Radiative transitions of the helium atom in highly magnetized neutron
  star atmospheres
Mon. Not. R. Astron. Soc. 000, 1–14 (2008) Printed 10 November 2018 (MN LATEX style file v2.2)
Radiative transitions of the helium atom in highly
magnetized neutron star atmospheres
Z. Medin1, D. Lai1, and A. Y. Potekhin1,2
1Center for Radiophysics and Space Research, Department of Astronomy, Cornell University, Ithaca, NY 14853
2Ioffe Physico-Technical Institute, Politekhnicheskaya 26, 194021 Saint-Petersburg, Russia
Accepted 2007 September 26. Received 2007 May 21; in original form 2007 April 12
ABSTRACT
Recent observations of thermally emitting isolated neutron stars revealed spectral fea-
tures that could be interpreted as radiative transitions of He in a magnetized neutron
star atmosphere. We present Hartree–Fock calculations of the polarization-dependent
photoionization cross sections of the He atom in strong magnetic fields ranging from
1012 G to 1014 G. Convenient fitting formulae for the cross sections are given as
well as related oscillator strengths for various bound-bound transitions. The effects
of finite nucleus mass on the radiative absorption cross sections are examined using
perturbation theory.
Key words: atomic processes – magnetic fields – stars: atmospheres – stars: neutron
1 INTRODUCTION
An important advance in neutron star astrophysics in the
last few years has been the detection and detailed studies
of surface emission from a large number of isolated neutron
stars (NSs), including radio pulsars, magnetars, and radio-
quiet NSs (e.g., Kaspi et al. 2006; Harding & Lai 2006).
This was made possible by X-ray telescopes such as Chan-
dra and XMM-Newton. Such studies can potentially provide
invaluable information on the physical properties and evolu-
tion of NSs (e.g., equation of state at super-nuclear densities,
cooling history, surface magnetic field and composition). Of
great interest are the radio-quiet, thermally emitting NSs
(e.g., Haberl 2006): they share the common property that
their spectra appear to be entirely thermal, indicating that
the emission arises directly from the NS surfaces, uncon-
taminated by magnetospheric emission. The true nature of
these sources, however, is unclear at present: they could be
young cooling NSs, or NSs kept hot by accretion from the
ISM, or magnetar descendants. While some of these NSs
(e.g., RX J1856.5−3754) have featureless X-ray spectrum
remarkably well described by blackbody (e.g., Burwitz et al
2003) or by emission from a condensed surface covered by
a thin atmosphere (Ho et al. 2007), a single or multiple ab-
sorption features at E ≃ 0.2–1 keV have been detected from
several sources (see van Kerkwijk & Kaplan 2007): e.g., 1E
1207.4−5209 (0.7 and 1.4 keV, possibly also 2.1, 2.8 keV;
Sanwal et al. 2002; De Luca et al. 2004; Mori et al. 2005),
RX J1308.6+2127 (0.2–0.3 keV; Haberl et al. 2003), RX
J1605.3+3249 (0.45 keV; van Kerkwijk et al. 2004), RX
J0720.4−3125 (0.27 keV; Haberl et al. 2006), and possibly
RBS 1774 (∼ 0.7 keV; Zane et al. 2005). The identifications
of these features, however, remain uncertain, with sugges-
tions ranging from proton cyclotron lines to atomic transi-
tions of H, He, or mid-Z atoms in a strong magnetic field (see
Sanwal et al. 2002; Ho & Lai 2004; Pavlov & Bezchastnov
2005; Mori & Ho 2007). Clearly, understanding these ab-
sorption lines is very important as it would lead to direct
measurement of the NS surface magnetic fields and composi-
tions, shedding light on the nature of these objects. Multiple
lines also have the potential of constraining the mass-radius
relation of NSs (through measurement of gravitational red-
shift).
Since the thermal radiation from a NS is mediated by
its atmosphere (if T is sufficiently high so that the surface
does not condense into a solid; see, e.g., van Adelsberg et al.
2005; Medin & Lai 2006, 2007), detailed modelling of ra-
diative transfer in magnetized NS atmospheres is impor-
tant. The atmosphere composition of the NS is unknown
a priori. Because of the efficient gravitational separation
of light and heavy elements, a pure H atmosphere is ex-
pected even if a small amount of fallback or accretion oc-
curs after NS formation. A pure He atmosphere results if
H is completely burnt out, and a heavy-element (e.g., Fe)
atmosphere may be possible if no fallback/accretion oc-
curs. The atmosphere composition may also be affected by
(slow) diffusive nuclear burning in the outer NS envelope
(Chang, Arras & Bildsten 2004), as well as by the bombard-
ment on the surface by fast particles from NS magneto-
spheres (e.g., Beloborodov & Thompson 2007). Fully ion-
ized atmosphere models in various magnetic field regimes
have been extensively studied (e.g., Shibanov et al. 1992;
Zane et al. 2001; Ho & Lai 2001), including the effect of vac-
uum polarization (see Ho & Lai 2003; Lai & Ho 2002, 2003;
c© 2008 RAS
http://arxiv.org/abs/0704.1598v2
2 Z. Medin, D. Lai, and A. Y. Potekhin
van Adelsberg & Lai 2006). Because a strong magnetic field
greatly increases the binding energies of atoms, molecules,
and other bound species (for a review, see Lai 2001), these
bound states may have appreciable abundances in the NS
atmosphere, as guessed by Cohen, Lodenquai, & Ruderman
(1970) and confirmed by calculations of Lai & Salpeter
(1997) and Potekhin, Chabrier & Shibanov (1999). Early
considerations of partially ionized and strongly magnetized
atmospheres (e.g., Rajagopal, Romani & Miller 1997) relied
on oversimplified treatments of atomic physics and plasma
thermodynamics (ionization equilibrium, equation of state,
and nonideal plasma effects). Recently, a thermodynami-
cally consistent equation of state and opacities for mag-
netized (B = 1012 − 1015 G), partially ionized H plasma
have been obtained (Potekhin & Chabrier 2003, 2004), and
the effect of bound atoms on the dielectric tensor of the
plasma has also been studied (Potekhin et al. 2004). These
improvements have been incorporated into partially ion-
ized, magnetic NS atmosphere models (Ho et al. 2003, 2007;
Potekhin et al. 2004, 2006). Mid-Z element atmospheres for
B ∼ 1012 − 1013 G were recently studied by Mori & Ho
(2007).
In this paper we focus on He atoms and their radiative
transitions in magnetic NS atmospheres. It is well known
that for B ≫ Z2B0, where Z is the charge number
of the nucleus and B0 = e
3m2e/h̄
3c = 2.35 × 109 G,
the binding energy of an atom is significantly increased
over its zero-field value. In this strong-field regime the
electrons are confined to the ground Landau level, and
one may apply the adiabatic approximation, in which
electron motions along and across the field are assumed
to be decoupled from each other (see Sect. 2.1). Using
this approximation in combination with the Hartree–Fock
method (“1DHF approximation”), several groups calcu-
lated binding energies for the helium atom (Pröschel et al.
1982; Thurner et al. 1993) and also for some other atoms
and molecules (Neuhauser, Langanke & Koonin 1986;
Neuhauser, Koonin & Langanke 1987; Miller & Neuhauser
1991; Lai, Salpeter & Shapiro 1992). Mori & Hailey
(2002) developed a “multiconfigurational perturbative
hybrid Hartree–Fock” approach, which is a perturba-
tive improvement of the 1DHF method. Other methods
of calculation include Thomas–Fermi-like models (e.g.,
Abrahams & Shapiro 1991), the density functional the-
ory (e.g., Relovsky & Ruder 1996; Medin & Lai 2006),
variational methods (e.g., Müller 1984; Vincke & Baye
1989; Jones et al. 1999; Turbiner & Guevara 2006),
and 2D Hartree–Fock mesh calculations (Ivanov 1994;
Ivanov & Schmelcher 2000) which do not directly employ
the adiabatic approximation.
In strong magnetic fields, the finite nuclear mass and
centre-of-mass motion affect the atomic structure in a non-
trivial way (e.g., Lai 2001; see Sect. 5). The stronger B
is, the more important the effects of finite nuclear mass
are. Apart from the H atom, these effects have been cal-
culated only for the He atom which rests as a whole, but
has a moving nucleus (Al-Hujaj & Schmelcher 2003a,b),
and for the He+ ion (Bezchastnov, Pavlov & Ventura 1998;
Pavlov & Bezchastnov 2005).
There were relatively few publications devoted to ra-
diative transitions of non-hydrogenic atoms in strong mag-
netic fields. Several authors (Miller & Neuhauser 1991;
Thurner et al. 1993; Jones et al. 1999; Mori & Hailey
2002; Al-Hujaj & Schmelcher 2003b) calculated oscillator
strengths for bound-bound transitions; Miller & Neuhauser
(1991) presented also a few integrated bound-free oscilla-
tor strengths. Rajagopal et al. (1997) calculated opacities
of strongly magnetized iron, using photoionization cross
sections obtained by M. C. Miller (unpublished). To the
best of our knowledge, there were no published calculations
of polarization-dependent photoionization cross sections for
the He atom in the strong-field regime, as well as the calcu-
lations of the atomic motion effect on the photoabsorption
coefficients for He in this regime. Moreover, the subtle effect
of exchange interaction involving free electrons and the pos-
sible role of two-electron transitions (see Sect. 3.2) were not
discussed before.
In this paper we perform detailed calculations of radia-
tive transitions of the He atom using the 1DHF approx-
imation. The total error introduced into our calculations
by the use of these two approximations, the Hartree-Fock
method and the adiabatic approximation, is of order 1%
or less, as can be seen by the following considerations: The
Hartree-Fock method is approximate because electron corre-
lations are neglected. Due to their mutual repulsion, any pair
of electrons tend to be more distant from each other than
the Hartree-Fock wave function would indicate. In zero-field,
this correlation effect is especially pronounced for the spin-
singlet states of electrons for which the spatial wave function
is symmetrical. In strong magnetic fields (B ≫ B0), the elec-
tron spins (in the ground state) are all aligned antiparallel
to the magnetic field, and the multielectron spatial wave
function is antisymmetric with respect to the interchange of
two electrons. Thus the error in the Hartree-Fock approach
is expected to be less than the 1% accuracy characteristic of
zero-field Hartree-Fock calculations (Neuhauser et al. 1987;
Schmelcher, Ivanov & Becken 1999; for B = 0 see Scrinzi
1998). The adiabatic approximation is also very accurate at
B ≫ Z2B0. Indeed, a comparision of the ground-state en-
ergy values calculated here to those of Ivanov (1994) (who
did not use the adiabatic approximation) shows an agree-
ment to within 1% for B = 1012 G and to within 0.1% for
B = 1013 G.
The paper is organized as follows. Section 2 describes
our calculations of the bound states and continuum states
of the He atom, and section 3 contains relevant equations
for radiative transitions. We present our numerical results
and fitting formulae in section 4 and examine the effects of
finite nucleus mass on the photoabsorption cross sections in
section 5.
2 BOUND STATES AND SINGLY-IONIZED
STATES OF HELIUM ATOMS IN STRONG
MAGNETIC FIELDS
2.1 Bound states of the helium atom
To define the notation, we briefly describe 1DHF calcula-
tions for He atoms in strong magnetic fields. Each electron
in the atom is described by a one-electron wave function
(orbital). If the magnetic field is sufficiently strong (e.g.,
B ≫ 1010 G for He ground state), the motion of an electron
perpendicular to the magnetic field lines is mainly governed
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 3
by the Lorentz force, which is, on the average, stronger than
the Coulomb force. In this case, the adiabatic approximation
can be employed – i.e., the wave function can be separated
into a transverse (perpendicular to the external magnetic
field) component and a longitudinal (along the magnetic
field) component:
φmν(r) = fmν(z)Wm(r⊥) . (1)
Here Wm is the ground-state Landau wave function (e.g.,
Landau & Lifshitz 1977) given by
Wm(r⊥) =
, (2)
where (ρ, ϕ) are the polar coordinates of r⊥, ρ0 =
(h̄c/eB)1/2 is the magnetic length and fmν is the longitudi-
nal wave function which can be calculated numerically. The
quantum number m (> 0 for the considered ground Landau
state) specifies the negative of the z-projection of the elec-
tron orbital angular momentum. We restrict our considera-
tion to electrons in the ground Landau level; for these elec-
trons, m specifies also the (transverse) distance of the guid-
ing centre of the electron from the ion, ρm = (2m+1)
1/2ρ0.
The quantum number ν specifies the number of nodes in
the longitudinal wave function. The spins of the electrons
are taken to be aligned anti-parallel with the magnetic field,
and so do not enter into any of our equations. In addition,
we assume that the ion is completely stationary (the ‘infinite
ion mass’ approximation). In general, the latter assumption
is not necessary for the applicability of the adiabatic ap-
proximation (see, e.g., Potekhin 1994). The accuracy of the
infinite ion mass approximation will be discussed in Sect. 5.
Note that we use non-relativistic quantum mechanics
in our calculations, even when h̄ωBe & mec
2 or B & BQ =
2 = 4.414 × 1013 G. This is valid for two reasons: (i)
The free-electron energy in relativistic theory is
1 + 2nL
)]1/2
. (3)
For electrons in the ground Landau level (nL = 0), Eq. (3)
reduces to E ≃ mec2 + p2z/(2me) for pzc ≪ mec2; the elec-
tron remains non-relativistic in the z direction as long as
the electron energy is much less than mec
2; (ii) it is well
known (e.g., Sokolov & Ternov 1986) that Eq. (2) describes
the transverse motion of an electron with nL = 0 at any field
strength, and thus Eq. (2) is valid in the relativistic theory.
Our calculations assume that the longitudinal motion of the
electron is non-relativistic. This is valid for helium at all field
strengths considered in this paper. Thus relativistic correc-
tions to our calculated electron wave functions, binding ener-
gies, and transition cross sections are all small. Our approx-
imation is justified in part by Chen & Goldman (1992), who
find that the relativistic corrections to the binding energy of
the hydrogen atom are of order ∆E/E ∼ 10−5.5−10−4.5 for
the range of field strengths we are considering in this work
(B = 1012 − 1014 G).
A bound state of the He atom, in which
one electron occupies the (m1ν1) orbital, and the
other occupies the (m2ν2) orbital, is denoted by
|m1ν1,m2ν2〉 = |Wm1fm1ν1 ,Wm2fm2ν2〉 (clearly,
|m1ν1,m2ν2〉 = |m2ν2,m1ν1〉). The two-electron wave
function is
Ψm1ν1,m2ν2(r1, r2) =
Wm1(r1⊥)fm1ν1(z1)
×Wm2(r2⊥)fm2ν2(z2)
−Wm2(r1⊥)fm2ν2(z1)Wm1(r2⊥)fm1ν1(z2)
. (4)
The one-electron wave functions are found using
Hartree–Fock theory, by varying the total energy with re-
spect to the wave functions. The total energy is given by
(see, e.g., Neuhauser et al. 1987):
E = EK + EeZ + Edir + Eexc , (5)
where
dz |f ′mν (z)|2 , (6)
EeZ = −Ze2
dz |fmν (z)|2Vm(z) , (7)
Edir =
mν,m′ν′
′ |fmν (z)|2 |fm′ν′(z′)|2
×Dmm′(z − z′) , (8)
Eexc = −
mν,m′ν′
m′ν′(z)fmν(z)
×f∗mν(z′)fm′ν′(z′)Emm′(z − z′) ; (9)
Vm(z) =
|Wm(r⊥)|2
, (10)
Dmm′ (z − z′) =
dr⊥dr
|Wm(r⊥)|2|Wm′(r′⊥)|2
|r′ − r|
, (11)
Emm′(z − z′) =
dr⊥dr
|r′ − r|
×W ∗m′(r⊥)Wm(r⊥)W
⊥)Wm′(r
⊥) . (12)
Variation of Eq. (5) with respect to fmν(z) yields
− Ze2Vm(z)
′ |fm′ν′(z′)|2Dmm′ (z − z′)− εmν
fmν (z)
mν (z
)fm′ν′(z
)Emm′(z − z′)fm′ν′(z) .
In these equations, asterisks denote complex conjugates, and
f ′mν(z) ≡ dfmν/dz. The wave functions fmν(z) must satisfy
appropriate boundary conditions, i.e., fmν → 0 as z → ±∞,
and must have the required symmetry [fmν(z) = ±fmν(−z)]
and the required number of nodes (ν). The equations are
solved iteratively until self-consistency is reached for each
wave function fmν and energy εmν . The total energy of the
bound He state |m1ν1,m2ν2〉 can then be found, using either
Eq. (5) or
εmν − Edir − Eexc . (14)
c© 2008 RAS, MNRAS 000, 1–14
4 Z. Medin, D. Lai, and A. Y. Potekhin
2.2 Continuum states of the helium atom
The He state in which one electron occupies the bound
(m3ν3) orbital, and other occupies the continuum state
(m4k) is denoted by |m3ν3,m4k〉 = |Wm3fm3ν3 ,Wm4fm4k〉.
The corresponding two-electron wave function is
Ψm3ν3,m4k(r1, r2) =
[Wm3(r1⊥)fm3ν3(z1)
×Wm4(r2⊥)fm4k(z2)
−Wm4(r1⊥)fm4k(z1)Wm3(r2⊥)fm3ν3(z2)] . (15)
Here fm4k(z) is the longitudinal wave function of the con-
tinuum electron, and k is the z-wavenumber of the electron
at |z| → ∞ (far away from the He nucleus).
We can use Hartree–Fock theory to solve for the ion-
ized He states as we did for the bound He states. Since the
continuum electron wave function fm4k(z) is non-localized
in z, while the bound electron wave function fm3ν3(z) is lo-
calized around z = 0, it is a good approximation to neglect
the continuum electron’s influence on the bound electron.
We therefore solve for the bound electron orbital using the
equation
− Ze2Vm3(z)
fm3ν3(z) = εm3ν3fm3ν3(z) . (16)
The continuum electron, however, is influenced by the bound
electron, and its longitudinal wave function is determined
− Ze2Vm4(z)
′ |fm3ν3(z
)|2Dm3m4(z − z
)− εf
fm4k(z)
)fm3ν3(z
)Em3m4(z − z
)fm3ν4(z) .
where εf = εm4k = h̄
2k2/(2me). Here, the bound electron
orbital |m3ν3〉 satisfies the same boundary conditions as dis-
cussed in Sect. 2.1. The shape of the free electron wave func-
tion is determined by the energy of the incoming photon and
the direction the electron is emitted from the ion. We will
discuss this boundary condition in the next section. The to-
tal energy of the ionized He state |m3ν3,m4k〉 is simply
E = εm3ν3 + εf . (18)
Note that the correction terms Edir and Eexc that appear
in Eq. (14) do not also appear in Eq. (18). The direct and
exchange energies depend on the local overlap of the elec-
tron wave functions, but the non-localized nature of the free
electron ensures that these terms are zero for the continuum
states.
3 RADIATIVE TRANSITIONS
We will be considering transitions of helium atoms from two
initial states: the ground state, |00, 10〉, and the first excited
state, |00, 20〉.
In the approximation of an infinitely massive, pointlike
nucleus, the Hamiltonian of the He atom in electromagnetic
field is (see, e.g., Landau & Lifshitz 1977)
j=1,2
Atot(rj)
j=1,2
|r1 − r2|
, (19)
where pj = −ih̄∇j is the canonical momentum operator,
acting on the jth electron, rj is the jth electron radius vec-
tor, measured from the nucleus, andAtot(r) is the vector po-
tential of the field. In our case, Atot(r) = AB(r) +Aem(r),
where AB(r) and Aem(r) are vector potentials of the sta-
tionary magnetic field and electromagnetic wave, respec-
tively. The interaction operator is Hint = H − H0, where
H0 is obtained from H by setting Aem(r) = 0. The un-
perturbed Hamiltonian H0 is responsible for the stationary
states of He, discussed in Sect. 2. The vector potential and
the wave functions may be subject to gauge transformations;
the wave functions presented in Sect. 2 correspond to the
cylindrical gauge AB(r) =
B × r. Neglecting non-linear
(quadratic in Aem) term, we have
Hint ≈
j=1,2
[πj ·Aem(rj) +Aem(rj) · πj ], (20)
where
π = p +
AB(r). (21)
is the non-perturbed kinetic momentum operator: π =
meṙ = me(i/h̄)[H0 r − rH0].
For a monochromatic wave of the form Aem(r) ∝ ǫ eiq·r ,
where ǫ is the unit polarization vector, applying the Fermi’s
Golden Rule and assuming the transverse polarization (ǫ ·
q = 0), one obtains the following general formula for the
cross section of absorption of radiation from a given initial
state |a〉 (see, e.g., Armstrong & Nicholls 1972):
σ(ω, ǫ) =
∣ǫ · 〈b|eiq·rj|a〉
δ(ω − ωba), (22)
where |b〉 is the final state, ω = qc is the photon frequency,
ωba = (Eb − Ea)/h̄, and j is the electric current operator.
In our case, j = (−e/me)(π1 + π2).
We shall calculate the cross sections in the dipole ap-
proximation – i.e., drop eiq·r from Eq. (22). This approxima-
tion is sufficiently accurate for calculation of the total cross
section as long as h̄ω ≪ mec2 (cf., e.g., Potekhin & Pavlov
1993, 1997 for the case of H atom). In the dipole approxi-
mation, Eq. (22) can be written as
σ(ω, ǫ) =
2π2e2
fbaδ(ω − ωba), (23)
where
fba =
h̄ωbame
|〈b|ǫ · π|a〉|2 = 2meωba
|〈b|ǫ · r|a〉|2 (24)
is the oscillator strength. In the second equality we have
passed from the ‘velocity form’ to the ‘length form’ of the
matrix element (cf., e.g., Chandrasekhar 1945). These rep-
resentations are identical for the exact wave functions, but
it is not so for approximate ones. In the adiabatic ap-
proximation, the length representation [i.e., the right-hand
side of Eq. (24)] is preferable (see Potekhin & Pavlov 1993;
Potekhin, Pavlov, & Ventura 1997).
To evaluate the matrix element, we decompose the unit
polarization vector ǫ into three cyclic components,
ǫ = ǫ−ê+ + ǫ+ê− + ǫ0ê0, (25)
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 5
with ê0 = êz along the external magnetic field direction
(the z-axis), ê± = (êx ± iêy)/
2, and ǫα = êα · ǫ (with
α = ±, 0). Then we can write the cross section as the sum
of three components,
σ(ω, ǫ) = σ+(ω)|ǫ+|2 + σ−(ω)|ǫ−|2 + σ0(ω)|ǫ0|2, (26)
where σα has the same form as Eq. (23), with the corre-
sponding oscillator strength given by
2meωbaρ
|Mba|2 =
|Mba|2, (27)
Mba = 〈b|ê∗α · r̄|a〉, (28)
where r̄ = r/ρ0 and ωc = eB/(mec) is the electron cyclotron
frequency.
3.1 Bound-bound transitions
Consider the electronic transition
|a〉 = |mν,m2ν2〉 = |Wmfmν ,Wm2fm2ν2〉
−→ |b〉 = |m′ν′,m2ν2〉 = |Wm′gm′ν′ ,Wm2gm2ν2〉. (29)
The selection rules for allowed transitions and the related
matrix elements are
σ0 : ∆m = 0, ∆ν = odd,
Mba = 〈gmν′ |z̄|fmν 〉〈gm2ν2 |fm2ν2〉, (30)
σ+ : ∆m = 1, ∆ν = even,
Mba =
m+ 1 〈gm′ν′ |fmν 〉〈gm2ν2 |fm2ν2〉, (31)
σ− : ∆m = −1, ∆ν = even,
Mba =
m 〈gm′ν′ |fmν〉〈gm2ν2 |fm2ν2〉, (32)
where ∆m = m′ −m, ∆ν = ν′ − ν. The oscillator strengths
for bound-bound transitions from the states |00, 10〉 and
|00, 20〉 are given in Table 1.
The selection rules (30) – (32) are exact in the dipole
approximation. The selection rules in m follow from the con-
servation of the z-projection of total (for the photon and two
electrons) angular momentum. Technically, in the adiabatic
approximation, they follow from the properties of the Lan-
dau functions (e.g., Potekhin & Pavlov 1993). The selection
rules in ν follow from the fact that the functions gm′ν′ and
fmν have the same parity for even ν
′−ν and opposite parity
for odd ν′ − ν.
In addition to these selection rules, there are approxi-
mate selection rules which rely on the approximate orthog-
onality of functions gm′ν′ and fmν (for general ν 6= ν′). Be-
cause of this approximate orthogonality, which holds better
the larger B is, we have
〈gm′ν′ |fmν 〉〈gm2ν2 |fm2ν2〉 = δν,ν′ + ε, (33)
where |ε| ≪ 1 and ε → 0 as ∆ν → ±∞. Therefore, the oscil-
lator strengths for transitions with α = ± and ∆ν = 2, 4, . . .
are small compared to those with ∆ν = 0. The latter oscilla-
tor strengths can be approximated, according to Eqs. (27),
(31), (32) and (33), by
ba ≈ 2(m+ 1)ωba/ωc, f
ba ≈ 2mωba/ωc (34)
(α = ∆m = ±1, ν′ = ν).
The same approximate orthogonality leads to the
smallness of matrix elements for transitions of the type
|mν,m2ν2〉 −→ |m′ν′,m2ν′2〉 with ν′2 6= ν2 for α = ± and the
smallness of cross terms in the matrix elements of the form
〈gm2ν2 |fmν〉〈gm′ν′ |fm2ν2〉 when m′ = m2 (i.e., the so-called
“one-electron jump rule”); we have therefore excluded such
terms from the selection rule equations above [Eqs. (30) –
(32)].
3.2 Photoionization
The bound-free absorption cross section for the transition
from the bound state |b〉 to the continuum state |f〉 is given
by Eq. (22) with obvious substitutions |a〉 → |b〉, |b〉 → |f〉,
→ (Lz/2π)
dk, (35)
where Lz is the normalization length of the continuum elec-
tron [
∫ Lz/2
−Lz/2
dz |gmk(z)|2 = 1] and k is the wave number of
the outgoing electron (Sect. 2.2). Therefore we have
σbf(ω, ǫ) =
2πe2Lz
mech̄
2ωfbk
∣〈fk|eiq·rǫ · π|b〉
∣〈f−k|eiq·rǫ · π|b〉
, (36)
where k =
2meεf/h̄ and |f±k〉 represents the final state
where the free electron has wave number ±k (here and here-
after we assume k > 0). The asymptotic conditions for
these outgoing free electrons are (cf., e.g., Potekhin et al.
1997) gmk(z) ∼ exp[iϕk(z)] at z → ±∞, where ϕk(z) =
|kz|+ (ka0)−1 ln |kz| and a0 = h̄2/mee2 is the Bohr radius.
Since we do not care about direction of the outgoing elec-
tron, we can use for calculations a basis of symmetric and
antisymmetric wave functions of the continuum – that is,
in Eq. (36) we can replace 〈fk| and 〈f−k| by 〈feven| and
〈fodd|. The symmetric state |feven〉 is determined by the
free electron boundary condition g′mk,even(0) = 0 and the
antisymmetric state |fodd〉 is determined by gmk,odd(0) = 0.
Since the coefficients in Eq. (17) are real, gmk,even(z) and
gmk,odd(z) can be chosen real. At z → ±∞, they behave as
gmk,(even,odd)(z) ∼ sin[ϕ(z) + constant] (where the value of
constant depends on all quantum numbers, including k). We
still have the normalization
∫ Lz/2
−Lz/2
dz |gmk,(even,odd)(z)|2 =
Similar to bound-bound transitions, we can decompose
the bound-free cross section into three components, Eq. (26).
Thus, using the dipole approximation and the length form
of the matrix elements, as discussed above, we have for (α =
±, 0)-components of the bound-free cross section
σbf,α(ω) =
× |〈f |ê∗α · r̄|b〉|
, (37)
where |f〉 = |feven〉 or |f〉 = |fodd〉 depending on the parity
of the initial state and according to the selection rules, and
σTh = (8π/3) (e
2/mec
2)2 is the Thomson cross section. The
selection rules and related matrix elements for the bound-
free transitions
|b〉 = |mν,m2ν2〉 = |Wmfmν ,Wm2fm2ν2〉
c© 2008 RAS, MNRAS 000, 1–14
6 Z. Medin, D. Lai, and A. Y. Potekhin
−→ |f〉 = |m′k,m2ν2〉 = |Wm′gm′k,Wm2gm2ν2〉 (38)
are similar to those for the bound-bound transitions [see
Eqs. (30) – (32)]:
σ0 : ∆m = 0, ∆ν = odd,
Mfb = 〈gmk|z̄|fmν 〉〈gm2ν2 |fm2ν2〉, (39)
σ+ : ∆m = 1, ∆ν = even,
Mfb =
m+ 1 (〈gm′k|fmν 〉〈gm2ν2 |fm2ν2〉
−δm′ν,m2ν2〈gm2ν2 |fmν 〉〈gm′k|fm2ν2〉) , (40)
σ− : ∆m = −1, ∆ν = even,
Mfb =
m (〈gm′k|fmν〉〈gm2ν2 |fm2ν2〉
−δm′ν,m2ν2〈gm2ν2 |fmν 〉〈gm′k|fm2ν2〉) , (41)
In this case, the condition ∆ν = odd means that gm′k and
fmν must have opposite parity, and the condition ∆ν = even
means that gm′k and fmν must have the same parity. The
oscillator strengths for bound-free transitions from the states
|00, 10〉 and |00, 20〉 are given in Table 2.
Note that in Eqs. (40) and (41), the second term in
the matrix element (of the form 〈gm2ν2 |fmν〉〈gm′k|fm2ν2〉)
corresponds to transitions of both electrons. This appears
to violate the “one-electron jump rule” and other approxi-
mate selection rules discussed in Sect. 3.1 [see Eq. (33)]. In
fact, these approximate rules are not directly relevant for
bound-free transitions, since the matrix elements involving
a continuuum state are always small: 〈gm′k|fmν 〉 → 0 as the
normalization length Lz → ∞. Rather, we use a different set
of selection rules to determine which of these ‘small’ matrix
elements are smaller than the rest. The first is that
〈gm′k|fmν 〉〈gm2ν2 |fm2ν2〉 ≫ 〈gm′k|fmν 〉〈gm2ν′2 |fm2ν2〉,
when ν′2 6= ν2. This selection rule is similar to the bound-
bound transition case as 〈gm2ν′2 |fm2ν2〉 involves a bound
electron transition, not a free electron transition. The second
approximate selection rule that applies here is more compli-
cated: terms of the form 〈gm′ν |fmν 〉〈gm2k|fm2ν2〉 are small,
unless m′ = m2 and ν2 = ν. This exception for m
′ = m2
and ν2 = ν is due to the exchange term in the differential
equation for the free electron wave function [Eq. (17)], which
strongly (anti)correlates the two final wave functions |gm′ν〉
and |gm2k〉. If m′ = m2 and ν = ν2, then since 〈gm′ν |fm2ν2〉
is not small (in fact, it is of order 1), 〈gm2k|fm2ν2〉 will not
be small but will be of the same order as other terms involv-
ing the free electron wave function. In particular, the second
selection rule means, e.g., that the matrix element for the
transition from |00, 10〉 to |00, 0k〉 is
M00,10→00,0k = 〈g0k|f10〉〈g00|f00〉 − 〈g00|f10〉〈g0k|f00〉, (43)
where the second term is non-negligible, but that the matrix
element for the transition from |00, 10〉 to |0k, 20〉, which is
M00,10→0k,20 = 〈g20|f10〉〈g0k|f00〉, (44)
is small compared to the other matrix elements and can be
ignored (see Fig. 1).
We make one final comment here about the effect of
exchange interaction on the free electron state. If the ex-
change term [the right-hand side of Eq. (17)] is neglected in
the calculation of the free electron wave function, then the
cross terms (i.e., those involving two-electron transitions) in
the matrix elements of Eqs. (40) and (41) are small and can
be neglected. One then obtains approximate photoionization
cross sections which are within a factor of two of the true val-
ues in most cases and much better for σ0 transitions. If the
exchange term is included in Eq. (17) but the cross terms in
the matrix elements are ignored, significant errors in the σ±
photoionization cross sections will result. To obtain reliable
cross sections for all cases, both the exchange effect on the
free electron and the contribution of two-electron transitions
must be included.
4 RESULTS
Tables 1 and 2 give results for transitions of helium atoms
from the ground state (|00, 10〉) and the first excited state
(|00, 20〉). Table 1 gives results (photon energies and oscilla-
tor strengths) for all possible bound-bound transitions with
∆ν 6 1, for the field strengths B12 = 1, 5, 10, 50, 100, where
B12 = B/(10
12 G). Transitions |a〉 → |b〉 for α = − are not
listed separately, being equivalent to transitions |b〉 → |a〉
for α = +. One can check that the oscillator strengths fba
presented in Table 1 for α = + are well described by the
approximation (34).
Table 2 gives results (threshold photon energies and
cross section fitting formulas, see below) for all possible
bound-free transitions. Figure 1 shows partial cross section
curves for all bound-free transitions from the ground state
of helium for B12 = 1. The transition |00, 10〉 → |0k, 20〉 is
an example of a ‘weak’ transition, whose oscillator strength
is small because of the approximate orthogonality of one-
electron wave functions, as discussed at the end of Sect. 3.1.
It is included in this figure to confirm the accuracy of our
assumption. Figures 2 and 3 show total cross section curves
for a photon polarized along the magnetic field, for B12 = 1
and 100 respectively. Figures 4 and 5 show total cross sec-
tions for the circular polarizations, α = ±, for B12 = 1.
Finally, Figs. 6 and 7 show total cross sections for α = ±
and B12 = 100.
4.1 Fitting Formula
The high-energy cross section scaling relations from
Potekhin & Pavlov (1993), which were derived for hydro-
gen photoionization in strong magnetic fields, also hold for
helium:
σbf,0 ∝
)2mi+9/2
σbf,± ∝
)2mi+7/2
, (46)
where mi is the m value of the initial electron that tran-
sitions to the free state. In addition, we use similar fitting
formulae for our numerical cross sections:
σbf,0 ≃
(1 +Ay)2.5(1 + B(
1 + y − 1))4(mi+1)
σTh (47)
σbf,± ≃
C(1 + y)
(1 +Ay)2.5(1 + B(
1 + y − 1))4(mi+1)
σTh (48)
where y = εf/h̄ωthr and h̄ωthr is the threshold photon en-
ergy for photoionization. These formulas have been fit to the
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 7
 1  10  100  1000  10000
εf (eV)
00,10->00,1k
σbf,0
 1  10  100  1000  10000
εf (eV)
σbf,0
00,10->0k,10
 1000
 10000
 1  10  100  1000  10000
εf (eV)
00,10->00,2k
σbf,+
 1000
 10000
 1  10  100  1000  10000
εf (eV)
σbf,+
00,10->10,1k
 0.001
 0.01
 1  10  100
εf (eV)
00,10->0k,20
σbf,+
 1000
 10000
 1  10  100  1000  10000
εf (eV)
σbf,-
00,10->00,0k
Figure 1. Partial cross sections σ(0,+,−) versus final ionized electron energy for photoionization of the ground state helium atom
((m1,m2) = (1, 0)). The field strength is 10
12 G. The transition |00, 10〉 → |0k,20〉 in the bottom left panel is an example of a ‘weak’
transition. We have ignored these transitions in our calculations of the total cross sections.
cross section curves with respect to the free electron energy
εf in approximately the 1 – 10
4 eV range (the curves are fit
up to 105 eV for strong magnetic fields B12 = 50 − 100,
in order to obtain the appropriate high-energy factor). The
data points to be fit are weighted proportional to their cross
section values plus a slight weight toward low-energy values,
according to the formula (error in σ) ∝ σ εf0.25.
Results for the three fitting parameters, A, B, and C,
are given in Table 2 for various partial cross sections over
a range of magnetic field strengths. For photoionization in
strong magnetic fields (B12 & 50) the cross section curves
we generate for the σ+ and σ− transitions have a slight
deficiency at low electron energies, such that the curves
peak at εf ≃ 10 eV, rather than at threshold as expected.
These peaks do not represent a real effect, but rather re-
flect the limits on the accuracy of our code (the overlap
of the wave function of the transitioning electron pre- and
post-ionization is extremely small under these conditions).
Because the cross section values are not correct at low ener-
gies, our fits are not as accurate for these curves. In Table 2
we have marked with a ‘∗’ those transitions which are most
inaccurately fit by our fitting formula, determined by cross
section curves with low-energy dips greater than 5% of the
threshold cross section value.
c© 2008 RAS, MNRAS 000, 1–14
8 Z. Medin, D. Lai, and A. Y. Potekhin
Table 1. Bound-bound transitions |a〉 → |b〉: The photon energy h̄ωba = Eb − Ea (in eV) and
the oscillator strength fα
for different polarization components α [see Eq. (27)]. All transitions
∆ν 6 1 from the initial states |00, 10〉 and |00, 20〉 are listed, for several magnetic field strengths
B12 = B/(10
12 G). The last two columns list the transition energies h̄ω∗
and oscillator strengths
, corrected for the finite mass of the nucleus, according to Sect. 5.1.
B12 σ |a〉 → |b〉 h̄ωba fba h̄ω
1 0 |00, 10〉 → |00, 11〉 147.5 0.234 – –
→ |10, 01〉 271.8 0.124 – –
+ → |00, 20〉 43.11 0.0147 44.70 0.0153
0 |00, 20〉 → |00, 21〉 104.4 0.312 – –
→ |20, 01〉 277.7 0.115 – –
+ → |00, 30〉 18.01 0.00930 19.60 0.0101
→ |20, 10〉 100.7 0.0170 102.3 0.0172
5 0 |00, 10〉 → |00, 11〉 256.2 0.127 – –
→ |10, 01〉 444.8 0.0603 – –
+ → |00, 20〉 66.95 0.00459 74.89 0.00512
0 |00, 20〉 → |00, 21〉 189.2 0.176 – –
→ |20, 01〉 455.0 0.0537 – –
+ → |00, 30〉 28.94 0.00299 36.88 0.00381
→ |20, 10〉 151.1 0.00512 159.0 0.00539
10 0 |00, 10〉 → |00, 11〉 318.9 0.0974 – –
→ |10, 01〉 540.8 0.0457 – –
+ → |00, 20〉 79.54 0.00273 95.42 0.00327
0 |00, 20〉 → |00, 21〉 239.4 0.136 – –
→ |20, 01〉 553.3 0.0405 – –
+ → |00, 30〉 34.84 0.00179 50.72 0.00261
→ |20, 10〉 177.0 0.00301 192.9 0.00328
50 0 |00, 10〉 → |00, 11〉 510.9 0.0557 – –
→ |10, 01〉 822.2 0.0266 – –
+ → |00, 20〉 114.2 7.85e−4 193.6 0.00133
0 |00, 20〉 → |00, 21〉 396.7 0.0776 – –
→ |20, 01〉 841.1 0.0235 – –
+ → |00, 30〉 51.92 5.37e−4 131.3 0.00136
→ |20, 10〉 246.5 8.41e−4 325.9 0.00111
100 0 |00, 10〉 → |00, 11〉 616.4 0.0452 – –
→ |10, 01〉 971.4 0.0221 – –
+ → |00, 20〉 131.4 4.52e−4 290.2 9.98e−4
0 |00, 20〉 → |00, 21〉 485.0 0.0626 – –
→ |20, 01〉 993.4 0.0195 – –
+ → |00, 30〉 60.57 3.13e−4 219.4 0.00114
→ |20, 10〉 280.7 4.80e−4 439.5 7.51e−4
5 FINITE NUCLEUS MASS EFFECTS
So far we have used the infinite ion mass approximation.
In this section we shall evaluate the validity range of this
approximation and suggest possible corrections.
It is convenient to use the coordinate system which con-
tains the centre-of-mass coordinate Rcm and the relative
coordinates {rj} of the electrons with respect to the nu-
cleus. Using a suitable canonical transformation, the Hamil-
tonian H of an arbitrary atom or ion can be separated
into three terms (Vincke & Baye 1988; Baye & Vincke 1990;
Schmelcher & Cederbaum 1991): H1 which describes the
motion of a free pseudo-particle with net charge Q and total
mass M of the ion (atom), the coupling term H2 between
the collective and internal motion, and H3 which describes
the internal relative motion of the electrons and the nu-
cleus. H1 and H2 are proportional to M
−1, so they van-
ish in the infinite mass approximation. It is important to
note, however, that H3 (the only non-zero term in the in-
finite mass approximation) also contains a term that de-
pends on M−10 , where M0 ≈ M is the mass of the nucleus.
Thus, there are two kinds of non-trivial finite-mass effects:
the effects due to H1 + H2, which can be interpreted as
caused by the electric field induced in the co-moving refer-
ence frame, and the effects due to H3, which arise irrespec-
tive of the atomic motion. Both kinds of effects have been
included in calculations only for the H atom (Potekhin 1994;
Potekhin & Pavlov 1997, and references therein) and He+
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 9
Table 2. Bound-free transitions |b〉 → |f〉: The threshold photon energy h̄ωthr (in eV) and
the fitting parameters A, B, and C used in the cross section fitting formulas [Eq. (48)]. All
transitions from the initial states |00, 10〉 and |00, 20〉 are listed, for several magnetic field strengths
B12 = B/(10
12 G).
B12 σ |b〉 → |f〉 mi h̄ωthr A B C
1 0 |00, 10〉 → |00, 1k〉 1 159.2 0.96 0.093 1.43e6
→ |10, 0k〉 0 283.2 0.89 0.20 8.83e5
+ → |00, 2k〉 1 159.2 0.70 0.061 7.95e2
→ |10, 1k〉 0 283.2 0.86 0.094 1.30e3
– → |00, 0k〉 1 159.2 0.62 0.030 8.89e2
0 |00, 20〉 → |00, 2k〉 2 116.0 1.00 0.062 1.78e6
→ |20, 0k〉 0 289.2 0.88 0.22 8.71e5
+ → |00, 3k〉 2 116.0 0.66 0.038 3.94e2
→ |20, 1k〉 0 289.2 0.54 0.14 6.48e2
– → |00, 1k〉 2 116.0 0.62 0.029 5.82e2
5 0 |00, 10〉 → |00, 1k〉 1 268.2 0.86 0.061 8.39e5
→ |10, 0k〉 0 456.4 0.69 0.16 4.60e5
+ → |00, 2k〉 1 268.2 0.68 0.036 1.14e2
→ |10, 1k〉 0 456.4 0.83 0.057 1.93e2
– → |00, 0k〉 1 268.2 0.60 0.020 1.36e2
0 |00, 20〉 → |00, 2k〉 2 201.2 0.92 0.039 1.11e6
→ |20, 0k〉 0 466.5 0.65 0.18 4.39e5
+ → |00, 3k〉 2 201.2 0.65 0.021 5.95e1
→ |20, 1k〉 0 466.5 0.54 0.084 9.13e1
– → |00, 1k〉 2 201.2 0.61 0.015 7.82e1
10 0 |00, 10〉 → |00, 1k〉 1 331.1 0.82 0.051 6.58e5
→ |10, 0k〉 0 552.5 0.63 0.15 3.51e5
+ → |00, 2k〉 1 331.1 0.67 0.029 4.94e1
→ |10, 1k〉 0 552.5 0.81 0.046 8.43e1
– → |00, 0k〉 1 331.1 0.59 0.016 6.00e1
0 |00, 20〉 → |00, 2k〉 2 251.6 0.88 0.033 8.77e5
→ |20, 0k〉 0 564.9 0.59 0.16 3.31e5
+ → |00, 3k〉 2 251.6 0.64 0.017 2.64e1
→ |20, 1k〉 0 564.9 0.53 0.069 3.97e1
– → |00, 1k〉 2 251.6 0.61 0.012 3.25e1
50 0 |00, 10〉 → |00, 1k〉 1 523.3 0.73 0.034 3.74e5
→ |10, 0k〉 0 834.2 0.54 0.11 1.96e5
+ → |00, 2k〉 1 523.3 0.63 0.020 7.15e0
→ |10, 1k〉 0 834.2 0.77 0.033 1.22e1
– → |00, 0k〉 1 523.3 0.57 0.012 8.94e0
0 |00, 20〉 → |00, 2k〉 2 409.1 0.79 0.021 5.02e5
→ |20, 0k〉 0 853.0 0.50 0.13 1.83e5
+ → |00, 3k〉 2 409.1 0.62 0.0104 4.04e0
→ |20, 1k〉 0 853.0 *0.52 0.052 5.88e0
– → |00, 1k〉 2 409.1 0.59 0.0058 4.13e0
100 0 |00, 10〉 → |00, 1k〉 1 628.8 0.69 0.029 2.96e5
→ |10, 0k〉 0 983.4 0.51 0.101 1.56e5
+ → |00, 2k〉 1 628.8 0.62 0.019 3.12e0
→ |10, 1k〉 0 983.4 0.75 0.031 5.33e0
– → |00, 0k〉 1 628.8 0.56 0.012 3.94e0
0 |00, 20〉 → |00, 2k〉 2 498.0 0.75 0.018 3.96e5
→ |20, 0k〉 0 1008 0.47 0.12 1.45e5
+ → |00, 3k〉 2 498.0 0.60 0.0092 1.81e0
→ |20, 1k〉 0 1008 *0.50 0.050 2.60e0
– → |00, 1k〉 2 498.0 0.58 0.0042 1.69e0
c© 2008 RAS, MNRAS 000, 1–14
10 Z. Medin, D. Lai, and A. Y. Potekhin
Figure 2. Total cross section σ0 versus photon energy for helium
photoionization, from initial states (m1,m2) = (1, 0) (solid lines)
and (2, 0) (dashed lines). The field strength is 1012 G. The dotted
lines extending from each cross section curve represent the effect
of magnetic broadening on these cross sections, as approximated
in Eq. (55), for T = 104.5 K (steeper lines) and 106 K (flatter
lines).
Figure 3. Total cross section σ0 versus photon energy for helium
photoionization, from initial states (m1,m2) = (1, 0) (solid lines)
and (2, 0) (dashed lines). The field strength is 1014 G. The dotted
lines extending from each cross section curve represent the effect
of magnetic broadening on these cross sections, as approximated
in Eq. (55), for T = 105.5 K (steeper lines) and 106 K (flatter
lines).
Figure 4. Total cross section σ+ versus photon energy for helium
photoionization, from initial states (m1,m2) = (1, 0) (solid lines)
and (2, 0) (dashed lines). The field strength is 1012 G. The dotted
lines extending from each cross section curve represent the effect
of magnetic broadening on these cross sections, as approximated
in Eq. (55), for T = 106 K.
Figure 5. The same as in Fig. 4, but for σ−.
ion (Bezchastnov et al. 1998; Pavlov & Bezchastnov 2005).
For the He atom, only the second kind of effects have been
studied (Al-Hujaj & Schmelcher 2003a,b).
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 11
Figure 6. The same as in Fig. 4, but for B = 1014 G.
Figure 7. The same as in Fig. 6, but for σ−.
5.1 Non-moving helium atom
The state of motion of an atom can be described by pseu-
domomentum K , which is a conserved vector since Q = 0
(e.g., Vincke & Baye 1988; Schmelcher & Cederbaum 1991).
Let us consider first the non-moving helium atom: K = 0.
According to Al-Hujaj & Schmelcher (2003a), there are
trivial normal mass corrections, which consist in the ap-
pearance of reduced masses me/(1 ± me/M0) in H3, and
non-trivial specific mass corrections, which originate from
the mass polarization operator.
The normal mass corrections for the total energy E of
the He state |m1ν1,m2ν2〉 can be described as follows:
E(M0, B) =
E(∞, (1 +me/M0)2B)
1 +me/M0
+ h̄Ωc
mj , (49)
where Ωc = (me/M0)ωc (for He, h̄Ωc = 1.588B12 eV). The
first term on the right-hand side describes the reduced mass
transformation. The second term represents the energy shift
due to conservation of the total z component of the angu-
lar momentum. Because of this shift, the states with suffi-
ciently large values of m1 +m2 become unbound (autoion-
izing, in analogy with the case of the H atom considered by
Potekhin et al. 1997). This shift is also important for radia-
tive transitions which change (m1 + m2) by ∆m 6= 0: the
transition energy h̄ωba is changed by h̄Ωc∆m. The dipole
matrix elements Mba are only slightly affected by the normal
mass corrections, but the oscillator strengths are changed
with changing ωba according to Eq. (27). The energy shift
also leads to the splitting of the photoionization thresh-
old by the same quantity h̄Ωc∆m, with ∆m = 0,±1 de-
pending on the polarization (in the dipole approximation).
Clearly, these corrections must be taken into account, unless
Ωc ≪ ωba or ∆m = 0, as illustrated in the last two columns
of Table 1.
The specific mass corrections are more difficult to evalu-
ate, but they can be neglected in the considered B range. In-
deed, calculations by Al-Hujaj & Schmelcher (2003a) show
that these corrections do not exceed 0.003 eV at B 6 104B0.
5.2 Moving helium atom
Eigenenergies and wave functions of a moving atom depend
on its pseudomomentum K perpendicular to the magnetic
field. This dependence can be described by Hamiltonian
components (e.g., Schmelcher & Cederbaum 1991)
H1 +H2 =
K · (B × rj), (50)
where
is the sum over all electrons. The dependence
on Kz is trivial, but the dependence on the perpendicular
component K⊥ is not. The energies depend on the abso-
lute value K⊥. For calculation of radiative transitions, it
is important to take into account that the pseudomomen-
tum of the atom in the initial and final state differ due to
recoil: K ′ = K + h̄q. Effectively the recoil adds a term
∝ q into the interaction operator (cf. Potekhin et al. 1997;
Potekhin & Pavlov 1997). The recoil should be neglected in
the dipole approximation.
The atomic energy E depends on K⊥ differently for
different quantum states of the atom. In a real neutron star
atmosphere, one should integrate the binding energies and
cross sections over the K⊥-distribution of the atoms, in or-
der to obtain the opacities.1 Such integration leads to the
specific magnetic broadening of spectral lines and ioniza-
tion edges. Under the conditions typical for neutron star at-
mospheres, the magnetic broadening turns out to be much
1 For the hydrogen atom, this has been done by
Pavlov & Potekhin (1995) for bound-bound transitions and
by Potekhin & Pavlov (1997) for bound-free transitions.
c© 2008 RAS, MNRAS 000, 1–14
12 Z. Medin, D. Lai, and A. Y. Potekhin
larger than the conventional Doppler and collisional broad-
enings (Pavlov & Potekhin 1995).
At present the binding energies and cross sections of
a moving helium atom have not been calculated. However,
we can approximately estimate the magnetic broadening for
T ≪ |(∆E)min|/kB, where (∆E)min is the energy difference
from a considered atomic level to the nearest level admixed
by the perturbation due to atomic motion, and kB is the
Boltzmann constant. In this case, the K⊥-dependence of E
can be approximated by the formula
E(K⊥) = E(0) +
, (51)
where E(0) is the energy in the infinite mass approxima-
tion and M⊥ = K⊥(∂E/∂K⊥)
−1 is an effective ‘transverse’
mass, whose value (M⊥ > M) depends on the quantum state
considered (e.g., Vincke & Baye 1988; Pavlov & Mészáros
1993).
Generally, at every value of K⊥ one has a different
cross section σ(ω,K⊥). Assuming the equilibrium (Maxwell–
Boltzmann) distribution of atomic velocities, the K⊥-
averaged cross section can be written as
σ(ω) =
E(0)− E(K⊥)
σ(ω,K⊥)
dE(K⊥)
, (52)
where Emin = −h̄ω.
The transitions that were dipole-forbidden for an atom
at rest due to the conservation of the total z-projection of an-
gular momentum become allowed for a moving atom. There-
fore, the selection rule ∆m = α [Eqs. (30)–(32)] does not
strictly hold, and we must write
σ(ω,K⊥) =
σm′(ω,K⊥), (53)
where the sum of partial cross sections is over all final quan-
tum numbers m′ (with m′ > 0 and m′ 6= m2 for ∆ν = 0)
which are energetically allowed. For bound-bound transi-
tions, this results in the splitting of an absorption line at a
frequency ωba in a multiplet at frequencies ωba + δmΩc +
−M−1⊥ )K
⊥/2h̄, where δm ≡ m′−m−α and M⊥,m′
is the transverse mass of final states. For photoionization, we
have the analogous splitting of the threshold. In particular,
there appear bound-free transitions at frequencies ω < ωthr
– they correspond to δm < K2⊥/(2M⊥h̄Ωc). Here, ωthr is
the threshold in the infinite ion mass approximation, and
one should keep in mind that the considered perturbation
theory is valid for K2⊥/2M⊥ ≪ |(∆E)min| < h̄ωthr. Accord-
ing to Eq. (53), σ(ω,K⊥) is notched at ω < ωthr, with the
cogs at partial thresholds ωthr + δmΩc − K2⊥/(2M⊥h̄) (cf.
Fig. 2 in Potekhin & Pavlov 1997).
Let us approximately evaluate the resulting envelope
of the notched photoionization cross section (53), assum-
ing that the ‘longitudinal’ matrix elements [〈. . .〉 con-
structions in Eqs. (30)–(32)] do not depend on K⊥. The
‘transverse’ matrix elements can be evaluated following
Potekhin & Pavlov (1997): in the perturbation approxima-
tion, they are proportional to |ξ||δm|e−|ξ|
2/2, where |ξ|2 =
0/(2h̄
2). Then
σ(ω < ωthr,K⊥) ≈ σ(ωthr, 0) exp
ωthr − ω
− h̄(ωthr − ω)
, (54)
where θ(x) is the step function. A comparison of this approx-
imation with numerical calculations for the hydrogen atom
(Potekhin & Pavlov 1997) shows that it gives the correct
qualitative behaviour of σ(ω,K⊥). For a quantitative agree-
ment, one should multiply the exponential argument by a
numerical factor ∼ 0.5–2, depending on the state and po-
larization. This numerical correction is likely due to the ne-
glected K⊥-dependence of the longitudinal matrix elements.
We assume that this approximation can be used also for the
helium atom. Using Eq. (52), we obtain
σ(ω) ≈ σ(ωthr) exp
ωthr − ω
− h̄(ωthr − ω)
for ω < ωthr. Here the transverse mass M⊥ can be evaluated
by treating the coupling Hamiltonian H2 as a perturbation,
as was done by Pavlov & Mészáros (1993) for the H atom.
Following this approach, retaining only the main perturba-
tion terms according to the approximate orthogonality rela-
tion (33) and neglecting the difference between M and M0,
we obtain an estimate
b(∆m=α)
ba/(2ωba)
1 + ωba/Ωc
, (56)
where |a〉 is the considered bound state (|00, 10〉 or |00, 20〉
for the examples in Figs. 2–7) and |b〉 are the final bound
states to which α = ± transitions |a〉 → |b〉 are allowed.
According to Eq. (34), the numerator in Eq. (56) is close to
m+ 1 for α = + and to m for α = −.
For the transitions from the ground state with po-
larization α = −, which are strictly forbidden in the
infinite ion mass approximation, using the same approx-
imations as above we obtain the estimate σ−(ω) ∝
σ+(ω)h̄ΩckBT/(kBT + h̄Ωc)
Examples of the photoionization envelope approxima-
tion, as described in Eq. (55) above, are shown in Figs. 2–7.
In Figs. 6 and 7 (for B = 1014 G), in addition to the mag-
netic broadening, we see a significant shift of the maximum,
which originates from the last term in Eq. (49). Such shift
is negligible in Figs. 4 and 5 because of the relatively small
Ωc value for B = 10
12 G.
Finally, let us note that the Doppler and collisional
broadening of spectral features in a strong magnetic field
can be estimated, following Pavlov & Mészáros (1993),
Pavlov & Potekhin (1995) and Rajagopal et al. (1997). The
Doppler spectral broadening profile is
φD(ω) =
− (ω − ω0)
, (57)
∆ωD =
]−1/2
, (58)
where θB is the angle between the wave vector and B. The
collisional broadening is given by
φcoll(ω) =
Λcoll
(ω − ω0)2 + (Λcoll/2)2
, (59)
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 13
h̄Λcoll = 4.8nea0r
= 41.5
1024 cm−3
eV, (60)
where ne is the electron number density and reff is an
effective electron-atom interaction radius, which is about
the quantum-mechanical size of the atom. The convolution
of the Doppler, collisional and magnetic broadening pro-
files gives the total shape of the cross section. For bound-
free transitions, the Doppler and collisional factors can
be neglected, but for the bound-bound transitions they
give the correct blue wings of the spectral features (cf.
Pavlov & Potekhin 1995).
6 CONCLUSION
We have presented detailed numerical results and fitting for-
mulae for the dominant radiative transitions (both bound-
bound and bound-free) of He atoms in strong magnetic fields
in the range of 1012 − 1014 G. These field strengths may
be most appropriate for the identification of spectral lines
observed in thermally emitting isolated neutron stars (see
Sect. 1).
While most of our calculations are based on the infinite-
nucleus-mass approximation, we have examined the effects
of finite nucleus mass and atomic motion on the opacities.
We found that for the field strengths considered in this paper
(B . 1014 G), these effects can be incorporated into the
infinite-mass results to obtain acceptable He opacities for
neutron star atmosphere modelling. For large field strengths,
more accurate calculations of the energy levels and radiative
transitions of a moving He atom will be needed in order to
obtain reliable opacities.
ACKNOWLEDGMENTS
This work has been supported in part by NSF grant AST
0307252 and Chandra grant TM6-7004X (Smithsonian As-
trophysical Observatory). The work of A.P. is supported in
part by FASI (Rosnauka) grant NSh-9879.2006.2 and RFBR
grants 05-02-16245 and 05-02-22003.
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	Introduction
	Bound states and singly-ionized states of helium atoms in strong magnetic fields
	Bound states of the helium atom
	Continuum states of the helium atom
	Radiative transitions
	Bound-bound transitions
	Photoionization
	Results
	Fitting Formula
	Finite nucleus mass effects
	Non-moving helium atom
	Moving helium atom
	Conclusion
