the

abdus

K/98/205

salam

international centre for theoretical physics

QUANTUM THEORY OF PARAMETRIC EXCITATION IN PLASMAS WITH THE DRIVING FIELD SPACE DISPERSION

Vo Hong Anh

..I.

“-.

Available

at: http://www.ictp.trieste.it/-pub-off

United

Nations

IC/98/205

Educational International

THE

ABDUS

SALAM

Scientific and Cultural and Atomic Energy Agency

INTERNATIONAL

QUANTUM THEORY IN PLASMAS WITH THE

CENTRE

Organization

FOR THEORETICAL

OF PARAMETRIC DRIVING FIELD

PHYSICS

EXCITATION SPACE DISPERSION

Vo Hong Anh’ Vietnam National Atomic Energy Commission, 59 Ly Thuong Kiet St., Hanoi, Vietnam2 and The Abdus Salam International Centre for Theoretical Physics,

Trieste,

Italy.

Abstract A development

of the quantum

to take into account equation

of motion

theory

of parametric

the effects of space dispersion method

with

equations

the excitable

waves region both in wave number

of dipole approximation

for the eigenmodes

considered

of the driving

the use of appropriate

of dispersion

wave excitation

of vibrations.

matrix

MIRAMARE

‘Senior Associate of the Abdus Salam ICTP. *Permanent address. FAX: 84-48-266133, E-mail:

formalism

values and directions

- TRIESTE

November

external

Calculations

earlier.

1998

vhanh98Qhotmail.com

in plasmas is presented fields. The quantum leads to the system

show the enlargement as compared

of

to the case

1. Introduction.

Formulation

The quantum usually

of the problem

theory

of parametric

with

an acceptance

Weak Coupling

Approximation

developed

phenomena

in plasmas

of the following

(solid state plasma

conditions

concerning

included)

is

the physics of the

systems (see [1,2]):

l

for particle-particle

interactions

in the system

consid-

ered, i. e.:

(1) where V is the interaction

potential

energy, K is the kinetic

energy of particles

and a is a con-

stant - 1. l

Random

Phase Approximation

(RPA)

in the collisionless

limit,

that

is:

UJI-p >> 1, where w, is the driving l

Dipole

external

Approximation

This means that

field frequency concerning

the wavelength

etc.)

the space dispersion

lengths

so that the external

field wave number

systems has been described

As for the third, mathematical

complications

small dimensions with

like thin

the wave number

conditions that,

Dipole

required

field.

than all the ones of internal

wave

or the ones of the region

x can be neglected.

in the treatment

However,

superlattices

of a set of dynamical

properties

of the theory

connected

some

under

wells, the case of wave excitation

consideration

does not satisfy

and the going out of it becomes represents

with

in many cases (e.g. for the systems with

and quantum

the system

by this Approximation

such a development

(driving)

the release of it is usually

and difficulties.

k N x etc.)

of the pumping

particles.

in [3,4].

Approximation,

films,

time between

(e.g. the system dimensions

A going out of the two first approximations of plasma

and rp is the collision

of this field is much larger

fields as well as other characteristic considered

(2)

certain

methodological

necessary. interest,

all the Besides

as one can

see below. So, in the present

the driving plasma,

treatment,

electromagnetic

the electric

in order

radiation

to take into account

(laser)

field and the vector

the effects of space dispersion

field on the parametric

potential

=

phenomena

in a

of this field are taken in the form:

E,(r, t) = -i aA$i’ t), Ao(r, t)

excitation

of

~[&‘~‘Pnt)

2

(3) + c.c.]

(4

Calculations

are carried

neous electron

out proceeding

from the basic Hamiltonian

H

=

H,+H,,+Ht,

He

=

;TT: %&iap

(5)

- 2

C(e-i”tai+x

47re2 9s

%a;+qaP+laP’aPr

=

P,P’,4

Ht

=

cu(q, P7q

where He is the Hamiltonian radiation

field,

representation

Furthermore, appropriate equations

matrix

Hee is the Coulomb

of an external

of Motion

which

quantum

Formalism

distribution

Heisenberg

term and U(q, t) in Ht is the space field.

method

will be developed

helps one to lead to the solvable

in Matrix

with

the use of the

system of dispersion

Form

above, we first derive the equation

of motion

for the so-called

function,

GJ + q, P, t> 3 in the well-known

electro-

of vibrations.

For the purpose just described generalized

“testing”

of motion

(7)

’

in the presence of an external

interaction

potential

equation

formulation

for the eigenmodes

2. Equation

subsystem

haq2

P2

$=G’

t)a,‘tqapl

of the electron

the quantum

(6)

+ ei”ta,+-xa,)p,

P

Hee= ; c

Fourier

homoge-

system of the solid or the plasma in general which consists of 3 terms:

P

magnetic

for an isotropic

representation

a~W,+,(t)

(9)

>

using the Hamiltonian

(5) - (8). The calculations

give:

(ep+q- ep- i-&)F(p + q, P,t) = c[ep(k,

t) + U(k, t)lJ’(p + q, P + k, t) -

k

x[ecp(k,

t) + U(k,

t)]F(p+ q - k, p,t) -

k 6 mcpie

eAo

%(P

-i”tF(p + q,p + x, t) + eZRtF(p + q,p - x, t>l+ + q)FWp

+ q - x, P, t) + eiRtF(p + q + x, P, t>l b&t)

= c

‘Pk < a,+-kaP

>).

(10) (11)

P

Writing

F(p +

q, p, t) in the form

F(p+q,p,t)

=

npWq,O

3

+ f(~

+ q, P, t),

(12)

rip(t)) f we come to the following time Fourier

equation

< up+(t)up(t) >,

for the perturbation

(13)

function

f(p

+ q, p, t) (in the space -

representation):

k*+q ‘P&P+,

-

nP> c P’

-

W(P

Ep - w -

+ % P, d> =

f(P’ + q, P’, WI + bp+q - ~PN%~ w) -

e-4, g& [Pf(P + cl, P + x, w-~2)-(P+q)f(P+q-xX,P,w--R)+ Pf(P+q-X,P,w+~2)-(P+q)f(P+q+X,P,w+R)l. Now, for further

consideration

of equation

=

f p,mn P p,mn

G

A p,mn

E

(14) we introduce

&p

-

Ep

E

With

G

- (w + nil)

+

as follows:

(17)

- i0 ’

w + T-AI),

(18)

4rre2 4q

notations

(15) (16)

++q+mx - np -(w+nR)-i0' Ep+q+mx 1

u mn = U(p+ mx,

A

the matrix

f(P + q + mx, P, u + nq,

Ep+q+mx

(Pm

(14)

(19)

mx) ’

A,=eE,. 2mc 2m!d

(20)

the use of (15) - (20) and the Poisson equation

Wr, PW)

t> = 4w+, =

(21) (22)

t),

Cf(P+%P>% P

equation

(14) obtains

the form:

p’,m,n

where

Bpmn p,m,n,

= Spp’6mm’6nn’

Cbfpmn ptmtnt AAprnn[P(Sp~p+x~rn~rn-l&/n-l

(P

(Pm

p

pmn

-

6

cPY pmn

4

(24)

6

m m ’ nn’ -

+ 6p~p-x~m~m+lb~n+l)

+ q)Spp’(6m’m-l&‘n-1

I)

+ ~m’m+l&‘n+l)l~

-

(25)

Otherwise,

one has:

f

pmn

CC

=

fjpnn -1p p’m ’n’ >

p’m ’n’

p’m ’n’ u m ’n’ 7

(26)

from which it follows:

= p(q+mx,w+nR) = Bpmn -ly p’m ’n’ = p’m ’n’ > U PP’

Pmn

Cf

pmn

P

(27)

u m/n’.

Now, setting

(28) PP’

one has:

P mn = c

D,“l”,t

(29)

U,t,t,

m ’n’

and

c (D,mT-,,)-’ m’n’

The system of equations dispersion

properties

3. Plasmon To consider

(30) is the basic one for the investigation

in the Two Modes

the wave excitation radiation

picture

persion equations Now, with

of different

dynamical

and

in a plasma

(Dm’nr)-l

system of equations for different

placed under the influence

of a strong

(30) to have:

Pm’n’ = 0.

determining

Pmn(q + mx, w + nn) from which the dis-

cases of wave propagation

the use of the known

the system of equations

Approximation

field we set U = 0 in the system of equations

C

This is the infinite

(30)

of the plasma system.

Excitation

electromagnetic

prnfn/ = Urn,.

and excitation

can be obtained.

tensor relation:

(31) can be expressed in the following

EmnPmn

-

C (E) m ’n’

5

flm’n’

more detailed

= 0,

form:

(32)

where

=

Emn

1 -Cpmpmn,

X ;Jnnnn)=

(As)

C

Pmn

= C

(33)

Ppmn,

ApmnPpm~n~(fiZm-~~n~n-~

+ Sm~m+lb~n+l)*

(34

P

Note that in the absence of external well-known

dispersion

equation

wave fields one has A = 0 and the system (32) turns to the

for plasma waves :

&(q,W) = 0. In the presence of the external approximation,

i.e.

retain

laser field, let us consider

in (32) the terms

with

POO = phw), p-l-1 = p(q- x,R - w) ), w h ic h quencies

(35)

w and w’ = (w - R] propagating

the lowest so-called

m = n = 0 and m = n = -1

means the consideration

under

from which one has in a more explicit

=

modes”

(note that

of two modes of fre-

the angle cy = L(q, q’) to each other

]q] = ]q’], g + g’ = X. In this case the system (32) yields the dispersion

EooE-1-1

“two

pmp-l-l

when

equation:

(36)

>

form:

(w2 - w,“)(w’2 -w,“,

= (Aq)2(w’ + w cosa)(w + W’COSQ).

(37)

Setting R = 2w,, and solving

Eq. (37) for the growth

w = wp + iy,

(38)

rate y, one comes to the result: (39)

It is interesting

to note that

0 0 This result excitation

if

is exactly in electron

4. Inclusion The theory

x=0,

E,=O, cu = lr,

so that

in agreement systems

if

with

one has

Y = 0,

one has

y = 0.

the conclusions

of the quantum

theory

of parametric

(see [1,2,5]).

of Phonons developed

in the previous

considers the mass m as the electron

sections can be applied

effective

mass derived 6

to the solid state plasma if one

in solid state physics in the so-called

effective mass approximation. However, interaction

more explicit

results

can be obtained

only

by including

effects which will be done in the present section.

the following

For this purpose

- phonon

we proceed from

Hamiltonian:

Hep

This is the Hamiltonian - phonon

the electron

interaction

=

xukQkxa;+kap, k

(42)

Qk

=

-=c bk + bz,p>,

(43)

pk

=

&.

(44)

(5) added by the phonon term

Hep.

(b,f and bk are respectively

the electron

- phonon

interaction

to see that

Hamiltonian

Here Qk and Pk are the phonon

respectively

It is not difficult

subsystem

the phonon

coefficient

the equation

creation

determined of motion

HP and the electron

coordinate

and annihilation

operators),

in the general lattice

for the phonon

and momentum,

coordinate

dynamics

is

Vk

theory.

can be written

in the form:

(45) or, in the Fourier

representation:

Q&4 = Q,(w)P,(w),

(46)

Dq(w)

(47)

where

is the phonon

propagator

= (w + iv:2 _ 0;

and

PSW

= c

f(P + q, P, w).

(48)

P

Using the Hamiltonian derive

the equation

combination obtain

with

equation

of motion Eq.

the corresponding

but transparent

(40) with

enough.

(6) - (8) and (41) - (44) taken into account,

for the function

F(p

+ q, p, t) in analogy

(45)) carry out all the following equations

of the matrix

Thus a procedure

placed under the action of a high intensity

formalism.

similar

for the case of two - mode interaction

transformations

and excitation

external

7

to (10) and (in the and simplifications

The calculations

to the electron

to

are cumbersome

case leads to the dispersion

in the electron

electromagnetic

we can now

radiation

- phonon field:

system

2?~~-1x,,l-l EooE-1-1=

poop~l~l

(49)

I

where

n;+q - n”p Eoo=Ehw) = 1 - [cps+ bq12Dq(41 T Ep+q_ Ep_ w _ irl is the dielectric

function

in the high frequency

of the electron

long wavelength

- phonon limit

(50)

system in the absence of external

obtains

the well-known

fields that

form:

E(q,w)=l-$-~ (w?yzwJ (1-Z).

(51)

w2

In this case the dispersion

equation

E (q, w) = 0 gives two known

longitudinal

modes wr , w2

(see,e.g.[1,23). In the presence of a strong laser radiation

we obtain

the growth

field, choosing

R

=

w1+ w2,

x

=

q1+q2,

the resonance condition

as follows:

(52) hl

(53)

= lq2l = 4,

rate for this pair of modes as follows

(a: = L(ql,qz)):

(Wlcoscu+wz)(w2 coso+w1)

(w? -w’ (w; -w,“) ,) w

+w2

(54)

WlW2

It is interesting

to find the CY-regions

0-y

(The notation

=

0

v

=

Im

07

>

0

for for for

with different

o=arccos

(

--%

>

values of the rate y . So we have:

(55)

,(7r>o>i);

(56)

7r>o>arccZ(-z),(--z>coso>-1); O