Octupole Contributions to the Generalized Oscillator Strengths of Discrete Dipole Transitio

Octupole Contributions to the Generalized Oscillator Strengths of Discrete Dipole

Transitions in Noble Gases.

M.Ya.Amusia,1,2L.V.Chernysheva,2Z.Fel?i,3and A.Z.Msezane3

1Racah Institute of Physics,The Hebrew University,Jerusalem91904,Israel

2A.F.Io?e Physical-Technical Institute,St.Petersburg194021,Russia

3Center for Theoretical Studies of Physical Systems,Clark Atlanta University,Atlanta,GA30314

(Dated:January2,2007)

The generalized oscillator strengths(GOS’s)of discrete excitations np→nd,both dipole(L=1)

and octupole(L=3)are studied,the latter for the?rst time.We demonstrate that although

the relevant transitions in the same atom are closely located in energy,the dependence of their

GOS’s on the momentum transfer squared q2,is remarkably di?erent,viz.the GOS’s corresponding

to L=3have at least one extra maximum as a function of q2and dominate over those of the

L=1,starting from about q2=1.5 a.u.The calculations were performed in the one particle

Hartree-Fock approximation and with account of many-electron correlations via the Random Phase

Approximation with Exchange.The GOS’s are studied for values of q2up to30a.u.

PACS numbers:32.80.Gc,34.50.-s,52.20.Hv,03.65.Nk

I.INTRODUCTION

Here we consider the lowest energy optically allowed transitions for the outer np subshells of the noble gas atoms np→nd and np→(n+1)s.The former series of transitions in Ne,Ar,Kr and Xe,namely np→nd can be dipole and octupole,while the latter can only be a pure dipole.The essential feature of the levels considered is

that those with the same con?gurations np→nd but di?erent total angular momentum L,L=1and L=3,are closely located and hardly separable in existing experiments.This means that they will be excited by electron(or other charged particle)impact simultaneously,but decay via photon emission separately;the decay of the octupole excitation being about eight orders of magnitude slower than the dipole.As a result,soon after excitation of a gas volume by incoming charged particles,only octupole levels will survive.Therefore,separate calculations of octupole GOS’s are of importance and interest,presenting the probability of populating and studying the long living excited levels.

The generalized oscillator strength,introduced by Bethe[1]and reviewed by Inokuti[2],characterizes fast electron inelastic scattering.It manifests directly the atomic wave functions and the dynamics of scattering.Since then, the GOS has received attention from a variety of perspectives:determination of the correct spectral assignment [3],exploring the excitation dynamics[4],probing the intricate nature of the valence-shell and inner-shell electron excitation[5],investigation of the in?uence of the angular resolution and pressure e?ects on the position and amplitude of the GOS minima[6],investigation of the GOS ratio[7]and various correlation e?ects[8-11],as well as multiple minima[12].

One of the most important utilities of the GOS concept in the limit q2→0is in the determination of optical oscillator strengths(OOS’s)from absolute di?erential cross sections(DCS’s)[13-16].Implicit in this is the extrapolation of the measured data through sometimes the unphysical region[17].The limiting behavior of the GOS as q2→0has been a subject of continuing interest because of the di?culty of measuring reliably the electron DCS’s for atoms,ions and molecules at and near zero scattering angles[3,14,18].This di?culty still plagues measurements of the DCS’s[19, 20],including the most recent measurements of the DCS’s[21,22].Thus a thorough understanding of the behavior of the GOS’s near q2=0is imperative to guide measurements.

To place the current investigation of the GOS’s for the np?nd and np?ns transitions in Ne,Ar,Kr and Xe in perspective,it is important to highlight some signi?cant new manifestations that have been uncovered in the recent studies of correlation e?ects in the GOS’s for atomic transitions:

1)Recently,GOS’s for monopole,dipole and quadrupole transitions of the noble gas atoms have been investigated in both Hartree-Fock(HF)and Random Phase Approximation with Exchange(RPAE)approximations as functions of q and the energy transferred,ω[8].There it was found that electron correlations,both intra-shell and inter-shell are important in the GOS’s for all values of q andωinvestigated and that the variation of the GOS’s with q andωis characterized by maxima and minima,arising entirely from many-electron correlations.These results have been used to understand and interpret[23,24]the?rst experimental observation of the absolute GOS for the nondipole Ar3p?4p transition[5].Of great signi?cance is that the calculated GOS’s for discrete transitions permitted the determination of their multipolarity quite reliably[10].The interpretation is of particular interest for nondipole transitions since they cannot be observed in photon absorption.Of even greater importance and accomplishment in

the recent measurement of the GOS’s for the valence-shell excitation of Ar is the separate measurement of the electric monopole and quadrupole of the GOS’s for the valence-shell excitation of Ar[25].

2)For the outer and intermediate shells of Xe,Cs and Ba,correlations have been discovered to produce impressive manifestations of intra-doublet interchannel interaction[26],yielding new structures in nondipole parameters and GOS’s[9].GOS results for Xe,Cs and Ba demonstrated the leveraging role of the spin-orbit interaction,viz.the strong interaction between components of the spin-orbit doublet of the3d electrons in Cs,Ba and Xe.This leads to the appearance of an additional maximum in the GOS for the3d5/2sub-shell,due to the action of the3d3/2electrons. The inter-doublet correlations were found to be very important in the monopole,dipole and quadrupole transitions.

3)Generalized oscillator strengths for monopole,dipole and quadrupole transitions in the negative ions I?and Si?have been investigated to assess the extent of importance of correlations[27].It was found that the GOS’s for monopole and dipole transitions,are generally characterized by two distinct sets of maxima as functions ofω, being most pronounced for the dipole transitions.For the negative ions,there are two interesting and signi?cant peculiarities.Firstly,contrary to the well-known behavior of atomic transitions,the limit of the GOS approaches zero as q2→0for the dipole transition.Secondly,in both the monopole and the quadrupole transitions,the GOS corresponding to q=2a.u.starts being zero at threshold and becomes dominant beyond aboutω=6Ry.

II.THEORY

In this paper,we consider a relatively simple case where the transition energy can be speci?ed almost entirely by the one-electron nomenclature,namely by the principal quantum number and angular momentum of the exciting electron in its initial and?nal states,nl and n l ,respectively and by the total angular momentum of the excitation L.Both the energiesωnl,n l and the GOS’s are a?ected by the multi-electron correlations,since the one-electron approximation is very often not accurate enough,even for qualitative,not to mention quantitative,description.With the above in mind,we have performed calculations both in the one particle Hartree-Fock approximation and with account of many-electron correlations via the RPAE[28].

The RPAE has proved to be very e?ective in describing the photoionization cross sections and GOS’s[8,23,24,28-30] including rather delicate features of the dipole and nondipole angular anisotropy parameters of photoelectrons,where impressive manifestations of electron correlations were recently observed in good accord with both experiment and calculations(see e.g.[26,31-36]).

The theoretical consideration in this paper of dipole and octupole excitations is similar to that of quadrupole and monopole excitations in[23,24].However for the convenience of the reader and for better understanding of the results we repeat here the main points of consideration of GOS in general and in the one-electron HF frame and with account of correlations in the RPAE frame.All necessary formulas are also presented.

One speci?c feature of the considered in this pasper discrete excitations require special https://www.sodocs.net/doc/9a10172403.html,ly,in Kr there are two closely located dipole excitations:4p-4d energy is0.93330Ry,while4p-6s energy is0.93403Ry and4p-5d energy is0.98429Ry,while4p-7s energy is0.98479Ry.As to the method implemented in the computing program in[38]and used in[23,24],it is suitable for the case of a single relatively isolated discrete excitation,for which all other act as perturbation.In our case the closely located levels are distorbing each other very strong.Therefore the corresponding interaction has to be taken into account accurately enough.

The inelastic scattering cross sections of fast electrons or other charged particles incident upon atoms or molecules are expressed via the GOS G(ω,q)[1,37]which is a function of the energyωand the momentum transferred q to the target in the collision process.The GOS is de?ned as[1](atomic units are used throughout this paper)

G fi(ω,q)=2ω

|ΣN j=1 ψ?f( r1,..., r N)exp(i q· r j)ψi( r1,..., r N)d r1...d r N|2(1)

where N is the number of atomic electrons andψi,f are the atomic wave functions in the initial and?nal states with energies E i and E f,respectively,andω=E f?E i.Because the projectile is assumed to be fast,its wave functions are plane waves and its mass M enters the GOS indirectly,namely via the energy and momentum conservation law

p2

?( p? q)2

=ω(2)

Here p is the momentum of the projectile.It follows from the GOS de?nition Eq.(1)that when q=0the GOS coincides with the OOS or is simply proportional to the photoionization cross section(see for example[37]),depending upon whether the?nal state is a discrete excitation or belongs to the continuous spectrum.The energyωenters the GOS either via a factor in Eq.(1)or indirectly,via the energy E f of the?nal state|f>.

In the one-electron Hartree-Fock approximation Eq.(1)simpli?es considerably,reducing to

g L fi (q,ωfi )=2ωfi | φ?f ( r )j L (qr )P L (cos?)φi ( r )d r |2

=2ωfi ||2(3)

where φf,i ( r )=R n f,i Y l f,i ,m f,i (θ r ,? r )χs f,i are the HF one-electron wave functions with their radial,angular and spin

parts,respectively,j L (qr )is the spherical Bessel function,P L (cos?)is the Legendre polynomial and cos?= q · r /qr .The excitation energy of the i →f transition is denoted as ωfi .The principal quantum number,the angular momentum,its projection and spin quantum numbers of the initial i and ?nal f states are denoted by n f,i ,l f,i ,m f,i and s f,i ,respectively.

To take into account of many-electron correlations in the RPAE the following system of equations was solved

=+(Σ

n ≤F,k >F ?Σ

n >F,k ≤F

)×<

k |A L (q,ωR fi )|n >

i >

ωR fi ? k + n +iη(1?2n k )

(4)

Here ≤F (>F )denotes occupied (vacant)HF states, n are the one-electron HF energies,η→+0and n k =l (0)

for k ≤F (>F );L =L ?L is the L component of the matrix elements of the Coulomb inter-electron interaction V .It is seen that the system of equations for each total angular momentum of an excitation L is separate.The procedure of solving this equation is described in details in [28,38].Note that the excitation energy of the i →f transition in RPAE ωR fi is di?erent from the HF value ωfi = f ? i .The procedure of calculating ωR fi is also described in [28,38].

A relation similar to Eq.(3)determines the GOS’s in RPAE G L fi (q,ωR

fi )

G L fi (q,ωR

fi )

=2ωR fi q

2||2

(5)

Here are the ?nal and initial HF states,https://www.sodocs.net/doc/9a10172403.html,ing these formulas the GOS’s were calculated

for dipole L =1and octupole L =3components.

The operator of the interaction between fast charged particles and atomic electrons can also be represented in

another form than ?A (q )=?A r (q )≡exp (i q · r ).This is anologous to the case of photoionization and can be called

length form.The other one is similar to the velocity form in photoionization and looks like [37]

?A v (ω,q )=[exp (i q ·r )(q ·??q ·←??)exp (i q ·r )]

(6)

where the upper arrow in ←?

?in Eq.(6)implies that the function standing to the left is being operated on.

For the speci?c case considered in this paper the explicit HF energies are ωnp →nd,(n +1)s ≡ nd,(n +1)s ? np with np , nd and (n +1)s being the HF one-electron energies.R np (r ),R nd (r )and R (n +1)s (r )are the radial parts of the one-electron wave functions in the HF approximation and L is the total angular momentum of the excitation,where in our case L =1or 3,n =2;3;4and 5for Ne,Ar,Kr and Xe,respectively.Symbolically,the RPAE equations can be presented as [28,38]

?T =?t +?T ?χU

(7)t L np →nd,(n +1)s (q )

≡

|nd,(n +1)s )>=

R np (r )j L (qr )R nd,(n +1)s (r )dr,

(8)

U is the Coulomb interelectron interaction,and

?χ=?1/(ω?ω +iγ)??1/(ω+ω )

(9)

with γ→0and ωbeing the excitation energy parameter of the relevant discrete excitation,while ω is the energy of

any other,including the considered discrete or continuous spectrum excitation of another electron,which is excited by the incoming electron.Its interaction via the potential U leads to the excitation of a given atom under consideration.

The RPAE values for the GOS’s F np →nd,(n +1)s (q )are connected to the matrix elements of ?T

similar to the connection of f np →nd,(n +1)s ,the HF GOS values,with ?t

.However,a complication arises for discrete excitations.This results

from the fact that one of the intermediate discrete excitation energies consistent with the energy of the excitation of the level under investigation and the corresponding element of?χbecomes in?nite.To circumvent this singularity an e?ective interaction matrix has to be created[28,38]:

Γ=U+U?χ Γ(10) where?χ excludes only a single term,with one of the transitionsω =ωnp→nd,(n+1)s,from summation over all intermediate states[see(4)].Then the total matrix of the e?ective interaction?Γis determined by a simple expression:

?Γ= Γ(ω?ωnp→nd,(n+1)s? Γ)?1.(11) This is correct only if the interaction between two adjaisent levels is weak enough and can be accounted for perturbatively.Then instead of Eq.(3),one can arrive at the following expression for?T

?T=?t+?t?χ?Γ(12) With the help of Eq.(11)we derive the GOS value in RPAE as

F np→nd,(n+1)s(q)=Z np→nd,(n+1)s 2πωnp→nd,(n+1)s

||2(13)

ωnp→nd,(n+1)s= nd,(n+1)s? np+ Γnp→nd,(n+1)s(14)

Z np→nd,(n+1)s= 1?? Γnp→nd,(n+1)s,

?ω|ω=ω

np→nd,(n+1)s

?1

.(15)

Here Z np→nd,(n+1)s is the spectroscopic factor of the discrete excitation level.

The equations(10)-(14)determine the RPAE values for both the GOS’s and the discrete excitation energies,while Eq.(3)gives the HF GOS,represented simply as f np→nd,(n+1)s with the appropriateωfi used.

However,as it was mentioned above,at least in Kr one has two very close levels.In this case one had at?rst to introduce an auhilary matrix of e?ective interelectron interaction?Γαβthat is a solution of equation similar to(10)

Γ=U+U?χ” Γ(16)

with?χ”that excludes two so-called"time-forward"terms,i.e.those with energy factors?1/(ω?ω”+iγ),whereω”are the energies of two strongly interacting transtions.In our case these are terms withω”=ω4p→4d andω”=ω4p→6s orω”=ω4p→5d andω”=ω4p→7s.However,the relation that determines?Γ,is not that simple as(11):it is instead of being an simple algebraic became a2×2matrix equation.

Let us consentrate on two?rst levels,4p→4d and4p→6s,denoting them as1and2,respectively.In this case the equation(16)looks as

Γ11Γ12Γ21Γ22 = Γ11 Γ12 Γ21 Γ22 + Γ11 Γ12 Γ21 Γ22 × (ω?ω1)?10

0(ω?ω2)?1 Γ11Γ12Γ21Γ22 (17)

In fact,(17)describes a two-level system that has two solutions,ω=ω 1,2=ω1,2.Such a system was considered in application to molecules in[36].The corresponding solution is also known:

ω 1,2=1

2

(ω1+ω2+ Γ11+ Γ22)± 4(ω1?ω2+ Γ11? Γ22)2+| Γ12|2(18)

As it should be,in absence of level mixing interaction( Γ12=0),ω 1,2=ω1,2.In principal,each level,1and2has its own spectroscopic factor Z1,2.But since these levels are close to each other,the corresponding Z values are close to each other and can be determined by(15)with Γtaken from(16).

III.RESULTS OF CALCULATIONS

The calculations were performed numerically using the programs and procedures described in[38]and for the case of Kr corrected with accord of the formulas(16-18).The results of the calculations are presented in the Table1and the?gures1-8below.It is important to bear in mind that at small q2(q2→0)the dipole GOS is absolutely dominant, since the GOS dipole component is non zero at q2→0,corresponding to the OOS.However,with increasing q2the situation very fast changes considerably and in some cases even dramatically,since f L=1

np→nd,(n+1)s

(q)rapidly decreases

with increasing q2while f L=3

np→nd,(n+1)s (q)rapidly increases as q2at least for small q.Then f L=3

np→nd,(n+1)s

(q)has to

reach its maximum with subsequent decrease.The q2-dependence of f proved to be more complicated,exhibiting maxima for all the cases considered.

Note that the results for the dipole components of the3p→3d level in Ar were obtained earlier[23].Here our previous results are complemented by those from the octupole contributions,calculated for the?rst time,to our knowledge.

IV.ACKNOWLEDGEMENTS

MYaA is grateful for support of this research by the Israeli Science Foundation under the grant174/03and by the Hebrew University Intramural fund.

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Table 1

GOS for discrete dipole levels with q=0.00001

----------------------------------------------------------

Ne

E,RY HF-L HF-V RPAE-L RPAE-V

1.58893 .2816E-01 .2682E-01 .2655E-01 .3551E-01 2p-3d

1.63788 .1558E-01 .1483E-01 .1470E-01 .1974E-01 2p-4d

1.66057 .8801E-02 .8378E-02 .8106E-02 .1092E-01 2p-5d

E,RY HF-L HF-V RPAE-L RPAE-V

1.34815 .1564E+00 .1444E+00 .1679E+00 .1731E+00 2p-3s

1.56349 .2779E-01 .2553E-01 .2944E-01 .3041E-01 2p-4s

1.62788 .9917E-02 .9101E-02 .1058E-01 .1096E-01 2p-5s

*********************************************************************** Ar

E,RY HF-L HF-V RPAE-L RPAE-V

.89702 .2964E+00 .2647E+00 .3158E+00 .3132E+00 3p-4s

1.06277 .5571E-01 .4947E-01 .3915E-01 .3877E-01 3p-5s

1.11637 .2055E-01 .1822E-01 .1488E-01 .1475E-01 3p-6s

1.06778 .1624E+00 .9771E-01 .1788E+00 .1792E+00 3p-3d

1.11821 .8241E-01 .4888E-01 .8760E-01 .8780E-01 3p-4d

1.14136 .4509E-01 .2657E-01 .4116E-01 .4129E-01 3p-5d

*********************************************************************

Kr without 4p-6s,4p-7s

E,RY HF-L HF-V RPAE-L RPAE-V

.93330 .2673E+00 .1525E+00 .2614E+00 .2607E+00 4p-4d

.98429 .1316E+00 .7385E-01 .1263E+00 .1258E+00 4p-5d

1.00760 .7116E-01 .3964E-01 .6358E-01 .6329E-01 4p-6d

without 4p-4d,4p-5d

.78002 .3752E+00 .3364E+00 .3576E+00 .3454E+00 4p-5s

.93403 .7049E-01 .6282E-01 .6592E-01 .6463E-01 4p-6s

.98479 .2621E-01 .2332E-01 .2351E-01 .2313E-01 4p-7s

*********************************************************************

Xe

E,RY HF-L HF-V RPAE-L RPAE-V

.67262 .4185E+00 .3589E+00 .4263E+00 .4043E+00 5p-6s

.80827 .8236E-01 .7005E-01 .1252E+00 .1190E+00 5p-7s

.85442 .3130E-01 .2656E-01 .4608E-01 .4376E-01 5p-8s

E,RY HF-L HF-V RPAE-L RPAE-V

.79777 .4839E+00 .2483E+00 .4573E+00 .4444E+00 5p-5d

.84984 .2309E+00 .1160E+00 .2123E+00 .2057E+00 5p-6d

.87351 .1235E+00 .6145E-01 .1069E+00 .1032E+00 5p-7d

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