Fluorescence of polarized atoms excited by polarized electrons

Nuclear Instruments and Methods in Physics Research B 267 (2009) 266–269

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Fluorescence of polarized atoms excited by polarized electrons A. Kupliauskiene˙ * Institute of Theoretical Physics and Astronomy of Vilnius University, A.Goštauto 12, LT-01108 Vilnius, Lithuania

a r t i c l e

i n f o

Available online 21 October 2008 PACS: 31.15 32.50.+d 34.10.+x 34.80.Dp 34.80.Nz

a b s t r a c t A general expression for the cross section describing properties of the fluorescence radiation from polarized atoms excited by polarized electrons is obtained in the distorted wave (DW) approximation. The expression is presented in the form of multiple expansion over the state multipoles of all particles taking part in the two-step process. It is used to derive the alignment parameters of excited atom and angular distribution of emitted fluorescence. The polarization degree of radiation from the lowest autoionizing states np5 ðn þ 1Þ s2 2 P3=2 following the excitation of non-polarized Na and K atoms by non-polarized electrons is calculated. Ó 2008 Elsevier B.V. All rights reserved.

Keywords: Fluorescence Atomic excitation by electron impact Spin dependence of cross section

1. Introduction The radiation emission is an important source of information about the processes taking part in laboratory and astrophysical plasmas. The simulated spectra of intensities and polarization are used for plasma diagnostics, e.g. tokamak [1]. The excitation of atoms by electrons is one of the most important processes leading to the fluorescence emission [2]. The collimated beam of electrons introduces a large degree of anisotropy into the atomic system. The alignment is manifested in the non-uniform angular distribution of radiation [3]. The use of the polarized atoms and electrons in the experiment can provide further information about the structure of atoms and the dynamics of the process. Therefore, general expressions for the cross sections involve description of the polarization state of all particles both in initial and final states are of importance. The main task of this work is to present such general expression for differential cross section describing the fluorescence following excitation of polarized atoms by polarized electrons in non-relativistic approximation. To derive expressions for the parameters suitable to characterize the polarization state of the particles participating in the electron–atom interaction, a density matrix formalism has been widely used [4], but also a number of alternative approaches was proposed [5–8]. In our method the cross section is expressed as a multiple expansion over the multipoles of the states of all particles * Tel.: +37052612723. E-mail address: [email protected] 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.038

taking part in the process instead of using the density matrix elements. The most general cases with respect to the number of polarized states were investigated. So far, the method has been applied to obtain a general expression for fluorescence radiation following recombination of polarized ions with polarized electrons [9], radiative [10] and Auger decay [11] following photoionization of polarized atoms, excitation [12,13] and ionization [14,15] of polarized atoms by polarized electrons. The method was also generalized for multi-step processes [8]. 2. General expression The process of the radiation characterized by the wave vector ^q which follows the excitation k01 and polarization unit vector  of polarized atom by polarized electron can be written as follows:

Aða0 J 0 M 0 Þ þ eðp1 m1 Þ ! A ða1 J 1 Þ þ eðp2 m2 Þ ! Aða2 J 2 M 2 Þ þ eðp2 m2 Þ þ hmð^q k01 Þ:

ð1Þ

Here ai denotes the configuration and other quantum numbers, while J i and Mi are the total angular momentum and its projection onto the chosen direction, respectively, for atoms in the initial (i ¼ 0), intermediate (i ¼ 1) and final (i ¼ 2) states, pi mi denote the electron momentum and spin projection. In two-step approximation, the expression of the differential cross section describing fluorescence radiation following the excitation of polarized atoms by polarized electrons can be written in the form of the expansion over the multipoles of the

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non-registered intermediate state of an excited atom Aða1 J 1 M 1 Þ by using the method proposed in [8] as follows: 2

d

rða0 J0 M0 p1 m1 ! a1 J1 p2 m2 ! a2 J2 M2 ^q k01 Þ

k1=2

ha2 J 2 kQ ðkÞ ka1 J 1 i ¼ k0

dXe dXf X drex ð a J 0 0 M 0 p1 m1 ! a1 J 1 p2 m2 Þ K 1 N1

¼

ð8Þ

r dW K 1 N1 ða1 J 1 ! a2 J 2 M 2 ^q k01 Þ

dXf

ð2Þ

:

Here dXe and dXf indicate solid angles of the scattered electron and emitted photon, respectively. The range of the summation over K 1 depends on the value of the total angular momentum J 1 and on second process. In the case of electric dipole radiation and J 1 P 1, it follows that K 1 ¼ 0; 1; 2. The expressions for the first and second factor above are presented in [12] and [9]. They are given by

dr

a

! a1 J 1 p2 m2 Þ ¼ 4pC½2K 1 þ 11=2 dX B ex ðK 0 ; K 00 ; K 1 ; K 01 ; K k1 ; K s1 ; K k2 ; K s2 ; KÞ

X K;K 0 ;K 00 ;K k1 ;K s1 K 01 ;K k2 ;K s2

"

X



N 0 ;N 00 ;Nk1 ;N s1 N 01 ;N k2 ;N s2 ;N

" 

K k1

K s1

K 00

Nk1

N s1

N00

K k2

K s2

K 01

Nk2

Ns2

N01

#"

K0

K 00

K

N0

N00

N

#"

K1

K 01

K

N1

N01

N

#

# 0 b ^1 ÞY K k2 Nk2 ðp ^2 ÞT K Y K k1 Nk1 ðp N0 ðJ 0 ; J 0 ; M 0 j J 0 Þ

dXf 

X  K1

Kr

K2

N1

Nr

N2

N r ;N2



¼ C1

ð3Þ

X K r ;K 2 ;k;k

0

AðK 1 ; K r ; K 2 ; k; k Þ 0

0 ^ 0 Þ: T KN22 ðJ 2 ; J 2 ; M 2 j^J 2 ÞT KNrr ðk; k ; qjk

ð4Þ

pffiffiffiffiffiffiffiffi The constant in (3) is C ¼ 4=p21 (p1 ¼ 2e1 in atomic units), C 1 ¼ 1=ð2pÞ, s ¼ 1=2 is the spin of an electron, Y KN ð^JÞ is the spherical function, the hat denotes the polar and azimuthal angles with respect to the chosen z axis, the square brackets stand for the Clebsh–Gordan coefficient, and

T KN ðJ; J; MjbJÞ ¼ ð1ÞJ0 M0



4p 2J þ 1

1=2 

J M

J M

 K Y KN ð^JÞ: 0

ð5Þ

The expressions for the parameters Bex and A in (3) and (4) are as follows [12,9]: X k0 þk0 Bex ¼ ð2J þ 1Þð2J 0 þ 1Þð2s þ 1Þð1Þ 1 2 k1 ;k01 ;k2 ;k02 ;j1 ;j01 ;j2 ;j02 ;J;J0

0

0



 ha1 J 1 ; e2 k2 ðj2 ÞJkVka0 J 0 ; e1 k1 ðj1 ÞJiha1 J 1 ; e2 k02 ðj2 ÞJ 0 kVka0 J 0 ; e1 k01 ðj1 ÞJ 0 i 0  ½ð2k1 þ 1Þð2k01 þ 1Þð2k2 þ 1Þð2k02 þ 1Þð2j1 þ 1Þð2j1 þ 1Þ 0 1=2 0 0  ð2j þ 1Þð2j þ 1Þ  ð2J þ 1Þð2J þ 1Þð2K þ 1Þð2K 0 1 2 0 1 þ 1Þ  2 0 k k K  1 1 k1 0 0 0 9 8 98   < J K 0 J =< k0 K k1 k1 = 0 0 1 k2 k02 K k2 0 0   j1 K 0 j1 s K s1 s 0 0 0 ; : 0 ;: 0 0 98 J K J 9 j1 K 0 j1 8 0 < k2 K k2 k2 =< J 1 K 1 J 1 = 0 ð6Þ j K0 j ;  s K s2 s ;: 20 1 2 ; : 0 J K J j2 K 01 j2 0

A ¼ ða2 J 2 kQ ðkÞ ka1 J 1 Þða2 J 2 kQ ðk Þ ka1 J 1 Þ

8  1=2 > < J1 ð2K 1 þ 1Þð2J 2 þ 1Þð2k þ 1Þ  k > 2K 2 þ 1 : J2

K1 K 0r K2

9 J1 > = 0 k : > ; J2

Q0kq ¼ r k C ðkÞ q

ð9Þ

and, for the magnetic multipole transition ðMkÞðp ¼ 1Þ, it is

Q1kq ¼

 iðkÞ h iðkÞ  iq 1 h ðk1Þ  Lð1Þ þ C ðk1Þ  Sð1Þ ½kð2k  1Þ1=2 r k1 C : q q c kþ1 ð10Þ

3. Applications

K s2 s1 ^ ^  T K N s1 ðs; s; m0 jsÞ  T N s2 ðs; s; m1 jsÞ; r dW K 1 N1 ða1 J 1 ! a2 J 2 M2 ^q k0 Þ

For the electrical multipole (Ek) transitions, p ¼ 0, and the transition operator in (8) is [10]

Here L and S are the operators of the orbital and spin angular momentum, respectively, C qðkÞ is the operator of the spherical function normalized to ½4p=ð2k þ 1Þ1=2 . The expression (2) together with (3), (4), (6) and (7) for the twostep process (1) is very general. It describes the polarization state of all particles taking part in the process and the angular distribution of the reaction products. It can be used for the derivation of some special cases applicable for specific experimental conditions with smaller number of polarization states specified.

ex K 1 N1 ð 0 J 0 M 0 p1 m1



k X k þ 11=2 i ha2 J 2 jQpk ja1 J 1 i: k ð2k  1Þ!! p¼0;1

dXe

K 1 N1



In (6) and (7), the braces denote 9j-coefficient. The reduced matrix element in (7) is expressed [10]:

In the case of electric dipole electromagnetic radiation 0 (k ¼ k ¼ 1), the cross section describing angular distribution of linearly polarized radiation which follows excitation of non-polarized atoms by non-polarized electrons can easily be obtained from the general expression (2) (K 0 ¼ N 0 ¼ K s1 ¼ N s1 ¼ K k2 ¼ N k2 ¼ K s2 ¼ N s2 ¼ K 2 ¼ N 2 ¼ N k1 ¼ N 1 ¼ 0, K 1 ¼ K r ¼ 2):

rða0 J0 ! a1 J1 ! a2 J2 ^q k01 Þ dXf ¼

2ð2J 0 þ 1Þ Z X  dXe

2

d

rða0 J0 M0 p1 m1 ! a1 J1 p2 m2 ! a2 J2 M2 ^q k01 Þ dXe dXf

M 0 ;M 2 ;m1 ;m2

rða0 J0 ! a1 J1 ! a2 J2 Þ ¼ ½1 þ bP2 ðcos hÞ: 4p

ð11Þ

Here the laboratory z axis coincides with the direction of the electron projectile, and h is the angle of emitted radiation with respect to the same axis, and P 2 ðcos hÞ is the Legendre polynomial. In (11),

b ¼ A2 ar

ð12Þ

is the asymmetry parameter of the angular distribution of emitted fluorescence. Here the first factor is the alignment of the non-polarized atoms excited by non-polarized electrons [12] along the direction of projectile electrons

A2 ¼

5Bex ð0; 2; 2; 0; 2; 0; 0; 0; 2Þ ; B ex ð0; 0; 0; 0; 0; 0; 0; 0; 0Þ

ð13Þ

and the second factor describes the asymmetry of spontaneous decay of the excited state of the atom

ar ¼ ð7Þ

1



1=2  1 3ð2J 1 þ 1Þ ð1ÞJ1 þJ2 þ1 2 J1

1

2

J1

J2

 ;

ð14Þ

where the braces f. . .g denote the 6j-coefficient. In the case of linearly polarized electric dipole radiation, the polarization degree is defined as follows: P ¼ 3b=ðb  2Þ.

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The alignment parameters A2 , the asymmetry parameter b and its polarization for the state np5 ðn þ 1Þs2 2 P3=2 of Na (n ¼ 2) and K (n ¼ 3) which are excited by non-polarized electrons were calculated by using our own computer programs to implement the general expression of the present work. The radial wave functions PðekjrÞ of the partial waves are numerical solutions of Hartree– Fock (HF) equations in the frozen field of the atom in the ground and excited states for the projectile and scattered electrons, respectively. The radial wave functions for discrete atomic states were obtained by using computer code by Froese Fischer [16]. The contribution of the exchange terms decreased rapidly with the increasing orbital momentum k, and at k ¼ 10 it became negligible. The transition amplitude was calculated as a sum of the direct and exchange terms of the electrostatic interaction operator. They are of opposite signs. The contribution of large lambda rapidly increased with the energy of projectile and scattered electron. The number of partial waves exceeds 100 at the electron energy 300 eV. The alignment parameters calculated in DW with exchange and in the plane wave Born (PWB) approximations are shown in Figs. 1 and 2 together with experimental data for Na measured for impact electron energies from the threshold to 80 eV [3] and for K measured in the range of incident electron energy 31.4–500 eV [18], respectively. Theoretical R-matrix and DW method results are also presented for Na [3] and K [18], respectively. For the excitation energies exceeding the threshold twice, the calculated alignment parameters for Na in the 2p5 3 s2 2 P3=2 state (Fig. 1) are in good agreement with experimental data [3] and values calculated by applying R-matrix approach [3]. Significant deviations of our alignment parameters can be noticed close to the excitation threshold indicating the importance of the correlations in the continuum. The results in Fig. 2 show that DW alignment parameters of 3 p5 4s2 2 P3=2 state of K excited by electron impact are in good agreement with experimentally determined ones [18] for the energies of the projectile electron larger than 70 eV. For comparison, two cases of calculated data from [18] are presented. DW-6 was obtained by using the optical potential V opt ¼ V st ðrÞ þ V ex ðE; rÞ and the ground state electron density. Here V st ðrÞ is the static potential of K, and V ex ðE; rÞ is the local dynamical exchange potential [17]. DW-7 was obtained with the same potential V opt , but the electron density was that of excited K. The results in Fig. 2 demonstrate that the use of the optical potential enables one to achieve better agreement between theoretical and experimental values of the alignment parameters than HF calculation in the region of incident electron energies from the threshold up to 70 eV. The PWB approx-

Fig. 2. The calculated DW and PWB alignment parameters of 3 p6 4s ! 3p5 4s2 2 P3=2 transition for K excited by non-polarized electrons. Experimental (squares) and theoretical (DW-6 and DW-7) data are taken from Matterstock et al. [18].

Table 1 The asymmetry parameter b of the angular distribution and polarization degree P of linearly polarized radiation following the excitation of Na and K atoms to np5 ðn þ 1Þs2 2 P3=2 state, calculated in the frame of DW with exchange and PWB approximations. E (eV)

Na ðn ¼ 3Þ

DW 20 26 31 40 50 100 200

0.24 0.15 0.15 0.16 0.09

K ðn ¼ 4Þ P

b PWB

0.49 0.40 0.35 0.20 0.10

DW

0.32 0.21 0.21 0.22 0.12

P

b PWB

DW

PWB

DW

PWB

0.59 0.50 0.32 0.27 0.14

0.09 0.12 0.14 0.16 0.16 0.12 0.06

0.47 0.40 0.35 0.30 0.25 0.14 0.06

0.13 0.17 0.19 0.22 0.22 0.17 0.09

0.58 0.50 0.45 0.38 0.33 0.20 0.09

imation can be used to calculate the alignment parameters for Na and K atoms only for very large incident electron energies. i.e. larger than 150 eV. The asymmetry parameter b as well as the polarization degree P of linearly polarized fluorescence radiation from 2 p5 3s2 2 P3=2 state of Na and 3 p5 4s2 2 P3=2 state of K exited by non-polarized electrons are presented in Table 1. They are calculated in DW with exchange and PWB approximations. The results show that both parameters for Na obtain the largest values at the threshold of excitation and decrease with increasing energy of exciting electrons. The results of PWB approximation are larger than those of DW approach, and the parameters decrease more slowly. In the case of K atoms, both the asymmetry parameter and polarization degree calculated in DW approximation achieve the maximum at about 50 eV electron impact energy whereas the maximum values of PWB approximation are at the excitation threshold. However, the values of b and P merge for higher electron impact energies, in full similarity with the case of Na. 4. Concluding remarks

Fig. 1. The DW (solid line) and PWB (long-dashed line) alignment parameters of 2p6 3s ! 2p5 3s2 2 P3=2 state for Na excited by non-polarized electrons. Experimental data (circles) and R-matrix (dashed line) calculations are taken from GrumGrzhimailo et al. [3].

In two-step approximation, the general expression for the cross section describing the fluorescence radiation following the excitation of polarized atoms by polarized electrons in DW approximation is obtained for the first time. Simpler formula for the cross sections describing the specific experimental conditions can be easy derived as a special case of the general expression. To demonstrate the approach, we have derived the cross section describing the angular distribution of fluorescence radiation in the case of

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the excitation of non-polarized atoms by non-polarized electrons is presented. The alignment parameters for Na and K atoms excited by electron impact to 2p5 3s2 2 P3=2 and 3p5 4s2 2 P3=2 state, respectively, were calculated by using the program implementing the general expressions (2)–(4). Their values are in good agreement with experimental data in the case of projectile electron energies larger than 70 eV. Acknowledgement The study was partially funded by the Joint Taiwan-Baltic Research Project, BalticGrid-II and LitGrid Projects. References [1] W. Mandl, R.C. Wolf, M. von Hellermann, H.P. Summers, Plasma Phys. Contr. Fusion 35 (1993) 1373. [2] S.C. McFarlane, J. Phys. B 5 (1972) 1908.

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