Simple analytical expressions may be derived for sound velocities in cubic crystals by using lattice harmonics or functions which are invariant under the crystal symmetry operations. These expressions are in good agreement with the exact results for typical crystals such as metallic iron and potassium fluoride

Park, H. Y.

Tsang, T.

English

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SOUND VELOCITY ANISOTROPY IN CUBIC CRYSTALS
T. Tsang and Hae-Young Park
Department of Physics and Astronomy, Howard University,
Washington, D. C. 20059, U.S.A.
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ABSTRACT-^26%t*
Simple analytical expressions may be derived for sound
velocities in cubic crystals by using lattice harmonics or
functions which are invariant under the crystal symmetry operations.
Thead expressions are in good agreement with the exact results
for typical crystals such as metallic iron and potassium
fluoride.
(NASA-CE-175344) SOUND 'VELOCITY ANISOTROPY
IN CUBIC CRYSTALS (Howard Univ.)	 10 p
HC A 02/MF A01	 CSCL 20L
N84- 170 16
Uncla s
G3/76 11622
Classification: 62
The propagation of sound waves in single crystals is governed
by a set of three simultaneous linear equations known as the
Christoffel equations. [ 1-3] The characteristic equation
relates the sound velocity v to the direction n of sound
wave and to the elastic constants of the crystal. When the
direction n lies along a crystal symmetry axis, simple solutions
may be obtained for the sound-velocities. In many cases, these
simple solutions may be readily reversed so that the measured
sound velocities may be used to calculate the elastic constants.
However, there are many situations where the solution of the
characteristic equation is required for arbitrary crystallographic
directions= for example, an argon crystal grown at low temperatures
may be inaccessible except for sound velocity measurements. [4,51
For these arbitrary directions, it is advantageous to have
simple approximate solutions, preferrably analytical in form.
Many experimental quantities, such as nuclear magnetic
resonance (NMR) second moments and sound velocities, are invariant
under the crystal symmetry operations, thus they may be expanded
in terms of lattice harmonics of the identity representation.
These lattice harmonics are specific linear combinations of
spherical harmonics which are invariant under the crystal symmetry
7	 operations. [6-8]	 The linear expansions are very convenient
for certain experimental quantities such as NMR second moments
where it can be rigorously shown that the expansion is limited
to only the zeroth, second and fourth order spherical harmonics.
[9] However, such limitations may be absent for the sound
velocities, thus the linear expansions many converge rather
- 1 -
_ w
1;Y
slowly. In this paper, the simple analytical solutions for sound
velocities along the major crystal symmetry axes have been
extended to arbitrary directions for cubic crystals. However,
we have not restricted ourselves to linear expansions alone.
Instead, we have used simple functional forms which are chosen
to fit the experimental data exactly along various directions
with high crystal symmetry. These approximate solutions have
been compared with the exact results, and the agreements are
satisfactory.
By choosing the reference or coordinate axes to be coincident
with the edges of a cubic cull, then the cubic crystal is
characterized by three elastic sti'fness constants, 
c11' c12
and c44 . The Kelvin-Christoffel stiffnesses are given by [ 1 -31
s	
r ii = ni2(c 11 	c44 ) + c44	 (1)
T' i j = nin j ( c12 + c44 )	 for i * j
	 (2)
where n1 , n2 and n3 are the components of the unit vector n.N
The sound velocities v are obtained by solving the characteristic
equation or determinant IT-i j - P v2 Sij l = 0 where P is the
density of the crystal. This equation inay be simplified by
introducing the dimensionless variables A=(p v2-c44)/(c11-c44)
and P=(c12+c44)/(c11-c44) so that the eigenvalue A depends
only on a single parameter P characteristic of the cubic
crystals
'ri se— X	 P n1 n 2	 P "n o *n3
t'	
z
Q -n i 11 2	'n1 '-	 r^2In3	 U
z
(3)
P"1 713
	 r r2'n3	 ^3 
_
- 2 -
12Y
On expanding the determinant, we get the cubic equation
,\3 - 2 + P(1 -^3 2 )A - Q(1 -
	
2 (1 + 20) = 0.	 (4)
where P = n1 2n22 + n 2 2 n 3 2 + n3 2n1 2 and Q = n1 2n2 2n3 2 are fourth
and sixth order functions which are invariant under the crystal
symmetry operations. These functions are equivalent to the lattice
harmonics [6,71 after orthogonalization and normalization= for
example, the fourth order lattice harmonic is given by
4•
(525/161t)I [P - (115)].	 From (3) or (4), the eigenvalues have
simple forms when n is parallel to one of the three major crystal
axes, the cubic edge [100], the face diagonal [110], and the body
diagonal [1111, [1 -3]
nil [100] s P=O, Q=O, AL =1 , AT=0,0.
NI) [110]: P =1/4, Q =O, XL
	
'\T=0' '\T=O, (1-0)/2.
nJJ [111] o P=1/3, Q=1/27, XL= (1+2p)/3, XT=(1-P)/3, (1-P)/^.
The largest eigenvalue XL refers to the longitudinal wave wher•as
the other two eigenvalues A T refer to the two transverse or shear
waves. It may also be seen from (3) or (4) that A T=O for one of
the transverse waves when n is perpendicular to one of the cube
edges (Q=0). Furthermore, simple results may also be obtained
when n is perpendicular to one of the face diagonals. By defining
= u(1-P), then (4) may be written as
[1-p]2[(u3- u2+Pu-Q) + R(-u3+Pu-2Q)] = 0.	 (5)
If u is independent of	 then both parentheses in (5) would be
zero. Solving u3-u2+Pu-Q=0 and -u3+Pu-2Q=0 for P and Q, we get
P = 2u - 3u2 ^	 Q = u2 (1-2u).	 (6)
When n is perpendicular to one of the face diagonals, then n 1 =n2N
and n3 2 = 1 - 2n1 2 . Thus we have
-3-
r4IN
P = 2n1 2
 - 3n 1 4 	Q = n14 (1 - 2n 1 2 ) .
	
(7)
Comparison with (6) indicates that u=n 1 2 is one of the solutions.
Thus one of the transverse wave eigenvalues is A T = ni2(1-p).
For ni =n2 , the characteristic equation is obtained by the substitution
of (7) into (4):
A3 - A2 + P(1-j32 ) (2n i 2 - 3n i 4 )X- ni 4 (1-2n1 2 ) (1_^) 2 (1+2.P) = 0.
Since one of the solutions is n i 2 (1-,3), this cubic equation may
be factorized into the product of A-n i 2 (1-p) and the quadratic
equation
^2 - [1+n 1 2y - 1)]A - (P-1) (2P +1)n 1 2 (1 - 2n1 2 ) = 0.	 (8)
The other two eigenvalues may be obtained by solving (8).
We will now extend these simple solutions to arbitrary
directions of n. For the longitudinal wave, the simple functionN
A L = 1 - 2(1-`3 ) P	 (9)
agrees with the eigenvalues for n along any one of the three
.v
major axes (100, 110, 111). The next order correction would be
proportional to an invariant function up to the eighth order.
This function would be a linear combination of P, Q and P 2 . By
imposing the condition that this function vanishes when n isN
along one of the three major crystal axes, the result is
4P 2_P-3Q. Thus we have the result
XL = 1 - 2(1-r )P + K(4P 2 -P-3Q) .	 (10)
The constant K will be chosen so that L agrees with the exact
value when n is along the [112] crystal axis (this crystal axisN
is perpendicular to both a face diagonal and a body diagonal),
where we have n1 2 = n2 2 = 116, n3 2 	 213, P = 1/4, Q = 1/54, and
E
-4-
12V
4P2-P-3Q = -1/18. From (8), we get /\ L = [5+p+(9-6'0+3392)1]/12.
Substitution into (10) gives K = (3/2) ` 1 +5p- (9-613+332)],
Thus we get
XL = 1 - 2(1-'P)P + (312)L1+5P-(9-6P +33
 
2 ) ^ J(4P 2 -P-3Q)
	
(11)
for the longitudinal waves.
From (3) or (4), the sum of the three eigenvalues is always
unity, hence it is necessary to evaluate /\ T for only one of the
transverse waves. We note that /\ T=0 when Q=0 (n perpendicular
to a cube edge) and that T = n1 2 (1-P) when n1 =n2 (n perpendicular
to a face diagonal). Because of these restrictions, it may not
be advant,
harmonics
diagonal,
n1 2 =P-
ageous to expand
or invariant fu
we get P2-3Q =
( P2-30 and it
A T as linear combinations of lattice
nctions. For 
^j perpendicular to a face
(n1 2 - 3n14 ) 2
 from (7), hence we have
follows that
AT = (1-p)[P- (P2-30 ]	 (12)
This relationship is also exact for n perpendicular to a cubeV
edge	 T=0 when Q =0). We note that from (7), n1 2 may also take
other forms. For example, we may also write n12=(2P2-6Q)/(P-9Q).
However, the expression ^ T = (1-p)(2P2-6Q)/(P-9Q) will not
vanish for Q=O.
The anisotropy (or the variation with the direction of the
sound wave) of the eigenvalues depends only on the dimensionless
parameter P. For A=1, the three eigenvalues are A =1 ,0, 0 for
any direction{ that is, there is no anisotropy for the sound
velocities. Usually (although not always), we have P< 1 for
ionic crystals and 
r 
>1 for metals; the extreme values are 
r 
=0.19
- 
5 -
ITY
for SnTe and	 =6 for metallic sodium. ^2, 101	 Typical values
are ^=2 .1866 for metallic iron (c 11 =23.30, c 12=13.92 and c44=11.62
in units of 10 11 dynes/cm 2 ) and 0=0.523 for potassium fluoride
(c 11 =6.58, c 12=1.49 and c 44 =1.28). The exact eigenvalues for these
crystals are shown in Fig. 1 and Fib;. 2 as the solid lines for n
in the (100), (110) and (111) crystal planes by solving (3) or (4)
numerically. The points in the figures are the approximate
eigenvalues obtained from the analytical expressions (11) and (12)
for the longitudinal and transverse waves. For all orientations
of the sound wave direction n, the agreement is quite good between
the exact and the approximate eigenvalues.
In conclusion, it may be seen that simple analytical expressions
may be used with good accuracy for sound velocities in cubic
crystals. These expressions are based on the lattice harmonics
or functions which are invariant under the symmetry operations of
the crystal. Whereas linear expansions can be used for the
longitudinal waves, other functional forms may be necessary for
the transverse waves because of the "cross-overs" of the two
branches. These simple results may be useful for fitting
experimental data of materials of unknown orientations crystallized
at low temperatures or for the inversions of sound velocity data
to obtain the elastic constants. [4,5]
The partial financial support by National Aeronautical and
Space Administration grant NAG-5-156 is gratefully acknowledged.
- 6 -
REFERENCES
[1] C. Kittel, Introduction to Solid State Physics (Wiley, New
York,	 1971),	 p.	 144.
[2] M. J. P. Musgrave, Crystal Accoustics (Holden-Day, San
Francisco,	 1970).
[3] J. F. Nye, Physical Properties of Crystals (Oxford University
Press, London,	 1972).
[4] D.	 Y. Chung and Y. Li,	 Phys. Lett.	 58A (1976)	 133-
f [5] D.	 Y. Chung and V. Konecny, Phys. Stat. Solidi 52A	 (1979)	 29•
[63 F. C. Von der Lage and H. A.	 Bethe, Phys. Rev.	 71	 (1947)	 612.
r [71 D.	 G.	 Bell,	 Rev.	 Mod.	 Phys.	 26	 (1954)	 311.
[81 T.	 Y.	 Thomas,	 Proc.	 Nat.	 Acad.	 Sci.	 U.	 S.	 55	 (1966)	 235,	 1011.
C91 D. E.	 O'Reilly and T.	 Tsang, Phys. Rev.	 128	 (1962)	 2639.
[10] R. Bechmann and R. F. S. Hearmon, Landolt-Bornstein: Numerical
Data and Functional Relationships in Science and Technology,
New Series, Vol. 	 III/1	 (Springer,	 Berlin,	 1966).
M:
FIGURE CAPTIONS
Fig. 1. Eigenvalues X L and	 T of the longitudinal and transverse
waves for metallic iron ( P= 2.1866) when n is in the (100),
(110) and (111) crystal planes. Solid lines are the exact
values from (4). The points are the approximate values
from the analytical expressions (11) and (12).
Fig. 2. Eigenvalues A L and AT of the longitudinal and transverse
waves for potassium fluoride ( P= 0.523) when n is in the (100),
(110) and (111) crystal planes. Solid lines are the exact
values from (4). The points are the approximate values
from the analytical expressions (11) and (12).
-7-
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Sound velocity anisotropy in cubic crystals

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