Kosevich A.M. The crystal lattice

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1.7 Soliton as a Particle in 1D Crystals

Within the main approximation for small perturbations, we can substitute

u = u

(x − ζ) on the r.h.s. of (1.7.10). ∂u/ ∂x = −∂u/∂ζ, and we obtain

= −

∂W

∂ζ

, W(ζ)=−

∞



−∞

f (x)u

(x − ζ) dx. (1.7.11)

Equation (1.7.11 ) has the form o f one of the Hamilton equations and W(ζ) plays

the role of the soliton potential energy in an external ﬁeld. If the force density f (x) is

of purely elastic origin, it can be written as f (x)=∂σ

/∂x,whereσ

are the elastic

stresses created in a 1D crystal by external loads. In this case

W(ζ)=−

∞



−∞

∂σ

∂x

dx = aσ

(−∞)+

∞



−∞

(x)u

(x − ζ) dx. (1.7.12)

The ﬁrst term on the r.h.s. of (1.7.12) is a constant that may not be taken into account

in writing the potential energy of a soliton W(ζ).

We assume further that the ﬁeld of stresses σ

(x) varies smoothly in space and the

characteristic distance of its variation is much larger than l (soliton width). Then we

may replace (1.7.12) by

W(ζ)=σ

(ζ)

∞



−∞

∂u

∂x

dx = aσ

(ζ). (1.7.13)

Comparing (1.7.11), (1.7.13) we conclude that the force acting on a soliton is deter-

mined by the gradient (space derivative) of elastic stresses created by external loads at

the point where the soliton center of mass is located.

On the other hand, the relation (1.7.13) makes it possible to generalize the expres-

sion for the Hamilton fun c tion that, for small soliton momenta, takes the form

H(ζ, P)=E

∗

+ aσ

(ζ). (1.7.14)

Using (1.7.14), a standard pair of Hamilton equations is constructed

= −

∂H

∂ζ

dζ

∂H

∂P

. (1.7.15)

The equations for soliton motion thus take the form of equations of the motion of

a particle with mass m

∗

and effective “charge” a, reﬂecting its interaction with the

elastic potential ﬁeld, where σ

(x) is the ﬁeld p otential.

The soliton, a collective excitation of a 1D crystal, has very different properties

from the collective small harmonic vibrations of a crystal lattice. The primary differ-

ence is that the soliton is generated by the nonlinear dynamics of a 1D crystal. Thus,

the usual superposition principle is inapplicable to solitons. One may expect that this

1 Mechanics of a One-Dimensional Crystal

restricts the possibility of using solitons to describe the excited states of a crystal.

Problems may arise when we try to consider many solitons in the crystal or the inter-

action between soliton perturbation and small harmon ic vibrations of the crystal.

The remarkable properties of (1.6.9) eliminate these problems. First, the asymp-

totic superposition principle is applicable to solitons. If at time t = 0 two solitons

with velocities V

and V

(assume them moving in the opposite directions) exist in a

1D crystal, then at t → ∞ the same two solitons with velocities V

and V

will remain

in the crystal. In other words, the asymptotic behavior of soliton solutions to the non-

linear equation (1.6.9) is analogous to the independent behavior of the eigensolutions

to a linear equation.

Then, the asymp totic superposition principle is valid also for the interaction of col-

lective excitations of different types: solitons and small harmonic vibrations. Without

going into details of the nonlinear mechanics to justify the ﬁrst remark, we illustrate

the validity of the second one by studying the properties of small harmonic v ibrations

in a 1D crystal containing one crowdion ( soliton).

1.8

Harmonic Vibrations in a 1D Crystal Containing a Crowdion (Kink)

We use the simplest (linear) perturbation theory to describe how a harmonic vibration

moves through a soliton (1.6.12). We assume a soliton to be at rest (results for a

moving soliton can be obtained by means of th e Lorentz transformation) and represent

the soliton to (1.6.9) in the form

u = u

+ u

, u

= 4arctan



exp



−



. (1.8.1)

We substitute (1.8.1) into (1.6.9) and linearize the equation obtained in u

.Then

−s

+ ω



1 − sech





= 0.

We seek for the solution to this linear equation in a standard form u

= ψ(x)e

−iωt

For the function ψ(x) we have

ψ = −s

+ ω



1 −sech





ψ. (1.8.2)

Equation (1.8.2) is a Schrödinger stationary equation with a reﬂectionless potential.

One localized eigenstate is among its solutions

loc

(x)=

sech





, (1.8.3)

1.8 Harmonic Vibrations in a 1D Crystal Containing a Crowdion (Kink)

corresponding to a zero frequency (ω = 0), and a set of harmonic vibrations of the

continuum spectrum

(x)=

√

2π

ω(k)



+ i tanh





ikx

, (1.8.4)

with the dispersion law (1.6.8).

Formula (1.8.3) gives a translational mode in the linear approximation

loc

(x)=

(x)

It reﬂects the homogeneity of a 1D crystal and the possibility to choose arbitrarily the

position of the cen ter of gravity of a soliton. Indeed, the linear approximation gives

(x)+δxψ

loc

(x)=u

(x)+δx

(x)

∼

(x + δx) .

The solutions (1.8.4) describe the harmonic vibrations with the background of a

crowdion at rest. Their form conﬁrms the asymptotic superposition principle accord-

ing to which the independent eigenvibrations are only slightly modulated near the

soliton center and change insigniﬁcantly. Each eigenvibration is still characterized by

the wave number k and the dispersion law (the vibration frequency dependence on k)

does not change.

A set of functions (1.8.3), (1.8.4), as a set of eigenfunctions of the self-adjoint

operator (1.8.2), forms a total basis in the space of functions of the variable x.This

is the most natural basis for the representation of p erturbations of a soliton solution,

as it allows one to give a clear physical interpretation. A translational mode describes

the motion of a soliton mass center, and the continuum spectrum modes refer to the

change in its form and the resulting “radiation” of small vibrations.

It follows from (1.8.4) that on passing through a soliton, the eigenvibration repro-

duces its standard coordinate dependence ∼ e

ikx

, but the vibration phase η(x, k) is

shifted by

= η(+∞, k) −η(−∞, k)=π −2arctan(kl

). (1.8.5)

The phase shift (1.8.5) affects the vibration density in a 1D crystal. In the absence

of a soliton, the expression for the density of states in the specimen of the ﬁnite dimen-

sion L follows from the requirement kL = 2πn, n = 1, 2, 3 . . ., that is a consequence

of cyclicity conditions (1.1.8) for the eigenvibrations. In the presence of a soliton an

additional phase shift (1.8.5) results in a change in the allowed wave-vector values:

kL + η(k)=2πn, n = 0, 1, 2, . . . . In the limit L → ∞, the spectrum of the values of

k becomes continuous. The vibration density (the distribution function for the wave

vector k) then equals

ν(k)=

2π

dη(k)

2π

−

1 +(l

k )

, (1.8.6)

1 Mechanics of a One-Dimensional Crystal

where L/2 π is the vibratio n density in the absence of a soliton. It is easily seen that

∞



−∞



ν(k) −

2π



dk = −1.

The number of plane-wave vibrations (with a continuous frequency spectrum) scat-

tered by the soliton is reduced by un ity because of the presence of one local state with

a discrete frequency ω = 0 (translational mode).

Analyzing the changes in the vibration distribution, it is of interest to ﬁnd the fre-

quency spectrum function ν(ω) in a crystal with a soliton. Since the dispersion law

(1.6.8) does not change in the presence of a soliton, the function ν(ω) is found by

multiplying (1.8.6) by dk/dω:

ν(ω)=ν

(ω) −



− ω

= ν

(ω)



1 −





, (1.8.7)

where ν

(ω) is the frequency spectrum in the absence of a soliton

(ω)=

2πs



−ω

, ω > ω

If a soliton moves with velocity V, the formula for the phase shift is obtained from

(1.8.5) by replacing l

→ l(V)=l



1 − (V/s

)

= π − 2arctan







1 −









. (1.8.8)

Finally, if there are several solitons (crowdions with different velocities) in the sys-

tem, the total phase shift of an eigenvibration is equal to the sum of the phase shifts

for each soliton

= Nπ −2

∑

α=1

arctan







1 −









, (1.8.9)

where V

is the soliton velocity with number α, N is the number of solitons. It be-

comes clear that a set of kinks and small harmonic vibrations can be considered as

“nonlinear normal modes”. It follows from the “asymptotic independence” of these

modes that the energy is the sum of soliton energies (1.7.1) and the energy of small

vibrations with the den sity of states determin ed by (1.8.6), (1.8.9).

1.9 Motion of the Crowdion in a Discrete Chain

1.9

Motion of the Crowdion in a Discrete Chain

According to the classical equations of crowdion motion, the crowdion Hamiltonian

function (1.7.6) is independent of the position of its center and this is a result of the

continuum approximation in which this function is d erived.

The simplest way to take into account the discreteness of the system concerned is

the following. We use the solution (1.6.12) to the continuous differential equation

(1.6.9) a nd calculate the static energy of a d iscrete atomic chain with a crowdion at

rest by using the formula

E =

∞

∑

n=−∞





n+1

− u

)

+ f (u



, (1.9.1)

where u

= u(x

) ≡ u(an).

It is easily seen that the static crowdion energy in the continuum approximation is

equally divided between the interatomic interaction energy and the atomic energy in

an external ﬁeld. It can be assumed that the same equal energy distribution remains in

a discrete chain, and instead of (1.9.1) we then write

E = 2

∞

∑

n=−∞

F(u

)=τ

∞

∑

n=−∞

sin



πu



We substitute here (1.6.12), assuming the point x = x

to be a crowdion center:

E = τ

∞

∑

n=−∞

sech



an − x



, (1.9.2)

where l

= as

√

m/(πτ )=a

√

/(πτ).

Using the Poisson summation formula,

∞

∑

n=−∞

f (n)=

∞

∑

m=−∞

+∞



−∞

f (k)e

2πimk

dk , (1.9.3)

we ﬁnd

E =

+∞



−∞

cosh





2τ

∞

∑

m=1

2πim(x

/a)

+∞



−∞

2πim(x/a)

cosh





dx. (1.9.4)

The ﬁrst term in (1.9.4) coincides with the energy E

of a crowdion at rest found

in the continuum approximation (1.7.1), and the second term is a periodic function of

1 Mechanics of a One-Dimensional Crystal

the coordinate x

with period a: E = E

+ U(x

),where

U(x)=

∞

∑

m=1

2πim(x/a)

+∞



−∞

cos



2πml



cosh

dξ

= 4α

∞

∑

m=1

sinh





cos



2πm



(1.9.5)

Sincewehaveassumedthatl

 a, it then sufﬁces to keep one term with m = 1

U(x

)=U

cos



2π



, U

= 8α

−π

/a)

. (1.9.6)

It is clear that the crowdion energy is periodically dependent on the coordinate of

its center x

that may be regarded as a quasi-particle coordinate. We set x

= Vt

in (1.6.15) using the ordinary relation between the coordinate and the velocity at

V = const = 0. The part of the energy (1.9.6) that is dependent on the coordi-

nate plays the role of the crowdion potential energy. Minima of the potential en-

ergy determine possible equilibrium states of the crowdion (x

= a/2 ± na,where

n = 0, ±1, ±2,...), an d the crowdion can oscillate relative to these equilibrium posi-

tions with the frequency

Ω



2π



∗

Vibration motion with such a freq uency can really be a free harmonic oscillation if

the quantum energy of the ground (zero) state of the oscillator ¯hΩ

is much smaller

than U

¯h

∗

 U

. (1.9.7)

Under such a condition the crowdion in a discrete structu re possesses an internal vi-

bration degree of freedom with the frequency Ω

However, in the case l

 a, the potential energy curve (1.9.6) creates very weak

potential barriers between the neighboring energy minima, and the crowdion may

overcome them through quantum tunneling. Thus, the crowdion migrates really in

a discrete atomic chain overcoming the potential relief (1.9.6).

The Hamiltonian

H = E

∗

+ U

cos



2π



(1.9.8)

is used for the quantum description of crowdion motion.

As both m

∗

and U

decrease with incr easing parameter l

/a ∼ a

√

/τ the phys-

ical situations, where

¯h

∗

 U

, (1.9.9)

1.10 Point Defect in the 1D Crystal

are quite reasonable. The inequality (1.9.9) means that the potential energy contribu-

tion that is dependent on the coordinate is a weak perturbation o f the kinetic energy

of free crowdion motion. In other words, the amplitude of zero crowdion vibrations in

one of the potential minima (1.9.9) greatly exceeds much the one-dimensional crystal

period and the crowdion transforms into a crowdion wave.

The energy spectrum of a crowdion wave with the Hamiltonian (1.9.7) is rather

complicated and consists of many bands in each of which the energy is a periodic

function of the quasi-wave number k with period 2π/a. However, small crowdion

wave energies for k  2π/a are not practically distinguished from the free particle

energy with the Hamiltonian (1.9.8) under the condition (1.9.9). I ndeed, if we calcu-

late quantum-mechanical corrections to the free particle energy in the second order of

perturbation theory in the potential (1.9.6), then

ε(k)= E

−U

∗

(2π¯h)

¯h

M = m

∗



1 +



∗

(π¯h )





Thus, in spite of the presence of a potential energy curve (1.9.6), the crowdion wave

moves through a crystal as a free particle with a mass close to the crowdion effective

mass.

1.10

Point Defect in the 1D Crystal

Any distortion of regularity in the crystal atom arrangement is regarded as a crystal

lattice defect. A point (from the macroscopic point of view) defect is a lattice dis-

tortion concentrated in the volume of the order of magnitude of the atomic volume.

The typical point defects in a 1D crystal are as follows: an interstitial atom is an atom

occupying position between the equilibrium po sitio ns of ideal lattice atoms ( a crow-

dion can be considered as the extende d interstitial atom); a substitutional impurity is

a “strange” atom tha t replaces the host atom in a lattice site (Fig. 1.14).

Fig. 1.14 Substitutional impurity in 1D crystal.

The simplest point defect arises in a monatomic species when one of the lattice

sites is o ccupied by an isotope of the atom making up the crystal. Since the isotope

atom differs from the host atom in mass only, it is natural to assume that the crystal

1 Mechanics of a One-Dimensional Crystal

perturbation does not change the elastic bond parameters. Let the isotope be situated

at the origin (n = 0) and have a mass M different from the mass of the host atom m.

With such a defect we get in the case of stationary vibrations the following set o f

equations

Mu(0) −α [u(+1) −2u(0)+u(−1)] = 0, n = 0;

mu(n) −α [u (n + 1) −2u(n)+u(n −1)] = 0, n = 0.

(1.10.1)

Equations (1.10.1) can be written more compactly as

mω

u(n) − α[u(n + 1) − 2u( n)+u (n − 1)] = (m − M)ω

, (1.10.2)

by introducing the Kronecker delta δ



We denote

∆m = M −m, U

= −

∆m

, (1.10.3)

and rewrite (1.10.2) in a form typical for such problems

u(n) −

[u(n + 1) − 2u(n)+u(n − 1)] = U

u(0)δ

. (1.10.4)

We write a formal solution to (1.10.4) as

u(n)=U

(n)u(0), (1.10.5)

where G

is the Green functio n for ideal lattice vibr ations, ε = ω

; u(0) is a constant

multiplier still to be deﬁned.

Setting n = 0 in (1.10.5), we ﬁnd that (1.10.5) is consistent only when

1 −U

(0)=0. (1.10.6)

Equation (1.10.6) is an equation to determine the squares of frequencies ε at which

the atomic displacements around an isotope have the form of (1.10.5). In the theory of

crystal vibrations with a point defect, an equation such as (1.10.6) was ﬁrst obtained

by Lifshits, 1947.

The expression (1.1.22) for the Green function for ε > ω

at n = 0 is substituted

into (1.10.6):

1 −



ε(ε − ω

)

= 0. (1.10.7)

Since ε > 0, a solution to (1.10.7) exists only at U

> 0. It is not difﬁcult to ﬁnd this

solution:

ε =

1 − (∆m/m)

. (1.10.8)

1.10 Point Defect in the 1D Crystal

Thus we obtain a discrete frequency corresponding to vibrations of the crystal with

the single point defect. For |∆m|m this frequency is slightly shifted relative to the

upper edge of the frequency spectrum:

ω −ω



∆m



. (1.10.9)

Crystal vibrations with the frequencies d escribed are called local vibrations,and

the discrete freq uencies are called local frequencies ω

. These names are attributed

to the fact that the amplitude of the corresponding vibration is only nonzero in a small

vicinity near the point defect. The local vibration amplitude is given b y (1.10.6),

implying its coordinate dependence is completely d etermined by the behavior of the

ideal crystal Green function. In order to obtain the Green function using (1.1.22) for

ε > ω

it is convenient to take the quasi-wave number in the complex form (1.1.24):

(n)=

(−1)

−κna



ε(ε − ω

)

, (1.10.10)

and take into account the connection of the frequency with the parameter κ (1.1.25)

for the discrete local frequency (ε

= ω

cosh

aκ

. (1.10.11)

Combining Eqs (1.10.8) and (1.10.11) one can get

κa = log

2m − M

. (1.10.12)

At m − M  m (1.10.12) can be simpliﬁed:

κa = 2

|∆m|

. (1.10.13)

We substitute the result (1 .10.12) in (1.10.10) and rewrite (1.10.5)

u(n)=u(0)(−1)



2m − M



. (1.10.14)

Thus, the local vibration amplitud e decreases if the distance na increases and this

decay of the amplitude conﬁrms the fact of the vibration localization.

Let us introduce a length of the localization region of vibrations l = 1/κ.Asitre-

sults from (1.10.12), under the condition |∆ m|m the length of localization is very

large (l = am/∆m  a). In this connection it is interesting to consider a long-wave

description of p roblems concerning with the localization of crystal vibrations near a

point defect. Returning to (1.10.4) and using (1.1.15) in the long-wave approxima-

tion the Kronecker delta δ

in the r.h .p. of (1.10.4) can be substituted by the Dirac

delta-function

= aδ(na)=aδ(x), x = na,

1 Mechanics of a One-Dimensional Crystal

and the ﬁnite differences in the l.h.p. can be substituted with the partial derivatives of

the function v(x) (see (1.1.16) written in the same approximation):

(ω

− ω

)v(x) −s

∂

v(x)

∂x

= aU

v(0) δ(x), s =

a ω

. (1.10.15)

The presence of the delta-function in the r.h.p. of (1.10.15) is equivalent to the

following boundary condition for (1.1.16)

−s



∂u(x)

∂x



−0

= aU

u(0), (1.10.16)

which determines a jump of the ﬁrst spatial derivative of the continuous function v(x)

at the point x = 0 (on the isotopic defect).

Take a solution to (1.10.15) in the form

u(x)=u(0) exp(−κ |x|), s

= ω

− ω

and ﬁnd the parameter κ from (1.10.16):

κa = 2

|∆m|





. (1.10.17)

In the long-wave approximation aκ  1 and ω − ω

 ω

, and then the simpliﬁ-

cation o f (1.10.17) is possible:

κa = 2

|∆m|

The result obtained coincides with (1.10.13).

Therefore, the long-wave approximation allows us to solve problems associated

with local vibrations in the frequency interval ω − ω

 ω

1.11

Heavy Defects and 1D Superlattice

In the previous section we analyzed the local vibrations and found that only a light iso-

tope defect could produce such a vibration with a frequency higher than all frequencies

of a defectless chain. Under the conditions m − M  m and ω − ω

 ω

de-

scription of the problems under consideration could be performed in the long-wave

approximation. It was explained why a heavy isotope defect could not produce a local

vibration. Nevertheless, the h eavy defect inﬂuences the continuous vibration spec-

trum. Obviously a heavy defect can inﬂuence low vibrational frequencies, because

heavier masses are associated with lower frequencies. One can expect to ﬁnd essential

effects at very low frequencies ω  ω

at M −m  m.

Consider a periodical array of isotope defects with M > m separated by the dis-

tance d = N

a in the linear chain. Suppose d  a (N

 1)andM − m  m;