A Transformation from density matrix kernel to distribution function for the electron in magnetic field

Relativistic kinetic equation for Compton scattering of polarized radiation in strong magnetic field

Abstract

We derive the relativistic kinetic equation for Compton scattering of polarized radiation in strong magnetic field using the Bogolyubov method. The induced scattering and the Pauli exclusion principle are taken into account. The electron polarization is also considered in the general form of the kinetic equation. The special forms of the equation for the cases of the non-polarized electrons, the rarefied electron gas and the two polarization mode description of radiation are found. The derived equations are valid for any photon and electron energies and the magnetic field strength below about G. These equations provide the basis for formulation of the equation for polarized radiation transport in atmospheres and magnetospheres of strongly magnetized neutron stars.

pacs:
52.25.Dg, 52.25.Os, 95.30.Gv, 95.30.Jx, 97.60.Jd

I Introduction

Observations of the soft gamma-ray repeaters and anomalous X-ray pulsars showed that these objects can be associated with the strongly magnetized neutron stars (NSs) with the magnetic field exceeding the Schwinger critical value of G Mereghetti et al. (2002); Woods and Thompson (2006). This has revived the interest in theoretical studies of the interaction processes between radiation and matter in such fields Harding and Lai (2006).

Compton scattering is an important process shaping the radiation spectra of the NS atmospheres. Its properties in the magnetic field differs substantially from the case when the magnetic field is absent. Even the classical non-relativistic limit of the scattering cross section has a resonance at the energy related to the Lorentz frequency and is strongly dependent on the photon energy, polarization and the B-field strength Canuto et al. (1971); Ventura (1979). While the classical description has been useful for understanding the approximate effects of energy, angle and polarization dependence of the cross section in the magnetic field, it does not include the possibility of the electron excitation to a higher Landau state corresponding to the resonances at higher harmonics, which required fully relativistic treatment. In the relativistic regime the recoil of the electron is important and the natural line width of the cyclotron resonances depends on the spin of the electron. The relativistic scattering cross section for the simplest case of ground-to-ground state scattering in the magnetic field was derived in Herold (1979). These results were extended to a more general case of scattering to arbitrary Landau states in Daugherty and Harding (1986); Bussard et al. (1986) and discussed further in Meszaros (1992). The derived expressions have been applied to modeling the cyclotron line formation in accreting neutron star atmospheres, but only for the case of one-dimensional thermal electron distribution because of the complexity of the expressions Alexander and Meszaros (1989, 1991); Harding and Daugherty (1991); Araya and Harding (1996, 1999). When the incident photons propagate along the magnetic field, the resonance appears only at the fundamental frequency and scattering to the higher Landau levels can effectively be neglected. This allows to simplify the expressions for the relativistic cross sections and to approximate them by analytical formulae Gonthier et al. (2000).

The transport of photons through the atmosphere involves multiple scattering, which have to be considered either by the Monte Carlo methods or using the kinetic equations. The former approach was used for a qualitative study of the line formation process in Her X-1 Yahel (1979, 1980), but it becomes impractical for a large optical depth and when the induced scattering has to be accounted for, and therefore has a limited field of applications. In the cold plasma approximation, assuming the coherent scattering, the radiative transfer equation can be formulated as a set of coupled equations for two normal polarization modes Gnedin and Pavlov (1974). The influence of the electron temperature on the radiation transport can be accounted by the Fokker-Planck approximation, for example, by modifying the Kompaneets equation Kompaneets (1956) to allow for the resonances in the scattering cross section Bonazzola et al. (1979). Such a treatment, however, does not account for the effects of the photon angular distribution and polarization. Photon polarization, however, influences the photon redistribution over the energy Nagel (1981); Pavlov et al. (1989).

In a sufficiently strong magnetic field, owing to the large Faraday depolarization, the radiation can be described in terms of two polarization modes. Under certain conditions (depending on the field strength, photon energy and propagation direction), however, the vacuum resonance is accompanied by the phenomenon of mode collapse and the breakdown of Faraday depolarization Zheleznyakov et al. (1983); Lai and Ho (2003); Pavlov and Shibanov (1979). In this case the two-mode description fails and instead the kinetic equations have to written in terms of the Stokes parameters or the coherency matrix. In the case when the induced scattering needs to be accounted for, the situation complicates further as there is no intuitive way to get such an equation.

The aim of this paper is to derive from first principles a general kinetic equation for Compton scattering in any magnetic field accounting simultaneously for photon polarization in terms of the Stokes parameters, for the induced scattering and the Pauli exclusion principle for electrons. We use methods of quantum statistics and follow an approach similar to that used for derivation of the kinetic equation for Compton scattering without magnetic field Nagirner and Poutanen (2001). The resulting equations are valid for any photon and electron energies, and for the magnetic field strength below about G. In the most general case, the electron polarization is also taken into account. We also consider several special cases and derive the kinetic equations when the electron gas is non-polarized and rarefied as well as when the radiation can be presented via two polarization modes. The derived equations provide the basis for construction of the models of radiation transport in atmospheres and magnetospheres of strongly magnetized neutron stars.

Ii Description of the electron and photon gases

We use the system of units where . We assume that the magnetic field is locally homogeneous, which is justified, because the scales of changes of the B-field are orders of magnitude larger than the microscopic magnetic scale for conditions in atmospheres of NS and even the geometrical depth of the atmosphere. The magnetic field is described by the dimensionless parameter . We choose the reference frame in any space-point so that the -axis coincides with the magnetic field direction. The following assumptions about the time scales are used:

  1. The typical time scale on which the distribution function changes (for electrons and photons) is much larger than the typical time scale between the interactions.

  2. The plasma is sufficiently rarified, so that we can use a generalization of the Bogolyubov method for the case of quantum statistics to derive the kinetic equation.

  3. The typical time scale of a single interaction is much smaller than the typical time scale between the interactions.

ii.1 Descriptions of single particles

The electron states are described by the wave-functions . Its arguments are the space-time coordinates, the momentum projection on the direction of the magnetic field, describing location of the center of electron gyro-orbit (its -coordinate), the Landau level , and the spin projection on the magnetic field direction ( in units). Dimensionless energy of an electron in this case,

(1)

is independent of . The energy levels are degenerate with the spin projection on the magnetic field direction, except for the ground Landau level with , where can have only one value . The full electron wave function is presented through the partial solutions of the Dirac equation for the electron in the magnetic field:

(2)

where are coefficients.

The photon state is described by four parameters: the wavenumber , the two angles and , which define the direction of the photon momentum, and the polarization state . The 3-dimensional photon momentum can be represented as

(3)

The corresponding photon 4-momentum is . Photon polarization is described by the polarization basis. It consists of two unit vectors, which are orthogonal to the photon momentum k:

(4)

The 4-vector potential can be defined as:

(5)

We note that the photons are described in the same manner as in the case when the magnetic field is absent. We assume that the dispersion relation for the photons in magnetized vacuum does not differ from the dispersion relation in the case when the magnetic field is absent. This approximation constrains the strength of the field and energies of photons. For estimations one needs to know vacuum dielectric tensor and the inverse permeability tensor for the case of magnetized vacuum Adler (1971); Potekhin et al. (2004). It is known that the indices of refraction differ from unity by more than only for the fields with strength Shaviv et al. (1999). This restricts application of the developed formalism to G.

ii.2 Description of the particle ensembles

Wave functions

We describe particle ensembles by density matrix using the rules of quantum statistics. Let us define the wave functions for the case of limited number of particles. These functions will be used for construction of the density matrix. The wave function for a limited number of particles with defined characteristics of each of them, can be found from the vacuum wave function by applying the operators of creation and annihilation. Let and be the creation and annihilation operators of a photon in the state with polarization and 3-momentum k. According to the methods of second quantization Bogoli’ubov and Shirkov (1959), these operators satisfy the relation

(6)

Let and be creation and annihilation operators of an electron on the Landau level in polarization state with momentum projections and . These operators satisfy the following relation

(7)

The system of photons with fixed parameters is described by the wave function

(8)

where is the vacuum wave function of the photon gas. Analogously, the system of electrons with fixed parameters is described by the wave function

(9)

where is the vacuum wave function of the electron-positron gas. The wave function for arbitrary state of particles can be presented as a sum of the wave functions with fixed particle parameters. For example, the state of photons is described by the function

(10)

where are the weight coefficients. The wave function for an arbitrary state of electrons can be represented as

(11)

The wave function for state with photons and electrons can be written as

(12)

Density matrix

The density matrix is defined as an averaged dyad product of the state vector with its conjugate. For the system consisting of photons and electrons it can be written in the form

(13)

The expressions in the triangle brackets are the elements of the density matrix kernel.

Algebra of the density matrix kernels

From now on we will operate only with density matrix kernels. All the equations and the final results are written through these kernels. It is easy to make a transformation from the simplest kernels to the distribution functions or to the coherency matrix. Let us write the density matrix kernel for the system of photons and electrons

(14)

Further let us write some useful relations for the kernels. For the sake of simplicity we consider only the photon gas. These relations can be generalized trivially to the case of the electron-photon gas. The kernel for the system of photons can be written through the density matrix:

(15)

Hereinafter we call it the -particle kernel. It is normalized to unity:

(16)

For any the -particle kernel can be calculated as

(17)

The 1-particle kernel can be expressed through the -particle kernel as

(18)

It is normalized to the total number of the particles:

(19)

The diagonal elements of 1-particle kernel compose the coherency matrix in the case of photon gas.

Transformation from the simplest kernel to the distribution functions

The transformation from 1-particle density matrix to the distribution function in the case of electrons in the case of field-free space can be made using the Wigner function. The Wigner function is defined as

(20)

where is a 1-particle density matrix in the coordinate representation. The momentum and coordinate representations are connected through the Fourier transforms:

(21)

Then one can rewrite the Wigner function using the density matrix in the momentum representation:

(22)

The inverse transformation from the Wigner function to the density matrix in momentum representation reads

(23)

The time scale of the electron-photon interaction is much smaller than the time scale of noticeable changes of the distribution functions. Therefore, in the last equation one can assume that the Wigner function does not depend on the space variables. In this case, the integration could be made easily and we can write

(24)

Then one can convert the 1-particle density matrix kernel to the distribution function in the case of spinless particles or to the coherency matrices in the case of particles with non-zero spin. In the last case a trivial generalization is used:

(25)

Equations (24) and (25) can be written immediately from the physical meaning of the density matrix and the assumption that the time scale of the interaction and the time between interactions are mush smaller than the time scale of noticeable changes of the distribution function.

One can also write similar relation for the case of electrons in the external magnetic field. If we consider a non-interacting electron in the B-field not accounting for cyclotron radiation (which should be described by another kinetic equation), then the electron should conserve its -momentum and the Landau level. This means that the kernel should be diagonal over both and , because it is not possible to have mixed states corresponding to different values of the -projection of momentum or the Landau level. Non-diagonal elements in the kernel can appear only if one accounts for interactions between particles, but because of the smallness of the interaction time scale the kernel should be diagonal over -projection of momentum and the Landau levels. In this case the relation will have the following form (a detailed derivation is given in Appendix A):

(26)

where and describe the electron spin-states, and are the Landau levels, and are the momentum projections and is the electron energy given by equation (1).

A transformation from the 1-particle density matrix to the distribution function in momentum space is trivial, but one must again assume that the typical time scale of changes of the distribution function is much larger than the typical time scales of interaction between the particles.

ii.3 Description of the interaction

Description of the single interaction

Let us mark parameters of the particles before the interaction with the subscript ”i” and particles after interaction with the subscript ”f”. There are three conservation laws for Compton scattering in the magnetic field. They are the energy conservation, the conservation of the momentum along the magnetic field and the conservation of the transversal momentum:

(27)

Let us use special designation for product of -functions which are describing these conservation laws:

(28)

A single interaction can be described by the -matrix. The elements of the -matrix can be calculated using methods of quantum electrodynamics. In the simplest case the elements of the -matrix can be obtained using second-order perturbation theory. In this case Compton scattering can be represented by two Feynman diagrams with two vertices in both of them and one can write an expression for the -matrix elements:

(29)

where is the Dirac inner product of a 4-vector and -matrix, and is a relativistic electronic propagator in the presence of a constant magnetic field, and are the electron wave-functions in coordinate representation, and is the fine-structure constant. The -matrix elements and the cross-sections for Compton scattering in magnetic field contain resonances which have to be regularized Nagirner and Kiketz (1993). The calculations are not trivial and have been performed only in special cases Herold (1979); Daugherty and Harding (1986); Bussard et al. (1986); Gonthier et al. (2000).

Evolution of the density matrix

The evolution of the density matrix can be described by equation:

(30)

where the Hamiltonian is

(31)

Equation (30) is written here in non-covariant form, but it can be transformed to the explicitly covariant form using the Tomonaga-Schwinger equation Bogoli’ubov and Shirkov (1959). It means that the form of the equation is covariant for the longitudinal Lorentz transformations (along the magnetic field direction). The solution of equation (30) can be presented by the operator of evolution :

(32)

On the other hand the operator can be represented in the following form:

(33)

where is the volume in Minkowski space and

(34)

The space integral of can be represented through the elements of the scattering -matrix:

(35)

then can be rewritten in the following form

(36)

Let us assume that the typical time scale of the density matrix changes is much larger than the typical time scales of a single interaction. In that case changes in the distribution on a macroscopically small times scale can be represented througth the -matrix because -matrix can be considered as the scattering -matrix divided by the fine-structure constant:

(37)

and the time interval is considered as a macroscopically small time. Equations (32) and (36) determine the solution formulated through the elements of the scattering matrix. We reformulate these equations below in terms of the kernels of the density matrix.

Iii Derivation of the kinetic equation for the photon gas

iii.1 Methodology of the kinetic equation derivation

Summary of our assumptions and the Bogolyubov method

We derive kinetic equation using a generalization of the Bogolyubov method (for the case of quantum statistics). At the first step we formulate Liouville’s theorem in terms of the kernels of density matrix. One can derive the equations for kernels of different orders (1-particle, 2-particle and other) by integrating over the parameters of different numbers of particles. We use this method to obtain the system of kinetic equations. If the full ensemble contains particles, the system of equations contains equations. In the case of the rarefied gas one can use only a few first equations from the Bogolyubov hierarchy. The criterion of rarefaction can be formulated through the “gaseous parameter”, which depends on the concentration of the particles and the cross sections of their interaction:

(38)

where is the Thomson cross section, and are the electron and the photon concentrations, correspondingly. According to the principle of weakening of correlations, which is satisfied for sufficiently rarefied gases, the correlations are accounted for only in the equation for the 1-particle matrix via kernel by entering the right-hand side (rhs) of the aforementioned equation. This kernel is assumed to characterize the electron and photon states after the interaction. It can be represented via the same kernel before the interaction and the correlation function. To derive the kinetic equation for the typical conditions in the neutron star atmospheres, it is enough to use only the first and the second equation from the Bogolyubov hierarchy.

Formulation of Liouville’s theorem and the equations of Bogolyubov hierarchy

We use the following notations: , etc. There are photons and electrons in the system. The equation, describing the change of -particle kernel during macroscopically small time is written as