● a 9*8 ● 9●9***.: ●9* ●● —0–00 ● ● ● *9● o*m* ● ** ●

UNCLASSIFIED● *9*

● meOmm

.* ● -* 9*. . .● 0*

● oeooee ::bm o... .

*

.

... ., ..- _...-1.

~ 4-;,,J4MC. 334e /.

“/’” @39.

COPY

.

is

●

ClassificationchmgcdtoUNCLASSIF~-byauthorityoftheU.S.AtomicEnergyConnnksio%.’

Per&f2KxL..&@%&d.d. W.J..,&&2-/5.6L

-C.ZL-L& A!

VERIFIED UNCLASSIFIEDb-=.-——. —

.’. +M t_.= _

1!

I

I 79

I ‘w”

!-A

-,. -

._

.1

-1 >

UNCLASSIFIED—. —

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

● e :00 ●. ..00.

&

●

.

.

.

.

.

aUIY23, 19.45

*

06 ●

●

●.s” ●

9***● e●:00

● ●** ●

● be

● 9*. ●● 9**

● ● 9* ●

● ● *-8● 00 ● ● ●

● 00 ● 00 ● *● 0

● . :::0 ● O.0

●:0 :00 ● 0

/

IJNCIASSIFIED

This clooument Contxainsa+ages

.

TRANSPORT PHENOMENA IN A MIXTURE OF ELECTRONS AND KUCLEX4

WORK D(XJEBY t

R. L@dshoff

R. Sember

Je Weinberg

REPORT WRITTEN BY :

..

R. Land$hoff

.

-.. .

4%/s’ -’ .+,

tainsInfoI .

●

,9 .e.mean- UNCIASSIFIED.4edtm.w$w~i++>dhl;Icdbyw.

● ● ●99●’***.. /● . “f i :“;●** ●

● ●**O● ::..: ● m.● 0000 ● ** 9*~-

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

.● e ● 9* ● * ‘● ** ●°0 : ●°0 ●

.99. *9D ● **S* *.. ● **● 9O ● *:m*::om*● * 9** ●

---● 08 i m-●io ***00 .

●

ABSTRACT d

, At extremely high temperatures atoms are stripped of all or most of

their eleotrons. The mean free path of the oleotrons moroover $s proportional

to the square of their kinetio energy. [!heeleotrons will therefore oause the

ionized gas to conduct both eleotrioitynnd heat quite ea8ily0 By solving the

Bol%zmann equation for assumed gradienta of dasity, eleotrio potential and

temperatures we find the velooity distribution of the eleotrons as an expansion

in Iagut3rrepolynomials. fiOmthO firSt tUW 00eff’iOiaXlt8Of this ~IM3iOn W13

find the elmtrio and thermal conduotivibies. The

leads to difficulties (divergent integrals) if one

binary oollisionu. This is avoidod by considering

rearrangement of the electrons in the ne@hborhood/’

.

.

4“

. .

,

● ☛ ● ea ● O* a ● ma● 0:2 @ a :00

● ●B**:* 9:**9*9*

. . .9. . . . ..: .:. . .8

9 ● ● .0 ● *9 ● *.** 9 ● ● ●*. ● **● ● ** ● ● ● O*

● c ● *9*● ::00: .-,● 9 ● ● ● ● 00 ● O

long range of Coulomb foraea

restriots the discussion to

a shieldtig effect due to the

of a uolliding Pairo

UNCLASSIFIE

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

.

.

.

.

9

.

.

● ☛ ● ☛☛ ● ☛

● OO .O. : ●** ●

●00- O.:.: .0● e***● *9 9*O ● ● ●.*

● 0 ● 89 . ● 9* ● ●

9* ● ,* ● D* ●:- :- :.*●

● :Scjti:o::● O

● : :0

9

1● 0 ●

UNCLASSIFIED

!I!RMWPORT PHENOMENA X\ A MIXTURE OF ELECTRONS AND NUCLEZI

1“ .

, [“At extremely high terneratures atom8 are 8tripped of all or most of

their eleotron8~ The free elec rons oause the ion$zed gas to conduot both

eleotr.ioItiyand’hea% quite easi yO This investigation uses the kinetio theory:.

of gases to find expression8 ft thp eleotria and thermal currents oarried by

the free electronm due to grad~.”ts of density$ potential and temperature o To1)

Ido this one has to determine IAj@vslosity distribution i’unotionof the eleotrona

from the Boltzmann equatiom IIeshall treat the problcmnin the -approximtion

wherein the heavy partioles shtP

no deviation from the 34xwell distribution Z*

is not moeseary

ture. We shall.

tim-independent

problem with all

to aspume the,eleotrons and the nuolei to be at the same tempera=

however~ not (.onsiderthe resulting heat exohange and assume

distribution ;hulutionso We Gonfine ourselvo8 to the Hnear!

gradients and currents in the x direotiono For the eleotron

MJCM$IFIED

(1)

(la)

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

.

● ☛ ● 9, ● 00 ●:0 :*. :~.●

●

:4”44 :“ ;:● :● ●:0 : Q:. :*0 ● 0

UIICIASSIFIED

.

J

The Boltzmmn equation oan be written as

where

anght in the

fmatteri.rigo

zJei(6.Fi)i

(2)

Similarly for the ~ollisions with nuclei we have

(44

element of solid

for e2eotron-elOotron

(9)

TIMOu%andard prouedure to obtain m approximate 6oUaWm of the Bo3tsmann

equation is to replaoe E@) by D(f) but leave 6 u@?Anged on the right&aad

side. H we do this the lef%-hmd aide can be expressed in the foll~g manner.

ITenote first that 8

1 3.!% {3/2 -jjw]1 M’n’z=;2” ~*&

andw

1 M~ ~.- = - 2/?i2%x

A

D@’) =[+ %“

On the right-hand

whi~h by combination Zead8 *Q

side of {b) We substitute 4 fl’OIS(1), and obtain t

~’ h(v *- Vx ‘h(vf)-Yf 1d%p.x,

(4)

that the heavy Parttoles

—

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

.

.

are muoh slower ‘thanthe eleotro.msso that wi oan bo replaced by vo In additions

the only important collisions are those for whioh 1 - cos 8<<1 so that

The integration with respect to vi can be oarried out at onoe asd leada tog

~,

Jei = Nivf(v)h[v) \ Uei@(vx - PxO)dfLb

,. We note again that V9S v and express Vxg thus:

4

e xV=9 =V(COS U 00S @ + SiSlcX SiIl e f308 ~)

The oross aeotion for collisions between eleotrons and nuclei

(Rutherford scattering) 5.ss

Vfeotm

.

9* ● ** ● ● *

● 00 ●°0:::0● 89.** ●

● *.9.** ●

:.O c ● 00 so. :● 0mm**9 m**

UNC1.AS$fFI

where

()zie2 2

‘ei = ~ (1 - 00s e)’2

thus reduoe Jei further tot

This last integration is carried out in Appencilx1. After we

and (~) into the Bo3tinaannequation (2) we are left with the

h(v) from ito This ocuabe done by expanQ~ h(v) in berms of

““”%+vwith ohargo Zi

(Em)

(6)

enimr (?Z)s O@) 8

problem to find

I.tagumrepolynomials2)

In parliiGukx we shall ui3ethe p01~oti818 of order 3/2 and write for brevitys

~(s) = Lr(3/2)(e)

::”::““;=gm=b UNCMSSIFIED.::ao: .**

● 0000 ● *O ● *

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

● O● *● *● m:.O

● m8

●

● :●

● *O ● ● *

● .0: ●.* ●

8em *m**● .0 ● O* ●

:00 ● @*b:*em**9e*0

.:. ●0: ● *O :Oe 9*

● --:/3 -: ● * ::● 00: ::

●:, : ●:0 9** ● e

UNCLASSIFIE

. The L= form a complete set and oan be derived from their orthogonality relations

(7)

and Lo(~) = lo MS note also that Ll(e) = #2 - so

We express (Sb) in the forn:

and substitutex

into f4b) and (~).

1% now multiply both sides of the Boltzmmn eqwatio~ by VIxLr(~2v~ s.nd

inteErate over d?e On the left-hand aide we ?btain, using (’7)x

,{

WJMO= w Bblr.,

where we have set2

On the right-hand side we obtainz - zC*I$B

where H SE ‘+ z irs rs i %8

and where the Er*e and ~,i are clefined&s follows$

(8)

(9)

(ma)

(lob]

● ● 9** ***●ge ● ● ● *a* ● ●8:● ● *, ● **99

● ● ● am*● :: .0: bob● **9* ● ** be

IJNCMMFIED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

.

and

● 0 ● ** ● o● ** ●°0 : ●°0 ●

.0000 ::.: .0● m**:-b ● ● bb ● o0.O

● ** ● ● .9 - ●

● m ● 9* 900 ●:0 :** ● .● 9*

● m● -+”7-4:”::.:*

● ● 00 ● ●:. ● .* ● e

The problem now consists

(13)

/

It is convenient

q=

to take

H =

UNCIMWIFI)

in solving the set of equations

= z C sHrs(14)

Aotually we will need only the first two ooeffioienteo

.

out of’the m%trix element8 and

% = @’s

to write J

written formlly in terms of the dimcmsionleas matrix

followsa

.

● ☛ ● O* ● bO . . . . .ba a... . .

9 ● ● ** ● e*99* ● ● ● 8**●

● ●°:9*9*9*8 ●

● ● ● *eb● ::..: 9**● O.*.* ● .*.

(15)

(16)

(171

(18)

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APPROVED FOR PUBLIC RELEASE

.

*

-1c.)~=v

00 hol %32‘e’

20 h21 ’22

-u-

-1s C3=P

However, these determinants are infinite and

used to get convergent results- We out both

determinants off beyond the row and column

ratio, and repeat with larger no To oa.rry

I~(n) = ‘Ooholoeoo”o”hti

I‘n&nl-.”e*ehmD&(n)which arewe abo use tb minors.

Ml Oolumno

UNCLASgFIED

’00 A

Llo B

120 0

’02““’

‘I? ““”

’22 ““”

’00 ’01 ho2 ““a

‘lo ’11hX? ““’

’20 ’21 ’22 0“’

(39)

the following

numerator and

limiting prooess is

denominator

carrying the index n~ take the

this through

..”

we write:

(20)

obtained by deleting the ith row and the

(21)

9* ●

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

,0 ● **● *

● e@ ●**: :’: ●

.000. : ●

● ** ● .9* ●

:.0 ●● 9* ● ● ●,*

● ** ● ● *9 ● ●

. . ,9* ● ** ●:* :9- :..● ::oooe . .

●● * ::

.: ●- $,L :● ●:0 ● . . ● *O ● O

-.

v

.

J

!\qcL~s$lFIED

By substitution into (19) we obtain;

co = (ARoo - BRIO)N-l, c1= (ARol i- BRJ P-l

Tha limiting prooess (21) can be carried through by means of a theorem on

3)~determinants by Sylvester From this theorem we obtain the relations

Jn)

~(n+l)

i~n+l

~(ni-1)

n+l,k SD (n+l) (n)D ik

(Ia+l)

‘ik

80 thIit

(22)

St is interesting to note a s5mplifioation which is itiioduoed if ’one imposes

the oondition j = O“or its equivalent Co = 00 In this ease we can eliminate

A from (2?) and obtain~

by another application of Sylvesteres theorem. We therefore do not need the

zeso row and column of our matrix if we only want the h= t oonduotivityo Xf&

however, we also xmnt to know the value of A, that is the eleotxio field In the

ease j = O or If we mat the ooeff’iolentsin the more general oaee when j A O

we have to use the oomplete matrix.

3) See e.g~ Xowabwski, W#d{MZt?M?@#!W..TEM, theorem 300

b-w b 6 -O *-.”*.*● ● ee ● ● ● **

b ● ●:,: ●

● :*: ● 0:● * ● 0 ● 00 ●:

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

w

.

.’

● O ● **● 9

● ** ●*. : ●*9 ●

.* *** ● 9*

.ee ● : ● 00 ●

● *O wee ● ● ●e.

● * ● ab o 900 ● ●

By oonibining(l(h), (lOb), (17) and (22) we are led to the equations%

(26)

Q%*+%,]

we are particularly interested in the.ease j = O where we find;

q=-$@)’(%)”2k(’$@&ROI’)~That is we get for the heat uonduotivity:

(’7)

(28)

(’9)

In this formula o was introduced to put the dimensions of k into evidacee For

the value of A see equation (~) o.

The

matrix %8 is

integrations (13) and [1~) are carried out in Appendix =. The

aeon to depend on an effeokive nuolear charge:

~dz= ~ (30)

The computations were oarrhd through for Z value cd!~,2, 2.5, 3 and 00. The

otme Z a m means that the term ~ae resulting from eleotron,.el~tr~ 8oattering

UNCMSIFIED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

.

ualso be solved in

table below shows

a closed form, so that we are able to

me first 4 terms and their ‘totalsof

ad‘n” Obviously we have RIO = Rol on acicountof the

IWCMSSIFIED

oheok our theoryo The

the series for ROO, ROl

symmetry. It alao shows

5 ‘1 find 75 %% - R012the important oombinationso 1 +3 ~ —16pnwhioh

00 Roo

eoour in (27) and (29).

z-+

‘erm ‘f ’001st

2nd3rd&h

total

2er.msof Rol

.. a.crt2nd3rd&th

total

‘mm ‘f ‘n18*2ndZrdbth

total

1

1.9320

001’p.011700o1+ll@6p

A21s. e0668Ooofi

● 00015Q5!%3

W@o-hooo21JAO0506665

l.~o

=

2

101590

4@OWj.0023.00U6

101654

04393-,0192

0001%700005vl$?o?

Q%9

4!

0250

000

IAO04

Q5443

t 733 4“

205

Q97M3.0002.0012.0002Q9763

03832-00063

00006-00003

03772

02555Owe0004oooo~.5002 ,

30966

3

08430.0000.Oooq.000108~36

03398.002700004

-00001

03428

022690236000003.000204630

2.026

a6CS?

a)

3.2500 z-ld@6 24.0039 z~.0005 z-~

3.39502-1

l.~o Z-l05625 Z-l

dx?34 z-l-.0014 z-~2.03772-1

205005

(31)

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

.

4/”

r

● ☛ ● ☛☛● ●°9 : ●“e ● ●°:

● ● ● ***● :0: ● : ● @b●*9*.... ● . .

● **.*8 ● a@ ● 9

● O ● 9* ● bm ● om.m. .0● *

●:::* !“*{ :“:. .

● 0 ●0:●:0 ● ●:c ●

For the eleotron and heat ourrent6 we have:

To aarry these

and similarly

integrations through we need x .

UNCLA$$IFIED

We substitute h(v) from (31) into (32) and (33) m-d use the integrals

Comparing these eqwations with (21j)and (26) we see that the follotig

(32)

{33)

(34)

(35)

.

R?-: Z-l= 30395,305 Z-X, ROZ =

32 .2 = ~Qo37x8~-100 = ‘n = Fz

● **●.**

“:: “i” kkiliib

{37)

(38]

Equalities

● m. ● 00 ● ●:00 ● ●**●● ** ● ● ●’*.Oeo: ● *e .

● m*: b

● O* ::9●. ●***@* ●* UN(IAS$IFIED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

● 9 ● **● .*8 : ●°0 ● ●°:

● ● ●●

.***:0: ●● 9.*..: ● @o● *.

● ● ● ● ob ● . . . . .

● e ●.: ● ** ● *. . . . . .● * 9*● *** : ●:0.● m ●● 0

●: : :-0

● O●0: ●Ao130eeoe●

derived quantities:

UNCLASSIFIED

‘o4

7%- 2 . 32 ~-1 ==Z-l~+$~=zos; 17 ROO%l - Roln“

’00n@i

These values agree very well with those listed in the prooeding table, so that

we aan be quite uonfident of the validity of our genera3.mathodo

.

\

● 9* ●●-e . ●:e . . . . .●

9:0

●::

: .0● *.

:000: ● *● * ● *b ● . . ● ** :.. ● *

“900:00 c● m* ● ●

9** ●● *,

● ● ●**

●ob. : ● ●** ●

● ** ●*: ●::0● 9*.*... ● *

--,

UNCLASSIFIED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

● ☛ ● *O● ●** : .O. ● .O:

‘* ●●

● O**:0: ● : 000● **.*.* ● **

● ● ***** ● e* ● 9

9. ● 9* ● ** ● 00 ● 9* ● *

● 0● .*:● D● .

: -L+”;j. :::● 0 ● *m ● O*

APPENDIX 1. CALCULATION OF A

Tho integral x given by (6)

WUSSIFIED

92

diverges M one uses 81 = O for the lower limit~ The reason is that the kinetio

liheory~as tt is used, restriots itself to the consideration of enoounfiersbetween

oIzlytwo particles at a fsimeP The Coulomb foroe law is$ however~ of sucsha nature

that the possibility of interactions between more than two partioles must not bo

exoludedo Another, less aatastrophio, difficulty arises out of the uncertainty

principle whioh exoludes the possibility of head-on Oollisionso beeauee one has to

oonsider an eleetron as being spread out over a region of the order of magnitude

of Its de BrogMe wave lengthO TQ remedy the situation we, first ofallo express

% in terms of the oollision parameter

[39)

The lower limit is~ ammrdtig to the uneertdnty prinai@eO the do Broglie mzive

lcx@ho .Thati8 we have;

hP@m

The expression;

ie

in

we

=2

()

#’J ,~~plz~

h our case coneidemably larger than one so that we

front out. The same is obviou~ly true at the upper

Oan wr5.%e:●*a ●●’. ●:0 ●e. ..

●●::. . ::

A;”&g:” “ ““ ‘:●:0 :.. ● .

.OJ+.C :00::.00:00 ●● *O● 90,:.0 ..’

‘0” ‘“” ““” ; -

(40

(44

UNCI=AMFIED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

● o ● m,

● ●*- : ●-e ● .O:● ● ● 9*.

● :0: ● : ● @e●-. ***D* ● ,.

● ● ● 9* ● ● *e ● 9 UNCLASSIFIED● ☛ ● 0* ● 69 ● ** ● 9* ● 0

● O● 00: : ●:● , ● m ‘%ulmiih“●.:●0: ●@ulye...

*

Yh chose the upper limit by excluding collisions whioh last longer thanw

the time during whioh the eleotron gas would be able to rearrange its

density distribution so as to give a shielding effect. The rate at which

“4)the plasma vibrations othis will take plaoe is determined by the frequenoy of

~~

fcd=

T

The Gollision parameter will thus be given to the right order magnitudeby .

the relation

.

Thus we obtain~

I

(42)

(43)

(44)

or, after introducing (@] and rearranging to put dimensions into evidenoes

“ ;45)

For oonvunicmoe we btroduoeAvegadro$s number 1?and obtaihs

(..-) - J@)h=loo88x +&q M) ●

,,Beoause of the slow variation with Te it will usually be sufficient to use

an average Te in this expression

U Sea Cobine, Gaseous Conductors (19M) pQ 1320

● ☛☛ ●✎☛✍

●;e ● . . . .

●● : :0 ::

● *..● *, :000: 6*

● * ● 99 ● 9* ● 00 :.. . .

9*9 ● 8* ● 9:00 ● ●be ● ● ● ●’e● 00● .*.:

8 SO* ●●

● a* ● 0: :.: ●80 ● ** ● . ● m bWSWIED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

● 0 ● ,8● ●e* : ●“e ● ●°:

● ● 9m. .● :0: b :● *e *m., ..! : :. ● .*... . .

UNCLASgFIED

s

Y

APPENDXX IL CALCULATION OF THE MMCMX ELEMENTS

In carrying through the integration (11) one oan replaoe

Vxfi(Vx~5) by $ ~ A(? L~) bemuse the rest of the integrand is spherically

symIuetrim2in the veloo ities. Let us furthermore express the veloeiti.esin

terxs of the velocity % of the oenter of mass and the re~ative velocities ~ and %P

of the two partioles before and after collisionc

●

e

We shall uolleot these four equations symbolically in one and wrMe:

We further

That means

introduce the angle@i W

The Jaoobian of the transfortration(L7) has the absolute value one so thah

d?d% = d~d~. Now let:

where

.=-$-1-$ ‘

y=+

Then, considering the generating function

● 98 ● ● ee ● m. ● .8°0 .●

●: :●

::: .0● *

:000: ● 9● m ● ** :*O ..0 :.. ● .

.

(47)

(49)

(!50

(50 .

of the Laguerre polynomialsx

(40

(!2)

● ☛☛ ● e, ● ●:00 ● ●ge ● ● ● ●a.● 00

● ***:● ● ** .

● 0:●

● O. ::0● ins..... .* UNCLASSIFIED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE

uNcLA5gFlm

.

● a ● ● 80. ●’9 b ●“e ● ●*:

● ● b**.

● :0: ● : 000● 0.0..0 ● *.

● ● ***** ● ** ● m

we an write:

in

(53)\

Hrs

e appears thus as a coefficid in expanding the expression (~)

powers of 5 and

~o Introducing

Our next step is therefore to d~ermine the integrals

we obtain:

(54)

Introducing the substitution (47) we rewriter

(55)where: .

Z=u+zibi?i2(@xly) (56).

d2(1 + x + y) + Xy(l - 00s @i)=-

“2(2 +x +y) (57)

We express

(5%with ; . .

(59)

(60)

(6x)● .9

. . . . . . . ..- u~cLA$$\f\ED● b ● 9* ● . . ●:0 :

●*● **● 000... . .9 . . . .● *O

● 9*● c:ooc9b .9,.*. ..

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.

.

9* ● **. ●*D : ●ee ● .O:

● ● ● **O● :0: ● : ● m@● **m **m ● e.

● . . ..0 . .0. . .

● * ● a8 ● 09 ● 09 ● ** ● e● e ●: } lq-: ●°::0● *

● 0 ●0:●:0: “:”‘-

Entering (55), (58) and (60) into (~, we oan oarry out

UNCUS9FIED

the integration

and obtain:

where ~. ● & oan be obtained from (5g) and (61]

“?,=-(x+y+w) +(.a+x+y+w)cos@J

?()~2

,2V?+X+Y)2

In order to oarry through the !integratiion(~) we sott

Thus we

x+y+-

3.(2+ x + y)‘$2

2+x+y ‘1-XyJ$2

3(2+x+Y)

2+,2x +&+xy 22(2+x+y) P

Xy

2(2+x+y) P

am write:

~ =A(l - EW2

Mow we form AM~ = ~ +

Aah.uilly8we will need

2

-(DA? c08 ~ i)w’+ c 00s@iw2) e

%-~”~ and qad in powers

(63)

. I

(64)

(65)

of Y= 00s e - 3.

only the M.neax’tarm of’ the expaneion beoause the mattering

-2 ‘orosa.seotien is proportional to v so that the qpadratio and higher terms give

small oontribution to the integral as oompared with the linear tonne

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.

.

● m ● *a● ●*. : ●°0 ● .O:

● ● ●**.● :0: ● : ● *b● **.... ● O*

. . . . . . ● . . . . . wcLhsYf\tD●*●00●9*●00●*O●*9* :-“w-:“:●::::.

● a ● 0●m●0:●**

““”””~and by 8ubtrao+ing “ ,

-(D-E)#~-~=AcJ B

Edl- Bw2)(l-e ] H??& (,+l]eE

9

= - #D-E)#El - BW2)E,V2+ CW2(1 -tEwe] w + 0(V2)

‘s

(67)

of C and EO The oross-

- E(B o C)W’J Wdw

.

and

.

so that:,

Bow we have to expres8 (~) as a funwttin of $ and

70 Zf ’weset

u = (1 - $)-1(1 - ~)-1 we findt\

●

● 80 ● ** ● ●:** . ●’* ● ● ● ●’9● 00 ●● **.

● *9 ●: ● **●0...:. ,.

●***.... ..

(69)

(n}

m)

UNCMSSIFIED

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*.

.

.

●

9* ●● ●*9 ● .O. ● 00

● .O:● ● *a ●

●● *em

bee ● ● ● *@.* **.** . . .

● ● om **m ● .* ● m

● e ● *9 ● 00 ● 9* ● O* ● *● , ● 4●: : ●:::0 ●

● * :0●0 ●.: .!s 2j *:. “.”

and autering this into (6$.):

E=; ‘+’+?92We enter these expressions

‘1 .5/2 42~(1-f) (1. %’() =

z z$r~”%.o=P@rs

into (71) and multiply aocording to (~) by

where p is defined by {17)~ We oan see immediately that all elements in the zero

row and zero oolumn ar6 zeroO By expanding the expression (74) in powers of!

symmetrical matrixs

0.’0 0 0 0 0-

1 3/22 15/25 3g29 --d

4%24 309/27 8@#2g .--

5657./21020349/212 .-e

49749/214 --

“

The integration (13) requires cons~dwrably less laboro The angular

a faotor ~~ and if we further set $2 V2 = c and use (Zk) we have x

● ☛ ● ☛☛ ● ☛☛ 9 ●● ** ●** ● ● 9 ●**● 0. ●● O**

9*- .:

● 90 ● 0: ●::0● .eee. .e ● e

integration gives

(76)

UNCLASSIFIED

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..

b

*

Y

-1

.

● m

● ●*8● ● 00● 9*.

● ● ** ● ●=:● ● **.: . :0: ● ob:4:

● ● O.... ●a: ●m:

.0: ●0: ●:e ● 90 .*. . . - ‘● 9

::.● 9 :- ‘jl ~ :*:

● * ●.: ... ● ... ,

U?flAS9FlED~m;?://?,

afiiiis’”’-’v”.<.-.;.-.........--..,..:-.-as before, we make use of the generating funotion, and write:

=.– ---- ,

By expanding this in powers off

and 7

%?=+ (

1 3A @ 35/24 315/27

13/22 69/24 “ 265/2~ 150928

433/26 ~077/27 looo#21*

@57/28 28257/211

28&#214

*

-\ ..’=

-->.

..- --- -

.

● ☛✎ ●●☛✎●:0 . . . . .

● : :●

● ● O::

● :0 ::. . .● .. . 9.. ● eb ●:0 :.. . .

%.

● b 8m,. ● **

● b* ● ●

J● *.

●*9 ● ● ● ●*.● *.. ●

:● -b ●

● 9. ●0: ●

● 9 . . .● a

, we obtain the symmetrical matrix s

I

UithilFltD “ ii’,,.

(78)

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UNCLASSIFIED

S-*”””

r I.

● ☛ ● O* 9** ● 800● 009°9:: : ● O9*99 : ●: ●

● e :0● . ●.: ●:0 ●.: ● 00 ● .

9 ● . ● 9m . 9.* ● *

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APPROVED FOR PUBLIC RELEASE