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The single and double ionization of helium atoms by fast nuclei and multicharged ions
239
Nuclear Instruments and Methods in Physics Research B36 (1989) 239-248 North-Holland, Amsterdam
THE SINGLE AND DOUBLE IONIZATION AND MULTICHARGED IONS V.S. NIKOLAEV
OF HELIUM ATOMS BY FAST NUCLEI
and V.A. SIDOROVICH
of Nuclear Physics, Moscow State University, Moscow II9899, USSR
Institute
Received 18 July 1988 and in revised form 25 October 1988
The cross sections for the single and double ionization of helium by nuclei with charges Z = l-100 at energies E = 0.05-4.5 MeV/amu have been calculated in the independent electron approximation with the use of the unitarized one-electron transition probabilities. It is shown that at E 2 0.2 MeV/amu the single ionization cross sections IJ&, calculated in the Born version of the decay model agree, within lo-15%, with the available experimental cross sections for nuclei and multicharged hydrogen- and helium-like ions with charges Z I 8, and the calculated double ionization cross sections IJ& at Z/u > 0.2 are 1.5-2 times greater than the experimental ui2, but their dependences on charge Z and ion velocity u are close to the experimental ones. It is found that the ratios of the calculated and experimental cross sections oil and ui2 to the total Born ionization cross sections ub at Z/u 2 1.5 depend, with the accuracy of 15’%, only on the parameter Z/U and at any Z-value these ratios should be described by the functions of Z/O’.~~ with + 15% accuracy. Taking into correct account the nondipole transitions, we have shown that for ions with Z >- 30 at \/22 5 u 5 Z/3 the single ionization cross sections uil should be a factor of 1.5 smaller than those for ions with Z = 4-8 at the same E/Z. It is found that at Z > 3u 2 10 the ratio of double to single ionization cross section R’ = CT’~/CT~’ much less sensitive to Z and u then in the region of Z = (0.3-1)~.
1. Introduction
per amu. The same conclusion lations of these cross sections
In connection mental
with
technique,
interests
in the last
more extensive
the development
the
practical
decade
there
and more exact,
on the single and double helium
ions with
the ionization lished
as point
energy
E > 200 keV,
whereas
the
with the Born v < 22
than
the
Janev’s
calculations as well
cross [lo]
sections
of which
rates
as the double
at a proton rate
[7-121,
0168-583X/89/$03.50 Physics
v 2 3 au, cross
and
coincide
coincide
smaller [7] with
of an approxi-
equations
[ll].
These
approxima-
cross
where
cross
E/Z
however,
sections
[5,7]. The
charged
ions
MeV/amu
in this method correct
and
Z
sections
ci2 cross
calculated
by
method
1.2-5
the experimental
Janev’s
than
theoretical
helium sistently
[3,7]. include
multipole
charges
However, only dipole
factor
hardly
which
when
these
for ions
for
modification
[16], are free of this limitation
and the latter
nation
electron
Publishers
Division)
B.V.
independent
of dipole
and its unitarized
of
the
high
due to distant
of
with
and this
sufficiently
prevalence
version
and
by the multi-
to 1.5 [ll]
is chiefly
of con-
monopole
with
approximation
the Born
of only
E = 0.13-2.31
sections
-
Thus,
calculations
transitions;
an overwhelming
The Born
the
cross sections
at energies
is set equal
the ionization
with
transitions.
in
a factor
ones.
are accounted
be justified
[6]
also
of
for multi-
[lo] are in good agreement
Z = 4-8
transitions
plicative can
are
are
which,
by a factor
sections
single ionization
by ions with
MeV/amu
[ll]
o”-values
with the experimental
Z = l-8 [5,15],
Andersen
Janev-Presnyakov higher
at
formulae
the cross
uil
[3,7].
of the single ioni-
v and Bohr
at the
the cross
are underrated
dependences
upon
overestimate
1.2-2.3
113,141. However,
> 0.3
as functions
E is the ion energy
Science
zation
(CTMC)
given by the approximate
collisions
divided by ion charge
ionization
Publishing
and
at Z = 4-8
in the dipole
0 Elsevier
sec-
v [8]. For nuclei
ui2 divided by Z, are represented E/Z
the ones
nuclei
by means
the en-values
act in
cross sec-
with
uil are markedly
sections
of
It is estab-
ionization
and lithium
obtained
were performed
(North-Holland
calculated
cross
coincide
of the close-coupling
the parameters
sections
The qualitatively
charges.
single
by helium
tion and, therefore, helium
new,
[l-9]
data
no more than two electrons,
Born
mate solution
energies
appeared
i.e. at the collision
the experimental
results
method
ion
approximation
ones at higher
ions containing and
Born
experimental
of helium
Carlo
reduced
scientific
single ionization
by protons
in the first
Monte
and
2 from 1 to 8 and by
Z 5 40, many
processes
of helium
calculated
Z
ionization
that the experimental oil
tions
have
of the experi-
experimental
by fast nuclei with charges
multicharged
tions
needs
is drawn from the calcuby the classical trajectory
of the decay
model
in combi-
approximation
240
V.S. Nikolaev,
V.A. Sidorovich / Single and double ionization of helium
was used rather successfully to calculate the single- and double-ionization cross sections of helium by fast ions with Z = 1-3 [16,19]. In the present paper we have calculated the singleand double-ionization cross sections of helium by nuclei with charges Z from 1 to 100 at energies of 0.05 to 4.5 MeV/amu using the independent electron approximation and the Born version of the decay model. In the above-noted approximation, double ionization results from the independent interaction of each atomic electron with the fast ion. This double ionization mechanism is dominant at an ion energy E from 0.1 to 2Z2 MeV/amu [16]. To elucidate the role of dipole transitions, the cross sections eil and oi2 have also been calculated in the dipole version of the decay model. Just as in the Janev-Presnyakov method, in the dipole version a”/Z and (J i2/Z depend only on E/Z. Our results are compared with experiment and with the results of other theoretical calculations. [17,18]
2. Basic formulae In the independent electron approximation [17,18] the total ionization cross section ei = eil + 2 IJ~~, the single and double ionization cross sections of helium by fast nuclei and ions uil and ui2 are given by the expression ua = 2a~;Z,-~
/
P”(b)b
db,
(1)
at P’(b) Pil(
P’*(b)
X
/
%(k
le-iqrlls)./p(bq,)
- 2Pi2(
b),
(2)
where s is the ion velocity; k is the electron momentum in the continuum, q is the momentum transfer, qL and + are its orthogonal component and the azimuthal angle with respect to the vector s; (k 1eeiq” 11s) is the Born matrix element of the ionization transition 1s + k; .$ is the uth order Bessel function. The quantities k, q and s are expressed in Coulomb units with respect to the nuclear charge Z, of the ionized particle. For example, s = v/Z,, where 0 is the ion velocity in atomic units. Some physical quantites are exposed shortly when using s and some when using 0 and, therefore, in what follows we use both s and 0 to denote velocity. From the numerical calculations of the probabilities wn( b) for hydrogen like systems by the formula (4) it follows that at not too large impact parameters (b 5 7) wn( b) can be approximated by the functions wn(b)=B,Nexp[-
(In N)‘+(b/bo)‘]
(5)
at B = ~~(0); N = 1.5-2.5;, b, - 1. At velocities s = 0.75, 1 and 2-10, B = 0.5 (Z/U)~, (Z/U)~ and 2 X(Z/U)2. Using the formulae (l)-(3) and (5), we get the Born total ionization cross sections and the Born double ionization cross sections of helium: In N),
0;” = (Ta&‘2)(b,/Z,)2B2(1
(6)
+ 2 In N),
and in the Born version of the decay model (BVDM) we have:
= [w(b)]‘,
where b is the impact parameter in Coulomb units; Pi (b) is the total ionization probability, P”(b) and P i2( b ) are the single and double ionization probabilities at a given impact parameter; Z, is the effective nuclear charge of the helium atom for the ejected electron; a, is Bohr’s radius; w(b) is the single electron ionization transition probability. In the ionization of atoms by multicharged ions the Born ionization probabilities tug(b) can exceed unity (the maximum possible value), and, therefore, as w(b) we used the unitarized Born probabilities wn,,( b) which correspond to the decay model [16,17] or, to be more exact, to the Born version of the decay model (BVDM): W’a,(b)=l-exp[-w,(b)].
eeipo 2,
(4)
0; = 4aa~(bo/Z,)2B(1+
= 2w(b), b) = P’(b)
In the 1s ionization of a hydrogen-like particle by nucleus with charge Z, wg(b) takes the form [20]
(3)
i OBD
= u;.P;,(
IS& = ok29’gD(
B, N),
(7)
B, N),
where
IE=O
P&,:D(B,N) =8
‘+I 5 (-l)kBk(2 k=O
- l)[l
+(k
(k+2)!(k+2)*(1+2ln
+ 2) In N]
N) ’
(8) The quantities Sg, and .YFD depend only on the two parameters, B and N, the N-dependence being much
V.S. Nikolaev,
V.A. Sidorovich / Single and double ionization
weaker. In this connection at collision rates s = 2-10, and N = 2, the BVDM-to-Born when B=2 (Z/u)’ ionization cross section ratios for ions with Z < (3-5)~ depend chiefly on Z/v and are approximated, with an accuracy of 10-2096, by the binomials gg,
= (1+ o.7z*/v*)-“.5,
y”& t: (1t
(9)
0.75z*/v*)-i?
At collision rates s > 2 and impact parameters b 2 7 the Born ionization probabilities differ by not more than 25% from the probabilities wr,( b) calculated in the dipole approximation that is obtained from the Born approximation by replacement of the exponent exp(- iqr) in the ionization transition matrix element with 1 - iqr. (See the appendix for the coincidence of asymptotic values of w a( b) and w u( b).) From eq. (4) in the dipole approximation we get w,(b)
= (2s/3)( Z/(ZG*))~T(~/~),
(10)
at exp i (1
+
1-
,p)3
- ; arctgk
exp(-2+)
where K, and K, are McDonald’s functions. Hence, at GC 1 and b/s B 1 we have, respectively,
b/s
1.1422 = m
w,(b)
4.9122
and w,(b)
= -e Z,2s2b2
a
_-b,s
.
(104 At any b/s the probabilities wu( b) are approximated within an accuracy of lo-15% by the formulae at b/s
wD(b)=(ZZ,/v2)*(s/b)*, w,(b)
= 4.9( ZZ,/v’)‘(
b2/s2
+ b/s)-’
at b/s>
e-b’s,
1.
(IOb) In the dipole version of the decay mode (DVDM) the ionization probability, in accord with eq. (3), is = 1-
exp(-w,(b)).
(12) Using eqs. (l), (2), and (lo)-(12), we get the total ionization cross sections and the double ionization cross sections w,,,,(b)
&,
= 2ZZL3Fi(
E/ZZ,)
a&
= ZZL3Fi2(
E,‘ZZ,).
F”(e)
= (2sa;c/25)/Da(r,
D’(r,
y) = 1 - exp[ -5.34
Di2(e,
u) = [D’(e,
(13)
y)
v)]‘,
dyy,
X 104e-2Qy)],
(14)
of helium
241
where the ion energy E is given in keV/amu. Thus, in the DVDM the reduced cross sections I$,,, = ub,Z,‘/Z Z and Gi2 or, = u&,Z,‘/Z depend only on the reduced energy e = E/ZZ,. The BVDM is a more correct approximation than the Born approximation (PWBA) because it accounts for the decrease in the initial state amplitude [16]. However, like the PWBA and semiclassical approximation (SCA), the BVDM does not include the binding energy variations and the electron wave function distortion in collisions and, also, the influence of the charge transfer upon ionization. Estimations of the influence of these factors on the ionization cross sections show that this influence is essential at v*/ZZ, 5 1. Therefore, it can be expected that the BVDM is applicable at v > 222 whereas the PWBA is valid, as it follows ~~ from ref. [7,8,12,15], at v > 2 Z.
3. Discussion of the results of the calculations In the Born and dipole versions of the decay model we have calculated the total ionization cross sections ei and the single and double ionization cross sections of helium CIi1 and cri2 by atomic nuclei with Z = l-100 and energies E from 0.05 to 4.5 MeV/amu. The ejected electrons were described by the Coulomb wave functions in the nuclear field with the effective charge Z, determined as in refs. [l&21], from the electron binding energy (in the calculations of e i and ai Z, = 1.345 and 1.69, respectively). Fig. 1 shows typical results of these calculations for E = 0.13, 0.64 and 2.31 MeV/amu and experimental ionization cross sections available for these energies of helium by nuclei with Z = l-8 and by high-charge hydrogen and helium-like ions, which act as nuclei with charge equal to the ion charge. (The latter will .be denoted by Z, as the charge of bare nuclei.) Since the Born values of 0; are proportional to Z*, all the cross sections shown in fig. 1 are divided by Z*. 3.1. Single
ionization
cross sections
Comparison between the results of calculation and experiment shows that at collision rates v 2 3 (E 2 200 keV/amu) the c &values obtained in the BVDM coincide, as a rule within lo-15%, with the available experimental cross sections for nuclei with Z = 1-X and also for hydrogen- and helium-like ions with Z = 2-7 and 4-6, respectively [7,8]. The dipole version of the decay model at v 2 3 gives the cross sections u&, which are smaller than the experimental and the BVDM cross sections by 5-20% for protons and by 25-35% for ions with Z = 3-10. For ions with Z 2 50 the cross sections e&, and e&, calculated in both the versions are practically coincident (fig. 1).
242
V.S. Nikolaeu, V.A. Sidorovich / Single and double ionization of helium
-----v.-.------..
10
y-+-Y “\
I
yy*
*“’
3v/
P
./L.-- -.>
+x0
/?
,+,.i2 ’ 0.1
x
-I*
\
’
.
6
x7 + 8
/ . 0.1 I
I
6
,
2
4
a
IO
1
1
20
40
a
0.1
I_
a0
I
2
I
I
4
810
I
I
20
40
b
I
I.
a0
I
2
a
4
z
2
IO
I
I
20
40
C, 80
2
Fig. 1. Single and double ionization cross sections of helium divided by Z2 are plotted versus ion charge Z at energies E = 130 (a), 640 (b) and 2310 (c) keV/amu. The curves represent our theoretical results obtained in the BVDM (l), the DVDM (2) and in the Born approximation (3) and also the results of Janev [IO] (4) and Bohr [5] (5). The symbols show the experimental data for nuclei (0,0), hydrogen-like ions (X) and helium-like ions (+): o from ref. 181, 0, X, + - from ref. [7]. The symbols in a circle @, @,
@correspond
to the one- and two-electron
Consideration of the cross sections calculated in the BVDM shows that in agreement with the approximate formulae (6)-(9) the u,, and o&values at all L: for
r-
loss cross sections
0” and (Y”.
ions with Z < (;/2 differ from an by not more than 10% and 20% respectively, and for ions with Z 2 u/2 the ratios B’ HI, = &,/o;, and Qi,, = ot,,/ok decrease
1
3
.;$i +. .\ 10-I P
@
>
$ \L =g
Io-2
5. ” a
co” 10-3 0.1
I
1.0
IO
Z/V
0.2
,
&II 1.0
b]
IO
z/v *25
Fig. 2. Reduced single and double ionization cross sections of helium by nuclei &, = ~&/Z20~(Hf) and &,“, = ‘~&/z*oi(H+) are plotted versus Z/o (a) and Z/u ’ 25 (b). Curves 1-4 correspond to nuclear energies E = 130,640,2310 and 4500 keV/amu, respectively. Curves (5) correspond to the formulae (15), (18) and curves (6), to (16), (19). The straight line (7) represent the Born approximation. The experimental data of 0” and Q12 from ref. [7,8] for nuclei with energies E = 130, 190, 1440 and 2310 keV/amu are marked, respectively, by A, v, n , +, and l .
monotonically with increasing ion charge 2. Since the Born values are proportional to Z2 and the difference between &I+) and &II’) for protons at v > 3 is not larger than _3%, .9$and Q&,-values are practically coincident with P;,, = u~,,/Z20~(HC) and Q& respectively, i.e. with the ratio of = u~n,/Z+&H+), uk, (or e&) to the Born single ionization cross section of helium by protons increased by a factor of Z2. and the experimental Qit = The @&-values u~~/Z~~~~~(H~)are plotted in fig. 2 as a function of Z/u. Calculations show that at 2 ( 3u for ail u > 2 the ratio @n depends mainly on Z/v and is described with an accuracy of K-20% by the binomial Q;n = (1 “l-l.5zz/0~)-0~5.
(151
This conclusion is supported by the available experimental data at Z/u < 2 [7,8] (fig. 2). Bohr’s formulae [l&5] obtained from the semiquantitative analysis of ionization with use of the Rutherford scattering cross section, give qualitatively the same results for Qi’. (In what follows, when we speak about Bohr’s formula, we mean the more exact formula which includes the finite value of the ma~mum energy transferred, i.e. formula (8) from ref. IS]). At K = 2Z/u < I, according to Bohr f155, perturbation theory is applicable and the ionization cross sections should be close to the ones calculated in a first Born approximation and at K > 1, when classical mechanics can be used, they should be less than the Born values. Though the approaches used in ref. [LS] and in the present paper are quite different, the ratios Qil = u”/Z2cri*(Hf) in these two cases are rather close and at Z = v 2 3 they are the same and coincide with the experimental data. For ions with Zk 20 the @i-values calcnfated by Bohr’s formufae are somewhat closer to unity than our Q&values. The oil-values obtained by Bohr’s formula are not less than a factor of 1.5-2 larger than the experimental and the BVDM values because at L < I this formula does not give correct (Born) ionization cross sections 0: (fig. 1). At s= 1.5-10 and Z/u >, 5 the Born ionization probabilities ~a( b) are less than unity only at relatively large impact parameters b 14 and, therefore, the collisions with b SC4 and b > 7, when fan and won(b) are close, account for not less than 50% of the cross sections &o and (I&,. In this connection relation (13) following from the DVDM is fulfilled rather well for the cross sections ~8, and the ratios 9$, = e&,/~~ and Q&, = a$,,/G~ depend critically not only on Z/u but on s as well. At given Z/u 2 3 these ratios are minimum at the reduced rates s = 1.6-2.0 and they increase with increasing and decreasing s as well (fig. 2). The increase of 9&, and Q&, values at s < 1.5 is caused by the general decrease of the Born reduced probabilities C, = we/( Z/V)~ at decreasing s (as a
result, the relative difference between the probabilities wrn-, and wg decreases). And at s > 2 increased P&, and Q&, values are due to the fact that the b-dependence of ws gets weaker with increasing s at large impact parameters b 2 7, where wn and ~a,-, are close. For ions with Z; 5 v at s = 1.5-10 .!P&, and Q&, depend chiefly on the parameter Z/v’.‘. In this case the Q-&-values are described with an accuracy of X0-15% by the function && = [ 1 + 1 .lS( Z/P)““]
-I--!
Pa)
and in the region of Z/U’.~ = l-10 they are about proportional to u’.~/Z. For ions with any Z from 1 to 100 at s = 2-10 the ratio Q& described with the accuracy of j;ZS% by the functions of Z,/$-*’ (fig. 2): Q;,, = [ 1 + 0.65( Z/U~,~~)~‘~]-I?
(16)
The experimental Qil data, which at u = 2-10 are now known for ions with Z I 8 in the region of Z/v cr:2 f7,8), coincide with those given by formulae (15) and (16) within 10% and 20%, respectively. Deviation of some experimental points from their averaged values, depending on Z/v and Z/U’-~~, is not larger than 5% and ‘lo%, respectively (fig. 2). But if the experimental pit data are considered as a function of E/Z, as is done in ref. [22], these deviations are larger, being of the order of 30%. The coincidence of the cross sections a&, and a&, for ions with Z > Sv at s = 2-10 shows that o&,/Z in these cases depends on E/Z, i.e.
4Adz=f~~fE/Z)+
(17)
at fon(E/Z)=2Z,-~~F’~E/ZZ,)-~F’~(E/ZZ,)~. where F’ and Fi2 are given by eqs. (14). For ions with Z= (OS-lS)u, for which at vz 3 a& = I.4 e&., the values of &/Z also satisfy, with an accuracy of - lO%, the scaling relation such as (17) but with the function far,{ E/Z) = l.4fr,,( E/Z). These conclusions are illustrated in fig. 3 where the calculated u&,/Z and a&,/Z are plotted versus E/Z and v2JZ. In the same figure are also given the experimental data rr”jZ and those obtained by Janev [lo] on the basis of an approximate solution of the close-coupling equations for three states with the direct inclusion of the dipole part of the interaction of a fast ion and the ejected electron and with the rem~ning part of this interaction being accounted for by the multiplicative factor equal to 1.5. It is seen from fig. 3 that at EjZ = 50-1000 keV/amu, which corresponds to u2/Z = 2-45, the a’“JZ-values calculated by Janev IlO], coincide with an accuracy of 10-U% with the BVDM-values for ions with Z = (0.5-1.5)~ and are 1.5 times as large as the values obtained in the DVDM. The latter means that if
244
VS. Nikolaev, V.A. Sidorovich / Single and double ionization of helium
,‘;,L IW
.
‘\
IWO
E/z,keV/mu Fig. 3. o”‘/Z and ui2/2 versus E/Z. The lines represent the theoretical calculation: BVDM (.and -.---), DVDM (- - -), Janev [lo] (-. . -), CTMC [13,14] (- - -); the numbers near the curves representingthe BVDM results indicate the ion charge 2. The symbols show the experimental data from ref. [7,8] for B5+ (O), C6+ (a), O*+ (9 ) nuclei and B4+ O), C5+ (X),
C4+ (+) ions. The symbols in a circle-@, 60 , &3,83 represent the electron loss cross sections 0” and (r12.
dipole transitions only are taken into account, the close-coupling method used in ref. [lo] gives the same oil values in the region of v2/Z = 2-40 as does the DVDM. The results of the present paper also show that the contribution of the direct nondipole tr~sitions to the ionization cross sections uii (and ui) can be regarded as roughly equal (maximum) only at Z = (0.5-1.5)u, whereas at Z > 3v( because of the strong predominance of collisions with large impact parameters) these transitions may be neglected. Therefore, for ions with Z 1 30 in the region of v2/Z = 2-Z/9 (i.e. at u = m-Z/S) the experimental ei’ data should be a factor of 1.5 less than these reported in ref. [lo] (fig. 3). This conclusion is in line with the results of the calculations of the general oscillator strengths of the ionization transitions according to which the contribution 6 of the nondipole transitions to the total oscillator strength at momentum transfers q < 5 and 4 is not larger than 4% and 12%, respectively [23,24]. The collisions with q - l/b ) 1, at which 6 are large (2 50%), for ions with Z > 3u at s z 5, as it follows from eqs. (3) and (5) provide a much smaller contribution to the formation of the cross sections &,, because of unitarization, than in the case of Z(V. In the region of v2/Z 2 0.5, eil calculated in ref. [lo] decrease rapidly with ion energy and are much less than the ones calculated in the present paper. At v2/Z
2 0.5 the single charge transfer cross sections ecl [8,18,25,26] are larger or comparable with the ionization cross sections uil and, therefore, the latter are markedly less than the electron loss cross sections by helium atoms cr’i = ui* + I+‘, whereas at u*/Z >> 1 a” and uil are coincident. The experimental data of eil, which are available at v2/Z from 1 to 2 for ions with Z 2 5 [7,8], coincide with the accuracy of 20% with the values obtained in ref. [lo], and the ~rresponding experimental data of err coincide with c& calculated in the present paper (figs. 1 and 3). The analogous coincidence of the Born cross sections u: and the experimental uli was reported in ref. [21] for the ionization of hydrogen and helium by single- and doubly-charged hydrogen, helium and lithium ions at D - 1.5-2. Therefore, we can suppose that, at least when ucl 5 a”, the electron bound states in the fast ion are formed chiefly from the continuum states of helium atoms which corresponds to statements of the strong potential Born approximation (SPB) [27] and of the continuum intermediate state appro~mation (CIS) [ZS]. This conception was used by Tan and Lee [29J in their binary encounter approximation calculations of the charge transfer and ionization cross sections in collisions of protons with atoms and, also, in the CTMC calculations of the same cross sections [13,14].
In accord with the formulae (6)-(9) at Z < 0.450 the double ionization cross sections IJ&, calculated in the BVDM are practically proportional to Z4 and differ from the Born ones by not more than 20%. Within Z 2 u 2 1 the ~t~zation has an effect and the ratio g& = cr~n/e~ decreases monotonically with increasing Z (figs. 1 and 2). At all u > 4 for Z < 2v the ratios @n and &$, = a&/Z2u,‘(H+) depend, with an accuracy of lo%, only on Z/v and at Z < 5v they are described within 15% by the approximate functions (9) and (18) @& = 0.178( z/*)2(1
+ 0.7522/@)
-=.
W
At Z > 5v these ratios depend chiefly on the parameter Z/V~.~‘. In this case for any Z > v’.*~ we have with an accuracy of lo-15% (fig. 2): &,
= 0.09[ 1 + 0.24( Z/&2’)“5]
-‘.
(19)
In accord with (l), (2) and (6)-(9) and also (15), (Isa), (18) and (19) the ratio of the double to single ionization cross section of helium Rk, = u&,/u&, (see fig. 4) at Z < v depends only on the parameter Z/v, and at Z 2 4u, on v. For Z < 0,/Z and Z > 4u we have, with an accuracy of 10% R& = 0.178( Z/v)”
and
lpi,, = 0.480-i/~,
(20)
respectively. And for all Z from 1 to 100 the R$,-val-
KS. Nikolaev, V.A. Sidorovich / Single and double ionization of helium
245
10-I
10-2
10-3
1
I
I
0.1
0.06
I
I 0.4
0.2
0.6
1.0
4.0
2.0
6.0
z/v Fig. 4. Ratios of the double to single ionization cross section R’ = oi2/ui* versus Z/v. Curves 1-5 represent the BVDM results for E = 130,190,640,2310 and 4500 keV/amu, respectively; 6 - the Born R’-values. The symbols (A),(v),(0) and ( X , + , 0) show the
experimental data: at Z/v 5 2 from refs. [7,8] for nuclei with E = 130,190,640 and > 1000 keV/amu, respectively; ions with E = 1400 keV/amu from ref. [9]. Curve 7 - the results obtained by the formula (21).
ues
are
described
with
an
accuracy
of
10%
by
the
binomial: RiD=
+ 8.7( Z/u)‘.‘]
The presently ions with 1.7-2.0
available
smaller
BVDM
dependence within
only at Z/V Qi2
caused
of
the
data
depend
the
Z/v
cross
within
values
a major
the
first
corresponds
term
ln(13.12@))
was
Z/v < 0.2
mechanism
contribution [30,18].
The
does
suggested
in ref.
[18,19,30]
since
ionization ionization
on
caused
Z/v
by shalce-
to ( Z/V)~
double
that
independent
ejection
of each of the atomic
and
is due electrons
and
for R’
as a function
charge
shake-up
from
are half as
This
two-fold
initial-state
1.4 times.
on the double
ionization
to
ui2
of the electron
up to the
cross
also
correlation
that at Z>_ 2r: the Z-dependence
nuclei
lations
was, apparently, with
as
compared
with
and
3
first established at
the
region
in the ex-
E = 200-300
energies
o’* of the singly-
ion production
ionization
in collisions
of
inde-
argon and krypton
in the recent
of R’
Z-dependence
by the experimental
R;1,, is
of R’ with increasing
of neon,
of the helium
weaker
u”
(20) and (20a)
CTMC
for ions with E = 2 and 5 MeV/amu
our calculation
firmed
Z/I
[33] and, besides,
by
on u’* at L’- 5. of the ratios
“saturation”
on the ionization
by data
a strong
and at Z/v > 4 they are practically
of Z. This
keV/amu
a:,,
of helium
suggest
It is seen from fig. 4 and the formulae
pendent
maximum
experimental
sections
[31,32]
at
of the
the cross section
The recent
and antiprotons
influence
by
de-
electrons
contribution
Z, = 1.60
not more than
[35]. The
of the
of the ejected the
to
with a fast ion [18]. Earlier,
values.
value Z, = 2 decreases
ion charge
depend
R’
Z/v > 0.4 is not larger than 15% and the increase
much weaker
experimental
(21)
not
term is proportional
expression
due to correlation
which
to the dou-
determined
i2 and R’-data compared in the BVDM seems to be mainly
with those calculated
periments
to the
the analogous
is
R’-values,
of Z/v = 0.25-0.9
of the cxpertmenta:‘:
protons
of Z/v 2 1 are de-
corresponds
of their interaction
crease
possible
Qi2
experimental
in the region
as the calculated
effective
in the BVDM
at
large
Z/v,
ratios
averaged
from (21),
2). However,
Z/u)‘,
to the double
up, and the second
a results
at Z/v
* 15% by the function
R’ = 2.2 x 1O-3 + 0.07( where
1 and 2).
of the experimen-
of the shake-up
of R’ = oi2/cri1 in the region
scribed
(fig.
The deviation
section
but with
only on
experimental
calculated
by the neglect
ionization
(figs.
15% with that calculated
at Z/v ,( 0.1 provides ble
Q&,
of
in the BVDM
v are in agreement
with
= 0.2-0.8.
from
[7,8] are a factor
data of Qi2 = oi2/Z2ai1(H+)
monotonically
Z/v
data on ui2 for
dependences
and the calculated
coincides tal
experimental
on Z and
The experimental
the
(2Oa)
than those calculated
the corresponding
increasing
The
-O.‘.
Z I 8 at Z/v = 0.2-0.8
dependences
5 1.5,
Z2/(E
171.
[2.5~‘.~l~
their
parameter
at Z/u L 2 for
cross at
is con-
on the cross
and doubly-charged of helium
atoms
sections
Z/v > 1, as
Z/c = 0.3-0.8,
data
calcu-
[34] and in
sections helium
with chlo-
246
V.S. Nikolaev,
V.A. Sidorovich / Single and double ionization of helium
0
i4
0 ;2
:i 2
4
6
a10
20
4
6 8 IO
20
a
6
8 IO
I
2
4
b ax0
?
2 WV Fig. 5. u’~/Z’ and o”/Z* versus ion charge 2 at E = 60 (a), 100 (b), 960 (c) keV/amu and the cross section ratios R’ = ,i2/~i1 versus Z/u. The Iines represent the BVDM calculations. The symbols show the experimental data of ,‘I, u12 and R’ for AuZf ions at E = 60 (a) and 100 (b) keV/amu from ref. [2] and for Cl’+ ions at E = 960 keV/amu (c) from ref. [l].
rine ions with Z = 6--13 at E = 0.96 Mev/amu (u = 6.2) R’-data obtained recently for ions with Z = 6-44 at E = 1.4 MeV/amu [9]. The experimental 0” and a12-data from ref. [l] are larger than the calculated u$, and u& by 20%, and the experimental data of R’ = (~‘*/a’~ coincide within 10% with the calculated values Rho = o&/o&, and are proportional to Z”-” in the region of Z/u = 1-2 (fig. 5). The experimental RI-data from ref. [9] for Z = 15-44 in the region of Z/u = 2-6 differ by 10-25s from the ones calculated in BVDM and are proportional to Z’.‘. The experimental RI-data at Z/U > 2 and u - 2 can also be obtained from the cross sections u” and a’* [2] for gold ions with Z = 2-20 at E = 100 and 60 keV/amu (u = 2 and 1.54). These values of R’ are given in fig. 5 together with the calculated RB,,. It is seen from fig. 5 that for ions with Z = 5-9 at u = 2 (Z/U = 2.5-4.5) the experimental H’ data coincide with the calculated Ro,,-values and are proportional to Z”-“. True, the parameter u*/Z is in these cases less than unity (c*/Z = 0.45-0.8) and the charge-transfer cross sections 0” are not small compared with the ionization cross sections ui’: as Z increases from 5 to 9, the ratios uc’/ui’ increase according to [8], from 0.7 to 1.6 and the ratios u’~/cI~~, from 1.2 to 5 ( uic are the simultaneous ionization and charge-transfer cross sections). In this connection the experimental cross sections ui’ and ui2 from ref. [2] are substantially smaller, and the experimental cross sections u” and u” are a factor of 1.2-1.4 larger a,,-values calculated in the BVDM than the u,‘, and u’* (fig. 5). Still greater influence of the charge-transfer causes the deviation of the experimental R’-values from the calculated ones (fig. 5) for the AuZt ions with Z = lo-20 at u = 2 (u*/Z = 0.2-0.4) and with Z = 4-20 at u = 1.54 ( u2/Z = 0.12-0.6). The oE,,- values calculated in the DVDM at Z = 1 are a factor of - 5 larger and at Z = 8-30 a factor of [l] and by the experimental
I .3-1.7 smaller than the BVDM results and are a factor of 1.2 smaller than the BVDM even at Z = 65 (fig. 1). The latter is due to the fact that the collisions with large impact parameters, when the dipole approximation is valid, provide a relatively smaller contribution to CJ~,, than to ~g,,. In this connection the spread in o&/Z values for ions with different charges of Z 2 4 at the same E/Z is wider than the spread in u&,/Z (fig. 3). This is confirmed by the experimental data for ions with Z = 4-8 171 (fig. 3). For ions with Z = 4 and 5 at r:*/Z = l-2, for which to experimental oi2 and u12-data differ drastically, the u&,-values (likewise the above-considered situation with ug,,) are close to the experimental two-electron loss cross sections ,12. For ions with Z = l- 3 at II = (2-5)Z the experimental u’*-data are much closer to o:‘, than to u&, because the dipole approximation over estimates strongly the ionization probabilities at small impact parameters. For ions with Z = 8 at u*/Z 2 8 the experimental oi2-data are closer to u$,, than to u&, because the DVDM neglects not only the correlation cffccts (which leads to increased CJ&,- and a&-values), but also the multipole transitions which leads to decrease of u;:‘,.
4. Conclusion The present calculations of the ionization cross sections of helium by fast nuclei with different Z show that the combination of the Born version of the decay model (BVDM) and the independent electron approximation enables one to reproduce the presently available experimental and the best theoretical results on the single ionization cross sections ui’ of helium by nuclei and multicharged ions with charges Z < 0*/2 at collision rates I:/ 2 3 ( E 1 200 keV/amu) and to de-
V.S. Nikolaev,
247
V.A. Sidorovich / Single and double ionization of helium
scribe the behaviour of the double ionization cross sections ei2 in the region Z/v > 0.3. The use of the decay model decreases the lower bound on the rates, at which the Born ionization probabilities can be used, from v=2Z down to v= &?. It is shown that for ions with Z 5 3v at v 2 3 the ratios of the single ionization cross sections uil to the total Born ionization cross sections CJ~ (which practically coincide with the single ionization cross sections of helium by protons increased by Z2 times) depend mainly on Z/v. For ions with Z > 3v, which ionize the helium atoms chiefly in far collisions due to the dipole part of the interaction with atomic electrons, the ratios of the &, to Born CJ~ cross sections depend mainly on Z/vi.‘. For ions with any Z from 1 to 100 at v = 2-15 these ratios are described within &25% by the functions of Z/V~.~‘. For ions with Z = 3-15 at v = (0.7-2) Z, if v > m, the single ionization cross sections u&, calculated in the BVDM coincide within 15% with Janev’s results obtained by approximate solution of the close-coupling equations, and roughly satisfy the scaling law, according to which the quantities an/Z depend only on the parameter E/Z. For ions with Z > 30 at v = &%-Z/3 (i.e. at v2/Z = 2-Z/9) the uil values should be a factor of - 1.5 less than for ions with Z = 4-8 at the same E/Z because of the negligible contribution of the nondipole transitions into ui*. The double ionization of helium atoms by ions with Z 1 v/3 occurs mainly due to independent interaction of fast ions with each of the atomic electrons. In this connection the ratios R’ = CJ~‘/IJ~’ increase with Z/v and at Z/v 5 2 depend practically only on Z/v. In the region of Z/v = 3-1, in accord with the results obtained in the Born approximation, the experimental R’-values are proportional to ( Z/V)~ and in the region of Z/v >, 4 at v > 4, according to the BVDM calculations, they should be proportional to v-l/3. To systematize the present conclusions, it is convenient to divide a set of values of Z = l-100 and v 2 3 into three regions, though the boundaries of these regions can be somewhat different when considering different features of cross sections. In the region of Z/v -C 0.5 the Born approximation is valid (with allowance for correlation and shake-up in case of the double ionization). Within 0.5 < Z/v < 1.5-3 the cross sections are all markedly less than the Born ones; the single ionization cross sections eil for ions with Z/v = 0.5-1.5 roughly obey the scaling law with the universal Janev function of the parameter E/Z, and the relations between different cross sections (a”, ui2, a:, 0:) depend practically only on Z/v. In the region of Z/v > 3 the ratios of the single ionization cross sections uil to the Born cross sections depend on Z/VI-~ and at 1 c Z/L&* < 10 they are about proportional to r&‘/Z; the ratios R’ of the double to single ionization cross sections depend weakly on Z and v and are about proportional
to ve113; the single ionization cross sections uil for ions with Z 2 30 at v > &%! should be a factor of - 1.5 smaller than for ions with Z = 4-8 at the same E/Z.
Appendix
We shall demonstrate that at the limiting values of the impact parameters b and of the collision rates s the Born ionization probabilities (4) coincide with the dipole probabilities (10) and (11). Since the Bessel function J,( bq,) entering into (4) oscillates rapidly at large values of the argument, at b >> 1 the major contribution to the integral over d2q, comes from the range of qL 5 l/b -=c 1. At large s the values of q,, = (1 + k2)/2s are also small and, therefore, the ionization probability is determined by the value of the matrix element M, = (k le-“’
11s) at q = /m
-K 1.
In this case the
matrix elements Ma and M, obtained in the Born and dipole approximations are connected by the asymptotic relation:
(A.11 where ai = (k+i)-‘, a,=
-2(2k-i)/(k2+1).
Substituting (A.l) into (4) and integrating over d2q, and over the angular variables of the vector k, we get at s2 > 1 + k2 and b > 7: - iarctgk [;)‘(2”,3.1’)/
W B=
k dk (1
+
,Q2)3
exp
I-
exp(
--2v’k)
x [ Kt(bq,,)+ K:(bq,,)+ h]T
(A.2)
where
h=
${
q1:[3K,2(bql,) +fG+q,)]
+[ q,,Ko(bq,,) + ;Kdbq,,)]2). Hence, at b > 4s, i.e. when we can use the asymptotic relations K,(x)
=K,(x)
we get s, = WB/WD- 1
= (~/2x)l’~
exp(-x),
KS. ~ikolaeu, VA. ~~doro~ic~/ Singleanddoubleionizationof helium
248 where Setting
X2=l
IS . the mean value of kZ for 4 < 1 [36].
kz e&al to a, we get from (A.3)
s ,,,=~-~+1.3,‘bs+2/b~. At s = 3 and b 2 12, we then get SW= l&11% s 2 5 and b > 20 8, < 6%.
(A.41
and for
References
PI CL. Cocke, Phys. Rev. A20 (1979) 149. PI P. Hvelplund, H.K. Haugen and II. Kuudsen, Phys. Rev.
A22 (1980) 1930. Haugen, L.H. Andersen, P. HveIphmd and H. Knudsen, Phys. Rev. A26 (1982) 1950. [41 H.K. Haugen, L.H. Anderson, P. Hvelplund and H. Knudsen, Phys. Rev. A26 (1982) 1962. [51 H. Knudsen, in: Invited Papers 12th Int. Conf. on the Physics of Electronic and Atomic Collisions, Gatlinburg (1981) ed. S. Datz (North-Holland, Amsterdam, 1982) p. 657. VI L.H. Andersen, IEEE Trans. Nucl. Sci. NS-30 (1983) 973. [71 H. Knudsen, L.H. Andersen, P. Hvelphmd, G. Astner, H. Cederquist, H. Danared, L. Liljeby and K.-G. Rensfelt, J. Phys. B17 (1984) 3545. 181 M.B. Shah and H.B. Gilbody, J. Phys. 318 (1985) 899. I91 J.H. M&&ire, A. Muller, B. Schuch, W. Grohand and E. Salzbom, Phys. Rev. A35 (1987) 2479. WY R.K. Janev, Phys. Lett. A83 (1981) 5. WI R.K. Janev and L.P. Presnyakov, J. Phys. B13 (1980) 4233. WI L.P. Presnyakov and D.B. Uskov, Zh. Eksp. Teor. Fiz. 86 (1984) 882. 1131 R.E. Olson, Phys. Rev. Al8 (1978) 2464. P41 S.J. Pfeifer and R.E. Oslon, in: Proc. 12th Int. Conf. on the Physics of Electronic and Atomic Collisions (Gatlinburg, 1981) ed. S. Datz (North-Holland, Amsterdam, 1981) p. 736. ll5l N. Bohr, K. Dan. Vidensk. Selsk, Mat.-Fys. Medd. 18 (1948) no. 8.
131H.K.
V.A. Sidorovich and V.S. Nikolaev, .I. Phys. B16 (1983) 3243. I171 J.H. McGuire and L. Weaver, Phys. Rev. Al6 (1977) 41. 1181 V.A. Sidorotich, J. Phys. B14 (1981) 4805. 1191V.A. Sidorovich, V.S. Nikolaev and J.H. McGuire, Phys. Rev. A31 (1985) 2193. I201 K. Taulberg, .I. Phys. B10 (1977) 314. WI V.S. Nikolaev, V.S. Senashenko, V.A. Sidorovich and V.Yu. Shafer, Zb. Tekh. Fiz. 48 (1978) 1399. [221 G.H. Gillespie, Phys. Lett. A93 (1983) 327. 1231A.R. Holt, I’roc. Phys. Sot. B2 (1969) 120.5. 1241 I.M. Kruglova, V.S. Nikolaev and V.I. Shulga, J. Phys. B10 (1977) 2971. 1251 VS. Nikdaev, I.S. Dmitriev, L.N. Fateeva and Ya.A. Teplova, Zh. Eksp. Tear. Fix 40 (1961) 989. VI T.R. Dillingham, J.R. McDonald and P. Richard, Phys. Rev. A24 (1981) 1237. ~71 J. Macek and S. Alston, Phys. Rev. A26 (1982) 250. [=I D. Be&c, R. Gayet and A. Salin, Phys. Rep. 56 (1979) 279. ~91 C.K. Tan and A.R Lee, J. Phys. B14 (1981) 3445. [301 J.H. McGuire, Phys. Rev. Lett. 49 (1982) 1153. [311 J.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Moller, K. Elsner, K.-G. Rensfeh and E. Uggerhej, Phys. Rev. Lett. 57 (1986) 2147. [321 J.F. Reading and A.L. Ford, Phys. Rev. Lett. 58 (1987) 543. 1331Yu.Z. Levchenko, Thesis, Fiz. Tekh. Inst. Ukr.SSR Akad. of Sciences, Kbarkov (1974). 1341 M.L. McKenzie and R.E. Olson, Phys. Rev. A35 (1987) 2863. [351 V.A. Sidorovich and V.S. Nikolaev, in: Lecture Notes in Physics, High-Energy Ion-Atom Collisions, Proc. 3rd Workshop on High-Energy Ion-Atom Collisions, Debrecen, Hungary, (August 3-5,1987) eds. D. Berenyi and G. Hock (Springer-Verlag, Berlin-Heidelberg, 1988) vol. 294, p. 437. [361 I.S. Dmitriev, Ya.M. Zhileikin and V.S. Nikolaev, Zh. Eksp. Tear. Fiz. 49 (1965) 500.
WI
Nuclear Instruments and Methods in Physics Research B36 (1989) 239-248 North-Holland, Amsterdam
THE SINGLE AND DOUBLE IONIZATION AND MULTICHARGED IONS V.S. NIKOLAEV
OF HELIUM ATOMS BY FAST NUCLEI
and V.A. SIDOROVICH
of Nuclear Physics, Moscow State University, Moscow II9899, USSR
Institute
Received 18 July 1988 and in revised form 25 October 1988
The cross sections for the single and double ionization of helium by nuclei with charges Z = l-100 at energies E = 0.05-4.5 MeV/amu have been calculated in the independent electron approximation with the use of the unitarized one-electron transition probabilities. It is shown that at E 2 0.2 MeV/amu the single ionization cross sections IJ&, calculated in the Born version of the decay model agree, within lo-15%, with the available experimental cross sections for nuclei and multicharged hydrogen- and helium-like ions with charges Z I 8, and the calculated double ionization cross sections IJ& at Z/u > 0.2 are 1.5-2 times greater than the experimental ui2, but their dependences on charge Z and ion velocity u are close to the experimental ones. It is found that the ratios of the calculated and experimental cross sections oil and ui2 to the total Born ionization cross sections ub at Z/u 2 1.5 depend, with the accuracy of 15’%, only on the parameter Z/U and at any Z-value these ratios should be described by the functions of Z/O’.~~ with + 15% accuracy. Taking into correct account the nondipole transitions, we have shown that for ions with Z >- 30 at \/22 5 u 5 Z/3 the single ionization cross sections uil should be a factor of 1.5 smaller than those for ions with Z = 4-8 at the same E/Z. It is found that at Z > 3u 2 10 the ratio of double to single ionization cross section R’ = CT’~/CT~’ much less sensitive to Z and u then in the region of Z = (0.3-1)~.
1. Introduction
per amu. The same conclusion lations of these cross sections
In connection mental
with
technique,
interests
in the last
more extensive
the development
the
practical
decade
there
and more exact,
on the single and double helium
ions with
the ionization lished
as point
energy
E > 200 keV,
whereas
the
with the Born v < 22
than
the
Janev’s
calculations as well
cross [lo]
sections
of which
rates
as the double
at a proton rate
[7-121,
0168-583X/89/$03.50 Physics
v 2 3 au, cross
and
coincide
coincide
smaller [7] with
of an approxi-
equations
[ll].
These
approxima-
cross
where
cross
E/Z
however,
sections
[5,7]. The
charged
ions
MeV/amu
in this method correct
and
Z
sections
ci2 cross
calculated
by
method
1.2-5
the experimental
Janev’s
than
theoretical
helium sistently
[3,7]. include
multipole
charges
However, only dipole
factor
hardly
which
when
these
for ions
for
modification
[16], are free of this limitation
and the latter
nation
electron
Publishers
Division)
B.V.
independent
of dipole
and its unitarized
of
the
high
due to distant
of
with
and this
sufficiently
prevalence
version
and
by the multi-
to 1.5 [ll]
is chiefly
of con-
monopole
with
approximation
the Born
of only
E = 0.13-2.31
sections
-
Thus,
calculations
transitions;
an overwhelming
The Born
the
cross sections
at energies
is set equal
the ionization
with
transitions.
in
a factor
ones.
are accounted
be justified
[6]
also
of
for multi-
[lo] are in good agreement
Z = 4-8
transitions
plicative can
are
are
which,
by a factor
sections
single ionization
by ions with
MeV/amu
[ll]
o”-values
with the experimental
Z = l-8 [5,15],
Andersen
Janev-Presnyakov higher
at
formulae
the cross
uil
[3,7].
of the single ioni-
v and Bohr
at the
the cross
are underrated
dependences
upon
overestimate
1.2-2.3
113,141. However,
> 0.3
as functions
E is the ion energy
Science
zation
(CTMC)
given by the approximate
collisions
divided by ion charge
ionization
Publishing
and
at Z = 4-8
in the dipole
0 Elsevier
sec-
v [8]. For nuclei
ui2 divided by Z, are represented E/Z
the ones
nuclei
by means
the en-values
act in
cross sec-
with
uil are markedly
sections
of
It is estab-
ionization
and lithium
obtained
were performed
(North-Holland
calculated
cross
coincide
of the close-coupling
the parameters
sections
The qualitatively
charges.
single
by helium
tion and, therefore, helium
new,
[l-9]
data
no more than two electrons,
Born
mate solution
energies
appeared
i.e. at the collision
the experimental
results
method
ion
approximation
ones at higher
ions containing and
Born
experimental
of helium
Carlo
reduced
scientific
single ionization
by protons
in the first
Monte
and
2 from 1 to 8 and by
Z 5 40, many
processes
of helium
calculated
Z
ionization
that the experimental oil
tions
have
of the experi-
experimental
by fast nuclei with charges
multicharged
tions
needs
is drawn from the calcuby the classical trajectory
of the decay
model
in combi-
approximation
240
V.S. Nikolaev,
V.A. Sidorovich / Single and double ionization of helium
was used rather successfully to calculate the single- and double-ionization cross sections of helium by fast ions with Z = 1-3 [16,19]. In the present paper we have calculated the singleand double-ionization cross sections of helium by nuclei with charges Z from 1 to 100 at energies of 0.05 to 4.5 MeV/amu using the independent electron approximation and the Born version of the decay model. In the above-noted approximation, double ionization results from the independent interaction of each atomic electron with the fast ion. This double ionization mechanism is dominant at an ion energy E from 0.1 to 2Z2 MeV/amu [16]. To elucidate the role of dipole transitions, the cross sections eil and oi2 have also been calculated in the dipole version of the decay model. Just as in the Janev-Presnyakov method, in the dipole version a”/Z and (J i2/Z depend only on E/Z. Our results are compared with experiment and with the results of other theoretical calculations. [17,18]
2. Basic formulae In the independent electron approximation [17,18] the total ionization cross section ei = eil + 2 IJ~~, the single and double ionization cross sections of helium by fast nuclei and ions uil and ui2 are given by the expression ua = 2a~;Z,-~
/
P”(b)b
db,
(1)
at P’(b) Pil(
P’*(b)
X
/
%(k
le-iqrlls)./p(bq,)
- 2Pi2(
b),
(2)
where s is the ion velocity; k is the electron momentum in the continuum, q is the momentum transfer, qL and + are its orthogonal component and the azimuthal angle with respect to the vector s; (k 1eeiq” 11s) is the Born matrix element of the ionization transition 1s + k; .$ is the uth order Bessel function. The quantities k, q and s are expressed in Coulomb units with respect to the nuclear charge Z, of the ionized particle. For example, s = v/Z,, where 0 is the ion velocity in atomic units. Some physical quantites are exposed shortly when using s and some when using 0 and, therefore, in what follows we use both s and 0 to denote velocity. From the numerical calculations of the probabilities wn( b) for hydrogen like systems by the formula (4) it follows that at not too large impact parameters (b 5 7) wn( b) can be approximated by the functions wn(b)=B,Nexp[-
(In N)‘+(b/bo)‘]
(5)
at B = ~~(0); N = 1.5-2.5;, b, - 1. At velocities s = 0.75, 1 and 2-10, B = 0.5 (Z/U)~, (Z/U)~ and 2 X(Z/U)2. Using the formulae (l)-(3) and (5), we get the Born total ionization cross sections and the Born double ionization cross sections of helium: In N),
0;” = (Ta&‘2)(b,/Z,)2B2(1
(6)
+ 2 In N),
and in the Born version of the decay model (BVDM) we have:
= [w(b)]‘,
where b is the impact parameter in Coulomb units; Pi (b) is the total ionization probability, P”(b) and P i2( b ) are the single and double ionization probabilities at a given impact parameter; Z, is the effective nuclear charge of the helium atom for the ejected electron; a, is Bohr’s radius; w(b) is the single electron ionization transition probability. In the ionization of atoms by multicharged ions the Born ionization probabilities tug(b) can exceed unity (the maximum possible value), and, therefore, as w(b) we used the unitarized Born probabilities wn,,( b) which correspond to the decay model [16,17] or, to be more exact, to the Born version of the decay model (BVDM): W’a,(b)=l-exp[-w,(b)].
eeipo 2,
(4)
0; = 4aa~(bo/Z,)2B(1+
= 2w(b), b) = P’(b)
In the 1s ionization of a hydrogen-like particle by nucleus with charge Z, wg(b) takes the form [20]
(3)
i OBD
= u;.P;,(
IS& = ok29’gD(
B, N),
(7)
B, N),
where
IE=O
P&,:D(B,N) =8
‘+I 5 (-l)kBk(2 k=O
- l)[l
+(k
(k+2)!(k+2)*(1+2ln
+ 2) In N]
N) ’
(8) The quantities Sg, and .YFD depend only on the two parameters, B and N, the N-dependence being much
V.S. Nikolaev,
V.A. Sidorovich / Single and double ionization
weaker. In this connection at collision rates s = 2-10, and N = 2, the BVDM-to-Born when B=2 (Z/u)’ ionization cross section ratios for ions with Z < (3-5)~ depend chiefly on Z/v and are approximated, with an accuracy of 10-2096, by the binomials gg,
= (1+ o.7z*/v*)-“.5,
y”& t: (1t
(9)
0.75z*/v*)-i?
At collision rates s > 2 and impact parameters b 2 7 the Born ionization probabilities differ by not more than 25% from the probabilities wr,( b) calculated in the dipole approximation that is obtained from the Born approximation by replacement of the exponent exp(- iqr) in the ionization transition matrix element with 1 - iqr. (See the appendix for the coincidence of asymptotic values of w a( b) and w u( b).) From eq. (4) in the dipole approximation we get w,(b)
= (2s/3)( Z/(ZG*))~T(~/~),
(10)
at exp i (1
+
1-
,p)3
- ; arctgk
exp(-2+)
where K, and K, are McDonald’s functions. Hence, at GC 1 and b/s B 1 we have, respectively,
b/s
1.1422 = m
w,(b)
4.9122
and w,(b)
= -e Z,2s2b2
a
_-b,s
.
(104 At any b/s the probabilities wu( b) are approximated within an accuracy of lo-15% by the formulae at b/s
wD(b)=(ZZ,/v2)*(s/b)*, w,(b)
= 4.9( ZZ,/v’)‘(
b2/s2
+ b/s)-’
at b/s>
e-b’s,
1.
(IOb) In the dipole version of the decay mode (DVDM) the ionization probability, in accord with eq. (3), is = 1-
exp(-w,(b)).
(12) Using eqs. (l), (2), and (lo)-(12), we get the total ionization cross sections and the double ionization cross sections w,,,,(b)
&,
= 2ZZL3Fi(
E/ZZ,)
a&
= ZZL3Fi2(
E,‘ZZ,).
F”(e)
= (2sa;c/25)/Da(r,
D’(r,
y) = 1 - exp[ -5.34
Di2(e,
u) = [D’(e,
(13)
y)
v)]‘,
dyy,
X 104e-2Qy)],
(14)
of helium
241
where the ion energy E is given in keV/amu. Thus, in the DVDM the reduced cross sections I$,,, = ub,Z,‘/Z Z and Gi2 or, = u&,Z,‘/Z depend only on the reduced energy e = E/ZZ,. The BVDM is a more correct approximation than the Born approximation (PWBA) because it accounts for the decrease in the initial state amplitude [16]. However, like the PWBA and semiclassical approximation (SCA), the BVDM does not include the binding energy variations and the electron wave function distortion in collisions and, also, the influence of the charge transfer upon ionization. Estimations of the influence of these factors on the ionization cross sections show that this influence is essential at v*/ZZ, 5 1. Therefore, it can be expected that the BVDM is applicable at v > 222 whereas the PWBA is valid, as it follows ~~ from ref. [7,8,12,15], at v > 2 Z.
3. Discussion of the results of the calculations In the Born and dipole versions of the decay model we have calculated the total ionization cross sections ei and the single and double ionization cross sections of helium CIi1 and cri2 by atomic nuclei with Z = l-100 and energies E from 0.05 to 4.5 MeV/amu. The ejected electrons were described by the Coulomb wave functions in the nuclear field with the effective charge Z, determined as in refs. [l&21], from the electron binding energy (in the calculations of e i and ai Z, = 1.345 and 1.69, respectively). Fig. 1 shows typical results of these calculations for E = 0.13, 0.64 and 2.31 MeV/amu and experimental ionization cross sections available for these energies of helium by nuclei with Z = l-8 and by high-charge hydrogen and helium-like ions, which act as nuclei with charge equal to the ion charge. (The latter will .be denoted by Z, as the charge of bare nuclei.) Since the Born values of 0; are proportional to Z*, all the cross sections shown in fig. 1 are divided by Z*. 3.1. Single
ionization
cross sections
Comparison between the results of calculation and experiment shows that at collision rates v 2 3 (E 2 200 keV/amu) the c &values obtained in the BVDM coincide, as a rule within lo-15%, with the available experimental cross sections for nuclei with Z = 1-X and also for hydrogen- and helium-like ions with Z = 2-7 and 4-6, respectively [7,8]. The dipole version of the decay model at v 2 3 gives the cross sections u&, which are smaller than the experimental and the BVDM cross sections by 5-20% for protons and by 25-35% for ions with Z = 3-10. For ions with Z 2 50 the cross sections e&, and e&, calculated in both the versions are practically coincident (fig. 1).
242
V.S. Nikolaeu, V.A. Sidorovich / Single and double ionization of helium
-----v.-.------..
10
y-+-Y “\
I
yy*
*“’
3v/
P
./L.-- -.>
+x0
/?
,+,.i2 ’ 0.1
x
-I*
\
’
.
6
x7 + 8
/ . 0.1 I
I
6
,
2
4
a
IO
1
1
20
40
a
0.1
I_
a0
I
2
I
I
4
810
I
I
20
40
b
I
I.
a0
I
2
a
4
z
2
IO
I
I
20
40
C, 80
2
Fig. 1. Single and double ionization cross sections of helium divided by Z2 are plotted versus ion charge Z at energies E = 130 (a), 640 (b) and 2310 (c) keV/amu. The curves represent our theoretical results obtained in the BVDM (l), the DVDM (2) and in the Born approximation (3) and also the results of Janev [IO] (4) and Bohr [5] (5). The symbols show the experimental data for nuclei (0,0), hydrogen-like ions (X) and helium-like ions (+): o from ref. 181, 0, X, + - from ref. [7]. The symbols in a circle @, @,
@correspond
to the one- and two-electron
Consideration of the cross sections calculated in the BVDM shows that in agreement with the approximate formulae (6)-(9) the u,, and o&values at all L: for
r-
loss cross sections
0” and (Y”.
ions with Z < (;/2 differ from an by not more than 10% and 20% respectively, and for ions with Z 2 u/2 the ratios B’ HI, = &,/o;, and Qi,, = ot,,/ok decrease
1
3
.;$i +. .\ 10-I P
@
>
$ \L =g
Io-2
5. ” a
co” 10-3 0.1
I
1.0
IO
Z/V
0.2
,
&II 1.0
b]
IO
z/v *25
Fig. 2. Reduced single and double ionization cross sections of helium by nuclei &, = ~&/Z20~(Hf) and &,“, = ‘~&/z*oi(H+) are plotted versus Z/o (a) and Z/u ’ 25 (b). Curves 1-4 correspond to nuclear energies E = 130,640,2310 and 4500 keV/amu, respectively. Curves (5) correspond to the formulae (15), (18) and curves (6), to (16), (19). The straight line (7) represent the Born approximation. The experimental data of 0” and Q12 from ref. [7,8] for nuclei with energies E = 130, 190, 1440 and 2310 keV/amu are marked, respectively, by A, v, n , +, and l .
monotonically with increasing ion charge 2. Since the Born values are proportional to Z2 and the difference between &I+) and &II’) for protons at v > 3 is not larger than _3%, .9$and Q&,-values are practically coincident with P;,, = u~,,/Z20~(HC) and Q& respectively, i.e. with the ratio of = u~n,/Z+&H+), uk, (or e&) to the Born single ionization cross section of helium by protons increased by a factor of Z2. and the experimental Qit = The @&-values u~~/Z~~~~~(H~)are plotted in fig. 2 as a function of Z/u. Calculations show that at 2 ( 3u for ail u > 2 the ratio @n depends mainly on Z/v and is described with an accuracy of K-20% by the binomial Q;n = (1 “l-l.5zz/0~)-0~5.
(151
This conclusion is supported by the available experimental data at Z/u < 2 [7,8] (fig. 2). Bohr’s formulae [l&5] obtained from the semiquantitative analysis of ionization with use of the Rutherford scattering cross section, give qualitatively the same results for Qi’. (In what follows, when we speak about Bohr’s formula, we mean the more exact formula which includes the finite value of the ma~mum energy transferred, i.e. formula (8) from ref. IS]). At K = 2Z/u < I, according to Bohr f155, perturbation theory is applicable and the ionization cross sections should be close to the ones calculated in a first Born approximation and at K > 1, when classical mechanics can be used, they should be less than the Born values. Though the approaches used in ref. [LS] and in the present paper are quite different, the ratios Qil = u”/Z2cri*(Hf) in these two cases are rather close and at Z = v 2 3 they are the same and coincide with the experimental data. For ions with Zk 20 the @i-values calcnfated by Bohr’s formufae are somewhat closer to unity than our Q&values. The oil-values obtained by Bohr’s formula are not less than a factor of 1.5-2 larger than the experimental and the BVDM values because at L < I this formula does not give correct (Born) ionization cross sections 0: (fig. 1). At s= 1.5-10 and Z/u >, 5 the Born ionization probabilities ~a( b) are less than unity only at relatively large impact parameters b 14 and, therefore, the collisions with b SC4 and b > 7, when fan and won(b) are close, account for not less than 50% of the cross sections &o and (I&,. In this connection relation (13) following from the DVDM is fulfilled rather well for the cross sections ~8, and the ratios 9$, = e&,/~~ and Q&, = a$,,/G~ depend critically not only on Z/u but on s as well. At given Z/u 2 3 these ratios are minimum at the reduced rates s = 1.6-2.0 and they increase with increasing and decreasing s as well (fig. 2). The increase of 9&, and Q&, values at s < 1.5 is caused by the general decrease of the Born reduced probabilities C, = we/( Z/V)~ at decreasing s (as a
result, the relative difference between the probabilities wrn-, and wg decreases). And at s > 2 increased P&, and Q&, values are due to the fact that the b-dependence of ws gets weaker with increasing s at large impact parameters b 2 7, where wn and ~a,-, are close. For ions with Z; 5 v at s = 1.5-10 .!P&, and Q&, depend chiefly on the parameter Z/v’.‘. In this case the Q-&-values are described with an accuracy of X0-15% by the function && = [ 1 + 1 .lS( Z/P)““]
-I--!
Pa)
and in the region of Z/U’.~ = l-10 they are about proportional to u’.~/Z. For ions with any Z from 1 to 100 at s = 2-10 the ratio Q& described with the accuracy of j;ZS% by the functions of Z,/$-*’ (fig. 2): Q;,, = [ 1 + 0.65( Z/U~,~~)~‘~]-I?
(16)
The experimental Qil data, which at u = 2-10 are now known for ions with Z I 8 in the region of Z/v cr:2 f7,8), coincide with those given by formulae (15) and (16) within 10% and 20%, respectively. Deviation of some experimental points from their averaged values, depending on Z/v and Z/U’-~~, is not larger than 5% and ‘lo%, respectively (fig. 2). But if the experimental pit data are considered as a function of E/Z, as is done in ref. [22], these deviations are larger, being of the order of 30%. The coincidence of the cross sections a&, and a&, for ions with Z > Sv at s = 2-10 shows that o&,/Z in these cases depends on E/Z, i.e.
4Adz=f~~fE/Z)+
(17)
at fon(E/Z)=2Z,-~~F’~E/ZZ,)-~F’~(E/ZZ,)~. where F’ and Fi2 are given by eqs. (14). For ions with Z= (OS-lS)u, for which at vz 3 a& = I.4 e&., the values of &/Z also satisfy, with an accuracy of - lO%, the scaling relation such as (17) but with the function far,{ E/Z) = l.4fr,,( E/Z). These conclusions are illustrated in fig. 3 where the calculated u&,/Z and a&,/Z are plotted versus E/Z and v2JZ. In the same figure are also given the experimental data rr”jZ and those obtained by Janev [lo] on the basis of an approximate solution of the close-coupling equations for three states with the direct inclusion of the dipole part of the interaction of a fast ion and the ejected electron and with the rem~ning part of this interaction being accounted for by the multiplicative factor equal to 1.5. It is seen from fig. 3 that at EjZ = 50-1000 keV/amu, which corresponds to u2/Z = 2-45, the a’“JZ-values calculated by Janev IlO], coincide with an accuracy of 10-U% with the BVDM-values for ions with Z = (0.5-1.5)~ and are 1.5 times as large as the values obtained in the DVDM. The latter means that if
244
VS. Nikolaev, V.A. Sidorovich / Single and double ionization of helium
,‘;,L IW
.
‘\
IWO
E/z,keV/mu Fig. 3. o”‘/Z and ui2/2 versus E/Z. The lines represent the theoretical calculation: BVDM (.and -.---), DVDM (- - -), Janev [lo] (-. . -), CTMC [13,14] (- - -); the numbers near the curves representingthe BVDM results indicate the ion charge 2. The symbols show the experimental data from ref. [7,8] for B5+ (O), C6+ (a), O*+ (9 ) nuclei and B4+ O), C5+ (X),
C4+ (+) ions. The symbols in a circle-@, 60 , &3,83 represent the electron loss cross sections 0” and (r12.
dipole transitions only are taken into account, the close-coupling method used in ref. [lo] gives the same oil values in the region of v2/Z = 2-40 as does the DVDM. The results of the present paper also show that the contribution of the direct nondipole tr~sitions to the ionization cross sections uii (and ui) can be regarded as roughly equal (maximum) only at Z = (0.5-1.5)u, whereas at Z > 3v( because of the strong predominance of collisions with large impact parameters) these transitions may be neglected. Therefore, for ions with Z 1 30 in the region of v2/Z = 2-Z/9 (i.e. at u = m-Z/S) the experimental ei’ data should be a factor of 1.5 less than these reported in ref. [lo] (fig. 3). This conclusion is in line with the results of the calculations of the general oscillator strengths of the ionization transitions according to which the contribution 6 of the nondipole transitions to the total oscillator strength at momentum transfers q < 5 and 4 is not larger than 4% and 12%, respectively [23,24]. The collisions with q - l/b ) 1, at which 6 are large (2 50%), for ions with Z > 3u at s z 5, as it follows from eqs. (3) and (5) provide a much smaller contribution to the formation of the cross sections &,, because of unitarization, than in the case of Z(V. In the region of v2/Z 2 0.5, eil calculated in ref. [lo] decrease rapidly with ion energy and are much less than the ones calculated in the present paper. At v2/Z
2 0.5 the single charge transfer cross sections ecl [8,18,25,26] are larger or comparable with the ionization cross sections uil and, therefore, the latter are markedly less than the electron loss cross sections by helium atoms cr’i = ui* + I+‘, whereas at u*/Z >> 1 a” and uil are coincident. The experimental data of eil, which are available at v2/Z from 1 to 2 for ions with Z 2 5 [7,8], coincide with the accuracy of 20% with the values obtained in ref. [lo], and the ~rresponding experimental data of err coincide with c& calculated in the present paper (figs. 1 and 3). The analogous coincidence of the Born cross sections u: and the experimental uli was reported in ref. [21] for the ionization of hydrogen and helium by single- and doubly-charged hydrogen, helium and lithium ions at D - 1.5-2. Therefore, we can suppose that, at least when ucl 5 a”, the electron bound states in the fast ion are formed chiefly from the continuum states of helium atoms which corresponds to statements of the strong potential Born approximation (SPB) [27] and of the continuum intermediate state appro~mation (CIS) [ZS]. This conception was used by Tan and Lee [29J in their binary encounter approximation calculations of the charge transfer and ionization cross sections in collisions of protons with atoms and, also, in the CTMC calculations of the same cross sections [13,14].
In accord with the formulae (6)-(9) at Z < 0.450 the double ionization cross sections IJ&, calculated in the BVDM are practically proportional to Z4 and differ from the Born ones by not more than 20%. Within Z 2 u 2 1 the ~t~zation has an effect and the ratio g& = cr~n/e~ decreases monotonically with increasing Z (figs. 1 and 2). At all u > 4 for Z < 2v the ratios @n and &$, = a&/Z2u,‘(H+) depend, with an accuracy of lo%, only on Z/v and at Z < 5v they are described within 15% by the approximate functions (9) and (18) @& = 0.178( z/*)2(1
+ 0.7522/@)
-=.
W
At Z > 5v these ratios depend chiefly on the parameter Z/V~.~‘. In this case for any Z > v’.*~ we have with an accuracy of lo-15% (fig. 2): &,
= 0.09[ 1 + 0.24( Z/&2’)“5]
-‘.
(19)
In accord with (l), (2) and (6)-(9) and also (15), (Isa), (18) and (19) the ratio of the double to single ionization cross section of helium Rk, = u&,/u&, (see fig. 4) at Z < v depends only on the parameter Z/v, and at Z 2 4u, on v. For Z < 0,/Z and Z > 4u we have, with an accuracy of 10% R& = 0.178( Z/v)”
and
lpi,, = 0.480-i/~,
(20)
respectively. And for all Z from 1 to 100 the R$,-val-
KS. Nikolaev, V.A. Sidorovich / Single and double ionization of helium
245
10-I
10-2
10-3
1
I
I
0.1
0.06
I
I 0.4
0.2
0.6
1.0
4.0
2.0
6.0
z/v Fig. 4. Ratios of the double to single ionization cross section R’ = oi2/ui* versus Z/v. Curves 1-5 represent the BVDM results for E = 130,190,640,2310 and 4500 keV/amu, respectively; 6 - the Born R’-values. The symbols (A),(v),(0) and ( X , + , 0) show the
experimental data: at Z/v 5 2 from refs. [7,8] for nuclei with E = 130,190,640 and > 1000 keV/amu, respectively; ions with E = 1400 keV/amu from ref. [9]. Curve 7 - the results obtained by the formula (21).
ues
are
described
with
an
accuracy
of
10%
by
the
binomial: RiD=
+ 8.7( Z/u)‘.‘]
The presently ions with 1.7-2.0
available
smaller
BVDM
dependence within
only at Z/V Qi2
caused
of
the
data
depend
the
Z/v
cross
within
values
a major
the
first
corresponds
term
ln(13.12@))
was
Z/v < 0.2
mechanism
contribution [30,18].
The
does
suggested
in ref.
[18,19,30]
since
ionization ionization
on
caused
Z/v
by shalce-
to ( Z/V)~
double
that
independent
ejection
of each of the atomic
and
is due electrons
and
for R’
as a function
charge
shake-up
from
are half as
This
two-fold
initial-state
1.4 times.
on the double
ionization
to
ui2
of the electron
up to the
cross
also
correlation
that at Z>_ 2r: the Z-dependence
nuclei
lations
was, apparently, with
as
compared
with
and
3
first established at
the
region
in the ex-
E = 200-300
energies
o’* of the singly-
ion production
ionization
in collisions
of
inde-
argon and krypton
in the recent
of R’
Z-dependence
by the experimental
R;1,, is
of R’ with increasing
of neon,
of the helium
weaker
u”
(20) and (20a)
CTMC
for ions with E = 2 and 5 MeV/amu
our calculation
firmed
Z/I
[33] and, besides,
by
on u’* at L’- 5. of the ratios
“saturation”
on the ionization
by data
a strong
and at Z/v > 4 they are practically
of Z. This
keV/amu
a:,,
of helium
suggest
It is seen from fig. 4 and the formulae
pendent
maximum
experimental
sections
[31,32]
at
of the
the cross section
The recent
and antiprotons
influence
by
de-
electrons
contribution
Z, = 1.60
not more than
[35]. The
of the
of the ejected the
to
with a fast ion [18]. Earlier,
values.
value Z, = 2 decreases
ion charge
depend
R’
Z/v > 0.4 is not larger than 15% and the increase
much weaker
experimental
(21)
not
term is proportional
expression
due to correlation
which
to the dou-
determined
i2 and R’-data compared in the BVDM seems to be mainly
with those calculated
periments
to the
the analogous
is
R’-values,
of Z/v = 0.25-0.9
of the cxpertmenta:‘:
protons
of Z/v 2 1 are de-
corresponds
of their interaction
crease
possible
Qi2
experimental
in the region
as the calculated
effective
in the BVDM
at
large
Z/v,
ratios
averaged
from (21),
2). However,
Z/u)‘,
to the double
up, and the second
a results
at Z/v
* 15% by the function
R’ = 2.2 x 1O-3 + 0.07( where
1 and 2).
of the experimen-
of the shake-up
of R’ = oi2/cri1 in the region
scribed
(fig.
The deviation
section
but with
only on
experimental
calculated
by the neglect
ionization
(figs.
15% with that calculated
at Z/v ,( 0.1 provides ble
Q&,
of
in the BVDM
v are in agreement
with
= 0.2-0.8.
from
[7,8] are a factor
data of Qi2 = oi2/Z2ai1(H+)
monotonically
Z/v
data on ui2 for
dependences
and the calculated
coincides tal
experimental
on Z and
The experimental
the
(2Oa)
than those calculated
the corresponding
increasing
The
-O.‘.
Z I 8 at Z/v = 0.2-0.8
dependences
5 1.5,
Z2/(E
171.
[2.5~‘.~l~
their
parameter
at Z/u L 2 for
cross at
is con-
on the cross
and doubly-charged of helium
atoms
sections
Z/v > 1, as
Z/c = 0.3-0.8,
data
calcu-
[34] and in
sections helium
with chlo-
246
V.S. Nikolaev,
V.A. Sidorovich / Single and double ionization of helium
0
i4
0 ;2
:i 2
4
6
a10
20
4
6 8 IO
20
a
6
8 IO
I
2
4
b ax0
?
2 WV Fig. 5. u’~/Z’ and o”/Z* versus ion charge 2 at E = 60 (a), 100 (b), 960 (c) keV/amu and the cross section ratios R’ = ,i2/~i1 versus Z/u. The Iines represent the BVDM calculations. The symbols show the experimental data of ,‘I, u12 and R’ for AuZf ions at E = 60 (a) and 100 (b) keV/amu from ref. [2] and for Cl’+ ions at E = 960 keV/amu (c) from ref. [l].
rine ions with Z = 6--13 at E = 0.96 Mev/amu (u = 6.2) R’-data obtained recently for ions with Z = 6-44 at E = 1.4 MeV/amu [9]. The experimental 0” and a12-data from ref. [l] are larger than the calculated u$, and u& by 20%, and the experimental data of R’ = (~‘*/a’~ coincide within 10% with the calculated values Rho = o&/o&, and are proportional to Z”-” in the region of Z/u = 1-2 (fig. 5). The experimental RI-data from ref. [9] for Z = 15-44 in the region of Z/u = 2-6 differ by 10-25s from the ones calculated in BVDM and are proportional to Z’.‘. The experimental RI-data at Z/U > 2 and u - 2 can also be obtained from the cross sections u” and a’* [2] for gold ions with Z = 2-20 at E = 100 and 60 keV/amu (u = 2 and 1.54). These values of R’ are given in fig. 5 together with the calculated RB,,. It is seen from fig. 5 that for ions with Z = 5-9 at u = 2 (Z/U = 2.5-4.5) the experimental H’ data coincide with the calculated Ro,,-values and are proportional to Z”-“. True, the parameter u*/Z is in these cases less than unity (c*/Z = 0.45-0.8) and the charge-transfer cross sections 0” are not small compared with the ionization cross sections ui’: as Z increases from 5 to 9, the ratios uc’/ui’ increase according to [8], from 0.7 to 1.6 and the ratios u’~/cI~~, from 1.2 to 5 ( uic are the simultaneous ionization and charge-transfer cross sections). In this connection the experimental cross sections ui’ and ui2 from ref. [2] are substantially smaller, and the experimental cross sections u” and u” are a factor of 1.2-1.4 larger a,,-values calculated in the BVDM than the u,‘, and u’* (fig. 5). Still greater influence of the charge-transfer causes the deviation of the experimental R’-values from the calculated ones (fig. 5) for the AuZt ions with Z = lo-20 at u = 2 (u*/Z = 0.2-0.4) and with Z = 4-20 at u = 1.54 ( u2/Z = 0.12-0.6). The oE,,- values calculated in the DVDM at Z = 1 are a factor of - 5 larger and at Z = 8-30 a factor of [l] and by the experimental
I .3-1.7 smaller than the BVDM results and are a factor of 1.2 smaller than the BVDM even at Z = 65 (fig. 1). The latter is due to the fact that the collisions with large impact parameters, when the dipole approximation is valid, provide a relatively smaller contribution to CJ~,, than to ~g,,. In this connection the spread in o&/Z values for ions with different charges of Z 2 4 at the same E/Z is wider than the spread in u&,/Z (fig. 3). This is confirmed by the experimental data for ions with Z = 4-8 171 (fig. 3). For ions with Z = 4 and 5 at r:*/Z = l-2, for which to experimental oi2 and u12-data differ drastically, the u&,-values (likewise the above-considered situation with ug,,) are close to the experimental two-electron loss cross sections ,12. For ions with Z = l- 3 at II = (2-5)Z the experimental u’*-data are much closer to o:‘, than to u&, because the dipole approximation over estimates strongly the ionization probabilities at small impact parameters. For ions with Z = 8 at u*/Z 2 8 the experimental oi2-data are closer to u$,, than to u&, because the DVDM neglects not only the correlation cffccts (which leads to increased CJ&,- and a&-values), but also the multipole transitions which leads to decrease of u;:‘,.
4. Conclusion The present calculations of the ionization cross sections of helium by fast nuclei with different Z show that the combination of the Born version of the decay model (BVDM) and the independent electron approximation enables one to reproduce the presently available experimental and the best theoretical results on the single ionization cross sections ui’ of helium by nuclei and multicharged ions with charges Z < 0*/2 at collision rates I:/ 2 3 ( E 1 200 keV/amu) and to de-
V.S. Nikolaev,
247
V.A. Sidorovich / Single and double ionization of helium
scribe the behaviour of the double ionization cross sections ei2 in the region Z/v > 0.3. The use of the decay model decreases the lower bound on the rates, at which the Born ionization probabilities can be used, from v=2Z down to v= &?. It is shown that for ions with Z 5 3v at v 2 3 the ratios of the single ionization cross sections uil to the total Born ionization cross sections CJ~ (which practically coincide with the single ionization cross sections of helium by protons increased by Z2 times) depend mainly on Z/v. For ions with Z > 3v, which ionize the helium atoms chiefly in far collisions due to the dipole part of the interaction with atomic electrons, the ratios of the &, to Born CJ~ cross sections depend mainly on Z/vi.‘. For ions with any Z from 1 to 100 at v = 2-15 these ratios are described within &25% by the functions of Z/V~.~‘. For ions with Z = 3-15 at v = (0.7-2) Z, if v > m, the single ionization cross sections u&, calculated in the BVDM coincide within 15% with Janev’s results obtained by approximate solution of the close-coupling equations, and roughly satisfy the scaling law, according to which the quantities an/Z depend only on the parameter E/Z. For ions with Z > 30 at v = &%-Z/3 (i.e. at v2/Z = 2-Z/9) the uil values should be a factor of - 1.5 less than for ions with Z = 4-8 at the same E/Z because of the negligible contribution of the nondipole transitions into ui*. The double ionization of helium atoms by ions with Z 1 v/3 occurs mainly due to independent interaction of fast ions with each of the atomic electrons. In this connection the ratios R’ = CJ~‘/IJ~’ increase with Z/v and at Z/v 5 2 depend practically only on Z/v. In the region of Z/v = 3-1, in accord with the results obtained in the Born approximation, the experimental R’-values are proportional to ( Z/V)~ and in the region of Z/v >, 4 at v > 4, according to the BVDM calculations, they should be proportional to v-l/3. To systematize the present conclusions, it is convenient to divide a set of values of Z = l-100 and v 2 3 into three regions, though the boundaries of these regions can be somewhat different when considering different features of cross sections. In the region of Z/v -C 0.5 the Born approximation is valid (with allowance for correlation and shake-up in case of the double ionization). Within 0.5 < Z/v < 1.5-3 the cross sections are all markedly less than the Born ones; the single ionization cross sections eil for ions with Z/v = 0.5-1.5 roughly obey the scaling law with the universal Janev function of the parameter E/Z, and the relations between different cross sections (a”, ui2, a:, 0:) depend practically only on Z/v. In the region of Z/v > 3 the ratios of the single ionization cross sections uil to the Born cross sections depend on Z/VI-~ and at 1 c Z/L&* < 10 they are about proportional to r&‘/Z; the ratios R’ of the double to single ionization cross sections depend weakly on Z and v and are about proportional
to ve113; the single ionization cross sections uil for ions with Z 2 30 at v > &%! should be a factor of - 1.5 smaller than for ions with Z = 4-8 at the same E/Z.
Appendix
We shall demonstrate that at the limiting values of the impact parameters b and of the collision rates s the Born ionization probabilities (4) coincide with the dipole probabilities (10) and (11). Since the Bessel function J,( bq,) entering into (4) oscillates rapidly at large values of the argument, at b >> 1 the major contribution to the integral over d2q, comes from the range of qL 5 l/b -=c 1. At large s the values of q,, = (1 + k2)/2s are also small and, therefore, the ionization probability is determined by the value of the matrix element M, = (k le-“’
11s) at q = /m
-K 1.
In this case the
matrix elements Ma and M, obtained in the Born and dipole approximations are connected by the asymptotic relation:
(A.11 where ai = (k+i)-‘, a,=
-2(2k-i)/(k2+1).
Substituting (A.l) into (4) and integrating over d2q, and over the angular variables of the vector k, we get at s2 > 1 + k2 and b > 7: - iarctgk [;)‘(2”,3.1’)/
W B=
k dk (1
+
,Q2)3
exp
I-
exp(
--2v’k)
x [ Kt(bq,,)+ K:(bq,,)+ h]T
(A.2)
where
h=
${
q1:[3K,2(bql,) +fG+q,)]
+[ q,,Ko(bq,,) + ;Kdbq,,)]2). Hence, at b > 4s, i.e. when we can use the asymptotic relations K,(x)
=K,(x)
we get s, = WB/WD- 1
= (~/2x)l’~
exp(-x),
KS. ~ikolaeu, VA. ~~doro~ic~/ Singleanddoubleionizationof helium
248 where Setting
X2=l
IS . the mean value of kZ for 4 < 1 [36].
kz e&al to a, we get from (A.3)
s ,,,=~-~+1.3,‘bs+2/b~. At s = 3 and b 2 12, we then get SW= l&11% s 2 5 and b > 20 8, < 6%.
(A.41
and for
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