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Excitation of helium by protons and electrons
Physica 42 (1969) 245-261
EXCITATION
0 North-Holland
Publishing
Co., Amsterdam
OF HELIUM BY PROTONS AND ELECTRONS J. VAN DEN BOS*
FOM-Instituut
VOOYAtoom-
en Molecuulfysica,
Received
Amsterdam,
Nederland
1 July 1968
synopsis Generalized oscillator strengths and cross sections for excitation of helium by protons and electrons have been calculated using Born approximation. Transitions from the ground state to all n = 2, 3, 4 singlet states have been considered, including some sub-state transitions. Three types of ground state wave functions have been employed : the one-parameter Hylleraas function, an exponential fit to the Hartree-Fock function, and a variationally determined Eckart function. The final states are linear combinations of products of hydrogenic wave functions. It is shown that the different wave function combinations give rise to largely different excitation functions, but that the combination with HF gives, except for nlS, results that complete favourable with results obtained using much more accurate wave function combinations.
1. Ivztrodz4ction Collisional excitation of helium has been considered theoretically by many authors. Most results are given in the form of excitation cross sections but some also gave generalized oscillator strengths or related quantities (see table I). Others only calculated oscillator strengths in the limit of zero momentum transfer (see table II). Usually the electron has been taken as the impinging particle, but also some calculations are available with a proton as the projectile. For electron kinetic energies from a few hundred eV upwards, the proton cross sections in the Born approximation may be obtained from the electron cross sections by comparing projectiles of equal velocity. For lower electron energies one may apply an approximate relationship between proton and electron energies one may apply an approximate relationship between proton and electron cross sections derived by Bates and Griffingl). For too low energies, however, this relationship breaks down. In this paper we present cross sections for excitation of helium by electrons and by protons. The results for the two projectiles are derived independently. * Present address: Institute of Computational Lincoln, Nebraska, USA. 245
Sciences, University
of Nebraska,
246
J. VAN DEN BOS TABLE I
Theoretical work on excitation of helium in the Born or related approximations Author
Approximation
Ground state used
H H H, HF, VE VE -
3iD-5iD,
Born Born Born Born Born Ochkur
HF
e e
3iD-6iD,
Born
Hyll.-6 par.
e
25-5000 eV
e
25-100 eV 40-260 eV
Final state
Massey and Mohr is) Altshuler 13914) Fox 15) Fox is) Lassettre 17) Ochkur and Brattsevis) Bell et al. 19) Percival and Seatonss) Rothensteinei) Moiseiwitsch and Stewart 22) Bell 23)
2rS, 3iS, 3iD-5iD, 2is, zip, 21%10% 3rD 21s 2iP-5iP, 4iF, 5iF 2lP-6rP, 2is, 31s
2rP-5iP, 4iF 31P
Projectile
Energy range
e
100-400 122-871 20-250 23-180 20-400 30-500
e
e e
eV eV eV eV eV eV
3iD 2iP
Born 2nd Born
H H
2iP 21P. 31P
Born Born, Distortion
H HF
10-1000 keV
Born Born
VE H
25-375 keV 10-1000 keV
Kim, Inokuti26)
3rD, 4iD 2iP-5iP, 3iD-5rD, 4rF 2is, 3is, 21P, 3iP
Born
Hyll.-53 par.
Bell et al. 27)
2iP-6rP,
Born
Hyll.-6 par
McDowell and Pluta 24) Gaillard 25)
3iD-6iD
The excitation of all upper states with principal 3, 4 has been considered.
quantum
e
3-600 keV
Born region P
10-2000 keV
number n = 2,
2. Theory With reference to the well-known quantity optical oscillator strength, Bethes) defined the generalized oscillator strength fin(K), for a collisioninduced transition from a state i to a state n, as 171
fin(K)
=
q$ I
eiK*rf IYi(Yl . . . Y,)>12,
(1)
where AE stands for the difference in energy between states i and n, K is the transferred momentum, ul, the initial state of the atom, ul, the final state and rj the electron coordinate of electron j. The summation is carried out over all electrons in the atom. All quantities are in atomic units, these units being employed throughout this paper unless otherwise mentioned. Taking the polar axis along K (K r = Kz)implies that only wave functions with Am = 0 (m stands for the magnetic quantum number) contribute to l
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TABLE IIa Values
of lim & K-+0
f0 12
fA
f2
ftl
12
Wave functions
21s
3%
21P
4%
H-BE
0.0268
0.0140
0.0674
L-BE
0.0558
0.0249
0.0116
B J-BE
0.0611
0.0266
0.0123
H-MYH
0.172
L-MYH
0.259
B J-MYH
0.269
VE-MYH
0.351
31P
H-GC
0.0523
L-GC
0.0728
B J-GC
0.0745
VE-GC
0.0879
41P
31D
41F
41D
H-E
0.187
0.0545
0.0228
0.315-2
0.172-2
0.222-4
L-E
0.280
0.0751
0.0320
0.801-2
0.419-2
0.100-3
BJ-E
0.290
0.0767
0.0340
0.907-2
0.469-2
0.1273
VE-E
0.376
0.0896
0.0352
0.192-l
0.932-2
0.495-3
eq. (1) ; the generalized oscillator strength is now only a function of the magnitude of K. Using this simplification (which was employed by us in most cases) the cross section for the mentioned transition is JLI,, 4X fnW cw
bn = -
AEvs
K
s
’
where we now have dropped the index i because only transitions from the ground state are further considered. In eq. (2) v denotes the initial velocity of the projectile and = Kzmx mill
M 2Mv= -
2AE rf 2Mv2 Jl
-
E];
M denotes the mass of the projectile. In the case of proton impact the limits of integration in eq. (2) may, to an entirely satisfactory approximation, be replaced by K,,, = CO and Kmin = AElv, down to 1 keV proton energy. The error in this approximation was shown to be maximally less than 1%. This simplification leads to the same final equations as the impact parameter treatment a+. 3. Wave fzmctions The general form of the final state wave functions %h
r2) =
$
hd2,
~1) W+Z)
+
9h(2,
~2)
used is given by:
~f(rl)l,
J. VAN DEN BOS
248
TABLE IIb Values of lim ri K-0
f0n.
fit Authors
21s
Altshuler I r3, 14) 0.0366 0.0557 Altshuler II l3, 14) Lassettre and Jones 30) Skerbele and Lassettre 31) 0.099 0.13 Garstang 32) 0.032 Foxr5); H 0.0715 Foxr5); HF 0.112 Fox is) ; VE Heideman and Vriens 33) 0.155 Kim and Inokuti26) 0.0836 Schiff and Pekeris 34) Dalgarno and Stewart 35) Skerbele and Lassettre 36) Moustafa Moussa 37) 0.11 Boersch et al. 3*) Weiss 39) Bell et al. 119) Bell et al. 1119) McDowell and Stauffer40) HI H II HF I HF II VE Correlated closed shell Correlated open shell
4%
3%
2iP
3iP
0.191 0.268 0.268
0.0564 0.0753
fA 4rP
3iD
4iD
0.021
0.043 0.0170
0.072
0.031 0.959-Z
0.276 0.275 0.268 0.270 0.276 0.288 0.299
0.0734 0.0746 0.0730 0.0730 0.0766 0.0732 0.0767 0.0788
0.0304 0.0300 0.0280 0.0359 0.0303 0.0311 0.0319 0.315-Z 0.567-Z 0.825-Z 0.903-Z 1.95 -2 0.333-Z 0.828-Z
0.172-Z 0.302-Z 0.419-Z 0.459-Z 1.01 -2 0.544-Z
where N is a normalization constant. Apart from the exceptions mentioned below the function yf(r) was taken equal to vf( 1, r), where the functions ~(2, r) are hydrogenic eigenstates with charge 2. In the case of final states 2P and 3P the variationally determined wave functions given by Morse et al. 5) and Goldberg et al. 6) were used, viz.
(5)
with c = 5Alj.4, A = l/(1 + Z/2/4,
N& = ,~7/(25As -
25A + 15/2), 2 = 0.97
EXCITATION
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and ,u = 0.325. Where eqs. (4) and (5) have been employed the results are labelled by MYH ans GC, resp. In the case of 2S, 3S, 4s hydrogenic wave functions have been employed, but in order to insure orthogonality to the ground state involved, the first node of the function has been shifted purposely; these results are labelled by BE (Van den Bos-Eckart). All other results are labelled by E (Eckart). Three initial (ground) state wave functions have been used, viz. I.
pt =
m4(f%
71) m&,
a =
r2),
27/16.
(6)
Results employing this ground state are denoted by H (Hylleraas exp. 1). 11. yg =
#o(r1) 90@2)
with
Me =
J$ (e-ay
+ q e-or).
The parameters are chosen such that $0 closely fits the Hartree-Fock wave function’). Two sets of parameters have been presented. The first by Lowdins) has (II= 1.455799, p = 2or,7 = 0.60 and Ni, = 1.48423; results employing this set will be denoted by HF(L). The second set has been given by Byron and Joachains) and has a = 1.41, /l = 2.61,~ = 0.799 and Nr, = = 1.302525; results with this set will be labelled HF(B J). III. !P$= 5
bls(l/t~l) 9%3(&r2)
+
m3(&r1)
w3(y,
r2)l.
This is a variationally determined wave function by Eckartia). The parameters are givenil) by y = 2.1832 and 6 = 1.1886. Results with this wave function will be denoted by VE. 4. Calculations
and reszclts
All expressions for generalized oscillator strengths and cross sections could be evaluated analytically. The complete expressions have been publised elsewhere 2%29). In th’is section we present some numerical and graphical results for generalized oscillator strengths and cross sections, together with results of other workers. 4.1. Generalized oscillator strengths. For small momentum transfer the generalized oscillator strength fn(K) can be expanded in powers of K2 (see ref. 2)) as follows : b(K)
= /: + /AK2 + f:K4
+ . . .,
(9)
250
J. VAN
DEN
B’S
TABLE IIIa Generalized
oscillator
strengths
as a function
for 4rP where
of Kz using as the ground HF(L)
f(K)lK2
Ks 21s
f(K) 4%
31s
state HF(B J), except
is employed.
fWlK2
2iP
31P
41P
f (W/K4 4rD
3iD
4iF
0
0.61 l-l
0.266-l
0.123-l
0.269
0.745-l
0.320-l
0.907-2
0.469-2
0.127-3
0.05
0.565-l
0.249-l
0.115-l
0.248
0.700-l
0.303-I
0.797-2
0.416-2
0.106-3
0.10
0.523-l
0.233-l
0.108-l
0.230
0.658-l
0.287-l
0.702-2
0.369-2
0.887-4
0.20
0.449-l
0.204-I
0.950-2
0.197
0.581-l
0.257-l
0.548-2
0.293-2
0.30
0.388-l
0.179-l
0.836-2
0.170
0.514-l
0.230-l
0.432-2
0.233-2
0.630-4 0.452-4 0.328-4
0.40
0,336-l
0.157-l
0.737-2
0.147
0.455-I
0.206-l
0.342-2
0.187-2
0.50
0.292-l
0.139-l
0.651-2
0.127
0.403-l
0,185-l
0.273-2
0.15 l-2
0.241-4
0.60
0,255-l
0,122-l
0.575-2
0.111
0.357-l
0.166-l
0.219-2
0.122-2
0.178-4
0.70
0.223-l
0.108-l
0.510-2
0.970-l
0.318-l
0.149-l
0.177-2
0.997-3
0.133-4
0.80
0.196-l
0.959-2
0.453-2
0.851-l
0.283-l
0.134-l
0.144-2
0.817-3
0.100-4
0.90
0.173-l
0.852-2
0.403-2
0.749-l
0.252-l
0.121-l
0.118-2
0.672-3
0.762-5
1.00
0.153-l
0.758-2
0.359-2
0.661-l
0,225-l
0.109-l
0.970-3
0.556-3
0.584-5
1.20
0.120-t
0.604-2
0.286-2
0.519-l
0.181-l
0.884-2
0.665-3
0.385-3
0.350-5
1.50
0.854-2
0.436-2
0.207-2
0.368-l
0.131-l
0.656-2
0.390-3
0.229-3
0.169-5
2.00
0.505-2
0.262-2
0.125-2
0.218-l
0.800-2
0.410-2
0.174-3
0.103-3
0.567-6
2.40
0.343-2
0.180-2
0.858-3
0.148-l
0.553-2
0.287-2
0.963-4
0.578-4
0.256-6
2.80
0.239-2
0.126-2
0.603-3
0.391-2
0.206-2
0.558-4
0.337-4
0.123-6
3.20
0.171-2
0.903-3
0.432-3
0.103-l 0.737-2
0.282-2
0.124-2
0.659-3
0.315-3
0.537-2
0.207-2
0.336-4 0.209-4
0.204-4 0.127-4
0.625-7
3.60
0.150-2 0. I1 l-2
0.332-7
4.00
0.920-3
0.489-3
0.234-3
0.398-2
0.154-2
0.832-3
0.134-4
0.819-5
0.183-7
TABLE IIIb Generalized
oscillator
strengths
as a function
formulation,
f(K) present
of Kz for different
II + velocity
groups;
for 2iP
Kim and Inokutise)
work
I + length
formulation
Bell et al. 19)
Ks
BJ-MYH
0.0
0.2691
0.2902
0.2754
0.2756
0.2882
0.2985
0.25
0.1828 0.6608-l 0.3982-2
0.1983 0.7270-l 0.4502-l
0.1859 0.6542-l 0.3448-2
0.1859
0.1973 0.7162-l 0.3991-2
0.2035 0.7412-l 0.4634-2
1.0 4.0
I
BJ-E
I
I
f(K)/Kz present Ks
II
0.6540-l 0.3431-2
for 3iD Bell et aZ.19)
work
BJ-E
I
I
I
0
0.9067-2
0.860-2
0.01 0.25
0.8835-2 0.4861-2
0.8399-2 0.4764-2
1.0 4.0
0.9697-3 0.1340-4
0.9980-3 0.1355-4
II
0.9147-2 0.5064-2 0.1004-2 0.1330-4
II
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f,(K)
f,(K)
K2
K*
I
0.06
I
1 ‘s-z’s
251
1
‘s-3
‘s
0.015
1
1 ‘s -4
‘s
0.030
0.06
\
\ \
\ .
1
1
0.5
-
1
I
0.5
1.0
1.0
K’
Fig. 1. Generalized oscillator strengths for is-1s transitions in He. this work Kim and Inokutiss) K,I ---a -- Lassettre et al. so) experimental , f,(K)
, ‘s-2 ‘p
O.‘O it
-it
I
0
I 0
, f,(K)
f,(K)
I
I
a5
1.0
1 ‘s-3
‘p -
ao4
1 ‘s -4
il
BJ-GC
I
I
Fbl
‘P
L-E
I
I
I
0.5
1.0
I
2
a5
I
1.0
-K
Fig. 2. Generalized oscillator strengths for iS-iP transitions in He. ___ this work. A Lassettre et aZ.30) experimental. v Heideman and Vriensss) experimental. Theoretical results of Kim and Inokutiss) almost coincide with our B J-MYH curve for 2iP and B J-GC curve for 3iP. Experimental data of Vriens et al. 3s) for 2rP coincide with theoretical data of ref. 26.
J. VAN DEN BOS fno 0.020
K2
a010
-I
I \
BJ-E Y
\ \
I I
-i-
\
1.0
a5 ----K
0
2
Fig. 3, Generalized oscillator strengths for %-1D and 1S-lF transitions in He.
where j”, is the usual optical oscillator strength (the superscript is often omitted). For optically non-allowed transitions /l = 0, for the limit K = 0. In table IIa we present our values for the first non-vanishing expansion coefficients in the limit of K = 0. In table IIb we present values of other groups. The 1abeIs I and II denote length and velocity formulation, resp.; where no label occurs the length formulation is meant. In table IIIa numerical values for the generalized oscillator strength as a function of Ks have been given only for the ground state HF(B J) (except in the 4P case where HF(L) has been given). This choice was made because as can be seen from table II this is in general the ground state which produces results very well comparable with the results using accurate but complicated ground states. This choice is supported by even more arguments when we cross sections compare cross sections (see below). The nS excitation are, however, an exception to this, but have been given with BJ for oscillator reasons of conformity. In figs. 1-3 we present generalized strengths and compare with experiment and some other theoretical results. In figs. 2 and 3 results of Kim and lnokuti26), Lassettre and Jonesao) and Bell et al. 19) have been omitted, because they almost coincide with our curves. An illustration of the good agreement of our calculations with the more extended ones of Kim and Inokuti26) (with 53 parameters Hylleraas wave functions) and of Bell et ~2.19) (with a six parameter ground state Hylleraas wave function) is given in table IIIb for 2iP and 3iD. These groups used both the length and velocity formulation of the matrix element. Kim and Inokuti26) obtained agreement better than 1% between both calculations.
EXCITATION
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TABLE IVa Excitation except
cross sections
in units zag for e on He using HF(B J) ground
in the case of 4rP HF(L),
and as final state wave
functions
state wave functions,
BE for niS, MYH
for 21P
GC for 3iP and E for the remainder 21s
E(eV)
31s
4%
21P
4iP
3iP
4iD
3lD
4iF
25
0.299-l
0.9341
0.170-l
0.3203
0.527-2
0.132
0.303-l
0.585-2 0.121-l
0.747-3
0.328-l
0.893-2 0.123-l
0.323-2
30
0.126-2
0.638-3
0.350-5 0.659-5
35
0.319-l
0.127-l
0.555-2
0.152
0.366-l
0,149-l
0.144-2
0.748-3
0.735-5
40
0.302-l
0.122-l
0.543-2
0.162
0.150-2
0.786-3
0.739-5
0.264-l
0.110-l
0.490-2
0.169
0.400-l 0,425-l
0.163-l
50
0.174-l
0.147-2
0.775-3
0.682-5
70
0.206-l 0.152-I
0.868-2
0.391-2
0.164
0.417-l
0.171-l
0.127-2
0.674-3
0.544-5
100
0.648-2
0.293-2
0.147
0.376-l
0.154-l
0.101-2
0.535-3
0.402-5
150
0.106-l
0.452-2
0.205-2
0.123
0.315-l
0.128-l
0.735-3
0.390-3
0.276-5
0.305-3 0.212-3
0.140-5
200
0.807-2
0.347-2
0.158-2
0.105
0.271-l
0.110-l
0.574-3
300
0.548-2
0.236-2
0.107-2
0,828-l
0.213-l
0.862-2
0.399-3
500
0.333-2 0.224-2
0.144-2
0.6553
0.592-l
0.152-l
0.1313
0.845-6
0.440-3
0.445-t
0.114-l
0.615-2 0.461-2
0.247-3
0.966-3
0.167-3
0.888-l
0.564-6
750
0.209-5
1000
0.168-2
0.728-3
0.331-3
0.361-l
0.925-2
0.373-2
0.126-3
0.671-4
0.423-6
5000
0.339-3
0.147-3
0.669-4
0.102-l
0.262-2
0.105-2
0.257-l
0.137-4
0.847-7
10000
0.170-3
0.735-4
0.335-4
0.577-2
0.148-2
0.593-3
0.12994
0.684-5
0.423-7
TABLE IVb Excitation
cross sections I --f length
in units nai for e on He for different
formulation,
II + velocity
o(2rP) present
work
E (eV)
BJ-MYH
50
I
in 7rui Kim and Inokutise)
BJ-E
groups:
formulation
I
Bell et al. is)
I or11
I
II
0.169
0.185
0.169
0.182
0.187
100 500
0.147 0.592-l
0.160 0.642-l
0.147
0.158
0.164
0.605-l
0.635-l
0.656-l
1000
0.361-l
0.391-l
0.368-l
0.386-l
0.400-l
5000
0.102-I
0.11 l-1
0.105-l
0.110-l
0.113-l
(I (3iD) _A E (eV)
present
in rcui
work
BJ-MYH
I
Bell et al. is)
-
I
II
50
0.147-2
0.149-2
0.152-2
100 500
0.101-2 0.247-3
0.101-2 0.246-3
0.105-l 0.2563
1000
0.126-3
0.125-3
0.1303
5000
0.257-4
0.254-l
0.266-l
254
J. VAN
DEN
BOS
TABLE V Excitation except
cross sections
in units ,a;
in the case of 4lP HF(L)
for p on He using HF(B J) ground
and as final state wave
functions
state wave functions,
BE for nrS, MYH
for 2rP
and E for the remainder E(keV)
2%
3’S 1
21P
41s
31P
41P
3rD
4rD
4rF 0.98 l-6
5
0.422-
0.128-l
0.511-2
0.365-l
0.785-2
0.322-2
0.249-3
0.12 1-3
10
0.664-l
0.233-l
0.976-2
0.107
0.252-l
0.106-l
0.102-2
0.5153
0.527-5
15
0.701-l
0.261-l
0.112-l
0.157
0.386-l
0.162-l
0.165-2
0.850-3
0.912-5 0.113-4
20
0.674-l
0.260-l
0.113-l
0.189
0.472-l
0.198-i
0.203-2
0.106-2
30
0.584-l
0.234-l
0.103-l
0.220
0.558-l
0.233-l
0.231-2
0.122-2
0.125-4
50
0,438-l
0.181-l
0.809-2
0.229
0.587-l
0.244-l
0.221-2
0.117-2
0.109-4
70
0.345-l
0.145-l
0.650-2
0.219
0.564-l
0.233-l
0.194-2
0.103-2
0.902-5
100
0.261-l
0.110-1
0.497-2
0.200
0.515-l
0.212-l
0.160-2
0.849-3
0.692-5
150
0.184-t
0.786-2
0.356-2
0.171
0.442-t
0.181-l
0.121-2
0.644-3
0.489-5
200
0.142-l
0.610-2
0.276-2
0.150
0.387-l
0.158-l
0.970-3
0.5153
0.376-5
250
0.116-l
0.498-2
0.226-2
0.134
0.345-l
0.140-l
0.8073
0.4293
0.304-5
300
0.979-2
0.421-2
0.121
0.312-l
0.127-l
0.6913
0.367-3
0.255-5
0.893-l
0.229-l
0.931-t
0.437-3
0.232-3
0.171-5
0.562-l
0.144-l
0.583-l
0.227-2
0.121-3
0.776-6 0.388-6
500
0.602-2
0.259-2
0.191-2 0.118-2
1000
0.307-2
0.175-2
0.603-3
2000
0.155-2
0.6693
0.305-3
0.340-l
0.871-2
0.351-l
0.116-3
0.617-4
5000
0.622-3
0.269-3
0.123-3
0.167-l
0.428-2
0.172-2
0.470-4
0.250-4
0.155-6
10000 0.312-3
0.135-3
0.6 14-4
0.955-2
0.244-2
0.982-3
0.236-4
0.125-4
0.777-7
TABLE VI Sub-level
excitation
functions
cross sections
in units xat for p on He using HF(B J) ground
and as final state wave 21Pr
E(keV)
2rPs
5
0.280-l
0.438-2
0.134-3
10
0.760-l
0.155-l
0.4583
3rDe
functions
MYH
state wave
for 2rP and E for the remainder
3lDa
4rFc
4rFr
4rFa
0.520-4
0.645-5
0.3806
0.235-6
0.640-7
0.660-8
0.242-3
0.388-4
0.159-5
0.130-5
0.48 l-6
0.645-7
3rDl
4rFs
15
0.106
0.258-l
0.641-3
0.423-3
0.820-4
0.227-5
0.222-5
0.104-5
0.170-6
20
0.122
0.337-l
0.695-3
0.545-3
0.123-3
0.243-5
0.265-5
0.151-5
0.289-6
30
0.132
0.439-l
0.645-3
0.650-3
0.157-3
0.223-5
0.261-5
0.203-5
0.495-6
50
0.124
0.520-l
0.459-3
0.635-3
0.2413
0.171-5
0.177-5
0.213-5
0.725-6
70
0.111 0.934-l
0.540-l 0.530-l
0.334-3
0.550-3 0.431-3
0.257-3 0.253-3
0.139-5
100
0.108-5
0.115-5 0.665-6
0.187-5 0.145-5
0.805-6 0.810-6
150 200
0.731-l 0,599-l
0.492-l 0.452-l
0.301-3 0.222-3
0.229-3
0.761-6
0.355-6
0.970-6
0.740-6
0.119-3 0.995-4 0.867-4
0.204-3 0.183-3
0.568-6 0.441-6
0.247-6 0.200-6
0.690-6 0.515-6
0.655-6 0.559-6
0.137-3
0.165-3 0.119-3
0.352-6 0.175-6
0.231-3 0.154-3
250
0.507-l
0.416-l
300 500
0.440-l
0.386-I
1000
0.288-l 0.155-l
0.303-l 0.203-i
0.376-4
0.700-4 0.254-4
2000 5000
0.812-2 0.335-2
0,129-l 0.670-2
0.221-4 0.102-4
0.850-5 0.186-5
10000
0.170-2
0.393-2
0.541-5
0.560-6
0.608-4
0.172-3
0.645-4 0.385-4
0.591-7
0.166-4
0.330-8 0.877-9
0.860-5
0.179-7
0.175-6
0.399-6
0.525-6
0.134-6 0.915-7
0.183-6 0.570-6
0.369-6 0.210-6
0.565-7 0.261-7
0.163-7 0.289-8
0.113-6 0.471-7
0.138-7
0.750-9
0.239-7
EXCITATION
OF HELIUM
4.2. Cross sections.
BY PROTONS
AND ELECTRONS
255
From the expression eq. (2) and a closely related one
employing eq. (1) we derived analytical expressions for the cross sections to all nlm states with n = 2, 3, 4. In the limit of infinite energy one derives, as expected,‘a, N E-1, crp N E-1 log E and CT~N E-l. For the m sub-states more complicated relationships were found, viz. op,,, gdo> odZ> uf,,
of,
-
E-l,
(10)
E-2,
(11)
E-1 log E,
(12)
bd, N E-2 log E.
(13)
no,
af,
UP1N
N
One also finds that in the high energy limit cd, = #ode and uf, = $uf, and that excitation to sub-states with quantum numbers I, m vanishes for l-m odd compared with even Z---m sub-states (see also ref. 20, eq. (6.15)). Results for excitation cross sections due to proton and electron impact have been given in tables IV, V and VI for a range of energies, and in figs. 4-8 together with some theoretical results of other authors. For comparison with experiment see for instance refs. 37, 29 and 41. In table 1Vb we illusstrate the good agreement of our calculations with more extended ones of TABLE VII Polarization fractions of transitions induced by collisions of protons on helium. Calculated using HF(BJ) ground state E (keV) threshold 5 10 15 20 30 50 70 100 150 200 250 300 500 750
1000 2000 5000 10000 00
17 (2iP-1%) 1.00 0.73 0.66 0.61 0.57 0.50 0.41 0.35 0.28 0.20 0.14 0.10 0.07 -0.03 -0.09 -0.13 -0.23 - 0.33 - 0.40 - 1.00
II (3’D-2iP) 0.60 0.44 0.40 0.36 0.33 0.27 0.19 0.14 0.07 0.00 - 0.05 -0.09 -0.12 -0.19 -0.24 -0.27 -0.33 -0.38 -0.40 -0.43
J. VAN DEN BOS
256 aEe1(10 -
20 keV cmz/atom
--102aEe1 4xaiR
I -5 _.-._.-4
-3
LAttshuler I -2
-1
o-
I
,.11.,
0.05
0.1
.
* ,‘..‘I
0.2
,
Q5
I 1 0
1
2
3
45
-
E,,
in keV
Fig. 4. Cross sections for 2% excitation by electrons (u&l versus In &I). R is the Rydberg energy.
dE,,(10~20keV cm*/atom)
lo3 d E, 4lT&R
JBJ-BE LL -BE
7 -8
C-Kim, Inokuti
_,_._._.-J._.-.---.-.-
I
H-BE
2-
0,
a05
0.1
1,
3’s
/dzohr
I...,
-6
o2
I . ‘7’1 0.5
1 0 1
__c
2 E,\
3
45
in keV
Fig. 5. Cross sections for 3% excitation by electrons (uEel vem4.s In Eel).
EXCITATION
OF HELIUM
BY PROTONS
AND ELECTRONS
cl
G
1.2
2’P
aa
0.4
I
0.1
i
I
0.2
,
I
,
1
a5
1
2
5
_
1
10
E,, in keV
Fig. 6. Cross sections for 21P excitation by electrons (uE,J4xai versus In Eel). A -+ Altshuler’s velocity formulation, using H-E. x + Altshuler’s length formulation using H-E. UEe,Ml-20keVcrn*/atom 1
lo36 &,
4nae2R
-1
I-
-4
-2
o-8 . . ..l a05 0.1
I 02
3
‘Is*.., cl5
1 -
I 2
at,-0 3 45 E,, in keV
Fig. 7. Cross sections for 31D excitation by electrons (u&l vewus In Eel).
257
J. VAN DEN BOS
258 oEel(eVxao2)
‘OI l4'F x10'
I 1
1
25
5
I
100
-
-*----
-
10000 E.(eV)
I
1
1000
Fig. 8. Comparison of cross sections for excitation by electrons and protons, using BJ-E wave function combinations (u&l versus In E,l). For protons E,. = (m/M) Ep.
Kim and Inokutis6) and of Bell et al. 19) for excitation of helium by electrons to 21P and 3rD. As mentioned before (see section 4.1) our B J-BE cross sections for 2% and 31s deviate much from those of Kim and Inokuti (see figs. 4 and 5), where the simple asymptotic Whittaker functions employed by Fox15) are seen to give much better results (see also the second article of ref. 19). Polarization fractions have been calculated (see table VII) for 1P-1s and ID-1P transitions in the case of proton impact, by using the method of reference 20 and the cross sections of table VI. In fig. 9 we illustrate that the ground state wave function has not a great influence on the polarization fraction in the case of ID--1P transitions. 5. Con&&on The different types of wave functions employed are seen to give very different results for the generalized oscillator strengths and the excitation cross sections, but reasonably agreeing results for the polarization fractions. Comparison with more extended and more accurate calculations for oscil-
EXCITATION
OF HELIUM
BY PROTONS
259
AND ELECTRONS
-il6
1 I
5
10
I
I
100
10.000
1000 -
E,+ in keV
Fig. 9. The polarization fraction n of 31D-21P radiation in the case of excitation by protons.
lator strengths as well as for cross sections shows, however, that for the description of the excitation of nrP and lziD states from the ground state the HF(BJ) wave function for the ground state is a very good substitute. The reason for the exceptional behaviour of the lziS cross sections might well be due to the excited state wave functions for rtS which are e.g. not mutually orthogonal. In this case the simple asymptotic Whittaker functions employed by Foxis) are seen to give much better results when compared with the results of Kim and Inokutisa) which are claimed to be correct within a few percent. In order to find theoretical results which are valid below the Born region one has to take into account the coupling between states other than initial and final. These kinds of calculations using the simple wave functions which can be accepted according to the results above are in progress.
260
J. VAN DEN BOS
Acknowledgments. We arevery grateful to Dr. F. J. de Heer for the cooperation in preparing this paper for publication. We also acknowledge the valuable discussions and remarks from Drs. M. Inokuti and L. Vriens and from Professors J. Kistemaker and C. Joachain. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research).
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)
Bates, D. R. and Griffing, G., Proc. Phys. Sot. 56 (1953) 961. Bethe, H. A., Ann. Physik 5 (1930) 325. Crothers, D. S. F. and Holt, A. R., Proc. Phys. Sot. 88 (1966) 75. Van den Bos, J. and De Heer, F. J., Physica 34 (1967) 333. Morse, P. M., Young, L. A. and Haurwitz, E. S., Phys. Rev. 48 (1935) 948. Goldberg, L. and Clogston, A. M., Phys. Rev. 56 (1939) 696. Roothaan, C. C. J., Sachs, L. M. and Weiss, A. W., Rev. mod. Phys. 32 (1960) 186. Lowdin, P.-O., Phys. Rev. 90 (1953) 120. Byron, F. W. and Joachain, C. J., Phys. Rev. 146 (1966) 1. Eckart, C., Phys. Rev. 36 (1930) 878. Shull, H. and Liiwdin, P. O., J. them. Phys. 25 (1956) 1035. Massey, H. S. W. and Mohr, C. B. O., Proc. Roy. Sot. A140 (1933) 6 13. Altshuler, S., Phys. Rev. 89 (1953) 1093. Altshuler, S., Phys. Rev. 88 (1952) 992. Fox, M. A., Proc. Phys. Sot. 86 (1965) 789. Fox, M. A., Proc. Phys. Sot. 88 (1966) 65. Lassettre, E. N., J. them. Phys. 43 (1965) 4479. Ochkur, V. I. and Brattsev, V. F., Opt. and Spectr. (USA) 19 (1965) 274. Bell, K. L., Kennedy, D. J. and Kingston, A. E., J. Phys. B. (Proc. Phys. Sac.) 1 (1968) 204; Bell, K. L. and Kingston, A. E., J. Phys. B. 1 (1968) 526. Percival, I. C. and Seaton, M. J., Phil. Trans. Roy. Sot. A251 (1958) 113. Rothenstein, W., Proc. Phys. Sot. 57 (1954) 673. Moiseiwitsch, B. L. and Stewart, A. L., Proc. Phys. Sot. (London) A67 (1954) 1069. Bell, R. J., Proc. Phys. Sot. 78 (1961) 903. McDowell, M. R. C. and Pluta, K. M., Proc. Phys. Sot. 89 (1966) 733. Gaillard, M., C. R. Acad. Sci. Paris B263 (1966) 549. Kim, Y.-K. and Inokuti, M., Phys. Rev. 175 (1968) 176. Bell, K. L., Kennedy, D. J. and Kingston, A. E., J. Phys. B 1 (1968) 218. Van den Bos, J., Phys. Letters 24A (1967) 329. Van den Bos, J., Thesis, University of Amsterdam, 1967 and FOM-report no. 24274, 1968. Lassettre, E. N. and Jones, E. A., J. them. Phys. 40 (1964) 1218;
EXCITATION Lassettre,
OF HELIUM
E. N., Krasnow,
BY
PROTONS
M. E. and Silverman,
AND
ELECTRONS
S. M., J. them.
Phys.
261 40 (1964)
1242. 31)
Skerbele,
32) 33)
Garstang, Heideman,
A. M. and Lassettre,
34)
Schiff,
35)
Dalgarno,
36)
Skerbele,
37)
MoustafaMoussa,
38)
Boersch,
B. and Pekeris,
39)
Weiss,
40)
McDowell,
A. and Stewart,
45 (1966)
1077.
46 (1967)
2911.
Geiger, J. Res.
Rev.
A. L., Proc.
Phys.
Sot.
Collisions, 71A
M. R. C. and Stauffer,
A640. 76 (1960)
Phys.
B., Proc.
Nauka, (196’7)
1271.
J., Physica
V Int.
Leningrad
49.
40 (1964)
F. J. and Schutten,
J. and Schroder, NBS
134 (1964)
E. N., J. them.
H. R., De Heer,
and Atomic
A. W.,
Phys.
44 (1966) 1308. L., J. them. Phys.
C. L., Phys.
A. M. and Lassettre, H.,
Electronic
E. N., J. them.
R. H., J. them. Phys. H. G. M. and Vriens,
Conf.
40 (1969)
on the Physics
Van den Bos,
J., Winter,
of
1967, pp. 481.
163.
A. D., Phys.
Letters
12 (1964)
207 and Private
Communication. 41)
517.
G. and De Heer,
F. J., Physica
40 (1968) 357.
EXCITATION
0 North-Holland
Publishing
Co., Amsterdam
OF HELIUM BY PROTONS AND ELECTRONS J. VAN DEN BOS*
FOM-Instituut
VOOYAtoom-
en Molecuulfysica,
Received
Amsterdam,
Nederland
1 July 1968
synopsis Generalized oscillator strengths and cross sections for excitation of helium by protons and electrons have been calculated using Born approximation. Transitions from the ground state to all n = 2, 3, 4 singlet states have been considered, including some sub-state transitions. Three types of ground state wave functions have been employed : the one-parameter Hylleraas function, an exponential fit to the Hartree-Fock function, and a variationally determined Eckart function. The final states are linear combinations of products of hydrogenic wave functions. It is shown that the different wave function combinations give rise to largely different excitation functions, but that the combination with HF gives, except for nlS, results that complete favourable with results obtained using much more accurate wave function combinations.
1. Ivztrodz4ction Collisional excitation of helium has been considered theoretically by many authors. Most results are given in the form of excitation cross sections but some also gave generalized oscillator strengths or related quantities (see table I). Others only calculated oscillator strengths in the limit of zero momentum transfer (see table II). Usually the electron has been taken as the impinging particle, but also some calculations are available with a proton as the projectile. For electron kinetic energies from a few hundred eV upwards, the proton cross sections in the Born approximation may be obtained from the electron cross sections by comparing projectiles of equal velocity. For lower electron energies one may apply an approximate relationship between proton and electron energies one may apply an approximate relationship between proton and electron cross sections derived by Bates and Griffingl). For too low energies, however, this relationship breaks down. In this paper we present cross sections for excitation of helium by electrons and by protons. The results for the two projectiles are derived independently. * Present address: Institute of Computational Lincoln, Nebraska, USA. 245
Sciences, University
of Nebraska,
246
J. VAN DEN BOS TABLE I
Theoretical work on excitation of helium in the Born or related approximations Author
Approximation
Ground state used
H H H, HF, VE VE -
3iD-5iD,
Born Born Born Born Born Ochkur
HF
e e
3iD-6iD,
Born
Hyll.-6 par.
e
25-5000 eV
e
25-100 eV 40-260 eV
Final state
Massey and Mohr is) Altshuler 13914) Fox 15) Fox is) Lassettre 17) Ochkur and Brattsevis) Bell et al. 19) Percival and Seatonss) Rothensteinei) Moiseiwitsch and Stewart 22) Bell 23)
2rS, 3iS, 3iD-5iD, 2is, zip, 21%10% 3rD 21s 2iP-5iP, 4iF, 5iF 2lP-6rP, 2is, 31s
2rP-5iP, 4iF 31P
Projectile
Energy range
e
100-400 122-871 20-250 23-180 20-400 30-500
e
e e
eV eV eV eV eV eV
3iD 2iP
Born 2nd Born
H H
2iP 21P. 31P
Born Born, Distortion
H HF
10-1000 keV
Born Born
VE H
25-375 keV 10-1000 keV
Kim, Inokuti26)
3rD, 4iD 2iP-5iP, 3iD-5rD, 4rF 2is, 3is, 21P, 3iP
Born
Hyll.-53 par.
Bell et al. 27)
2iP-6rP,
Born
Hyll.-6 par
McDowell and Pluta 24) Gaillard 25)
3iD-6iD
The excitation of all upper states with principal 3, 4 has been considered.
quantum
e
3-600 keV
Born region P
10-2000 keV
number n = 2,
2. Theory With reference to the well-known quantity optical oscillator strength, Bethes) defined the generalized oscillator strength fin(K), for a collisioninduced transition from a state i to a state n, as 171
fin(K)
=
q$ I
eiK*rf IYi(Yl . . . Y,)>12,
(1)
where AE stands for the difference in energy between states i and n, K is the transferred momentum, ul, the initial state of the atom, ul, the final state and rj the electron coordinate of electron j. The summation is carried out over all electrons in the atom. All quantities are in atomic units, these units being employed throughout this paper unless otherwise mentioned. Taking the polar axis along K (K r = Kz)implies that only wave functions with Am = 0 (m stands for the magnetic quantum number) contribute to l
EXCITATION
OF HELIUM
BY
PROTONS
AND
ELECTRONS
247
TABLE IIa Values
of lim & K-+0
f0 12
fA
f2
ftl
12
Wave functions
21s
3%
21P
4%
H-BE
0.0268
0.0140
0.0674
L-BE
0.0558
0.0249
0.0116
B J-BE
0.0611
0.0266
0.0123
H-MYH
0.172
L-MYH
0.259
B J-MYH
0.269
VE-MYH
0.351
31P
H-GC
0.0523
L-GC
0.0728
B J-GC
0.0745
VE-GC
0.0879
41P
31D
41F
41D
H-E
0.187
0.0545
0.0228
0.315-2
0.172-2
0.222-4
L-E
0.280
0.0751
0.0320
0.801-2
0.419-2
0.100-3
BJ-E
0.290
0.0767
0.0340
0.907-2
0.469-2
0.1273
VE-E
0.376
0.0896
0.0352
0.192-l
0.932-2
0.495-3
eq. (1) ; the generalized oscillator strength is now only a function of the magnitude of K. Using this simplification (which was employed by us in most cases) the cross section for the mentioned transition is JLI,, 4X fnW cw
bn = -
AEvs
K
s
’
where we now have dropped the index i because only transitions from the ground state are further considered. In eq. (2) v denotes the initial velocity of the projectile and = Kzmx mill
M 2Mv= -
2AE rf 2Mv2 Jl
-
E];
M denotes the mass of the projectile. In the case of proton impact the limits of integration in eq. (2) may, to an entirely satisfactory approximation, be replaced by K,,, = CO and Kmin = AElv, down to 1 keV proton energy. The error in this approximation was shown to be maximally less than 1%. This simplification leads to the same final equations as the impact parameter treatment a+. 3. Wave fzmctions The general form of the final state wave functions %h
r2) =
$
hd2,
~1) W+Z)
+
9h(2,
~2)
used is given by:
~f(rl)l,
J. VAN DEN BOS
248
TABLE IIb Values of lim ri K-0
f0n.
fit Authors
21s
Altshuler I r3, 14) 0.0366 0.0557 Altshuler II l3, 14) Lassettre and Jones 30) Skerbele and Lassettre 31) 0.099 0.13 Garstang 32) 0.032 Foxr5); H 0.0715 Foxr5); HF 0.112 Fox is) ; VE Heideman and Vriens 33) 0.155 Kim and Inokuti26) 0.0836 Schiff and Pekeris 34) Dalgarno and Stewart 35) Skerbele and Lassettre 36) Moustafa Moussa 37) 0.11 Boersch et al. 3*) Weiss 39) Bell et al. 119) Bell et al. 1119) McDowell and Stauffer40) HI H II HF I HF II VE Correlated closed shell Correlated open shell
4%
3%
2iP
3iP
0.191 0.268 0.268
0.0564 0.0753
fA 4rP
3iD
4iD
0.021
0.043 0.0170
0.072
0.031 0.959-Z
0.276 0.275 0.268 0.270 0.276 0.288 0.299
0.0734 0.0746 0.0730 0.0730 0.0766 0.0732 0.0767 0.0788
0.0304 0.0300 0.0280 0.0359 0.0303 0.0311 0.0319 0.315-Z 0.567-Z 0.825-Z 0.903-Z 1.95 -2 0.333-Z 0.828-Z
0.172-Z 0.302-Z 0.419-Z 0.459-Z 1.01 -2 0.544-Z
where N is a normalization constant. Apart from the exceptions mentioned below the function yf(r) was taken equal to vf( 1, r), where the functions ~(2, r) are hydrogenic eigenstates with charge 2. In the case of final states 2P and 3P the variationally determined wave functions given by Morse et al. 5) and Goldberg et al. 6) were used, viz.
(5)
with c = 5Alj.4, A = l/(1 + Z/2/4,
N& = ,~7/(25As -
25A + 15/2), 2 = 0.97
EXCITATION
OF HELIUM
BY
PROTONS
AND
ELECTRONS
249
and ,u = 0.325. Where eqs. (4) and (5) have been employed the results are labelled by MYH ans GC, resp. In the case of 2S, 3S, 4s hydrogenic wave functions have been employed, but in order to insure orthogonality to the ground state involved, the first node of the function has been shifted purposely; these results are labelled by BE (Van den Bos-Eckart). All other results are labelled by E (Eckart). Three initial (ground) state wave functions have been used, viz. I.
pt =
m4(f%
71) m&,
a =
r2),
27/16.
(6)
Results employing this ground state are denoted by H (Hylleraas exp. 1). 11. yg =
#o(r1) 90@2)
with
Me =
J$ (e-ay
+ q e-or).
The parameters are chosen such that $0 closely fits the Hartree-Fock wave function’). Two sets of parameters have been presented. The first by Lowdins) has (II= 1.455799, p = 2or,7 = 0.60 and Ni, = 1.48423; results employing this set will be denoted by HF(L). The second set has been given by Byron and Joachains) and has a = 1.41, /l = 2.61,~ = 0.799 and Nr, = = 1.302525; results with this set will be labelled HF(B J). III. !P$= 5
bls(l/t~l) 9%3(&r2)
+
m3(&r1)
w3(y,
r2)l.
This is a variationally determined wave function by Eckartia). The parameters are givenil) by y = 2.1832 and 6 = 1.1886. Results with this wave function will be denoted by VE. 4. Calculations
and reszclts
All expressions for generalized oscillator strengths and cross sections could be evaluated analytically. The complete expressions have been publised elsewhere 2%29). In th’is section we present some numerical and graphical results for generalized oscillator strengths and cross sections, together with results of other workers. 4.1. Generalized oscillator strengths. For small momentum transfer the generalized oscillator strength fn(K) can be expanded in powers of K2 (see ref. 2)) as follows : b(K)
= /: + /AK2 + f:K4
+ . . .,
(9)
250
J. VAN
DEN
B’S
TABLE IIIa Generalized
oscillator
strengths
as a function
for 4rP where
of Kz using as the ground HF(L)
f(K)lK2
Ks 21s
f(K) 4%
31s
state HF(B J), except
is employed.
fWlK2
2iP
31P
41P
f (W/K4 4rD
3iD
4iF
0
0.61 l-l
0.266-l
0.123-l
0.269
0.745-l
0.320-l
0.907-2
0.469-2
0.127-3
0.05
0.565-l
0.249-l
0.115-l
0.248
0.700-l
0.303-I
0.797-2
0.416-2
0.106-3
0.10
0.523-l
0.233-l
0.108-l
0.230
0.658-l
0.287-l
0.702-2
0.369-2
0.887-4
0.20
0.449-l
0.204-I
0.950-2
0.197
0.581-l
0.257-l
0.548-2
0.293-2
0.30
0.388-l
0.179-l
0.836-2
0.170
0.514-l
0.230-l
0.432-2
0.233-2
0.630-4 0.452-4 0.328-4
0.40
0,336-l
0.157-l
0.737-2
0.147
0.455-I
0.206-l
0.342-2
0.187-2
0.50
0.292-l
0.139-l
0.651-2
0.127
0.403-l
0,185-l
0.273-2
0.15 l-2
0.241-4
0.60
0,255-l
0,122-l
0.575-2
0.111
0.357-l
0.166-l
0.219-2
0.122-2
0.178-4
0.70
0.223-l
0.108-l
0.510-2
0.970-l
0.318-l
0.149-l
0.177-2
0.997-3
0.133-4
0.80
0.196-l
0.959-2
0.453-2
0.851-l
0.283-l
0.134-l
0.144-2
0.817-3
0.100-4
0.90
0.173-l
0.852-2
0.403-2
0.749-l
0.252-l
0.121-l
0.118-2
0.672-3
0.762-5
1.00
0.153-l
0.758-2
0.359-2
0.661-l
0,225-l
0.109-l
0.970-3
0.556-3
0.584-5
1.20
0.120-t
0.604-2
0.286-2
0.519-l
0.181-l
0.884-2
0.665-3
0.385-3
0.350-5
1.50
0.854-2
0.436-2
0.207-2
0.368-l
0.131-l
0.656-2
0.390-3
0.229-3
0.169-5
2.00
0.505-2
0.262-2
0.125-2
0.218-l
0.800-2
0.410-2
0.174-3
0.103-3
0.567-6
2.40
0.343-2
0.180-2
0.858-3
0.148-l
0.553-2
0.287-2
0.963-4
0.578-4
0.256-6
2.80
0.239-2
0.126-2
0.603-3
0.391-2
0.206-2
0.558-4
0.337-4
0.123-6
3.20
0.171-2
0.903-3
0.432-3
0.103-l 0.737-2
0.282-2
0.124-2
0.659-3
0.315-3
0.537-2
0.207-2
0.336-4 0.209-4
0.204-4 0.127-4
0.625-7
3.60
0.150-2 0. I1 l-2
0.332-7
4.00
0.920-3
0.489-3
0.234-3
0.398-2
0.154-2
0.832-3
0.134-4
0.819-5
0.183-7
TABLE IIIb Generalized
oscillator
strengths
as a function
formulation,
f(K) present
of Kz for different
II + velocity
groups;
for 2iP
Kim and Inokutise)
work
I + length
formulation
Bell et al. 19)
Ks
BJ-MYH
0.0
0.2691
0.2902
0.2754
0.2756
0.2882
0.2985
0.25
0.1828 0.6608-l 0.3982-2
0.1983 0.7270-l 0.4502-l
0.1859 0.6542-l 0.3448-2
0.1859
0.1973 0.7162-l 0.3991-2
0.2035 0.7412-l 0.4634-2
1.0 4.0
I
BJ-E
I
I
f(K)/Kz present Ks
II
0.6540-l 0.3431-2
for 3iD Bell et aZ.19)
work
BJ-E
I
I
I
0
0.9067-2
0.860-2
0.01 0.25
0.8835-2 0.4861-2
0.8399-2 0.4764-2
1.0 4.0
0.9697-3 0.1340-4
0.9980-3 0.1355-4
II
0.9147-2 0.5064-2 0.1004-2 0.1330-4
II
EXCITATION
OF HELIUM
BY PROTONS
AND ELECTRONS
f,(K)
f,(K)
K2
K*
I
0.06
I
1 ‘s-z’s
251
1
‘s-3
‘s
0.015
1
1 ‘s -4
‘s
0.030
0.06
\
\ \
\ .
1
1
0.5
-
1
I
0.5
1.0
1.0
K’
Fig. 1. Generalized oscillator strengths for is-1s transitions in He. this work Kim and Inokutiss) K,I ---a -- Lassettre et al. so) experimental , f,(K)
, ‘s-2 ‘p
O.‘O it
-it
I
0
I 0
, f,(K)
f,(K)
I
I
a5
1.0
1 ‘s-3
‘p -
ao4
1 ‘s -4
il
BJ-GC
I
I
Fbl
‘P
L-E
I
I
I
0.5
1.0
I
2
a5
I
1.0
-K
Fig. 2. Generalized oscillator strengths for iS-iP transitions in He. ___ this work. A Lassettre et aZ.30) experimental. v Heideman and Vriensss) experimental. Theoretical results of Kim and Inokutiss) almost coincide with our B J-MYH curve for 2iP and B J-GC curve for 3iP. Experimental data of Vriens et al. 3s) for 2rP coincide with theoretical data of ref. 26.
J. VAN DEN BOS fno 0.020
K2
a010
-I
I \
BJ-E Y
\ \
I I
-i-
\
1.0
a5 ----K
0
2
Fig. 3, Generalized oscillator strengths for %-1D and 1S-lF transitions in He.
where j”, is the usual optical oscillator strength (the superscript is often omitted). For optically non-allowed transitions /l = 0, for the limit K = 0. In table IIa we present our values for the first non-vanishing expansion coefficients in the limit of K = 0. In table IIb we present values of other groups. The 1abeIs I and II denote length and velocity formulation, resp.; where no label occurs the length formulation is meant. In table IIIa numerical values for the generalized oscillator strength as a function of Ks have been given only for the ground state HF(B J) (except in the 4P case where HF(L) has been given). This choice was made because as can be seen from table II this is in general the ground state which produces results very well comparable with the results using accurate but complicated ground states. This choice is supported by even more arguments when we cross sections compare cross sections (see below). The nS excitation are, however, an exception to this, but have been given with BJ for oscillator reasons of conformity. In figs. 1-3 we present generalized strengths and compare with experiment and some other theoretical results. In figs. 2 and 3 results of Kim and lnokuti26), Lassettre and Jonesao) and Bell et al. 19) have been omitted, because they almost coincide with our curves. An illustration of the good agreement of our calculations with the more extended ones of Kim and Inokuti26) (with 53 parameters Hylleraas wave functions) and of Bell et ~2.19) (with a six parameter ground state Hylleraas wave function) is given in table IIIb for 2iP and 3iD. These groups used both the length and velocity formulation of the matrix element. Kim and Inokuti26) obtained agreement better than 1% between both calculations.
EXCITATION
OF HELIUM
BY
PROTONS
AND
ELECTRONS
253
TABLE IVa Excitation except
cross sections
in units zag for e on He using HF(B J) ground
in the case of 4rP HF(L),
and as final state wave
functions
state wave functions,
BE for niS, MYH
for 21P
GC for 3iP and E for the remainder 21s
E(eV)
31s
4%
21P
4iP
3iP
4iD
3lD
4iF
25
0.299-l
0.9341
0.170-l
0.3203
0.527-2
0.132
0.303-l
0.585-2 0.121-l
0.747-3
0.328-l
0.893-2 0.123-l
0.323-2
30
0.126-2
0.638-3
0.350-5 0.659-5
35
0.319-l
0.127-l
0.555-2
0.152
0.366-l
0,149-l
0.144-2
0.748-3
0.735-5
40
0.302-l
0.122-l
0.543-2
0.162
0.150-2
0.786-3
0.739-5
0.264-l
0.110-l
0.490-2
0.169
0.400-l 0,425-l
0.163-l
50
0.174-l
0.147-2
0.775-3
0.682-5
70
0.206-l 0.152-I
0.868-2
0.391-2
0.164
0.417-l
0.171-l
0.127-2
0.674-3
0.544-5
100
0.648-2
0.293-2
0.147
0.376-l
0.154-l
0.101-2
0.535-3
0.402-5
150
0.106-l
0.452-2
0.205-2
0.123
0.315-l
0.128-l
0.735-3
0.390-3
0.276-5
0.305-3 0.212-3
0.140-5
200
0.807-2
0.347-2
0.158-2
0.105
0.271-l
0.110-l
0.574-3
300
0.548-2
0.236-2
0.107-2
0,828-l
0.213-l
0.862-2
0.399-3
500
0.333-2 0.224-2
0.144-2
0.6553
0.592-l
0.152-l
0.1313
0.845-6
0.440-3
0.445-t
0.114-l
0.615-2 0.461-2
0.247-3
0.966-3
0.167-3
0.888-l
0.564-6
750
0.209-5
1000
0.168-2
0.728-3
0.331-3
0.361-l
0.925-2
0.373-2
0.126-3
0.671-4
0.423-6
5000
0.339-3
0.147-3
0.669-4
0.102-l
0.262-2
0.105-2
0.257-l
0.137-4
0.847-7
10000
0.170-3
0.735-4
0.335-4
0.577-2
0.148-2
0.593-3
0.12994
0.684-5
0.423-7
TABLE IVb Excitation
cross sections I --f length
in units nai for e on He for different
formulation,
II + velocity
o(2rP) present
work
E (eV)
BJ-MYH
50
I
in 7rui Kim and Inokutise)
BJ-E
groups:
formulation
I
Bell et al. is)
I or11
I
II
0.169
0.185
0.169
0.182
0.187
100 500
0.147 0.592-l
0.160 0.642-l
0.147
0.158
0.164
0.605-l
0.635-l
0.656-l
1000
0.361-l
0.391-l
0.368-l
0.386-l
0.400-l
5000
0.102-I
0.11 l-1
0.105-l
0.110-l
0.113-l
(I (3iD) _A E (eV)
present
in rcui
work
BJ-MYH
I
Bell et al. is)
-
I
II
50
0.147-2
0.149-2
0.152-2
100 500
0.101-2 0.247-3
0.101-2 0.246-3
0.105-l 0.2563
1000
0.126-3
0.125-3
0.1303
5000
0.257-4
0.254-l
0.266-l
254
J. VAN
DEN
BOS
TABLE V Excitation except
cross sections
in units ,a;
in the case of 4lP HF(L)
for p on He using HF(B J) ground
and as final state wave
functions
state wave functions,
BE for nrS, MYH
for 2rP
and E for the remainder E(keV)
2%
3’S 1
21P
41s
31P
41P
3rD
4rD
4rF 0.98 l-6
5
0.422-
0.128-l
0.511-2
0.365-l
0.785-2
0.322-2
0.249-3
0.12 1-3
10
0.664-l
0.233-l
0.976-2
0.107
0.252-l
0.106-l
0.102-2
0.5153
0.527-5
15
0.701-l
0.261-l
0.112-l
0.157
0.386-l
0.162-l
0.165-2
0.850-3
0.912-5 0.113-4
20
0.674-l
0.260-l
0.113-l
0.189
0.472-l
0.198-i
0.203-2
0.106-2
30
0.584-l
0.234-l
0.103-l
0.220
0.558-l
0.233-l
0.231-2
0.122-2
0.125-4
50
0,438-l
0.181-l
0.809-2
0.229
0.587-l
0.244-l
0.221-2
0.117-2
0.109-4
70
0.345-l
0.145-l
0.650-2
0.219
0.564-l
0.233-l
0.194-2
0.103-2
0.902-5
100
0.261-l
0.110-1
0.497-2
0.200
0.515-l
0.212-l
0.160-2
0.849-3
0.692-5
150
0.184-t
0.786-2
0.356-2
0.171
0.442-t
0.181-l
0.121-2
0.644-3
0.489-5
200
0.142-l
0.610-2
0.276-2
0.150
0.387-l
0.158-l
0.970-3
0.5153
0.376-5
250
0.116-l
0.498-2
0.226-2
0.134
0.345-l
0.140-l
0.8073
0.4293
0.304-5
300
0.979-2
0.421-2
0.121
0.312-l
0.127-l
0.6913
0.367-3
0.255-5
0.893-l
0.229-l
0.931-t
0.437-3
0.232-3
0.171-5
0.562-l
0.144-l
0.583-l
0.227-2
0.121-3
0.776-6 0.388-6
500
0.602-2
0.259-2
0.191-2 0.118-2
1000
0.307-2
0.175-2
0.603-3
2000
0.155-2
0.6693
0.305-3
0.340-l
0.871-2
0.351-l
0.116-3
0.617-4
5000
0.622-3
0.269-3
0.123-3
0.167-l
0.428-2
0.172-2
0.470-4
0.250-4
0.155-6
10000 0.312-3
0.135-3
0.6 14-4
0.955-2
0.244-2
0.982-3
0.236-4
0.125-4
0.777-7
TABLE VI Sub-level
excitation
functions
cross sections
in units xat for p on He using HF(B J) ground
and as final state wave 21Pr
E(keV)
2rPs
5
0.280-l
0.438-2
0.134-3
10
0.760-l
0.155-l
0.4583
3rDe
functions
MYH
state wave
for 2rP and E for the remainder
3lDa
4rFc
4rFr
4rFa
0.520-4
0.645-5
0.3806
0.235-6
0.640-7
0.660-8
0.242-3
0.388-4
0.159-5
0.130-5
0.48 l-6
0.645-7
3rDl
4rFs
15
0.106
0.258-l
0.641-3
0.423-3
0.820-4
0.227-5
0.222-5
0.104-5
0.170-6
20
0.122
0.337-l
0.695-3
0.545-3
0.123-3
0.243-5
0.265-5
0.151-5
0.289-6
30
0.132
0.439-l
0.645-3
0.650-3
0.157-3
0.223-5
0.261-5
0.203-5
0.495-6
50
0.124
0.520-l
0.459-3
0.635-3
0.2413
0.171-5
0.177-5
0.213-5
0.725-6
70
0.111 0.934-l
0.540-l 0.530-l
0.334-3
0.550-3 0.431-3
0.257-3 0.253-3
0.139-5
100
0.108-5
0.115-5 0.665-6
0.187-5 0.145-5
0.805-6 0.810-6
150 200
0.731-l 0,599-l
0.492-l 0.452-l
0.301-3 0.222-3
0.229-3
0.761-6
0.355-6
0.970-6
0.740-6
0.119-3 0.995-4 0.867-4
0.204-3 0.183-3
0.568-6 0.441-6
0.247-6 0.200-6
0.690-6 0.515-6
0.655-6 0.559-6
0.137-3
0.165-3 0.119-3
0.352-6 0.175-6
0.231-3 0.154-3
250
0.507-l
0.416-l
300 500
0.440-l
0.386-I
1000
0.288-l 0.155-l
0.303-l 0.203-i
0.376-4
0.700-4 0.254-4
2000 5000
0.812-2 0.335-2
0,129-l 0.670-2
0.221-4 0.102-4
0.850-5 0.186-5
10000
0.170-2
0.393-2
0.541-5
0.560-6
0.608-4
0.172-3
0.645-4 0.385-4
0.591-7
0.166-4
0.330-8 0.877-9
0.860-5
0.179-7
0.175-6
0.399-6
0.525-6
0.134-6 0.915-7
0.183-6 0.570-6
0.369-6 0.210-6
0.565-7 0.261-7
0.163-7 0.289-8
0.113-6 0.471-7
0.138-7
0.750-9
0.239-7
EXCITATION
OF HELIUM
4.2. Cross sections.
BY PROTONS
AND ELECTRONS
255
From the expression eq. (2) and a closely related one
employing eq. (1) we derived analytical expressions for the cross sections to all nlm states with n = 2, 3, 4. In the limit of infinite energy one derives, as expected,‘a, N E-1, crp N E-1 log E and CT~N E-l. For the m sub-states more complicated relationships were found, viz. op,,, gdo> odZ> uf,,
of,
-
E-l,
(10)
E-2,
(11)
E-1 log E,
(12)
bd, N E-2 log E.
(13)
no,
af,
UP1N
N
One also finds that in the high energy limit cd, = #ode and uf, = $uf, and that excitation to sub-states with quantum numbers I, m vanishes for l-m odd compared with even Z---m sub-states (see also ref. 20, eq. (6.15)). Results for excitation cross sections due to proton and electron impact have been given in tables IV, V and VI for a range of energies, and in figs. 4-8 together with some theoretical results of other authors. For comparison with experiment see for instance refs. 37, 29 and 41. In table 1Vb we illusstrate the good agreement of our calculations with more extended ones of TABLE VII Polarization fractions of transitions induced by collisions of protons on helium. Calculated using HF(BJ) ground state E (keV) threshold 5 10 15 20 30 50 70 100 150 200 250 300 500 750
1000 2000 5000 10000 00
17 (2iP-1%) 1.00 0.73 0.66 0.61 0.57 0.50 0.41 0.35 0.28 0.20 0.14 0.10 0.07 -0.03 -0.09 -0.13 -0.23 - 0.33 - 0.40 - 1.00
II (3’D-2iP) 0.60 0.44 0.40 0.36 0.33 0.27 0.19 0.14 0.07 0.00 - 0.05 -0.09 -0.12 -0.19 -0.24 -0.27 -0.33 -0.38 -0.40 -0.43
J. VAN DEN BOS
256 aEe1(10 -
20 keV cmz/atom
--102aEe1 4xaiR
I -5 _.-._.-4
-3
LAttshuler I -2
-1
o-
I
,.11.,
0.05
0.1
.
* ,‘..‘I
0.2
,
Q5
I 1 0
1
2
3
45
-
E,,
in keV
Fig. 4. Cross sections for 2% excitation by electrons (u&l versus In &I). R is the Rydberg energy.
dE,,(10~20keV cm*/atom)
lo3 d E, 4lT&R
JBJ-BE LL -BE
7 -8
C-Kim, Inokuti
_,_._._.-J._.-.---.-.-
I
H-BE
2-
0,
a05
0.1
1,
3’s
/dzohr
I...,
-6
o2
I . ‘7’1 0.5
1 0 1
__c
2 E,\
3
45
in keV
Fig. 5. Cross sections for 3% excitation by electrons (uEel vem4.s In Eel).
EXCITATION
OF HELIUM
BY PROTONS
AND ELECTRONS
cl
G
1.2
2’P
aa
0.4
I
0.1
i
I
0.2
,
I
,
1
a5
1
2
5
_
1
10
E,, in keV
Fig. 6. Cross sections for 21P excitation by electrons (uE,J4xai versus In Eel). A -+ Altshuler’s velocity formulation, using H-E. x + Altshuler’s length formulation using H-E. UEe,Ml-20keVcrn*/atom 1
lo36 &,
4nae2R
-1
I-
-4
-2
o-8 . . ..l a05 0.1
I 02
3
‘Is*.., cl5
1 -
I 2
at,-0 3 45 E,, in keV
Fig. 7. Cross sections for 31D excitation by electrons (u&l vewus In Eel).
257
J. VAN DEN BOS
258 oEel(eVxao2)
‘OI l4'F x10'
I 1
1
25
5
I
100
-
-*----
-
10000 E.(eV)
I
1
1000
Fig. 8. Comparison of cross sections for excitation by electrons and protons, using BJ-E wave function combinations (u&l versus In E,l). For protons E,. = (m/M) Ep.
Kim and Inokutis6) and of Bell et al. 19) for excitation of helium by electrons to 21P and 3rD. As mentioned before (see section 4.1) our B J-BE cross sections for 2% and 31s deviate much from those of Kim and Inokuti (see figs. 4 and 5), where the simple asymptotic Whittaker functions employed by Fox15) are seen to give much better results (see also the second article of ref. 19). Polarization fractions have been calculated (see table VII) for 1P-1s and ID-1P transitions in the case of proton impact, by using the method of reference 20 and the cross sections of table VI. In fig. 9 we illustrate that the ground state wave function has not a great influence on the polarization fraction in the case of ID--1P transitions. 5. Con&&on The different types of wave functions employed are seen to give very different results for the generalized oscillator strengths and the excitation cross sections, but reasonably agreeing results for the polarization fractions. Comparison with more extended and more accurate calculations for oscil-
EXCITATION
OF HELIUM
BY PROTONS
259
AND ELECTRONS
-il6
1 I
5
10
I
I
100
10.000
1000 -
E,+ in keV
Fig. 9. The polarization fraction n of 31D-21P radiation in the case of excitation by protons.
lator strengths as well as for cross sections shows, however, that for the description of the excitation of nrP and lziD states from the ground state the HF(BJ) wave function for the ground state is a very good substitute. The reason for the exceptional behaviour of the lziS cross sections might well be due to the excited state wave functions for rtS which are e.g. not mutually orthogonal. In this case the simple asymptotic Whittaker functions employed by Foxis) are seen to give much better results when compared with the results of Kim and Inokutisa) which are claimed to be correct within a few percent. In order to find theoretical results which are valid below the Born region one has to take into account the coupling between states other than initial and final. These kinds of calculations using the simple wave functions which can be accepted according to the results above are in progress.
260
J. VAN DEN BOS
Acknowledgments. We arevery grateful to Dr. F. J. de Heer for the cooperation in preparing this paper for publication. We also acknowledge the valuable discussions and remarks from Drs. M. Inokuti and L. Vriens and from Professors J. Kistemaker and C. Joachain. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research).
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