Issue 
A&A
Volume 518, JulyAugust 2010
Herschel: the first science highlights



Article Number  A24  
Number of page(s)  12  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201014781  
Published online  25 August 2010 
Cyclotron harmonics in opacities of isolated neutron star atmospheres
A. Y. Potekhin^{1,2,3}
1  CRAL (UMR CNRS No. 5574),
École Normale Supérieure de Lyon,
69364 Lyon Cedex 07, France
2 
Ioffe PhysicalTechnical Institute,
Politekhnicheskaya 26, 194021 St. Petersburg, Russia
3 
Isaac Newton Institute of Chile,
St. Petersburg Branch, Russia
Received 13 April 2010 / Accepted 5 May 2010
Abstract
Some Xray dim isolated neutron stars (XDINS) and central
compact objects in supernova remnants (CCO) contain absorption features in
their thermal soft Xray spectra. It has been hypothesized
that this absorption may relate to periodic peaks in
freefree absorption opacities, caused by either Landau quantization
of electron motion in magnetic fields
G or analogous
quantization of ion motion in magnetic fields B>10^{13} G.
Here, I review
the physics behind cyclotron quantum harmonics in freefree
photoabsorption, discuss different approximations for their calculation,
and explain why the ion cyclotron harmonics (beyond the fundamental)
cannot be observed.
Key words: stars: neutron  stars: atmospheres  opacity  magnetic fields  Xrays: stars
1 Introduction
Thermal radiation from neutron stars can provide important information about their physical properties. Among neutron stars with thermallike radiation spectra (see, e.g., reviews by Kaspi et al. 2006; and Zavlin 2009), there are two classes of objects of particular interest: central compact objects (CCOs; see, e.g., de Luca 2008) in supernova remnants and Xray dim isolated neutron stars (XDINSs, or the Magnificent Seven; see, e.g., review by Turolla 2009).
The CCOs are young, radioquiet isolated neutron stars with relatively weak magnetic fields G (e.g., Halpern & Gotthelf 2010, and references therein). The XDINSs are older and are believed to have much stronger fields G (van Kerkwijk & Kaplan 2007; Turolla 2009; Haberl 2007). For some CCOs and XDINSs, there are estimates of B, and in some cases only upper limits to B are available.
In the past decade, broad absorption lines have been detected in the thermal spectra of several isolated neutron stars (see, e.g., van Kerkwijk & Kaplan 2007; van Kerkwijk 2004; Haberl 2007, and references therein). In all but one case, the energies of the absorption are centered on the range 0.20.7 keV and the effective blackbody temperatures are keV. Here and hereafter, the Boltzmann constant is suppressed, and the superscript ``'' indicates a redshifted value. In particular, it has been found that: (i) the spectrum of RX J1605.3+3249 with eV has a broad absorption at 0.5 keV and a possible second absorption at 0.55 keV (van Kerkwijk 2004; van Kerkwijk et al. 2004); (ii) RX J1605.3+3249 exhibits an absorption feature at keV (Haberl et al. 2004) and a possible second absorption at 0.57 keV (Hambaryan et al. 2009), while the effective blackbody temperature varies over years across the range 95 eV (Hohle et al. 2009); (iii) the spectrum of RX J1605.3+3249 (RX J 1308.6+2127) was reproduced by a model with eV and two absorption lines at keV and keV (Schwope et al. 2007); and (iv) the spectrum of RX J1605.3+3249 (1RXS J 214303.7+065419) with eV shows indications of a line at 0.4 keV and an absorption edge at 0.730.75 keV (Kaplan & van Kerkwijk 2009; Cropper et al. 2007; Schwope et al. 2009). The first discovered isolated neutron star with absorption lines, CCO RX J1605.3+3249, has two absorption features centered on keV and 1.4 keV (Sanwal et al. 2002) and an effective blackbody temperature (which may be nonuniform) of 0.32 keV (de Luca et al. 2004; Zavlin et al. 1998). For this object, two more harmonically spaced absorption features (at keV and 2.8 keV) were tentatively detected (de Luca et al. 2004; Bignami et al. 2003), but were later shown to be statistically insignificant (Mori et al. 2005). We note that realistic values of the effective temperature , obtained using atmosphere models, can differ from by a factor 23 (see, e.g., Zavlin 2009, and references therein).
Many authors
(e.g., de Luca et al. 2004; Sanwal et al. 2002; Bignami et al. 2003)
have considered the theoretical possibility that the absorption
lines in the thermal spectra of the CCOs and XDINSs
may be produced by cyclotron harmonics,
formed because
of quantum transitions between different Landau levels of charged
particles in strong magnetic fields.
Zane et al. (2001) discussed
this possibility prior to the observational discovery
of these absorption features.
The fundamental cyclotron
energy equals
for the electrons and eV for the ions, where m and M are the electron and ion masses, respectively, Z and A are the ion charge and mass numbers, and G. In the following, we consider protons, whose cyclotron energy is
Beginning with the pioneering work of Gnedin & Sunyaev (1974), numerous papers have been devoted to the physics and modeling of cyclotron lines in Xray spectra of accreting neutron stars (e.g., Nishimura 2005; Daugherty & Ventura 1977; Pavlov et al. 1980; ArayaGóchez & Harding 2000; Wang et al. 1993; Nishimura 2008; Araya & Harding 1999). These emission lines have been observed in many works following their discovery by Trümper et al. (1978). Cyclotron harmonics have been found in spectra of several Xray pulsars in binaries (e.g., Pottschmidt et al. 2004; Enoto et al. 2008; RodesRoca et al. 2009, and references therein), and up to four harmonics were registered for one of them (Santangelo et al. 1999).
In the photospheres of isolated neutron stars, unlike Xray binaries, the typical energies of charged particles are nonrelativistic. In this case, firstorder cyclotron transitions of free charged particles are dipoleallowed only between neighboring equidistant Landau levels and form a single cyclotron resonance with no harmonics. Special relativity and nondipole corrections at the energies of interest can be estimated to be for the electrons and for the protons.
Beyond the first order in interactions, transitions between distant Landau states are also allowed in the nonrelativistic theory. They are, in particular, caused by Coulomb interactions between plasma particles. Thus cyclotron harmonics appear in freefree (bremsstrahlung) crosssections. To obtain 1 keV, one may assume either the electron cyclotron harmonics at 10^{11} G, according to Eq. (1), or proton cyclotron harmonics at 10^{14} G, according to Eq. (2).
Pavlov & Shibanov (1978) presented the calculations of spectra for isolated neutron stars with prominent electron cyclotron harmonics due to the freefree absorption in the atmosphere. Suleimanov et al. (2010b) performed a similar atmosphere modeling and concluded, in agreement with Zane et al. (2001), that electron cyclotron harmonics could be observed in CCO spectra. Proton cyclotron harmonics cannot be calculated based on the assumption of classical proton motion, used by these authors.
In this paper, I review the physics of freefree photoabsorption in strong magnetic fields, discuss restrictions on different published approximations for freefree opacities, and present numerical results that demonstrate the relative strengths of the electron and proton cyclotron resonances under the conditions characteristic of the atmospheres of isolated neutron stars with strong magnetic fields. This gives a graphic explanation of the smallness of the ion cyclotron harmonics. I also demonstrate that the contribution of boundbound and boundfree transitions to the opacities of neutron stars with B>10^{13} G is much larger than that of the proton cyclotron harmonics.
In Sect. 2, quantum mechanical integrals of motion and wave functions of a charged particle in a magnetic field are recalled for subsequent use. Section 3 is devoted to the properties of an electronproton system in a magnetic field that is quantizing for both particles: general equations for calculation of wave functions are given, and the Born approximation is considered in detail. In the same order, general expressions and Born approximation are considered in Sect. 4 for photoabsorption matrix elements and crosssections. Section 5 gives numerical examples of cyclotron harmonics in freefree photoabsorption with discussion and comparison of various approximations. Consequences for the CCOs and XDINSs are discussed in Sects. 6, and 7 presents our summary.
2 Charged particles in a magnetic field
Since special relativity effects are of minor importance in the atmospheres of isolated neutron stars, we use nonrelativistic quantum mechanics.
We assume that the magnetic field vector
is
along the z axis and consider its vector potential
the cylindrical gauge to be
with an arbitrary center in the xy plane.
We recall the description
of a charged particle
in a uniform magnetic field
(e.g., Johnson et al. 1983; Johnson & Lippmann 1949; Landau & Lifshitz 1976).
The Hamiltonian equals the kinetic
energy operator
where m is the mass,
is the kinetic momentum, Q is the charge, and is the canonical momentum conjugate to . In Eq. (4) and hereafter, ``'' denotes the ``transverse'' part, related to the motion in the xy plane.
A classical particle moves along a
spiral around
the normal to the xy plane at the guiding center
.
In quantum mechanics,
is an operator,
related to the pseudomomentum operator
where . Its cartesian coordinates commute with , but do not commute with each other: . Another important integral of motion is the zprojection of the angular momentum , where is the cyclotron frequency.
The eigenvalues of are given by , where is the Landau quantum number. The simultaneous eigenvalues of are ( with integer , and eigenvalues of the squared guiding center equal to , where is the socalled magnetic length.
In general, should be supplemented by , where is the intrinsic magnetic moment of the particle, is the spin operator, and is the spin gfactor ( and 5.5857for the electron and the proton, respectively). In most applications, one can choose the representation where the electron and proton spins have definite zprojections and set the electron gfactor to 2, thus regarding the excited electron Landau levels as double degenerate.
The form of a wave function depends on a choice of the
gauge for
.
We consider the cylindrical gauge given by Eq. (3) centered
on the coordinate origin (
). The
eigenfunctions of H^{(1)} and in the coordinate representation are
where is the particle wave number along the field, L_{z} is the normalization length, ,
is Landau function, the asterisk denoting a complex conjugate, and I_{n'n}(u) is a Laguerre function (e.g., Sokolov & Ternov 1986).
We define cyclic components of any vector
as
and
a_{0}=a_{z}.
The transverse cyclic components of the
kinetic momentum operator given by Eq. (5)
transform one Landau state
,
characterized by
,
into another Landau state
where , if Q<0, and if Q>0.
3 Electronproton system in a magnetic field
The Hamiltonian of the electronproton pair
(i.e., of H atom) is
where . The kinetic part can be written as
(11) 
where m and M are the electron and ion masses, is the center of mass, is the total mass, and is the reduced mass.
Since the electron and the proton have opposite
charges, their orbiting in the transverse plane is
accompanied by a
drift across the magnetic field lines with
velocity
depending on
the distance between their guiding centers
or equivalently on the total pseudomomentum
(12) 
where is the canonical momentum conjugate to .
In quantum mechanics, it is not only true that
the pseudomomentum operator
commutes with H,
but also that its cartesian components
(K_{x},K_{y},K_{z})commute with each other.
Therefore, all components of
can be determined
simultaneously. Coordinate eigenfunctions of
the pseudomomentum operator with eigenvectors are given by (Gor'kov & Dzyaloshinskii 1968)
From the general Schrödinger equation , one can derive an equation for , which has the form
where the effective Hamiltonian depends on .
3.1 Exact solution
Solutions of Eq. (14) for arbitrary
in strong
magnetic fields in the cylindrical gauge
represented by Eq. (3) were
obtained by Vincke et al. (1992) and Potekhin (1994) for bound states,
and by Potekhin & Pavlov (1997) for
continuum states of the electronproton system.
Potekhin (1994) used the variable
as an independent argument
of the wave function
and found that the most convenient
parametrization in Eq. (3) is
Then
where
is the Hamiltonian of the harmonic motion in the xy plane, is the momentum conjugate to ,
We note that H_{K}=0 when .
The first term in Eq. (15) is the total kinetic energy along z, uncoupled from the relative electronproton motion, therefore we set P_{z}=0 without loss of generality.
The eigenvalues of equal where and are the electron and proton Landau numbers, respectively, and are eigenvalues of the relative angular momentum projection operator ( ).
We construct numerical solutions of Eq. (14)
in the energy representation
for
(
)
in the form
where is the composite quantum number enumerating quantum states. One retains in Eq. (19) as many terms ( ; ) as needed to reach the desired accuracy. We choose a principal (``leading'') term (n,s) and define ``longitudinal'' energy of the state as The functions are computed from
where
,
denotes the sum over all pairs (n',s')except
(n'', s''),
and
are effective potentials (see Potekhin 1994 for calculation of these potentials and matrix elements ).
3.1.1 Bound states
Bound states of the H atom can be numbered as , where enumerates energy levels for every fixed pair (n,s) and controls the zparity according to the relation . The longitudinal energies are determined from the system of equations in Eq. (20) together with the longitudinal wave functions .
The atomic states are qualitatively different for small and large values. For small , the electron remains mostly around the proton, the energy dependence on is nearly quadratic, so that the transverse velocity is nearly proportional to , i.e., . The effective mass exceeds and increases with increasing B (Vincke & Baye 1988). For large , the atomic state is decentered (Gor'kov & Dzyaloshinskii 1968): the electron finds itself mostly around , rather than around the proton. In the latter case, decreases with increasing . The two families of states are separated by the critical value of the pseudomomentum, , where the electron wave function is mostly asymmetric, while the transverse velocity of the atom reaches a maximum (Potekhin 1994,1998; Vincke et al. 1992).
3.1.2 Continuum
Wave functions of the continuum are computed using
the same expansion, Eq. (19), and system of equations
in Eq. (20), as for the bound states, but
for a given energy E for every zparity.
The solution is based on a
translation of the usual Rmatrix formalism
(e.g., Seaton 1983) to the case of a strong magnetic field.
Now
,
where
``'' reflects the symmetry condition
.
Numbers n and smark a selected open channel,
defined for
by asymptotic conditions
at
where the pairs , as well as (n,s), relate to the open channels ( ),
is the zdependent part of the phase of the wave function at , and is the wave number. For the closed channels, defined by the opposite inequality , one should select at . If is the total number of open channels at given E, then the set of solutions, defined by Eqs. (20) and (22), constitute a complete set of independent real basis functions. The quantities constitute the reactance matrix , which has dimensions . If the wave functions are normalized according to the condition
then the reactance matrix satisfies the relation
k_{n's'} R_{ns; n's'} = k_{ns} R_{n's'; ns},  (25) 
which differs from the usual symmetry relation (Seaton 1983).
The representation with must be used for continuum states, to ensure that the righthand side (r.h.s.) of Eq. (20) vanishes at , which is required by the asymptotic condition of Eq. (22).
For a final state of a transition,
one should use
wave functions describing outgoing waves.
The basis of outgoing waves with definite zparity
is defined by the asymptotic
conditions
where are the elements of the scattering matrix . The matrix is unitary, but (again unlike conventional theory) asymmetrical. The basis of outgoing waves is obtained from the real basis by transformation
Here, pairs (n,s) and (n',s'), being respective parts of the composite quantum numbers and , run over open channels, but (n'',s'') run over all (open and closed) channels. From the unitarity of the scattering matrix it follows that the wave functions that satisfy the asymptotic condition given by Eq. (26) should be multiplied by a common factor (2L_{z})^{1/2}, to ensure the normalization expressed by Eq. (24). As the initial state of a transition, one should use the basis of incoming waves, .
After the
orthonormalized outgoing waves have been constructed
for each zparity, with symmetric
and antisymmetric longitudinal coefficients
in expansion (19),
solutions for electron waves propagating
at
in a definite open channel (n,s)with a definite momentum
are given by the expansion in Eq. (19)
with coefficients
where the sign + or  represents electron escape in the positive or negative z direction, respectively, and we have suppressed in the subscripts. Waves incoming from with a definite momentum are given by the complex conjugate of Eq. (28).
3.2 Adiabatic approximation
In early works on the H atom in strong magnetic fields, a socalled
adiabatic approximation was widely used (e.g.,
Canuto & Ventura 1977; Gor'kov & Dzyaloshinskii 1968, and references
therein), which neglects all terms
but one in Eq. (19), i.e.,
This approximation reduces the system (20) to the single equation with (n'',s'')=(n,s) and zero r.h.s.
The accuracy of the adiabatic approximation for bound states can be assessed by comparing with the distance between the neighboring Landau levels that are coupled by the r.h.s. of Eq. (20). For an atom at rest (K=0), all the channelcoupling terms become zero for . In this case, the relevant Landau level distance is , while the longitudinal energies of the states with (``tightly bound states'') are 0.3 keV at 10^{14} G , so that the adiabatic approximation is accurate to within a few percent or better. It becomes still better for the ``hydrogenlike states'' with , which have keV.
From comparison of with , one can conclude that the adiabatic approximation is generally inapplicable to a moving atom. However, the accuracy remains good for sufficiently slow atoms, that is when either or (provided that ) (Potekhin 1994,1998). Otherwise, since offdiagonal effective potentials V_{n''s'',n's'}(r_{B},z) in Eq. (20) decrease at more rapidly than diagonal ones, this approximation accurately reproduces wave functions tails at large z, provided that .
For continuum states, the reactance and scattering matrices are diagonal in the adiabatic approximation, with a separate scattering coefficient S_{ns}=S_{ns; ns} for every open channel.
3.3 Born approximation
In the Born approximation, the potential V in a Hamiltonian H=H_{0}+V, which acts on particles in the continuum states, is treated as a small perturbation. We define to be the nonperturbed function, which satisfies the equation . Then from the Schrödinger equation , one obtains the continuum wave function in the first Born approximation in the form , where is determined by the equation
Since we consider the continuum states corresponding to definite Landau numbers (n,N) at (Sect. 3.1.2), the zeroorder wave function is given by the adiabatic approximation with replaced by plane waves.
3.3.1 Two forms of solution
We now consider continuum states.
We first choose the nonperturbed wave function
in the representation where the zprojections of
angular momentum operators,
,
have definite values
for
the electron
and
for the proton. Then
and is governed by the equation
Using an expansion of over the complete set of , we obtain in the standard way
where (since P_{z}=0) and
In the limit , we replace by .
Potekhin & Chabrier (2003) obtained a simpler solution, based on the
representation of quantum states with definite .
In this
case, there are no separate quantum numbers
and .
After applying the
transformation in Eq. (13),
is given by Eq. (29) with
.
Using Fourier transform
(34) 
we obtain from Eq. (20) in the first Born approximation
(35) 
with
where can be presented as a single integral of a combination of elementary functions (Appendix B of Potekhin & Chabrier 2003).
3.3.2 Approximation of infinite proton mass
The neglect of the proton motion is equivalent to the
assumption that
.
In this approximation,
depends only on
in Eq. (30)
without
on the r.h.s.
Then Eq. (32) simplifies to
Taking into account the definition in Eq. (21), we see that this solution is identical to the solution provided by Eqs. (37)(39) in the particular case where , after the obvious replacement of by and by m. The zero value of naturally reflects the condition .
4 Electronproton photoabsorption
4.1 General expressions
The general nonrelativistic formula for the differential crosssection
of absorption of radiation by a quantummechanical system
is (e.g., Armstrong & Nicholls 1972)
where and are the initial and final states of the system, is the density of final states, is the photon energy, is the polarization vector, , is the electric current operator, and is the photon wave number. In our case,
where the velocity operators and are given by Eq. (5).
Equation (42) does not yet include either the photon interaction with electron and proton magnetic moments or . For transitions without spinflip, the latter interaction can be taken into account by adding to the operator the term (cf. Kopidakis et al. 1996), whereas operators are responsible for spinflip transitions (cf. Wunner et al. 1983).
We consider the representation where
and
are definite in
the initial and final states. For an initial state with fixed
n_{i}, s_{e,i}, N_{i}, s_{p,i}, and k_{z}=k_{i} in Eq. (32), and for a final state with either a fixed
zparity or a fixed sign of k_{z}=k_{f}, we
have in Eq. (41)
.
Therefore, the crosssection of
photoabsorption for a pure initial
quantum state
is
where ,
the sum is performed over those n_{f} and N_{f} which are permitted by Eq. (44), and ``'' means the sum over the zparity of the final state (in the case where the parity is definite) or the sum over the signs of k_{f} (in the case where k_{z} in the final state is definite).
In the alternative representation with definite
cartesian components of pseudomomentum
,
using the transformation
in Eq. (13),
one can express the crosssection in terms of the interaction
matrix element between the initial and final
internal states of the electronproton system
(Bezchastnov & Potekhin 1994).
The result has the same form as Eq. (41), but now
is the density of final states at fixed
,
initial and final states
are described by wave functions
,
and the effective current operator in the
conventional representation with
(
)
is given by
where operator is defined by Eq. (17) and The transverse cyclic components of operators and act on the Landau states as
where and .
Changes in
and
induce
transformations of operator
,
studied by Bezchastnov & Potekhin (1994).
In the particular case where for both initial and final states,
the representation with
(
,
)
is used, their result reads
In this representation, instead of Eq. (43), we have
where the sum is performed over those n_{f} and s_{f} that are permitted by Eq. (44). For the solution described in Sect. 3.1, the matrix element in Eq. (49) becomes
Using Eqs. (46) and (47), we can express the transverse matrix elements in terms of Laguerre functions. Hence, Eq. (50) presents a sum of overlap integrals over z. For instance, Eqs. (A7)(A12) of Potekhin & Pavlov (1997) provide an explicit expression in terms of this overlap integrals for the matrix elements of the operator in the approximation where small terms O((m/M) q) are neglected, but separate terms O(m/M) and O(q) are retained.
4.2 Dipole and Born approximations
Hereafter, we use the dipole approximation ().
Then
vanishes, and the total effective
current in Eq. (42) reduces to
(51) 
while the transformed effective current in Eq. (48) becomes
By substituting Eqs. (46) and (47), the sum in Eq. (50) reduces to
(53) 
where
(54)  
(55)  
(56)  
(57) 
and is the proton Landau number. In the first Born approximation (Sect. 3.3),
In the representation where and are definite, using Eq. (30) for and Eq. (32) for , and taking into account the relations in Eq. (9), we can derive the explicit expression for the matrix element in Eq. (43) of
(59) 
where
=  (60)  
=  
(61) 
and .
In the representation where
cartesian components of
have definite values,
using Eqs. (37)(39), (46), and (47),
one can derive the matrix element in Eq. (49)
in the form
where
Substituting Eqs. (62)(64) into Eq. (49), assuming Maxwell distribution of k_{i}, and taking the average over the initial states, we obtain (Potekhin & Chabrier 2003; Potekhin & Lai 2007)
where f^{e}_{n} and f^{p}_{N} are the electron and proton number fractions at the Landau levels n and N,
is the partial crosssection for transitions between the specified electron and proton Landau levels for polarization ,
is the effective partial collision frequency,
is a partial Coulomb logarithm, and
where , , is the Heaviside step function, and
Since , two different Coulomb logarithms and describe all three basic polarizations.
Terms that are proportional to with are absent in Eq. (65), because, for every pair of pure quantum states and , only one of the three basic polarizations provides a nonzero transition matrix element in the dipole approximation.
Potekhin & Lai (2007) mentioned that Debye screening might be taken into account by using as the arguments of in Eq. (68), being the inverse screening length. However, Sawyer (2007), following Bekefi (1966), showed that scattering off a Debye potential is not a valid description of the screening correction for photoabsorption; instead, the integrand in Eq. (69) should be multiplied by where is the electron contribution to the squared Debye wave number .
4.3 Damping factor
Equation (66) gives divergent results
at
for
and at
for
,
because it ignores damping effects due to the finite
lifetimes of the initial and final states of the transition.
A conventional
way of including these effects consists of adding
a damping factor to the denominator in Eq. (66),
which results in Lorentz profiles (e.g., Armstrong & Nicholls 1972).
The damping factor can be
traced back to the accurate treatment of the complex dielectric
tensor of the classical magnetized plasma (Ginzburg 1970).
This treatment allows one to express the
complex dielectric tensor
in terms of the effective collision
frequencies related to
different types of collisions in the plasma.
Imaginary parts of the refraction indexes,
calculated from the complex dielectric tensor,
provide complicated expressions for the freefree
photoabsorption cross sections
for the basic polarizations
.
Based on the assumption that the effective collision
frequencies are small compared to ,
the latter expressions
greatly simplify and reduce to (Potekhin & Chabrier 2003)
where
and being the effective damping factors for protons and electrons, respectively, not related to the electronproton collisions. In general, and may also depend on and . Ginzburg (1970) considers and for collisions of electrons and protons with molecules, whereas Potekhin & Chabrier (2003) take into account damping factors due to both the scattering of light by free electrons and protons and protonproton collisions. The derivation of Eq. (72) from the complex dielectric tensor of the plasma assumes that , , and .
Although the general expressions given in Eqs. (71),
(72) can be established
in frames of the classical theory,
accurate values of the effective frequencies are provided
by quantum mechanics. In our case,
where is provided by Eqs. (67)(70). In the second equality, and are, by definition, Coulomb logarithms for and . Parallel and transverse Gaunt factors (e.g., Mészáros 1992) equal and , respectively.
Since different quantum transitions contribute to the cyclotron resonance at the same frequency ( or , depending on ), their quantum amplitudes are coherent. Therefore it is important that the same damping factor be used in all the transitions (cf. the discussion of radiative cascades in quantum oscillator by CohenTannoudji et al. 1998). Moreover, the same given by Eq. (72) should be used for the absorption and scattering processes. This ensures that the cyclotron crosssection, being integrated across the resonance, provides the correct value of the cyclotron oscillator strength (e.g., Ventura 1979), otherwise the equivalent width of the cyclotron line would be overestimated.
In the electron resonance region, where and , one can neglect , because it is much smaller than 1, and the term that contains , because it is small compared to the other terms. The result coincides with the conventional expression for the electron freefree crosssection without allowance for proton motion with . In the proton resonance region, where and , the denominator in Eq. (71) becomes where In this approximation, Eq. (71) becomes formally equivalent to a simple oneparticle cyclotron crosssection (cf. Eq. (14) of Pavlov et al. 1995, or Eq. (47) of Sawyer 2007), apart from a difference in notations and the difference in (the latter being discussed in Sect. 5.2).
The treatment that leads to Eq. (72) predicts a small shift in the position of the resonance due to the damping. This shift is unimportant for applications and therefore neglected in Eq. (71).
5 Cyclotron harmonics
In addition to the fundamental cyclotron resonances, the quantum treatment of the freefree absorption identifies electron and proton cyclotron harmonics at integer multiples of and , respectively. They appear because of the increase in the partial Coulomb logarithms at . Thus, lth electron cyclotron harmonics (in addition to the fundamental at ) arises at due to the terms with n'n=l+1, and each lth proton cyclotron harmonics (additional to the fundamental at ) is formed by the terms with N'N=l+1 in Eq. (73). Unlike the classical electron and proton cyclotron resonances, the quantum peaks of contribute to at any polarization and are the same for and 1.
The relative strengths of the harmonics depend on the distribution numbers f^{e}_{n} and f^{p}_{N}. In this paper, we assume local thermodynamic equilibrium (LTE) and thus use the Boltzmann distributions, as in most of the previous papers (but see Nagel & Ventura 1983 and Potekhin & Lai 2007 for nonLTE effects on the electron and proton cyclotron radiation rates, respectively).
We calculate freefree crosssections in magnetized neutronstar atmospheres using Eqs. (65)(73). Examples of opacities and/or spectra calculated with the use of these crosssections can be found, e.g., in Potekhin & Chabrier (2003,2004), Potekhin et al. (2004), Ho et al. (2008), Suleimanov et al. (2009,2010a). In previous studies, various additional simplifications have been made in addition to the nonrelativistic, dipole, first Born approximations described above for the freefree crosssections. Below we assess the applicability ranges of these simplifications by comparing with our more accurate results.
5.1 Electron and muon cyclotron harmonics
5.1.1 Fixed scattering potential
In early works (e.g., Mészáros 1992, and references therein),
freefree (or bremsstrahlung) processes were treated assuming
scattering off a fixed Coulomb center, which is equivalent to
the approximation of
,
described in
Sect. 3.3.2. In this approximation, one can set
and explicitly perform the summation over N' in Eq. (65) using the identity
.
Taking damping (Sect. 4.3) into
account, we obtain
where and Assuming Boltzmann distribution ( at , where the factor 2 takes account of the electron spin degeneracy), one can reduce this result to Eq. (27) of Pavlov & Panov (1976) (as corrected by Potekhin & Chabrier 2003).
Figure 1: Transverse Coulomb logarithm as a function of at for different approximations: the model of fixed Coulomb potential (dotted line), approximate account of proton recoil (solid line), adiabatic approximation (longdashed line), and the first postadiabatic approximation (short dash  long dash). The divergent peaks are trimmed at ( ). For comparison, the nonmagnetic Coulomb logarithm (short dashes; in this case the horizontal axis displays ) and the model with approximate account of proton recoil in the muonic atom (dotdashed line) are shown. 

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Figure 2: The same as in Fig. 1 but for the longitudinal Coulomb logarithm. In this case, lines with the approximate account of proton recoil almost coincide with the line corresponding to the fixedpotential approximation. 

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This approximation was used in all models of the spectra of strongly magnetized neutron stars until the beginning of the 21st century (e.g., Zane et al. 2000; Pavlov et al. 1995, and references therein). It is validated by the large value of the mass ratio M/m. In addition, it requires that , as seen directly from the comparison of Eq. (74) with Eq. (71).
5.1.2 Approximate account of proton recoil
Pavlov & Panov (1976) proposed an approximate treatment of proton recoil, which assumes that and does not take into account Landau quantization of proton motion. In Fig. 1, the dotted line shows the perpendicular Coulomb logarithm calculated according to Eqs. (74)(77), while the solid line takes the approximate account of proton recoil. As an example, we show the case where . The familiar nonmagnetic Coulomb logarithm in the first Born approximation (e.g., Bethe & Salpeter 1957) is shown by the shortdashed line, assuming the same along the horizontal axis as for the other curves ( ).
To enhance the difference caused by the recoil, we replace the electron by the muon . All the above formulae and discussion remain unchanged, but now the mass ratio is M/m=8.88. The result of the approximate treatment of the recoil is shown by the dotdashed line.
In Fig. 2, the same approximations are shown for . In this case, the lines related to the cyclotron harmonics are smoothed, because the factors quench the nearthreshold growth of the integrand in Eq. (76). The same smoothing results in the infinite proton mass approximation being even more applicable (under the necessary condition ): the dotted, solid, and dotdashed lines almost coincide in Fig. 2.
5.1.3 Adiabatic and postadiabatic approximations
Several authors (Virtamo & Jauho 1975; Nagel & Ventura 1983; Mészáros 1992) used the adiabatic approximation not only for the unperturbed wave function , but also for . This was done in addition to assuming the infinite proton mass (Sect. 5.1.1). In other words, they kept only one (n,s) term in the sum given by Eq. (40). The result is shown in Figs. 1 and 2 by longdashed lines. We see that this approximation works well at , but becomes inaccurate at .
Sawyer (2007) analyzed the
photoabsorption problem
by using the method of field theory.
In the region
,
he considered
account two electron Landau levels n=0 and 1
and applied a perturbation theory assuming the parameter
to be small.
His result is identical to the results discussed in
Sect. 5.1.1 expanded in powers of
,
which
we can write as
Here and in the next equation, are modified Bessel functions, and Equation (78) differs from Eq. (28) of Sawyer (2007) in two respects: first, we have restored the factor 2 at , and second, we have dropped a term proportional to , because it is of the same order as the contribution from the level n=2, and therefore should be treated together with the latter contribution in the next order of the perturbation theory.
In the same way, we obtain
Equations (78) and (79) can be obtained by the first iteration in the perturbation expansion, starting from the adiabatic aproximation.
5.2 Proton cyclotron harmonics
Proton cyclotron harmonics in the photoabsorption coefficients at are superimposed on the peaks related to the electron cyclotron harmonics. However, for the H atom the two series of harmonics are separated because of the large value of M/m=1836.1. To observe the superimposition and the qualitative differences of various approximations, it is instructive to consider, in place of the H atom, the muonic atom (the system), which has a smaller mass ratio M/m=8.88. The transverse Coulomb logarithm of photoabsorption by such system is shown in Fig. 3. The solid line displays the result of a calculation made according to Sect. 4. The other lines, as well as in Fig. 1, show the results of different approximations: a fixed Coulomb center (Sect. 5.1.1, dotted line), the approximate account of proton recoil (Sect. 5.1.2, dotdashed line), and a nonmagnetic Coulomb logarithm (dashes).
Figure 3: Transverse Coulomb logarithm as function of at in the model of fixed Coulomb potential (dotted line) and the approximate account of proton recoil (dotdashed line), compared to the nonmagnetic Coulomb logarithm (dashed line) and the accurate calculation for the muonic atom (solid line). 

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The smaller peaks in the solid curve correspond to the proton cyclotron harmonics. They are superimposed on the largescale oscillations, which correspond to the muon cyclotron harmonics. Although the approximate recoil treatment (dotdashed line) improves the agreement with the exact calculation compared to the infinite proton mass model (dotted line), both that approximate models that neglect proton Landau quantization differ significantly from the precise result.
In Fig. 4, we compare the proton cyclotron harmonics for different relative masses of the positive and negative particles. Here the proton cyclotron parameter is fixed to , and the horizontal axis displays the ratio . The solid lines show the transverse Coulomb logarithm for the muonic atom (the lower curve) and the H atom (the upper curve). The dashed lines show calculated for the same and the same as the solid curves, but for the approximation of a fixed Coulomb potential in the electron or muon scattering. By comparison, the dotted line shows calculated for proton scattering off a fixed Coulomb center, which can be regarded as a model where . We see that the approximate models are unable to reproduce the proton cyclotron features correctly. It is also noteworthy that the larger the ratio M/m, the smaller the proton cyclotron peaks. In addition, the cyclotron resonance strength decreases with increasing harmonics number l. These properties of the cyclotron harmonics allow us to conclude that the solid lines in Figs. 1 and 2 are precise (proton cyclotron harmonics are negligible on their scale).
Figure 4: Transverse Coulomb logarithm as function of at . The accurate calculation (solid line) for the systems (lower lines) and ep (upper lines) is compared to the approximation of a fixed Coulomb potential for the electron or muon scattering (dashed lines) or for the proton scattering (dotted line), and to the first postadiabatic approximation (short dash  long dash). 

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In the early models of magnetized neutron star atmospheres (e.g., Shibanov & Zavlin 1995; Pavlov & Shibanov 1978; Shibanov et al. 1992), the authors considered moderate magnetic fields 10^{12} G, where the proton Landau quantization is unimportant. More recently, observational evidence has accumulated that some of the isolated neutron stars are probably magnetars, which have fields of G (see, e.g., the review by Mereghetti 2008 and references therein). According to Eq. (2), the proton cyclotron lines of magnetars are in an observationally accessible spectral range, which has encouraged theoretical modeling of these features. In the absence of an accurate quantum treatment, several authors (Ho & Lai 2001; Zane et al. 2000,2001; Ho & Lai 2003; Özel 2001) employed the scaling previously suggested for this purpose by Pavlov et al. (1995), according to which the freefree crosssection for protons equals , where is given by Eq. (74). The latter equation differs remarkably from the correct expression in Eq. (71). At photon frequencies , the difference roughly amounts to a factor of .
In addition, the Coulomb logarithm that determines cannot be obtained from this scaling. An example is shown in Fig. 4, where the dotted line corresponding to the fixedpotential model is compared with the accurate calculations displayed by the solid lines. We see that the fixedpotential model strongly overestimates the strength of the proton cyclotron harmonics. The origin of the discrepancy is clear: while considering a collision of a proton with an electron, one cannot assume the electron to be a nonmoving particle.
Sawyer (2007) employed a representation with definite and and analyzed the first protoncyclotron peak of , in a way similar to his analysis of the first electron cyclotron peak (see Sect. 5.1.3), by taking into account the ground electron Landau level n=0 and two proton Landau levels, N=0 and 1. The result (his Eq. (30)) is quite accurate close to the fundamental cyclotron frequency, as we illustrate wuth the lines of alternating short and long dashes in Fig. 4. In the case of hydrogen (higher M/m), it almost coincides with the accurate result (solid line) at and with the result obtained by neglecting the Landau quantization of protons (dashed line) at higher values.
6 Discussion
6.1 Corrections beyond Born approximation
The formulae presented in Sects. 3.1 and 4.1 in principle allow one to perform an accurate calculation of photoabsorption rates in the electronproton system in an arbitrary magnetic field, taking into account the effects of Landau quantization of the electron and proton motion across the field and the transverse motion of the center of mass. For boundfree absorption, this calculation was presented by Potekhin & Pavlov (1997). For freefree processes, we apply the first Born approximation and the dipole approximation. We plan to perform calculations of the freefree opacities beyond Born approximation in future work. An approximate estimate of the nonBorn corrections can be obtained (Potekhin & Lai 2007) by introducing correction factors into the integral of Eq. (68), where and The accuracy of the approximation is ensured by the smallness of and the additional condition , which is the usual applicability condition for a Born approximation without a magnetic field.
We have checked that these corrections are sufficiently small for the electron cyclotron harmonics at G (relevant to CCOs) and negligible for the proton cyclotron harmonics at B>10^{13} G (relevant to XDINSs).
6.2 Importance of bound states
Figure 5: Opacities for the two normal electromagnetic waves propagating at the angle to the magnetic field direction in a hydrogen atmosphere of a neutron star with G and T=120 eV at density g cm^{3} (which is in the middle of the photosphere at these B and T). The results are shown for fully ionized (dotted lines) and partially ionized (solid and dotdashed lines) plasma models. In the latter model, the nonionized atomic fraction equals 0.0066. The solid line shows the opacity obtained with the accurate calculation of the freefree Coulomb logarithm, and the dotdashed line demonstrates the result of the approximate treatment that corresponds to the dashed line in Fig. 4. 

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Freefree absorption contributes only a part of the total opacities in the atmospheres of neutron stars. A second constituent is the familiar scattering, and a third the absorption by bound species (see, e.g., Canuto & Ventura 1977; Pavlov et al. 1995). It was realized long ago (Ruderman 1971) that in strong magnetic fields the increase in the binding energies of atoms and molecules can lead to their nonnegligible abundance even in hot atmospheres. With increasing B, the binding energies and abundances of bound species increase at any fixed density and temperature T (Potekhin et al. 1999; Lai 2001), so that even the lightest of the atoms, hydrogen, provides a noticeable contribution to the opacities at the temperatures of interest, if the magnetic field is strong enough. Even a small neutral fraction can be important, because the boundbound and boundfree crosssections are large close to certain characteristic spectral energies.
For electron cyclotron harmonics to appear at keV, we should ensure that G according to Eq. (1). At these relatively weak magnetic fields and the characteristic temperature eV, the assumption of full ionization may be acceptable. However, at B>10^{13} G, which is required for ion cyclotron harmonics, the situation is different. An illustration is given in Fig. 5. The solid curves show true absorption opacities for two normal electromagnetic waves propagating at the angle to the magnetic field lines at G and T=120 eV. The upper and lower curves correspond to the ordinary and extraordinary waves, respectively. The density in this example is chosen to be g cm^{3}, which is a typical atmosphere density at G and eV (at this density the thermodynamic temperature T approximately equals the effective temperature ). According to our ionization equilibrium model (Potekhin et al. 1999), at these B, T, and values, 0.66% of protons in the plasma are comprised in the groundstate H atoms that are not too strongly perturbed by plasma microfields so that they contribute to the boundbound and boundfree opacities (the ``optical'' atomic fraction), and only 0.1% of protons are in excited bound states. Even though the groundstate atomic fraction is small, it is not negligible. In Fig. 5, at keV, the opacities in two normal modes, calculated with allowance for partial ionization (solid and dotdashed curves), are significantly higher and have more characteristic features than the opacity calculated under the assumption of complete ionization (dotted lines). In particular, the broad feature on the lower curve near 0.4 keV is produced by the principal boundbound transition between the two lowest bound states ( ), and the increased value of the opacity at higher energies is due to the transitions to other bound and free quantum states. The wavy shape of the lower solid curve (for the extraordinary mode) at keV is explained by boundfree transitions to different open channels, each having its own threshold energy. All the boundbound absorption features and photoionization thresholds are strongly broadened by the effects of atomic motion across the magnetic field lines (``magnetic broadening'', see Potekhin & Pavlov 1997 and references therein).
In the insets, we zoom in on the regions of the first and second proton cyclotron harmonics. Both of them are visible, but negligible compared to the effect of partial ionization on the opacities.
6.3 Other possibilities for CCOs and XDINSs
Apart from the cyclotron harmonics, a number of alternative explanations of the observed absorption features in CCOs and XDINSs have been suggested in the literature.
Mori & Ho (2007) constructed models of strongly magnetized neutron star atmospheres with midZ elements and compared them to the observed spectra of the neutron stars 1E 1207.45209 and RX J1605.3+3249. They demonstrated that the positions and relative strengths of the strongest absorption features in these neutron stars are in good agreement with a model of a strongly ionized oxygen atmosphere with B=10^{12} G and B=10^{13} G, respectively. This explanation seems promising, but unsolved problems remain: the effects of motion across the field have been treated approximately, based on the assumption that they are small, and detailed fits to the observed spectra have not yet been presented.
Among other hypotheses about the nature of the absorption features, there was a suggestion that they could be due to boundbound transitions in exotic molecular ions (Turbiner & López Vieyra 2006). However, our estimates show that the abundance of these ions in a neutron star atmosphere would be negligible compared with the abundance of H atoms. Suleimanov et al. (2009) proposed a ``sandwich'' model atmosphere of finite depth, composed of a helium slab above a condensed surface and beneath hydrogen, and demonstrated that this model can produce two or three absorption features in the range of 1 keV at G, although a detailed comparison with observed spectra was not performed. One cannot also rule out that some absorption lines originate in a cloud near a neutron star, rather than in the atmosphere (see Hambaryan et al. 2009).
7 Summary
We have considered the basic methods for calculation of freefree opacities of a magnetized hydrogen plasma. Our emphasis has been on the case where not only electron, but also proton motion across the magnetic field is quantized by the Landau states. We have derived general formulae for the photoabsorption rates and considered in detail the dipole, first Born approximation. We have presented numerical examples, compared them with the results of previously used simplified models, and analyzed the physical assumptions behind the different simplifications and conditions of their applicability. We have demonstrated that the proton cyclotron harmonics at a given value of the parameter are much weaker than the respective electron cyclotron harmonics at the same value of , and explained this difference in terms the large (nonperturbative) effects of proton motion in the case of proton cyclotron harmonics, in contrast to the case of the electron cyclotron harmonics.
AcknowledgementsI am pleased to acknowledge enlightening discussions with Gilles Chabrier, Gérard Massacrier, Yura Shibanov, and Dima Yakovlev, and useful communications with Ray Sawyer. This work is partially supported by the RFBR Grant 080200837 and Rosnauka Grant NSh3769.2010.2.
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All Figures
Figure 1: Transverse Coulomb logarithm as a function of at for different approximations: the model of fixed Coulomb potential (dotted line), approximate account of proton recoil (solid line), adiabatic approximation (longdashed line), and the first postadiabatic approximation (short dash  long dash). The divergent peaks are trimmed at ( ). For comparison, the nonmagnetic Coulomb logarithm (short dashes; in this case the horizontal axis displays ) and the model with approximate account of proton recoil in the muonic atom (dotdashed line) are shown. 

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In the text 
Figure 2: The same as in Fig. 1 but for the longitudinal Coulomb logarithm. In this case, lines with the approximate account of proton recoil almost coincide with the line corresponding to the fixedpotential approximation. 

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In the text 
Figure 3: Transverse Coulomb logarithm as function of at in the model of fixed Coulomb potential (dotted line) and the approximate account of proton recoil (dotdashed line), compared to the nonmagnetic Coulomb logarithm (dashed line) and the accurate calculation for the muonic atom (solid line). 

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In the text 
Figure 4: Transverse Coulomb logarithm as function of at . The accurate calculation (solid line) for the systems (lower lines) and ep (upper lines) is compared to the approximation of a fixed Coulomb potential for the electron or muon scattering (dashed lines) or for the proton scattering (dotted line), and to the first postadiabatic approximation (short dash  long dash). 

Open with DEXTER  
In the text 
Figure 5: Opacities for the two normal electromagnetic waves propagating at the angle to the magnetic field direction in a hydrogen atmosphere of a neutron star with G and T=120 eV at density g cm^{3} (which is in the middle of the photosphere at these B and T). The results are shown for fully ionized (dotted lines) and partially ionized (solid and dotdashed lines) plasma models. In the latter model, the nonionized atomic fraction equals 0.0066. The solid line shows the opacity obtained with the accurate calculation of the freefree Coulomb logarithm, and the dotdashed line demonstrates the result of the approximate treatment that corresponds to the dashed line in Fig. 4. 

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In the text 
Copyright ESO 2010
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