HEAT PULSE INTERACTION

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HEAT PULSE INTERACTION

M. Ribbands, D. Osborne

To cite this version:

M. Ribbands, D. Osborne. HEAT PULSE INTERACTION. Journal de Physique Colloques, 1972, 33

(C4), pp.C4-119-C4-122. �10.1051/jphyscol:1972425�. �jpa-00215102�

JOURNAL DE PHYSIQUE

Colloque C4, supplkment au no 10, Octobre 1972, page C4-119

HEAT PULSE INTERACTION

M. S. RIBBANDS and D. V. OSBORNE

School of Mathematics and Physics, University of East Anglia, Norwich, England

Rksumk.

Nous dCcrivons des expbiences ou les probabilites de diffusion phonon-phonon sont estimkes en Btudiant les interactions de pulses de chaleur, dans les cristaux dielectriques.

Les rksultats preliminaires suggerent des probabilites de diffusion trks faibles, ce qui est en accord avec le travail thkorique et experimental sur l'interaction ultrasonore de Taylor et Rollins. La propagation du deuxieme son dans NaF est considerke a la lumiere de ces rbultats.

Abstract.

Experiments are described in which phonon-phonon scattering probabilities are estimated by studying heat pulse interactions in dielectric crystals. Preliminary results suggest very low scattering probabilities, in agreement with the theoretical and experimental work of Taylor and Rollins on ultrasonic interaction. The propagation of second sound in sodium floride is considered in the light of these results.

Introduction. - Several measurements [I] have been made of the 3 phonon normal process scattering rate in dielectric solids from thermal conductivity data fitted to theoretical models [2].

Ultrasonic methods have been used to measure 3rd order elastic constants [3], phonon lifetimes for normal processes 141, r57 and also to observe 3 phonon processes directly [ 6 ] . Because of the large frequency dependence of the transition probability for such events, the use of thermally generated phonons, as in heat pulses [7], of much higher frequency, w - ^loi2 sC1, should make direct observation of phonon-phonon scattering more accessible.

The work described here is an attempt to repeat the ultrasonic experiment of Taylor and Rollins [6], using heat pulses in place of acoustically generated phonons.

General theory.

The deformation of a solid can be described by w , ~ , the deformation tensor [8]

ua being the displacement of a point in the xa direction.

The elastic energy density can then be written, assum- ing all terms of 4th order and above in w are negligible

=

C

+ 4

Cijklm,r wij

(3)

i j k l

where cijk, and cijklm, are 2nd and 3rd order elastic constants.

Taking into account the reduction of cijkl and cijkIm, caused by symmetry, for cubic media

where C i j and Cijk are the more usual Voigt notation The Hamiltonian density is now written as the for Cijkl and Cijklmn. sum of two Hamiltonian densities, one quadratic Defining two new tensors in uup, the other cubic in uup and vup The latter is the perturbation Hamiltonian density responsible for

U a p =

Huu,p

up,u) ( 5 ) transitions between available phonon states in a three

vap

Q(ua,p - ~ p , a ) - ⁽⁶⁾ phonon normal process and is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972425

C4- 1 20 M. S. RIBBANDS A N D D. V. OSBORNE

The total displacement vector u(r) can be written as the sum of the displacement vectors associated with each of three phonons involved in a three pho- non process.

en is the polarisation vector and k , the propagation vector associated with phonon n.

The perturbing Hamiltonian density can now be written in terms of the products of the displacements associated with each of the three phonons. The result- ing expression contains 145 terms, it does however simplify for specific interactions, for three longitu- dinal phonons collinear in the x , direction.

superscripts refer to phonon 1, 2 and 3 in the interac- tion.

In the work considered here, phonons 1 and 2 are in the x , and x2 directions and only 2 types of interactions are considered, namely t + t

I and I + t - t z .

For the first interaction

for transverse phonon polarisation vectors in the plane x1 x2.

For the second interaction

x sin (8 + ^y) + 2 ( C 1 2 + C4,) sin (8

y ) ) (I 1) again for transverse phonons in the plane x, x2.

8 is the angle between the directions of phonons 1 and 3, y is the direction of the phonon 3 polarisation vector. In general

differs from 8 because of elastic anisotropy but for longitudinal phonons, putting

y =

8 only introduces a small error. This reduces eq. (10) and (1 1 ) to t + t

1

~ ( ~

+

(I2) l + t + l

X'

4 k t k 2 k 3 ( C l l + ^{C 1 2} + ^{4 C44} + ^{4 Ct66)}

x sin 8 cos 8 (13) The amplitudes of eq. (8) are taken to be the annihi- lation and creation operator of the linear harmonic oscillator whose only non zero matrix elements are

where N is the initial number of phonons, ho is the phonon energy, and m the mass of the volume of interaction V. The matrix elements of the components of uap and v,, can be obtained as

- -

- & e-ik-r(e, k, + e, k,) < ^N + 1 I ^a* I N > (15) and similarly for vap.

The transition probability per unit time is then given by [9]

zi is the energy of the initial state, Df(.si) is the density of final states about e i H' is obtained by integrating over the volume of interaction. Using for the density of states

the transition probability can be written

HEAT PULSE INTERACTION

el,

e, are the phonons emissivities of heaters 1 and 2,

is the phonon velocity, p the density and A is a linear combination of 2nd and 3rd order elastic constants which depends on the interaction. For the case I + t

I, A is given by eq. (14)

This has been calculated assuming that phonon energy and momentum is conserved in the interaction, i. e. k, + k,

k3 and o, + ^w2

^o3 .

Ignoring dispersion, for a given set of directions, these two conditions will only be satisfied by one set of values of energy [lo]. Taking into account the elastic anisotropy the ratio o,/o, which satisfies these condi- tions can be calculated as a function of the direction of k, for k, fixed.

The result of such a calculation for potassium chlo- ride and for an isotropic material is shown in figure 1.

FIG. 1. -

A

+

I

chloride), (broken line, isotropic material).

Phonon-phonon scattering experiments.

In order to test the preceding theory, cubic materials in the form of the alkalihalides were chosen for the experi- ments. The experimental geometry used for the two interacting phonon beams, took full advantage of the cubic symmetry, see figure 2.

FIG. 2. - Arrangement of heater and detector films on crystal.

A groove was cut into the crystal to provide some degree of shielding for detector 2 from heater 1. It was expected that phonon beam 1 would interact with and scatter phonon beam 2 thereby reducing the height of the signal pulse received a t detector 2.

The heaters were evaporated films of constantan or bismuth and the detectors were aluminium super- conducting bolometers, the whole crysta lwas immers- ed in liquid helium at about 1.3 OK. The heater areas were varied from 0.15 mm2 to 10 mm2 depending on the phonon frequency required.

The signals from detector 2 were amplified and dis- played on a sampling scope used as a gated amplifier and variations in the amplitudes measured using a phase sensitive detector.

The sensitivity of the electronics was such that an attenuation of the phonon beam 2 of 5 % was visible directly on the scope screen and an attenuation of 0.5 % was detectable on the phase sensitive detector output. The latter would correspond to a mean free path for phonon-phonon scattering of about 100 cms.

So far no such attenuation has been detected at this sensitivity when this experiment was performed in sodium chloride, potassium chloride, sodium flo- ride and lithium floride.

Interpretation of results.

The frequency of the phonons in the heat pulse was estimated using the acoustic mismatch theory for thermal boundary resistance [ I l l to calculate the heater temperature and so find the frequency of the peak in the phonon frequency distribution. For a constantan heater on potassium chloride 10 watts into 0.15 mm2 gives o -- loL3 s-l. There have been several recent expe- riments supporting this method [12], [13], [14] to the accuracy required here.

For the t + 1

l interaction in KC1 with w , = 5 x loi1 s-', conservation of momentum and energy requires o,

1 . 2

1012 and

03 =

1.7 x 10''

.

Before eq. (19) can be used to calculate the expect- ed transition probability for this interaction, some adjustment of the phonon numbers is required. Some reduction from that given by Bose-Einstein occupation numbers, will be produced simply by the inverse square law. This will add a factor A/2 zd2 to N, and N2, where A is the heater area and d the distance of the heater from the region of interaction.

Incorporating these extra factors and appropriate values for the emissivity into eq. (19), we find a value for the transition probability for the t + I

I inter- action specified in KC1 of

- ³ 1 0 4 ~ - l

using values for the 2nd order elastic constants mea-

sured by heat pulse experiments and agreeing with

those in the literature [15].

C4-122 M. S. RIBBANDS AND D. V. OSBORNE

No published values for the 3rd order elastic constants of KC1 could be found but these are typi- cally of the same order as 2nd order constants in other materials and so would not greatly affect the result.

This value of z i l suggests a value for the mean free path for this scattering process for the longitu- dinal phonons in beam 2 of 1 , - ^{10 cm.}

This is within the limit set by the sensitivity of our experiment.

The observed lack of scattering cannot be explained by the values of the 3rd order elastic constants. It is felt that the most likely cause is either the incorrect interpretation of phonon numbers, or over optimistic

estimation of the phonon frequencies, particularly the latter because of the

dependence of

Recent observations of second sound in NaF 16 imply 1 , -- 1 cm in order for second sound pulses to have developed within the dimensions of the crys- tals used. I t is difficult to compare this with the 2, calculated from elastic theory which applies to a specific interaction situation, although one would expect at least order of magnitude agreement between the two methods.

Experiments are continuing with attempts being made to increase the available heater temperature and the purity, chemical and isotopic, of the crys- tals.

References

[I] BERMAN (R.) and BROCK (J. C. F.), Proc. R. SOC. [9] ZIMAN (J. M.), 1962, Electrons and Phonons, Oxford

A., 1965, 289, 46. University Press.

[2] CALLAWAY (J.), Phys. Rev., 1959, 113, 1046. [lo] PEIERLS (R. E.), 1955, Quantum Theory of Solids, [3] BATEMAN (T.), MASON (W. P.) and MCSKIMIN (H. J.), Oxford University Press.

J . Appl. Phys., 1961, 32, 928.

[4] LANDAU (L.) and RUMER (G.), Physik

Sovjetunions, [ l l ] LITTLE (W. A.), Can. J. Phys., 1959, 37, 334.

1937. 11. 18. [12] NARAYANAMURTI (V.), Phys. Lett., 1969,30A, 521.

[5] SIMONS (s.),

Cambridge Phil. Soc., 1957, 53, [13] WIGMORE (J. K.) to be published.

702. [14] HERTH (P.) and WEISS (O.),

Angew. Phys., 1970,37,

[6] TAYLOR (L. H.) and ROLLINS (F. R.), Phys. Rev., 101.

1964, 136, 591.

[73 V O ~ GUTFELD (R. J.) and NETHERCOTT (A. H.), ^[I51 HVNT~NGTON (H. B.1, 1958, Elastic Constants of Phys. Rev. Lett., 1964, 12, 641. Crystals, New York Academic Press.

[8] SLONIMSKII (G. L.), Zh. Eksp.

Teor. Fiz., 1937, 12, [16] MCNELLEY (T. F.), ROGERS (S. J.) et al, Phys. Rev.

1457. Lett., 1970, 24, 100.