and L-Shell Ionization Cross Sections in the Plane

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Author(s)
Citation
Issue Date
URL
A Computer Code for K- and L-Shell Ionization Cross Sections
in the Plane-Wave Born Approximation
Mukoyama, Takeshi; Sarkadi, László
Bulletin of the Institute for Chemical Research, Kyoto
University (1980), 58(1): 60-66
1980-03-31
http://hdl.handle.net/2433/76859
Right
Type
Textversion
Departmental Bulletin Paper
publisher
Kyoto University
Bull. Inst. Chem.
Res., Kyoto
A Computer
Sections
Univ.,
Vol. 58, No. 1, 1980
Code for K- and L-Shell
in the Plane-Wave
Ionization
Cross
Born Approximation
Takeshi MUKOYAMA*
and Laszlo SARKADI#
Received
December
7, 1979
A computer
codeDEKYhasbeenwrittento calculate
theK- andL-shellionization
crosssections
by heavy charged-particle
impactin the plane-waveBorn approximation.Correctionsfor
binding-energy
andCoulomb-deflection
effectsas wellasrelativistic
effectare takenintoaccount.
KEY WORDS:
K- and L-shell ionization cross section / Plane-wave
Born approximation /
Binding-energy, Coulombdeflection, and relativistic effects /
I. INTRODUCTION
In recent years inner-shell ionization of atoms by impact of heavy charged particles, such as protons, a particles and carbon ions, has been studied both theoretically
and experimentally. Extensive data on K- and L-shell ionization cross sections have
been reported and theoretical cross sections have been calculated in the plane-wave
Born approximation (PWBA).1)
For low-velocity projectiles, two effects become important;
(1) the increase in
the binding energy of the target electron due to penetration of the projectile inside the
inner shell during collision and (2) the deflection of the projectile by the Coulomb
field of target nucleus. The PWBA theory modified for both effects has been successfully used to predict the direct Coulomb ionization cross sections. Furthermore, in
the case of targets with large atomic number, the relativistic effect should be taken
into consideration.
There have been already published numerical tables for the PWBA calculations,2-6) and the ionization cross sections can be obtained from these tables by interpolation technique. However, it is useful to prepare a computer code to calculate
ionization cross sections in the PWBA modified for the effects described above.
The computer code DEKY has been written to calculate the PWBA cross sections
for K- and L-shell ionization by impact of heavy charged particles. The corrections
for the binding-energy and Coulomb-deflection effects as well as the electronic relativistic effect are taken into account. The program has been written originally for
the PDP-11 /40 computer in the Institute of Nuclear Research of the Hungarian
* la7flj
: Laboratoryof Nuclear Radiation, Institutefor ChemicalResearch,KyotoUniversity,
Kyoto.
# Institute of Nuclear Research of the Hungarian Academyof Sciences(ATOMKI), Debrecen,
Hungary.
(60 )
A Computer
Code for K- and L-Shell
Ionization
Academy of Sciences (ATOMKI) and then modified for the FACOM M-190 computer in the Data ProcessingCenter of Kyoto University.
H. THEORY
A. Ionization Cross Section
The s-shellionization cross section in the PWBAcan be written as
87;Z12)2()
s4yl2LA, 72sa0,1
where Zle is the charge of the projectile, Z8 is the effectivenuclear charge seen by the
s-shellelectron, and ao denotes the Bohr radius of the hydrogen.
When the projectile velocity is v1,the scaled projectile velocity parameter, 9„
is defined as
sZ
Yls=
22-----(hvi/e2)2,(2)
S
and the scaled target-electron binding energy is given by
Os= Is n2/(Zs2R.),(3)
where I, is the measured ionization potential, R,. the Rydberg energy, and n is the
principal quantum number of the s shell.
The functionf,(B„ )28)is calculated as
s(sS)—IF
'!W m1n~Qmin
WQmaxdQ2Ws
F (Q) 12,(4)
f8y~/1'wmaxd
where Z,Q112/q0
is the momentum transfer and WZ,2RO.
is the energy transfer. The
lower limit of the integral with respect to W is taken to be Wm,n=0,/n2,while Wmax
=(M1/m)rt,where M1 and m are the masses of the projectile and the target electron,
respectively. On the other hand, Qm,nand Qmax
are given by:
Qmin= (311/74)277s
{1 [1 m lW/(M177s)1112}2,
Qmax=(M1/m)277s{1
+[1—mW/(MOls)1112I}2.
The form factor, F,,,,(Q), is given explicitly for K shell and each L subshellin Refs. 2
and 3.
B. Binding-Energy Effect
Basbas et al.7)estimated the influence of the projectile on the initial state of the
K-shell electron by the use of the perturbed-stationary-state theory. They incorporated this result in the PWBA and developed a simple method to include the effect of
increased binding energy. According to them, the factor by which the scaled binding
energy is increased is given by
ss(Es)=1 +2(Z1/Zses) gs(es)•(5)
The parameter e. is defined as
( 61 )
T. MUKOYAMA and L. SARKADI
gSv~/(2esys)(6)
where vs=Zsvo/nand vo=e2/h,is the Bohr velocity. The correspondingformula for
L shellhas been derived by Brandt and Lapicki.8)The binding-energyeffect can
be takeninto accountby increasingOsto esB,.
The functiongs(es)is estimatedby the use of screenedhydrogenicwave functions
and by averagingthe binding-energyincreaseover the impact-parameter-dependent
ionization crosssections. The values are obtained by numerical integrationand
expressedin the analyticalapproximations
7,8)
gx(E)_ (1+5E+7.14e2+4.27E3
+0.947E4)/(1+E)5;(7)
gi,(e) = (1 +9e + 30.2e2+ 66.8E3+ 100E4
+ 94.1E5
+51.3E6+ 15.2E7+1:89c8)/(1
+ e)9,(8)
gL2,,(e)_ (1+ 95+ 35.9E2+ 84.5E3
+ 110E4
+91.9E5
+ 42.7E6+ 12.2E7
+ 1.51E8)/(1+s )9.(9)
C. Coulomb Deflection
The Coulomb-deflection
effectis taken into considerationthrough the method
of Basbaset al.7) They used the differentialionizationcrosssectionfor hyperbolic
trajectory of the projectile in the semiclassicalapproximation,9,10)
and derived a
simplecorrectionformulaby integrationover the energytransfer. The crosssection
with Coulomb-deflection
effect,680,can be expressedas
6KL,=9E1o(ndgo)6K
LBA,(10)
6E2
,1,3
=11E12(7C
dg0)6iwsA,(11)
wherehigois the minimummomentumtransferfor ionization,d is the half-distanceof
closestapproach in a head-oncollision,E„(x) is the exponentialintegralof order n,
and c8PwBA
denotesthe PWBAcrosssectioncalculatedfrom Eq. (1).
D. Relativistic Effect
The relativisticcorrectionisincorporatedthroughtheusualmethodof Merzbacher
and Lewis.1) The relativisticscaledbindingenergyis definedas
BsR=1—(/R—Is)/IsNR,(12)
where I, is the observedionizationpotential, IsR.the ionizationenergy for s shell
calculatedby the relativisticscreenedhydrogenicmodel,and /„NRthat obtainedfrom
the nonrelativisticscreenedhydrogenicmodel. The relativisticcrosssectionsare obtained fromEq. (1) by replacingOsof Eq. (3) by 0,Rdefinedin Eq. (12).
III. DESCRIPTIONOF THE PROGRAM
The program is designed to calculate the PWBA cross section, the PWBA modified
for the binding-energy effect (PWBA—B), the PWBA-B including the Coulombdeflection effect (PWBA—BC),and the PWBA—BCcorrected for the relativistic effect
( 62 )
A Computer
Code for K and L-Shell
Ionization
(PWBA—BCR).
Input for each case consists of two cards; a card for the projectile and a card for
the target. After input data are read, the scaled binding energies, O$and OsR,are
determined. In the next step, the scaled projectile velocity, Y7s,is estimated for a
given projectile energy. Then the program starts to compute asPwBA. For this purpose, two subroutines are used. Subroutine F calculate the form factor IF„(Q )12
for a fixed values of W and Q, and numerical integration over Q is performed in Subroutine FI. The numerical integration with respect to W is made and the cross
section asPWBA
is obtained from Eq. (1).
The binding-energy factor s, is estimated from Eq. (5) by using gs(e,) function
in Eqs.(7)—(9)and the scaledbindingenergy0s is replacedby EsOs.The procedures similar to those used for asPwBAare repeated and the PWBA—Bcross section,
aZWBA`B,
is obtained. The factor for the Coulomb-deflection effect is determined
from 08, ss, and r)s, and the PWBA—BCcross section is obtained by multiplying this
factor to o PWBA-B
Finally the relativistic correction is made. The scaled binding energy 0, is
replaced by 0,R of Eq. (12) and calculation of the cross section is made for s,O,R By
multiplying the Coulomb-deflection factor, the PWBA cross section including bindingenergy, Coulomb-deflection, and relativistic effects (PWBA—BCR) is obtained.
The projectile energy is increased by the energy step given in the input data and
all the cross-section calculations are made for this energy. The whole procedures
are repeated until the projectile energy exceed the maximum value given in the input.
Then the program reads the next input cards.
IV. INPUT DATASPECIFICATION
In the following we give details of the input required, including FORMAT
specifications and explanations of the variables and options.
Card 1.
IZ2
A2
L
N
BE
Card 2.
IZ1
IAI
EMIN
EMAX
DE
FORMAT (I10, F10.0, 2I5, F10.0)
Atomic number of the target.
Mass number of the target.
Identifying label for the atomic shell. For K shell L=0, and L=i
corresponds to Le-shell electron.
Number of integration points. Usually 101.
Measured binding energy of the target electron (keV).
FORMAT (2I5, 3F10.0)
Atomic number of the projectile.
Mass number of the projectile.
Minimum projectile energy (MeV).
Maximum projectile energy (MeV).
Energy step size (MeV).
( 63 )
T. MUKOYAMA
and L. SARKADI
A blank card will terminate the run or another input data may follow.
V. THE TEST RUN
The test run has been made for the K-shell ionization cross sections by protons
on silver (Z2=47). The energy range is taken to be between 0.5 and 1.0 MeV with
the energy step of 0.5 MeV. Thus the cross sections are calculated for two projectile
energies, 0.5 and 1.0 MeV.
ACKNOWLEDGMENTS
One of the authors (T. M.) would like to acknowledge his appreciation of the
hospitality extended him by Professor D. Berenyi and members of Nuclear Atomic
Group of ATOMKI during the time when the work reported here was carried out.
He also wishes to thank the Hungarian Institute for Cultural Relations and the Japan
Society for Promotion of Science for award of Research Fellowship.
REFERENCES
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
E. Merzbacherand H. Lewis, "Handbuch der Physik,"Vol. 34, ed. by S. Flugge,Springer,
Berlin,(1958),p. 166.
G. S. Khandelwal,B.-H.Choi,and E. Merzbacher, Atomic
Data, 1, 103 (1968).
B.-H.Choi,E. Merzbacher,and G. S. Khandelwal, Atomic
Data,5, 291 (1973).
R. Rice,G. Basbas,and F. D. McDaniel, Atom.Dataand Nucl.Data Tables,20, 503 (1977).
O. Benkaand A. Kropf, Atom.DataandNucl.Data Tables,22, 219 (1978).
T. Mukoyamaand L. Sarkadi, Bull.Inst.Chem.
Res.,KyotoUniv.,58, 95 (1980).
G. Basbas,W. Brandt,and R. Laubert, Phys.Rev.A, 7, 983 (1973).
W. Brandtand G. Lapicki, Phys.Rev.A, 10, 474 (1974). The coefficients
in gL,(e)and gL,(e)
givenin this referenceare in error. The valuesin Eq. (9) are calculatedby the presentauthors.
The correctformulais alsogivenby G. Lapickiand W. Losonskyin Phys.Rev.A, 15, 896 (1977).
J. Bangand J. M. Hansteen, Kgl.Dan. Vidensk.
Selsk.Mat.-Fys.Medd.,31, No. 13 (1959).
J. M. Hansteenand O. P. Mosebekk, Z. Phys.,234,281 (1970).
( 64 )
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