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Autoionization from stark-mixed states
284
Nuclear
Instruments
and Methods
in Physics Research B23 (1987) 284-286
North-Holl~d,
Amsterdam
let us keep only the Coulomb and the Stark contributions in the last term of eq. (2) which reads:
1. Introduction
What happens to a doubly excited multicharged ion moving in the field of a distant charge’? Such a question arises after a double electron capture collision like: N”
+He-Ns’**+He?+,
(1)
It has already been shown on the system Li ’ + He** 111 that the Stark effect may affect strongly the autoionizing states (AIS) even at rather large internuclear distances. Now, double capture resulting from collisions between heavy ions and atoms has become a subject of extensive experimental works [2]. Consequently the study of PC1 effects as well as the determination of resonances in two electron ions need improved theoretical treatments. Indeed, the desexcitation pattern of the doubly excited multicharged ion N5’ ** may be alfccted by the residual (Y particule, depending on both the A.1.S lifetime and the impact velocity. It has been shown experimentally [2] that a non negligible number of doubly excited states N’ ’ ** (31, 31’) are produced in the reaction (1); hence we shall focus ou this series of resonances in what follows.
2. Theory 2.1. Itrrroductiott The total nonrelativistic is written: w = &,(l)
+ H,,(2) + $
Hamiltonian
of the system
z - Z,, c --L , ,=, Ir,-RI’
P=P,-t
P2-P,P2,
,/v-1 ,>
p,=
(41
1 n,=
+i
nz=
-/
c z: c
,1=1
I-0
I~,~,,,,(i))(~,,,,,,,(i)I 3
&,,,,(i) is the hydrogenic wave function for the electron i and tl, I, tn are the usual quantum numbers. N specifies the ionization threshold of the series of AIS under consideration. Q is defined as the complement of P, Q spans a subspace which contains the doubly excited states under consideration. When Z,, # 0 the Stark effect induces couplings between intcrshcll hydrogenic states. Therefore the validity of eq. (5) becomes questionable in this context. It can be shown easily that, when 2, becomes large, the Stark effect mixes predominantly the degenerate states $,,,,,,(i). Indeed, if we consider the term (3) as the perturbation. the first order correction to the wave function (p,,,,,,(i) due to the presence of other nondegenerate states varies as l/Z&. Therefore, we retain the effect of the Stark mixing between the substates of given n, in the context of this approximation the definition (5) for the projector P is valid.
i2)
Fl,,( i) is the one electron Hamiltonian relative to the electron i and the charge Z,: R is the internuclear distance; r, is the distance between the electron i and the nuclear charge 2,; Z,, is the distant perturbative charge; r,? is the interelectronic distance. We use the Feshbach formalism [3]. When R 5 r,, 0168-583X/87/$03.50 (0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
where 8, is the angle between J+,and H. When Z,, = 0 (“pure atomic” case) a projection operator P is introduced [4]:
B.V.
We consider a given total spin S. Due to the prcsence of the distant charge, only the projection M of the total orbital momentum on the internuclear axis (the z-axis) is conserved. The function x,” is expanded on a
H. Bachau, P. Galan / Automnization
basis of hydrogenic x:“=
from Stark-mixed
285
.stutes
functions:
&&‘,
(6)
K
with (7)
r/,” = A( %,,,1,,1(1)+,,,,,,“~ ._ 2(2)). This configuration the antisymmetrisor
is labelled and
(n,l, m,, m,l,m,)
M=m,+m,.
A is
(8)
The coefficients uK and the unshifted determined by a diagonalization method
energy {s are through:
Then ~7 is expanded on the solutions “,+L,’’ of the “pure atomic” problem (when Z, = 0), L, and n, are respectively the total orbital momentum and the parity. We have:
of the width
The determination of the continuum (nonresonant) wave function is the principal difficulty of the kind of problem we are interested in. To simplify it, we make three assumptions. (1) The bound electron (after deexcitation) makes a full screening of the charge Z, (2) We neglect the Stark effect on the ejected electron. Thus only the Coulomb contribution is considered (the ejected electron needs to be represented only in the region where x” is non-negligeable). (3) The Stark mixing occurs predominantly between quasi degenerate “pure atomic” AIS states. Consequently the energy of the ejected electron is close to the value obtained with Z, = 0 (in the region where x,” is non-negligeable). Assumption (2) and (3) result in a width r given by (see [l] for a particular application)
1a, 12=,p
r = c
12.
16.
R (A.U)
(9)
(QHQ-L>x;=o.
2.3. Culcuiution
8.
Fig. 1. The dominant mixing coefficients ah (see [lo]) relative to the pcrturbcd AIS ’ F” (I).
I
(11)
number of states to be determined (11 here). Furthermore, it allows us to compare our results with our molecular calculations [5]. For example, at R = 10 a.u. the energy of the state ‘Ge(l) is -5.2312 a.u. in [5], while it is - 5.2278 au. in the present work. When the quadrupolar term is included in eq. (3) the energy is -5.2304 a.u. The main correction to the energy comes from the configuration interaction with the states (3/m, 41’m’) [6]. We have a strong mixing between the unperturbed states ‘De(3), ‘F’(l) and ‘Ge(l) since their energies differ only by a few eV. The results relative to the state ‘F’(l) are presented in figs. 1 and 2. The Stark mixing takes place in the region R -C 20 a.u. It is clear that, for a projectile energy about a few keV/amu, the Stark effect may be important to evaluate PC1 effects (depending on the AIS produced in the double capture process). The method, particularly adapted for heavy multicharged ions, permits to calculate rapidly the influence of the Stark mixing. Thanks are due to Dr. R. Gayet for useful comments on the draft manuscript.
P (,o’%?),
Where n,r,L, is the width associated to the resonant state “,$:I called 2s+‘L~~( n) in the standard notation, n being its position number in the series. The approximation (1) permits the use of coulomb functions to represent the continuum. Therefore n,cL, is obtained quasianalytically.
3. Application and conclusion We calculate the ‘Z states using all configuration 31’m’). This restriction is convenient to limit the
(3/m,
-4 6.
12.
16.
Fig. 2. The partial width of the perturbed ated to the flnal state Nh+ *
RCA.
J)
AIS ‘F”(I)
asxx-
(II = 2).
VIII. SPECTROSCOPY
286
H. Bachau, P. Galan / Autoronm~tion from Stark-mixed
states
References [I] N. Stoltcrfoht, I>. Brandt and M Prost. Phy Re\ Lett. 43 (IY7Y) 16.54. [2] S Tsurubuchl. T. Iwai. Y. Kancko, M. Kimura. N. Kobakaaahi. A. Matsumoto. S Ohtanl. K. Okuno. S. Takagi and H. Tawara. J. Phv.\. 13 1.5 (IYX2) L733; A RordenaveMontesquicu. P. R&it-Cattin. A. <;leizch, A.I. Mnrrukchi. S. Dous\on and D. Hitz. J. Phw I3 17 (1984) L127
[3] II. Feshbach. Ann. Phqs. 19 (lY67) 2X7. [4] P.L. Altick and E. Neal Moore. Phys Rev. Lctt. 15 (1Y65) 100. [S] H. Rachau. Communication to: lie Colloque \ur la Physique dcs ColliGons Atomlqucs ct Elcctroniquc> (Mctz X6). paper auhrmttcd [6] H. Bachau. J. Phy\. 11 17 (19X4) 1771.
Nuclear
Instruments
and Methods
in Physics Research B23 (1987) 284-286
North-Holl~d,
Amsterdam
let us keep only the Coulomb and the Stark contributions in the last term of eq. (2) which reads:
1. Introduction
What happens to a doubly excited multicharged ion moving in the field of a distant charge’? Such a question arises after a double electron capture collision like: N”
+He-Ns’**+He?+,
(1)
It has already been shown on the system Li ’ + He** 111 that the Stark effect may affect strongly the autoionizing states (AIS) even at rather large internuclear distances. Now, double capture resulting from collisions between heavy ions and atoms has become a subject of extensive experimental works [2]. Consequently the study of PC1 effects as well as the determination of resonances in two electron ions need improved theoretical treatments. Indeed, the desexcitation pattern of the doubly excited multicharged ion N5’ ** may be alfccted by the residual (Y particule, depending on both the A.1.S lifetime and the impact velocity. It has been shown experimentally [2] that a non negligible number of doubly excited states N’ ’ ** (31, 31’) are produced in the reaction (1); hence we shall focus ou this series of resonances in what follows.
2. Theory 2.1. Itrrroductiott The total nonrelativistic is written: w = &,(l)
+ H,,(2) + $
Hamiltonian
of the system
z - Z,, c --L , ,=, Ir,-RI’
P=P,-t
P2-P,P2,
,/v-1 ,>
p,=
(41
1 n,=
+i
nz=
-/
c z: c
,1=1
I-0
I~,~,,,,(i))(~,,,,,,,(i)I 3
&,,,,(i) is the hydrogenic wave function for the electron i and tl, I, tn are the usual quantum numbers. N specifies the ionization threshold of the series of AIS under consideration. Q is defined as the complement of P, Q spans a subspace which contains the doubly excited states under consideration. When Z,, # 0 the Stark effect induces couplings between intcrshcll hydrogenic states. Therefore the validity of eq. (5) becomes questionable in this context. It can be shown easily that, when 2, becomes large, the Stark effect mixes predominantly the degenerate states $,,,,,,(i). Indeed, if we consider the term (3) as the perturbation. the first order correction to the wave function (p,,,,,,(i) due to the presence of other nondegenerate states varies as l/Z&. Therefore, we retain the effect of the Stark mixing between the substates of given n, in the context of this approximation the definition (5) for the projector P is valid.
i2)
Fl,,( i) is the one electron Hamiltonian relative to the electron i and the charge Z,: R is the internuclear distance; r, is the distance between the electron i and the nuclear charge 2,; Z,, is the distant perturbative charge; r,? is the interelectronic distance. We use the Feshbach formalism [3]. When R 5 r,, 0168-583X/87/$03.50 (0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
where 8, is the angle between J+,and H. When Z,, = 0 (“pure atomic” case) a projection operator P is introduced [4]:
B.V.
We consider a given total spin S. Due to the prcsence of the distant charge, only the projection M of the total orbital momentum on the internuclear axis (the z-axis) is conserved. The function x,” is expanded on a
H. Bachau, P. Galan / Automnization
basis of hydrogenic x:“=
from Stark-mixed
285
.stutes
functions:
&&‘,
(6)
K
with (7)
r/,” = A( %,,,1,,1(1)+,,,,,,“~ ._ 2(2)). This configuration the antisymmetrisor
is labelled and
(n,l, m,, m,l,m,)
M=m,+m,.
A is
(8)
The coefficients uK and the unshifted determined by a diagonalization method
energy {s are through:
Then ~7 is expanded on the solutions “,+L,’’ of the “pure atomic” problem (when Z, = 0), L, and n, are respectively the total orbital momentum and the parity. We have:
of the width
The determination of the continuum (nonresonant) wave function is the principal difficulty of the kind of problem we are interested in. To simplify it, we make three assumptions. (1) The bound electron (after deexcitation) makes a full screening of the charge Z, (2) We neglect the Stark effect on the ejected electron. Thus only the Coulomb contribution is considered (the ejected electron needs to be represented only in the region where x” is non-negligeable). (3) The Stark mixing occurs predominantly between quasi degenerate “pure atomic” AIS states. Consequently the energy of the ejected electron is close to the value obtained with Z, = 0 (in the region where x,” is non-negligeable). Assumption (2) and (3) result in a width r given by (see [l] for a particular application)
1a, 12=,p
r = c
12.
16.
R (A.U)
(9)
(QHQ-L>x;=o.
2.3. Culcuiution
8.
Fig. 1. The dominant mixing coefficients ah (see [lo]) relative to the pcrturbcd AIS ’ F” (I).
I
(11)
number of states to be determined (11 here). Furthermore, it allows us to compare our results with our molecular calculations [5]. For example, at R = 10 a.u. the energy of the state ‘Ge(l) is -5.2312 a.u. in [5], while it is - 5.2278 au. in the present work. When the quadrupolar term is included in eq. (3) the energy is -5.2304 a.u. The main correction to the energy comes from the configuration interaction with the states (3/m, 41’m’) [6]. We have a strong mixing between the unperturbed states ‘De(3), ‘F’(l) and ‘Ge(l) since their energies differ only by a few eV. The results relative to the state ‘F’(l) are presented in figs. 1 and 2. The Stark mixing takes place in the region R -C 20 a.u. It is clear that, for a projectile energy about a few keV/amu, the Stark effect may be important to evaluate PC1 effects (depending on the AIS produced in the double capture process). The method, particularly adapted for heavy multicharged ions, permits to calculate rapidly the influence of the Stark mixing. Thanks are due to Dr. R. Gayet for useful comments on the draft manuscript.
P (,o’%?),
Where n,r,L, is the width associated to the resonant state “,$:I called 2s+‘L~~( n) in the standard notation, n being its position number in the series. The approximation (1) permits the use of coulomb functions to represent the continuum. Therefore n,cL, is obtained quasianalytically.
3. Application and conclusion We calculate the ‘Z states using all configuration 31’m’). This restriction is convenient to limit the
(3/m,
-4 6.
12.
16.
Fig. 2. The partial width of the perturbed ated to the flnal state Nh+ *
RCA.
J)
AIS ‘F”(I)
asxx-
(II = 2).
VIII. SPECTROSCOPY
286
H. Bachau, P. Galan / Autoronm~tion from Stark-mixed
states
References [I] N. Stoltcrfoht, I>. Brandt and M Prost. Phy Re\ Lett. 43 (IY7Y) 16.54. [2] S Tsurubuchl. T. Iwai. Y. Kancko, M. Kimura. N. Kobakaaahi. A. Matsumoto. S Ohtanl. K. Okuno. S. Takagi and H. Tawara. J. Phv.\. 13 1.5 (IYX2) L733; A RordenaveMontesquicu. P. R&it-Cattin. A. <;leizch, A.I. Mnrrukchi. S. Dous\on and D. Hitz. J. Phw I3 17 (1984) L127
[3] II. Feshbach. Ann. Phqs. 19 (lY67) 2X7. [4] P.L. Altick and E. Neal Moore. Phys Rev. Lctt. 15 (1Y65) 100. [S] H. Rachau. Communication to: lie Colloque \ur la Physique dcs ColliGons Atomlqucs ct Elcctroniquc> (Mctz X6). paper auhrmttcd [6] H. Bachau. J. Phy\. 11 17 (19X4) 1771.