Two-dimensional dendritic growth at arbitrary Peclet number

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Two-dimensional dendritic growth at arbitrary Peclet

number

E.A. Brener, V.I. Mel’Nikov

To cite this version:

157

Two-dimensional dendritic

_growth

at

_arbitrary

Peclet number

E. A. Brener

₍₁₎

and V. I. Mel’nikov

₍₂₎

(1)

Institute for Solid State

_{Physics, Academy}

of Science of the U.S.S.R., 142432

_{Chernogolovka,}

Moscow Distr., U.S.S.R.

(2)

L. D. Landau Institute for Theoretical

_{Physics, Academy}

of Science of the U.S.S.R.,

142432

_{Chernogolovka,}

Moscow _Distr., U.S.S.R.

(Reçu

le 29 mai 1989, révisé le 6

_septembre

1989,

accepté

le 3 octobre

₁₉₈₉₎

Résumé. 2014 La sélection de vitesse due à

l’anisotropie

de

_l’énergie

de surface dans la croissance

dendritique

à deux dimensions est considérée. Dans la limite de

_{petite anisotropie, l’équation}

intégro-différentielle

du _spectre de vitesse se réduit à une

_équation

différentielle. La solution de

cette

_{équation correspondant}

à la vitesse maximale est _{stable alors que les} autres solutions

présentent

des modes instables. Une

équation

différentielle pour le spectre des taux de croissance

est

_également

dérivée. La vitesse de croissance v se _comporte comme

03B13/4 f(p03B11/2)

où p est le nombre de Péclet et 03B1 ~ 1, un

_{petit paramètre d’anisotropie.}

Pour

_{p03B11/2 ~}

1, la fonction

_{f est}

calculée

_{numériquement.}

Les taux de croissance des modes instables 03A9 se _comportent comme

03B13/2 _g

(p03B1

1/2).

Pour _p03B11/2 ~ 1, g se _comporte comme

_(p03B1

1/2)3

et comme

_(p03B1

1/2)-

4/5 dans le cas

contraire.

Abstract. 2014

Velocity

selection in two-dimensional dendritic

_growth

caused

by anisotropy

of the surface energy is considered. In the limit of low

anisotropy

the

_{integro-differential}

_equation

for the

velocity

spectrum is reduced to a differential

_equation.

The solution of this

equation,

corresponding

to the maximal

_velocity,

is stable, whereas the other solutions have unstable modes. A differential

equation

for the spectrum of

_growth

rates is also derived. The

growth

velocity

v ~

03B13/4.

_f(p03B1

1/2),

_{where p}

is the Peclet number, 03B1 ~ 1 is a small

_anisotropy

_parameter.

At

_{p03B11/2 ~}

1 the function

_f

is calculated

_numerically.

The

growth

rates of the unstable modes

03A9 ~

03B13/2 g (p03B11/2).

At

_{p03B11/2 ~ 1}

the

function g

~

(p03B11/2)3,

and in the

_opposite

limit g ~

(p03B1

1/2)-4/5.

J.

_Phys.

France 51

₍₁₉₉₀₎

157-166 15 JANVIER 1990,

Classification

Physics

Abstracts 68.70

1. Introduction.

Lately

the mechanism of

_velocity

selection in dendritic

_{crystallization}

has been

_investigated

in detail. It has been shown that the crucial role in this selection is

played by

the

_anisotropy

of the surface _{energy. The}

_velocity

_spectrum

of a needle

_crystal

is

discrete,

and the

_only

stable solution is the one with maximal

velocity.

An

analytical approach

to this

_problem

is in the

general

case

complicated

due to the

_necessity

to solve a nonlinear

_{integro-differential}

equation.

Some progress can be achieved in the low

_anisotropy

limit,

when the Ivantsov

parabolic

solution for the dendritic front

_[1]

can be taken as a

_{starting point.}

Then the

anisotropic

surface energy acts as a

singular

_{perturbation.}

To calculate the

_{growth velocity}

spectrum

one should solve an

_equation

near the

_{singular point}

in the

_{complex plane.}

Amar and Pomeau have been the first to derive a nonlinear differential

equation

of this kind

[2].

From dimensional

_analysis

of this

_equation

the

_{following expression}

was obtained for the

growth velocity

where D is the thermal

_diffusivity,

_do

is the

_{capillary length, p}

is the Peclet

number,

and

a « 1 is a small

_anisotropy

_parameter.

The Peclet number is determined

_{by undercooling}

and,

at small dimensionless

_{undercoolings}

d,

we

have p = .d2/ ’TT.

To derive the above result

for the

_{growth velocity,}

_{in papers}

_[2-4]

the small Peclet number

_{approximation}

was

_formally

employed. Actually

this result is valid in a rather broad interval of

p’s

[5].

The

genuine

criterion of

_validity

of

_equation

₍₁₎

for the

_growth

_velocity

follows from

_estimating

the size of the

_{singular region}

and is

_{given by}

the

_{inequality p}

a - 1/2

_[6].

The

_opposite

limit of

_large

Peclet numbers can be considered for

_{arbitrary anisotropy a}

[7, 8].

The limit of small

a « 1 then

_gives

where

/3 -

1.

_Comparison

of limit

_expressions

for the

_{growth velocity}

shows that their orders of

_magnitude

are

_comparable

at pa l2 -

1.

_Thus,

it can be

_conjectured

that in a

_general

case at

a « 1 the

_{growth velocity}

is

_{given by}

the

_expression

In section 2 of the

_present

_paper we consider the

_problem

of selection of the

_{growth velocity}

of a two-dimensional needle

_crystal

at

_arbitrary

Peclet

numbers,

exploiting only

the smallness of

_anisotropy

a. The

equation

for the

growth velocity

_spectrum

depends only

on

pa 1/2.

Structurally

it is

_quite

similar to that derived

_previously

in reference

_[2]

in the limit of small Peclet numbers. From this

_equation

follow the limit

_expressions

₍₁₎

and

_(2),

whereas at

pa 1/2 --

1 the function

_{f, entering equation}

_(3),

is found

_numerically.

In section 3 we derive an

equation

for the

_growth

rates of unstable modes f2 and show that

where

2. Sélection of the dendritic

growth velocity

The

_dynamics

of the

_{crystallization}

front

_{y (x, t )}

of a

_{freely growing}

two-dimensional needle

crystal

is

_governed

in the

_{symmetric model, by}

the

_equation

_[9]

In order to consider in what follows the

_{steady-state growth}

of a needle

_crystal

of an almost

parabolic

form with a

_tip

radius p and

_velocity

v, and also to

_analyze

the

_stability

of the

steady-state

_growth,

we have introduced in

_equation

₍₉₎

the

_following

dimensionless

_parameters.

All

lengths

are measured in units of p, time in units

_{p /v, p}

=

v p /2 D

is the Peclet

number,

159

To

the melt

temperature

far away from the

crystallization

front

_(To

_{Tm ),}

_cp

is the

_specific

heat

_(thermal

_parameters of the

_crystal

and melt are assumed to be the

_same),

L is the latent heat. The second term in the left-hand-side of

_equation

₍₄₎

describes the shift of the

_melting

temperature

caused

_by

the curvature of the front

_{K (x, t ) (Gibbs-Thomson effect),}

The

_{capillary length}

_d(0)

=

Y E(0)

Tm

cp

L-2,

YE (0)

=

y ( 0)

+

d2 y

(e)/d 82,

_{y ( e)}

is the

anisotropic

surface energy,

tg 0

=

dy/dx.

Usually

for the

anisotropy

of the

capillary length,

the

_simplest

model

_expression

is

used,

which

_corresponds

to a four-fold

_crystal

_symmetry.

Neglecting

the surface energy

(do

=

0 )

Ivantsov derived a

_steady-state

solution of

_equation

(4)

in the

_shape

of a

_{parabola, moving}

with a constant

_velocity

_[1]

The Peclet number p is then

relàted

to

_{undercooling by}

the

_expression

The

_anisotropic

_{surface energy} acts as a

_{singular perturbation. Consequently,}

the role of the small

_anisotropy

_{parameter a}

is two-fold. On the one

_hand,

it

_gives

a small correction to

the

_parabolic

front

_shape

which enables one to linearize the

_integral

term in

_equation

₍₄₎

about Ivantsov solution. On the other

hand,

the size of the

_{singular region}

near the

_singular

point

is also small at small a. As a

_result,

derivatives of the correction are not small in the

singular region,

and the left-hand-side of

_equation

₍₄₎

after all

_{simplifications}

remains nonlinear

_[2].

To find the corrected

_steady-state

_solution,

we

_represent

the front

_shape

as

and linearize the

_integral

term in

_equation

₍₄₎

over

C.

Then we

_get

where R is the distance between two

_points

on the

_parabolic

front of

_{crystallization.}

If we assume

_that

_l"

₁

«

_1,

from the left-hand-side of

_equation

₍₇₎

it follows that the

_singular

points

of this

_equation

are x = _± i. We consider _a close

vicinity

of the

_{singular point}

x = i. In the

limit

lx - i

[

1 _we

have A (@) = 1

+ 2 a 1 (x - i )2.

_Therefore,

the actual size

It follows

then,

that for

_{pa 1/2 $ 1}

the

_argument

of the Bessel function is small. When pa 1/2 » 1 in the

_region

where the

exponential

is

_{non-negligible,}

the

_argument

of the Bessel functions

_p

_{1 RI --}

_{p -1/2 «}

1. We shall

_analyze

contributions of different terms of the

_expansion

of the Bessel functions.

We start with the term

_1/pR

in the

_expansion

of the function

_{Ki. Calculating}

the

_integral

in

equation

(7) with ’(x’)

an

_arbitrary

smooth function

_along

the real axis of

x’,

we find that at

x ;, i the contribution of the term is of order

_{exp (- p 12)}

and should be

_neglected

when

p > 1. The next terms of the

_expansion

contain

_{log R}

and have

_quite

different

_analytical

properties.

We shall show that at x - i the main contribution to the

_integral

comes from the

branching point

of the

_logarithm

at x’ = x. From the

expression

In

(pR )

only

In

lx - x’

1

should be

retained,

since the other factors under the

_logarithm

have no

_singularity

on the real

axis,

and their contributions could also be

_neglected

for the reasons

_given

above for the case

of

_1/pR-term.

We write

The

_integral

with the first term can be

_directly

continued

_analytically

_{into the upper half of the}

complex plane

of x with the

_integration

contour maintained on the real axis of x’. This contribution can also be

_neglected

at x = i for the reasons stated above. The second

term

_corresponds

to the

_integration

contour

_passing

above the

_point

x. The

_shifting

of x into

the upper

half-plane

leads

_inavoidably

to the deformation of the

_integration

contour

_{(Fig. 1).}

Fig. 1.

- The

integration

contours in the x’

_{complex plane :}

_Ci)

for

_{ln (x - x’ - i £ ); C2)}

for

In

_{(x - x’}

+

_{i £).}

_{Point x lies in the upper}

half-plane.

Even for x = i no

_{exponential suppression}

of this contribution is

_observed,

as it comes

_mainly

from the

_vicinity

of the

_{branching point}

at x’ = x. To describe the situation

_completely

we

take a cut

_going

from the

_{point x’}

= x

_vertically

down. The values of the

_logarithm

on the different sides of the cut differ

_by

2 7Ti.

_Accordingly,

to transform the

_integral

term in

equation

(7)

we can use the relation

The upper limit of

_integration

in

_{equation (8)}

is taken to be

_infinite,

which

_implies

the

assumption

that

_{f decays}

_{rapidly along}

the

_imaginary

axis.

_Retaining

in the Bessel functions

161

in the small a limit we

_get

from

₍₇₎

the

_equation

Equation

(10)

is derived under the

_{assumption that p >}

1,

but it is valid at

_arbitrary

_{p. The}

point

is that the coefficient

of e in

the

_{right-hand-side}

of

₍₁₀₎

does not

_depend

on p, whereas the

_integral

term becomes

_{important only}

at p >

1 due to the smallness of the

_anisotropy

parameter

a.

In the limit

_{pa 1/2}

« 1 the

_integral

term in

_equation

₍₁₀₎

can be

_neglected,

then the

_equation

is reduced to the differential

_equation,

derived in

_[2].

It means that this differential

_equation

together

with the known

_scaling

relation

are valid in a rather broad interval of Peclet

_{numbers p a - 1/2.}

.

In the

_general

case of

_arbitrary

Peclet numbers one must solve the

_{integral-differential}

equation

(10).

Fortunately,

the

_simple

structure of the

_integral

term makes it

_possible

to

reduce this

_equation

to a differential

_{equation. Introducing}

the function

and

_{differentiating}

twice

_{equation (10),}

we

_get

a

_system

of differential

_equations

These

_equations

have no

_{explicit dependence}

on z, so

_they

are

_equivalent

to a nonlinear second order differential

_equation.

In order to

_{investigate properties}

of these

_equations

we

consider their

_asymptotic

behavior at

_large

_{1 z 1.}

In this limit we _can, in the first

approximation, neglect

the terms with derivatives in

_equation

_(12).

Then we

_get

After substitution T = z we consider the linear differential

_equation

₍₁₂₎

in the limit

1 z 1

> 1. Solutions of the uniform

_equation

have an

_asymptotic

behaviour

and

_diverge

most

_{rapidly along}

_{the rays arg}

_(z)

=

0,

± 4

’TT /7.

To ensure an

_asymptotic

behaviour

₍₁₃₎

for the function F we need to kill these

_divergences

_along

_{the three rays. To} fulfill the condition of boundedness of F we have two

_integration

constants and the

_parameter

A,

so that À = À

_{(pa 1/2)}

is the

_eigenvalue

of the

_problem.

To find the function À

_(pa

_1/2)

we have solved the

_system

of

_equations

₍₁₂₎

_numerically,

taking

F to be real and bounded at

_large

real z and

_requiring

that F be bounded at _{the ray}

arg

(z )

= 4

’TT /7.

This solution is then

automatically

bounded _on the ray arg

following

iterative

_procedure.

At a

_given

_T(z)

the linear

_equation

for the function

F(z)

was solved and the

_eigenvalue

À determined. To kill the

_{solutions, growing}

at

_large

Izl,

we solve the

_equation

for

_F(z),

_going,

_{e.g., from}

_points

_lying

on the rays

arg

(z )

= 4

7r/7

and arg

(z )

= 0

at

1 z 1 -

7,

into the inner

region

1 z 1 -

1. The condition of

matching

of the

solutions,

originated

from different rays, at some z

( 1 z 1 -

1 )

determines the

spectrum

of A. Our numerical calculations confirmed that the results for À do not

_depend

on

the location of the

_{matching point}

within the inner

_region.

The function

_{F (z )}

found in this way was then used to calculate

_{T (z)}

from the second of

_{equations (12).}

_{The convergence of} this iteration was

_fairly

fast : the value of À becomes stable after 6-7 iteration

_cycles.

_Hence,

we write the

_{growth velocity}

as

where

_Ao

is minimal value of À for a

_given

pa 1/2.

Fig.

2. -

Dependence

of the function

_{f on}

_{pa 1/2.}

For

_{pa 1/2,>}

10 the function

_{f varies}

rather

_slowly,

its

asymptotic

behaviour

_{being given by equation}

_(19).

f (y)

is

_plotted

in

_figure

2.

_Numerically

_{Ào(O) = 0.42.}

At

_{large y > 1,}

_f

saturates to a

constant value. This limit was treated in

[7, 8].

The

_asymptotic

behaviour

of f can

be derived

from

_equations

₍₁₂₎

in the

_following

_{way. From the first of}

_equations

₍₁₂₎

one can _guess that

at _{p a}

1/2 »

1 the

_eigenvalue

À is of

_{order p2}

a. It follows then that almost

_everywhere

in the

complex

z

_plane

the derivatives in the first of

_equations

(12)

can be

_neglected.

In this

approximation

F is

_{given by}

The second of

_{equations (12)}

can then be used to find T as a function of z.

It can be

_easily

verified that at

_{arbitrary À -- p2 a}

there are two

_points,

where the above

163

To

_justify

the

_procedure

described above and calculate a correction

h

1 to

the

eigenvalue

a

= k 0

+ À

_1,

we have to take into account the terms with derivatives in

_{equations (12).}

The second of

_equations

(12)

in the

_vicinity

of To is written as

In the first of

_equations

₍₁₂₎

we

_neglect

the term with second derivative

_compared

to the term

with first

derivative,

i.e. we assume that

d Fldz2

pa 112

_{dF /dz (it}

will be shown below that

dFIdz - (pa

1/2)3/5

F).

This

_corresponds

to

_dropping

out the

_asymptotics

_{F cn exp}

_{(p a}

1/2

_{z ),}

which

_diverges

at

_large

real z. The

_resulting

first-order

equation

for F can be transformed with the use of

_equation

(16)

into an

_equation

with the

_only

variable T,

In

_writing

down this

_equation

use has been made of the fact that T:= _To and

A 1/ A ° 1.

Substituting

then

_gives

rise to the dimensionless

_equation

for the

_spectral

_{parameter 8}

From

_equations

_{(14), (15)}

and

₍₁₇₎

we find the final

_expression

for the

_{growth velocity,}

where

₁₃₀

is the minimal value of the

_spectral

_parameter

₁₃

which can be found from the solution of

_equation

_(18).

For a distant

_region

of the

_spectrum,

it follows from

_{equation (18)}

that

_{13 n en n 2/5,}

where n is an

_integer,

n » 1.

In conclusion of this section we note that its main results are

₁₎

the

_reproduction

of

previously

obtained

_asymptotics

of the

_{growth velocity}

in the limit

_{p a 112 >}

1,

and

₂₎

the calculation of the

_{growth velocity dependence}

on the Peclet number in the intermediate

region

pa 1/2 --

1

_(see

_Fig.

_2).

These results were obtained in the limit a « 1 and therefore

_they

only slightly overlap

with

_existing

numerical results. The

_point

is

that,

in order to determine the

_{growth velocity by}

direct solution of

_equation

₍₄₎

at real x, one must take into account

singularly

small terms with relative

_magnitude

of order of

_{exp (-}

_cla7’8)

_(at

p a 1/2

1),

,

where c is number of order

_unity.

So,

when a

_decreases,

one should increase the numerical accuracy

accordingly,

which

_prevents

one from

_reaching

the

_region

of small a. That is

_why

the

scaling

relation

₍₁₎

was

_compared

in the literature with numerical results

_only

relevant to the

dependence

v en p 2.

Our results allow us to _go

_beyond

this

_scaling

relation

_only

for p >

a -1/2,

i.e.,

at

_large

Peclet numbers. To our

_knowledge,

_up to now no numerical results were obtained

_{for p >}

1.

3.

Stability

of the

_steady-state

solutions.

In the

_preceding

section we have derived

_equation

₍₁₀₎

and the

_equivalent

_system

of

this purpose we set

The

_equation

for the

_spectrum

of fi can be found

_by

linearization of

₍₄₎

in

_en.

It has been

pointed

above that the

_steady-state

_{correction (x)}

is

small,

but its derivative near the

singular point

is not small. Therefore when

_{linearizating equation}

₍₄₎

in

_en

we can also

neglect C (x)

as

_compared

to

x2/2

in the

_integral

term of

_equation

_(4).

The

_integral

_operator

in the

_equation

for

_en

differs from that in

_equation

₍₁₀₎

_by

terms linear in fi. One of them

originates

from i

in

_equation

_(4),

and the other results from the

_expansion

of the Bessel function

_K,

near the

_{singular point,}

if one takes into account

_that,

in the

_argument

of the Bessel

_function,

_p2

is substituted

_by

_{p (p}

+

_{2 il).}

_Omitting

details of the derivation of the

integral

term, which are similar to the derivation of the

_integral

term in

_equation

_(10),

and

linearizing

in

_Ca

the left-hand-side of

_equation

_(10),

which contain

_derivatives,

we can write the

_equation

for

_en

near the

_{singular point}

_(compare

to

₍₁₀₎₎

where

The

_parameter

À and the function

_T(z)

_{entering equation}

₍₂₁₎

are determined

_by

the solution

of the

_{steady-state problem}

_(see

_Eqs.

₍₁₀₎

and

_(12)).

Note that

_equation

₍₂₁₎

has been derived without the conventional

_{quasistationary approximation.}

This

_corresponds

to

_discarding

the

term with 2 co in

_(21).

It is clear from

_equation

₍₂₁₎

that this is

_{justified only}

in the limit

pa 1/2 « 1 .

In close

_analogy

with the case of

_equation

_(10),

the

_simple

structure of the

_integral

term

enables one to reduce

_equation

₍₂₁₎

to a third order differential

_equation.

Let us denote the

right-hand-side

of

_equation

₍₂₁₎

_by

R.

_Taking

the second derivative of this

_expression

we find

that R

_obeys

the

_equation

Substituting

for the

_{function cp}

from

_equations

₍₂₁₎

and

₍₂₃₎

we obtain the third order

_equation

where

_{B (z )}

is determined

_{by expression}

₍₂₂₎

via the solution of the

_{steady-state problem.}

For

1 z

[

>

_1,

_B(z)--00FF

(21/2

_Àz3/2)-1,

and the three

_linearly

_independent

solutions of

_equation

₍₂₄₎

are

165

needs,

similarly

to the

_steady-state

_problem,

to _{suppress the}

_exponential

_growth

of

’1’ 1,2

along

the rays arg

(z )

=

0,

_± 4 ’TT

/7.

With these

_boundary

conditions taken into account,

equation

(24)

determines the

_spectrum

of increments w and the

_{eigenfunctions}

cp.

It is known that the

_steady-state

solution with the maximal

_velocity

is

_stable,

whereas solutions with smaller velocities have unstable modes

( w > 0)

whose number is

_equal

to the

sequential

number of the unstable solution.

From

_equation

₍₂₄₎

it follows that the

_growth

rates a)

_depend

on the

_parameter

Pa

1/2 .

Atpa 1/2

«

1,

the

growth

rates w take some constant values. In this limit the

problem

of

growth stability

in terms of a linear version of the

_{steady-state problem}

was considered in

_[10]

(see

also

_{[11, 12]).}

_{Solving equation}

₍₂₄₎

_{numerically together}

with the nonlinear

_steady-state

system

(12)

at

p« lr2

= 0 _we have made _sure that for the

steady-state

solution with maximal

velocity

there are no

_{positive eigenvalues}

w. For the solution next in terms of

_velocity

one

unstable mode has been found with w = 0.87.

Similarly

to

_equation

₍₁₄₎

for the dimensional

_{growth velocity,}

we can write an

_expression

for the dimensional

_growth

rates fi.

_Having

in mind that the time was measured in units of

plv,

we obtain

The function g

_(y)

is

_expressed

in terms of the function

_f

introduced in

_equation

₍₁₄₎

and of

w

(y),

For

_{p a 1/2 «}

_1,

,

f cn y 2

and w = const, so that

finally,

in the small

p a 1/2

limit,

,

Consider now the

_{limit p a}

n2 > 1.

_{Simplifications,}

which are

_justified

in this case, are

_quite

similar to those introduced in the case of the

_{steady-state problem}

in the course of the

derivation of

_equation

_(18).

As in section

_2,

_only

T close to _To are

_important,

The order of

_equation

₍₂₄₎

can be reduced

_by

one, as has

_already

been done in the

steady-state

_problem.

After

_setting

we

_neglect

in the

_equation

for w some small terms,

taking

for

_granted

that

but

_noticing

that

Exploiting

the

_expansions

of B and

À,

after substitutions

This

_equation

determines the

_spectrum

of

_growth

rates for the

_steady-state

_solution,

specified by

the value of the

_spectral

_{parameter f3}

and

_by

the

_shape

function

_{x(z) (see}

Eq.

(18)).

In the nearest

_region

of the

_velocity

_spectrum,

when

_{/3 -}

1,

W -- 1. In the distant

region

of the

_spectrum,

when

_{13n --}

n2/5,

, the

analysis

of

equation (30)

yields

for the most

unstable mode

The

_dependence

of the increment w on the

_parameter

p a 1/2 is

given by

relation

(29),

w ~

(pa

1/2)1/5.

At pa

1/2 >

1 the

_{growth velocity}

as well as the function

_{f (pa}

_1/2)

in

_equation

(27)

become

_{p-independent.}

For the function

_{g (pa 1/2)}

_describing

the

_dependence

of the

growth

rate on the Peclet

number,

we obtain the

_{following asymptotic}

behaviour

Comparing expressions

(28)

and

_(31),

one can see that the increment

_{f2 depends}

on the Peclet

number

_{nonmonotonously, exhibiting}

a maximum at

pa 1/2 --

1.

References

[1]

IVANTSOV G. P., Dokl. Acad. Nauk. SSSR 58

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567.

[2]

BEN AMAR M. and POMEAU Y.,

Europhys.

Lett. 2

₍₁₉₈₆₎

307.

[3]

BARBIERI A., HONG D. and LANGER J. S.,

Phys.

Rev. A 35

₍₁₉₈₇₎

1802.

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BRENER E. _A., ESIPOV S. E. and MELNIKOV V. I., Pisma ZhETF 45

₍₁₉₈₇₎

95.

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547.

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