the froble>1 of runaway electrons in p~o · the froble>1 of runaway electrons in p~o...
TRANSCRIPT
THE FROBLE>1 OF RUNAWAY ELECTRONS IN P~o
CERN-PS/JGL 2 December 1957
We shall analyse some of the more important effects resulting from the application
of an electric field E to an .infinite volume of uniform plasmao As the electrons
have e. much smaller mass than the positive ions their velocity will be affected by the
field E much more than that of the ionso Let us observe the motion of a single
electron through plaamao lts equation of motion is
where
F ie. the friction force due to collisions with electrons e F ie the friQtion force due to collisions with positive ionao
p
(l)
These two forces are velocity-4epen.dent as shown in figo lo '!'hey can be represented
by the following formulae (1 - 3)a
where
n s F .-;;8-n;A-e 2
v
"'if F i;:: 19 A n -=v e e s:J
a
"llf l!' i:: 19 A n -
3 p Ps p
Ac 0
for v < S e
for v < S p
(2 )
.'$:Jg . ..J.
·=· 2 '-'"
T'ne forca F is not e. truo friction p
force as it causes a deflection of the
velocity 1recto1•, :iather than diminishing
its magnitude o
The criterion for de{loupling an electi·on in the velocity space follows from
eqfla. ti on l and it is
~.2.
0
where lJ' :ts tbe Bilgl& between v and
i (figo 2)o Let us consider the case in
vhic.~ v > S and n "' n =: m ,, Thl!!IJD.. a e p equa t-lon 4 may be written
w .,,,,,_---t:;rr- m. (abo volts/om) 'iP ~o/ ) ( 2
·" x y
(5)
This electric rield strength is therefore capable o~ decoupli~ all electrons·
whose 1'.'8Presentative points in the velocity space lie above the contour given by
equation 5 (see a.1.ao fig,, 3~ contour C)c,
or takiilg ln A N lO)
tet us ovalu&te equation 5 for
"tt """ S ~ "f -;=; Ori One obtains x ~ y
E ) 12 ~:. .• Maf' ~ (abo V/@m) 2k 'r e
(vol ta/om) (6)i&
;z) ----A similar formW.a is derived by conaideri~ the growth of the mean free path of am. electron dUe to the acceleration (eE/m) {Appo l)o
I
Let us now calculate the portion An of n which is decoupled by a given :C~-eld Ea
Thie ia
n (v v ) 2n v o dv dx x y y y
where v fal·l,l)WS from equation 5o Assuming that n( v w ) was or:lginally Maxwellian . xy
one gets
21' n !.:>. n ·.r.:: ---="-=---
(l!L kT ) 3/2 m
The double integral ean be written as follows~
27 Y. rQQ 2
~xp (= ~-) ....:X s s 0
JO!c:\ 2
l' 'f
exp (= ~) d ...J. 6
2 . s v (v ) .
y s
w dv dv y y JC
'IF d~ s
n ie obtained if v(v ) i;; T(o) : v o y 0
As
2 .e3
ln /\ " 'f' ::::; 12 'It - 'Z<~ o m E
one gets
~n :y4 [ l - ¢ d-t; ] where E · follows f'roD1 equation 6"
0
0 dx
(8)
(7a)
" a'iJ -
O'l . .
()'/ ..
21.-'f-
\
I /f,. F41.+..
Let ua auppase that n is a function of x and E :=. E (x)o It :\s inforesting to x: calculate 6. :!\(:r.:) as tM.s shows the departure from contim.ti ty of the decouple~1 electrrJgi,
dendi ty a.nd th•:;, corresponding positive space charge a.cownulationo 0Ile has
'l'he der.iva.tive of ~n waro to x must be zero in order that a. c:ontinuoue stream
<Uif decoupled eltictrone should be obtainedo Thus
Ji.t::. ·, '). , • .£.. ..;:RL, i. ·- 0 x (9)
'l'hus :\.:f r~{ ;:) is g:tYenP tlie fonn of E(x) follows fr.om oquatic•n 9o The value of E rx fo1Jows -~·::-om J
0w F.(1': ) dx ;;~ xE
0., If :i.nit:L~l!y Ef~) dous not sat:i.;;:.fy the equatiolrn 9~
it seemu p~u11.rn:lble that the non·~uniformtilty in .An will lead ttJ spacl7 c~harge ac(;:'l..'\DlUl=
lab.on whfrh will modify E(x) oo that the latter becomes a so.luticn of equatio~ 9o
1..ot iJ.s ,~onsider~n(x). = n1
:1~ H(o) " (~ ~ n1
)o Then if iinJL '~ .01!12
one- has
l ~ ¢ {a \) f> 1. . (10)
l ~¢(a 6'
(10 a)
J.iet us write E2
:::i 'i; Elf:l2 0 where E~2 ie given by equation 6 in plasm whoae
density is n2
o Then
(10 b)
Thia equatim has been solved numerically for different values crf the parameter po
'l"he soluUona are plotted as graphs of ~(Cl) in figo 5o Thus H can. be saen thatp
aogo, for a step in density of 1:10 and a field strength which is }~ E02 ~ the
corresponding step in the field strength is 1:5 (point A in fi.go 5)o
The decoupling process will be well established within a time equal trri a few mean
collision times for the electronso Ae the mean collis:i..on time is
t f'V .L coll 4.0
3/2 J_e n
(sec)
the decoupling time ie of the order of
{sec)
#Ll l ( ) 'V .,..._ llAA"' 10 II'"""~"" 0
tu)
The length of i;;he a.ecoupling time corresponds to a certa'ln spread in 1.reloci ty tl v of
the decoupled electron streamo This is
1t eE AW,.._, 12 -
m {om/see) (U b)
"•
CBHN-PS/ JGL 2
Dsin.g· tiqU6\tions 6 and n. o:no gets
In most cases Te rv 104 and therefore
(cm/sec) (n d)
5 0 '11his corres1,on.ds to an elect?·on energy of 7'0 eV or a kinetic tempertdaire of 5ol0 ~"
Owing to collisions the velocity distribv.tion of those. electr-ons that did not run
away :will not remain cut~·off by the curve C as shown in figo 3~ t.heee electrons will
cross the boundary C and they too will be decoupled by the field Eo The rate at which
electrons cross C is difficult to calculate, but one may assume that the Max:weUian
distribution will be reestablished within a time
T ,~ T t .rvt Cl m r
_ _ J; y_
'I' ~}· 'l' (sec) (12)1!:1
x y
where \. ;t.s tha energy relaxation time~ which is nearly equal to \on (given by -equation 11)? T -is the kinetic temperature in the direction of E and T that x . y ;~n the direction perpendicular to E o The kinetic temperature T is obviously smaller x . than TY as the originally symmetric velocity distribution contain.i.ng n electrons
was robbed o:r An electrons with a large wx o Let ua again put approximately
whioh is a good app1·oxit11a ti on if E > E ©
~) Ttds iB appro:dmateJ.y the relaxation time from a SchwarzechU.d distributiol!t
2 .,,.y2 [ ( T T ) exp "" .. ~~
x y 2k ..
2 2 -j <t ~~ 2 ~) x y
to a Maxwellian diatributiono
,-, '?' •.=i CERN~, PS/ JGL 2
Therl!l e~ul1.tion 12 becc;mes
(12 a)
and the rate at which ele<~trons cross the bo'Ulllidary C is
• l'Zl ~ -
20 .-=-.,. 2
·~2 Te
(13)
Let us assume that 'l1e rv consta This is a good approximation as~ j_n spite of the
loss of particles whose v is large and therefore T becomir.1g smal lerp the electric · . x . x field hel:i ts 'those electrons that· did not run away thus increasing T o As a reaul t of
y th ts
Te ~ ~ ~ (T {- 2 T ) l'V consto .J x y
The solution of equation 13 is thellll
:Let us find the time t c
in which n c'Y2 n eJl.•c;trons cross C,, Thia 0
.,,.1 t i::: 1rs3/ 2 (n ,gQ, ) (see)
t) 0 ll
J(f one applies the field given by equation 6 it follows that a e: 0°4 and
c~ec)
( 14) .
(14 a)
Consequently the velocity spread in the decoupled stream is dependent o~ ~ as well (l.'l
as on t~o
Let us now calculate the time in which the velocity distribution of the decoupled
stream becomes Maxwellizeda In, absence of collisions and any other fields apart frum
E the cylinder-like velocity distribution, wldch ie generated directly after dec.c)\lpling~
•
CERN-PS/JGL 2o
• eE 11 would prog:I'ess along the axis v· in the velocity space with a speed v = ~ \.figc,6)0 x x m
Elecfa.·on·-electron collis:tons t~nd to transfom this cylinder into a aphere in auch
a manner that
T -,~2'11 =3~ 0
x y
If T » T then this transformation is accomplised in a tine ( coJD!tare with x y equation 9) o
T 3/2 x
n (sec)
The ini.tial kinetic temperature in the x...clireotion is
where Av ;.s the velocity spread due to the delay in decoupling"
(15)
(15 a)
!n order to show the orders of magnitude let us neglect
consider A'~' as due only to the delay td o Then
t (equation 14 a) and 0
12 m 2 TX f,'C: 16 " 10 k= Te:::; l"lolO a Te
Substituting this into eq1mtion 15 aF there ia
rv t ::::'::: 3 0 10 r
(sec) (15 b)
If the stream constd~~ta~ n may be by several orders larger thun the initial n 0
and t i:nay become ehorta The rise in T desribed by equation 15 results then r Y
in an increase in propc)rtiono This is a danger similar to the beat:tng of the elec~r stream due to electron-ptr0ton collisions and the subsequent blow up of the beamo However,
the riae in T due to the redistribution in the electron velocity space is a pheno11=> y l""" 1
enon whose duration is limited only to t ~the final T ; T ~"i'T o · r v y ,., x 2
e 2 N ~
Using the pinch relation T ll.':: .......,c ____ and putting .l T ( T one gets 3 x 2 k
2/3 k T ~-~
2 LN
2 c
~· lo~ ~' 014 I rTe"' - JI.
0 •• VI
The time in which this v is reached is
t rv a
(eec)
CERN. .. ps/ JGL 2
(16)
(17)
where mA is a' longitudinal electron-mass defined in the report CEP.N-Ps/JGL 1 9 page llo
This time must be shorter than t and therefore using equation.a 6 and 12a ~ r gets
4 < 3 0 10 Te3/2 n
From this follows the minimum mean pinab radius
aee repo ci to)~
m~ r (assuming - rv 3°5 p m m
r > r m o ~ ... Y4 N
"r 0
This will be satisfied in all cases of interest t;i'uso
----i~IAil---·-~ Vy
! . I
(18)
(19)
= 10 = CEBN .. PS/JGL 2
CONCLUSIONo
lt followe 9 mainly from equations 6 and e, that the extraction of runaway electrom
from plasma is feasible ·dth moderate field strengths (i.,eo a few Vi/cm at discharse
plasma densities of nNl012 el/om'?J)o It is also shown that small non-unifol'llities
(a few o/o) in plasma density oould be tolerated owing to the self-regulat~ influence
ot space charge fieldso l t ie argued that the velocity spread in the run-away electron
beam-is relatively emall and that this velocity spread is evened out after a few
microseconds, especially in pinched run-away beamso
"'' 11 = CERN-PS/ JGJ.J 2
The expression. for the mean free path of 8ll electron in plasma is given by
2 2 Cm v ) __
ne4 nlnl\ (cm)
The velocity which the electron acquires in an electric field E is
(cm/sec)
The length of path traversed due to this accelerated motion is
x ~ Y2 v t {om)
Jet is evident that the mean free pa.th A. of an electron in accelerated motion
grows as t 4 ~ whereas the path actually traversed only as t 2 o lf ~ therefore 9 an
average particle does not suffer a collision before 'tA/x retiches unity it will not be
involved in a oolliaion ~t any time after thaiha Thus
1 _r. t2 = l 5 e:: m n
from which
Y2 5 1t m t ·-;;; (a } n ) E e
lf thia t is shorter than
(sec)
then the electrons will be decoupled., The decoupling field E follows from Cl