Radiative recombination of twisted electrons with bare nuclei: going beyond the Born approximation
We present a fully relativistic investigation of the radiative recombination of a twisted electron with a bare heavy nucleus. The twisted electron is described by the wave function which accounts for the interaction with the nucleus in all orders in . We use this wave function to derive the probability of the radiative recombination with a single ion being shifted from the twisted electron propagation direction. We also consider more realistic experimental scenarios where the target is either localized (mesoscopic) or infinitely wide (macroscopic). The situation when the incident electron is a coherent superposition of two vortex states is considered as well. For the nonrelativistic case we present analytical expressions which support our numerical calculations. We study in details the influence of the electron twistedness on the polarization and angular distribution of the emitted photon. It is found that these properties of the outgoing photon might be very sensitive to the total angular momentum and kinematic properties of twisted beams. Therefore, the recombination of the twisted electrons can serve as a valuable tool for atomic investigations as well as for the diagnostics of the vortex electron beams.
Since the theoretical prediction (1), the
twisted (or vortex) electrons have become one of the most attractive
objects of interest in the contemporary physics.
They are characterized by the energy , one of the
momentum components which sets the propagation direction,
and the projection of the total angular momentum on this
The interest to such particles is caused mainly by the non-zero value
of this projection, being an additional degree of freedom.
Moreover, the growth of leads to the increase of the twisted
electron magnetic moment ( is the Bohr
magneton) along the propagation direction.
This fact points on the sensitivity of the electron vortex beams to
magnetic properties of matter (2); (3); (4); (5).
First experimental realizations of these electrons were performed just
half a decade ago (6); (7); (8).
In these experiments the twisted electrons possessing were
Presently, the twisted electrons with the momentum projection up to
can be routinely produced at electron microscopes (9); (10).
Electrons with such a large value of the total angular momentum
projection can be used for the detection of the polarization radiation (11); (12).
In addition, the vortex electrons provide a new opportunity to get a
deeper insight in the role of the spin-orbit interaction in various
Despite a great interest, there are only few works presented in the literature being dedicated to the investigation of the processes involving ionic (or atomic) targets and twisted electrons (13); (14); (15); (16). In all these articles the interaction of the twisted electrons with targets was considered perturbatively in the framework of the first Born approximation. This approximation stays valid only for light systems with relatively small nuclear charge and at rather large projectile velocities. Meanwhile the manifestation of the twistedness is expected to become the most pronounced in heavy systems where the spin-orbit interaction increases drastically. In order to investigate the processes involving heavy systems one needs to account for the interaction of twisted electrons with the targets in all orders in . This can be achieved via the construction of the twisted electron relativistic wave function in the long-range Coulomb field of the nucleus. In the present paper, we construct such a wave function and utilize it for the description of the radiative recombination (RR) of a twisted electron with a bare heavy nucleus. Two types of the targets are considered, namely the infinitely extended one (macroscopic) and the target with a finite spatial distribution (mesoscopic). For the macroscopic target, we compare our nonrelativistic results with the ones obtained within the first Born approximation (14). We also consider the case when the twisted electron is a superposition of two coherent vortex states. For the second type of the target we investigate the dependence of the experimentally measurable quantities on the position and size of the target. Besides, we present the analytical nonrelativistic expressions which allow one to check the results obtained by the numerical calculations and to get a deeper insight into physics beyond them.
The relativistic units () and the Heaviside charge unit () are used in the paper.
Ii Basic Formalism
The radiative recombination being the time-reversed photoionization is the process in which a continuum electron is captured into an ion bound state with the simultaneous emission of a photon. The relativistic theory of the plane-wave electron RR is well established and vastly presented in the literature (see, e.g., Refs. (17); (18); (19); (20)). In the case of the twisted incident electron, only the nonrelativistic study within the first Born approximation was performed (14). Here we are focused on the systematic relativistic description of the twisted electron RR with bare nuclei beyond the Born approximation. Since the main aspects of this description are rather similar to those for the plane-wave case, we start with the brief recall of the plane-wave (conventional) electron RR theory.
ii.1 Radiative recombination of plane-wave (conventional) electrons
The probability of the asymptotically plane-wave electron RR can be represented as follows
where and are the asymptotic momentum and the helicity of the incident electron, respectively, denotes the final bound state, and the emitted photon is characterized by the energy , the momentum , and the polarization . The amplitude of the RR process is given by
where and are the wave functions of the electron in the initial and final states, respectively. The transition operator in the Coulomb gauge has the following form
Here is the vector incorporating the Dirac matrices and is the photon polarization vector. The wave function of the incident electron is constructed as the solution of the Dirac equation in the external nucleus field with the following asymptotic behaviour
Here and are the unit vectors in the and directions, respectively, is the bispinor amplitude, and the plane-wave solution of the free Dirac equation expresses as
where is the Dirac quantum number with and being the total and orbital angular momenta, respectively, is the Clebsch-Gordan coefficient, is the phase shift being induced by the potential of the extended nucleus, is the Wigner matrix (24); (25), and is the partial wave solution of the Dirac equation in the nucleus field (21). Let us note here that at large distances the flux corresponding to the wave function (6) coincides with the flux of the free electron and equals
This fact is clearly seen from Eq. (4). The RR cross section reads
Here we would like to stress that in Eq. (8)
the momentum direction of the incident electron is arbitrary with respect
to the axis which is not yet fixed.
The presented formulas completely describe the process of the plane
In what follows, we will often refer to the nonrelativistic theory of the radiative recombination into the state. In this case, utilizing the dipole approximation one can obtain the following formulas for the process probability and the cross section (21)
where and . In the Born approximation (), the function (11) is given by
ii.2 Radiative recombination of asymptomatically twisted electrons
Let us now switch to the description of the twisted electron RR. As already been mentioned, a free twisted electron is characterized by the following set of quantum numbers: the energy , the helicity , and the projections of the momentum and the total angular momentum on the propagation direction. Here and throughout the axis is fixed along this direction. Besides, the twisted electron possesses a well-defined absolute value of the transverse momentum . The vortex electron can be represented as a coherent superposition of the plane waves with momenta forming the surface of a cone with the opening (conical) angle . The explicit expression for the wave function of the free twisted electron is given by (16)
where and are the longitudinal and perpendicular components of momentum , respectively. In the plane-wave limit, this wave function behaves as
From Eq. (13) it is seen that the density and the flux of the twisted electron are not the homogeneous functions of the space variables. In particular, the density equals
where is the perpendicular component of and . Therefore, in contrast to the plane wave case the relative position of the twisted electron and the target is important. For the target ion being shifted from the axis on the impact parameter (see Fig. 1) the amplitude of the RR process is given by
where is the wave function of the twisted electron. For practical calculations it is more convenient to change the integration variable in Eq. (16) as follows . For such the geometry, the wave function of the twisted electron is to be taken as the solution of the Dirac equation in the central field with the following asymptotics (28)
The corresponding solution is given by
Utilizing Eq. (19) one can obtain the following expression for the amplitude of the process under consideration:
Then the probability of the twisted electron RR is given by
Since all measurable quantities can be expressed in terms of the probability (21) we regard the theoretical description of the twisted electron RR as completed. Here it should be emphasized that the wave function, which is introduced in Eqs. (19) and (19), accounts for the interaction of the asymptotically twisted electron with the target ion in all orders in . Thus, utilizing this wave function one obtains the results beyond the Born approximation.
Presently, the experiments with a single ion, especially heavy and highly-charged one, are very difficult and time consuming. Therefore, in the present paper, we focus on the analysis of the twisted electron RR with various targets. The distribution of the ions within the target can be considered as the classical one and is assumed to be given by the function with the following normalization condition
In this case, all measurables are determined via the integral of this function with the probability of the RR with the ion located at the distance from the axis
Here stands for the coordinates of the target centre.
One of the main process characteristics, the cross section, cannot be determined for the twisted electrons as a ratio of the probability to the flux density of the incoming particles. Indeed, it is clear from the free twisted electron wave function (13) that, in contrast to the plane wave case, the flux is neither a homogeneous function nor even positively defined. Nevertheless, it is very useful to have an “effectively” defined cross section. For example, it can be used for the estimation of the experimental feasibility. In the present paper, we propose the following expression for the cross section
is the density of the free twisted
electron (15) with .
Both the cross section (24) and the
flux (25) goes to the well known conventional
expressions (8) and (7), respectively, in the
plane-wave (paraxial) limit.
This fact is regarded as the main argument in favour of the
However, it is worth noting that one can easily present a set of
different cross section determinations possessing the same limit.
In addition to the cross section, one can characterize the twisted electron RR by the relative measurable quantities. One of them is normalized on average angular probability
The relative variables also include the Stokes parameters
Here denotes meanwhile designates the probability of the photon emission with the linear polarization .
Iii Results and Discussions
The radial parts of the bound- and continuum-state wave functions being
the solutions of the Dirac equation in the central field of the extended
nucleus are numerically found utilizing the modified RADIAL package (29).
The Fermi model of the nuclear charge distribution is employed.
In order to reach the convergence of the results the partial waves with
up to are taken into account.
There are two distinct types of experiments. In the first one, the target has a macroscopic size and therefore can be regarded as infinite. A target, which consists of a finite number of ions (up to a single ion), forms the second type and is referred to as the mesoscopic one. In order to describe the ion distribution for both types of the targets we choose the Gaussian distribution which is generally realized in ion traps. Then, the function reads
where corresponds to the centre of the target and the dispersion characterizes the size of the target. The macroscopic target corresponds to the limit , while at one obtains the case of a single ion. For this distribution, the flux defined by Eq. (25) takes the following form
iii.1 Macroscopic target
The macroscopic target is the simplest one for the experimental realization as well as for the theoretical investigation. As it was mentioned above, such a target can be described by the function (29) with . This corresponds to the infinite spatial size with the uniform distribution of ions inside the target. With this in mind, one can utilize with (the radius of the cylindrical box) instead of the Gaussian distribution (29). Repeating the calculations of Ref. (16) we obtain the differential cross section for the macroscopic target in the simple form
where is defined by Eq. (8). Note, that this cross section is and independent. In addition, from Eq. (31) one can obtain the following relation for the total cross section being averaged over and summed over and
Comparison of the Born approximation with the exact treatment
Let us first consider the RR into the state of a H-like ion. In this case, the exact nonrelativistic expression for the differential cross section is given by Eq. (10). In order to investigate the importance of the calculations beyond the Born approximation we introduce the following parameter
From Eq. (31), it is clearly seen that the
parameter takes the same values for both the plane-wave and twisted electrons.
Additionally, one can conclude that the ratio (33) does not depend on
the parameters of the outgoing photon.
It equals to at and rapidly decreases with the growth of the
For the process discussed in Ref. (14), where
the keV twisted electron RR into the state of the hydrogen ion
() was studied, one gets .
This corresponds to the 23% difference between the results obtained within the Born approximation and beyond it.
In the case of the recombination with the argon ()
ion at the same electron energy !
This means that the Born approximation does not provide reliable results
for the absolute value of the differential cross section.
The situation differs for the relative values of the measurables (26) and (28). The explicit nonrelativistic expression for the angular distribution is
and the Stokes parameters are given by
where is an arbitrary unit vector.
The corresponding expressions for the conventional case can be obtained by
As an example, the degree of linear polarization .
From Eq. (34) one can see that the
angular distributions being calculated within and beyond the Born
approximation coincide with each other.
The same is valid for the Stokes parameters (35).
The results obtained by the usage of Eqs. (34)
and (35) are in excellent agreement (up to the terms
of order ) with the ones presented in
Here it is worth stressing that the coincidence of the relative measurable values being calculated within and beyond the Born approximation occurs only in the nonrelativistic framework. This is not the case in the relativistic formalism. In Fig. 2 we present the normalized angular distribution for the RR of the twisted electron into the state.
The kinetic energies were chosen to provide the same parameter () for all the ions. Fig. 2 demonstrates the difference between the results obtained with the usage of the Born approximation and beyond it. The comparison indicates the importance of the exact relativistic calculations for the systems with middle and high . Indeed, for the uranium ion at there is a qualitative difference in the behaviour of the differential cross sections. Specifically, the forward photon emission becomes preferable in this case. In Fig. 3 we present the differential cross section for the RR of the twisted electron into the state.
From this figure one can see that the role of the electron
twistedness increases with the growth of .
Let us now consider the outgoing photon polarization. For an initially plane-wave electron, the degree of the linear polarization takes only positive values. In the case of the twisted electron, the Stokes parameter becomes negative at (see Eq. (35)). This means that the emitted photon is linearly polarized in the direction perpendicular to the scattering plane. A similar effect has been observed in Ref. (32) where the Vavilov-Cherenkov radiation by twisted electrons has been studied. The Stokes parameter, which was calculated using the relativistic formalism beyond the Born approximation, is presented in Fig. 4.
From this figure one can see that in the case of the argon () ion the photon polarization becomes negative at . This is in a good agreement with the predictions by Eq. (35). For the much heavier thorium () ion changes its sign already at . Such a shift to smaller conical angles at higher is due to the more pronounced manifestation of the electron twistedness.
The RR of the electron being in a superposition of two vortex states
It is of special interest the situation when the twisted electron is not an eigenstate of the operator but a coherent superposition of such states. As an example, let the superposition consists of two twisted waves with different (33). In order to obtain the wave function of such an incident electron one has to perform the following substitution in Eq. (19)
where the complex coefficients satisfy the normalization condition . As a result of this substitution the differential cross section (31) takes the form
In Ref. (14), it was pointed out that the presence of this additional factor leads to a modification of the angular distribution and the Stokes parameters and . Here we will focus only on the modification of the differential cross section. It can be shown that after the summation and averaging over the final and initial states projections, respectively, the differential cross section (38) can be written in the following form
where is the differential cross section (31) being averaged over and summed over and . In the nonrelativistic case, substituting Eq. (9) into Eq. (38) and summing over the polarization of the emitted photon one can obtain the explicit expression for the azimuthal asymmetry parameter
From this expression it is clearly seen that the differential cross section possesses the azimuthal asymmetry only at or (we assume that ). Here it is worth mentioning that these selection rules originate from the dipole approximation which was used to derive Eq. (9). However, these rules partly take place in the exact relativistic calculations too. This can be explained as follows. The azimuthal asymmetry appears due to the interference of the RR amplitudes being related to different partial waves in the decomposition (19). The higher , the higher are required. The partial amplitudes decrease with the growth of that leads to a decrease of the asymmetry. As a result, the manifestation of the asymmetry is more prominent at and less at . In Fig. 5, the azimuthal asymmetry parameter being obtained within the relativistic framework is depicted.
From this figure it is seen that the asymmetry is the most pronounced at . Nevertheless, the parameters for and become comparable with each other at large conical angles .
iii.2 Mesoscopic target
Let us now consider the targets of the limited size.
In this case, the measurables appear to be sensitive to the total angular
momentum projection on the propagation direction.
These targets are also sensitive to the spatial structure of the
incoming vortex particles (37); (36); (34); (35).
Here we present the results only for the case of the bare argon
Mesoscopic target consisting of such ions can be, in principle, created
Let us start from the consideration of the total cross section
where is the target position and is defined by Eq. (24) with the flux being given by Eq. (30). From Eq. (42) it can be seen that the total cross section is independent of . The ratio of this cross section to the plane-wave one is depicted in Fig. 6 as a function of the target position.
From this figure it is seen that for the total cross
section appears to be less sensitive to and, as a result, to the
spatial structure of the incoming electron state.
Therefore, in what follows we will consider only the case .
At the ratio which is presented in
Fig. 6 goes to that corresponds to the case of a macroscopic target (see Eq. (32)).
In addition, the dependence becomes much less pronounced.
The situation changes at fixed and .
In this case, the ratio
equals to at and zero at .
This can be explained as follows.
At the transverse momentum .
Therefore, the projection of the total angular momentum on the propagation
direction equals to the spin projection on the momentum ().
As a result, in the averaging over the helicities ()
in Eq. (42) only one term with contributes
and the factor remains.
The ratio which is depicted in Fig. 6 goes exactly to
the factor at the limit .
In order to get the “correct” paraxial limit, namely , one has to put simultaneously
The Stokes parameters are depicted in Fig. 7 as functions of the target position for different values.
From this figure one can observe a strong correlation between the target
position and the polarization of the emitted photon.
It can also be seen that the correlation increases with the growth of .
Thus, one can investigate the spatial structure of the
twisted electron via measuring the Stokes parameters of the RR photon
for different target positions.
Alternatively, the target position can be determined by studying the
polarization of the emitted radiation.
Here it is worth mentioning that for the keV twisted electron with corresponds to the target size about nm. Therefore, one needs to utilize focused twisted electron beams of a sub-nanometer size. The possibility of generating such beams was demostrated in Refs. (39); (40); (41).
In the present work, the fully relativistic description of the twisted
electron radiative recombination with a bare nucleus was presented.
The interaction of the incident electron with the ionic target was
taken into account to all orders in .
It was done by determining the vortex electron wave function as the
solution of the Dirac equation in the central field.
The solution was constructed in such a way that its asymptotic has the
form of the superposition of the free twisted and outgoing spherical
The resulting wave function was used for the description of two
different experimental scenarios, namely with macroscopic and mesoscopic
targets, beyond the Born approximation.
In the case of the macroscopic target, the comparison of the results, which were obtained within the Born approximation and with a usage of the developed formalism, has been conducted. For the sake of comparison clarity, the analytical nonrelativistic expressions for both approaches were also considered. It was found that the total cross section for the keV vortex electron RR into the state of the hydrogen atom being calculated within the Born approximation differs from the exact value by . This discrepancy increases very rapidly with the growth of the parameter and for the recombination with the bare lithium nucleus amounts to . Contrary to the cross section, the normalized angular distribution and the Stokes parameters being obtained within the Born approximation coincide with the exact values in the nonrelativistic case. In the framework of the relativistic formalism, however, this result is no longer valid.
For the macroscopic target it was also found that the linear polarization of the emitted photon becomes negative at certain conical angles. This means that the photon is polarized perpendicular to the reaction plane. In the conventional plane-wave case, the degree of linear polarization is strictly positive, i.e., the photon is polarized in the reaction plane.
Additionally, the situation when the incident electron is a coherent superposition of two vortex states with different was studied. In this case, the asymmetry of the angular distribution was calculated. It was found that the asymmetry becomes most pronounced at and decreases rapidly with the growth of . The analytical nonrelativistic expression for the angular distribution was also presented.
For the mesoscopic target the dependence of the total cross section on the distance between the target center and the twisted electron propagation direction has been investigated. The dependence of the Stokes parameters on the target position has been also studied. It has been found that both the total cross section and the Stokes parameters are sensitive to the spatial structure of the incoming electron state, i.e. to . However, this dependence vanishes with the growth of the target size.
At the end, let us add that the developed formalism can be utilized for the description of other processes involving twisted electrons and heavy ionic or atomic targets beyond the Born approximation.
The authors are grateful to A. I. Milstein for useful discussions. This work was supported by RFBR (Grants No. 16-02-00334, No. 16-02-00538, and No. 15-02-05868), and SPbSU (Grants No. 11.38.269.2014 and No. 220.127.116.115). VAZ acknowledges financial support from the government of St. Petersburg.
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