The influence of a magnetic field on dielectric relaxation processes

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The influence of a magnetic field on dielectric relaxation

processes

D.A. Garanin, A.P. Luchnikov, V.S. Lutovinov

To cite this version:

1229

The influence of

a

magnetic

field

on

dielectric relaxation

processes

D. A.

_Garanin,

A. P. Luchnikov and V. S. Lutovinov

Moskow Institute of

Radioengineering,

Electronics and Automation, Pr.

_Vernadskogo

78,

117454 Moskow, U.S.S.R.

(Reçu

le 17 mai _1989, révisé le 5

_février

1990,

accepté

le 7

_{février 1990)}

Résumé. 2014 Nous étudions l’influence du

champ magnétique

sur les pertes

diélectriques

dans des substances peu

polaires.

En utilisant la solution de

_{l’équation}

de Fokker-Planck décrivant la diffusion de rotation des groupes

polaires

à travers une barrière de

potentiel,

nous montrons

qu’avec l’augmentation

du

_{champ magnétique}

le maximum de la courbe des _pertes

diélectriques

tg 03B4 se

_déplace

vers les hautes

températures

et _diminue, en accord avec les résultats

expérimentaux [3, 4].

Abstract. 2014 The influence of

a

_magnetic

field on the dielectric losses in

_weak-polar

dielectric

compounds

is

_{investigated. By}

the solution of the Fokker-Planck

_equation

_describing

the rotational diffusion of the

polar

group over a

_potential

barrier it is shown that with the

_increasing

of a

_magnetic

field the maximum of the dielectric loss curve _tg 03B4 is shifted to the

higher

temperatures and diminished, in accordance with the

_experimental

data

_{[3, 4].}

J.

_Phys.

France 51 ₍₁₉₉₀₎ 1229-1238 1 er JUIN 1990,

Classification

Physics

Abstracts 77.40 - 61.40

The list of

_symbols.

U(O)

the

_potential

_energy of a

_particle

on a

_sphere

which models

_polar

groups in

polytetrafluoretilen.

0 : the

_{polar angle}

of the

_particle

with

respect

to the « easy

polarization »

axis

z.

(P : the azimuth

_angle

of the

_particle

under consideration.

Ç

: the

_angle

between the vector of

_magnetic

field induction B and the

_polar

axis

z.

f3

: the

_angle

between B and the

_alternating

electric field

_E(t)

=

Eo

_cos

(lût).

m : the mass of the

_particle.

q : the electric

_charge

of the

particle.

r : the radius of the

sphere

on which the

_particle

is

_moving.

y : the

_{phenomenological}

friction coefficient of the

_particle

_(the

viscous relaxation

frequency).

f2 : the

_frequency

associated with the

_top

of the barrier :

WA : : the

_{« attempt »}

_{frequency :}

i

ca T : the thermal

_{angular velocity :}

_w2T

= 2

TI mrL..

Wc : the

_cyclotron

_frequency

of the

_{particle :}

_wc =

qB /m .

: see

_{equation (2.6).}

(21T

:

_{(1/2 7T)}

Îo

g ( cp ) dcp; [

contains the

_dependence

of a

_polar

_group rotational

0

diffusion

_{pre-exponential}

factor

_To

on the ratio a =

y/f2

and on the

_magnetic

field

induction B. p =

(w ci n )

sin f/J

cos cp. k =

(w ci n)

sin

x

_{cos tb.}

y

sin e

cos cp. a =

w ci )’ .

v =

w ci n .

1. Introduction.

It is well-known that dielectric materials

_{containing polar}

groups demonstrate one or several

peaks

of dielectric losses at

_frequencies

_{wi -}

_T

_{where Fi}

is a relaxation rate of a

_polar

group of the i-th

_type.

In

_{weakly polar compounds containing}

small amounts of electric

_dipoles,

the interaction between

_relaxators

is

_negligible

and the

_dynamics

of the latter _may be understood in the framework of

_one-particle

models such as the

_Debye

model

_(rotating

with friction

electric

_dipoles)

_[1],

a model of a

_{charged particle}

in a double-well

_potential

considered

_by

Kramers

_[2],

etc. In relevant cases

_{(isotropic Debye}

model or

_high-barrier

Kramers

model)

the relaxation of electrical

_polarization

is described

_by

a

_single

_exponential

which leads to the

Debye-type frequency dependence

of dielectric losses.

Recent

_experiments

on various

polymer compounds

revealed the influence of a

magnetic

field on _{dielectric relaxation processes}

_{[3, 4].}

In all cases

the

maxima of the curve

tg

8 ( T) (tg 8

= e" 1 e’)

diminished and shifted to the

_higher

temperatures

with the increase

of the

_magnetic

field induction B. An

analysis

of

_experimental

data showed the activation

temperature

dependence

of the relaxation rate r with the

_{pre-exponential}

factor which diminishes with B

_depending

on mutual orientation of the vectors B and

_{E( t ).}

As an

example,

for

_{weakly polar polytetrafluoretilen}

_(PTFE)

_{F=Foexp(-1670/T),}

whereas

Fo = 3.2 x

105 Hz at B = 0;

Fo = 2.6 x

105 Hz at B =

1.4 T

_(B//E)

and

_ro

= 1.9 _x

10 5

Hz

at B = 1.4 T

(B

₁

E) [4] (see Figs.

1,

2).

The maximal value of

_{tg 8}

in PTFE is of the order

10-4

and the

_change

of the real

_part

of dielectric

_{permeability (e’ = 2.1 ) crossing}

the

_{dispersion region (T- 190 K)}

is very small. This indicates a very low concentration of

polar fragments

in PTFE and allows us to consider them

_{independently}

from each other. It is well established that the dielectric relaxation

_peak

at T - 190 K

is

due to

_{amorphous regions}

of

_{semicrystalline samples}

of PTFE and the

_dipole

moments are formed

_by

the ends of

_polymer

chains

_containing

5-10 monomer units

CF2 (see,

f.e.,

[5]).

The motion of these

_{large fragments}

may be considered

classically,

but it is

too difficult to handle actual multi-dimensional

_{dynamical equations}

with a

_{complex potential}

energy function which is difficult to extract from

_experiment.

Thus,

one has to resort to some

simplified

model of dielectric relaxation in

_weakly

_{polar polymer compounds}

sensible to the influence of a

magnetic

field.

Obviously,

the Kramers model

_describing

the activation

_dependence

of the relaxation rate

T of a

_polar

_group on

_temperature

cannot take into account the influence of a

_magnetic

field

1231

Fig.

1. - The

temperature

dependence

of the value of dielectric losses _{tg 8} for the PTFE film at the

frequency f

= 60 Hz for the different values of the

magnetic

field induction :

0) B

= 0 ;

1)

B = 1.4 T

_{(B//E) ;}

_2)B=

1.4 T

_{(B 1 E ).}

Fig.

2. -

one _may

_put

forward the so-called «

_{Debye-Kramers}

model » of

_rotating

electric

_dipoles

with

two

_equilibrium

states

_{separated by}

an _energy barrier. In a

magnetic

field there is the Lorentz

force

_acting

on effective

_{charges moving}

on a

sphere

which causes

curving

of

trajectories

and

interferes with them to cross a

potential

barrier. As a

result,

the

pre-exponential

factor

Fo

for such relaxators diminishes with the

_increasing

of a

magnetic

field.

In the next _{section of the paper} we will write down and solve the Fokker-Planck

_equation

describing

the diffusion of

_{charged particles through}

a

_potential

barrier in a

_magnetic

field for

the model mentioned above and calculate the

_{corresponding}

relaxation rate. To describe the dielectric losses observed in bulk materials it is necessary to _carry out the

_averaging

over the

orientations of the

_principal

axes of

polar impurities.

This is done in the final section of the

article.

2. Thermoactivation rate for the

_{Debye-Kramers}

model in a

_magnetic

field.

Let us consider a relaxator as a

particle

with mass m and

charge q moving

on a

sphere

of

radius r. The

equation

of motion of such a

particle interacting

with the bath reads

where U is an

angular

dependent potential

_{energy, N} is a reaction force

_confining

the

_particle

to the

sphere,

y is a friction coefficient

and C(t)

is the

_Langevin

_{force: ( , i (t)}

_{’j (t’)}

=

2 y m TS i & (t - t ’).

After

_projecting

out the normal

_component

of

_(2.1)

the force N

_disappears

and the

_quantity

_[vB

_{] is}

_{replaced by}

_(rB)[vr]/r2.

Now one can

proceed

from a

stochastic

_{equation (2.1)}

to the Fokker-Planck

_equation

for the

_{probability density}

G(r,v,t):

which is more convenient for calculations. In

(2.2) v

is a

_{two-component}

vector

_tangential

to

the

_sphere

at a

_{given point.}

For

_simplicity

we will restrict ourselves to the

symmetric

potential

energy

dependent

on the

polar angle 0

and

_parabolic

near the

top

of the barrier and the bottoms :

I.e.,

for

U = - K cos2 B

one has

_{Uo = ko = k =}

K.

In the

_{low-temperature region}

T «

_Uo

the

_{quasiequilibrium}

solution of

_(2.2)

is of the form

where g

is a kink-like function

_changing

in a narrow

_region

about the

_top

of the barrier

(Ao -

(T/k) 1/2

1 ).

Having

solved the

_{equation for g following}

from

_(2.2)

one can relate the difference of

_{populations N2 - N1 ~ g2 - g1 to}

the flow

_{of particles}

from one well to the other which is

_proportional

to the derivative

N.

So,

since the derivative

_{ag / a t is}

_{exponentially}

small in the limit T «

_Uo,

and in the relevant

_region

near the

_top

of the barrier one _may

_expect

the

1233

solved

_{analytically.}

In terms of dimensionless units the

equation

for the function g in the

region

_{x = 8 - 7r /2 1}

reads

where p

=

vo/vo;

Pcp =

vcp/vo;

vo =

r{J; n

is the

frequency

associated with the

top

of the

barrier :

{J 2 =

2

_{k/mr2 ; a}

=

y/ ;

_{p ==}

(wc/f2 ) -

sin

b -

cos _{cp ; Wc} is the

_cyclotron

fre-quency : Wc =

qB /m ;

is the

angle

between the vector B and the

_polar

axis.

As far as the variables _{x, p,}

_{p cp}

enter the left-hand side of

_{(2.5) linearly}

one _may search for

the solution in the form

where the

_{quantities e}

_{and e,,}

are chosen so as to

_give

_Auf"

in the left.

_Equating

the coefficients at x, p and

p,,

one

_{gets A}

= - e

and the

_system

of

_equations

_{for e}

and

({J:

where from the cubic

_equation

_for

follows :

Now

(2.5)

reduces to the

_ordinary

differential

_equation

The first

integral

of this

_equation

is

Now the

change

of the

function g

across the barrier may be calculated

directly :

This

_jump

is

_proportional

to the difference of

_populations

_{N2 -}

_N

1 where

The time derivatives

_{N1, 2}

are determined

_by

the flow of

_particles

over the

barrier,

i.e. :

The

_integration

in

_(2.13)

is carried out with the use

_{of (2.4), (2.6)}

and

(2.9) ;

uT is the thermal

velocity :

vT

= 2

Tlm.

Now

combining equations (2.10)-(2.13)

one arrives at the balance

equation

with the relaxation rate

where

_{£ù 2}

= 2

T /mr2

and _{lù A} is an

_{« attempt}

_{frequency » :}

wA2

= 2

ko/mr2.

The

quantity

e in

(2.15)

is the

_positive

solution of the cubic

_{equation (2.8) averaged}

over the azimuth

_angle

(p. In zero

_magnetic

field

_(p

=

0 )

_one

gets é7 = e = Co = 2 / (a + V a2 + 4).

In this case the

formula

_(2.15)

differs from that of Kramers

_by

an additional factor 2

w Aw T

related to

the

_geometry

_{of potential}

minima

_{(see (2.11)).}

Generally,

the

_positive

solution of

_(2.8)

_may be written in the form

where

and

In the case p = 0 the discriminant

Q

0 and the _upper

_part

of

(2.16) reproduces

the

zero-field result

_eo

cited above. With the

increasing

of p the

quantity Q

rises and after some

p = Pc becomes

positive ;

the

solution e

monotonously

decreases with p. The transition

between two

_regions

in

_(2.16)

becomes

_sharp

in the

_{small-damping}

limit a =

y / f? «

1 :

This result means

_that,

for a

given

azimuth

angle

_ç, the

permeability

of the barrier falls

abruptly

to values as small as y at some critical value of the

magnetic

field :

Be = mfJ/(q. sin I/J cos ’P).

With the

_increasing

of the

_{quantity a}

the

_boundary

value decreases from the

_unity

and the

curve e

(p )

is

_being

smoothed. In the

_damping

controlled limit a > 1 one

_{gets :}

This result _may also be obtained

_by

the solution of the

_{appropriate Smoluchovsky equation}

following

from

_{(2.2) (see}

Ref.

_[4]).

The

_high-field

_asymptote

of the

_general

solution

_(2.16)

is

1235

The

_averaging

of the

_{quantity e}

over the azimuth

_angle

cp may be carried out

_analytically

only

in the

_limiting

cases

_{corresponding}

to

_{(2.18)-(2.20).}

For _y » f2 with the use

_{of (2.19)}

one

obtains

In the

_opposite

limit y « f2 the

integration

of

_{(2.18) gives}

where K and E are

_{complete elliptic integrals}

of the 1 st and the 2nd kinds. The low- and

high-field limits of

_{expression (2.22)}

_{are é = 1 - k 2/4}

_{and i = 1/2 k}

_{correspondingly.}

The

derivative of the relaxation rate T with

_respect

_p to the

_magnetic

_g field

_{ar laB -}

de - -

oo at

dk

k = 1.

And,

at

last,

the low-field

expression

for )

for the

arbitrary

y/f2

is

easily

calculated

with the use of

_{(2.20) :}

3. The dielectric losses in bulk materials.

In the

previous

section of this paper we have calculated the relaxation rate T of

_polar

impurities

in

_weak-polar

dielectric materials in a

_magnetic

field within the framework of the

high-barrier symmetric Debye-Kramers

model. It was shown that the influence of a

_magnetic

field is to diminish the

_{pre-exponential}

factor

_ro

for thermoactivated transitions of

_charged

groups

through

a

_potential

barrier

_{(see (2.15)}

_{and, f.e., (2.21), (2.22)).}

The effect becomes

large

for

_magnetic

fields

_{corresponding}

to _wc

_{sin gi}

> max

_{( y,}

_{f2 ) (the}

definitions were

explained

in the text below

_Eq.

_(2.5))

and,

as it _may be seen,

depends

on the

_angle

gi

between the vector B and the

polar

axis

_{corresponding}

to the

_{lowest-energy}

orientations of the electric

_dipole.

In order to calculate the dielectric losses in non-structured bulk materials

(i.e.,

polymers)

one has to _carry out the

_averaging

over the orientations of the «

_{easy-polarization »}

axis

z. As the dielectric

susceptibility

in the direction

perpendicular

to the axis z is

negligibly

small,

the

_{susceptibility}

of the

_sample

may be written in the form

where O is the

_{body angle}

circumscribed

_by

the axis z,

0E

is the

_angle

between the axis

z and the electric field

_E(t),

_{X o} is the static

_{susceptibility}

of the

_sample.

For the calculation of

(3.1)

it is convenient to choose the

_{« laboratory »}

coordinate

_system

XYZ with the axis Z

_along

the vector B and the axis Y

_along

_[EB].

In this coordinate

_system

cos

_{0 E}

=

_{cos «/J}

cos

_B

+

sin «/J

sin

_/3

cos _{’P z} where

_B

is the

_angle

between the vectors B and E and _çoz is the azimuth

_angle

of the « easy axis ». Now after the

integration

over

where x =

cos «fi.

As it _may be seen from

_(2.15)

and

(3.2),

with the

_increasing

of a

_magnetic

field the initial

Debye

curve of dielectric losses

_{tg 5 ’" X"}

=

Xo

r w / ( r 2 + w

2)

is shifted to

lower

_frequencies

or

_higher

_temperatures

and smeared

_becoming

a

_{superposition}

of

_Debye

curves with a wide distribution of relaxation rates. The latter leads to the

_diminishing

of

(tg 5 )max

with a

_magnetic

field which was observed

_{experimentally [3, 4].}

The

_dependence

of dielectric losses on the mutual orientation of the vectors B and

_E(t)

is also contained in formula

_(3.2).

One _may check with the use of

_(2.23)

_that,

in the lowest order in

B2,

the curve

_{tg 5}

remains undeformed and the effect of a

_magnetic

field is to renormalize the relaxation rate :

Similarly

to

_experimental

observations

[4],

the effect is about two times

_larger

for the

perpendicular

orientation of the vectors B and E than for the

_parallel

one.

For the

_large

values of a

magnetic

field

(wc > max (y, f2 »

one has to _carry out the numerical calculation of the

_{integral (3.2)}

where the

_{quantity r(x)}

is itself in the

_general

case

y - f2

the result of the numerical

_{integration (see (2.15)).}

To illustrate the

_possible

behavior of dielectric losses at the

_{arbitrary magnetic}

fields let us consider the

_{high-frequency}

limit

w » T. In this case the situation

_simplifies

since one _may

_change

the order of the

_integrations

and choose the more convenient

_{variable y}

=

sin gl ,

_cos

entering

the basic

equation (2.8)

(p =

(wc/il)

y ).

In such a _way the result reduces to the

_{ordinary quadrature :}

where

_{T (y)}

is the « relaxation rate » not

_averaged

over the

_{angle cp}

which is

_{given by}

formula

(2.15)

with the

_{substitution i => e ;}

_r(O)

is the zero-field result for the relaxation rate of a

polar impurity.

The coefficients

_{g l, 2}

containing

the

_magnetic

field

_dependence

of the effective relaxation rate in

_(3.4)

may be calculated

analytically

in the limits y » Q and y « f2. For

y » f2 the results obtained with the use of

_(2.19)

are as follows :

1237

where v =

w,lf2.

It is

easily

checked that at _zero

magnetic

field

(a

or v =

0)

g l, 2

= 1. In the

high-field

limit the functions

g 1, 2

vanish as

1 IB

_{(g1,2 ~}

1 1 a, 1/v )

the

quantity

g2

being

two _{times smaller than gl. The latter correlates with the} low-field result

_(3.3)

and the

experimental

observations

_[4].

It is seen from

_(3.6)

that,

in the

_{small-damping}

limit

y «

_il,

the

_averaging

over the orientations of the

_principal

axes of

polar impurities

smears

the

_{logarithmic singularity}

of the derivative

_{aT /aB}

manifested in

_(2.22)

at k = 1.

Never-theless,

the

_vestiges

of the

_singularity

at the critical value of a

_magnetic

field

_(see

also

_(2.18))

are

_clearly

discernable in

_figure

3 where the

_{analytical dependences (3.5)}

and

_(3.6)

are

represented.

One can see that the

_magnetic

field

_dependence

of the relaxation rate of

_polar

impurities

in the

_strong

_{damping ( y}

>

_il)

and in the weak

_{damping ( y}

«

_f2)

cases are

substantially

different which may be used for the identification of the

regime

realized in the

experiment.

Fig.

3. - The

magnetic

field

_dependence

of the relaxation rate of

_{polar impurities (the}

coefficients

g1,2 in

(3.4)-(3.6)

as the functions of the _parameters a or

_{v) :}

₁₎

and

2) ’Y n,

B//E and

B 1 E ;

3)

and

4)

y « f2, B//E and B 1 E.

Finally,

we have calculated

_numerically

the

_imaginary

_part

of the dielectric

_{susceptibility}

(3.2)

in the

limiting

cases considered above as a function of

_temperature

for the zero-field

parameters

taken from the

_{experiment [4] (see}

the

_{Introduction).}

The results

_represented

in

figure

4 show that the theoretical model

_developed

in the

present

paper

reproduces

the main

qualitative

features of the influence of a

_magnetic

field on the dielectric losses.

_However,

some

_{discrepancies (i.e.}

on the

_{high-temperature wing}

of the relaxation

_curve)

between the

theory

and the

_experimental

data

_{(see Fig. 1)}

are observed. This is not

_surprising

because the

simplest

of theoretical models was

_adopted.

In

_particular,

we have

_neglected

the

_possible

dependence

of the

_potential

energy

U(O,

cp)

on the azimuth

angle

cp, the effect of the

orientation of molecular chain

fragments

in a

_magnetic

_field,

the

temperature

dependence

of the

_{phenomenological}

friction coefficient y, etc. The clarification of these factors

require

more extensive and

_{purposeful experimental investigations}

which are in progress.

One more

_question

concerns the behavior of a thermoactivation rate T in the limit of _very small

_damping

constant : y s

_{OT /vo.}

Here,

in the one-dimensional case, a

_particle

with the

energy close to the

_top

of a

potential

barrier

(AE - T )

can

perform multiple

oscillations

multi-Fig.

4. - The theoretical results for the

temperature

dependence

of then value of dielectric losses tg

& - X " in a weak-polar compound in the strong damping limit

( y

>

_{f2 ) : 0)}

a = 0

(a

=

qB / ym ) ;

1)

a = 2,

B//E ;

_{2) a}

_{= 2, B 1 E ;}

_{3) a}

= 10,

B//E ;

4) a

= 10, B -1. E. In the weak

damping

limit

(y

) the results are

_visually

the same.

dimensional case, because the

_particle

which overcomes the saddle

point

with small kinetic energy can come back

_only

if a

_potential

_energy

_profile

has some

_special

form

_{(for example,}

allowing splitting

of

_variables).

One may say that a multi-dimensional

potential

well is in some sense similar to the

_cavity

which is used for

_modelling

the

_absolutely

black

body.

Acknowledgments.

The authors thank A. T. Shermuchamedov for the discussion of the results of the _paper and A. S.

_Sigov

for the

_support.

References

[1]

DEBYE _P., Polar molecules

_{(Chemical Catalog}

_Co.,

_New-York)

1929.

[2]

KRAMERS H. A.,

Physica

7

₍₁₉₄₀₎

284.

[3]

LUCHNIKOV A. P., SHERMUCHAMEDOV A. _T., KAMILDJANOV B. I. in : The electric effect and the electrical relaxation in solid dielectric materials

_{(Moskow 1986),}

_p. 90.

[4]

GARANIN D. A., LUTOVINOV V. S., LUCHNIKOV A. P., SIGOV A. S., SHERMUCHAMEDOV A. T.,

Fiz.

_Tverdogo

Tela 32

₍₁₉₉₀₎

N° 4.