Born series

It is well known that the conventional Born approximation can be applied iteratively, generating A -th order Born approximations. This approximation can be treated as the sum of N terms of the Born (or Neumann) series. However, the convergence of the Born series is questionable and depends on the norm of the integral equation (Green s) operator. It seems to be very attractive to construct similar series on the basis of the QL and QA approximations. 

Green s operator with norm less than 1 Gg 1. 

We begin with a short review of the method of constructing an always convergent Born series, following mainly Zhdanov and Fang (1997). In the subsequent sections we introduce QL and QA series based on QL and QA approximations. 

Equation (9.125) can be treated as an integral equation with respect to the anomalous field E . The solution of this integral equation has to be a fixed point of the operator C (See Appendix B). In other words, the application of the operator C to the anomalous field E should not change this field. This solution can be obtained using the method of successive iterations which is governed by the equations 

It is well known that successive iterations converge, if operator C is a contraction operator (see Appendix B, Banach theorem), that is 

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One must expect the presence of mixed terms of the form k B in the expansion. The term of lowest order a —2, d = l), contributing oczf to the stopping cross section, would indicate a difference between the Barkas-Andersen correction evaluated from the Born series and the Bohr model, respectively. While such a comparison has not been performed in general terms, a numerical evaluation for the specific case of Li in C revealed a negligible difference [24]. 

At intermediate and higher energies it is appropriate to expand the forward elastic scattering amplitude in a Born series ... 

Byron Jr., F.W., Joachain, C.J. and Potvliege, R.M. (1981). Unitarisation of the eikonal-Born series method for electron- and positron-atom collisions. 

The approximations to be discussed all treat at least one two-body pair interaction fully. Different kinematic regions depend differently on the amount of detail necessary in the treatment of particular pair interactions. Some success in isolated cases has been achieved by calculations based on low-order terms of the Born series. They are not considered here. 

The Born scries would be a powerful tool for EM modeling if they were convergent. However, in practice the condition (9.130) does not hold, because in a general case the L2 norm of the Green s operator is bigger than 1. That is why the Born series has not found a wide application in EM modeling. 

However, inside D they are different. Actually, it is this difference that makes the new Born series always convergent. 

We have noticed that the background of the modified Born approximation and the new first order QL approximation is the same. The main difference is that in the case of the Born approximation the starting point (zero order approximation) for the iteration process is the zero anomalous field, while in the QL approach we start with the anomalous field proportional to the background field. In principle we can extend our approach to computing all iterations bj (9.149). In this case we will obtain a complete analog of the Born series. For example, the second order QL approximation is equal to... 

Comparison of the inequalities (9.147) and (9.169) clearly demonstrates that the accuracy of the Born approximation depends only on the order N, while the accuracy of QL approximation of the same order can be increased by a proper selection of A. This circumstance makes the QL approximation a more efficient tool for EM modeling than a conventional or modified Born series. 

Now we can start iterations by the modified Born series with a quasi-analytical approximation for the anomalous field ... 

In 1933, Bloch revisited the problem [15], and found that, in the dipole approximation, the odd powers in the Born series vanish, but that there is a non-negligible third term in the Born series, giving S(v) oc Zp This term is now referred to as the Bloch correction and corresponds to in the Bom series (equation (2)),... 

In 1963, Barkas noticed the difference in stopping powers when measured with positively and negatively charges projectiles [16], which implied an odd power Lj contribution to the Born series, and the Zj contribution to the stopping cross-section is now referred to as the Barkas correction. 

Higher order terms in the Born series, relevant to relativistic interactions, are seldom, but occasionally (cf. e.g., Refs. [2,17]) discussed. 

Born series, an expansion in powers of — w), is violated in this... 

The convergence of this series is better than the convergence of the standard Born series because the singularities caused by the resonances have been transferred into the resonant term Ti z). To produce results of a general validity (spectroscopy and dynamics) we derive an exact expression of the transition operators that extend the model investigated in Section 2. The unperturbed part of the Hamiltonian is chosen in the form... 

As the effective gz is now zero, naturally an on-shell result, the Born approximation, would be obtained. But there is a further complication. An intemuclear potential is added to Vp. This is variously justified as making sure that the effective potential is finite-range, or to improve convergence of the Born series. However in the SCA we should not change the total cross section when we add an... 

As we turn up the interaction, EIMP becomes important. We must certainly modify our treatment of the K-shell electron and treat its excitation to all orders in the Born series. However, now we have to worry about the role that the other electrons play. Should we not introduce Pauli blocking operators in the Bom series Now we can make a hole in the L-shell, should we not allow this as a possible final state for the K-shell electron The answer to these questions is yes, if we wish to calculate exdusive cross sections, in perturbation theory. However it is not necessary to use many-electron perturbation theory, and we are only interested in an inclusive cross section. 

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