Fundamentals of the Physics of Solids / 11-Dynamics of Crystal Lattices

	11.3 The General Description of Lattice Vibrations								361
(λ)	√
If the quantity eν,β	(0)/ Mν is constant and independent of ν for the three
Cartesian coordinates β, that is
			eν,β(λ) = C			eβ(λ) ,		(11.3.33)
					Mν
where the normalization factor
		C = #		p			1/2
				μ=1 Mμ$−				(11.3.34)

ensures that the orthogonality relation of the polarization vectors is satisfied by vectors e(λ) of unit length, then (11.1.8) implies the vanishing of the righthand side of (11.3.32), so the frequency on the left-hand side must also be zero.

In vector form this condition reads

e(νλ) = Mν e(λ). (11.3.35)

Since three mutually perpendicular vectors can be chosen for the e(λ), there are three solutions with vanishing frequency at q = 0. For such vibrations the atomic displacements

1					1
uα(m, μ) =				eμ,α(λ)	Qλ(0) =	√		eα(λ)Qλ(0)	(11.3.36)
		N Mμ					N

are independent of the label μ; in other words, all the atoms of the primitive cell oscillate in phase, with equal vibrational amplitudes. Owing to continuity, there are three similar vibrations for small values of q. Atoms in the primitive cell vibrate almost in phase with almost equal amplitudes – just as in a sound wave (long-wavelength elastic wave) propagating in a solid. The dispersion relation of these vibrations give the three acoustic branches.

Optical Vibrations

In addition to the three acoustic branches, there are 3p −3 further vibrational branches in which frequencies generally start at some finite value. These are the branches of optical modes. To understand the character of such vibrations, consider a crystal with a diatomic basis. It follows from (11.3.35) that for vibrations of polarization λ in the acoustic branch

	e(λ)		=	e(λ)		(11.3.37)
	√	1		√	2
		M1			M2

at the center of the Brillouin zone. Whichever optical branch with polarization λ is chosen,

e1(λ) · e1(λ ) + e2(λ) · e2(λ ) = 0

(11.3.38)

362 11 Dynamics of Crystal Lattices

because of the orthogonality of eigenvectors. From this pair of equations

		&e1						' = 0 .
e1	·		+	/		M1	e2		(11.3.39)
(λ)		(λ )				M2 (λ )

This condition must hold for each of the three acoustic branches, consequently

		e1(λ ) +		e2(λ ) = 0	(11.3.40)
	M1		M2

for each optical branch. Equation (11.3.14) permits us to set up a relation between the polarization vector and the atomic displacements due to vibrations of wave number q = 0. The above equation then implies that for the displacements due to optical vibrations

M1u1(m, 1) + M2u2(m, 2) = 0 ,

(11.3.41)

that is, in long-wavelength optical vibrations atoms in the primitive cell move in opposite directions in such a way that the center of mass remains stationary. Therefore these vibrations may be regarded as internal oscillations of the system of particles inside a primitive cell.

Among optical vibrations one may distinguish longitudinal and transverse modes as well – at least in directions of high symmetry. Depending on the symmetry in question, frequencies may become degenerate. Figure 11.12 shows the spectra of lattice vibrations in some characteristic directions of the Brillouin zone for crystalline silicon, which contains two atoms per primitive cell in a diamond structure.

Fig. 11.12. Dispersion relations in some characteristic directions of the Brillouin zone for crystalline silicon, which contains two atoms per primitive cell. The solid line is a calculated curve fitted to experimental data. [G. Dolling, in Inelastic Scattering of Neutrons in Solids and Liquids, IAEA, Vienna, 1963, Vol. II., p. 37]

11.4 Lattice Vibrations in the Long-Wavelength Limit

363

Besides experimental results, theoretical calculations are also shown in the figure. Assuming central forces, a relatively large number of force constants have to be introduced among distant neighbors to ensure a good fit. The reason for this is that approximating highly directional covalent bonds by springs is di cult to justify. Another drawback of the picture of springs connecting atoms is that the polarization of electron clouds due to the motion of ions cannot be appropriately taken into account in calculations of the vibrational spectrum. The proper treatment of such e ects requires much more sophisticated methods that cannot be discussed here.

11.4 Lattice Vibrations in the Long-Wavelength Limit

In the previous section we saw that in long-wavelength acoustic modes the atoms in a primitive cell oscillate in phase and with equal amplitudes. The displacement of the atoms in the mth primitive cell – characterized by the lattice vector Rm – is independent of the label μ in the first approximation, therefore the internal structure of the primitive cell (the basis) is irrelevant from the viewpoint of vibrations. Moreover, the common displacement u(Rm) varies little between adjacent cells. For this reason, a slowly varying continuous function u(r) may be introduced, whose value in the lattice point Rm is the displacement of the atoms of the primitive cell. This quantity specifies the displacement with respect to the equilibrium position in an arbitrary point r of the solid when atomic details are ignored – i.e., when matter is considered to be continuous. In this approximation the solid can be regarded as a continuous elastic medium characterized by a handful of elastic constants, in which classical elastic waves may propagate. Below we shall examine the relationship between these waves and atomic oscillations.

This approach cannot be applied to optical vibrations. Instead, a simple relation will be established between the frequency of optical vibrations and dielectric polarizability in ionic crystals.

11.4.1 Acoustic Vibrations as Elastic Waves

The starting point for our investigations into the long-wavelength limit of acoustic vibrations is (11.1.10), a formula of general validity when interatomic interactions are treated in the harmonic approximation. As atoms in a primitive cell move together – i.e., u(m, μ) is essentially the same for all atoms in the primitive cell at Rm –, and Φμναβ (m, n) depends only on the di erence of vectors Rm and Rn, the harmonic term can be written in a simple form, similar to that in crystals with a monatomic basis:

Uharm = −14 [uα(m) − uα(n)] Φαβ (m − n) [uβ (m) − uβ (n)] , (11.4.1)

m,n

α,β

364 11 Dynamics of Crystal Lattices

where Φαβ (m − n) is an e ective force constant between the primitive cells at Rm and Rn.

If atomic displacements vary little over the range of the force constants then the di erence in the displacement of the atoms in the primitive cells at Rm and Rn is well approximated by the first derivative of the continuous displacement field u(r):

u(m) = u(n) + ((rm − rn) · r ) u(r)|r=Rn .

Using this expansion, the harmonic term of the interaction is

Uharm = 21 n,α,β	∂rσ uα(r) r=Rn		∂rτ uβ (r) r=Rn Eαβστ		(Rn) ,
		∂		∂
σ,τ

where

Eαβστ (Rn) = −12 [Rm − Rn]σ Φαβ (m − n)[Rm − Rn]τ .

m

(11.4.2)

(11.4.3)

(11.4.4)

In a homogeneous crystal this quantity is independent of the position vector Rn. Since the function u(r) varies slowly, the above sum is well approximated by the integral

Uharm = 2v	στ		dr	∂rσ uα(r)		∂rτ uβ (r) Eαβστ	,	(11.4.5)
1					∂	∂

αβ

where v is the volume of the primitive cell. This expression gives the energy of the elastically deformed crystal in terms of a four-index tensor Eαβστ , whose elements could be taken as the elastic constants. Due to some symmetry properties, Eαβστ does not have 81 di erent components even in the most general case: the number of independent elastic constants is much smaller.

To prove this and to find the appropriate constants, we first make use of the property that the antisymmetric combination

1		∂		∂
			uα −		uβ	(11.4.6)
2		∂rβ		∂rα

corresponds to a rigid rotation of the solid, and so it gives a vanishing contribution to the energy. Therefore only the symmetric combination

			∂		∂
εαβ =	21			uα +		uβ		(11.4.7)
			∂rβ		∂rα

may appear in the energy expression. Among the elements εαβ of the strain tensor, diagonal ones (εxx, εyy, and εzz ) are related to the relative change in length along the coordinate axes, while o -diagonal elements describe shear strains. Elastic energy is a homogeneous second-order expression of these

11.4 Lattice Vibrations in the Long-Wavelength Limit

365

quantities. Which combinations appear is determined by the symmetries of the system, since the energy expression must be invariant under all symmetry operations that take the system into itself. This requirement imposes severe restrictions on the number of independent elastic constants.

As an example, consider a homogeneous and isotropic (therefore noncrystalline) material. In this case only those second-order expressions may appear that are invariant under arbitrary rotations of the coordinate system. By its definition, the strain tensor is symmetric: εαβ = εβα. Such a tensor has one linear invariant, the sum of its diagonal elements (its trace):

I1 = εxx + εyy + εzz ,

(11.4.8)

which is equal to the relative variation of the volume. Besides I12, there is another (independent) quadratic invariant,

I2 = εxxεyy + εyyεzz + εzz εxx − 12 ε2xy + ε2yx + ε2yz + ε2zy + ε2zx + ε2xz .

(11.4.9) Using the combination I12 − 2I2 instead, only two parameters are needed to write down the elastic energy in the isotropic case:

Uharm =	λ	[εxx + εyy + εzz ]2	(11.4.10)
	2

+μ ε2xx + ε2yy + ε2zz + ε2xy + ε2yx + ε2yz + ε2zy + ε2zx + ε2xz

where λ and μ are the Lamé constants:3 μ is the usual shear modulus G, while λ is related to the compressibility κ and the bulk modulus K via

1	= K = λ +	2	μ .	(11.4.11)
κ		3

In elasticity, stresses generated by elastic strains are characterized by the symmetric stress tensor σαβ , and the force per unit volume of the deformed body is given by the derivative

Fα = ∂σαβ . (11.4.12)

β

∂rβ

The vibrations of an elastic medium of density ρ are therefore governed by the equation of motion

ρ u¨α = ∂σαβ . (11.4.13)

β ∂rβ

To put this to use, another relation is needed: a generalization of Hooke’s law4 that establishes the connection between the stress tensor and the strain tensor. This can be written as

3G. Lamé, 1852.

4R. Hooke, 1676.

366 11 Dynamics of Crystal Lattices

σαβ = δαβ λ [εxx + εyy + εzz ] + 2μ εαβ ,

(11.4.14)

derived from the relation

	σαβ =	∂Uharm	.	(11.4.15)
		∂εαβ

We are now in the position to determine the propagation velocity of waves in an isotropic elastic medium. Writing out the equation of motion for the component ux in full detail,

ρ u¨x =

∂σxx

+

∂σxy

+

∂σxz

∂x

∂y

∂z

∂

∂ux

∂uy

∂uz

∂2ux

= λ

+

+

+ 2μ

∂x

∂x

∂y

∂z

∂x2

(11.4.16)

+ μ

∂2ux

∂2ux

∂2uy

∂2uz

+

+

+

∂y2

∂z2

∂x∂y

∂x∂z

= (λ + μ) ∂x

∂x + ∂y +

∂zz + μ 2ux .

∂

∂ux

∂uy

∂u

Similar equations apply to other components. The three components can be written concisely in the vector form

ρ u¨ = (λ + μ) grad div u + μ 2u .

(11.4.17)

Solutions that are periodic in space and time are sought in the form

u(r) = e ei(q·r−ωt) ,

(11.4.18)

which leads to a relationship between e, q and ω:

ρ ω2e = (λ + μ) q(q · e) + μ q2e .

(11.4.19)

Decomposing the vector e that represents the displacement amplitude into components parallel to the vector q specifying the direction of propagation (longitudinal wave) and perpendicular to it (transverse wave),

ρ ω2eL = (λ + μ) q2eL + μ q2eL ,

(11.4.20)

ρ ω2eT = μ q2eT .

It is readily seen that the dispersion relation is linear for longitudinal (compressional) and transverse (torsional) waves alike:

ωL = cL q ,

ωT = cT q ,

(11.4.21)

however, their propagation velocities are di erent:

cL =	7		ρ	,	cT =				(11.4.22)
						/	ρ	.
			λ + 2μ				μ

11.4 Lattice Vibrations in the Long-Wavelength Limit

367

Table 11.1. Propagation velocities of long-wavelength longitudinal and transverse waves in some metals at room temperature

	Element	cL	cT	Element	cL	cT
		(m s−1) (m s−1)			(m s−1) (m s−1)
	Ag	3640	1690	Mg	5700	3170
	Al	6360	3130	Ni	5810	3080
	Au	3280	1190	Pb	2050	710
	Be	12720	8330	Pt	4080	1690
	Cr	6850	3980	Sn	3300	1650
	Cu	4760	2300	Ti	6260	2920
	Fe	5920	3220	V	6000	2780

In crystals, the velocity of longitudinal waves (sound) is usually around several thousand m/s. Transverse waves usually propagate more slowly, roughly at half their speed, mostly in the range 700 to 3500 m/s. Values measured in some metals are listed in Table 11.1.

A vibration may be considered of “long wavelength” if its wavelength is at least one order of magnitude larger than the interatomic spacing. On such length scales solids can be described to a good approximation as elastic continua. The discrete character of the lattice becomes important only for vibrations of shorter wavelengths. In terms of frequencies, the interpretation of acoustic lattice vibrations as elastic waves is justified for frequencies up to the order of 1011 Hz.

11.4.2 Elastic Constants of Crystalline Materials

The model of elastic continua may also be applied to the discussion of the elastic properties of crystals. However, instead of (11.4.14) – Hooke’s law, establishing a linear relationship between stress and strain for isotropic systems

– an even more general relationship is required between the stress tensor and the strain tensor:

σαβ =	Cαβ,γδεγδ .	(11.4.23)
	γδ

Instead of (11.4.5), the elastic energy is now expressed as

Uharm = 21	εαβ Cαβ,γδεγδ .	(11.4.24)
	αβγδ

Elastic properties are thus characterized by a four-index tensor. However, since both the stress and strain tensors are symmetric,

Cαβ,γδ = Cβα,γδ = Cαβ,δγ = Cγδ,αβ .

(11.4.25)

368 11 Dynamics of Crystal Lattices

Depending on the symmetries of the crystal, further relationships may exist among the tensor elements.

Instead of the four-index elastic constants, the Voigt elastic constants5 are used in the literature. These are obtained by using the notation

εxx = ε1 ,	εyy = ε2 ,	εzz = ε3	,
2εyz = ε4 ,	2εzx = ε5 ,	2εxy = ε6	(11.4.26)

for the elements of the symmetrical strain tensor. Using these quantities, the homogeneous quadratic expression for the elastic energy density reads

	6
Uharm = 21		(11.4.27)
	cij εiεj ,

i,j=1

where the coe cients cij are the Voigt elastic constants. A similar notation is introduced for the elements of the stress tensor:

σxx = σ1 ,	σyy = σ2 ,	σzz = σ3	,
σyz = σ4 ,	σzx = σ5 ,		(11.4.28)
		σxy = σ6 .

The generalized Hooke’s law, which establishes the relation between the stress tensor and the strain tensor, now takes the form

		∂Uharm	6
	σi =		= cij εj .	(11.4.29)
		∂εi
			j=1

By its definition, the matrix of elastic constants is also symmetric, therefore in the most general case the elastic properties of triclinic crystals are characterized by 21 independent constants.

In crystals of higher symmetry the number of independent elastic constants is lower. For example, when the z-axis is chosen as the twofold rotation axis in the monoclinic crystal system, elastic energy must be invariant under the transformation x → −x, y → −y, z → z. In such rotations ε4 and ε5 change sign but the four other elements of the strain tensor do not. Since the elastic energy has to remain invariant,

c14 = c24 = c34 = c15 = c25 = c35 = c46 = c56 = 0 .

(11.4.30)

Therefore in monoclinic crystals there are 13 independent elastic constants. In the orthorhombic crystal system elastic energy must be invariant under

180◦ rotations around each of the three axes. This requires

c16 = c26 = c36 = c45 = 0 ,

(11.4.31)

so elastic behavior is characterized by 9 elastic constants.

5 W. Voigt, 1910.

11.4 Lattice Vibrations in the Long-Wavelength Limit

369

In tetragonal crystals rotation through 90◦ around the z-axis appears as a new symmetry. Invariance of the elastic energy under the transformation x → y, y → −x, z → z requires that

c11 = c22 ,

c13 = c23 ,

c44 = c55 .

(11.4.32)

This reduces the number of independent elastic constants to 6. The expression for the elastic energy is then

Uharm = 12 c11 ε2xx + ε2yy + 12 c33ε2zz + c12εxxεyy

(11.4.33)

+ c13 [εxxεzz + εyyεzz ] + 2c44 ε2yz + ε2zx + 2c66ε2xy .

In cubic crystals rotations around the x- and y-axes through 90◦ (x → x, y → z, z → −y, and y → y, x → z, z → −x) are also symmetries. These imply further restrictions:

	c33 = c11 ,		c13 = c12 ,	c66 = c44 .	(11.4.34)
In terms of the three remaining elastic constants,
Uharm = 21 c11	εxx2 + εyy2 + εzz2		+ c12 [εxxεyy + εxxεzz + εyyεzz ]
	2	2			(11.4.35)

+ 2c44 εxy + εyz + ε2zx .

It can be shown in much the same manner that the number of independent elastic constants is 6 for the rhombohedral and 5 for the hexagonal crystal system.

Further connections called Cauchy relations exist among the elastic constants when interatomic forces are all central. For materials of cubic crystal structure

c12 = c44

(11.4.36)

in this case. Elastic constants determined at room temperature are shown in Table 11.2 for some materials with cubic crystal structure. Values measured at low temperatures are somewhat higher but their deviation from the listed ones does not exceed ten to twenty percent.

As the table shows, the Cauchy relation is almost perfectly satisfied in ionic and covalent crystals, while significant deviations are observed in the majority of metals. This indicates the importance of taking noncentral forces into account in the theoretical determination of elastic constants.

It should be noted that instead of the Voigt constants elastic behavior is often characterized by four elastic moduli – Young’s modulus or the tensile modulus (also known as the modulus of elasticity) E, the bulk modulus or compression modulus K (the inverse of the compressibility κ), Poisson’s ratio ν related to lateral contraction, and the shear modulus or modulus of rigidity G. The relationships among them and their connections to the Lamé and

370 11 Dynamics of Crystal Lattices

Table 11.2. Voigt elastic constants (in units of GPa) for some materials of cubic crystal structure

Material	c11	c12	c44	Material	c11	c12	c44
Na	7.59	6.33	4.30	C	1079	124.5	578
K	3.69	3.18	1.90	Si	165.6	63.9	79.6
Rb	2.96	2.44	1.60	Ge	129	48.3	67.1
Cr	348	67	100.0	Pb	48.8	41.4	14.8
Mo	465	163	109	NaCl	49.47	12.88	12.87
W	523	203	160	KCl	40.69	7.11	6.31
Fe	230	135	117	CsCl	36.44	8.82	8.04
Ir	580	242	256	NaBr	39.7	10.01	9.98
Ni	247	153	122	GaAs	118.8	53.7	59.4
Pt	347	251	76.5	ZnS	104.6	65.3	46.1
Cu	169	122	75.3	Fe3O4	273	106	97.1
Ag	122	92	45.5	TiC	500	113	175
Au	191	162	42.2	MgO	297.1	95.4 156.1
Al	108	62	28.3	MgAl2O4	298	154	158

Voigt constants are presented in most textbooks on elasticity. For example, in isotropic materials the elastic moduli can be simply expressed in terms of the Lamé constants:

K = λ + 2	μ, G = μ, ν =	λ	, E =	μ(3λ + 2μ)	, (11.4.37)
3		2(λ + μ)		λ + μ

or conversely, the Lamé constants can be given in terms of two independent moduli, E and ν or K and ν as

λ =	νE			= 3K		ν	,
	(1 + ν)(1 − 2ν)				1	+ ν				(11.4.38)
μ =	E		= 3K	1 − 2ν		.
	2(1 + ν)			2(1 + ν)
For cubic crystals
K = 31 (c11 + 2c12) ,	G = 21 (c11 − c12) ,					ν =		c12	,	(11.4.39)
								c11 + c12

and, depending on the direction relative to the crystallographic axes, Young’s modulus is

	E[100]	=	(c11 − c12)(c11 + 2c12)			,
			c11 + c12
	E[110]	=	4c44	(c11 − c12)(c11 + 2c12)			,	(11.4.40)
			2c11c44	+ (c11 − c12)(c11 + 2c12)
	E	= 3c44(c11 + 2c12) .
	[111]		c11 + 2c12 + c44