Time delay in electron collision with a spherical target as a function of the scattering angle

We have studied the angular time delay in slow-electron elastic scattering by spherical targets as well as the average time delay of electrons in this process. It is demonstrated how the angular time delay is connected to the Eisenbud-Wigner-Smith (EWS) time delay. The specific features of both angular and energy dependences of these time delays are discussed in detail. The potentialities of the derived general formulas are illustrated by the numerical calculations of the time delays of slow electrons in the potential fields of both absolutely hard sphere and delta-shell potential well of the same radius. The studies conducted shed more light on the specific features of these time delays.


Introduction
In the first experiments, the purpose of which was to study the time delays of electrons in atomic photoeffect, electrons with the wave vector k emitted along the polarization vector e of the absorbed photon are recorded [1][2][3]. With this experimental technique, the delay times of the electrons escaping at an arbitrary angle to the vector e were unknown. Now, investigation of time delays as a function of the emission angle ϑ have become available [4][5][6][7], and the corresponding calculations have been able to reproduce this dependence for different atoms [8][9][10][11][12][13]. The electron delay time is a function depending on both the photoelectron emission angle ϑ with respect to the radiation polarization vector e and the photoelectron energy E. In most calculations of the time, its dependence on the energy E is analyzed at fixed values of the angle ϑ, revealing the pronounced angle dependence for large emission angles.
The angular dependence of the time delay of the wave packet scattered (or emitted) by a spherical target was obtained by Froissard, Goldberger, and Watson in [14], where the following expression for the angular time delay of the packet scattered in the direction ϑ was derived: (1) Here ) , ( ϑ k f denotes the amplitude of electron elastic scattering by a target [15] ∑ − + = δ l (k) is the partial scattering phase shifts, P l (cosϑ) are the Legendre polynomilas. According to (1) the forward scattering 0 = ϑ must be excluded due to the interference effects between the forward scattered wave and the incident wave that give rise to the optical theorem [15].
The domain of applicability of the angular time delay ) , (1) is considerable broader than that of the Eisenbud-Wigner-Smith (EWS) partial-wave time delay [16][17][18] In particular, Eq. (1) serves as the basis for describing the temporal picture of atomic photoionization processes [17][18][19][20][21][22][23][24][25]. Eq. (1) in this case needs not to be modified to exclude 0 ϑ = , since the problem of the interference with the unscattered wave does not exist in the case of photoionization. The scattering amplitude ) , ( ϑ k f for this process must be replaced in Eq. (1) by the photoionization amplitude ( , ) ph f ω ϑ where ω is the photon energy The dipole selection rules in photoionization of l-states of atom A lead to emission into continuum of the pair of electronic spherical waves 1, ( ) The time delay (4) at some electron emission angles ϑ was studied in the series of works on photoionization [19][20][21][22][23][24]. To the best of our knowledge, the angular dependence of the time delay in elastic electron scattering (1) received no attention so far. Our goal in this article is to close somewhat the gap in the area of investigation of the angular time delay in electron scattering (1) by spherical targets.
We will see further that when only one scattering phase is different from zero in the scattering amplitude (2), the angular time delay (1) does not depend on the scattering angle. Here we analyze the scattering amplitude ) , ( ϑ k f containing two Legendre polynomials only, i.e., we will consider model targets, in which, as in the case of the dipole photoelectric effect, only one pair of phase shifts is different from zero. In the next Section 2, the angle dependence of the angular time delay ) , ( ϑ k t ∆ for some fixed electron momenta k is investigated. In Section 3, the time delay is studied as a function of k for some fixed polar angles ϑ of the scattering of an incident plane wave train. Finaly, the function ) , is averaged over the distance of the order of the de Broglie wavelength, and the average angular time delay ) (k t ∆ is obtained in the Section 4.

2.
Angular ϑ-dependence of the function ) , The argument of the amplitude ) , ( ϑ k f is determined by the ratio of the imaginary part of function (2) ) , ( Im whereas the angular time delay (1) is described by the general expression Here and everywhere below, we use the atomic system of units. Let us first consider the case when all the phase shifts in (2), with the exception of δ l (k), are equal to zero. In this case .
It is seen that the angular time delay does not depend on the scattering angle ϑ, and it is equal to half of the EWS-partial time delay (3). Suppose that only two scattering phases ) ( 0 k δ and ) ( 1 k δ are nonzero. In this case, the scattering amplitude and its argument are represented as Differentiating the argument of the scattering amplitude (8), we obtain the expression for the time delay It is easy to demonstrate that when only two scattering phases ) (k Explicit expressions for the time delays for selected nonzero scattering phase pairs (11) are given in [26], where the results of the calculations of the ϑ-and Edependences of the corresponding angular time delays are also given. We use further the hard sphere-and delta-shell potentials as potential functions for the model targets. For these potentials, the analytical expressions for the scattering phases are known. When an electron is scattered by the model target in the form of an ideally repulsive solid sphere of radius R, the phase shifts of the electron are determined by the formula [27] ) ( where ) (kR j l and ) (kR n l are the spherical Bessel functions. The scattering phase shifts of an electron for another model target taken in the form of an attractive delta-shell (delta-shell potential well [28]) are determined by the expression (see Eq. (10) in [29]) where the variable kR x = . The parameter L ∆ in (13) is the jump of the logarithmic derivative of the electron wave functions at the point r = R where the delta-shell potential is infinitely negative. In the numerical calculations of phase shifts (12) and (13), the radii R and the parameter L ∆ have the same values as those used in our article [29], where the EWS time delay of slow electrons scattered by a C 60 cage was calculated. Figure 1 shows the results of the calculation by formula (9)  The only exceptions are for the curves at k = 0.68. In the case of hard sphere, the graph of the angular dependence is almost a straight line, passing from a positive to a negative half-plane at the angle of about 60 o . Whereas on the right panel, this curve almost coincides with the x-axis. According to both panels, at low electron energies (k = 0.17 and 0.34), the time delay of the scattering packet is negative at all the scattering angles. The rest of the curves are alternating for both targets. At the momenta k = 0.51 and k = 1.0, the time delays on the right panel reach a maximum (~ 298 atomic units (au) at ϑ = 95 o in the first case and ~ 140 au at the same angle in the second one). The appearance of these sharp peaks in the curves in Figure1 is due to the almost vanishing of the denominator in the expression (9). The curves at k = 0.85 and k = 1.0 on the left panel cross the x-axis into the positive half-plane in the region of 90 o , forming a peak with a height of about 30 atomic units. Figure 2 depicts the curves corresponding to the pair of polynomials P 0 (cosϑ) and P 2 (cosϑ). We see here the results of the calculation with formula (10)  Let us pay attention to the similarity of the curves in Figure 2 and the angular spectra in Figure 1  Summarizing, we note that according to Figures 1 and 2, the angular ϑdependences of the function ) , are represented by nontrivial rapidly oscillating curves lying at low electron energies mainly in the negative half-plane. The situation changes with increasing the electron energy.

3.
k-dependence of function ) , , that contain the derivative of s-phase shift, for k → 0 tends to infinity: For the orbital moments l > 0 the derivative of phase shifts should be vanishing at the threshold as The left column of the figures corresponds to the electron scattering by the hard sphere potential. The figures in the right column correspond to scattering by the deltashell potential. In the upper right panel of Figure Figure 3, we observe strong resonance behavior of all curves, except for the one at ϑ = 90 o , and energy 0.4 E ≈ atomic units. In the second and third sections, we limited ourselves to the specific examples of two nonzero phases in the expansion of the wave function of a scattered electron (2) into partial waves. We see that even this simplest example leads to a very difficult for interpretation, rapidly oscillating dependence of the time delays upon the energy E and scattering angle ϑ. An increase in the number of included essential scattering phases significantly affects the picture of the angular time delay, making it so rapidly oscillating that its averaging over the energy of incident electrons and the scattering angle becomes inevitable in order to make the angular time delay ) , ( ϑ k t ∆ observable in an experiment.

4.
Average time delay of the scattering process. The average angular time delay ) (k t ∆ is obtained from (1) by averaging over the energy spectrum of the incident wave packet, as well as over the directions weighted by the differential cross section is not defined at 0 = ϑ . It was shown in [31] that the contribution to the integral from forward scattering of an electron is determined by the real part of the scattering amplitude at zero angle. As a result of such averaging, Nussenzweig [31][32][33] obtained the expression The second term on the left-hand side of the equation (14) eliminates the contribution of the forward scattering into the average angular time delay. Thus, the average time delay for the plane wave train ) (k t ∆ is a linear combination of the EWS time delays The results of the calculation of the function ) (k t ∆ (14) in the case of electrons scattered by the hard sphere target are shown in Figure 4. Figure 4 also shows the dependencies calculated under the assumption that the statistical weight of ) (k l τ in the sum (14) is not equal to 2 / ) 1 2 ( k l tot σ π + , but it is the ratio of the electron elastic scattering partial cross section ) (k l σ to the total cross section ) About this assumption see, for example, Eq. (10) in [10] or Eq. (8) in [29]. The deep peak of the curve corresponding to the combination of the Legendre polynomials P 0 and P 2 is due to the the resonant behavior of curves at E ~ 0.4 au in Fig. 3.

5.
Concluding remarks Using the instructive soluble example of electron scattering by the hard sphere potential and delta-shell potential well, we for the first time explicitly obtained the angular time-delay ( , ) t k ϑ ∆ in terms of the scattering phase shifts ) (k l δ and their energy derivatives ) (k l δ ′ . We demonstrate the complexity of ( , ) t k ϑ ∆ as a function of the incoming electron energy E and the scattering angle ϑ . We see that ( , ) t k ϑ ∆ and the function ( ) t k ∆ , even averaged over proper intervals of E and ϑ , are more sensitive to the scattering phases than the absolute cross section ) (k tot σ and even the differential in angle scattering cross section that is proportional to . This is because the time delay functions depend not only on the cross-section phases, but upon their energy derivatives. This makes theoretical and experimental investigation of time delays a promising direction of research in the area of atomic scattering.   as a function of the electron energy E for some fixed values of the polar angle ϑ. P 0 (cos ϑ)-P 1 (cos ϑ) is the pair of Legendre polynomials in the upper panels. P 0 (cos ϑ)-P 2 (cos ϑ) are the polynomials used in both lower panels.   14) in the case of electrons scattered by the hard sphere target. P 0 +P 1 and P 0 +P 2 are the pairs of Legendre polynomials P l (cos ϑ). Note that Equation (10) corresponds to Formula (10) in paper [10].