• 5

COLLISIONS AND TRANSPORT

Temperatures are in eV; the corresponding value of Boltzmann's constant is k = 1.60 X 10-12 erg/eV; masses \i' are in units of the proton mass; ea = Zae is the charge of species a. All other units are cgs except where noted.

Relaxation Rates

Rates are associated with four relaxation processes arising from the in­teraction of test particles (labeled a) streaming with velocity va through a background of field particles (labeled (3):

i .                                 i                                            dva               QA q

slowing down                                                          ------ = — v va

transverse diffusion                            — (va — va)|_ =

dt d

dt

parallel diffusion                                                     —(va — va)n =

dt

i     d 2                                                                          a\(3 2

energy loss                                                              — va = -v                        ,

dt

where the averages are performed over an ensemble of test particles and a Maxwellian field particle distribution. The exact formulas may be written19

uol\P = (1 + moc/mf3)^(xoc^f3)u^ X/3;

"(1 - l/2xaX/3)^(xaX/3) +

= 2

cx\(3

Vn

M 1

1J V 1

'

v^P = 2

oc\(3

ij v

'

where

a\0 , 2 2 x                                                    / 2 3                         a\/3                   2/n1rr

uQ = 4Tree e/3 Xa(3n(3/ma va ; x =                                                                    /2/cT^;

•X

^(x) = —— I dtt1//2e ip'(x) = ——, \fir I                                                   dx

Jo

and \oc[3 = InAayg is the Coulomb logarithm (see below). Limiting forms of zys, zy_L and ^y are given in the following table. All the expressions shown

have units cm3 sec-1. Test particle energy e and field particle temperature T

are both in eV; \i — rrii/mp where mp is the proton mass; Z is ion charge state; in electron-electron and ion-ion encounters, field particle quantities are distinguished by a prime. The two expressions given below for each rate hold

for very slow (x0^ <C 1) and very fast (x01^13 > 1) test particles, respectively.

Slow                                                                                                            Fast

Electron-electron

Ine, Aee, » 5.8 X 10-6T"3/2                                               — 7.7 x 10"6e"3/2

fne,\ee, » 5.8 X 10-6T-i/2£-i                                                     7.7 x 10-6e~3/2

v*W/ne,\ee, « 2.9 X 10-6T-i/2£-i                                                3.9 x 10-6Te-5/2

Electron-ion

vl^/mZ2Xei « 0.23m3/2T"3/2                                                      3.9 x lO"6^2

v^/mZ2Xei » 2.5 X 10"V1/2T-1/2e-1^ 7.7 x IO"6-"3/2

 

e

i/®XVniZ2Aei « 1.2 X 10-4M1/2T-1/2e-1—► 2.1 x lO"^1^ Ion-electron

i\e / ry2 X

Aie

 

1.6

X

10"

9 -

M

1T-

3/2

1.7

X

10"

■4

Ae / ^2 x

/neZ Aie

 

3.2

X

10"

9 -

M

1T-

1/2 -1 z e —

1.8

X

10"

■7

Ae / r^.2 x z^H /neZ Aie

 

1.6

X

10"

9 -

1T-

1/2 -1 z e —

1.7

X

10"

4

Te

-3/2

-3/2 e z

5/2

Ion—ion

—---------------------- ~ 6.8 X 10 8-------------- 1 + —                     rji — 3/2

nifZ2Z^\iif

i\i'

jJL \                fJL

^------------------------ - 1.4 x 10-7At,1/2M-1T-1/2e

 

iA, x

n^Z' aA

_8 Zl 1 \ /i /

4 9.0 x 10 - H------------------------------------------------------------------------------------------- ) t—r-

Vm h' ) e3/2

* 1.8 x 10-7m"1/2£-3/2

6.8 x lO-V^V'T-^e"1

^9.0x 10-V1/2M,_1Te-5/2

In the same limits, the energy transfer rate follows from the identity

ve = 2vs — — v\\,

except for the case of fast electrons or fast ions scattered by ions, where the leading terms cancel. Then the appropriate forms are

zyeeV —► 4.2 x 10~9niZ2\ei

e 3/2 p 1 - 8.9 X 104O/T)1/2e-1 exp(-1836^e/T)

-l

sec

and

v

i\i'

->1.8x10 rni,Z2Z'2\ii,

e~3/2p1/2/p - l.l(M7T)1/2e~1exp(-p/e/T)

sec

In general, the energy transfer rate                         is positive for e > ea* and nega­

tive for e < ea *, where x* = (™>p\m>oi)£ot.*/Tp is the solution of (x*) = (ma\myg)'0(x*). The ratio ea*/Tg is given for a number of specific a, (3 in the following table:

a\{3

i\e e\e, e\p e\D e\T, e\He3 e\He4

f *

1.5 0.98 4.8 x 10~3 2.6 x 10~3 1.8 x 10~3 1.4 x 10~3

When both species are near Maxwellian, with Ti ^ Te, there are just two characteristic collision rates. For Z — 1,

ve = 2.9 X 10~6nATe~3/2 sec-1; ^ = 4.8 x 10~8nATi~3/2)Li~1/2 sec-1.

Temperature Isotropization

Isotropization is described by

dt 2 dt where, if A = T±_ /Ty - 1 > 0,

1 dT\\

"                                                                                                   -Tn).

z/

2^/7vea2 ep2naXap 2

T

ma1/2(/cT||)3/2

A"

-3 + (A+ 3)

tan-1 (A1/2) AV2

If A < 0, tan-1(A1/2)/A1/2 is replaced by tanh-1 (-A)1/2/(-A)1/2. For Tl « Ty = T,

= 8.2 x 10~7nAT~3/2 sec-1; uzT = 1.9 X 10~8nAZ2p~1/2T~3/2 sec-1.

Thermal Equilibration

If the components of a plasma have different temperatures, but no rela­tive drift, equilibration is described by

dTr

 

dt

13

where

= lgx 1q-19 (mamf3)1/2Za2Zf32nf3Xaf3

For electrons and ions with Te « = T, this implies

v^/rii = ^Xe/ne = 3.2 x 10~9Z2\/pT3^2cm3 sec-1

Авторы: 1379 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 1908 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я