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Transcript of APPROVED FOR PUBLIC RELEASE 9***.: 9*8 9* 9 …**.: 9*8 9* 9 —0–00 *9 o*m* ** *9* UNCLASSIFIED...
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APPROVED FOR PUBLIC RELEASE
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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 .
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● ●**O● ::..: ● m.● 0000 ● ** 9*~-
APPROVED FOR PUBLIC RELEASE
APPROVED FOR PUBLIC RELEASE
.● e ● 9* ● * ‘● ** ●°0 : ●°0 ●
.99. *9D ● **S* *.. ● **● 9O ● *:m*::om*● * 9** ●
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ABSTRACT d
, At extremely high temperatures atoms are stripped of all or most of
their eleotrons. The mean free path of the oleotrons moroover $s proportional
to the square of their kinetio energy. [!heeleotrons will therefore oause the
ionized gas to conduct both eleotrioitynnd heat quite ea8ily0 By solving the
Bol%zmann equation for assumed gradienta of dasity, eleotrio potential and
temperatures we find the velooity distribution of the eleotrons as an expansion
in Iagut3rrepolynomials. fiOmthO firSt tUW 00eff’iOiaXlt8Of this ~IM3iOn W13
find the elmtrio and thermal conduotivibies. The
leads to difficulties (divergent integrals) if one
binary oollisionu. This is avoidod by considering
rearrangement of the electrons in the ne@hborhood/’
.
.
4“
. .
,
● ☛ ● ea ● O* a ● ma● 0:2 @ a :00
● ●B**:* 9:**9*9*
. . .9. . . . ..: .:. . .8
9 ● ● .0 ● *9 ● *.** 9 ● ● ●*. ● **● ● ** ● ● ● O*
● c ● *9*● ::00: .-,● 9 ● ● ● ● 00 ● O
long range of Coulomb foraea
restriots the discussion to
a shieldtig effect due to the
of a uolliding Pairo
UNCLASSIFIE
APPROVED FOR PUBLIC RELEASE
APPROVED FOR PUBLIC RELEASE
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9
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● ☛ ● ☛☛ ● ☛
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● :Scjti:o::● O
● : :0
9
1● 0 ●
UNCLASSIFIED
!I!RMWPORT PHENOMENA X\ A MIXTURE OF ELECTRONS AND NUCLEZI
1“ .
, [“At extremely high terneratures atom8 are 8tripped of all or most of
their eleotron8~ The free elec rons oause the ion$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$IFIED
(1)
(la)
APPROVED FOR PUBLIC RELEASE
APPROVED FOR PUBLIC RELEASE
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● ☛ ● 9, ● 00 ●:0 :*. :~.●
●
:4”44 :“ ;:● :● ●:0 : Q:. :*0 ● 0
UIICIASSIFIED
.
J
The Boltzmmn equation oan be written as
where
anght in the
fmatteri.rigo
zJei(6.Fi)i
(2)
Similarly for the ~ollisions with nuclei we have
(44
element of solid
for e2eotron-elOotron
(9)
TIMOu%andard prouedure to obtain m approximate 6oUaWm of the Bo3tsmann
equation is to replaoe E@) by D(f) but leave 6 u@?Anged on the right&aad
side. H we do this the lef%-hmd aide can be expressed in the foll~g manner.
ITenote first that 8
1 3.!% {3/2 -jjw]1 M’n’z=;2” ~*&
andw
1 M~ ~.- = - 2/?i2%x
A
D@’) =[+ %“
On the right-hand
whi~h by combination Zead8 *Q
side of {b) We substitute 4 fl’OIS(1), and obtain t
~’ h(v *- Vx ‘h(vf)-Yf 1d%p.x,
(4)
that the heavy Parttoles
—
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APPROVED FOR PUBLIC RELEASE
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.
are muoh slower ‘thanthe eleotro.msso that wi oan bo replaced by vo In additions
the only important collisions are those for whioh 1 - cos 8<<1 so that
The integration with respect to vi can be oarried out at onoe asd leada tog
~,
Jei = Nivf(v)h[v) \ Uei@(vx - PxO)dfLb
,. We note again that V9S v and express Vxg thus:
4
e xV=9 =V(COS U 00S @ + SiSlcX SiIl e f308 ~)
The oross aeotion for collisions between eleotrons and nuclei
(Rutherford scattering) 5.ss
Vfeotm
.
9* ● ** ● ● *
● 00 ●°0:::0● 89.** ●
● *.9.** ●
:.O c ● 00 so. :● 0mm**9 m**
UNC1.AS$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
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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APPROVED FOR PUBLIC RELEASE
.
.
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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APPROVED FOR PUBLIC RELEASE
.
.
● 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 - $)-1(1 - ~)-1 we findt\
●
● 80 ● ** ● ●:** . ●’* ● ● ● ●’9● 00 ●● **.
● *9 ●: ● **●0...:. ,.
●***.... ..
(69)
(n}
m)
UNCMSSIFIED
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APPROVED FOR PUBLIC RELEASE
*.
.
.
●
9* ●● ●*9 ● .O. ● 00
● .O:● ● *a ●
●● *em
bee ● ● ● *@.* **.** . . .
● ● om **m ● .* ● m
● e ● *9 ● 00 ● 9* ● O* ● *● , ● 4●: : ●:::0 ●
● * :0●0 ●.: .!s 2j *:. “.”
and autering this into (6$.):
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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APPROVED FOR PUBLIC RELEASE
..
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
28Ö
*
-\ ..’=
-->.
..- --- -
.
● ☛✎ ●●☛✎●: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