The thermoactivation rate of charged particles in a magnetic field

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The thermoactivation rate of charged particles in a magnetic field

D. Garanin

To cite this version:

D. Garanin. The thermoactivation rate of charged particles in a magnetic field. Journal de Physique

I, EDP Sciences, 1991, 1 (7), pp.1019-1028. �10.1051/jp1:1991186�. �jpa-00246382�

J. Phys. I France 1

(1991)

1019-1028 JUILLET1991, PAGE 1019

Classification

Physics

Abstracts

77.40 61.40

The thermoactivation rate of charged particles in

_a

magnetic

field

D. A. Garanin

Moscow Institute of Radio

Engineering,

Electronics and Automation, 117454, Moscow, Prospect

Vernadskogo

78, U-S-S-R-

(Received19

November1990, reviseds March 1991,

accepted11

March

1991)

Abstract. We consider _a diffusion of _a

charged particle

_{over a} three-dimensional barrier in _a

magnetic

field.

By

the solution of the Fokker-Planck

equation

it is shown that the

magnetic

field

causes the diminishing ^{of the}

pre-factor ro

^{in the} expression for the particle's therrnoactivation

rate _: r

=

ro

_exp (-

Uo/T~,

T _« Uo. ^{Unlike the} two-dimensional _case considered earlier [3,

4],

in the general situation this diminishing is non-uniform with respect to the orientation of the vector

B, with _a saturation _{at some non-zero} level for _some range of orientations. The latter _causes the

splitting

of the relaxation

peak

in the bulk materials

containing randomly

oriented two-well relaxators at

high magnetic

fields.

The fist of

symbols

m, q the _mass and the electric

charge

of the

particle

B the

magnetic

field

y the

phenomenological

friction coefficient of the

particle (the

viscous relaxation

frequency)

w~ the

cyclotron frequency

vector of the

particle

_w~

=

qB/m

n the

frequency

» associated with the saddle

point

of the

potential

energy

U(r

_: n ^~

=

3~U/32~[

~~~ ~~~ m

w ~ the _«

attempt

»

frequency ml

=

3~U/32~[

~~~,~~~

m

w~_~ ^the characteristic

frequencies

^{in the directions}

perpendicular

to the

« reaction coordinate

Z,

associated with the saddle

point

w_[,~ ^{the same associated with the minima of}

U(r)

x, y, z the dimensionless coordinates _x

=

Xii,

_etc.,

f

_is

some

length

unit

(e.g. ^I

=

Zo-the

coordinate of the

potential minimum)

p the dimensionless _{momentum : p}

=

v/u

o, vo =

in

_{is the} characteris- tic

velocity

w~ the thermal

frequency

_w~

=

v~/f,

_v~

=

(2 T/m)~'~

is the thermal

velocity

~~,~ ^{the saddle}

point

form-factors

~~,~

=

(w~,~/n)~

a the dimensionless

damping

_{: a}

=

yin

p

W~lil

f

_see

equations (2.10)

and

(2.18)

y ^» n the strong

damping

limit

n _{» y »} n

(T/Uo)

the moderate

damping

_case

(absolute

_or

Eyring regime) n(T/Uo)

» y the weak

damping (Kramers) regime.

1. Introduction.

Since the

pioneering

work

by

Kramers

[I]

the

problem

of noise activated _{escape processes} had drawn attention of _many researches. It _was shown that the model

considering

the diffusion of

a

particle

in _an external

potential

is _an abstraction of such different

phenomena

as chemical

reactions, optical bistability,

ionic

conductivity,

relaxation of

polar

_groups in

dielectrics, dynamic

_processes in

Josephson junctions,

etc.

(see,

_e.g.,

[2]). Usually

it is

possible

to choose

one relevant «reaction» coordinate and consider the

problem

_as one-dimensional. Even

within this

simplified approach

the situation remains rather

complex, especially

if the quantum ^effects are taken into account.

However,

there _{are some cases,} in which the thermoactivation process can not be considered _as one-dimensional. One of

them,

the consideration of which is the aim of this

investigation,

is the _case of

charged particles crossing

a barrier in the presence of _a

magnetic

field

(ionic magnetoresistance,

dielectric relaxation in _a

magnetic

field

[3,4], etc.).

It is understood that the Lorentz force

acting

_on

moving charged particles

_causes

curving

of their

trajectories

and diminishes the

pre-exponential

factor

ro (r

=

roexp(- UOT),

T«

Uo).

This

picture

_was confirmed

by

concrete calculations in the

particular

two-dimensional _case of

particles

confined to a

sphere (rotating

electric

dipoles)

in the strong

damping

_[3] and the

;irong to moderate

damping [4] regions.

If'charged particles subject

to the influence of _a

magnetic

field _move in _a

general

three- dimensional

potential,

_{some new} features of their behaviour appear. In

particular,

there is _a saturation of the decrease of

ro

with the

increasing

of the

magnetic

field induction B

taking place

in _{some range} of

angles

between the vector B and the reaction coordinate

axis _» Z. The latter _{may cause} such _an effect _as the

splitting

of the relaxational

peak

ⁱⁿ

high magnetic

fields in materials

containing

two-well relaxators with directions of the Z axes

distributed

randomly.

In the next

part

of the paper we write down and solve the

corresponding

Fokker-Planck

equation

and determine the thermoactivation rate of _a

charged particle

in _a

magnetic

field for the

general paraboloid

form of _a

potential

barrier. In the final section the influence of _a

magnetic

field _on the relaxation

peak

due _to the

randomly

oriented two-well

species

is

discussed.

2. The thermoacfivatiton rate of _a

charged partitcle.

The motion of _a

particle

with _{a mass m} and the

charge

_q in _an external

potential

U(r)

and _a

magnetic

field B is described

by

the Fokker-Planck

(FP) equation

of the form

where

G(r,v,

_t is the

particle's

distribution

function,

_y is _a

phenomenological

friction

M 7 THERMOACTIVATION IN A MAGNETIC FIELD 1021

coefficient

(viscous frequency),

T is

temperature.

^The

potential U(r)

is of _a two-well form

quadratic

_near the saddle

point

~z2 k~X~ k~Y~

U(r

m + +

(2.2)

and the minima

k( (Z

_±

Zo)~ k( X~ kj

Y~

U(r)

m

Uo

+

~ ⁺

j

⁺

j (2.3)

At small

temperatures (T

«

Uo)

the distribution function of the

particle

is

quasiequilibrium

characterized

by

the

equilibrium

in each well and the absence of that between two.

So,

_{as we} will _see, the solution of the FP

equation (2.I)

_may be

represented

_as

G

(r,

_v, t

)

=

Go(r,

_v

)

_g

(r,

_v, t

)

,

(2.4)

where

Go

_{= exp}

(- E/T),

E

= mv

~/2

+

U(r)

and _{g is a} kink-like function

changing

in _{a narrow}

region

about the saddle

point, 3g/3t being exponentially

small.

For the

subsequent

consideration it is convenient to introduce the

frequencies

characteri-

zing

the

potential U(r)

n ^~

=

Kim,

_w

j,~

₌

k~,~/m, ml

=

k(/m

_(«

attempt frequency), w()~

=

k( _~/m.

We introduce also _a

length

unit

f (e.g. ^f

=

Zo)

and pass ^to dimensionless

coordinat~s

_and

momenta : x =

Xii,

_y

=

Y/f,

z =

Z/f,

_p~

=

vJvo, _p~

= 1~~/l~o, p~ = 1~~/vo where l~o =

in

_is

a

velocity

unit. With these notations the FP

equation (2.I)

^for the function g

(2.4)

is written _as

lpx)+py)+p=)

_g+

(-ljxx)-ljyy)+z)

g+

X Y Z Pz

Py

Pz

+

(p _zpy

_p

ypz)

+

(p

_{xpz p}

z

px)

+

(pypx

_p

xpy)

Px

Py

Pz

+alPxk+Py]+Pil ~-lot(+(+()

~,

(2.5)

where

3g/3t

is

neglected,

_~~,~

=

(w~,~/n)~

m

k~ ~/K,

a =

yin,

_p~

= w~,

In,

_w~

=

qB/m

is the

cyclotron frequency

vector,

Q= (y/n)(w~/n)~, w~=1~~/f

is the thermal

frequency,

1~~ =

(2 T/m)~/~

is the thermal

velocity.

As the variables _{x, y, z, p~}

p~,

p~ ^enter the left-hand side of

(2.5) linearly,

_{one can} search its solution in the form

g

(x,

_{Y, z, P}

x, P

y, P

z

)

₌

f (u

,

(2.6)

where

u = z + v

~

x + v~ y

fp~ f~

_p~

f~ _p~ ^(')

^and _v~ ^and

f~

are unknown parameters

(we

will _see below that

f

^{contains the}

magnetic

field

dependence

of the

pre-exponential

factor

ro

of the relaxation

rate).

The substitution of

(2.6)

into

(2.5) gives

R

~~

=

Q (f

^~ ₊

fj

₊

fj) _~~(, (2.7)

au 2 _au

(~) Such _a method

applies

_as well to the

original

Kramers

problem

and

gives

the solution in the strong and moderate

damping

cases y D » D

(T/Uo) [1,

5].

where

R

= ~p= + v~ p~ + vy

py) (fz

₊

f~

_~~ x +

f~

_~~

^{y) a(fp=}

+

f~

_p~ +

fy py)

+

+

t (py

px P

xPy)

+

fx(PzPy _P>,Pz)

+

fy(PxPz PzPx) (2.8)

The constants v~ and

f~

must be chosen _so that to make the function R

proportional

to

u R

=

D _u.

Equating

coefficients at each of the

variables,

_{one gets} D

=

f

and the system

of

algebraic equations determining adjustable

_constants

I,<'lx#-ivx, iy'ly#-ivy, I-at+p~t~-p~t~=t~,

Vx-aix+P=iy-Pyi#ii~ Vy-aiy+Pxi-Pzi<#iiy. ~~~~

Eliminating

_v~

~

and

f~,~

^from

^(2.9)

one arrives _at the

resulting equation

^for the

quantity f

that is of the most

significance

in

subsequent

consideration _:

i(i~+ _at

₊

'l>)(i~+ at

_{+ 7~~) +}

pjt~i(t~+ _at

I ₊

+

t~ip i(t~

₊

_at

₊

7~

x)

+ p

j(t~

₊

_at

₊

7~

~)i

= 0

(2.10)

An

analysis

shows that

(2.10)

has in all _cases the _one and

only

_one

physically acceptable positive

solution.

The 6-th power

algebraic equation (2.10) plays

the main role in the

description

of the thermoactivation rate of _a

charged particle

in _a

magnetic

field. The details of its solution with

particular

and

limiting

_cases will be considered

below,

and here _we will

proceed

the main line of calculations.

So,

with the

adjustable

constants

f~

^and _v~ ^determined

by (2.9)

_or

(2.10)

the

differential

equation (2.7)

becomes the

ordinary

_{one :}

~"

~

~

~~~''

_~

~

[

f2

₊

li

+

ii

^~~.~~~

having

the first

integral

f'

₌ C _exp

(- pu~) _(2.12)

Now _{we can} calculate

directly

the

change

of the function g

(see (2.4)

and

(2.6))

_across the barrier _:

W

~ l/2

g~ gj m

f~(u)

du

=

C

,

(2.13)

-w R

which is

proportional

to the well

population

difference

N~ Nj.

~~'~

~_~'~ ~~~

~~w

~~ ~ ~ ~~ ~~~

~~

~~

_~

~~~~ ~ ^~~

~

~~ ~~

~

dl~ dl~ du exp

"~~~

= gj _~

"~ ~~

exp

"

(2 14)

~~-w

^{~ ~ ~} 2 T

w(wjw~

T

The time derivative

fit

_{is determined}

by

the flow of

particles

from the I-st to 2-nd well

w w

N

=

jj

dX dY

jjj

dl~~ dl~~ dl~~1~~ G

(r,

_v, t

(2.15)

w w

(Z=0)

M 7 THERMOACTIVATION IN A MAGNETIC FIELD 1023

The

integration

in

(2.15)

with the _use of

(2.4), (2.6)

and

(2.12) gives

the result

ii

=

C

I

^{~~~ ~ ~~}

f, (2,16)

2 _p

~~~

wherefrom, employing (2,13)

and

(2.14)

_{one gets} the closed-form kinetic balance

equation

flip ₌

r(N2 Nj) (2.17)

with the relaxation rate

W ~

W( Wj

Uo

~

~

~ ~

X ~

y

~~~ ~ ~~ ~~

In

(2,18)

_{w are} the

particle's

oscillation

frequencies

in the saddle and minima

points

of the

potential U(r) (see

the text above the

equation (2.5)),

and the dimensionless constant

f containing

the

dependence

of the thermoactivation rate on the

magnetic

field and characteristic

frequencies

_y and tl is

given by

the

positive

solution of the

algebraic equation (2.10).

Now _we will

proceed

to the

analysis

of the factor

f

in the

expression

for the thermoactivation rate of _a

charged particle

in _a

magnetic

field

(2.18).

In the _case B=0 the

components

_p~~

~=0,

and the

positive

root of the reduced

equation f~

+

at

= 0 reads

[Ii

~ ~°

a +

a~+

4)~'~

_y +

(y~~4

_n

~)~/~'

~~ ~~~

which

simplifies

to

f

_m I

la

m

n/y

in the strong

damping

limit

(y

» n and to

f

_m I in the moderate

damping

limit

(y

_« tl

).

Remember that _our

approach

is

inadequate

in the weak

damping

_{case y s} tl

(T/Uo), though, just

_as in the one-dimensional _case, it is difficult _to

point

out in the present calculation where the condition _{y S} n

(T/Uo)

_was used

(see,

_e.g.

[5]

and references

therein).

In the _case B # 0 the value of

f

is less then

to

and diminishes with _a

magnetic

field. If the motion of the

particle

is confined to the

plane

_y

=

0(~~

- cc then the basic

equation (2.10)

reduces to the

quartic

_one

(f~

₊

at

_{+ ~}

)(f2

₊

_at

i ₊

f 2p

²

=

o, (2.20)

where _~

m ~~ and _p

m p~.

If,

in

addition,

in

(2.20)

_~

- 0

(the

wide pass

case),

_one arrives at the cubic

equation

studied in

[4].

^{In the}

opposite

limit _~

- cc

(the

_narrow

pass)

the zero-field

solution

to (2.19)

is recovered. This is _seen also from

(2.10)

and is

physically plausible

because in the _{narrow pass} situation the

problem

becomes

virtually one-dimensional,

and the

magnetic

field has _no influence _on the

particle's dynamics.

The two-dimensional model described

by (2.20)

allows to trace

analytically

the behaviour of the thermoactivation _rate for the

arbitrary

values of _a

magnetic

field in the strong

damping (y

»

tl)

and moderate

damping (y

« n

)

limits. If _a

m

yin

_» I then the solution of

(2.20)

has the form

I

" 12

(a~

_{+ P ~)}

i~ ia(1

7~

)

₊

(a~(1

₊

7~)~ + 4

7~p ~)~/~i

(2.21)

with the _zero field value

to

=

I

la

and the

high

^field

asymptote f

m ~ ~'~/p

independent

of the viscous

frequency

_y. In the wide pass case

(

_~

=

0)

the decrease

off

with the

magnetic

field is faster _:

f

=

a/(a~+ p~) [3, 4].

I-f am I then

(2.20)

results in _an

expression f~

=

Ii

_{~ p} ² ₊

((1

_{~ p 2)2 +} 4 _~

)i/2j (2.22)

having

the _zero field value

to

₌ ^{I and the}

high

field asymptote

just

the _{same as} in the _case ah I

fm~~'~/p.

In the wide pass case

(~ =0)

the

magnetic

field

dependence

of

f

is

peculiar f

=

(I

_{p ~)~'~}

(p

_< I and

f

=

0(p

~ l

),

I.e. the

permeability

of the

potential

barrier turns to _zero at some critical value of the

magnetic

field

corresponding

to

w~ m

qB/m

=

n

[4].

The consideration of the

general

three-dimensional model described

by (2.10)

reveals _some

new

interesting

features. These _{are a} nontrivial

dependence

of r _on the orientation of the

magnetic

field and the _saturation of r decrease with

increasing

the

magnetic

field induction B in _some

region

of its orientations.

Being

unable to write down the

analytical

solution to

(2.10)

in the whole range of the

magnetic field,

_we resort to the consideration of the low-field and

high-field limiting

_cases. At low fields the calculation

gives

~ ~

~ to P](1

₊

_~~)

₊

pj(1

₊

_~~)

~

~ °

(a~

₊

4)~'~ (l

+

l~x)(1

+

l~y)

^'

'~~~

where

to

^is

given by (2.19).

At

high

fields the result is _more

interesting

~~~~ _5~0;

f

_m

~~

_{~ ~}

~~

_{~ ~~~}

(2.24)

j'lx'ly _jl'21

S _p

~"~'

where S

= n ~

~

n

j

_~

~ n

j

is _an orientation

dependent quantity,

_n~,

~,~ ⁼ p

~,~,~

/p,

p~= pj+ pj+ PI.

Whereas the second

asymptote

ⁱⁿ

(2.24)

is _a

generalization

of that

occurring

ⁱⁿ the two-dimensional model

(see

the

high-field

_asymptote of

(2.21)

and

(2.22)),

the first _one the saturated

value,

not

longer dependent

on the

magnitude

of the

magnetic

field is

specific

for the three-dimensional _case.

The

physical interpretation

of the thermoactivation _rate behaviour demonstrated

by (2.24)

is the

following.

At

high magnetic

fields

charged particles

_{can move}

only along

its lines of force. If the

potential

_energy

along

such _a line

going through

the saddle

point

has _a

maximum,

then _a

particle

_{can cross a} barrier

moving along

this line. If there is _a minimum of the

potential

energy

along

the

line,

then the

particle

_can not

get

^from one

potential

well to the other

moving

this way and needs to _cross the

magnetic

field lines of force _near the saddle

point.

This leads to the _severe

diminishing

of the

potential

barrier

permeability (see

the second

asymptote

^of

(2.24)).

The

geometrical meaning

of the

quantity

^{S in}

(2.24)

is

supported by

the

following

fact _: the

equation

S

= 0 determines the

straight

lines of _constant

potential

_energy

going through

the saddle

point.

Complementary

to the consideration of the

limiting

_cases

(2.23)

and

(2.24)

in

figure

I _are

represented

the

dependencies

of the

quantity f (and, hence,

the thermoactivation _rate r

itselo

_on the

angle

o between the vector B and the reaction _» axis _z in the

symmetric

model ~~=

~~

= ~ for different values of the

magnetic

field in the limits _{y «} n and

y » n. It is _seen that in the

high-field

limit the thermoactivation rate r vanishes if o exceeds

some critical value

o~, tg (o~)

= ~ ~'~

3.

Appficatiton

to bulk materials.

The results for the thermoactivation _rate of _a

charged particle

in _a

magnetic

field obtained in the

previous

section _:

(2.17), (2.10)

_may be relevant for

understanding

such

phenomena

_as

ionic

magnetoresistance,

dielectric relaxation in _a

magnetic field,

etc. In

amorphous

compounds,

where the directions of the _« reaction coordinate _{axes z} of individual barriers

M 7 THERMOACTIVATION IN A MAGNETIC FIELD 1025

r/ r( oi= t/ to

i

i

a

o-e

DA

z

0.2

3 4

~ 5

0 O-I O-Z 0.3 DA 0.5 0.6 0.7 O-e 0.9

sind

(a)

r/ r(

0

)= f/ to

0.8

b

o-e

DA

0.2

3

o

0 O-1 0.2 0.3 DA 0.5 0.6 0.7 O-e 0.9 1

sin~i

(b)

Fig.

I. The

dependence

of the therrnoactivation rate of _a

charged particle moving

over 3-dimensional

symmetric (~~

= ~~ = ~

) potential

barrier in a

magnetic

field _on the

angle

0 between the vector B and reaction axis _z for different values of the

magnetic

field.

(a)

_~

= l, _{y «} D (I) _{p w}

qB/mD

= ;

(2)

p = 3 ;

(3)

_p

= 10_;

(4)

_p

= 30 _;

(5)

_p

=

100 _;

(6)

_p

= co.

(b)

_~

= l, _y ha _;

(I)

_a

m

qB/my

=

1;

(2)

a =

3

(3)

_a =10 _; (4) _{a =} 30

(5)

_a =100 _; (6) _a

= co.

and the

angles

between them and the _vector B _are

random,

the situation is characterized

by

some

spectrum

^{of relaxation times. Whereas the} calculation of the

conductivity

_tensor in such conditions is _a collective

problem,

the consideration of the relaxation

peak

in dielectrics due to _an ensemble of

non-interacting

two-well

species

reduces _to the

averaging.

Concretely,

_as the dielectric

susceptibility

of _an individual relaxator in the direction

perpendicular

to the axis

z is

negligible,

the

dynamic susceptibility

of the

sample

_may be written _as

3

~3~

^~

Cos~ (0E),

_{~~ ~~}

x = X0

@

_{T iw}

where _xo is the static

susceptibility, o~

is the

angle

between the vector

E(t)

and the axis z, the

integration

is carried out over the Euler

angles

o, _q~ and

ji characterizing

the orientation of the coordinate _{axes x, y, z} of relaxators with respect to the

laboratory

coordinate system

x', y',

z' with e~,^[

B,

the relaxation

frequency

r is

given by (2,17), (2.10)

with _p

=

qB/(mn )

and p~ = p cos

(o ),

_p~

= p sin

(o

_cos

(ji ),

_p~

= p sin

(o )

sin

(ji ).

With the _use of the

identity

_cos

(o~)

= cos

(o)

_cos

(p)

₊ sin

(o)

sin

(p

_{cos (q~}

),

where

p

is the

angle

between the vectors B and

E(t)

and

q~ is the azimuth

angle

of the axis z, one gets ^{the final}

expression

X = Xo

~

j~

^sin

^(o

^do

_j~

^~

^dji

^~

^{cos~ (o cos~ (p )}

⁺

sin~ (o) sin~ (p )j

4 w ~ ~

r iw 2

(3.2)

This

general

result reduces _to

simple quadratures

ⁱⁿ two

particular

_cases. In the

symmetric

case

(~~

= ~~. =

~)

the

integration

over is

trivial,

and

(3.2)

reduces to

r 2

~~~2 ( p )

₊ l

x~)

Slll~

(~ j

'

~~ ~~

~ = ~ ~0

~

~~

~ ~

~ ~

where T

=

T(x),

_x

= cos

(o)

=

B~/B.

In the

plane

case

(~~

= cc, ~

~ ⁼ ~

)

the relaxation

rate r is the function of

only

the y component ^{of the}

magnetic

field:

B,.

= B. sin

(o)

sin

(ji).

After _some

rearrangement

ⁱⁿ

(3.2)

_one

gets

~ _

j'~~ lj ^dy

~

~~~ i(~

^Y~~

^{~°~~ ~~}

^{~ ~~ ~ ~~~ ~~~~}

^{~~ l}

~

~~~~

where r

=

r(y),

_{y =}

B~/B.

In the

plane

case, where the thermoactivation rate of _a

charged particle

diminishes with the

increasing

of the

magnetic

field induction _{more or} less

uniformly

with respect to its

orientation,

the relaxation

peak

in

X"

W Im

(X)

is shifted _to the lower

frequencies

_or

high temperatures

and is somewhat smeared _as described

by (3.4).

A similar situation _was

investigated

in

[3, 4],

where the model of

rotating

electric

dipoles (the charged particles

_{on a}

sphere)

_was

adopted.

In the

general

three-dimensional _case the situation is much _more

interesting

because of the non-uniform decrease of the relaxation rate r with B

(see (2.24)

and the discussion

below).

Whereas for _some orientations of the coordinate _axes of the relaxator with respect to the vector B the

quantity

r diminishes

infinitely

with the

increasing B,

for other orientations r saturates at some level.

Accordingly,

the relaxation

peak

_may

split

at

high magnetic

fields in two

parts

^the

high frequency (low temperature)

saturated

part

^and the low

frequency (high temperature)

part

displacing

_more and _more with

increasing

B

(see Fig. 2).

The parameters ^of the relaxation

peak (in particular,

the

weights

of its two

M 7 THERMOACTIVATION IN A MAGNETIC FIELD 1027

X

a ^{i o .}

z

3

4 5

10~ 10~ 10~

_{0.01 O-i I IO 100}

~~

ml r(

°

X

b

_{i o}

z

3

4

~ 5

10~~

10~

_{10~~ a-al o-I}

I lo loo

w/r(O) b)

Fig.

2. The relaxational absorption

peak

due to the two-well

charged

relaxators for different values of the

magnetic

field. ~~ = ~~ = ~ =

l,~«Q

(0) _p

=

0

(1)

_p 1;

(2)

_p

=

3

(3)

_p

= 10_;

(4)

p = 30 _;

(5)

_p

=

100 _; (6) _p

= 300

(7)

_p

= 000.

(a)

p

= 0

(b)

p

w/4,

p is the angle between the electric and

magnetic

fields,

r(0)

is the therrnoactivation rate for B

=

0.

parts) depend

_on the

angle p

between the electric and

magnetic

fields and _on the form of the

potential

_{energy near} the saddle

point (2.2)

described

by

the dimensionless

quantities

~~~. Therefore,

measurements of the thermoactivation rate of

charged particles

in

high

mignetic

_{fields and of their} contribution into the

dynamic susceptibility

of _a

sample

_may be

used,

in

principle,

for

testing

the form of the

potential

barrier.

The results of this paper are valid for

arbitrary

values of the

parameter yin (strong

and

moderate

damping cases). However,

the situation in the _case of small

damping:

y s a

(T/Uo) (the

Kramers

regime)

is not _so clear.

Evidently,

the method of the solution of the Fokker-Planck

equation

used does not describe the nullification of the relaxation rate in the _{case y}

=

0,

which is understood from the

general

_arguments

(the coupling

between the

particle

and the thermostat

vanishes). Here,

in the one-dimensional _case, various alternative

approaches

were used. In

particular,

Kramers

[I]

considered slow diffusion

along

the _energy

coordinate,

while the fast variables _were

averaged

out. It is _seen that the

generalization

of this

approach

to the multi-dimensional _case is nontrivial

by geometrical

_reasons. The other («

naive») approach (see,

_e.g.

[6])

considers

multiple

oscillations between two wells with successive loss of energy

performed by

a

particle

with the energy close to the top ^{of the}

potential

barrier. Such _an

approach

does not suite the multi-dimensional _case because for the

general

form of the

potential

relief

(or

in the presence of _a

magnetic field)

the

particle overcoming

the saddle

point

with small kinetic _energy will not _come back.

So,

the behaviour

of the thermoactivation rate in the multi-dimensional small

damping (Kramers)

_case is

nontrivial and deserves _a separate

investigation.

In the absence of definite results in the small

damping limit,

in

typical experimental

situations

(say, T/Uo

l

/4)

the condition of the

validity

of the

present approach

_y » a

(T/Uo) (note

that this condition _was obtained in the _one-

dimensional _case and _may be

replaced

in the multi-dimensional

one)

_may be rather

restrictive, making

the moderate

damping region

a » y » a

(T/Uo)

too narrow.

Acknowledgments.

The author thanks Prof. V. S. Lutovinov for the discussion of the results of the paper.

References

[1] ^{KRAMERS H.} A.,

Physica

7

(1940)

284.

[2] ^{GARDINER C.} W., Handbook of Stochastic Methods

(Springer, Berlin, 1983).

[3] ^{GARANIN D.} A., LUTOVINOV V. S., LUCHNIKOV A. P., SIGOV A. S., SHERMUCHAMEDOV A. T.,

Fiz. Tverd. Tela 34

(1990)

l172.

[4] ^{GARANIN D.} A., LUTOVINOV V. S., LUCHNIKOV A. P., J. Phys. France 51 (1990) ^1229.

[5) DEKKER H.,

Physica

135A

(1986)

80.

[6] DEKKER H.,

Physica

136A

(1986)

124.