The ponderomotive force of a high frequency field acting on an

ENSE:'iA:"ZA
REVISTA MEXICANA DE FíSICA 45 (3) 322-325
JUNIO 1999
The ponderomotive force of a high frequency field acting on an isotropic plasma
C. Gutiérrez- Tapia
Instituto Nacional de !lIrestigaciones Nucleares. Departamento de F(sica
Apartado posta//B-/027,
//BO/ México, D.F., Mexico
Recibido el 20 de marzo de 1998; aceptado el 12 de diciembre de 1998
Using the hyJrodynamic approach lO describe a plasma, an expression for the ponderomotivc. force ~f.a high frequency field acti~g on an
isotropic plasma is obtaillcd and shown in a simple way. Non stationary (crms and the cffcctlvc COl1l51011 [requency belween partlcles are
lakcn ioto account. Thc collisionless approximation is analyzed.
Kl'.Y\\'ordc Pondcromo!ivc
force; isotropic plasma
Usando el modelo hidrodinámico para describir un plasma, se muestra en forma simple la obtención de una exprcsión para la fu~rza po~dero.
motriz de un campo de alla frecuencia que actúa sobre un plasma isótropo incluyendo lérminos no estacionarios y una frecuencia efectiva de
colisioncs. Se analiza e; límile no colisiona!.
Descril'fores: Fuerza pondcromolriz;
plasma isótropo
rAes: 52.35.Mw
I. Introduction
Two non.linear mechanisms, excluding non-linear wave interaetions, rcsponsihle 01' high frequency (IIF) field mornenlum and energy transfer lo lhe plasma particles can be clas.
sificJ. rhe ¡¡rsl one consisls in lhe transfer of momenlum
p = f¡k and cnergy n" = n\.JJ by a mechanism rclated ro
lhe inhomogcneily 01' lhe HF ficld amplitud~. Here, k, w and
ti are lhe wavc vector. Ihe frcquency, and the Planck conslant
diviJed hy 217. rcspcclively. J( is known lhat a time averaged
1<1I'cc1I.2J
.,
(fon)
e-
'J
= -,---"'--¡- vi Eo 1- ,
'1J11nW
(I)
ac!s on plasma particles. where eo ami mo are the dlarge
anJ Ill.:J.SS nf lhe particles. respectively.
Ea is Ihe HF field
ampli[ude.
Al Ihis point i[ is important to spccify I\\'o concepls used
in Ihis \\'ork: the valid frequcncy mnge 01' lhe melhod used
and Ih\.' validilY of Ihe.:assurned linear rclation bCI\\'ccn the
currenl and lhe HF clcctrie field. The appro.xirnation of a
HF ficld assumed in our (heory is: Ihe oscillation frequency
-,-,'01" lhe elcctromagnctic
field is largc compared wilh Ihe
Laflnor (q:clotron) frequcncy O
cJJ/lne.c ami wilh the
reciproi.:al transit time t'/ L, where v is the particle vcloc.
ity anJ L is the characteristic
length 01' Ihe de"ice 13J. In
(hc Cl.lllisinnal case, il can be imposed Ihe additional condirion ..•.' » /1, whcre 1I is rhe effeclive eollision frequency
hetwcL'n paniclcs. There cxisl several applicalions,
whcre
lhe study 01 lile motion of charged particles in a HP fields
is importanl as found in focusing and Irapping 01' particles.
I"or L'\,-llllplc. These processes mean Ihat particles are elaslically hound lO an axis or a coordinate in spacc ir a binJing force acls on thcl11 increasing Iinearly with dislance r.
=
i.e., F = -er, where e is a constant (4]. In practice, 1'0cusing is ohtained using a paraholic potential 01' the forrn
,p '" (el]:' + (3y' + 'yz') , in eartesian coordinates [3,4). The
components, along Ihe x, y and z directions, of lhe equation of motion of lhe partieles moving in a parabolic potenlial as abo ve can be written in tcrms of the Malhieu equalions
[lA]. The analysis 01' the cocfficients 01' the Mathieu equations determines hoth the stahle regions (focusing and trapping) and the instahle regions (acceleration) of partiele motion respective1y [51. A particular case of particle motion in
n paraholic potential is the 2-dimensional
quadrupole speclroscopy where the focusing of particles is rcalized using a
HF e1eelric field [4 J. In these de\'ices, the rnost rele\'ant frequeney region ocellrs when -.! > (l/ro) J2e ¡'/m, where V
is the HF \'oltage applted nnd ro is the half-distnnee hetween
elcclrodcs. To achieve a stl'Ong I"ocusing (trapping) of partieles (3-dirnensional quadrupole field), is tecommended
(6)
lo use rOlating magnetic flclds 13. l. 8] whcre. the frequency
01' rotalion (..¡.. is comparahle Wilh rhe Larrnor frcquency. An.
olhcr areJ of application of HF fields is found in wave guides
alle! n:son<ltors theory [5J. If .•: «
Le - \I/here :..JLe =
-1i711¡(,:.? /1I1e
is Ihe electron plasma frequency, within Ihe
framcwork of Ihe linear theory-the
HF tleld can be introduccd as far as the skin penctration depth amilhe approxima.
lion 01' [he ponderomorive force is no longer va:id.
rhe assumption 01' a linear relation berwecn the HF field
and Ihe curren! is valid when rhe influence ol' lhe HF field 0/1
lhe rlasmJ is slll,-lll. In ;'.n isolropic plasma, this condilion is
e.xpressed as Eo « E,. [9J. where
"",".t
Ec
Ep
= -;:¡::- :.!:l/2
.
Herc k = :-,.J/I.: and E1I -= 31~11If'..•..
,l/e'2. Te is Ihe elcctron
rcmpcraturc and 11I,. is the clcctron mass. In Ihe opposilc
THE PONDEROMOTlVE
FORCE OF A HIGH FREQUENCY FIELO ACTING O;>;AN ISOTROPIC PLASMA
case. for a large-amplitude
HF field (Ea» E,). we eross
inlO Ihe region of non linear interaetion of a HF field with a
plasma where lhe interaction between wavcs beco mes impor(ao1. One of lhe most intcrcsting processes of lhe non-linear
interaction of wa\'cs is lhe parametric resonance, This proces s is related to the aetion of a strong (Iarge-amplitude)
HF
field inducing a strong periodie lime variation of the plasma
pararnetcrs. These nuctuations, in conjunction with lhe externally imposed HF field can resonantly couple (parametric
resanance). When the plasma parameters oscillations beeome
sufficiently strong, sorne plasma mode is driven unstable
by the externally imposed HF field at a different frequency
(parametrie instability). Parametrie instabilities of this kind
have been investigatcd intensivcly in relation to lhe 1a5erplasma inleraelions [la].
To show the influence of the force on a plasma. it will be
considered lhe force (1) per unit volume of plasma (praetieally on eleetrons).
=_
2
wLo
16rrw2
vlEol2•
(2)
where Xo. and ;",;L} = 4rrXo.e~/mo. are lhe dcnsity and
Ihe plasma frequency of the speeies a respeetively. In a
cold isotropie plasma the dielectric permittivity has the
fmm [9. 11]
W2
[-l--!&.
-
w2
(fo)
= -- 16rr (e -
.,
1) vIEol-.
In a more general form lhe cxpression
(fo)
=
E:ij -
16rr
óij
lasers and masers capable of delivering peak powers [13. 14].
exceeding the tera-walt level the coneept of lhe ponderomo(¡ve force has aequircd a renewed interest. Reeently, (he
problem for ohtaining an cxprcssion for the non s(ationary
ponderomotive force of a HF field has been diseussed exlensively [15J. Among the most imponant applieations are
the following: the beat-wave and weak field partiele aceeleration methods [16-18J. the ion eyelotron isolope separation [19-22J. lhe generation of intense magnetie fields using
the in\'erse Faraday effeet [23-25J. lhe generalion of driven
currenlS by non induelive methods [26-28] and recently a
melhod has been pro po sed using the ponderomotive
force.
to produce a plasma cdge rolation in tokamaks in order lo
supprcss, by a forming thermal harrier, the turbulence resulting [ram the anomalous transport and incrcasing the energy
confinement time in lhese machines [29-31].
An express ion for the pondcromotivc force can be realized using single parlicle [32-34]. hydrodynamie [15. 35. 36]
or kinclic approaehes [15.37]. As can be found in lhe lileraturc, the cxprcssions for the non.stationary ponderomotive force ohlained in lhese approaehes differ. This constilutes the origin of controversy between the expressions for
lhe cnergy-l11olllentulll tensors proposed by }'1inkowski and
Abraham [38.391.
This paper is organized as follows: in Sect. 1 the basic
cquations arc obw.incd. In Sect. 2 an cxpression for the non
s(ationary pondcromotive force is obtained where the colli.
sionlcss limit is analyzcd.
.
In this case {he exprcssion for lhe pondero motive force acting
on a unir volume of plasma has lhe form
1
323
(3)
(3) can be written as
vEo,Eoj.
2. Basic equations
Thc interaClion 01"a HF f1cld with plasma particles \ViII be
dcseribed using hoth Ihe hydrodynamie plasma model proposeo in Ree 40. considcring a collisional tcrm and Maxwell
cquations:
(4 )
where E: ij and óij are the dielectric and unit tensors respectivel\'. As can be seen from (1). (3) m (4) this force has a
pote~tial formo In general. the dieleetrie tensor slrongly depcnds 011 geometry 01' the electric and magnetic fields. Se\"eral rnethods to ohlain lhis tensor are shown in Refs, 9. 11,
and 12.
Thc second non linear mechanism is reIated to the time
dependence of the HF field amplitude. In panicular. the force
aeting on the plasma particles arises when the HF field amplitude increases \\'ith time. Duc to the (ime dependen ce of the
amplitudc in lhe cxpression for the stationary pondcromotivc
force (1). (3) or (4) new lerms will appear. In the lilerature.
lhe expression for the ponderomotive force, which ineludes
lime Jepcndcnt tcnns. is known as the non-stationary pond~rornotivc force.
1I is a faet that the magnitudc of the ponderomotive force
is srnall hUI with the auvent of Bew radiation sourccs likc
1 DE
4rreo
--D + --noVa,
e t
e
1 DB
V x E= ---o
e 81
v
x TI
=
(7)
(8)
(9)
whcrc flo ilOlhe cffcctivc eollision frequency between parti.
eles. Assuming Ihal eleetrie E and magnetie B fields. lhe
veloci(ics Vo and the densi(ies /lo have all a slow and rapid
time dependcllccs. lhc)' can be v.'riuen in the form
E
=
Rel'. Mex. Fú. 45 (3) (1999) 322-325
(E)
+ E.
B
= (B)
+ B,
(J O)
C. GUTIÉRREZ-TAPIA
324
whcrc (he anglc hrackcts
denote an avcraging
(.r(l))=-
1
[1+10
lo
,
over time [40]
the following
cquations
for lhe fast variables
are obtaincd
fmm (5). (6) amI (8)
1'(1')
<lt'.
'0
Thc time inlcrval
is largc comparcJ with Ihe characlcristic
lime T 01' varialioll 01' lhe fast variables and is small comparcd
with the charilCtcristic time of varialion of lhe slow variables.
Il will he considcrcu lllal lile slnw ano fasl variables vary
sllloolhly in spacc. i,e.,
suflkicntly
\7 x
SolUlions
-
E
01' lhese equations
lOE
---o
c al
=
within lhe approximation
mcn.
lioneJ ahove are the following:
whcrc L and .\ tJcnolc lhe charaClcristic distanccs over which
thcsc quanlitics
vary, rcspcclivcly. In this case, il can be
showo 1.IOJ wilh respeel 10 lhe IIF field amplilude Eo. ¡hal
in lhe cqualiolls for lhe slow variahlcs it is sufficicnt lO cnnsiLlcr lhe linear approximiltion ami in lile cqualions for lhe fast
variables il is Ilcccssary lO use Ihe quatJratic approximation.
SUhSliluting (10) inlo Eqs. (5)-(9) anu averaging in lime.
the following
equations
for the slow quantities
Vo =
c'_' __
21110
(1.10
+ ((v,,),
\7) (vo)
= ~
\7 .
(E) - ((v" . \7) vo}
( I 1)
+ \7.
((11
0)
(vo)
1
\7x(E}=-~OI(I3},
\7 x (13)
-111'
'\'
= -;:- L..('''
+ (¡¡o v,,)) =
le
13=--\7x
O,
( 12)
V
~
o
=
. (n ) r;l.,
• _
1-
I ., ("')
J cOI
,- III~W
is Ihe magnetizalion
= O.
v(t
(14)
( 15)
Eqllalions (11 )-(15) wcre ohtained using Ihe cuasi-neUlrality
conditioll. Also. il can
notkcd that the inlluence 01' the
rapidly varying motion on the slow 1ll0tiOll is takell ¡nto acCOllnt in lenns 01' Ihe high frequency prcssure and the non.
linear force in (1 1) (second and Ihird lerms on (he righl hand
side) and in lerms 01' Ihe urag Ilow (¡¡o v,,) in ( 12) ami ( 14).
oc
-
=
( 18)
+ e.e.
exp (-iwl)
( 19)
introduced
in
+ r 0'
l.
-J
eo
r el
-
+ w-
(20)
currenl
•
\7 x (Eo x Eo) ,
{II)(,;~
4m2w (1/:!. + w:!.)
o
"
x {1'(E
v(;
O'
i I/n
+w:!'
Vl~
"')E'
V
o
1 o
-:E
( \7. 75t
DEI;)
ó) + DEo (\7 . Eó)]
[Eo (\7 . DE
al.
al
1
Z[Eo(r,t.)('xp(-iwl.)+c.c.),
( 16)
D~to (\7 . Eó)]
u
(22)
Inlroducing Ihe rclativc
conuuclivity currenl
where Eo (r, t.) is the slowly varying HF tielLl ampliludc.
Considcring
lhe tcnns to first orde!" in tcmporal derivalivcs,
R('l'.
Eo (\7 8~tó) + w2 [
U
+c.c.} .
as
(21 )
anu
w
3. Non stationary pOlldcromotive force
E(r,l)
(17)
D~o]
+ e.e.
(-iwt)
- -)
(I/.(lVn
+ e.e.,
whcrc
+ (¡¡ovo))
lhal lhe IIF field is descrihed
iW)
v
(U)
+~~(E),
It \ViII be assumed
Dt
Using (17) amI (18) as \Vell as Ihe notation
Rcf. 35. lhe drag t10w (ño n) lnkes the form
"
\7 . (E)
+
i OEo)
(Eo - : 75t
O
((110) (v,,)
OEo}
iw
Va -
[Eo - (~
X (,Xl'
~.:.u
o
1
_
are oOlaincd:
Illn
01 (11,,)
{Eo
x exp (-iwt)
x~
0 (v,,)
01
iw)
-
Mi'x. Fú. 45 (3) (1999) 322-325
vclocity
1I,,=(vo)+-(
[35] which descrihes
1
"o
}ro•
the
(23)
THE PONDEROMOTIVE
= O. the equalion
and assurning tha! (E)
[D;:"
(n,,)
11I"
FORCE OF A HIGH FREQUENCY FIElD ACTING ON AN ISOTROPIC PLASf\.-1A
Iakes the fa/m
(11)
+ ""U" + (u" . v)
=
325
U,,]
-1Itn (un) ((Vn . \7) Vn) + (~(:\(Un)
(vn X ñ) +/1/.
0
D;~n
+ l/()IIl(\f n :=
fo.
(24)
This cxpression constitutes the general form ol"the ponderomolivc force. Suhstiluting (17)-( 19) in (24), it (akes the form
(llo)e~
., 2
.,
") vlEol
1
(
'11In
v(; + w-
+
1
'1Jlo
(n,,)e~
(2
Vo
+ w 2)
{i""
I
-
_ --"" +2iwEI)
X
('"' v
X
DDEto)
+ iW)2 DE"
i (u"
w
o
--
--Eov'- DEo
w
DI
-.
/1'1
el
iu" . D (Eov'E')-.,
+ w2 Df o
X
DI
W(I/~ +w'l)
[1
1'"
+w
1',,-iwE
+ Eo. vEol- --O'
I/~ + w'l
Eox (v x Ea)
W
w
11o
+ w,2.
(v
X
Ea)
DEo
v-
-
Df.
/In
.,
11;:'
+ iw DEo
+w
2 D
t
•
. vEO
+ -i -D (E" . vEo)
wDI
( Eov.---v.E'
DEo
DE"
Df
DI
)]
}
+C.C ..
o
(25)
=
In Ihe collision!ess limit (vo
O), the exprcssion (25) is retluccLl to that ohtained in Refs. 35, 36 (in the case of an isotropic
plasma). Thc f¡rs1 1erm in (25) is (he known stationary ponderomotivc force (3).
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19