Kosevich A.M. The crystal lattice

Подождите немного. Документ загружается.

4.4 Dependence of Frequency Distribution on Cr ystal Dimensionality

137

The Distribution Functions of 2D Crystal Vibrations. The distribution functions of

2D crystal vibrations have singularities similar to 3D crystal singularities. They are at

the edges of the eigenfrequency spectrum and coincide with the frequencies that divide

the regions of closed and open isofrequency curves (Fig. 4.3). Full analysis of the sin-

gularities in a 2D case is little different from th a t presen ted in Section 4.3. Therefore

we restrict ourselves only to a qualitative ch aracterization of the singularities, empha-

sizing their difference from those in a 3D crystal. A low-frequency (long-wave) limit

of the dispersion law for each branch in the isotropic approximation is

ε = s

= s

+ k

). (4.4.1)

The behavior of the density of vibrational states g(ε) for ε → 0 is estimated by

(4.4.1) using the following chain of equations

g(ε)dε =

(2π)

k =

(2π)

∼ a

kdk∼





dε ,

where S

is the unit cell area of a 2D lattice. Thus, in the two-dimensional case

g(ε) → g(0)=const ∼





at ε → 0. (4.4.2)

Taking into account the relation between the two frequency distribution functions

(4.2.5), we see that

ν(ω) → 2Nωg(0) ∼ N





ω as ω → 0. (4.4.3)

The behavior of the density of states (4.4.2) and the frequency spectrum (4.4.3) in

the extremely low-frequency region differs from (4.3.2) and (4.3.3) for a 3D crystal.

As follows from (4.3.4) near the edge of the high-frequency spectrum we can write

ε = ε

− γk

, to obtain g(ε)dε ∼ a

kdk ∼





dε. Consequently, the limiting

behavior of the vibration density is analogous to (4.4.2): g(ε) → g(ε

)=constant ∼





at ε → ε

, i. e., the function g(ε) has a break at the point ε = ε

,since

g(ε) ≡ 0 at ε > ε

. This singularity of the vibration density also differs from the

dependence (4.3.6) that is typical for a 3D crystal.

We examine the vicinity of the frequency ω

corresponding to the analog of a con-

ical point in a 3D crystal, a point near which the isofrequency cu rves differ little fro m

hyperboli (Fig. 4.8a). The dispersion law near the frequency ω

can be written, similar

to (4.1.5), as

ω = ω

+ γ

−γ

, γ

> 0. (4.4.4)

We repeat the reasoning used in calculating the functions g(ε) and ν(ω) near the

frequency ω

in a 3D cry stal. Let us draw two straight lines k

= ±Q that cut off part

of the space near the point we are interested in and calculate a 2D volume (the area Σ

138

4 Frequency Spectr um and Its Connection with the Green Function

in a 2D k-space) limited by these straight lines and the curve ω( k)=ω = constant.

We set for deﬁniteness γ

> 0 and γ

> 0.Forω < ω

(curve 1 in Fig. 4.8a), we

then get

(ω)=



= 4







− k

√

Q/k





−1 dy ,

(4.4.5)

where k

=(ω

− ω)/γ

Near the singular point, when k

 Q we have from (4.4.5):

(ω)=2





−k



. (4.4.6)

For ω > ω

(curve2inFig.4.8a)

(ω)=4







+ k

√

Q/q





+ 1 dy ,

where q

=(ω − ω

)/γ

and, thus, near the singular point (q

→ 0), we have

(ω)=2





+ q



. (4.4.7)

The terms in (4.4.6), (4.4.7) that lead to a singular behavior at |ω − ω

|→0 are

(ω)=2





−

− ω

log

√

−ω



, ω < ω

;

(ω)=2





ω −ω

log

√

ω − ω



, ω > ω

(4.4.8)

Since the frequency spectrum is

ν(ω)=

(2π)

dΣ

dω

where S is the crystal area, from (4.4.8) we directly derive the logarithmic singularity

at ω → ω

ν(ω)=

(2π)

√

log



|ω − ω



. (4.4.9)

4.4 Dependence of Frequency Distribution on Cr ystal Dimensionality

139

Fig. 4.8 Singularities of the frequency distribution in a 2D crystal:

(

) isofrequency lines near the critical point of the 2D dispersion law;

(

) density of states in a 2D lattice with nearest-neighbor interaction.

Using (4.2.5), we see that the vibration density of a 2D crystal g(ε) also has a

logarithmic singularity at ε = ε

= ω

and its singular part is symmetric in the

vicinity of this point:

g(ε)=

(2π)

√

log



|ε −ε



1/2

. (4.4.10)

As indicated in Chapter 3, there should exist no less than two such singular points.

If γ

= γ

two singular points merge to form one singularity.

For the simplest models the vibration density of a 2D crystal can be calculated

explicitly, e . g., in a scalar model for a crystal lattice with interaction of nearest neigh-

bors only. The dispersion law of vibrations is simple in this case and, for a rectangular

lattice with period a along the x-axis and period b along the y-axis, can be written as

ε = ω

sin

+ ω

sin

. (4.4.11)

We write the deﬁnition of the de nsity of vibration states

g(ε)=

(2π)

dε



(2π)

dε



(ε, k) dk, (4.4.12)

where k

(ε, k

) is the solution to (4.4.11) considered as an equation relative to k

The 2D integral in (4.4.12) is calculated in the area limited by the isofrequency curve

ε = constant and situated in one unit cell of the k-plane. The integration limits of

the last integral in (4.4.12) depend on whether the isofrequency curve intersects the

boundaries of the reciprocal lattice unit cell. If such a curve is closed and does not

intersect the unit cell boundaries the integration limits are obtained from the condition

140

4 Frequency Spectr um and Its Connection with the Green Function

(ε, k)=0. The change of the integration variables in (4.4.12) results in

g(ε)=

(πω

)

(ε)



−x

(ε)



(ξ

−sin

x)(η

−ξ

+ sin

, (4.4.13)

where ξ

= ε/ω

, η

=(ω

/ω

)

and sin x

= ξ.

We know that the isofrequency curve is closed at frequencies that are near edges of

the vibration spectrum. Thus, (4.4.13) describes the behavior of the vibration distri-

bution in these parts of the spectrum. The density of states g(ε) is shown in the form

of a total elliptic integral of the ﬁrst kind K(k):

g(ε)=





ε(ε

− ε)



, (4.4.14)

where ε

= ω

+ ω

.AsK(0)=π/2, (4.4.14) agrees with (4.4.2). Calculating

the vibration distribution in the middle part of the eigenfrequency spectrum, we set

for deﬁniteness η > 1. Then for ω

< ε < ω

the isofrequency curve intersects

the reciprocal lattice unit cell boundaries k

= ±π/a, and the density of states is

determined by (4.4.13) with x

= π/2. Simple calculation shows that

g(ε)=



ε(ε

− ε)





ε(ε

−ε)



. (4.4.15)

We note that (4.4.5) and (4.4.15) are symmetrical relative to permutation of ω

and

. Thus, their applicability conditions are: (4.4.14) is valid for 0 < ε(ε

− ε ) <

, and (4.4.15) – for ε(ε

− ε) > ω

(Fig. 4.8b).

The elliptic integral K(k) has a logarithmic singularity at the point k = 1.This

singularity is typical for the function g(ε) at ε = ω

and ε = ω

. The behaviors of

the functions (4.4.14), (4.4.15) near these points can be considered as a partial case of

the general relation (4.4.10).

One-Dimensional Structure. In the case of a linear chain where only the nearest

neighbors interact, an analog of (4.4.11) is the dispersion law

ε = ε

sin

, (4.4.16)

where ε

is the upper-band boundary of the possible frequency squares. It is important

that the dependence of the frequency ω on k given by (4.4.16), determines a monotonic

function in the range of values of the quasi-wave vector (0, π/a) . This means that in

the internal points of the interval (0, ω

), the frequency distribution functions ν(ω)

and g(ε) of a 1 D crystal with the dispersion law (4.4.16) have no singularities. Indeed,

from the deﬁnition of the vibration density g(ε) and (4.4.16) it follows that

g(ε)=



ε(ω

− ε)

. (4.4.17)

4.5 Green Function for the Vibration Equation

141

Thus, the spectral characteristics of a 1D crystal differ from those of a 3D (and 2D)

one in that they have no singularities inside the continuous spectrum band. At the

ends of the continuous spectrum interval the singularities are more pronounced than

those of 3D and 2D crystals. It is clear from (4.4.17) that for ε → ε

∗

,whereε

∗

is a

continuous spectrum edge,

g(ε) ∼



|ε − ε

∗

. (4.4.18)

The frequency spectrum function of a 1D crystal with the nearest-neighbor interac-

tions is

ν(ω)=



−ω

, (4.4.19)

where N is the number of atoms in the chain.

If the dispersion law of a lD crystal is a nonmonotonic function ω(k) such as in

Fig. 4.9a the spectral characteristics of 1D crystal vibrations at the points ω = ω

and ω = ω

, will have distinctive singularities. Indeed, suppose that near k = k

and

ω = ω

the dispersion law is then given by

= ω

− γ

(k − k

)

The contribution of this part of the dispersion law to the density of states is then

easily determined to be

δgε =

dε











2πγ

√

− ε

, ε < ε

;

0, ε > ε

Similarly, if we write the dispersion law near the points k = k

and ω = ω

the form of ω

= ω

− γ

(k − k

)

, the corresponding contribu tion to the vibration

density is

δgε =











0, ε < ε

= ω

;

2πγ

√

ε − ε

, ε > ε

A plot of the density of states for a 1D crystal in the case when the interval (0, ω

)

has two singular points ε = ε

(maximum) and ε = ε

(minimum) is given in Fig. 4.9b.

4.5

Green Function for the Vibration Equation

The singularities of the spectrum of crystal vibrations can be analyzed, apart from

the geometric method, by the analytical one, which is based on the mathematical

structure of the equations of crystal lattice motion. The so-called Green dynamic

142

4 Frequency Spectr um and Its Connection with the Green Function

Fig. 4.9 Spectrum of vibrations of a 1D crystal: (

) dispersion curve;

(

) singularity of the density of states.

function is used for this purpose. It is deﬁned as follows. The equations of free

vibrations of a monato mic crystal lattice (1.1.12) are represented as

u(n, t)

∑



A(n − n



)u(n



, t)=0. (4.5.1)

The Green function for unbounded crystal vibrations or the Green function of

(4.5.1) is called a solution to an inhomogeneous equation vanishing at inﬁnity

(n, t|n

∑



(n − n



, t|n

)=−mδ

δ(t)δ

, (4.5.2)

where δ(t) is th e Dirac δ-function.

By virtue of space homogeneity of an unbounded crystal, the solution to the equa-

tion d epends on the difference n − n

: G(n, t|n

)=G(n − n

, t). The Green func-

tion of any linear equation allows one to ﬁnd a partial solution to an inhomogeneous

equation describing the induced motion of the system concerned. If the motion of

crystal atoms is determined by the force mf

(n, t):

(n, t)+

∑



(n − n



, t)=mf

(n, t), (4.5.3)

the partial solution to this equation is

(n, t)=−

∑





(n − n



, t −t



) f



, t) dt



. (4.5.4)

However, use of the Green function is not limited to (4.5.4) alone. The Green

function gives rich information on the properties of a system whose free motion is

described by a homogeneous equation (4.5.1).

In a scalar model, the Green function G(n, t) is a solution to the equation

G(n, t)+

∑



α(n − n



)G(n



, t)=−δ(t) δ

. (4.5.5)

4.5 Green Function for the Vibration Equation

143

We represent G(n, t) as the Fourier integral

G(n, t)=

2π

+∞



−∞

(n)e

−iωt

dω;

(n)=

+∞



−∞

G(n, t)e

iωt

dt,

(4.5.6)

and call the function G

(n) ,whereε = ω

, the Green function of stationary crystal

vibrations. This function is deﬁned as an appropriate solution to the equation with

behavior at inﬁnity

εG

(n) −

∑



α(n − n



)=δ

. (4.5.7)

The Green function of stationary vibrations is dependent on ε, i. e., it depends only

on the frequency squared. Thus, judging from (4.5.6), (4.5.7), the frequency sign

in the argument of the time Fourier transformation of the Green function is of no

importance. However, in fact this is n ot so. By an a lyzing (4.5.1), (4.5.7), it is easy

to obtain the Green function for the vibrations of an ideal crystal. We perform the

following Fourier transformation of the function G

(n)

(n)=

∑

G(ε, k)e

ikr (n)

;

G(ε, k)=

∑

(n)e

−ikr (n)

(4.5.8)

determining the Green func tion in (ε, k) representation. We now perform an analo-

gous transformation of (4.5.7)



ε − ω

(k)



G(ε, k)=1, (4.5.9)

where, as before, the function ω(k) gives the dispersion law of a crystal. It follows

from (4.5.9) that the Fourier components o f the Green function are

G(ε, k)=

ε − ω

(k)

. (4.5.10)

The fo r mu la (4.5.1 0 ) determines th e Green fu nction in (ε, k) representation. For

some methods of theoretical calculation of ideal crystal properties, this representation

is fundamental. As (4.5.10) is important, we write it in the general case when various

vibration branches in a crystal exist

(ε, k)=

∑

α=1

(k, α)e

(k, α)

ε − ω

(k)

. (4.5.11)

144

4 Frequency Spectr um and Its Connection with the Green Function

The form of the functions (4.5.10), (4.5.11) and, in particular, the appearance of

characteristic denominators is not speciﬁc only for the crystal vibrations but reﬂects

the general features of the Green function for systems with collective excitations. We

thus analyze the function G(ε, k).

The function G(ε, k) regarded as a function of the variable ε has a pole at the point

ε = ω

(k) , i. e., at a point where ω coincides with the frequency o f one, the eigen-

vibrations. We, therefore, arrive at the following important property of the Green

function. The poles of the Fourier components of the Green function are determined

by the spectrum of crystal eigenfrequencies or, in other words, by its dispersion law.

Using (4.5.8), we obtain the Green function for stationary vibrations (in a scalar

model) in the form

(n)=

∑

ikr (n)

ε − ω

(k)

. (4.5.12)

Assuming quasi-continuity of the spectrum of the k-vector values we may rewrite

(4.5.12) in the form of an integral:

(n)=

(2π

)



ikr (n)

ε − ω

(k)

, (4.5.13)

where V

is the unit cell volume.

The generalization of (4.5.12) is obvious in view of (4.5.11):

(n)=

∑

k,α

(k, α)e

(k, α)

ε − ω

(k)

ikr (n)

. (4.5.14)

Changing (4.5.14) to an integral form is accomplished analogously to the transfor-

mation from (4.5.12) to (4.5.13).

We now return to a scalar model. If the value of the parameter ε does not get into

the band of crystal eigenfrequency squares (in our case ε > ω

) the formula (4.5.13)

unambiguously determines a certain function n dependent on the parameter ε.

However, the case 0 < ε < ω

, i. e., when the frequency ω is in the continuous

spectrum interval, is more interesting. As the Fourier components of the Green func-

tion have a pole, the integral (4.5.13) is meaningless (it diverges). More exactly, it

is senseless in its literal interpretation when the parameter ε is considered to be real.

However, this singularity of the vibrational system behavior is typical for any reso-

nance system when damping (dissipation) of the eigenvibrations is neglected, and it

results in inﬁnitely large amplitudes of vibrations as soon as the frequency of the ex-

citation force coincides with one of the eigenfrequencies of the system. It is known

how to overcome this difﬁculty. It is necessary to take into account at least small

damping of the eigenvibration existing in the system. For unbounded systems with

distributed parameters, whose eigenvibrations have the form of waves with a contin-

uous frequency spectrum, equations such as (4.5.13) can be regularized by choosing

special conditions at inﬁnity. Formally, this reduces to the fact that the parameter ε

4.6 Retarding and Advancing Green Functions

145

should be regarded as complex rather than real, but with a vanishingly small imaginary

part. Adding even a small imaginary part to ε removes the divergence of the integral

(4.5.13), but generates some new problems

In discussing the general properties of the function G(ε, k ), we focus on one purely

mathematical point. Let us represent the equation for stationary vibrations (1.1.17) in

the matrix (operator) form εu −

Au = 0. Equation (4.5.7) for the Green function

of stationary vibrations can be written in the same style: εG −

AG = I,whereI is

the unit operator.

The solution to this equation can be easily found

G(ε)=



εI −



−1

, (4.5.15)

if we introduce an inverse operator notation (the operator M

−1

is inverse to M).

The Green function in the matrix (or operato r) form (4.5.15) is sometimes called a

resolvent.

The formula (4.5.6) does not specify the representation where the Green function

is written , but it implies that the matrix G(ε) is diagonal in a representation in which

the operator A is diagonal. For a vibrating crystal this is a k-representation where

(4.5.10), (4.5.11) are derived.

4.6

Retarding and Advancing Green Functions

We return to (4.5.6), substitute (4.5.13) in it, and write the Green function as a

function of time

G(n, t)=

(2π)



ikr (n)−iωt

− ω

(k) −iγ

dωd

k. (4.6.1)

To regularize the integral (4 .6.1) we h ave just explicitly separated a small imaginary

component iγ .

Let us consider only the frequency transformation in (4.6 . 1), resulting in the integral

(t)=

∞



−∞

−iωt

dω

−ω

(k) −iγ

. (4.6.2)

The coordinate representation of the Green function is determined by an obvious

transformation involved in (4.6.1)

G(n, t)=

(2π)



(t)e

ikr (n)

k. (4.6.3)

2) In particular, the Fourier component of the Green function becomes depen-

dent on the sign of ω.

146

4 Frequency Spectr um and Its Connection with the Green Function

The integral (4.6.2) is easily calculated by the residues method. The integration

should be performed over a closed path consisting of the real axis of the variable ω

and an arc of the inﬁnitely remote semicircle. As the convergence o f the integral and

its vanishing for an inﬁnite semicircle is provided by the exponential factor, for t < 0

we should close the path above the real axis, and for t > 0 below it. As a result,

for t < 0 the integral (4.6.2) is determined by residues in the upper half-plane of a

complex variable ω,andfort > 0 this integral equals the sum of residues in the lower

half-plane.

Since the imaginary shift in the complex plane ω is determined by p arameter γ,

the integration result in (4.6.2) is determ ined completely by the sign of the param eter

γ. The relation between th e sign s of γ and ω is then very important. In (4.6.1) and

(4.6.2), no assumptions were made with regard to the sign o f γ, so that the ambiguity

in choosing the dependence of the sign of γ on the sign of ω indicates the a mbiguity

in determination of the Green function by (4.6.1).

Let us assume that the sign of γ changes together with the sign of the frequency,

i. e., we admit γ = γ

sign(ω). The Fourier component of the Green function for

which a new notation G(ω, k) was used, to avoid confusion with (4.5.10), then takes

the form

G(ω, k)=

− ω

(k) −iγ sign(ω)

. (4.6.4)

It is readily seen that for γ

< 0 (4.6.4) determines a function of the complex

variable ω analytic in the upper semi-plane and at γ

> 0, in the low semiplane.

Integrating in ( 4.6.2), we ﬁnd the time dependence of the Green functions generated

by the Fourier components (4.6.4). For γ

< 0,wehave

(t)=











0, t < 0;

−

sin [ω(k)t]

ω(k)

, t > 0.

(4.6.5)

By using the transformation (4.6.3), we get a retarding Green function G

(n, t).

This function describes the perturbation arising at any crystal point, with an instan-

taneous force applied at the coordinate origin. At inﬁnity, it corresponds to waves

diverging from the origin.

When γ

> 0,wehavetodetermine

(t)=











−

sin [ω(k)t]

ω(k)

, t < 0;

0, t > 0

(4.6.6)

to derive by means of (4.6.3) an advancing Green function G

(n, t). The advancing

Green function describes the crystal perturbations for t < 0 that generate a force con-

centrated at the origin at time t = 0. At inﬁnity, it corresponds to waves converging to