSHINO (S.), Progr. Theoret. Phys., Kyoto, 1960, 24, 484.
[13] KOSHINO (S.), Progr. Theoref. Phys., Kyoto, 1960, 24, 1049.
[I41 KOSHINO (S.), Progr. Theoret. Phys., Kyoto, 1963, 30, 415.
1151 TAYLOR (P. L.), Proc. Phys. Soc., 1962, 80, 755.
[16] TAYLOR (P. L.), Phys. Rev., 1964, 135, A 1333.
[17] ABRAGAM (A.), LOW Temperature Physics, edited by C. de Witt, B. Dreyfus and P. G. de Gennes (Gordon and Breach, New York, 1962), p. 479.
[IS] FRAUENFELDER (H.), The Mossbauer Effect (Benja- jamin, New York, 1962).
[I91 ZIMAN (J.), Electrons and Phonons (Oxford Univer- sity Press, London, 1960), chapter VII, particu- larly eq. (7.1.9) and (7.4.4).
[20] KLEMENS (P. G.), J. Phys. Soc. Jap., 1963, 18, Suppl. 11, 77.
[21] DAMON (D. H.), MATHUR (M. P.) and KLEMBNS (P. G.), Phys. Rev., 1968, 176, 876.
[22] HERTH (P.) and WEIS (O.), 2. Angew. Phys., 1970, 29, 101.
[23] WIGMORE (J. K.), Phys. Rev., 1972, 5 , 700.
[24] TAYLOR (B.), Ph. D. Thesis, Brown University 1970, unpublished.
DISCUSSION C . J. ADKINS. - AS regards the collimation of
phonons on transmission from a solid t o liquid He, the best evidence is from resonance experiments in thin superfluid films. The first of this sort were done by C. H. Anderson using cleaved halide crystals.
With good surfaces the resonances were still visible as the film was made thicker. Blackford did similar experiments at Cambridge using phonons generated by superconducting tunnel junctions at higher fre-
quencies ( N 150 GHz). Again very good surfaces were necessary (he used cleaved mica). The number of resonances observed corresponded to collimation to within some 100, which is of the order one would calculate from the acoustic impedance change.
H. BUDD. - I would like to first point out that agreement with different interface models cannot generally be inferred from resistance measurements,
which are not very sensitive to the form of the phonon distribution.
It's also amusing to note that the usual single tem- perature model is a reasonable approximation for those phonons which don't escape from the film. The ones that do readily escape can't be characterized by a single temperature.
Going to thick films raises other difficulties, since the spatial problems become important.
WIGMORE. - In the context of Professor Budd's comments, I wish to report that some data has been taken with the paramagnetic spectrometer using a constantan film only 80 A thick. The phonon distri- bution is certainly not as calculated from the acoustic mismatch model, and appear that these is a conside- rable excess of higher frequency phonons compared with the thick film case. More details of these results will be given in the paper later in the programme.
