the partial wave treatment of electron-hydrogen atom collisions. Explicitly antisymmetrized wave functions are used throughout. The boundary conditions are ...
L 654 ] THE PARTIAL WAVE THEORY OF ELECTRON-HYDROGEN ATOM COLLISIONS B Y I. C. PERCIVAL AND M. J . SEATON. Communicated by It. A. BUCKINGHAM Received 17 January 1957 ABSTRACT. The paper is concerned with the solution of the algebraic problems arising in the partial wave treatment of electron-hydrogen atom collisions. Explicitly antisymmetrized wave functions are used throughout. The boundary conditions are written in ,
(16)
Q{nl-Lkn,n'l\]cn) = £ Q(nl-J,Jcn,n'VxVJcn)LS,
(17)
where
(2L+ +1)
l2-
lf
(18)
It follows from the conservation of charge that the £-matrix is unitary, so that
2 | r#r"
and from (18)
S (inelastic)QinlJ^, n'V&k^ nhh
^ - ^ ^+,}}^i+^• ^n'
( 2 °)
2
( '1+1)
To obtain the differential cross-section for transitions between the two levels, (13) may be averaged over the initial and summed over the final magnetic and spin quantum numbers. Using the method of Blatt and Biedenharn (2) we obtain A(£»-£»'),
(21)
A
where l l ^ '
[Z(l2Lr2L"; ^A) Z(l'2Ll'2" L"'; l[ LL'U\i',i;
JJz, n'kn.1ir2)LS T*(nkM
n'kn.l{l2")L.s]
(22)
and Z(abcd;ef) = ( - l)4tf-«+c>Cg$[(2a+ 1)(26+ l)(2c+ l)(2d+ 1)]* W(abcd; ef). (23) Tables of the Racah coefficients W ((l), (6),(ii)) and of the coefficients Z (10) are available. In practice the summation may be reduced by considering the symmetry of the summand. t In speaking of nl levels of hydrogen the accidental Z-degeneracy is disregarded.
Partial wave theory of electron-hydrogen atom collisions
657
If unorientated atoms are initially in the level n'%, and a beam of unpolarized electrons is incident in the Oz direction, the total cross-section for excitation of the state nlxm{7n% will be denoted by Qin^m^mlhn, n'J^kJ). To calculate this, (13) should be summed over m|, integrated over da>(k.n) and averaged over m^m^m^. Using *(«*', 0),
(24)
where z is a unit vector in the Oz direction, one obtains Qinl^nirr^,
n'l[kn.) = \{Q{nlxm{kn, n%kn.)8^ + %Q{nlxm\kn, n'l^kn,)s=1},
(25)
where
(26)
X S
rn'iUm1, l\LML
3. The radial equations. In practical calculations only a finite number of terms can be included in the expansion (7); it is then no longer possible to satisfy the Schrodinger equation (9). Instead one may take the function xF(r1 crx r 2 " 4= v', then S(0(v, v') = 0 and r) [ F(v, V ') £ - W(v, v')^] F%s(y'LS \ r) dr. (46) REFERENCES (1) BIEDBNHABN, L. C, BLATT, J. M. and ROSE, M. E. Rev. mod. Phys. 24 (1952), 249. (2) BLATT, J. M. and BIBDENHABN, L. C. Rev. mod. Phys. 24 (1952), 258. (3) BLATT, J. M. and WEISSKOPF, V. F. Theoretical nuclear physics (New York, 1952), p. 517. (4) CONDON, E. U. and SHOBTLEY, 6. H. The theory of atomic spectra (Cambridge, 1953).
(5) Kora, W. Phys. Rev. 74 (1948), 1763. (6) OBI, S., ISHIDZTT, T., HOEIE, H., YANAGAWA, S., TANABE, Y. and SATO, M. Ann. Tokyo
(7) (8) (9) (10)
Astr. Obs. 3 (1953), 89. RACAH, G. Phys. Rev. 61 (1942), 186. RACAH, G. Phys. Rev. 62 (1942), 438. RACAH, G. Phys. Rev. 63 (1943), 367. SHABPE, W. T., KENNEDY, J. M., SEABS, B. J. and HOYLE, M. G. Tables of coefficients for angular distribution analysis (Report C.R.T.-556, Atomic Energy of Canada Ltd., Chalk River, Ontario, 1954).
(11) SIMON, A., VANDER SLUIS, J. H. and BLBDENHABN, L. C. Tables of Racah coefficients
(U.S. Atomic Energy Commission, Report No. ORNL-1679, Oak Ridge, Tennessee, 1954).
DEPARTMENT OF PHYSICS UNIVERSITY COLLEGE LONDON
41-2
660
I. C. PERCIVAL AND M. J. SEATON
Table 1. Aihhlik, L) and hdhhZk L) = (2A + 1 ) 0 A ( M G : L)for (h + l2-L) evenj h 0
h L r.
A
n 0
A £
0 1
T
1
1
n
1
r a. i
1
r\
o
r
o
A
2
r
A.
2
r
9
2 t
o
o
1
L-l
1
£-1
0
1
L-l
1
£-1
2
1
L-l
1
£+1
2
1
L+l
1
L+l
0
1
L+l
1
L+l
2
1
L-l
2
L-2
1
1
L-l
2
L-2
3
1
L-l
2
L
1
i
T
L+2
i
o
1 1
L+l
1
L+l
1
L+l
1
L+l
Q
2
L+2
1
A
h
+1
+1 V i
1
' V3(2£+l) V(^+i) V3(2£+l) V3(£-l)£ V2.5(2£-l)(2£+l) Vi(i+1) V5(2£-l)(2£ + 3) V3(£+D(£ + 2) V2-5(2Z,+ l)(2£+3) +1
7"
T J_ 1
7"
O
r
/•
V3£
1
i 9
V(2£ + l) V3(£+l) V(2£+l) V3.5(£-l)£ 1 V2(2£-l)(2£ + lj V5£(£+l) V(2£-l)(2£ + 3) V3.5(£+l)(£ + 2) ' V2(2£+l)(2£ + 3) 3(£-l) (2£-l)
£-2
(£-1) 5(2£+l) 5(2£+l)
1
£
.(2£+l)
£
+1 (L + 2)
5(2£+l) J2{L-1) y/5(2L-l) 3(£-2)V(£-l)
7(2L+1)J2.5(2L-1) V(£+l)(2i + 3) V3.5(2i-l)(2i+l) 3(i-l)V3(L+l) 7V5(2£-l)(2£+l)(2L + 3) 3j5L(L+l){L + 2) 1 7(2£+l)V2(2£ + 3) 3V5(£-I)£(i> + 1) 7(2£+l)V2(2£-l) JL(2L-1) 1 V3.5(2£+l)(2i+3) 3(£ + 2)V3£ ' 7 J5(2£-l)(2£+l)(2i+3) V2(£ + 2) V5(2£ + 3) 3(£ + 3)V(£ + 2) 7(2Z,+ l)V2.5(2L + 3)
1
£ +2 £-3 £-1 £-1
T -L. 1
r
I
r
I
r
3
(2£-l)(2£ + l) 3V£(£+1)
i_ i
£+1 T J_ Q
3
(2£ + l)(2£ + 3) 3(£ + 2) ' (2£ + 3) 3(£-2)V5(£-l) (2£-3)V2(2£-l) 3V2.5(£-1) + (2£-3)(2£+l)V(2£-l) (£-l)V3.5(£+l)(2£ + 3) (2£+l)V(2£-l)(2£+l) 3V3.5(£+1) (2£+l)V(2£-l)(2£+l)(2£ + 3) 3V5£(£+l)(£ + 2) 1 (2£+l)V2(2£ + 3) 3V5(£-1)£(£+1) (2£+l)V2(2£-l) 3V3.5£ 1 (2£+l)V(2£-l)(2£+l)(2£+3) 1
(£ + 2) N /3.5£(2£-l) (2£+l)V(2£+l)(2£ + 3) 3^2.5(£ + 2) (2£+l)(2£ + 5)V(2£ + 3) 3(£ + 3)V5(£ + 2) (2£ + 5)V2(2£ + 3)
t In each numerator and denominator take the square root of all quantities following the symbol,
Partial wave theory of electron-hydrogen atom collisions
661
Table 1 {ccmt.)
n
A
2
L 2
0
£-2
2
L-2
2
1
9
Z,-2
2
L 2
4
(
2
L 2
2
2
2
L 2
2
4
2
£ 2
2
T.A-1
4
2
Z,
2
L
0
2
2
T,
2
2
2
T,
4
2
L+2
2
h
h
2
L 2
2
A
fx +1
^A
i
2(£-2) 7(21,-1) (L-S)(L-2) j2.3(L-l)(L+l){2L+3) 5(L-2)J(L-1)(L+1) l(2L-l)j2.3(2L+l)(2L+3) 5s/(L-l)L{L+l)(L + 2)
+1
4
L-2
T.
1
9
r. r
£-2
(2L-3){2L + 5)
r.
1 2
2
L
3 (L~-lHL++2)
r,-i- 9
T.
2
r i o
•7/ O T* _i Q\ // O 7" _L 1 \ pr/ 7" i Q\ IT 1 T _L_ 9 \
4 n(of
o
r i o
9
T -i- 9
n
2
£ +2
2
L+2
2
j _ Q\ / 9 *\(Q.T — 1 ^ ^97" -l-1 ^
1 1
r,-i- 9 j.
L+2 7 { 2I±J -f~ o )
2
£ +2
2
L+2
4
(JC + 3)(X, + 4)
2.7(2£+l)(2Z, + 3)
3.5(£-3)(£-2) 2(2£-5)(2£-3) 2.3.5(£-2) (21,-5) (21,-I) 2 32.5 5(£- 2) V3(£- 1) (£+ 1) (2£ + 3)
(2£-l) 2 V(2£+l)(2£ + 3) 3.5V(£-i)£(£+l)(£ + 2) 32.5(£-l) 1 (2£-l)2(2£ + l) 5[4£2(£ + 1 )2 - 39£(£ + 1) + 63] 32.5(£ + 2) Q K /O Q7"/7" _i_ 0\ O . O2\ & . Qi/^JJ ~J" ji^
In Q T i r i o \ / O 7" 1\ \j £ • OJU\-LJ ~p ^ ^ ^ZiJL/ ~ ~ X^ I ( ^ JL/ ~J~ *5 J v V • " • " "T" •!• ^
o
f 1
L+4
(2L+3) V(2£-1)(2L+1) 5(1, + 3) V3£(£ + 2) (2£ - 1) /cyj~ 1^ Q\2 /9/07" _i_ 1 \
32.5
(2£+3) 2 (2£ + 7)
662
I. C. PEBCIVAL AND M. J. SEATON
) and h&l,^;
Table 2. h
h
H
A 0
+1
£-1
2
1 5
£+1
1
L
1
1
L
1
L
T,
2
T
1
i
T.
2
7"
1
Q
1
1
L
2
L+l
1
j
T.
2
T -U 1
o
9
T
2
r
1
n
2
L-l
2
r
i
2
2
Tj
1
2
r, i
4
2
T,
1
2
L+l
2
2
£-1
2
L+l
4
2
L+l
2
L+l
0
2
L+l
2
L+l
2
2
L+l
2
L+l
4
1
L) = (2A+ l)gAhWz\ L)for A
h
V(£-i)
T,
9
r
1 s/5(2L+ 1) V(£ + 2) V5(2£+l) o // 7" _L o^
1 1
(£ + 5) 7(2£+l) (L-2) 7(2£+l) 3V(£— l ) ( £ + 2)
7(21,+1) 5J(L— l)(L + 2)
3.7(2£ + l) +1 (£ — 4) 7(2£+l) (£+3) 7{2L+1)
•)• See footnote to Table 1.
3
(2L+1) 3
' (2£+l) 3V5(£-1)
(aL
s"jliS)+1> l(a Vv^> + 1 )
/ o r _i_ Q\ / / o y _i_ i \
T. 4- 9
r
Q
r
I
£+1
r,
I
L+l L - l
L+l i+a
3^/5(^ + 2)
3.5(£-2) 3.5(2i 2 -JD-7) 1 (2£-3)(2£+l) 2 32.5 (2£-l)(2£+l) 2 3.5 J(L— l)(L + 2)
(2£+l) 2 3.5y/(L—
1)(L + 2 )
(2£+l) 2 32.5 (2£+l) 2 (2£ + 3) 3.5(2£2 2+ 5iS —4) (2£+ l) (2£ + 5) 3.5(£ + 3) (2£ + 3)(2£ + 5)
