SOUND WAVES

1. Introduction

In an isotropic engineering material, e.g. a texture-free piece of poly-crystalline iron, there are two kinds of sound waves, corresponding to longitudinal and transverse vibrations. The longitudinal mode has the velocity CL = {[K + (4/3)G]/p}^{1 / 2} and the transverse mode has the velocity Cj = {G/p}^1/2. Here p is the mass density of the medium.

In this case there is no distinction between phase velocity and group velocity. The transverse mode is degenerate; its vibrations can be along any of two (arbitrary) directions perpendicular to the velocity vector.

In a single crystal there are still three modes for sound waves, with velocities expressed in the elastic constants of the medium and the crys-tallographic direction of the wave vector. However, they can no longer be classified as pure longitudinal and transverse waves, and the phase velocity is not parallel to, or equal in magnitude to, the group velocity except in certain symmetry directions of the lattice. Our approach is related to the treatment of lattice vibrations in Chapter 6, and sound waves may be viewed as the long-wavelength limit of phonons.

2. Formulation of the secular equation

Let the displacement vector u = (ux,uy, uz) of a sound wave be

Uj = Aj exp[i(qix + q2y + q^z - cot)]. (5.1) The index j refers to Cartesian coordinates. We will write j — x,y,z

or j = 1,2, 3, alternatively. The displacement in eq. (5.1) is a complex quantity, while the actual displacement is, of course, real. However, both

70

Formulation of the secular equation 71

the real and imaginary parts of u are solutions to the equation of motion.

It is convenient to lump the two solutions together, as in eq. (5.1). The sound wave has the frequency co and the wave vector q = (#i, g₂, g₃).

The (phase) velocity of the mode s is

Cs(q) = co(q,s)/\q\. (5.2)

When there is no risk of confusion with the Cartesian index j , we let the label s have any of the values 1, 2 or 3. The wave properties are obtained from the equation

T n - p ^

²

r

12

r

13

\ (A

X

\

r

2

i r

2 2

-

P

co

²

r

2 3

A

2

= o. (5.3) r

3

i r

3 2

r

3 3

- pa? )

\A₃J

This is known as the Christojfel (1877) (also Christoff el-Kelvin or Green-Christoffel) equation. The quantities r/y are related to q and the elastic coefficients by

3

r,7 = ^2 (^l/²)(^ckiji + ckiij)qkqu (5.4)

k,l=\

and p is the mass density of the material. There are non-trivial solutions (A\, A₂, A₃) to eq. (5.3) only if co is a solution to the secular equation

\r_ij-p(o%\=0. (5.5)

Here |... | is a 3 x 3 determinant and 8tj = 0 for / ^ j and 1 for / = j . Often, eq. (5.5) is written as an equation for the sound velocity C, i.e. with co² replaced by C²q². We shall now consider the solution to the secular equation, first in a general mathematical formulation, then in terms of the engineering elastic constants for an elastically isotropic system, for a single crystal of cubic symmetry, and finally for hexagonal lattice symmetry.

3. General solution of the secular equation

Since eq. (5.5) is a cubic equation in C² (or co²), it has solutions in a closed mathematical form, for any lattice symmetry. A convenient expression for C_s is (Every 1979, 1980)

ZpC] = T + 2VG cos[* + (2TT/3)(J - 1)]; s = 1, 2, 3. (5.6) The quantities 7\ G and * contain the elastic constants and the direction cosines of q, and s also serves as a mode index. If 4> = 0, the cosine term in the last part of eq. (5.6) is the same for s = 2 and s = 3, i.e.

these modes then have equal velocities. Appendix D gives the explicit solution for cubic lattice symmetry.

From a well-known mathematical relation between the sum of eigenvalues, £&>², and the trace of the determinant we get from eq. (5.5)

3

3(C₅²) = J2 ^Cl = <Tn + T₂₂ + r_{3 3})/pq², (5.7)

s=l

for any lattice symmetry. In a cubic lattice (cf. eq. (5.14)).

<C²) = (cn+2c44)/3p. (5.8)

Explicit expressions for Cs in various lattice symmetries are given by Every (1980). The relation (eq. (5.8)) for cubic lattice symmetry is isotropic, i.e. it holds for each direction q. Still, the separate veloci-ties C_s for s = 1, 2, 3 may be more (Fe) or less (Al) anisotropic, as exemplified in table 4.2.

4. Secular equation for isotropic polycrystalline materials

In an isotropic material we can identify (cn + 2ci₂)/3 and C44 with K and G, respectively (cf. eqs. (3.37) and (3.42)). Then the secular equation yields, for the longitudinal wave,

2 4 3*(1 - v) G(4G - E)

pCL = K + -G= x + v = 3 G E

= E{\ - v) = 2G(1 - v) = 3K(3K + E)

(l + v ) ( l - 2 v ) l - 2 v 9K-E

Secular equation for cubic symmetry 73

The velocity Cj of transverse sound waves is given by

pC\ = G. (5.10) Since K > 0 and G > 0,

CL>J(4/3)CT. (5.11)

This inequality may be sharpened because, almost universally, the Pois-son ratio v > 0 and then CL > V2Cj. Also, when K > G, one has C_L > V(7/3)C_T « 1.53C_T. The result that the longitudinal sound velocity is higher than the transverse sound velocity holds for isotropic materials, e.g. polycrystalline materials without texture, but not neces-sarily for each direction in an anisotropic lattice (see the example in

§5).

Example: an important average of sound velocities. In the Debye theory of the lattice heat capacity, one encounters an average sound velocity C_SOund,D defined by

3 1 2

—, = — + — (5.12)

^ sound, D ^ L ^ T

By eqs. (5.9) and (5.10), the average velocity CSOUIKI,D can be written

1 3 / 2 ) "¹/³ CSound,D — Cj { - + - 1 - 2 v

3 3 | _ 2 ( l - v ) . (5.13)

The factor in the parenthesis varies slowly with the Poisson ratio v, e.g.

CSOUnd,D = (1.12 ± 0.02)CT if v = 0.31 ± 0.14. Since CT = (G/p)¹/², one may in this way connect G with the Debye temperature (see Chapter 6, §4, and Schreiber et al. 1973).

5. Secular equation for cubic symmetry

Among all c,-^/, (i.e. c_ap) there are only three independent parameters for a cubic lattice structure; en, c\i and C44. The explicit form of eq.

(5.5) is

Table 5.1

a Polarised along [001].

b Polarised along [110].

( c n - C44)?? + Q 4 ?² - P&>²

When the solution to this equation is expressed in the general form eq. (5.6), the angle 4> only depends on the direction cosines of q and on the combination A_E = (cn — cyi — 2c\\)l{c\\ — C44) of the elastic parameters. For this reason, AE is a natural parameter to measure the anisotropy in cubic lattices (Chapter 3, §8). The solutions to eq. (5.14) have very simple forms when q is along the principle crystallographic directions. Table 5.1 gives the quantity pC² = pco²/q² for these cases.

Numerical results for Al and Fe are given in table 4.2.

Example: transverse sound velocity being highest. We noted in §4 that in a material described by the isotropic engineering elastic constants K and G, the sound velocity CL of the longitudinal mode is always larger than Cj of the transverse mode. From table 5.1 we see that in the [111] direction, Cj > CL if c\i < — C44. Lattice stability requires that C44 > 0, so Cj > CL implies that c\i < 0. This is extremely unusual, but happens for the so called intermediate-valence compound Smo.75Yo.25S (cf. table 4.1). Table 5.2 gives the sound velocities, calculated from data in Tu Hailing et al. (1984) (cf. with CL and CT in Al and Fe, table 4.2).

Similarly, in the [100] direction the transverse mode has the highest velocity if C44 > c\ \. This inequality has been reported for some Mn-Cu and Mn-Ni alloys (see Chapter 4, §5).

Secular equation for hexagonal symmetry 75 Table 5.2

Longitudinal and transverse sound velocities (unit m/s) in Smo.75Yo.25S Mode

Longitudinal Transverse

[100]

4567 2292

[110]

3390 2292; 3823

[HI]

2894 3390

6. Secular equation for hexagonal symmetry

The secular equation for an hep lattice is easier to solve than for a cubic lattice, since it separates into a linear and a quadratic equation in co². The sound velocities (phase velocities) are (Hearmon 1961, Musgrave 1970)

p [ C_u]² = c₄₄ + (l/2)[n²P + (l-n²)Q]

±{\/2){[n²P + {l-n²)Qf

+4n²(l - n²)(R² - PQ)}^1/2, (5.15) p[C3]² = C44 + (1/2)(1 - n²)(cn - cl2 - 2c44), (5.16) with n being related to the angle 9 between q and the crystallographic

oaxis; n = 93/lql = cos#. Furthermore,

P = c33 - c44; Q = cn- c44; R = cu+ c44. (5.17) In the basal plane of the hep lattice, n = cos 9 = 0. The sound velocities

are isotropic, with

p[Ci]² = cn\ PiC2]² = c44; P[C3]² = (en - c12)/2. (5.18) When q is parallel to the crystallographic oaxis, we have n = 1 and

p[Ci]² = c33; p[C2]² = p[C3]² = c44. (5.19) The sound velocities C\(2) and C3 in eqs. (5.15) and (5.16) are isotropic,

i.e. independent of the angle 6, if

c1 1- c1 2- 2 c4 4 = 0; P = Q; PQ = R². (5.20)

These relations follow from the isotropy conditions (3.63).

Example: sound velocities in zinc. The hep single crystal of Zn is in many respects very anisotropic (see figs. 7.3 and 15.3). The elastic con-stants are cn = 165, c\i — 31.1, c\z = 50.0, C33 = 61.8, C44 = 39.6 (unit GPa; data from Every and McCurdy 1992). This gives the sound velocities 2940 and 2360 m/s along the crystallographic c-axis. In the basal plane, perpendicular to the c-axis, the velocities are 4810, 3060 and 2360 m/s.

7. Phase and group velocity We define phase velocities

Cphase(q, s) = qo)(q, s)/\q\, (5.21)

and group velocities

Cgroup(q, s) = Vqco(q, s). (5.22)

When Cphase and Cgroup are isotropic, the label q can be dropped.

Then the angle </> between CPhaSe(q> s) and the displacement vector u(q, s) = (ux,uy,uz) for the atomic vibrations is either zero (the longitudinal mode), or 90° (the two transverse modes). In this case, one speaks of pure modes. The longitudinal pure mode has the phase velocity CL = (cn/p)^{1 / 2} and the two degenerate transverse pure modes have velocities Cj = (c44/p)^1/2.

A single crystal often has quite anisotropic elastic properties. Then the three vectors Cphase(q, L), Cgr0Up(q, L) and u(q, L) are not parallel (except for q-vectors in certain symmetry directions) and CPhase ^a^d Cgroup also differ in magnitude. Still it is customary to call the modes

"longitudinal" (or quasilongitudinal) and "transverse" (or quasitrans-verse). The label of a branch is then retained as one moves with the q-vector away from a symmetry direction where the mode is pure. Since eigenvectors referring to different eigenvalues are normal to each other, the vectors u(q, s) for s = 1, 2 and 3 are orthogonal. Brugger (1965a) has listed cystallographic directions of pure modes, and the correspond-ing phase velocities expressed in elastic constants, for cubic, hexagonal, orthorhombic, tetragonal and rhombohedral lattice symmetries.

Energy transport by sound waves 11

[010] 1 3

4

5 [100]

6

Fig. 5.1. A schematic illustration of "phonon focussing". The solid curve gives co(q, T2)

= constant for the low-velocity transverse mode (T2) in the (001) q-plane in Ge.

Cgroup(q, T2) is exemplified by arrows 1-6.

Every (1980) has shown that the proper anisotropy parameter for sound waves in a cubic lattice structure is AE = (en — ci₂ — 2c^)l{c\\ — C44), rather than the often used parameter due to Zener (1948), Az = 2c44/(cn - C12) (see Chapter 3, §8).

8. Energy transport by sound waves

The energy transport by an elastic wave can be described by a ray velocity, analogous to the Poynting vector in electromagnetism. In a non-dissipative medium, the ray velocity equals the group velocity (cf.

Every 1980). The group velocity Cgroup can be derived from eqs. (5.21) and (5.22), and the explicit relations for the phase velocity given by Every (1980). However, Cgr0up(q, s) does not have as simple a form as Cphase- The wave vector q, the phase velocity CPhase and the group velocity Cgroup are, for small q, related by

Cgroup * q ⁼ I q l I cphase I- (J.ZJ)

Hence, |Cgroup| > |CphaSel> with equality only in pure modes.

Calculations of group velocities have been performed for, e.g., LiF, KC1, and A1203 (Taylor et al. 1971), af-Si02 and A1203 (Farnell 1961, Rosch and Weis 1976a), and diamond, Si and Ge (Rosch and Weis

Ch. 5. Sound waves

1976b). Among experiments, emphasis has been on germanium (e.g., Hensel and Dynes 1979, Dietsche et al. 1981).

Example, anisotropic group velocities and phonon focussing. The group velocity Cgr0Up(q, s) = Vq&>(q, s) is a vector normal to the surface co(q, s) = constant. Figure 5.1 shows the shape of co(q, T2) for the low-velocity transverse mode (T2) in germanium when q is in the (001) plane. Cgroup tends to be directed along "channels", when q lies near a point where the curve &>(q, s) = constant has an inflection point in q-space. The energy flow is either decreased (arrows 1-3), or increased (4-6) in certain q-directions. This is the physical basis for "phonon focussing" in crystals (Wolfe 1980).

CHAPTER 6

In document THERMOPHYSICAL PROPERTIES OF MATERIALS (pagina 91-100)