Physica 145B (1987) 209-214 North-Holland, Amsterdam

DAMPING OF ACOUSTIC WAVES IN METALS AT LOW TEMPERATURES

KAILASH and S.K. KOR Department of Physics, University of Allahabad, Allahabad 211 002, India

Received 17 October 1986 Revised manuscript received 22 January 1987

The theory for studying the damping of acoustic waves in metals is evolved in the low temperature region and zero magnetic field starting from primary physical constants and assuming basic potentials. The theory is tested for some highly conducting and transitional metals, viz. aluminium, nickel, copper and lead. It is observed that a sharp change in damping occurs if the temperature is elevated and damping is very high around 10 K. The experimental evidences are found in good agreement with present values. It is concluded that the damping is one of the characteristic properties of metals.

1. Introduction

In this decade there has been considerable interest in studying the internal structure and inherent properties of solids using ultrasonic de- vices [1-7]. The study of elastic and inelastic properties in crystals of different kinds, viz. metallic, semiconducting and dielectric, has es- tablished on a firm footing that the background loss of all materials and sometimes the principal loss in single crystals is related to conversion of acoustic phonons into thermal phonons in the high temperature region [8-13]. The magneto- acoustic effect [14-19] has been used in studying the electronic properties of metallic and semimetallic substances. Acoustic waves prop- agating in metals or semimetals interact with conduction electrons or holes in these substances at low temperatures and suffer a damping of the power which is well documented [20-26]. How- ever, temperature or frequency dependence of damping in metallic single crystals have not been studied in detail in the absence of an external magnetic field.

In the present work, the theory for studying damping in conducting solids is evolved in the low-temperature region starting from primary physical constants: the nearest-neighbour dis- tance (r) and the repulsive parameter (q) and using basic potentials: the electrostatic potential (+Z2e2/r) and the Born-Mayer potential

(Q(r) = A exp(- r /q) ) [27]. Section 2 deals with the derivation of the theory. In section 3, the theory is tested for some highly conducting and transitional metals. The results obtained are widely discussed in section 4.

2. Theory

The theory evolved in the present work deals with elastic properties, electron viscosity and damping from 0 to 80 K for conducting materi- als. This is restricted to the face centered cubic crystals only. Assuming the electrostatic poten- tial (+Z2e2/r) and repulsive [27] potential (Q(r) = A e x p ( - r / q ) ) ; Z , e, r, A and q being valency, electronic charge, nearest-neighbour distance, some constant and repulsive parame- ters, respectively (+ sign in electrostatic poten- tial is an indication for like-positive charges) and recalling Briigger's [28] definition for elastic con- stants, the second order elastic constants (SOEC) at absolute zero (C °) are derived as expressed below:

C~l = - 1.56933(Z2e2/r~)

+ (1 /qro) (1 / r o + 1 /q)O(ro)

+ ( 2 / q r o ) ( V ~ / 2 r o + 1/q)Q(roX/2 ) (1)

and

0378-4363/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

210 Kailash and S.K. Kor / Damping of acoustic waves

6 0 4 = 0.347775(Z2e2/r 4)

+ (1/qro)(X/2/2r o + 1 / q ) Q ( r o V ~ ) , (2)

where superscript or subscript '0' is to indicate the absolute zero. Q(r) is the B o r n - M a y e r potential [27[. When the temperature is raised the vibrational energy is affected and the total energy of the crystal tends to increase. At any temperature, the resulting elastic constant is ob- tained by an addition of the vibrational energy contribution ((-~.v.ib-} to SOEC at absolute zero. \ - - t ] J

Using the anharmonic theory for lattice dynamics [29-31], one derives the vibrational energy contribution as shown further:

c v i b I , / 2 11 = f G I + fZG2

and

c v i b 2 44 = f GI,I '

(3)

where the fn 's and Gn's are:

f2 =f~coth X,

f1,1 = fo ( (X / s inh ZX) + coth X ) / 1 2 , (4)

gives the SOEC at any temperature which will be used for determination of ultrasound velocities later on.

For correlation with the viscosity of an elec- tron gas in metals, one uses the following for- mula for the viscosity [32]:

ne = N m J V / 3 , (8)

where T~e , N, m e, [ and I7 stand for electron viscosity, volume charge density, mass of an electron, mean f r ee path and mean velocity re- spectively. [ and V can be derived from the electron gas [33] theory and can be expressed as:

f ~ -- 2 meVo' /Ne (9)

and

9 = 3hZ(37rZN)2/3/5m2e, (10)

where o- is the electrical conductivity in CGS units which is related with electrical resistivity as follows:

or = 9 x 101J/R ,

G I = 2((2 + 2 % - qZ)Q(ro)

+ 2(x/2 + 2q0 - V~q2)Q(roX/-2))/H, (5)

G 2 = 2 ( ( - 6 - 6 % - q~ + q~)Q(ro)/H) + 2G,,I ,

G,,, = (-3X/2 - 6q0 - X/2q~ + 2q~)Q(roX/--2) /H,

with

fo = h°)o/8r3 X = h w o / 2 k T qo = ro/q

H = (q0 - 2)Q(r0) + 2 (% - x / 2 ) Q ( r o X / 2 ) , (6)

where k is Boltzmann's constant, h is Planck's constant divided by 2"rr, T is absolute tempera- ture and w 0 is natural angular frequency of the lattice vibration obtained from 6o~ = 2 H / q r o M , M being the mass of the positive ion. Thus the expression

Ci/( T ) = C o vib ij -1- Cij (7)

R being the electrical resistivity in Ohm/cm. Hence eq. (8) transforms to

~1~ = 9 × 1 0 1 1 h 2 ( 3 " r r 2 N ) 2 / 3 / 5 e 2 R , (11)

which will help in the determination of damping as is described below.

At low temperatures the mean free path of the electron becomes large and is comparable to the wavelength of ultrasonic waves and the momen- tum given to it by the vibrating lattice is not returned at once to the lattice in metals [20, 34, 35] and viscous loss occurs. However , in the superconducting state, the lattice is not able to transfer momentum to the electron gas and the damping disappears. Ultrasonic measure- ments done by B6mmel [36] and McKinnon [37] made it possible to observe the interaction of free electrons and acoustical phonons at low temperatures in the study of Fermi surface, elec- tronic properties of metallic crystals and mea- surements of energy gap in semiconducting ma-

Kailash and S.K. Kor Damping of acoustic waves 211

terials. Pippard [20] considered the free electron model and used a semiclassical approach to ex- plain the damping of ultrasonic waves. The basic idea of this approach is that if the electrons in metal are supposed to be a degenerate Fermi gas enclosed in a box and are subjected to periodic distortion, then in the absence of collisions the Fermi surface will be distorted from a spherical to an ellipsoidal shape. Collision of conducting electrons with the ionic lattice tend to restore its spherical shape. There is a steady dissipation of energy due to the total electronic energy which on the average will be somewhat greater than its equilibrium value. The damping can then be expressed as follows:

For compressional waves:

Ot = (Nme[(p2{ 2 arctan pD/3

x ( p [ - arctan p[)] - 1) /dVlz . (12)

For transverse waves:

a = 2NmepZ-[Z/3dVsz[(pZ-[Z + 1)

x arctan p [ - p[], (13)

where a is damping coefficient, p is the wave number associated with elastic wave, ~- is relaxa- tion time of electron, d is density of substance, V~ and V s are compressional and transverse wave velocities, respectively. Mason [34, 35] described that the damping caused by the energy loss due to the compressional and transverse viscosities of the lattice at low frequencies are as given below:

(oz/f2), = (27rz)(4,r/e/3 + x) /dV~ (14)

and

= 2w ~e/dV~, (15)

Eqs. (14) and (15) are obtained through simplifi- cation of eqs. (12) and (13) for the case pl ~ 1. Thus from the primary physical parameters, one may get the knowledge about elastic and acoustic properties of conducting substances.

3. Evaluation

The theory is tested for four metals which possess a face centered cubic crystal structure. Starting with nearest neighbour distance [33, 38] (AI = 2.861, Ni = 2.489, Cu = 2.558, Pb = 3.470) and repulsive parameter (A1 = 0.425, Ni = 0.355, Cu = 0.395 and Pb = 0.495), all in .Angstr6ms; the Cij are computed. The two parameters A and q in the Bo rn -May e r [27] potential, Q(r)= A e x p ( - r / q ) , are determined using the total free energy (U) of a crystal in equillibrium which should be minimal, namely, (OU/O~ij)ro=O, where r 0 is the lattice parameter and ~gj are Cartesian coordinates. In cubic crystals, it can be shown that 0U/0~11 =0U/0~22 =0U/0~33-" 0 and OU/O~I2 = 0U/3~23 = OU/O~3 ~ =0 . There- fore, the equilibrium condition is (OU/O~ll)r o = 0 and the explicit form is obtained as follows:

+0.58253(Z2e2/ro) - (2ro/q) Q(ro)

-(4X/2ro/q) Q(roV~ ) + hw o coth X / 4 = O. (17)

Taking densities [33, 39, 40] with a variation as d = d0/(1 + 3 a ' T ) , where d, d o are densities at T and 0 K, a ' is the thermal expansion coefficient [38], compressional and transverse wave vel- ocities are computed at different temperatures. The electrical resisivities are taken from litera- ture [39, 41-43] to evaluate electron viscosity (table I). Finally, the temperature variation of damping is obtained as shown in figs. 1-4.

where f is the ultrasonic frequency, "0e is the transverse viscosity (electron viscosity), X(=0) is the compressional viscosity [34]. 1 and s are used for compressional and transverse waves. The V~ and V s are obtained by:

V~ = (C, , /d ) ~/2 and V s = ( C 4 4 / d ) 1/2 . (16)

4. Discussion

Nickel and copper have partially filled d-shells and are different from alkali metals. These have their d-shells nearly in contact and act as rigid spheres while the alkali metals have a small ion embedded in a large atomic volume. The com-

212 Kailash and S .K. Kor / Damping o f acoustic waves

pressibility of metals which have partially filled d-shells is much less than that of the alkali metals owing to interaction between the d-shells. The magneto-acoustic effect [44] has been used to show the Fermi surface of potassium which is spherical to 0.3%. The other alkali metals have a spherical Fermi surface whereas these metals have a Fermi surface which is somewhat compli- cated. It has been shown in the case of copper that the Fermi surface is roughly spherical, ex- cept where it is pulled out to touch the eight hexagonal { l l l ) - f a c e s of the Brillouin zone. Many predictions have been put forward for non-spherical Fermi surfaces but the factors con- trolling damping have not been precisely accoun- ted for. However , assuming a spherical Fermi surface for aluminium, nickel, copper and lead, the results are obtained.

From the figures it can be seen that the damp- ing is very high below 10 K. The nature of the curves is the same for aluminium and copper (figs. 1 and 3) as these are highly conducting. Nickel and lead seem to be of the same type up to a certain extent (figs. 2 and 4). A rapid change is observed if the tempera ture is lowered which occurs due to increasing l. It is obvious from table I also that the ratio of maximum and

- o - - - o . \

16c \

12C \

~. 8C

\

/

20 40 60 80 TEMPERATURE ( K )

Fig. 1. Damping values for aluminium from 4.2 to 80 K in l0 ]~Nps2/cm ( 0 compressional; - - - O - - - trans- verse).

12.0

8.0

4.0

i

I

4

i

-f

2O 4O 6O TEMPERATURE (K)

Fig. 2. Damping values for nickel from 4.2 to 40K in 10 ~SNps2/cm ( O c o r r i p r e s s i o n a l ; - - - © - trans- verse).

minimum value of % is 1.5 (A1), 165(Ni), 1.4(Cu) and 56(Pb) at 22 K. For Ni and Pb the damping is almost negligible above a certain tempera ture which is due to a sharp change in electrical conductivity. However , it has a consid- erable value in the case of A1 and Cu. The contribution of compressional waves to damping is much less than that of transverse waves which is due to the small value of transverse wave velocity. It is observed that the compressional

20 /40 60

TEMPERATURE (K)

Fig. 3. Damping values for copper from 4.2 to 70K in 10 ]6Nps2/cm ( O compressional; - - - O - - - trans- verse).

27.0

18.0

9.01

Kailash and S .K . Kor / Damping of acoustic waves 213

6.0

fl I1 I I

- i I i I

i j i I i I

t i I

4..0 -~ I I

2.C

20 40 60

TEMPERATURE (K)

Fig. 4. Damping values for lead from 2 to 22K in 10-16Nps2/cm ( O - - compressional; - - - O - - - trans- verse).

wave velocity is 1.3 to 1.7 times more than the transverse velocity. It is seen that the wave velocities are increasing as the temperature is increased which is 0.20 to 0.25% only, i.e. wave velocity is also a prominent factor which causes a reduction in damping along with the resistivity of the substance above 2 0 K in metals. It can be seen from the figures that the ratio of transverse and compressional waves damping is 1.3 (A1), 3.8 (Ni) , 2.5 (Cu), 3.5 (Pb). In this way, one may also say that nickel and lead are showing the same nature. It may be pointed out that damping increases (decreases) as the atomic number changes in decreasing (increasing) order in trans- itional (highly conducting) metals and the nature of the curves becomes very sharp as the atomic number becomes higher as can be seen for these substances (AI = 13, Ni = 28, Cu = 29 and Pb =

Table I

Electron viscosity, r/o temperatures

(in poise), for aluminium, nickel, copper and lead at low

Temperature (K) Aluminium Nickel Copper Lead

2 - - - 0.56 4.2 7.459 209.14 5.02 0.82

5 7.469 212.56 5.05 1.53 6 7.478 215.10 5.07 0.78

7 7.475 210.48 5.08 0.42 8 7.470 168.39 5.08 0.26 9 7.465 84.19 5.09 0.20

10 7.458 42.09 5.11 0.13

11 7.316 21.04 5.09 0.08 12 7.195 11.22 4.94 0.07

13 6.952 9.90 4.88 0.04

14 6.899 8.41 4.85 0.03 15 6.882 4.20 4.76 0.02 16 6.619 3.36 4.68 0.02

17 6.357 2.80 4.61 0.02 18 6.048 2.40 4.55 0.01 19 5.832 2.10 4.50 0.01

20 5.672 1.87 4.42 0.01 21 5.215 1.53 4.12 0.01 22 4.893 1.29 3.71 0.01 25 4.533 0.99 3.46 -

30 3.394 0.56 2.53 - 40 1.616 0.24 1.31 - 50 0.485 0.11 0.71 60 0.181 0.07 0.47 - 70 0.121 0.04 0.35 - 80 0.098 0.03 0.21 -

214 Kailash and S .K. Kor / Damping o f acoustic waves

82). The peak observed below 10 K is due to electrical conductivity (Table I). It also shifts from right to left and down to up as the atomic number increases. The experimental results [45] are found in well agreement with present data for lead as the values are 0 . t l N p / c m (ex- perimental) and 0.15 Np/cm (present) for com- pressional waves and 0.06 Np/cm (experimental) and 0.07 Np/cm (present) for transverse waves. The authors propose to study Ca, Sr, Rh, Ir, Pd, Pt, Ag and Au with this theory for obtaining more fruitful results in the future. At last, one may conclude that the damping is the charac- teristic property of the metals.

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