Scattering polarization in the presence of magnetic and electric fields

Journal of Quantitative Spectroscopy &

Radiative Transfer 108 (2007) 161–179

Scattering polarization in the presence of magnetic and electric fields

Yee Yee Oo

, M. Sampoorna

, K.N. Nagendra

, Sharath Ananthamurthy

, G. Ramachandran

aDepartment of Physics, Mandalay University, Mandalay, Myanmar

bIndian Institute of Astrophysics, Bangalore 560 034, India

cDepartment of Physics, Bangalore University, Bangalore 560 056, India

dJoint Astronomy Program, Department of Physics, IISc, Bangalore 560 012, India Received 6 February 2007; received in revised form 24 April 2007; accepted 29 April 2007

Abstract

The polarization of radiation by scattering on an atom embedded in combined external quadrupole electric and uniform magnetic fields is studied theoretically. Limiting cases of scattering under Zeeman effect, and Hanle effect in weak magnetic fields are discussed. The theory is general enough to handle scattering in intermediate magnetic fields (Hanle–Zeeman effect) and for arbitrary orientation of magnetic field. The quadrupolar electric field produces asymmetric line shifts, and causes interesting level-crossing phenomena either in the absence of an ambient magnetic field, or in its presence. It is shown that the quadrupolar electric field produces an additional depolarization in the Q=I profiles and rotation of the plane of polarization in the U=I profile over and above that arising from magnetic field itself. This characteristic may have a diagnostic potential to detect steady-state and time-varying electric fields that surround radiating atoms in solar atmospheric layers.

r2007 Elsevier Ltd. All rights reserved.

Keywords:Atomic processes; Polarization; Scattering; Magnetic field; Line profiles

1. Introduction

Scattering of polarized radiation by an atom is a topic of considerable interest to astrophysics ever since Hale [1] first observed polarization related to Zeeman effect in spectral lines originating in Sun spots. The polarized radiation is usually expressed in terms of the Stokes parameters. The concept of scattering matrix connecting the Stokes vector S⁰ of incident radiation to the Stokes vector S of scattered radiation was introduced quite early in the context of Rayleigh scattering[2]. Polarized radiation in spectral lines formed in the presence of an external magnetic field has been studied widely and a comprehensive theoretical framework has been developed [3–15]. The Hanle effect is a depolarizing phenomenon which arises due to ‘partially

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doi:10.1016/j.jqsrt.2007.04.009

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E-mail address:gr@iiap.res.in (G. Ramachandran).

overlapping’ magnetic substates in the presence of weak magnetic fields, when the splitting produced is of the same order or less than the natural widths. Favati et al.[16] proposed the name ‘second Hanle effect’ for a similar effect in ‘electrostatic fields’. Casini and Landi Degl’Innocenti[17]have discussed the problem in the presence of electric and magnetic fields for the particular case of hydrogen Lymanaline. It was followed by a more recent paper by Casini[18]. The relative contributions of static external electric fields, motional electric fields and magnetic fields in the case of hydrogen Balmer lines, have been studied by Brillant et al.[19]. A historical perspective and extensive references to earlier literature on polarized line scattering can be found in Stenflo[12], Trujillo Bueno et al.[20]and Landi Degl’Innocenti and Landolfi[21].

A quantum electrodynamic theory of Hanle–Zeeman redistribution matrices has been developed by Bommier[10,11]and Landi Degl’Innocenti and co-workers (see the book by[21]). The formulation presented in[10,11]includes the effects of partial frequency redistribution (PRD) in line scattering for a two-level atom.

It is a perturbation theory, in which PRD effects appear in the fourth order. The theory presented in[21]and references cited therein, considers only complete frequency redistribution (CRD) in line scattering.

A classical theory of line scattering PRD for the Hanle–Zeeman effect has been formulated by Bommier and Stenflo[15]. This theory is non-perturbative and describes the scattering process in a transparent way. The classical theory for Hanle–Zeeman scattering developed by Stenflo[14]considered only coherent scattering in the laboratory frame. In [15] the redistribution matrices were derived in the atomic rest frame. The corresponding laboratory frame redistribution matrices have been derived in[22]. The equivalence between the classical (non-perturbative theory) and quantum electrodynamic (perturbative theory) redistribution matrices for the triplet case is established in[23]. In all these papers only the dipole type line scattering transitions in the presence of pure magnetic fields is considered. Taking into account all higher order multipoles as well, polarization of line radiation in the presence of external electric quadrupole and uniform magnetic fields was studied[24,25], where scattering of radiation by atoms, however, was not considered.

The purpose of the present paper is to develop a quantum electrodynamical approach to scattering processes in the presence of external electric and magnetic fields of ‘arbitrary strengths’, taking also into consideration all other multipole type transitions apart from the usually dominant electric dipole transition.

The atomic electron is represented using non-relativistic quantum theory including spin. The radiation field is described in terms of its electric and magnetic multipole states, in a second quantized formalism. The external electric field is assumed to be ‘quadrupolar’ in nature, while the magnetic field is uniform and arbitrarily oriented with reference to the principal axes frame (PAF) of the electric quadrupole field. This general formalism can be employed also to solve the scattering problems involving linear steady-state electric fields at the radiating atom.

In Section 2 we describe the theoretical formulation. In Section 3 the scattering matrix for the general physical situation is derived. The particular case of the dipole transitions for a triplet is also considered, for the purpose of comparison with Stenflo[14]in the pure magnetic field limit. Section 4 contains numerical results and discussions. Conclusions are presented in Section 5.

2. Theoretical formalism for scattering

We consider polarized radiation incident on an atom along an arbitrary direction ðy⁰;f⁰Þ and getting scattered into a direction ðy;fÞ with respect to a conveniently chosen right-handed Cartesian coordinate system, referred to as the astrophysical reference frame (ARF) and shown asðX;Y;ZÞinFig. 1. Ifn⁰and n denote, respectively, the frequencies of the incident and scattered radiation, we may define wave vectorsk⁰and kwith polar co-ordinatesðk⁰;y⁰;f⁰Þandðk;y;fÞwherek⁰¼2pn⁰¼o⁰andk¼2pn¼oin natural units with _¼1;c¼1 and mass of the electron me¼1. The atom is exposed to an external magnetic field B with strengthB, directed alongðyB;f_BÞand an electric quadrupole field characterized by strengthAand asymmetry parameterZin its PAF, which is denoted byðXQ;YQ;ZQÞinFig. 1. The transformation to PAF from ARF is achieved by a rotationRðaQ;b_Q;g_QÞthrough Euler anglesðaQ;b_Q;g_QÞas defined by Rose[26]. The magnetic fieldB is directed along ðey_B;fe_BÞwith respect to PAF. Following Rose[26], we define left and right circular states of polarization ^m (m¼ 1), respectively, which are mutually orthogonal to each other and to k.

We use here the symbol^m¼1instead ofu^_p¼1employed in Rose[26]. Likewise^⁰_m0¼1, which are orthogonal to

k⁰. Any arbitrary state of polarization ^⁰ of the incident radiation may then be expressed as ^⁰¼ c⁰_þ1^⁰_þ1þc⁰₁^⁰₁ using appropriate coefficients c⁰₁, which are in general complex and satisfy jc⁰_þ1j²þ jc⁰₁j²¼1.We, therefore, denote the orthonormal states of polarized incident radiation by jk⁰;m⁰i, with m⁰¼ 1. We seek the probability for scattering into two polarized states of scattered radiationjk;mi, m¼ 1 on an atom which is initially in a statec_iwith energyE_ibefore scattering and makes a transition to a final statec_f with energyE_f, in the process of scattering of polarized radiation.

2.1. Energy levels of an atom in electric quadrupole and uniform magnetic fields

The energy levels of an electron in an atom are primarily determined by the Hamiltonian

H0¼ ¹₂r²þVðrÞ, (1)

whereVðrÞdenotes its Coulomb interaction with the nucleus. If we start with the Dirac equation[17]and use its non-relativistic reduction, terms like spin–orbit interaction may also be included inH0. In the absence of external fields, the energy levels of the atom are determined by

H^A₀ ¼X^Z

i¼1

H₀ðiÞ þ X^Z

i4j¼1

e²

r_ij, (2)

whereedenotes the charge of the electron,r_ij¼ jr_ir_jjandZdenotes the atomic number. IfEdenotes an energy level and cthe corresponding wave function of the atom with total angular momentum J, it is well known in the context of Zeeman effect that Egets split intoð2Jþ1Þequally spaced levels E_M ¼EþgBM with corresponding energy eigenstatesjJMi_B,M¼J;J1;. . .;Jþ1;J, when the atom is exposed to an external uniform magnetic fieldBwith strengthB. The statesjJMi_Bare defined with the axis of quantization chosen alongBandgdenotes the magnetic-gyro ratio or Lande´g-factor. ForBo100 gauss, whengBis of the same order as the width of a line, Hanle effect [27] takes place. For a line in the optical range, the region 100oBo1000 gauss is generally referred to as the Hanle–Zeeman regime. For Bo1 gauss, one has to pay attention to the interaction of electron with the magnetic and electric moments of the nucleus, which give rise

Y Z

Y_Q

Z_Q B

k′ θB

θB θ

k

θ′

φ φ′

~

X X_Q

Fig. 1. The scattering geometry:ðX_Q;Y_Q;Z_QÞrefers to the principal axes frame (PAF) characterizing the electric quadrupole field. The radiation is incident alongðy⁰;f⁰Þand scattered alongðy;fÞwith respect to the astrophysical reference frame (ARF) denoted byðX;Y;ZÞ.

The magnetic fieldB~is oriented alongðeyB;fe_BÞwith reference to PAF andðyB;f_BÞwith reference to the ARF (the azimuthal anglesfe_Band f_Bare not marked in the figure).

to hyperfine splitting[28,29]. If the atom is exposed to an external electric quadrupole field either by itself or in combination withB, the splitting of the energy levels is not, in general, equally spaced [24,25,30,31]and in such scenarios, the atomic Hamiltonian in PAF is given by

H_A¼H^A₀ þgJBþA½2J²_zJ²_xJ²_yþZðJ²_xJ²_yÞ. (3) The split energy levels may be denoted byEs, wherestakes valuess¼1;2;. . .;ð2Jþ1Þstarting from the lowest levelðs¼1Þfor a givenJ. The corresponding energy eigenstates may be denoted byjJ;si, which are expressible as

jJ;si ¼ X^J

M¼J

a^s_MðA;Z;B;ey_B;fe_BÞjJMi_Q; s¼1;2;. . .;ð2Jþ1Þ, (4) wherejJMi_Q are defined with the quantization axis chosen along theZ-axis,ZQ of the PAF. The notationcⁱ_mu was used in[25]forJ ¼1;3=2 to denote the expansion coefficients, without any specified convention for ordering of the levels. We may rewrite Eq. (4) as

jJ;si ¼ X^J

m¼J

c^s_mjJmi, (5)

in terms of thejJmistates, which are defined with respect to theZ-axis of ARF chosen as the quantization axis.

Clearly, c^s_m¼ X^J

M¼J

D^J_mMða_Q;b_Q;g_QÞa^s_MðA;Z;B;ey_B;fe_BÞ, (6) where D^J_mM denotes Wigner’s D—functions defined in [26]. Hence c^s_m depend on aQ;b_Q;g_Q;A;Z;B. If the magnetic field is absent, thec^s_mdepend only onaQ;b_Q;g_Q;A;Zsincea^s_M in that case[24]depend only onAandZ.

It may be noted that the frame of reference employed in[24,25]was PAF itself, i.e.,aQ;b_Q;g_Q¼0; henceyB;f_B was used there instead of theey_B;fe_Bhere and the Euler anglesða_Q;b_Q;g_QÞfind no mention there. It may be added thatD^J_mMð0;0;0Þ ¼d_mM. On the other hand, if the electric quadrupole field is absent and the atom is exposed only to a magnetic fieldBdirected alongðy_B;f_BÞ, it is clear that

c^s_m¼ X^J

M¼J

D^J_mMðf_B;yB;0ÞdM;sJ1, (7)

which reduces to

c^s_m¼ds;Jþmþ1, (8)

if the fieldBis along theZ-axis of ARF itself.

In general, therefore, when the energy levels of an atom are defined throughHAc_n¼Enc_n, the atomic wave functionsc_n are of the form given by Eq. (5), which specialize appropriately tojJMi_B orjJmi if Eq. (7) or Eq. (8) is used instead of Eq. (6). Thus, in general, the complete set of orthonormal energy eigenstates of an atom in a combined external electric quadrupole and uniform magnetic field environment may be denoted by fc_ng, wheren is used as a collective index, which includes the serial numbers_n along with the total angular momentum J_n and all other quantum numbers which may be needed to specify each c_n uniquely. In the presence of a pure magnetic fieldB, the magnetic quantum numberM_n replacess_n through thed-function in Eq. (7). Moreover, ifBis along the Z-axis of ARF itself,s_n gets replaced bym_n through thed-function in Eq. (8).

In general, a summation overnas inP

njc_nihc_nj ¼1, implies a summation with respect tosnas well. This summation over sn may be replaced by a summation with respect to Mn or mn in some particular cases as mentioned above. The initial and final states of the atom before and after scattering are denoted byc_iandc_f. They also belong tofc_ng. We use the short-hand notation

jii ¼ jc_i;k⁰;m⁰i; jfi ¼ jc_f;k;mi. (9)

2.2. Interaction of atom with the radiation field

It is well known that the local minimal coupling, i.e.,cg¯ _ncA_n(with implied summation overn) of the Dirac field c and the electromagnetic field represented by the four potential A_n;n¼1;. . .;4 is the fundamental interaction responsible for all electrodynamical process involving photons and electrons [32,33]. In the interaction representation,candAnsatisfy the free field equations of Dirac and Maxwell, respectively. The quantityc¯ ¼c^yg₄, wherec^ydenotes the hermitian conjugate ofcandg₁;g₂;g₃;g₄are 44 Dirac matrices. To facilitate calculations using the atomic wave functionsfc_ng, we may use the non-relativistic two componental forms ofcandc¯ incnumber theory for electrons, retain the Maxwell field inqnumber theory and represent the interaction of the radiation field in the Coulomb gauge with the atom as

H_int¼e^iHAtX^Z

j¼1

e iAðr_j;tÞ r_jþ1

2s_j ðr_jAðr_j;tÞÞ

e^iHAt, (10)

wheresjdenote the Pauli spin matrices of the electron labeledjlocated atr_jandZdenotes the atomic number.

The quantum field variableAðr;tÞin interaction representation may be expressed as Aðr_j;tÞ ¼ 1

ð2pÞ³⁼²

Z d³k⁰⁰ ffiffiffiffiffiffiffiffi 2o⁰⁰

p X

m⁰⁰

½a_k⁰⁰_m⁰⁰A_k⁰⁰_m⁰⁰ðrÞe^io00tþa^þ_k00

m⁰⁰A_k⁰⁰_m⁰⁰ðrÞe^io00t, (11) whereo⁰⁰¼ jk⁰⁰jand the creation and annihilation operators, denoted bya^þ_kmanda_km, respectively, satisfy the commutation relation

½a_km;a^þ_k0

m⁰ ¼dðkk⁰Þd_mm⁰ (12)

for any pairk;mandk⁰;m⁰ in general, while

A_kmðrÞ ¼^_me^ikr (13)

denotes a cnumber and A_kmðrÞ denotes its complex conjugate. In particular, the operators are also used to generate the initial and final states of radiation in Eq. (9) through

jk⁰m⁰i ¼a^þ_k0m⁰ji₀; hkmj¼₀hja_km, (14)

whereji₀ denotes the vacuum state of the radiation field.

2.3. The scattering process

TheS-matrix for scattering may be defined, as usual[32–34], by S¼ lim

t!1 t0!1

Uðt;t₀Þ, (15)

where the evolution operator satisfies Uðt;t₀Þ ¼1i

Z t t0

dt⁰H_intðt⁰ÞUðt⁰;t₀Þ, (16)

which on iteration leads to the perturbation series S¼1þX¹

N¼1

ðiÞ^N Z 1

1

dt1

Z t1

1

dt2 Z tN1

1

dtNHintðt1Þ HintðtNÞ. (17) SinceH_intðtÞgiven by Eq. (10) is linear inA(see Eq. (11)), the first-orderðN¼1Þterm can contribute to either absorption through the first term in Eq. (11) or to emission through the second term in Eq. (11) and the integral over dt₁, from1 ! 1leads to the respective energy conservation criteria of Bohr. In the scattering problem under consideration, the lowest order (ine) contribution tohfjSjiiis obtained from theN¼2 term, which we may denote as hfjS^ð2Þjii. We introduce P

njc_nihc_nj ¼1 between Hintðt1Þ and Hintðt2Þ, neglect contribution from two photons in the intermediate state and employ the notationjni ¼ jc_niji₀. This leads, on

using Eqs. (11) and (12), to

hnjH_intðt₂Þjii ¼Aniðk⁰;m⁰Þe^{½iðEnEio0Þt2}, (18) hfjH_intðt₁Þjni ¼Efnðk;mÞe^{½iðEfþoEnÞt1}, (19) whereAniðk⁰;m⁰ÞandEfnðk;mÞdenote amplitudes for absorption and emission, involvingAk⁰m⁰ðrjÞandAkmðrjÞ respectively, which are independent of time variable, instead of Aðr_j;tÞ. We may change the variable of integration from t2 to t⁰₂¼t2t1, ranging from 1 !0, associate a width Gn with c_n by introducing a factor expðGnt⁰₂Þ(see[35,36]) and obtain after completing both the integrations, the expression

hfjS^ð2Þjii ¼2pidðE_f þoE_io⁰ÞT_fiðkm;k⁰m⁰Þ, (20) where the on-energy-shellT-matrix element is of the form

Tfiðk;m;k⁰;m⁰Þ ¼X

n

Efnðk;mÞf_nAniðk⁰;m⁰Þ, (21) and the profile function is given by

f_n¼ ðo_nf oiG_nÞ¹; o_nf ¼E_nE_f, (22)

on making use of EnEio⁰¼onio⁰¼onf oby virtue of the energy d-function in Eq. (20). Using Eq. (5) and observing thatfc_ngare completely antisymmetric with respect to the labels 1;2;. . .;j;. . .;Zof the electrons, we have

Aniðk⁰;m⁰Þ ¼ X

m⁰_n;m⁰_i

c^s_mⁿ0 nc^s_mⁱ0

ihJ_nm⁰_njAðk⁰;m⁰ÞjJ_im⁰_ii;

Efnðk;mÞ ¼ X

m⁰_f;m⁰⁰_n

c^s

f

m⁰_fc^s_mⁿ00

nhJfm⁰_fjEðk;mÞjJnm⁰⁰_ni, ð23Þ

where the matrix elements on the right-hand side satisfy

hJ_um_ujAðk;mÞjJ_lm_li ¼ hJ_lm_ljEðk;mÞjJ_um_ui (24) between any pair of lower and upper atomic states and

Eðk;mÞ ¼ Ze ð2pÞ³⁼² ffiffiffiffiffiffi

p2o iA_k;m r þ1

2s ðr A_k;mÞ

(25) with respect to an electron in the atom. Since atomic transitions during absorption and emission conserve total angular momentum and parity, we use the standard multipole expansion[26]forA_k;mgiven by Eq. (13), viz,

A_k;m¼e^ikr^_m¼ ð2pÞ¹⁼²X¹

L¼1

X^L

M¼L

ðiÞ^L½LD^L_Mmðf;y;0Þ½A^ðmÞ_LMþimA^ðeÞ_LM, (26) where ½L ¼ ð2Lþ1Þ¹⁼² and ðy;fÞ denote polar angles of k, while A^ðmÞ_LM and A^ðeÞ_LM denote, respectively, the

‘magnetic’ and ‘electric’ 2^L-pole solutions of the free Maxwell equations. Using Eq. (26) we may write Aðk;mÞ ¼X¹

L¼1

X^L

M¼L

D^L_Mmðf;y;0Þ½J^ðmÞ_LMðoÞ þimJ^ðeÞ_LMðoÞ, (27) where the notation

J^ðm=eÞ_LM ðoÞ ¼Zei^L½L 2p ffiffiffiffiffiffi

p2o iA^ðm=eÞ_LM r þ1

2s ðr A^ðm=eÞ_LM Þ

(28) is used. Noting thatJ^ðm=eÞ_LM ðoÞis an irreducible tensor of rankL, we may apply the Wigner–Eckart theorem to write

hJumujJ^ðm=eÞ_LM ðoÞjJlmli ¼CðJl;L;Ju;ml;M;muÞhJukJ^ðm=eÞ_LM ðoÞkJli. (29)

Thus we obtain

hJumujAðk;mÞjJlmli ¼Aðk;mÞ_muml ¼X

L

CðJl;L;Ju;ml;M;muÞJ_LðoÞðimÞ^gþðLÞD^L_Mmðf;y;0Þ, (30) where the reduced matrix elements are given by

J_LðoÞ ¼ hJ_ujjJ^ðmÞ_L ðoÞjjJ_ligðLÞ þ hJ_ujjJ^ðeÞ_LðoÞjjJ_lig_þðLÞ, (31) in terms of the projection operators

gðLÞ ¼¹₂½1 ð1Þ^Lpupl. (32)

In the above equationpu;pldenote the parities of the upper and lower levels. Using Eq. (24), we have hJ_lm_ljEðk;mÞjJ_um_ui ¼Eðk;mÞ_mlmu ¼X

L

CðJ_l;L;J_u;m_l;M;m_uÞJ_LðoÞðimÞ^gþðLÞD^L_Mmðf;y;0Þ. (33) Thus, we may express Eq. (21) as

Tfiðkm;k⁰m⁰Þ ¼X

n

f_nX

m⁰_fm⁰_i

c^s

f

m⁰_fc^s_mⁱ⁰

i½Eðk;mÞG^snAðk⁰;m⁰Þ_m⁰

fm⁰_i, (34)

where the summation P

n implies summation with respect to sn as well, and Aðk⁰;m⁰Þ and Eðk;mÞ denote matrices, whose elements

hJnm⁰_njAðk⁰;m⁰ÞjJim⁰_ii ¼Aðk⁰;m⁰Þ_m⁰

nm⁰_i; hJfm⁰_fjEðk;mÞjJnm⁰⁰_ni ¼Eðk;mÞ_m⁰

fm⁰⁰_n, ð35Þ

may be written explicitly using Eqs. (30) and (33) andG^sn denotes a hermitianð2Jnþ1Þ ð2Jnþ1Þmatrix, which is defined in terms of its elements

G^s_mⁿ⁰⁰

nm⁰_n¼c^s_mⁿ00 nc^s_mⁿ0

n. (36)

Clearly, the summation overn on right-hand side of Eq. (34) indicates a summation with respect to all the atomic states fc_ng, which constitute the complete orthogonal set. Sincen is a cumulative indexP

n includes P

Jn

P2Jnþ1

sn¼1 , apart from summation with respect to other quantum numbers. The left-hand side of Eq. (34) is written for a givenc_iandc_f with energiesEiandEf, respectively. The quantitiessiandsf are specified by left- hand side of Eq. (34) and hence they are fixed entities on right-hand side of Eq. (34).

In the absence of the electric quadrupole field, the sf;sn;si may be replaced, respectively, by appropriate Mf;Mn;Mi which are determined by the Kroneckerd-function in Eq. (7), when the magnetic fieldBalone is present and is directed alongðy_B;f_BÞ. Thus, c^s

f

m⁰_f andc^s_mⁱ0

i are replaced, respectively, by D^J_m^f0

fMfðf_B;y_B;0Þ and D^J_mⁱ0

iMiðf_B;y_B;0ÞwithM_f andM_ibeing fixed by left-hand side. It may be noted thatf_ndepends onM_nand the summation overn includesP

Jn

PJn

Mn¼Jn, withG^sn replaced now byG^Mn whose elements are given by G^M_m⁰⁰_nⁿ_m⁰_n¼D^J_mⁿ00

nMnðf;y;0ÞD^J_mⁿ0

nMnðf;y;0Þ. (37)

If the magnetic fieldBis along theZ-axis of ARF itself, thesf;sn;siin Eq. (34) may, respectively, be replaced bymf;mn;midetermined by the Kroneckerd-function in Eq. (8). Thus,c^s

f

m⁰_f andc^s_mⁱ⁰

i are replaced, respectively, bydm⁰_fmf anddm⁰_imi, wheremf andmi are fixed by left-hand side of Eq. (34). Therefore, the summation with respect tom⁰_f andm⁰_ion right-hand side of Eq. (34) drops after making the replacementsc^s_m^f0

f ¼1 andc^s_mⁱ0

i ¼1.

The c_n depends on m_n and the summation over n includes P

Jn

PJn

mn¼Jn, withG^sn replaced byG^mn, whose elements are given by

G^m_mⁿ⁰⁰_nm⁰_n¼d_m⁰⁰_nmnd_m⁰_nmn, (38) i.e.,G^sn gets replaced by a diagonal matrix with zeros everywhere exceptG^m_mⁿ_nmn ¼1 in Eq. (34).

It may be noted that the atomic transitions fromc_i toc_n following absorption of o⁰ and fromc_n to c_f consequent to the emission ofoare virtual transitions, which do not satisfy the celebrated Bohr criteria. This is in contrast to absorption or emission represented by theN¼1 term. They are real transitions which satisfy the Bohr criteria as already pointed out. The summation overn includes all atomic statesc_n with different energy eigenvalues E_n. However, all of them do not contribute equally to Eq. (34). The presence of f_n on right-hand side of Eq. (34) indicates that one has to pay more attention to contributions coming from those states c_n with En close to Eiþo⁰¼Ef þo. If there is an En such that En¼Eiþo⁰¼Ef þo, the contribution from this state alone overshadows all other contributions. The scattering is then referred to as resonance scattering. In particular, ifEi¼Ef the terminology ‘two-level resonance scattering’ is employed.

This is shown as (a) inFig. 2, whereo⁰¼o. On the other hand, ifEf4Ei as in (b) ofFig. 2, the resonance scattering is referred to as three-level resonance or fluorescence.

If there is no electric quadrupole field and the atom is exposed only to a pure magnetic fieldB, such that the ð2Jnþ1Þ states jJnMni refer to distinctly separated energy levels as in Zeeman effect, one can envisage resonance scattering taking place individually with each one of these taking the role of the upper level, as shown in (a) and (b) ofFig. 2, if the conditionE_Mn ¼E_iþo⁰¼E_f þois satisfied. On the other hand, if gBoG_n, the levels are not distinct and all of them contribute coherently to form a single line. This is referred to as quantum interference in the context of Hanle scattering, which is shown as (c) inFig. 2. In contrast to Hanle effect where interference occurs between magnetic substates with the sameJ_n, interference effects between states with differentJ_nhave also been observed in polarization studies of solar Ca II H-K and Na I D₁and D₂ lines[13], wherein it is mentioned that this can take place even when the lines are 3.5 nm apart. The general terminology, ‘Raman scattering’ has been employed [12,13]to denote scattering, where contributions from several intermediate states are involved. In general, therefore, we may rewrite Eq. (21) in the form

Tfiðk;m;k⁰;m⁰Þ ¼ hc_fjEðk;mÞjc_vi, (39)

wherejc_vi represents a virtual state defined by jc_vi ¼X

n

c^v_njc_ni; jc^v_ni ¼f_nhc_njAðk⁰;m⁰Þjc_ii, (40)

ψr

E_r

E_i = E_f ψi = ψf

E_r

E_f

E_i

ψf

ψr

ψi

J_i, m_i

J_f, m_f ψH

E_i ψi

ψ

E_f ψf

Fig. 2. Level diagrams showing the atom–radiation interaction processes discussed in this paper.ðaÞTwo-level resonance scattering process,ðbÞthree-level fluorescence scattering process,ðcÞHanle scattering process in weak magnetic fields andðdÞthe general case of Raman scattering.

which is clearly not an eigenstate of energy. In Raman effect, shown as (d) inFig. 2, the lines corresponding to E_foE_i are referred to as anti-Stokes lines, in contrast to those withE_f4E_i referred to as Stokes lines.

2.4. The dipole approximation

If we neglect the spin-dependent second term in Eq. (25) and employ dipole approximation e^ikr1 in A_kmðrÞgiven by Eq. (13), then we may express Eq. (24) as

hc_ujAðk;mÞjc_li hc_uj^_mpjc_li hc_ljEðk;mÞjc_ui. (41) In the above equation the momentum operator p¼ ir may be replaced by½r;H0, to obtain

hc_uj^_m ½r;H₀jc_li ðE_lE_uÞhc_ujð^_mrÞjc_li, (42) ifEu;Eldenote the energy eigenvalues ofc_u;c_lwhen considered as eigenstates of Eq. (1). We, thus, realize the Kramers–Heisenberg form represented by Eq. (1) of [14].

3. The scattering matrix for atoms in external electric quadrupole and uniform magnetic fields

The central result of the previous section is the derivation of the general expression for the on-energy-shell T-matrix element T_fiðm;m⁰Þ. If the incident radiation is in a pure state

^i¼X

m⁰

cⁱ_m0^⁰_m0, (43)

the amplitude for detecting the scattered radiation in a pure state ^f ¼X

m

c^f_m^m, (44)

is given by

T_fið^_f;^_iÞ ¼X

mm⁰

c^f_mcⁱ_m0T_fiðm;m⁰Þ, (45)

where P

m⁰jcⁱ_m0j²¼1¼P

mjc^f_mj². On the other hand, it is more convenient to employ the density matrix formalism[24,25,37]to describe the states of polarization of the incident and scattered radiation, as it is more general and can handle mixed states of polarization as well.

3.1. The density matrix for polarized radiation

The density matrixr for polarized radiation may be written as r¼1

2½Iþs^g_xQþs^g_yUþs^g_zV ¼1 2

X³

p¼0

s^g_pSp, (46)

in terms of the well-known[2]Stokes parametersðI¼S0;Q¼S1;U¼S2;V¼S3Þand Pauli matricess^g_x¼ s^g₁;s^g_y¼s^g₂;s^g_z¼s^g₃ and the unit matrixs^g₀ whose rows and columns are labeled by the left and right circular polarization statesjm¼ 1iof radiation. Clearly,

S_p¼trðs^g_prÞ; p¼0;1;2;3, (47)

wheretrdenotes the trace or spur. A column vectorSwith elementsS_p;p¼0;1;2;3 is referred to as the Stokes vector for polarization. If we consider T in Eq. (21) as a 22 matrixTwith elementsT_mm⁰ T_fiðm;m⁰Þ, the density matrix rof scattered radiation is given by

r¼Tr⁰T^y, (48)

wherer⁰denotes the density matrix of polarized radiation incident on the atom. Using Eq. (47), we have Sp¼1

2 X³

p⁰¼0

trðs^g_pTs^g_p0T^yÞS⁰_p0 (49)

for the Stokes parameters of the scattered radiation, in terms of the matrixT, its hermitian conjugateT^yand the Stokes parametersS⁰_p0 characterizing the radiation incident on the atom.

3.2. The scattering matrix

If the Stokes vector S⁰ with elements ðI⁰¼S⁰₀;Q⁰¼S⁰₁;U⁰¼S⁰₂;V⁰¼S⁰₃Þ, characterizes the radiation incident on the atom, the Stokes vectorScharacterizing the scattered radiation may be expressed as

S¼RS⁰, (50)

where the 44 matrix R is referred to as the scattering matrix. Comparison of Eqs. (49) and (50) readily identifies the elements ofRas

Rpp⁰ ¼1 2

X

mm0 m00m000

ðs^g_pÞ_m⁰⁰_mTmm⁰ðs^g_p0Þ_m⁰_m⁰⁰⁰ðT^yÞ_m⁰⁰⁰_m⁰⁰, (51) where we may use Eq. (34) for T_mm⁰ and note that ðT^yÞ_m000m⁰⁰ ¼T_m00m⁰⁰⁰, for which we may use the complex conjugate of Eq. (34). We may thus write

Tmm⁰¼X

n

f_nX

m⁰_fm⁰_i

c^s

f

m⁰_fc^s_mⁱ0

iMm⁰_fm⁰_iðm;m⁰Þ;

ðT^yÞ_m000m⁰⁰ ¼T_m00m⁰⁰⁰¼X

n⁰

f_n0

X

m⁰⁰_fm⁰⁰_i

c^s_m^f00 fc^s_mⁱ00

iMm⁰⁰_fm⁰⁰_iðm⁰⁰;m⁰⁰⁰Þ, ð52Þ

where

Mm⁰_fm⁰_iðm;m⁰Þ ¼ ½Eðk;mÞG^snAðk⁰;m⁰Þ_m⁰

fm⁰_i; Mm⁰⁰_fm⁰⁰_iðm⁰⁰;m⁰⁰⁰Þ¼ ½Eðk;m⁰⁰ÞG^sn0Aðk⁰;m⁰⁰⁰Þ_m⁰⁰

fm⁰⁰_i ¼ ½A^yðk⁰;m⁰⁰⁰ÞG^sn0E^yðk;m⁰⁰Þ_m⁰⁰

im⁰⁰_f, ð53Þ

sinceG^sn0 is hermitian. Using the above in Eq. (51), we have Rpp⁰ ¼ 1

2 X

mm0 m00m000

ðs^g_pÞ_m00mðs^g_p0Þ_m0m⁰⁰⁰

X

nn⁰

f_nf_n0

X

m0 fm0

i m00

fm00 i

½Eðk;mÞG^snAðk⁰;m⁰Þ_m⁰

fm⁰_i

c^s_mⁱ⁰

ic^s

i

m⁰⁰_i½A^yðk⁰;m⁰⁰⁰ÞG^sn0E^yðk;m⁰⁰Þ_m⁰⁰

im⁰⁰_fc^s_m^f00

fc^s

f

m⁰_f. ð54Þ

Following Eq. (36), we may define hermitian matricesG^si andG^sf through their elements G^s_mⁱ⁰

im⁰⁰_i ¼c^s_mⁱ⁰

ic^s

i

m⁰⁰_i; (55)

G^s_m^f00

fm⁰_f ¼c^s_m^f00 fc^s

f

m⁰_f, (56)

so that we may rewrite Eq. (54) as Rpp⁰ ¼1

2 X

mm0 m00m000

ðs^g_pÞ_m00mðs^g_p0Þ_m0m⁰⁰⁰

X

nn⁰

f_nf_n0Tr½Eðk;mÞG^snAðk⁰;m⁰ÞG^siA^yðk⁰;m⁰⁰⁰ÞG^sn0E^yðk;m⁰⁰ÞG^sf, (57) where Tr P

m⁰_f denotes the trace or spur of theð2Jf þ1Þ ð2Jf þ1Þ matrix within the square brackets, which is defined through matrix multiplication of the eight matrices, each of which is well defined through

Eqs. (30), (33), (36), (55), (56)) for any specified atomic transition from an initial statec_iwith energyE_i and total angular momentumJ_ito a final statec_f with energyE_f and total angular momentumJ_f, when the atom is exposed to a combined external electric quadrupole field and a uniform magnetic field. It may be noted that s_i ands_f are fixed and the summation overn;n⁰includes summation overs_n;s_n⁰.

3.3. The particular case of resonance scattering via electric dipole transitions between Ji¼Jf ¼0 and Jn¼1 In this important particular case, which has often been investigated in the presence of pure magnetic fields, it is clear thatG^si ¼G^sf ¼1 in Eq. (57) and L¼L⁰¼1 in Eqs. (30) and (33), so that we may write the trace appearing in Eq. (57) as

tr½Aðk⁰;m⁰ÞA^yðk⁰;m⁰⁰⁰ÞG^sn0E^yðk;m⁰⁰ÞEðk;mÞG^sn

¼ X

m0 n m00

n m000

n m0000 n

½Aðk⁰;m⁰ÞA^yðk⁰;m⁰⁰⁰Þ_m0

nm⁰⁰⁰_nðG^sn0Þ_m000

nm⁰⁰⁰⁰_n½E^yðk;m⁰⁰ÞEðk;mÞ_m0000

nm⁰⁰_nðG^snÞ_m00

nm⁰_n. ð58Þ

We may use Eq. (30), in combination with Eq. (35), to write

½Aðk⁰;m⁰ÞA^yðk⁰;m⁰⁰⁰Þ_m⁰

nm⁰⁰⁰_n ¼ jJ₁ðo⁰Þj²m⁰m⁰⁰⁰D¹_m0

nm⁰ðf⁰;y⁰;0ÞD¹_m000

nm⁰⁰⁰ðf⁰;y⁰;0Þ, (59)

and Eq. (33), in combination with Eq. (35), to write

½E^yðk;m⁰⁰ÞEðk;mÞ_m⁰⁰⁰⁰

nm⁰⁰_n ¼ jJ₁ðoÞj²mm⁰⁰D¹_m0000

nm⁰⁰ðf;y;0ÞD¹_m00

nmðf;y;0Þ. (60)

Using Eq. (36) for ðG^snÞ_m00

nm⁰_n andðG^sn0Þ_m000

nm⁰⁰⁰⁰_n, we may attachc^s

n

m⁰_nc^s_mⁿ⁰⁰⁰⁰

n along withðs^g_p⁰Þ_m0m⁰⁰⁰ to Eq. (59), while we may attachc^s_mⁿ⁰⁰

nc^s

n0

m⁰⁰⁰⁰_n along withðs^g_pÞ_m00m to Eq. (60), so that X

m⁰m⁰⁰⁰

ðs^g_p0Þ_m0m⁰⁰⁰m⁰m⁰⁰⁰ X

m⁰_nm⁰⁰⁰_n

c^s_mⁿ0 nc^s_mⁿ⁰000

nD¹_m0

nm⁰ðf⁰;y⁰;0ÞD¹_m000

nm⁰⁰⁰ðf⁰;y⁰;0Þ

¼X²

la¼0

X^la

m_a¼la

X^la

ma¼la

f_p0ðl_a;m_aÞF_nn⁰ðl_a;m_aÞD^l_m^a_am

aðf⁰;y⁰;0Þ; ð61Þ

X

mm⁰⁰

ðs^g_pÞ_m00mm⁰⁰m X

m⁰⁰_nm⁰⁰⁰⁰_n

c^s_mⁿ00 nc^s

n0

m⁰⁰⁰⁰_n D¹_m0000

nm⁰⁰ðf;y;0ÞD¹_m00

nmðf;y;0Þ

¼X²

le¼0

X^le

m_e¼le

X^le

me¼le

f_pðl_e;m_eÞF_n;n⁰ðl_e;m_eÞD^l_m^e_em

eðf;y;0Þ. ð62Þ

Thus, we have Rpp⁰ ¼ 1

2jJ₁ðoÞj²jJ₁ðo⁰Þj²X

nn⁰

f_nf_n0

X²

la¼0

X²

le¼0

Fn;n⁰ðla;maÞFn;n⁰ðle;meÞ

f_pðl_e;m_eÞf_p0ðl_a;m_aÞD^l_m^e_em

eðf;y;0ÞD^l_m^a_am

aðf⁰;y⁰;0Þ, ð63Þ

where

Fn;n⁰ðl;mÞ ¼X

m⁰_n

Cð1;1;l;m⁰_n;m⁰⁰⁰_n;mÞð1Þ^m000nc^s_mⁿ⁰

nc^s_mⁿ⁰⁰⁰⁰

n; (64)

f_pðl;mÞ ¼X

l

Cð1;1;l;m⁰;m⁰⁰⁰;mÞð1Þ^m000m⁰m⁰⁰⁰ðs^g_pÞ_m⁰_m⁰⁰⁰. (65)