Bethe stopping-power theory for heavy-target atoms.

In spite of its applicability to relativistic incident particles it is well known that the Bethe stopping power theory limited to nonrelativistic target elements with eigenstates isfying the Schro ̈dinger equation for many-electron atom This limitation arises mainly from the derivation of the orig nal Bethe theory, which has applied the various nonrela istic sum rules~the Bethe and TRK sum rules ! @1#. For heavy elements, one would expect a nontrivial correction to Bethe theory due to the fast motion of the inner shell el trons. This problem was first pointed out by Fano in 1964 a review of the outstanding unsolved problems in stopp power theory which existed at that time @2#. Since then, to the knowledge of the author, not much effort has been voted to the study of this problem until recently @3–5#. As also pointed out by Fano in the same review @2#, the difficulty in solving this problem lies right in the possible gene alization of the various sum rules to the relativistic doma Indeed, the relativistic generalization of various atom sum rules has been an intriguing problem over the pas years since the first work on the generalization of the T sum rule@6#. It has been studied extensively in the literatu using both the single-particle and many-particle ~fieldtheoretic! approaches@7–11#. In a previous attempt, we hav used a semirelativistic single-particle approach to obtain leading relativistic correction terms to the Bethe sum rule @9# and applied the results to derive corrections to the Be stopping power theory for heavy target atoms @3#. Unfortunately, it was pointed out later @10# that in most of these previous works based on the same approach @6,7,9,11#, there exists an inconsistency in that the transformation of the erator was not included in the Foldy-Wouthuysen transf mation performed, which leads to the semirelativistic corr tion terms for the sum rules. Very recently @12#, this error has been corrected and it was found that while the prev corrections to the TRK sum rule were not affected by t error, those for the Bethe sum rule have to be modified. It is the purpose of this paper to apply these latest c rected results for the semirelativistic sum rules to amend previous work published in the correction to the Bethe st ping power theory@3#. As before, we shall limit ourselves t the single-particle case and apply the results to a real a by adopting the independent-particle, local-potential desc tion. Though this seems to be an oversimplified picture does have some success in the literature in the analys x-ray scattering data using the TRK sum rule @13#. In any case, our preliminary attempt will at least give a first es