A Quantum Mechanical Treatment of Electron Broadening in Strong Magnetic Fields. II. Large Enhancements due to Exchange Interactions

We present a quantum treatment of atom–electron collisions in magnetic fields, demonstrating the significant importance of including the effect of exchange that arises from two interacting electrons. We find strange behaviors that are not encountered in collisions without a magnetic field. In high magnetic fields, exchange can lead to orders of magnitude enhancements of collision cross sections. Additionally, the elastic collision cross sections that involve the ground state become comparable to those involving excited states, and states with large orbits have the largest contribution to the collisions. We anticipate significant changes to spectral line broadening in neutron star surfaces and atmospheres.


Introduction
In Gomez et al. (2023), hereafter Paper I, we began the process of developing a quantum mechanical model for electron broadening in collisions in the presence of a large magnetic field.The ultimate goal of this work is to provide accurate diagnostics for magnetic white dwarfs and neutron stars.
Paper I laid the groundwork for constructing an electronbroadening model, but neglected to include a detailed treatment of the motional Stark effect and the important property that the colliding electrons are indistinguishable fermions.When accounting for the indistinguishability of electrons, the wave function that describes the total system is antisymmetric with exchange of coordinates, resulting in exchange interactions; this effect is sometimes referred to as "Pauli repulsion."The impact of exchange interactions on electron-atom collisions in magnetic fields is unknown, but is expected to be different due to the change in the geometry of the scattering problem, i.e., going from spherical to cylindrical symmetry.In the absence of a magnetic field, the strongest scattering events take place when the perturbing electron is the closest to the atom.Exchange processes are also strongest for short-range collisions, but with a different symmetry and atomic structure in a high magnetic field, this may no longer be the case.
Our aim in this paper is to explore how exchange affects the electron-electron part of the scattering problem in the presence of a magnetic field.The results that we present here will not be completely representative of the physical system in neutron star atmospheres.More specifically, we ignore the detailed treatment of the motional Stark effect due to its complexity in the collision problem, but plan to consider it in a future study.Despite the omission of this effect in the present study, we are able to provide important insights into the collision problem in magnetic fields.
The rest of the paper is organized as follows.Section 2 discusses atomic structure in a magnetic field.The collision process is reviewed in Section 4, and our results are presented in Section 5. We discuss how the motional Stark effect will modify our results in Section 6. Last, Section 7 presents our conclusions.

Atomic Structure in a Large Magnetic Field
The total nonrelativistic Hamiltonian of a two-body system in a magnetic field is given by (Equation B11 in Johnson et al. 1983) where μ, m e , m N , and M 0 are the reduced, electron, nuclear, and total masses; the coordinates denoted by R and r are the centerof-mass coordinates and the relative coordinates, respectively.The first term is the center-of-mass kinetic-energy operator, the second term is the canonical momentum of the relative coordinates, the third term is the nuclear potential, the fourth is the linear Zeeman effect, the fifth is the diamagnetic term, and the final term is the Hamiltonian term due to the motional Stark effect.Here, the center-of-mass operators are given as capital letters, and the relative operators are set in lower case.It is important to note that the ratios of the electron mass to the nuclear mass become important at high field values and high values of the angular momentum, and they can substantially change the energy (Ruder et al. 1994).
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In a large magnetic field, the motion of the center of mass is coupled to the internal coordinates.It is common practice (e.g., Johnson et al. 1983) to perform a pseudo-separation of coordinates, and then perturbatively solve for the coupling.For a hydrogenic system of arbitrary nuclear charge, the correction for the motional Stark effect, which connects the true mechanical momentum to the internal states, is given in the final term e N e N s MSE 0 where It is customary to solve Equation (1) without the motional Stark correction first and to then include it at a later step.The presence of a strong magnetic field requires solutions to the Schrödinger equation (even without accounting for the motional Stark effect) that are no longer separable in any given coordinate system.There are multiple ways to solve this problem.In the case where the nuclear charge dominates, the wave function solutions can be built up as a linear combination of spherical states.When the magnetic field dominates, the wave function solutions are built up from cylindrical Landau states.The wave function solution is then given by where f nm (ρ, j) are the Landau states, and where corrections for m e /m N have been suppressed for brevity.This latter method is particularly effective for higher m states, but it is problematic for m = 0 states due to the inability of the method to accurately capture the singularity at the origin.We solve this issue by using a numerical 2D Schrödinger solver based on the method of Gomez et al. (2018).Corrections due to the motional Stark effect must be made perturbatively.
As mentioned previously, for this particular study, the motional Stark effect is ignored.This omission means that the results do not fully account for all important physical effects.We further discuss this issue in Section 6 after reviewing the results.

Line Broadening and Collisions
Our line-broadening model follows from the relaxation theory of Fano (1963).Under the usual approximations, the width and shift operator from the plasma electrons is given by a combination of T-matrices in the tetradic notation, where the first two terms are the upper-and lower-state broadening terms, the last two terms arise from interference between the upper and lower states, and the Greek letters denote atomic states and lowercase Roman letters indicate perturbing electron states.To calculate the width, a thermal average needs to be performed, which usually is taken to be a Boltzmann distribution.This formulation makes the independent-particle approximation, which ignores the correlations between plasma particles, but preserves the time-dependence of the collisions.In practice, correlations are accounted for by screening atom-plasma interactions (Smith & Hooper 1967).However, since the atomic structure in a magnetic field is in the isolated-line limit (i.e., the separation between state energies is large), screening is relatively unimportant, and for the purposes of the study here, we ignore it.We note that in the original derivation, Fano (1963) produced a formula that included a projection operator.Due to the N-body nature of the problem, the projection terms cancel, and it is often justified in line broadening without a magnetic field (Iglesias & Gomez 2024).The terms that arise from the projection operator will not cancel when there is a magnetic field.This is because for a single radiator in a plasma, the motional Stark effect term arises from a single particle rather than from the ensemble bath.
In a large magnetic field, there will not be any degeneracy in the states for H-like systems (which is our sole focus in this study), and the broadening will be in the isolated-line limit.In this limit, the T-matrices in Equation (7) will only be composed of elastic T-matrices.Therefore, for our study here, we consider the impact of improved collision physics on elastic T-matrices.The elastic T-matrix is proportional to the state's total collision cross section (via the optical theorem), and it therefore indicates the important collision interactions and processes.

Electron-Atom Collisions in a Magnetic Field
The classical description of scattering that is found in textbooks describes the wave function as an incoming plane wave and an outgoing spherical wave, ikz ikr where f (θ, j) is the collision amplitude.This picture does not apply in a large magnetic field, where electron-atom collisions more closely resemble concentric cylinders.In a strong magnetic field, the magnetic field significantly alters the trajectories of the projectile electrons, limiting the free motion perpendicular to the field while still propagating freely parallel to the field.This effectively reduces scattering in a magnetic field to a 1D problem rather than the usual 3D problem.In this effective 1D problem, we have an incoming wave and outgoing reflected and transmitted waves.

Behavior of Free Electrons in a Large Magnetic Field
The Hamiltonian for a free particle in a magnetic field (which is conventionally aligned in the z-direction) contains a harmonic oscillator potential in the x-y direction (Ventura 1973;Clark 1983), where we have used atomic units throughout (m e = ÿ = e = a 0 = 1 and β = B/B 0 , where B 0 = 2.35 × 10 9 G), the superscript p denotes the "projectile" in the scattering problem, and the subscript 0 denotes that the particle does not interact with other electrons.Here, L z and S z are orbital and spin angular momentum operators with eigenvalues m and m s , respectively, and V nuc (z, ñ) is a long-range nuclear potential.The only direction in which electrons can freely propagate is the z-direction.The magnetic field states and energies are described by respectively, and where k is the momentum in the z-direction, n is the radial quantum number, σ is the sign of the charge, and m is the azimuthal/magnetic quantum number, defined the same way as in the field-free case, and m s is the quantum number describing the component of the spin along the z-axis.Here, N is a normalizing factor, and is an associated Laguerre polynomial.The presence of the magnetic field creates wave functions that decay to zero as ñ becomes large, i.e., the wave functions are bound in the x-y direction.The n quantum number describes the number of nodes of the wavefuction in the ñ direction.The m quantum number indicates the number of nodes of the wave function in the j direction, but is also analogous to angular momentum in the spherical case.This means that larger |m| corresponds to cylindrical orbits with larger radii.Throughout, we refer to states with large |m| as wide orbits.
There are other representations that can be used, such as in Pavlov & Bezchastnov (2005), where instead of using the nm basis, the ˆns basis is used (where the hat is used to avoid confusion).The energy in the latter representation is The conversion between these two representations is (Johnson et al. 1983) The ˆns representation, which has been used in bare Coulomb scattering (Ventura 1973;Brandi et al. 1978;Ferrante et al. 1980), does not provide any particular advantage over the nm representation, and both deal with infinite degeneracies (Johnson et al. 1983).Because our study focuses on the Coulomb problem, which is diagonal in the total azimuthal quantum number (Mott & Massey 1949;Cowan 1981) where from here on out, m a is the azimuthal quantum number from the target atom, and m p is the azimuthal quantum number from the projectile/plasma electron.Therefore, due to this explicit dependence on m a and m p , the n p m p representation is most convenient for the atomic collision problem.
For scattering, the fact that the projectile electron can scatter into different n p or m p quantum numbers that describe bound states means that the problem of scattering in a magnetic field is effectively reduced to a 1D problem.Electrons can therefore only scatter freely in the z-direction, where the resulting collision will result in either reflected or transmitted waves.
Fortunately, the formulation for scattered waves is unchanged by the presence of a magnetic field.The difference is that the set of wave functions to construct the scattered wave are those in Equation (9) instead of 3D plane waves.The scattered wave is defined as in Bray & Stelbovics (1992), where c ñ | i is the set of perturber wave functions defined in Equation (10), y ñ | i is a set of target atomic states that is the solution to Equation (1) ignoring motional Stark, and T(E) is the collision "transition" or T-matrix, defined in Lippmann & Schwinger (1950) where V is the interaction potential, and H 0 is the Hamiltonian of the noninteracting system.The T-matrix is closely related to the collision amplitude (Baranger 1958), and the imaginary part of an elastic T-matrix is a proxy for the total collision cross section through the optical theorem,

Antisymmetry of the Total Wave Function
An important property of identical quantum particles is that they are indistinguishable.For electrons (which are fermions), this means that their wave functions are antisymmetric with respect to an exchange of coordinates, where S is the total spin of the atom+projectile system; S = 0 corresponds to singlet scattering where the spins of the particles are antisymmetric, and S = 1 is triplet scattering where the spatial wave functions are antisymmetric.Since electrons are fermions, they cannot occupy the same state and are described by antisymmetric wave functions; the result is a repulsion between electrons named after Wolfgang Pauli; this is sometimes known as Pauli repulsion or Pauli exclusion (Bethe & Salpeter 1957).Explicit inclusion of exchange in the scattering problem leads to nonuniqueness for the scattered wave that is addressed by imposing (Bray & Stelbovics 1992;Bray 1994) Equation ( 16) is implemented by modifying the exchange terms in the atom-projectile interaction, where V ap and V p N are the direct atom-projectile and nuclearprojectile potentials, respectively, H and E are the total Hamiltonian and energy, respectively, and P is the space exchange operator.For the new form, is a unit-like operator acting on the projectile electron space, and θ is an arbitrary nonzero scalar.The T-matrix solutions are independent of the value of θ when it is nonzero.We note that in low magnetic field cases, |θ| needs to be large to satisfy the property in Equation ( 16).This requirement is likely due to the competing symmetries: at low fields, the target atom is more spherical, but the projectile is still cylindrical.At high magnetic fields seen in neutron stars, such as 10 12 G, both target and projectile are cylindrical, and nearly any nonzero value of θ can be used.

Results
Since the elemental composition of neutron star atmospheres is unknown, we explore a couple of different cases.Predictions for the composition of nonaccreting neutron star atmospheres range from light elements, such as hydrogen or helium (Ho et al. 2003;Güver et al. 2011), to mid-Z elements, such as carbon, oxygen, or neon (Mori & Hailey 2006;Ho & Heinke 2009;Alford & Halpern 2023), and Fe (Rajagopal et al. 1997;Nättilä et al. 2015).Therefore, for this study, we examine how magnetic fields affect electron collisions of both neutral hydrogen and highly ionized oxygen.
Due to the novelty of collisions in magnetic fields, our examples are limited to one-electron atoms.An extension to two or more electron atoms will be the subject of future study.Currently, due to the lack of electron-atom collisions in a magnetic field, we can learn much about collision processes in one-electron atoms.
We specifically examine the imaginary part of the elastic Tmatrix, as it is representative of the total cross section through the optical theorem (Mott & Massey 1949;Lippmann & Schwinger 1950;Gomez et al. 2021).In Figure 1, we compare the 1s state of neutral hydrogen elastic imaginary T-matrices from the direct-only, singlet, and triplet interactions (black, blue, and red lines, respectively, in Figure 1) for β = 0, 1, and 10 3 with different physics included.These conditions were chosen to sample a range of relative importance between Coulomb and magnetic interactions.At β = 0, the Coulomb interaction between particles alone dictates electron motion.At β = 1, the magnetic energy is on the same order as the nuclear energy; this is at the lower end of the range of magnetic fields expected to be found in neutron stars.Finally, at β = 10 3 , the magnetic energy dominates the Coulomb interactions and is a "typical" field found on neutron star surfaces.
Figure 1 shows that for low and high energies of the projectile, exchange is a modest correction to the T-matrices for all values of β.However, for k values that correspond to wavelengths on the order of the atom's dimension, exchange becomes large compared to direct interactions, and T-matrices that include exchange are larger by more than an order of magnitude than T-matrices that include direct-only interactions for β 1.This increase appears as a broad contribution and is not due to dielectronic resonances.These significant increases in collision amplitudes/T-matrices are largely due to the confining nature of the magnetic fields.In the absence of a magnetic field, when an electron starts to experience exchange, the electron can scatter into a nonzero angle.In the presence of a large magnetic field, however, scattering into an angle away from the z-axis is extremely difficult.In order to scatter into an excited Landau state, the electron must have a kinetic energy in excess of the magnetic energy.Therefore, when constrained to a particular trajectory, the electron is forced to interact with the atom, and its only scattering options are to transmit to the other side or completely change its direction of propagation in the form of a reflected wave.Therefore, the electrons are forced to experience the Pauli repulsion, which then becomes the dominant collision process, which means that collisions at wide orbits (large |m p |) are particularly effective.
1. Importance of exchange for the collision T-matrices of the 1s state of neutral hydrogen in high magnetic fields.We have three cases: zero field (dotted line), β = 1 (dotted-dashed line), and β = 10 3 (solid line).For these three cases, we show the imaginary part of the elastic T-matrix as a representation of the total cross section through the optical theorem.We perform three types of calculations: direct-only, singlet scattering, and triplet scattering.At β = 1, the T-matrices decrease, but as the field goes up, the Tmatrices increase over their field-free values.

O VIII in a β = 10 3 Field
In Figure 2, we present the 1s m=0 and 2p m=−1 imaginary Tmatrices of H-like oxygen in a β = 10 3 field.We note the unusual behavior in the collision strengths of different levels.For one, ordinarily, the T-matrices of excited states are usually much larger than the ground state; this behavior is the reason why line-shape calculations can neglect the collisions of the lower state for K-shell transitions (Gomez et al. 2022).This behavior is less extreme in the high magnetic field case when only direct interactions are included with 2p about 4-5 × larger than 1s.When Pauli repulsion is included, however, Figure 2 shows that the elastic collisions of 1s are stronger than 2p m=−1 up to ε knm ∼ 100 eV, a behavior that is only seen in these high magnetic fields.
There is a stark difference in the amount of increase seen in neutral hydrogen versus ionized H-like oxygen.The latter is demonstrated in Figure 2, where Pauli repulsion leads to increases in the imaginary part of the T-matrices of four orders of magnitude.Elastic scattering off ions has an additional complication over neutrals: the bound states of the projectile have to be included (Bray 1994;Gomez et al. 2020).In H I and even more so in O VIII, much of the T-matrix enhancement is due to the slow convergence of states with high m values.In Figures 1  and 2, we included |m| up to 180 and 250, respectively.

Slow Convergence of the T-matrix Solution with m
When we break down the T-matrix solutions by individual m, we see strange behavior.Namely, the values of the T-matrices increase as the partial waves increase (see Figure 3).In the absence of a magnetic field, we expect that higher partial waves, m, contribute progressively less to the total T-matrix elements.In a large magnetic field, that is not the case, however.Rather, we see significant enhancements of Tmatrices at higher partial waves at varying impact energies.It appears that many of these enhancements come in the form of broad resonances.The widths of these resonances are broader than the changes in the resonances with m p , and they therefore appear as the flat behavior in Figure 2.
We find that the reason for this enhancement is largely due to exchange interactions.When m p is large, exchange interactions of the type become the dominant type of collision interaction, and more m states (both atomic and projectile) are needed to achieve convergence.Figure 4 demonstrates the slow rate of convergence for H I at β = 10 3 , and the calculation did not converge until m ≈ −3000.The convergence of the collision cross sections is dependent on the strength of the magnetic field.For β = 10 2 field, the cross sections converge around an m ≈ −300.Further, at β = 10, convergence occurs around m ≈ −40.With this trend, we determine empirically that for high magnetic fields, to ensure convergence.The position operator in the direction perpendicular to the magnetic field scales roughly as b m (Ruder et al. 1994).Therefore, we can use the criteria in Equation (20) to create another empirical constraint that the expectation value of the radius of the projectile electron has to exceed We make a further note that the resulting T-matrices for the high m states are strongly dependent on the energy of the target atom.The energy of the target states can significantly change depending on the mass of the nucleus (Johnson et al. 1983;Pavlov & Meszaros 1993; see Equation (1)).Changing the mass terms in the Hamiltonian for deuterium changes the energy, and since energy is included in the exchange part of the 3. Breakdown of the collision T-matrices by m for the 1s state of O VIII in a β = 10 3 field.The enhancement at low energies comes from a resonance that moves with each m, but the width of the resonance is more significant than the movement of the resonance.This causes the relatively flat behavior in Figure 2. interactions, the resulting T-matrix solutions are therefore dependent on the mass of the nucleus.

Applicability
One consideration that is important for high magnetic fields that generally is not considered for collision problems is the motion of the atom.Transformation into center-of-mass coordinates usually results in negligible corrections.These corrections, when multiplied by a large magnetic field, are now no longer negligible, however.Further, the motion of the atom is coupled to the motion of the electrons relative to the atom's center of mass.

Mass Correction for the Projectile in the Relative
Coordinate System In Equation ( 9), the Hamiltonian used to describe the motion of the projectile electron was performed in the laboratory frame.The proper collision problem would need to be performed in the reference frame of the atom.This involves a transformation into relative coordinates.The transformation into relative coordinates, worked out in Appendix B of Paper I, results in a motional Stark correction for the projectile electron coordinates (elaborated in Section 6) as well as mass corrections to the projectile Hamiltonian (Equation ( 1 Ordinarily, these mass corrections would be a negligible correction, but because of large β, they cannot be ignored (Ruder et al. 1994).The effect of the first term would systematically shift the projectile energies toward positive values.The resulting shifts, for instance, for m p = −100 at β = 10 3 would result in a 3.4 hartree (∼92.5 eV) shift in oxygen and a 54.4 hartree (∼1480 eV) shift in hydrogen.However, accounting for the mass correction in the dielectric term cancels the shift from the first term, leaving the results presented above unaltered.

Motional Stark Effect
It is imperative that we acknowledge that we have omitted the motional Stark effect.This effect is an additional electric field that is applied on the atomic electron as a result of the atom moving through the magnetic field, which was given in Equation (2).Proper inclusion of the motional Stark effect is complicated, and a full treatment is reserved for a future publication.There are two major complications, the first being that m a is no longer a good quantum number to describe the motion of the electron and atom, and the second complication is that there is a motional Stark component coming from the plasma electrons (see Appendix B of Paper I).The details of both effects will be explored in a future publication.For now, it is sufficient to discuss the constants of motion for the problem in order to gauge the applicability of the results presented here.
The total wave function of a hydrogenic atom where motional Stark is omitted can be described by the set of where the ion and electronic quantum numbers, ˆk n s c c c and ν a n a m a , respectively, are independent of each other.From here on out, we suppress the ŝc quantum number as it plays no role in the motional Stark effect.To include the motional Stark effect, the quantum number = -ˆˆ( ) p n m 24 c a is introduced, which is a constant of motion of the combined system (Johnson et al. 1983), and the matrix elements of Equation (2) are diagonal in p.The quantum number p can take any positive or negative integer value, but the only restriction is that nc cannot be lower than zero.
In the collision problem, p is no longer a conserved quantity of the motion of the system.a a , these results are virtually unchanged, as is the case for oxygen when nc is small.Preliminary calculations of the atomic structure of the isolated atom with motional Stark result in relative shifts of only some eV in the energies of the 1s → 2p m = −1 and 1s → 2p m = 0 transitions, a minor correction that does not significantly change the results presented here.However, wide orbits such as those described in Equation (26) will be shifted by a substantially greater amount, and the wave function needs to be built from a linear combination of states, ; .27 i p j ij c i i i , i For example, the lowest-energy level of the m = −100 state of oxygen without motional Stark is −11.1 hartree, but with the motional Stark term of the Hamiltonian, this state is shared between about 17 states ranging between −12.7 and −9.27 hartree, resulting in effectively no change in the energy of that state.The level of mixing tell us that the motional Stark effect caused m a to no longer be a good quantum number.However, the energy of the states has been largely unchanged, meaning that the results described here are still applicable to a physical system, with motional Stark likely changing the collision cross sections presented here only moderately.

Conclusions
The present calculations demonstrate that magnetic fields can cause exchange collision processes to dominate direct interactions.Inclusion of exchange results in large (orders of magnitude) increases in the elastic collision T-matrices, which are representative of the total cross section.These enhancements cause some unusual behavior in the T-matrix solutions that is not seen in nonmagnetic cases, such as the cross section of the ground state having similar total cross sections (if not a larger cross section) than those of excited states.These enhancements will have a major effect on the spectral line broadening (Gomez et al. 2023).
In addition, this increase may result in collisions that are much stronger than previously thought, although more work is needed before emergent spectral models are mature enough to be used.The motional Stark effect needs to be included in the collision process for both the projectile and the target atom.The work presented here is an essential aspect of studying the spectra of neutron stars and-with measurements with a high enough resolution-allows us to potentially determine the gravity of neutron stars and constrain the equation of state of the interior.

Figure
Figure of Pauli repulsion on the collision T-matrices on the 1s m=0 (red lines) and 2p m = −1 (black lines) state of H-like oxygen at β = 10 3 .The dotted lines neglect Pauli repulsion, and the solid lines include it.The most extreme increase in the T-matrices is in excess of a factor of 10 4 .Pauli repulsion causes the elastic T-matrices of the ground and excited states to become comparable.The sum over m goes from 0 to −250.

4.
Breakdown of the collision cross section by a partial wave of the 1s state of H I in a β = 10 3 field at T eff = 400 eV.The individual contributions of the total T-matrix are black crosses connected by a solid black line.The dotteddashed gray line indicates the cumulative cross section up to that partial wave (i.e., at m = −20, and the gray line indicates the sum of partial waves between 0 and −20).It is clear that the convergence as a function of partial wave is particularly slow.
)). Probably the most important mass correction is on the second term of the projectile Hamiltonian.The mass-corrected term becomes dominate the large increases in the T-matrices, which involve no change in m a − m p .Therefore, preserving pT involves no change in nc .Therefore, in the limit that y