7 A general expression for the excitation cross section 0 0 of polarized atoms by polarized electrons 2 n a J A Kupliauskiene˙ 5 Institute of Theoretical Physics and Astronomy of Vilnius University, A.Goˇstauto 12, 2 01108 Vilnius, Lithuania ] h E-mail: [email protected] p - m Abstract. The general expression for excitation cross section of polarized atoms by o polarized electrons is derived by using the methods of the theory of an atom adapted t for polarization. The special cases of the general expression for the description of the a . angular distribution and alignment of excited atoms in the case of polarized and non- s c polarizedatoms as well as the magnetic dichroism of the total ionization cross section i s of polarized atoms are obtained. The cross sections and alignment parameters for the hy excitation of the autoionizing states 2p53s2 2P3/2 for Na and 3p54s2 2P3/2 for K are p calculated in distorted wave with exchange approximation. [ 1 v PACS numbers: 34.80.Df, 31.50.df, 29.25.Pj 3 9 2 1 0 7 0 / s c i s y h p : v i X r a Excitation cross section of polarized atoms by polarized electrons 2 1. Introduction Polarized atoms and ions presents in plasma where directed flows of charged particles takeplace. Thedirected movement oftheseparticlesisveryimportantinlaboratoryand astrophysical plasmas resulting in the distortion of Maxwellian distribution of electrons [1]. The non-equilibrium population of magnetic sub-levels or the ordering of angular momentum of atomic particles that is called a self-alignment is causes the polarization of the emitted electromagnetic fluorescence radiation that could be used for the fusion plasma diagnostics [2]. Recently, the methods of the theory of an atom were applied for the derivation of the general expressions describing the interaction of polarized photons and electrons with polarized atoms and ions [3-9]. The probability (cross section) of the interaction was expressed as multiple expansion over the multipoles (irreducible tensors) of the state of all particles taking part in the process both in initial and final states. The applied approachwas analternative tothe density matrix method[10]where thedensity matrix elements were expressed via multipoles or statistical tensors. The density matrix formalismwas used forthestudy of somespecial cases of thepolarizationof theparticles presented in the excitation process. These expressions were applied for the calculations of the alignment of autoionizing states of alkaline atoms excited by electrons [11, 12, 13], positrons [11] or other charged particles [12] The expressions for the description of the polarization and angular distribution of the radiation from unpolarized atoms excited by polarized electrons were derived by Bartschat et al [14] in a general case. These expressions were applied for the calculation of the electron impact excitation of the 4 1P state of calcium [15, 16]. 1 The main task of the present work was the derivation of a general expression describing the polarization state of all particles taking part in the excitation of polarized atoms by polarized electrons with the help of the method based on the theory of an atom [3]. The next section of the present work is devoted to obtaining of the general expression. Its special cases as well the calculations of total cross section and alignment parameters are presented in Section 3. The inequality fine structure splitting line width hyperfine structure splitting ≫ ≫ is also assumed. The modifications enabling to take into account hyperfine structure splitting can be easily made [3, 5]. 2. General expression − For the excitation of an atom A in the state α J M by an electron e moving with the 0 0 0 momentum p and spin projection m 1 1 − − A(α J M )+e (p m ) A(α J M )+e (p m ), (1) 0 0 0 1 1 → 1 1 1 2 2 Excitation cross section of polarized atoms by polarized electrons 3 the cross section can be written in atomic system of units as follows: dσ(α J M p m α J M p m ) p 0 0 0 1 1 → 1 1 1 2 2 = 2 α J M p m V α J M p m dΩ (2π)2p h 1 1 1 2 2| | 0 0 0 1 1i 2 1 ∗ α J M p m V α J M p m δ(E E ). (2) 1 1 1 2 2 0 0 0 1 1 0 1 ×h | | i − Here V is the operator of the electrostatic interaction between projectile and atomic electrons, E and E are the energies of the system atom+electron in the initial and 0 1 final states, p is the absolute value of the momentum,p = √2ε , ε is the energy of i i i i the projectile electron in the initial (i = 1) and final (i = 2) states, α J M and p m 2 2 2 2 2 describe the states of the excited atom and scattered electron, respectively. The wave function of the projectile and scattered electrons can be expressed via expansion over Spherical harmonics ∗ pm = 4π R (r)Y (rˆ)Y (pˆ)ξ (σ) | i λ λµ λµ m λµ X = 4π(2λ+1)R (r)C(λ)(rˆ)Y∗ (pˆ)ξ (σ). (3) λ µ λµ m Xλµ q Here ξ (σ) is the spin orbital of an electron, C(λ)(rˆ) is the operator of the spherical m µ function [17], where rˆdenotes the polar and azimutal angles of the spherical coordinate system, R∗(r) = iλexp[i(σ (p)+δ )]r−1P(ελ r) (4) λ λ λ | ′ is the radial orbital of the electron in a continuum state normalized to δ(ε ε). For the − electron moving in the field of an ion, the asymptotic of the Hartree orbital P(ελ r) is | P(ελ r ) (πp)−1/2sin(pr λπ/2+Z ln(2pr)/p+σ (p)+δ ) (5) ef λ λ | → ∞ ∼ − with Coulomb phase σ (p) = arg Γ(λ+1 i(Z 1)/p) (6) λ ef − − and effective nuclear charge Z = Z N + 1. Here Z is the nuclear charge, N is ef − the number of electrons. In (5), δ is the phase arising due to the deviation of the λ self consistent field from Coulomb one. In the case of neutral atom, the asymptotic expression of Hartree orbital obtains the following expression: P(ελ r ) (πp)−1/2sin(pr λπ/2+δ ). (7) λ | → ∞ ∼ − The substitution of (3) into (2) leads to the following expression for one transition matrix element: α J M p m H α J M p m = [(2λ +1)(2λ +1)]1/2 1 1 1 1 1 0 0 0 1 1 1 2 h | | i M˜ ,M˜X,m˜ ,m˜ , 0 1 1 2 λ ,µ ,λ ,µ 1 1 2 2 α J M˜ ε λ µ m˜ H α J M˜ ε λ µ m˜ DJ0 (Jˆ) D∗J1 (Jˆ) Dλ1 (pˆ ) ×h 1 1 1 1 1 2 1| | 0 0 0 1 1 1 1i M˜0M0 0 M˜1M1 1 µ10 1 Excitation cross section of polarized atoms by polarized electrons 4 D∗λ2(pˆ ) Ds (sˆ) D∗s (sˆ). (8) × µ20 2 m˜1m1 m˜2m2 Here the possibility of the registration of the orientation of angular momentum of all particles with respect to different quantization axes is taken into account. For the evaluation of the matrix element, the wave functions of all particles taking part in the process were transformed to the same system of coordinates by using the transformation procedure jm˜ = Dj (α,β,γ) jm . (9) | i mm˜ | i m X Here Dj (α,β,γ) is the Wigner rotation matrix. mm˜ The integration over orbital and summation over spin variables was performed with the help of the graphical technique of the angular momentum [17, 18]. The angular momentum diagrams used for the derivation of the general expression of the excitation cross section of polarized atoms by polarized electrons are represented in figures 1 and 2. The angular part of the matrix element (8) is represented in diagram E (see figure 1). 1 Here the rectangle with (kk) indicates the orbital and spin parts of the electrostatic interaction operator which orbital part is defined as follows: rk V = < (C(k)C(k)). rk+1 1 2 k > X Hereand1and2showtheatomicandprojectileelectrons, respectively. Otherrectangles in diagram E represent the orbital and spin parts of the configuration and other 1 quantumnumbers. ThecircleswithD inside areWignerrotationmatrices. Theelectron was excited from the shell which orbital quantum number is l to the shell which orbital 0 quantum number is l , and s = 1/2. 1 To extract the reduced matrix element from diagram E , one needs to cut of the 1 Wigner rotation matrices from the open lines of each angular momentum (circles with D) and to choose the order of the coupling of open lines. Let us choose the following order of coupling: J ,λ s(j )J and J ,λ s(j )J. Then the procedure of the extraction 0 1 1 1 2 2 of the reduced matrix element may be written by using the diagrams from figure 1 ′ E = (2J +1) E E , (10) 1 2 3 j1X,j2,J where E is the angular part of the diagram of reduced matrix element 2 ′ α J ,ε λ (j )J H α J ,ε λ (j )J and E is the generalized Clebsch-Gordan h 1 1 2 2 2 || || 0 0 1 1 1 i 3 coefficient [17] represented by the left side of diagram E . The right side of diagram E 3 3 comes from the angular part of the complex conjugate matrix element from (2). For further simplification of the part describing the space rotation dependence, the following expansion can be used: ′ DJ (Jˆ) D∗J′ (Jˆ) = T∗K(Jˆ) J K J . (11) M˜M M˜′M N M˜′ N M˜ K,N (cid:20) (cid:21) X Excitation cross section of polarized atoms by polarized electrons 5 Figure 1 The angular momentum diagrams used for the derivation of the general expression of the excitation cross section of polarized atoms by polarized electrons. In(11), the angle brackets with angular momenta inside show the Clebsch-Gordan coefficient [17], and the tensor is defined [8] as T∗K(Jˆ) = ( 1)J′−M 4π 1/2 J J′ K Y∗ (θ,φ). (12) N − 2J +1 M M 0 KN (cid:20) (cid:21) (cid:20) − (cid:21) Excitation cross section of polarized atoms by polarized electrons 6 Figure 2 The angular momentum diagrams used for the derivation of the general expression of the excitation cross section of polarized atoms by polarized electrons. Six Clebsch-Gordan coefficients obtained by applying (12) for all angular momenta λ ,λ ,J ,J andtwo spinssoffreeelectrons areusedtoperformthesummationover the 1 2 0 1 projections M˜ ,M˜′,M˜ ,M˜′, µ˜ ,µ˜′,µ˜ ,µ˜′,m˜ ,m˜′,m˜ ,m˜′ of the matrix element in (8) 0 0 1 1 1 1 1 1 1 1 2 2 and its complex conjugate. This summation was performed graphically, and the result is represented by diagram E (see figure 2) that looks rather complicated. But further 4 simplifications of E are still possible. The closing of the open lines from diagram E 4 4 produces diagram E invariant under space rotation and diagram E describing rotation 5 6 Excitation cross section of polarized atoms by polarized electrons 7 properties of the cross section (2). The bow at the end of each open line stands for the tensor (12) [5] in the case of J ,J , spin s and spherical function for λ , and λ . 0 1 1 2 Four 9j-coefficients [17] can be obtained from diagram E by cutting it trough the lines 5 ′ ′ ′ ′ ′ (J,K,J ),(j ,K ,j ) and (j ,K ,j ). 1 0 1 2 1 2 The final expression for the cross section (2) obtained by using diagrams E , E 2 5 and E is as follows: 6 dσ(α J M p m α J M p m ) 0 0 0 1 1 1 1 1 2 2 → dΩ = 4πC ex(K ,K′,K ,K′,K ,K ,K ,K ,K) B 0 0 1 1 λ1 s1 λ2 s2 ′ K,K ,K ,K ,K ,K 0 ′ 0Xλ1 s1 1 K ,K ,K 1 λ2 s2 ′ ′ ′ K K K K K K K K K λ1 s1 0′ 0 0′ 1 1′ × N ,N′,N ,N ,N (cid:20) Nλ1 Ns1 N0 (cid:21)(cid:20) N0 N0 N (cid:21)(cid:20) N1 N1 N (cid:21) 0 ′ 0 Xλ1 s1 1 N ,N ,N ,N 1 λ2 s2 ′ ×(cid:20) KNλλ22 KNss22 KN11′ (cid:21)YK∗λ1Nλ1(pˆ1) YKλ2Nλ2(pˆ2) TN∗K00(J0,J0,M0|Jˆ0) TNK11(J1,J1,M1|Jˆ1) T∗Ks1(s,s,m sˆ) TKs2(s,s,m sˆ), (13) × Ns1 1| Ns2 2| ex(K ,K′,K ,K′,K ,K ,K ,K ,K) B 0 0 1 1 λ1 s1 λ2 s2 = (2J +1)(2J′ +1)(2s+1)( 1)λ′1+λ′2 − λ1,λ′1,λ2,λ′2X,j1,j1′,j2,j2′,J,J′ ′ ′ ′ ′ ′ ′ ∗ α J ,ε λ (j )J H α J ,ε λ (j )J α J ,ε λ (j )J H α J ,ε λ (j )J ×h 1 1 2 2 2 || || 0 0 1 1 1 ih 1 1 2 2 2 || || 0 0 1 1 1 i ′ ′ ′ ′ [(2λ +1)(2λ +1)(2λ +1)(2λ +1)(2j +1)(2j +1)(2j +1)(2j +1) × 1 1 2 2 1 1 2 2 ′ ′ (2J +1)(2J +1)(2K′ +1)(2K′ +1)]1/2 λ1 λ1 Kλ1 λ2 λ2 Kλ2 × 0 1 0 1 0 0 0 0 0 0 (cid:20) (cid:21)(cid:20) (cid:21) J K J λ′ K λ λ′ K λ J K J ′0 0′ 0 1 λ1 1 2 λ2 2 ′1 1′ 1 j K j s K s s K s j K j .(14) × J1′ K0 J1  j′ Ks′1 j  j′ Ks′2 j  J2′ K1 J2    1 0 1  2 1 2   Taking into account the equation (3) and asymptotics (5) and (6), the constant in (2)      is C = 4/p2. 1 The expression (13) represents the most general case of the cross section describing the excitation of polarized atoms by polarized electrons and enabling one to obtain information on the angular distributions and spin polarization of scattered electron and the alignment of excited atom in non relativistic approximation. 3. Special cases 3.1. Total cross section for the excitation of unpolarized atoms In the case when the polarization state of excited atoms and scattered electrons are not registered, the total cross section describing the excitation of unpolarized atoms by unpolarized electrons can be obtained from the general expression (13) by summation Excitation cross section of polarized atoms by polarized electrons 8 over the magnetic components of the particles in the final state and averaging over them in the initial state as well as integration over the angles of scattered electron. The expression under consideration was obtained by applying the formulas [3] TK(J,J,M Jˆ) = δ(K,0)δ(N,0), (15) N | M X π 2π sinθdθdφY (θ,φ) = √4πδ(K,0)δ(N,0) (16) KN Z0 Z0 and is as follows: 1 dσ(α J M p m α J M p m ) 0 0 0 1 1 1 1 1 2 2 σ(α J α J ) = dΩ → 0 0 1 1 → 2(2J +1) dΩ 0 Z M0,mX1,M1,m2 4π = ex(0,0,0,0,0,0,0,0,0). (17) (2J +1)ε B 0 1 Here ex(0,0,0,0,0,0,0,0,0) B = (2J +1) α J ,ε λ (j )J H α J ,ε λ (j )J 2. (18) 1 1 2 2 2 0 0 1 1 1 |h || || i| λ1,j1X,λ2,j2,J 3.2. The angular distribution of scattered electrons following the excitation of unpolarized atoms Theexcitationofunpolarizedatomsby unpolarizedelectronsisusualandoftenoccurred process in plasmas. To obtain the expression of the differential cross section suitable for the description of the angular distribution of scattered electrons one needs to performer the summation of the general expression (13) over the magnetic components of the particles in the final state and averaging over them in the initial state. The application of the expression (15) and the choice of the laboratory quantization axes along the direction of the projectile electron, that means Y (0,0) = [(2K + 1)/4π]1/2δ(N,0), KN leads to the expression dσ(α J α J p ) 1 dσ(α J M p m α J M p m ) 0 0 1 1 2 0 0 0 1 1 1 1 1 2 2 → = → dΩ 2(2J +1) dΩ 0 M0,mX1,M1,m2 σ(α J α J ) 0 0 1 1 = → 1+ β P (cosθ) . (19) K K 4π " # K>0 X Here the asymmetry parameters of the angular distribution of the scattered electrons are defined as follows: (2K +1) ex(0,K,0,K,K,0,K,0,K) β = B . (20) K ex(0,0,0,0,0,0,0,0,0) B ′ In (19), the summation parameter K can acquire the values max λ λ , λ {| 1 − 1| | 2 − ′ ′ ′ λ K min λ +λ ,λ +λ for each set of the partial wave momenta which can 2|} ≤ ≤ { 1 1 2 2} be very large depending on the energy of the projectile electron. Several partial waves are enough only for the energies close to the excitation threshold. Excitation cross section of polarized atoms by polarized electrons 9 3.3. The angular distribution of scattered electrons following the excitation of polarized atoms In the case of the atoms prepared in polarized state, the expression for the differential cross section describing the angular distribution of the scattered electrons can be also obtained by the summation over the magnetic components of the particles in the final stateandaveragingoverthestatesofthespinintheinitialstateinthegeneralexpression (13). In the case of the choice of the laboratory quantization axes along the direction of the projectile electron, the expression for the cross section is as follows: dσ(α J M α J p ) 1 dσ(α J M p m α J M p m ) 0 0 0 1 1 2 0 0 0 1 1 1 1 1 2 2 → = → dΩ 2 dΩ m1,XM1,m2 C√4π = [2K +1]1/2 ex(K ,K ,0,K ,K ,0,K ,0,K ) 2 λ1 B 0 λ1 λ2 λ1 λ2 λ2 Kλ1,KX0,Kλ2,N0 K0 Kλ1 Kλ2 Y (pˆ ) 4π 1/2( 1)J0−M0 J0 J0 K0 × N0 0 N0 Kλ2N0 2 2J +1 − M0 M0 0 (cid:20) (cid:21) (cid:20) 0 (cid:21) (cid:20) − (cid:21) Y∗ (Jˆ). (21) × K0N0 0 This expression becomes more simple in the case of special geometry of the experiment. If the atom is polarized along the direction of the projectile electron, ∗ then N = 0, M = J , Y (0,0) = (2K +1)/4πδ(N ,0), Y (pˆ ) = 0 0 0 K0N0 0 0 Kλ20 2 (2K +1)/4πP (cosθ), and the angle θ isqmeasured from the direction of the λ2 Kλ2 qprojectile electron. Then the expression (21) transforms into the following expression: dσ(α J M =J α J p ) σ(α J α J ) 0 0 0 0 1 1 2 0 0 1 1 → = → 1+ B P (cosθ) .(22) dΩ 4π  Kλ2 Kλ2  KXλ2>0   Here B = ex(0,0,0,0,0,0,0,0,0)−1 ex(K ,K ,0,K ,K ,0,K ,0,K ) Kλ2 B B 0 λ1 λ2 λ1 λ2 λ2 K0X,Kλ1 K K K J J K [(2J +1)(2K +1)(2K +1)(2K +1)]1/2 0 λ1 λ2 0 0 0 (23) × 0 0 λ1 λ2 0 0 0 J0 J0 0 (cid:20) (cid:21)(cid:20) − (cid:21) is one from the set of the asymmetry parameters of the angular distribution of the scattered electrons. The difference of the expression (23) and that with opposite directions of J is equal to the magnetic dichroism in the electron-impact excitation 0 cross section of polarized atoms describing the angular distribution of the scattered electrons. 3.4. Magnetic dichroism in the total electron-impact excitation cross section of polarized atoms The total cross section of the excitation of polarized atoms by unpolarized electrons can be easily obtained by integration of (21) over the angles of the scattered electrons. Excitation cross section of polarized atoms by polarized electrons 10 Then K = N = 0, K = K . This cross section depends on the direction of the λ2 λ2 0 λ2 polarization of atoms and is as follows: σ(α J M p α J ) = 2π C 1 ( 1)K0+J0−M0 J0 J0 K0 0 0 0 1 → 1 1 [(2J +1)(2K +1)]1/2 − M0 M0 0 KX0,N0 0 0 (cid:20) (cid:21) ex(K ,K ,0,0,K ,0,0,0,0)Y (pˆ )Y∗ (Jˆ). (24) ×B 0 0 0 K0N0 1 K0N0 0 In the case of the choice of the quantization axis z along the direction of the projectile electron, the expression (24) becomes more simple: σ(α J M α J ) = 2π C ( 1)K0+J0−M0 2K0 +1 1/2 J0 J0 K0 0 0 0 → 1 1 − 2J +1 J0 J0 0 XK0 (cid:20) 0 (cid:21) (cid:20) − (cid:21) ex(K ,K ,0,0,K ,0,0,0,0)P (cosθ), (25) ×B 0 0 0 K0 where the angle θ of the orientation of the total angular momentum J of an atom is 0 measured from the direction of the projectile electron. The degree of magnetic dichroism can be defined by the parameter a which equals to σ(α J M α J ) σ(α J M α J ) 0 0 0 1 1 0 0 0 1 1 a = → − − → σ(α J M α J )+σ(α J M α J ) 0 0 0 1 1 0 0 0 1 1 → − → B(K )P (cosθ) = K0=odd 0 K0 , (26) B(K )P (cosθ) PK0=even 0 K0 where P J J K B(K) = ( 1)K√2K +1 0 0 ex(K,K,0,0,K,0,0,0,0). (27) J J 0 − 0 0 B (cid:20) − (cid:21) Here the values of the summation parameter are K 2J . If the total angular 0 0 ≤ momentum of an atom J is directed along and opposite directions of the projectile 0 electron, then M = J , P (0) = 1, and the parameter of the diamagnetic dichroism is 0 0 K0 as follows: B(K ) a = K0=odd 0 . (28) B(K ) PK0=even 0 For small valuPes of the total angular momentum of an atom, the expression of the magnetic dichroism is presented as an example. In the case of J = 1/2, it is as follows: 0 ex(1,1,0,0,1,0,0,0,0) a = √3B . (29) − ex(0,0,0,0,0,0,0,0,0) B For J = 1, the magnetic dichroism is defined by the expression: 0 (3/√2) ex(1,1,0,0,1,0,0,0,0) a = − B . (30) ex(0,0,0,0,0,0,0,0,0)+ 5/2 ex(2,2,0,0,2,0,0,0,0) B B q