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