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

1.2 Motion of a Localized Excitation in a Monatomic Chain 25

(major) perturbation maximum comes at the point n(n 1) at a time t1 determined by the condition ωm t1 ≈ 2n (Fig. 1.3). The velocity of motion of a perturbation maximum is na/t1 ≈ (1/2)ωm a = s, i. e., it is practically the same as the sound velocity in a 1D crystal. Thus, the effective velocity of the transfer of the perturbation is not different from that of the sound velocity, i. e., it is the same as the limiting group velocity that follows from the dispersion relation (1.1.6) or (1.1.9).

Fig. 1.3 The time dependence of the perturbation coming to the point located at a distance an from the displaced atom (z = ωm t, z1 = ωm t1 ≈ 2n).

Having clarified the role of the dispersion in an excitation signal moving along the 1D discrete chain, we now describe the velocity dispersion in the long-wave continuous approximation. It is known that competition of the higher dispersion with nonlinearity is very important in the dynamics of complex media. Hence, deriving a dynamic equation in partial derivatives for the 1D crystal anew, we take into account unharmonic terms in the interaction energy of the nearest neighbors. We assume the

potential energy of a crystal to have the form of a sum
U = ∑ϕ(ξn) = ∑ϕ(un − un−1 );		ξn = un − un−1 ,	(1.2.4)
n	n

so that in the harmonic approximation

ϕ(ξ) = 12 ϕ (0)ξ2 , ϕ (0) = α > 0. (1.2.5)

Generalizing the equations of motion of a 1D crystal, we assume the function ϕ(ξ) to be different from (1.2.5) and reduced to the parabolic dependence (1.2.5) only at

ξ → 0.

The equation of crystal motion with interactions between nearest neighbors only has the form

	d2 un
m		= ϕ (un+1 − un) − ϕ (un − un−1 ).			(1.2.6)
	dt2
For small relative displacements, one can take
	ϕ (ξ) = ϕ (0)ξ +		1	ϕ (0)ξ2 ,	(1.2.7)
			2

including only the so-called cubic anharmonicity.

26 1 Mechanics of a One-Dimensional Crystal

We substitute (1.2.7) into (1.2.6) and compare the result with (1.1.1); it is seen that α = mϕ (0). Just this relation connects the elements of the force matrix introduced phenomenologically with the parameter of the interatomic interaction potential.

However, (1.2.6) allows one to avoid the restrictions that arise in considering the harmonic vibrations by means of (1.1.4). To enable a description of inhomogeneous crystal states varying weakly in space we consider (1.2.6) in the long-wave approximation. Assume the characteristic distance ∆x of the change in the field of displacements to be large (∆x a). This makes it possible to pass to a continuum treatment, i. e., to replace the functions of a discrete argument n by the function of a continuous coordinate x and use the expansions

ξn+1 =	un+1 − un =	au +	1	a2 u +	1	a3 u +	1	a4 u ;
			2		3!		4!
							(1.2.8)
ξn =	un − un−1 =	au −	1	a2 u +	1	a3 u −	1	a4 u .
			2		3!		4!

In (1.2.8), the terms with the fourth-order derivatives remain, since the corresponding terms in the equations of motion may compete with terms generated by nonlinearity.

The nonlinear terms in (1.2.6) will be calculated to the first nonvanishing approximation with the lowest orders of the derivatives. Therefore, using (1.2.8), we take

ξ2	= a2 u 2 + a3 u u ,	ξ2	= a2	u 2	−	a3 u u .	(1.2.9)
n+1		n

Substitute (1.2.8), (1.2.9) into (1.2.7), (1.2.6) to obtain

m

d2 u

= a2

ϕ (0)

∂2 u

+

1

a2

∂4 u

+ a3

ϕ (0)

∂u ∂2 u

.

(1.2.10)

dt2

∂x2

12

∂x4

∂x ∂x2

Introduce the notations

	a2			a2			a3
s2 =		ϕ (0),	B2 =		s2 ,	Λ = −		ϕ (0),	(1.2.11)
	m			12			m

and write (1.2.10) as a nonlinear wave equation (the Boussinesq equation):

utt = s2 uxx + B2 uxxxx − Λ2 uxuxx,

(1.2.12)

where utt is the second derivative in time; ux, uxx, uxxxx are the derivatives of the corresponding order with respect to the coordinate x.

Since, when the atoms approach each other their mutual repulsion increases and when they separate from each other their attraction decreases, it may be assumed that ϕ (0) < 0. This was used in introducing the notation of (1.2.11). Besides that, although a simple relation between the parameters B and s results from the assumption of interaction of nearest neighbors only, it is quite natural to regard the coefficient of ∂4 u/∂x4 to be positive. Indeed, the dispersion law of harmonic vibration of a 1D

1.2 Motion of a Localized Excitation in a Monatomic Chain 27

crystal is generally such that the group velocity dω/dk decreases with increasing k for small k.

For the equation of motion (1.2.12) the long-wave (ak 1) dispersion relation for small (harmonic) vibrations has the form

ω2 = s2 k2 − B2 k4 ,

(1.2.13)

and the group velocity for ak 1 is v = s − (3/2)(B2 k2 /s). Thus, the coefficient discussed should really be positive to provide a decrease in v with k increasing.

An harmonic approximation describes well small crystal perturbations. But in some cases there arises the necessity to describe the motion of crystal atoms, which is accompanied with their large displacements. It is natural to pose the question whether the motion of crystal atoms is possible in which a strong perturbation will move along the crystal without changing the form of this perturbation?

If the displacement gradients connected with this perturbation are small there exists a positive answer to this question within the harmonic approximation. As we have noted, in this case the atom displacement is described by the linear wave equation (1.1.14) whose solution is any double-differentiated function depending on the argument x − st (if the wave runs in the positive direction of the axis x) or x + st (if the wave runs in the opposite direction). We remind readers that s is the sound velocity.

But if the harmonic approximation is insufficient, the answer to the question posed is no longer obvious. We study this question using the Boussinesq equation (1.2.12). Since a transfer of any deformation impulse in a 1D crystal is connected with motion of a local compression, the analysis of dynamics of the derivative p = ux makes sense supposing that the plot of p(x) has a form similar to Fig. 1.4.

Thus, we find a stationary solution to (1.2.12), moving along the axis x, i. e., a solution of the form u = u(x − Vt), where V is an arbitrary parameter (the perturbation velocity). As the desired function is, actually, a function of one argument ξ = x − Vt (1.2.12) is transformed into the ordinary differential equation

B2 uxxxx − Λ2 ux uxx − γuxx = 0,

(1.2.14)

where γ = V2 − s2 . We introduce new notations α = Λ/B, β = γ/B2 and rewrite (1.2.14) with respect to p:

pxxx − α2 ppx − βpx = 0.

(1.2.15)

We detract from a specific value of the coefficient B given by the definition (1.2.11) and seek for a formal solution to (1.2.15) for all possible values of β. Integrating (1.2.15) over x twice, taking into account the boundary conditions at infinity, we obtain

dp	2	−	1	α2 p3 − βp2	= 0.	(1.2.16)
dx			3

28 1 Mechanics of a One-Dimensional Crystal

Fig. 1.4 Solitary wave of 1D crystal deformation.

Equation (1.2.16) is easily integrated, and we obtain the solution that is interesting for us

p = −p0 sech2	1	β(x − Vt) ,	(1.2.17)
	2

where p0 = 3β/α2. It follows from the solution (1.2.17) that it exists only for p0 > 0, i. e., for V > s.

As the quantity p has the meaning of 1D crystal strain, the solution (1.2.17) obtained by us really describes a local compression moving with velocity V along a 1D crystal. A similar solution for the deformation is called a solitary solution, or a soliton. This is a singular (isolated) solution to the equation concerned, which may move along the

	crystal without changing its form.
	Rewrite (1.2.17)					x − Vt
	p =	−	p	0 sech	2		,	(1.2.18)
						l

where l is the soliton width determining the transition region ∆x in Fig. 1.4. The value of l can be found from asymptotes of the solution u exp(−x/l) of a corresponding linear equation (obtained from (1.2.14) or (1.2.15) with Λ = 0), and it is

l =	1		=	√		B			.	(1.2.19)
		β			V	2	− s	2

The velocity of the motion of the perturbation concerned is connected with the

	amplitude p0:	1
	V2 = s2 +		Λ2 p0.	(1.2.20)
		3

Thus, the nonlinear differential equation (1.2.14) has the desired solution (1.2.17) only for the velocity determined by (1.2.20).

In conclusion, it should be noted that substituting a specific value B2 = a2 s2 /12

that follows from (1.2.11) into the formula for the overall			width l yields:
l = as/	12(V2 − s2 )	. Thus, the long-wave approximation (l	a) is actually
valid only in a narrow interval of velocities V near s.

1.3 Transverse Vibrations of a Linear Chain 29

1.3

Transverse Vibrations of a Linear Chain

We consider a special linear analog of a simple crystal lattice assuming the atoms to be positioned periodically along a certain line in 3D space. Let a be a lattice constant and n the atom number counted from any point of the chain. We direct the x-axis along the undeformed straight line chain and denote by v the vector of transverse atom displacements (perpendicular to the x-axis), retaining the notation u for the longitudinal component of the displacement vector. The great interest in studying the vibrations of the 1D system proposed is explained by the fact that this problem is an excellent model of the dynamics of homopolymer molecules.

Keeping in mind possible applications to vibrations of the homopolymer molecules we restrict ourselves to the long-wave approximations. In the harmonic approximation the longitudinal and transverse vibrations are independent, and we analyze each form of motion separately. If the atoms are displaced along the x-axis, the elastic energy is determined by their relative displacements. The relative displacement of neighboring atoms is ξn = un − un−1 , and, in the nearest-neighbor approximation, the crystal potential energy equals a sum such as (1.2.4), so that in the harmonic approximation, it is possible to employ only the expansion (1.2.5). The forces generating the potential energy (1.2.4), (1.2.5) provide, between the neighboring atoms, a certain analog of spring coupling with the elasticity coefficients α. Such forces are called central forces.

In going over to the long-wave vibrations with the replacement un → u(x) can be effected, the leading term when expanding the difference ξn in powers of a/λ (1.2.8), where λ is the characteristic wavelength, is proportional to the first derivative of u(x) with respect to x. The crystal potential energy (1.2.4) then becomes

U =	ϕ		au (x)		dx	,	(1.3.1)
					a
and according to (1.2.5) the energy density is
ϕ =	1		αa2 u	2 .			(1.3.2)
	2

Equation (1.1.14) with s2 = αa2 /m is obtained in a standard way from (1.3.1), (1.3.2). If the atoms in a linear chain are displaced perpendicular to the x-axis, in the harmonic approximation central interaction forces do not arise and the crystal energy depends on the relative rotations of the segments connecting atoms in neighboring pairs rather than on the relative displacement of neighboring atoms. We assume that the transverse displacements of all atoms lie in one plane and denote the transverse displacement in this plane as vn, the angle of similar rotation by θ (Fig. 1.5). Then, as

seen from the figure, for small θ one can write for the nearest neighbors

	1	1			1
θn =		(vn+1 − vn) −		(vn − vn−1 ) =		(vn+1 + vn−1 − 2vn) .
	a		a		a

30 1 Mechanics of a One-Dimensional Crystal

Thus, the crystal energy with noncentral interaction forces of the nearest atom pairs taken into account will have an additional term

V = ∑ψ(aθn) = ∑ψ (vn+1 + vn−1 − 2vn) ,

n

and, in the harmonic approximation,

ψ(η) = 21β η2 .

In the long-wave limit, when vn is replaced by a continuous function of the coordinate v(x), the leading term of the θn angle expansion in powers of a/λ is proportional to the second derivative of v(x) with respect to x:

aθn = vn+1 + vn−1 − 2vn = a2	∂2 v
		.	(1.3.3)
	∂x2

The crystal energy V takes the form of an integral over the whole length of a linear chain

	V = ψ a2 v (x)				dx	,
					a
	where the energy density is	1
	ψ =		βa4	v 2 .		(1.3.4)
		2

Fig. 1.5 Atom configuration in a 1D chain with transverse (bending) vibrations.

The potential energy density (1.3.4) leads to the following equation

	m	∂2 v	+ aA2	∂4 v	= 0,	(1.3.5)
		∂t2		∂x4

where A2 = βa2 /m. Equation (1.3.5) describes the so-called bending vibrations. Comparing (1.1.14) and (1.3.5) shows that the term with the fourth-order derivative

in (1.3.5) includes a small parameter of the order (a/λ)2 unavailable in the term with the second-order derivative in (1.1.14). With such small terms preserved, the linear approximation may be insufficient. The nonlinearity should be taken into account, in

1.3 Transverse Vibrations of a Linear Chain 31

particular, in the presence of static stretching forces applied to chain ends. Under the action of such forces there arises a homogeneous longitudinal deformation ∂u/∂x =0 = const dependent on stretching load, so that it can be large.

In constructing an elementary nonlinear theory of vibrations of the chain concerned, anharmonicity should be taken into account only in the terms associated with central forces and the potential energy of small noncentral forces V can be calculated in the ordinary harmonic approximation.

The main nonlinearity is generated by the anharmonicities of central forces. If the nonlinearity is small, the crystal energy can be described by (1.2.4), (1.2.5) having determined more exactly the relative distance between neighboring atoms δln, with account taken of displacements in a direction perpendicular to the x-axis (Fig. 1.6)

δln = a + unz − unz −1	1	− a = (a + ξn)2 + η2n		1	− a,
	2 + (vn − vn−1 )2 2			2
where v(uny , unz ) is the 2D transverse displacement vector, ξn			= unx − unx−1

=un − un−1; ηn = vn − vn−1.

To describe the nonlinear bending vibrations of a chain, we use the expression for

	δln, written with sufficient accuracy
	δln2 = ξn2 +	1	ξnηn2	+	1	ηn2 ηn2 .	(1.3.6)
		a			4a2

Fig. 1.6 A scheme of displacements at transverse 1D chain vibrations (the displacement of an atom with the number n − 1 equals zero).

Comparison of the first two terms on the r.h.s. of (1.3.6) reveals that under bending vibrations, the value of ξn is commensurate with (l/a)ηn2 ; therefore, we retain the last term proportional to ηn4 .

Taking into account the above arguments one should write instead of (1.2.4)

	U = ∑ϕ(δln) =	1	α ∑δln2 ,
		2
	n
	conserving, for the noncentral interaction energy, the expression

V = ∑ψ (vn+1 + vn−1 − 2vn) = ∑ψ (ηn+1 + ηn) =		1	β ∑(ηn+1 + ηn)2 .
		2
n	n		n

32 1 Mechanics of a One-Dimensional Crystal

It is seen that the total potential energy of the chain W in the nonlinear approxima-

tion can be divided into three parts
W = U + V = U + U + Uint,	(1.3.7)

where U is the total energy of longitudinal vibrations given by (1.2.4); U is the energy of transverse (bending) vibrations

U =	α	∑ηn4 +	1	β ∑(ηn+1 − ηn)2 ,	(1.3.8)
	8a2		2
		n		n

and Uint is the energy of the interaction between transverse and longitudinal vibrations

	Uint =	α	∑ξnηn2 .	(1.3.9)
		2a
			n

The presence of the energy (1.3.9) means that in a nonlinear approximation the

independence of longitudinal and transverse vibrations in a chain vanishes.

We compose the nonlinear equations of motion by using

d2 un

∂W

d2 vn

∂W

m

=

−

,

m

=

−

.

dt2

∂un

dt2

∂vn

Simple calculations lead to

d2 un

α

η2n+1 − η2n ,

m

= α (ξn+1 − ξn) +

(1.3.10)

dt2

2a

m

d2vn

= α

(ξn+1 ηn+1

− ξnηn) +

α

η2n+1

ηn+1 − η2nηn

(1.3.11)

dt2

2a2

−β (ηn+2 − 3ηn+1 + 3ηn − ηn−1 ) .

These equations allow us to describe the bending vibrations, taking into account the influence of longitudinal chain vibrations. In order to derive the nonlinear equations of vibrations in a continuous approximation we make use of the expansions such as (1.2.8) and (1.3.3), retaining the higher space derivatives and the leading nonlinear terms containing the functions u(x) and v(x):

∂2 u

= s2

∂2 u

+

1

s2

∂

∂v

2

;

(1.3.12)

∂t2

∂x2

2 ∂x ∂x

∂2 v

= s2

∂

∂u ∂v

+

1

s2

∂

∂v

2

∂v

− a2 A2

∂4 v

. (1.3.13)

∂t2

∂x

∂x ∂x

2

∂x

∂x

∂x

∂x4

The total potential energy of a linear chain (1.3.7), in the long-wave approximation

corresponding to (1.3.12), (1.3.13), is equal to

W =

1

a2

α

∂u 2

+ α

∂u

∂v

2

2

∂x

∂x

∂x

α

∂v

2

∂v

2

∂2 v

2

dx

+

+ a2

β

. (1.3.14)

4

∂x

∂x

∂x2

a

ω = √ 0 (sk) = s k.

1.4 Solitons of Bending Vibrations of a Linear Chain 33

Using the rules of functional differentiation it is easy to find from (1.3.14) the forces acting on vibrating atoms and leading to (1.3.12), (1.3.13).

Neglecting anharmonicities of transverse displacements reduces (1.3.12) to (1.1.14) with the dispersion law ω = sk and simplifies (1.3.13). If the linear chain is free (static stresses are absent), the nonlinear term retained in (1.3.13) that contains ∂u/∂x is also small. If the action of external forces generates a considerable static homogeneous deformation ∂u/∂x = 0 = constant, (1.3.13) in a linearized form is reduced to

vtt = 0 svxx − a2 A2 vxxxx.	(1.3.15)
Equation (1.3.15) is associated with the dispersion relation
ω2 = 0 s2 k2 + a2 A2 k4 .	(1.3.16)

It is interesting that the relation (1.3.16) is similar to the dispersion law (1.2.13) but with the opposite sign in front of the term proportional to k4 .

For ak 0 (s/ A) the dependence typical for the acoustic branch follows from (1.3.16)

(1.3.17)

Its sound velocity s = s√ 0 is small. Thus, the long-wave dispersion law of bending vibrations of a linear chain that experienced a static longitudinal stretching does not differ qualitatively from the dispersion law of longitudinal vibrations of this chain.

In the region of wavelengths for which √ 0 (s/ A) ak 1, we obtain the

dispersion law

ω = aAk2 ,

which is typical for bending waves.

1.4

Solitons of Bending Vibrations of a Linear Chain

We analyze (1.3.12), (1.3.13) to clarify their purely nonlinear properties. Recall that the interest in the above equations is explained by the fact that they model the dynamics of homopolymer molecules.

To simplify the system of nonlinear equations (1.3.12), (1.3.13), we introduce, instead of time, a variable τ = st. The one-dimensional parameter aA/s will then remain in (1.3.12), (1.3.13). Estimating it we assume that with regard to order of magnitude, A s. The equations will be written in the form

uττ =

∂

ux +

1

v2

;

(1.4.1)

∂x

2 x

vττ =

∂

ux +

1

vx2

aA

2

vx −

vxxx .

(1.4.2)

∂x

2

s

= ∂2 v/∂x2 ,

34 1 Mechanics of a One-Dimensional Crystal

We remind ourselves of the notations ux = ∂u/∂x, vx = ∂v/∂x, vxx

etc. It is easy to find the solutions to (1.4.1), (1.4.2) in the form of a stationary profile whose dependence on the coordinate and time is represented through a combination ζ = x − Vτ. The waves of a stationary profile are the solutions to a system of ordinary

differential equations

1 d

(1

− V2 )uxx +

vx2 = 0,

(1.4.3)

2

dx

V2 vxx +

aA

2 vxxxx =

d

ux +

1

v2

vx .

(1.4.4)

dx

s

2 x

The equations given admit one trivial integration. To perform this, we assume the linear chain experiences a static longitudinal stretching (longitudinal strain) equal to ε0 (ε0 1). Besides that, as we are interested primarily in the solitary waves, we assume all velocities and all gradients vanish at infinity. Under such boundary conditions we get from (1.4.3)

v2

ux = ε0 −

x

;

ε0 = constant.

(1.4.5)

2(1 − v2 )

It follows then from (1.4.4) that

2

aA

2

V2

v2 vx

(V

− ε0 )vx +

vxxx =

x

.

(1.4.6)

s

2

V2 − 1

We use the fact that (1.4.6) involves only the derivatives of the vector function v and denote w = vx. Equation (1.4.6) will then take the form

				wxx + (γ + βw2 )w = 0,						(1.4.7)
where	1		s V2					s
β =							γ =			V2 − ε0 .
						,
	2		aA		1 − V2			aA

We introduce the amplitude and the phase ϕ of the transverse motion velocity by means of the relation

w = w(i1 cos ϕ + i2 sin ϕ),

where i1 , i2 are the unit vectors of two coordinate axes perpendicular to the direction of a nondeformed chain. The vector equation (1.4.7) will then be reduced to the two

equations
wxx − wϕ2x + βw3 + γw = 0;					(1.4.8)
	d
		w2	ϕx	= 0.	(1.4.9)
	dx

Equation (1.4.9) has the form of the area conservation law (if by the variable x we

understand the time) and gives the integral of motion
I = w2 ϕx = const.	(1.4.10)