Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

0.4 Use of Penetrating Radiation to Determine Crystal Structure 13

Fig. 0.6 Graphical solution of Problem 3.

Solution. n = 2, 3, 4, 6.

Part 2 Classical Dynamics of a Crystal Lattice

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich Copyright c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

1

Mechanics of a One-Dimensional Crystal

1.1

Equations of Motion and Dispersion Law

The physics of condensed media concerns itself generally with periodic structures of lower dimensions, in particular, one-dimensional (1D). One-dimensional problems are attractive mainly because of the simplicity of the mathematics and the possibility of obtaining exact solutions in many cases, not only for small (harmonic) vibrations, but also for more complex situations in which the nonlinear (anharmonic) crystal properties may be involved. Qualitative aspects of nonlinear dynamics can be understood most easily in one-dimensional systems; therefore, these systems are still being intensely studied. Finally, some phenomena in three-dimensional crystals can be modelled by one-dimensional problems. Thus mechanics of the one-dimensional crystal can be considered as a spring-board for studying dynamics of three-dimensional structures.

In our study of 1D crystal mechanical vibrations, we focus on the existence of two physical objects each of which can be called a one-dimensional crystal. First, there are periodic linear structures in a three-dimensional space. A long macromolecule of any homopolymer is the best example. We shall call such crystals linear chain. The second object is a periodic structure that enables one to study the motion in onedimensional space. For this purpose, we should imagine the physical phenomena and processes in a “Straight Line Land: Lineland”.

We start with equations of mechanics in 1D space, i. e., we consider the onedimensional crystal itself. Consider a periodical array of particles (atoms, moleculas or other units whose internal structure can be neglected) situated along the x-axis with period a. The position of an arbitrary site is equal to

xn ≡ x(n) = na,

(1.1.1)

where n is an integer. Suppose Eq. (1.1.1) gives the coordinates of particles (“atoms”) in the equilibrium state of the crystal. A real coordinate of an atom differs from

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich Copyright c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

18 1 Mechanics of a One-Dimensional Crystal

the corresponding lattice site (1.1.1) when the atoms are displaced relative to their equilibrium positions. Such a situation appears undoubtedly during vibrations of the crystal. We denote the displacement of the atom with number n in a monatomic (with a single atom per unit cell) from its equilibrium position by u(n).

In order to write dynamic equations of motion of crystal atoms we need to describe an interatomic interaction. First restrict ourselves to the model of nearest-neighbor interaction, i. e., consider the 1D crystal to be similar to the array of small balls connected with elastic springs. Secondly take into account the fact that the relative atomic displacements being considered in crystal dynamics are small compared with a. Therefore, it is natural to begin studying the lattice dynamics with the case of small harmonic vibrations. Assuming the crystal to be in equilibrium at u(n) = 0 we can write the potential energy U in the harmonic approximation as

U = U0 +	1	∑α(un − un−1 )2 ,	(1.1.2)
	2
		n

where U0 = constant and the summation is over all crystal sites. In the simplest model the interatomic interaction is characterized by only one elastic parameter α.

With the expression for the potential energy (1.1.2) one can easily write down the equation of motion of every atom:

m	d2 u	= −	∂U	,	(1.1.3)
	dt2		∂u(n)

where m is the atomic mass. Equation (1.1.3) leads to the following dynamic equation

d2 u(n)	=	ωm	2	[u(n + 1) − 2u(n) + u(n − 1)] ,
dt2		2

where ωm2 = 4α/m.

Take the solution to (1.1.4) in the form

u(n) = ueikx(n).

(1.1.4)

(1.1.5)

The parameter k is analogous to a wave number of vibration and is regarded as a quasi-wave number.

The stationary crystal vibrations for which the displacement of all atoms are time dependent only by the factor e−iωt are of special interest. For such vibrations substituting (1.1.5) into (1.1.4), we obtain

ω2 u − 12 ωm2 (1 − cos ak)u = 0,

hence

	2		1	2	2	2 ak
ω		=		ωm (1	− cos ak) = ωm sin			.	(1.1.6)
			2				2

1.1 Equations of Motion and Dispersion Law 19

In mechanics the dependence of frequency on the wave number is called the dispersion law or dispersion relation. Thus, (1.1.6) gives the dispersion relation

ω2 = ω2 (k)

for the lattice vibrations.

From (1.1.6) we note that the dispersion law determines the frequency as a periodic function of the quasi-wave number with a period of a reciprocal lattice

ω(k) = ω(k + G), G =	2π	.
	a

This periodicity is the basic distinction between the dispersion law of crystal vibrations and that of continuous medium vibrations, since the monotonic wave-vector dependence of the frequency is typical for the latter. The difference between the quasiwave number k and the ordinary wave number is also observed in the fact that only number k values lying inside one unit cell of a reciprocal lattice (−π/a < k < π/a) correspond to physically nonequivalent states of a crystal.

When the lattice period a tends to zero, the Brillouin zone dimension becomes infinitely large and we return to the concept of momentum and its eigenfunctions in the form of plane waves.

To clarify the available restrictions on the region of physically nonequivalent k values we note that k = 2π/λ always, where λ is the corresponding wavelength. We consider, for simplicity, a one-dimensional crystal (a linear chain) with a period a for which the reciprocal lattice “vector” G = 2π/a. Choose the interval −π/a ≤ k ≤ π/a as the reciprocal lattice unit cell. The limiting value of the quasi-wave number k = π/a will then respond to the wavelength λ = 2a. It follows from the physical meaning of wave motion that this wavelength is the minimum in the crystal, since we can observe the substance motion only at points where material particles are located. A wave of this length is shown as a solid curve in Fig. 1.1 (the dark points are the equilibrium positions of particles, the light ones are their positions at a certain moment of the motion). A wave with wave number larger than the limiting one reciprocal lattice period namely, k = π/a + 2π/a = 3π/a, is shown as a dashed line. Both waves correctly reproduce the crystal motion but the introduction of the wavelength λ = 2a/3, carrying no additional information on the particle motion, is not justified physically.

We now propose a short analysis of the dispersion relation of the one-dimensional crystal. According to (1.1.6), possible vibration frequencies fill band (0, ωm) where ωm is the upper boundary of the band of possible vibration frequencies. To continue the analysis one needs to know a spectrum of quasi-wave number values inside the Brillouin zone. In order to define such a spectrum consider the one-dimensional crystal containing N atoms (N 1) and having the length L (L = Na). The spectrum mentioned depends on the boundary conditions at the crystal ends. We formulate the simplest boundary conditions supposing that the atomic chain is closed up into a ring

20 1 Mechanics of a One-Dimensional Crystal

Fig. 1.1 Profile of the wave with λ = 2a (solid line) and the wave with

3λ = 2a (dashed line).

and N+n atom coincides with the n-th atom:
u(xn) = u(xn + Na).	(1.1.7)

The cyclicity condition (1.1.7) is called the Born–Karman condition. Combining (1.1.7) and (1.1.5), we obtain

ka =	2π	p,	or k =	2π	p,	(1.1.8)
	N			L

where p = 0, 1, 2, . . . , N. Usually it is convenient to choose symmetrical conditions p = 0, ±1, ±2, . . . , ± N 2+ 1 .

Since in the macroscopic case (N 1) the discrete values of the k numbers are divided by the very small interval ∆k 1L and the spectrum of k values can be regarded as quasi-continuous. Therefore one can analyze the dispersion relation considering the frequency as a continuous function of quasi-wave number.

Let us begin from the vibrations with small k. The 1D-crystal long-wave vibrations

(ak 1) have the ordinary acoustic frequency spectrum
ω2 = s2 k2 or w = sk,	(1.1.9)

where s is a sound velocity (2s=aωm).

To describe the dispersion law at the upper edge of the band of possible frequencies (when ωm − ω ωm) it is convenient to introduce q = k − π/(2a) and consider |q| 1. Then the atom displacements assume the form

u(n) = u0 eikan = (−1)nu0 eiqan.

(1.1.10)

It is interesting to note that the neighbor atoms vibrate with the opposite phases in the limiting case ω → ωm. And the following dependence is obtained for the dispersion

law	1			a	2
ω = ωm −		γq2 ,	γ = ωm
					.	(1.1.11)
	2			2

A dispersion relation of the type (1.1.11) is known as a quadratic dispersion law.

1.1 Equations of Motion and Dispersion Law 21

According to (1.1.6) the dependence of the frequency on k is characterized by the monotonic plot in the interval (−π/a < k < π/a). However, this fact is a result of using a model of the nearest-neighbor interaction. Taking into account the interaction of the next neighbors leads to a possibility to obtain a nonmonotonic dependence of ω on k with a diagram similar to Fig. 1.2 (see Problem 1 at the end of the section). Therefore maxima inside the Brillouin zone and minima on its boundaries can appear on a graphical representation of the dispersion relation.

Fig. 1.2 One-dimensional dispersion diagram.

Having discussed the behavior of the frequence spectrum inside a band of possible free vibrations in an unbounded 1D crystal and at the edges of the band one should say that vibrations with frequencies outside of this band are possible in a bounded chain (see Problem 2 at the end of the section).

Note in conclusion that the acoustic dispersion relation (1.1.9) is a natural consequence of the long-wave approximation for an equation of motion of the crystal. Actually, in such an approximation the function u(n) of a discrete argument n can be considered as a continuous function of the argument x ≡ xn =na and the following expansion can be used

	un±1 − un = ±au +			1		a2 u + · · · ,			(1.1.12)
					2
	where	∂u(n)					∂2 u(n)
	u =		, u	=				,	(1.1.13)
		∂x					∂x2

and so on. Then (1.1.4) can be transformed into the following differential equation in

partial derivatives
	∂2 u(x)	− s2	∂2 u(x)	= 0,	(1.1.14)
	∂t2		∂x2

where x is a continuous coordinate that determines the position in a 1D crystal. Obviously, the acoustic dispersion law (1.1.6) is typical for the solutions to the wave

equation (1.1.14).

22 1 Mechanics of a One-Dimensional Crystal

Another form of a dynamic differential equation in partial derivatives is typical for vibrations with frequencies close to the upper edge of the frequency spectrum (ωm − ω ωm). Taking into account (1.1.10), in this region the following approximation

can be used
u(n) = (−1)nv(na),	(1.1.15)

where v(x) is a continuous function of x. Now, using the expansion (1.1.18) for the function v(na) = v(x), one can transform (1.1.4) into the following

	∂2 v(x)	+ ωm2 v(x) + s2	∂2 v(x)	= 0.	(1.1.16)
	∂t2		∂x2

Obviously, the dispersion relation (1.1.11) follows from (1.1.16).

Proceeding further from quasi-continuity of the spectrum of k values we change the summation over the discrete values of a quasi-wave number for the integration. Taking (1.1.8) into account it is easy to obtain the rule governing this transition to the

	integration	L
	∑ f (k) =		f (k) dk,	(1.1.17)
		2π
	k

where the integration is carried out over the interval of a single unit cell in k-space (or the Brillouin zone).

Having analyzed the spectrum of eigenvalues of the crystal vibrations let us consider a set of eigenfunctions of this problem. The crystal eigenvibrations (1.1.5) are

numbered by k. Introduce normal vibrations in the form
1			eikna,
φk(n) =	√			(1.1.18)
		N
which provides the normalization condition
∑φk (n)φk (n) = δkk .				(1.1.19)
n

The normal eigenvibrations (1.1.19) are often called the normal modes of the vibrations.

The set of normal modes allows us to construct easily the so-called Green function for crystal vibrations. According to the definition the Green function for vibrations of the unbounded chain in the site representation has the form

Gω2 (n, n ) = ∑

φ (n)φk (n )

1

∑

eika(n−n )

k

=

.

(1.1.20)

2

2

(k)

N

ω

2

2

(k)

k

ω

− ω

k

− ω

The function Gε(n) where ε = ω2 is called the Green function of stationary crystal

	vibrations. Expression (1.1.20) can be rewritten as
	Gε(n) =	1	∑G(ε, k)eikna;		(1.1.21a)
		N
			k
	G(ε, k) =		1	,	(1.1.21b)
		ε − ω2 (k)

determining the Green function in the (ε, k) representation.

1.1 Equations of Motion and Dispersion Law 23

It is not difficult to calculate the Green function for the vibrations with the dispersion law (1.1.6):

Gε(n) =		i	e−iakn,	n < 0;
				(1.1.22)
		ε(ωm2 − ε)	eiakn,
				n > 0,

where ak = 2 arcsin(√ε/ωm) = 2 arcsin(ω/ωm).

1.1.1

Problems

1. Find the dispersion law for a 1D crystal vibrations taking into account interactions between nearest neighbors and next-but-one neighbors. Find a possibility of a nonmonotonous dependence ω = ω(k) within the interval 0 < ak < π.

Solution. The required dispersion relation

ω

2

=

4α1

sin

2 ak

+

4α2

sin

2

ak,

(1.1.23)

m

2

m

where α1 and α2 are parameters of the elastic interaction between the nearest and nextnearest neighbor atoms, respectively. A required nonmonotonic dependence appears under the condition 4 |α2 | > α1 and the plot of the (1.1.18) becomes similar to Fig. 1.2.

2. Find the wave number values for the frequencies exceeding the maximum frequency of harmonic 1D crystal vibrations with interaction of nearest neighbors only. Interpret the result.

Hint. Proceeding from the fact that in (1.1.4) ω is real, find the complex k values corresponding to ω > ωm.

Solution. Complex values
k = ±π/a + iκ ,			(1.1.24)
where κ is determined by	aκ
ω = ωm cosh
		.	(1.1.25)
	2

The solutions exponentially decreasing (or increasing) with the distance can describe the vibrations of a bounded 1D crystal that are localized near its free edge and not penetrating inside.

3. Find the Green function for stationary vibrations, accounting for the interactions of not only the nearest neighbors, when ω = ω(k) is a nonmonotonic function in the interval 0 < k < π/a.

241 Mechanics of a One-Dimensional Crystal

1.2

Motion of a Localized Excitation in a Monatomic Chain

It is customary to think that an excitation in the crystal moves with the sound velocity. What meaning can be associated with such an assertion? A concept of “sound” is connected with the acoustic vibrations, i. e., with the long-wave approximation of the crystal dynamics. Motion of small vibrations in the long-wave approximation is

described by the wave equation (1.1.14). Any function of the argument x − st is the solution to (1.1.14), and an arbitrary perturbation u = u(x − st) with small enough gradients a(∂u/∂x) 1 moves with velocity s along the 1D crystal without chang-

ing its shape. This is a consequence of the fact that the dispersion relation (1.1.9) is dispersionless, i. e., it determines a wave phase velocity independent of k or ω.

The dispersion law (1.1.6) does not possess such a property and the phase velocities of corresponding waves depend on k, and a localized excitation should expand moving along the crystal. The unique stationary solution to (1.1.4) not being deformed when moving along a 1D crystal is the harmonic wave whose frequency and wave number are related by the dispersion relation (1.1.6). However, it is said generally that a perturbation in a crystal moves with sound velocity, with the character of the perturbation remaining unspecified. We shall clarify the meaning of such an assertion.

We analyze (1.1.4) that coincides with the recursion relation for a Bessel function of the first kind Jν(z):

d2 Jν(z) = 1 [ Jν+2(z) − 2 Jν(z) + Jν−2(z)] . dz2 4

It is clear that the solution to (1.1.4) can be expressed directly through Bessel functions. Assuming a 1D crystal of infinite length (−∞ < n < ∞) and taking into account the boundedness of displacements, we can write

un(t) = const · J2(n−p)(ωm t),

(1.2.1)

where p is an arbitrary integer and n an independent integer.

If the initial displacements u0n and the initial velocities v0n of all atoms are given at t = 0 the corresponding solution to (1.1.4) for t > 0 reads as

∞	∞	t
		J2(n−p)(ωm τ) dτ.
un(t) = ∑ u0p J2(n−p)(ωm t) +	∑ v0p		(1.2.2)
p=−∞	p=−∞	0

We assume that at the initial time only one atom is displaced from the equilibrium position (u0n = unδn0 , vn = 0). It then follows from (1.2.2) that

un(t) = u0 J2n(ωm t), t > 0.

(1.2.3)

The perturbation (1.2.3) has no pronounced propagation front even far from the onset (n 1). But it follows from the properties of Bessel functions that the first