###### Abstract

We consider the Dirac equation in cylindrically symmetric magnetic fields and find its normal modes as eigenfunctions of a complete set of commuting operators. This set consists of the Dirac operator itself, the -components of the linear and the total angular momenta, and of one of the possible spin polarization operators. The spin structure of the solution is completely fixed independently of the radial distribution of the magnetic field which influences only the radial modes.

We solve explicitly the radial equations for the uniform magnetic field inside a solenoid of a finite radius and consider in detail the scattering of scalar and Dirac particles in this field. For particles with low energy the scattering cross section coincides with the Aharonov–Bohm scattering cross section. We work out the first order corrections to this result caused by the fact that the solenoid radius is finite. At high energies we obtain the classical result for the scattering cross section.

PACS numbers: 03.65.Bz, 11.80-m

Scattering of scalar and Dirac particles

by a magnetic tube of finite radius

[1cm]
Vladimir D. Skarzhinsky^{1}^{1}1e-mail:
and
Jürgen Audretsch^{2}^{2}2e-mail:
Juergen.A

[0.5cm] Fakultät für Physik der Universität Konstanz, Postfach 5560 M 673,

D-78434 Konstanz, Germany

[0.5cm] P. N. Lebedev Physical Institute, Leninsky prospect 53, Moscow 117924, Russia

[1cm]

## 1 Introduction

The behavior of relativistic charged particles in external magnetic fields has been the subject of many investigations in QED (see, for example, [2, 6]). Most of them has been concerned with the synchrotron radiation emitted by charged particles moving in cyclic accelerators and storage rings [16]. This effect has found wide applications in different fields of physics, biology and technology.

By its origin the synchrotron radiation is the classical effect. It has been shown that quantum corrections became important only for relativistic particles or intense magnetic fields [16, 10]. This happens at high energies and high field strengths leading to a characteristic product [14]

(1) |

of the order 1. Other QED processes, such as pair production and pair annihilation, have no classical counterparts. In these cases too, intensities of the processes become relevant if particle energies and magnetic fields fulfil Eq.(1). Therefore, in QED processes, the external magnetic field can not be treated as a perturbation. The relativistic particles must instead be described by exact solutions of the Dirac equation in an external magnetic field. This will also be done in this paper. For the comprehensive review of exact solutions of relativistic wave equations in external fields see [5].

Much work has been devoted to solutions of the Dirac equation in an uniform electric and magnetic fields [7, 13] and in the electromagnetic plane wave field [17]. Different QED processes have been studied for the corresponding physical situations [7, 13].

A new field of research was initiated with the study of the Aharonov–Bohm (AB) effect (for the latter see [1, 12]). QED processes in the presence of a magnetic string were elaborated in detail; the differential cross sections of the bremsstrahlung of an electron passing by the magnetic string [4] and the pair production by a single photon in this potential [15] have been worked out. In these cases it surprisingly turned out that the cross sections do not become small even for low particle energies. This fact can be interpreted as follows: for a finite flux the string magnetic field becomes infinitely strong in the limit of a string, so that the criterion of Eq.(1) is formally fulfilled for all energies.

To describe realistic situations the magnetic string has to be replaced by a thin solenoid which contains an intense but finite magnetic field. It is now very important to know how the results obtained in the string limiting case are modified for a solenoid of finite radius. We will attack this problem in this and following papers where we are going to investigate in detail different QED processes in cylindrical magnetic fields. Below we concentrate on the scattering of scalar and Dirac particles.

In the Sections 2 and 3 we define the normal modes of the Dirac equation in cylindrical magnetic field of arbitrary radial dependence as eigenfunctions of a complete set of commuting operators. The set consists of the Dirac operator itself, the -components of linear and total angular momenta, and also, of one of the possible spin polarization operators. We fix completely the spin structure of the solution independently of the radial distribution of the magnetic field which influences only the radial modes. In Sec.4 we solve explicitly the radial equations for the model of a solenoid of finite radius with an uniform magnetic field. With the tube radius going to zero this model can be considered as a realistic model replacing the AB magnetic string. This approach allows to solve correctly the Dirac equation for the AB potential. Then we consider in detail (Sec.5) the scattering of scalar and Dirac particles by the magnetic field of the solenoid of the finite radius. At low energies of incident particles the scattering cross section coincides with the AB scattering cross section; we derive the first order corrections caused by the finite tube radius. At high energies we obtain the classical result for the scattering cross section.

Throughout we use units such that

## 2 The Dirac equation in a cylindrically symmetric magnetic field

The Dirac equation in an external magnetic field reads in cylindrical coordinates

(2) |

where , is the charge of the Dirac particle ( for the positron and for the electron). and are the known matrices written in the cylindrical coordonates, and the kinetic momenta are given by

(3) |

The vector potential for an arbitrary cylindrically symmetric magnetic field of a fixed direction (along -axis) has a nonzero angular component and the magnetic field reads in terms of the potential

(4) |

The dependence of the solutions on and the azimuthal angle can be fixed in demanding that is an eigenfunction of the operators of linear momentum projection and total angular momentum projection

(5) |

Here is the integral part of the half-integral eigenvalue

We find

(6) |

where

(7) |

is the energy, denotes the radial momentum and are the spin polarization coefficients.

## 3 Spin polarization states

In the case of cylindrical magnetic fields the spin polarization coefficients can be fixed by one of two spin operators or [16] which commutes with operators and but do not commute with each other. The helicity operator,

(11) |

describes longitudinal polarization (the projection of the spin onto the velocity direction of the Dirac particle). The -component of the operator

(12) |

defines a transverse polarization (along the direction of the magnetic field) for nonrelativistic motion, or for the motion in the plane perpendicular to the magnetic field. The operators and are not independent. However, to make things easy for the reader we will discuss them separately.

### 3.1 The helicity states

For the solution of the Dirac equation (2) which is the eigenstate of the helicity operator (11),

(13) |

the Eqs.(2) and (11) connect the and spinors according to

(14) |

It means that and that the coefficients are coupled by Eq. (14) to correspondingly. Choosing and to be

(15) |

and determining coefficients and from Eq.(14),

(16) |

we obtain from Eq.(2) the following set of the equations for the independent radial components

(17) |

The solution of the Dirac equation with the quantum numbers has then the form of Eqs.(6), (7) with and the coefficients (15), (16).

### 3.2 The transverse polarization states

On the other hand, for the eigenstate of the operator (12),

(18) |

we find from Eqs.(2) and (12) that

(19) |

This means that and that the coefficients are defined by (19) from correspondingly. Fixing the coefficients and by the expressions

(20) |

where is the sign of and determining the coefficients the and from Eq.(19),

(21) |

we obtain from Eq.(2) again the radial equations (3.1). The solution of the Dirac equation with the quantum numbers has therefore the form Eqs.(6), (7) with and the coefficients (20), (21).

### 3.3 The charge conjugate states

The solutions (6) of the Dirac equation (2) represent electron () and positron () wave functions of positive energy, . The complete set of solutions of the Dirac equation (2) includes the positive and negative energy electron (or positron) states. Instead of negative energy electron (or positron) states one can use as well the charge conjugated positron (electron) states which can be obtained by the charge conjugation operation,

(22) |

For the helicity and transverse states we obtain

(23) |

where

(24) |

with the coefficients (15), (16) and (20), (21), correspondingly.

With reference to these states, the electron-positron field operator reads

(25) |

where and are the annihilation operators for electrons and positrons with quantum numbers This field operator obeys Eq.(2) with Of course, we could use the charged conjugated field operator which obeys this equation with

### 3.4 The radial equations

The set of the radial equations (3.1) is equivalent to the following second order equation for the first component

(26) |

the second component is then defined by the equation

(27) |

Alternatively one could start with the second order equation for and define the first component by the equation (3.1). Both ways are completely equivalent. We would like to stress that because of the first order constraints (3.1) between the components and the Dirac equation differs from the Klein–Gordon equation for relativistic scalar particles.

The radial functions can be normalized by the condition

(28) |

The Eqs. (26), (27) together with the normalization condition (28) define the radial functions and up to a common phase factor.

The nonrelativistic limit: The positron and electron radial and angular functions conserve their forms in the nonrelativistic approximation () with nonzero spin coefficients

(29) |

## 4 Solenoid of a finite radius and the magnetic string

Our basic aim is to study the AB scattering of scalar and Dirac particles for experimentally realistic situations. We consider therefore a solenoid of finite radius with the uniform magnetic field inside. This model contains the AB magnetic string in the limit of zero radius and constant magnetic flux. This limit can alternativly be based on a different model [9] for which the magnetic field is concentrated on the surface of a cylinder of radius It is obvious that this second model being more simple for the investigation of zero radius limit does not describe a realistic experimental set up.

### 4.1 The uniform magnetic field inside a solenoid of finite radius

The uniform magnetic field restricted to the interior of a solenoid of finite radius can be described by the vector potential

(30) |

where is the magnetic flux in the solenoid, is the magnetic flux quantum and is the sign of the charge ( for positrons and for electrons). We take so that the constant magnetic field inside the solenoid

(31) |

points in the positive direction. To obtain the uniform magnetic field in the whole space one may perform the limiting procedure together with

Let us consider first the internal solution. The general radial equations (26) and (27) read in this case

(32) | |||||

(33) |

Introducing the new variable and the new functions according to

(34) |

we obtain from Eq.(32) the confluent hypergeometric equation [8]

(35) | |||

Then from Eqs.(33) - (35) we find the following confluent hypergeometric equation for the function

(36) | |||

Solutions of Eqs.(35) and (36) which are regular at are the confluent hypergeometric functions and correspondingly, so that we obtain

(37) |

The coefficients and are connected with each other by the Eq.(33).

The external solution obeys the radial equations (26) and (33) at

(38) |

Solving these equations by Bessel functions of the first kind with positive and negative order and using Eq.(27) we find the external radial components

(39) |

where

(40) |

The coefficients and can be obtained from the matching conditions at the surface or of the solenoid for the internal and external solutions. Since radial components are connected with each other by Eq.(33) one can use any couple of the matching conditions for components and their first derivatives . Choosing the conditions for and we obtain

(41) | |||

Eliminating the coefficient from these equations we find the matching conditions in the form

(42) | |||

which fixes all coefficients up to a normalization constant. The same results can be obtained for instance from the continuity conditions for and at

### 4.2 The zero radius limit (magnetic string)

The vector potential for the infinitely thin, infinitely long straight magnetic string lying along -axis

(43) |

can be obtained from the vector potential of the solenoid (30) for keeping the magnetic flux constant. It is singular on -axis and produces the singular magnetic field

(44) |

which is concentrated on the -axis and points to the positive direction. We separate the integral number from the flux parameter , since it is the fractional part of the dimensionless magnetic flux which produces all physical effects. Its integral part will appear as a phase factor in the solutions to the Dirac equation.

In the case of the AB potential the radial solutions are given by Eqs.(4.1), but have another domain of definition. We note that the component contains modes with coefficients which are singular at for all The component contains singular modes with coefficients and The appearance of the singular modes is usually forbidden by the normalization condition (28) for radial functions. But for unbounded potentials, and for the AB potential (43) in particular, this condition is less restrictive. The normalization condition (28) requires that the integrals of the type

(45) |

were convergent at small what takes place at For the positron solution the normalization condition (45) for allows nonzero coefficients and while this condition for allows nonzero coefficients and For the Dirac equation both components and must satisfy the normalization condition (45). It means that only the coefficient can be nonzero presenting one singular mode with in the positron solution of the Dirac equation (while two singular modes are allowed for scalar wave equations). For the electron solution the normalization condition (45) for allows nonzero coefficients and while the condition for allows nonzero coefficients and Therefore only one singular mode with can be present in the electron solution of the Dirac equation.

The problem of the singular modes is related to the fact that the Dirac operator as well as any Hamilton operator for charged particles is not selfadjoint in the presence of the AB potential. This would cause many problems for the unitary evolution of quantum systems unless selfadjoint extensions of these Hamilton operators exist. The selfadjoint extension procedure applied to this problem gives results which can be obtained by the direct calculation of the normalization integrals (for a detailed discussion see [3]). However, this procedure does not fix the extension parameters which determines the behavior of the wave function at the origin. This situation is not satisfactory from the physical point of view. It can be overcome in turning to better defined models [9], in particular, to the model of a solenoid of a finite radius with the uniform magnetic field [4].

From Eq.(42) we obtain for

(46) |

where

(47) |

and

(48) |

The asymptotic behavior of for depends on the behavior of the denominator in Eq.(48). Since we have for unless the following equality is valid,

(49) |

This can happen only if

(50) |

In this case the parameter goes to zero,

(51) |

unless

(52) |

The Eqs.(50) and (52) are fulfilled for the positron () singular mode only. It means that and that accordingly the first positron component with is singular and the second component is regular at

(53) |

In this case the interaction between positron magnetic moment and the string magnetic field is attractive for the first (upper) component of the wave function and it is repulsive for the second (down) component. For the electron () solution all coefficients It means that the singular electron mode has the regular first component and the singular second component

(54) |

Introducing new notations we can rewrite the positron and electron solutions of the Dirac equation in the presence of the magnetic string in the following final forms which are valid both for regular and singular modes,

(55) |

with

These radial solutions satisfy the normalization conditions (28).

Redefinition of The matrix elements of the QED processes (bremsstrahlung of electrons and positrons, pair production and annihilation) in the presence of the AB potential contain integrals over the products correspondingly, where and are the electron and charge conjugate positron functions. The integer number disappears from all these matrix elements after the redefinition of the angular quantum number , This means that the matrix elements of the QED processes are independent of the integer part of the magnetic flux in units of the magnetic quantum. One can foresee this fact beforehand since the Dirac equation (2) with can be transformed to the Dirac equation with by means of the gauge transformation

All observable quantities are conserved under this gauge transformation.

## 5 Scattering of scalar and Dirac particles by a solenoid of finite radius

The expressions obtained above present the partial positron and electron wave functions in terms of cylindrical modes. These states do not describe outgoing particles with definite linear momenta at infinity. In order to calculate the cross section of QED processes we need the positron and electron scattering wave functions. In external fields there exist two independent exact solutions of the Dirac equation, which behave at large distances like a plane wave (propagating in the direction given by ) plus an outgoing or ingoing cylindrical wave, correspondingly. Because of the damping of the cylindrical waves at large distances we may use these superpositions instead of plane waves. To evaluate correctly the matrix elements in the presence of external fields we have to take wave functions for ingoing (outgoing) particles which contain outgoing (ingoing) cylindrical waves. We turn first to a discussion of scalar particles.

### 5.1 Low energy scattering of scalar particles

The wave function for scalar particles can be found as the eigenfunction of the complete set of the commuting operators (2):

(57) |

Its radial modes obey the radial Klein–Gordon equation,

(58) |

which is similar to the Eq.(26) but does not contain the term which describe the spin-magnetic field interaction.

For realistic models the internal radial solutions are some regular functions For model (30) of a solenoid with constant magnetic field this is

(59) | |||||

The external solutions are similar to the first line of Eqs.(4.1). It is convenient to rewrite them as follows

(60) | |||||

where are the Hankel function. The corresponding scattering wave functions can be obtained by the superposition of these cylindrical modes.

Since the plane wave term of the scattering wave function at is defined only by the external solution, the scattering wave function can be written as follows

(61) | |||

with the coefficients

(62) |

The wave function (61) behaves asymptotically as follows