In view of the inapplicability of the asymptotic expressions for the stopping number available in the literature at high energies, an alternative approach is taken to compute the shell correction to the stopping number of K electrons. Anholt\u27s formula (1979) for the K-shell ionization has been used to calculate the excitation function for longitudinal interaction and numerical integration over energy has been carried out to evaluate the shell correction. Comparison with other theoretical calculations is made. It is proposed that, with the inclusion of relativistic effects, an asymptotic expansion of the stopping number with a leading-term logarithmic in the energy of the incident particle would be more meaningful and might enable one to extract the relativistic contribution to the shell correction from it

Leung, P.T.

Long, S. A.

Rustgi, M. L.

PDXScholar (Portland State University)

Portland State UniversityPDXScholarPhysics Faculty Publications and Presentations Physics4-1-1986Shell correction for the stopping power of K electronsP.T. LeungPortland State UniversityM. L. RustgiS. A. LongLet us know how access to this document benefits you.Follow this and additional works at: http://pdxscholar.library.pdx.edu/phy_facPart of the Physics CommonsThis Article is brought to you for free and open access. It has been accepted for inclusion in Physics Faculty Publications and Presentations by anauthorized administrator of PDXScholar. For more information, please contact pdxscholar@pdx.edu.Citation DetailsLeung, P. T., Rustgi, M. L., & Long, S. T. (1986). Shell correction for the stopping power of K electrons. Physical Review A (GeneralPhysics), 33(4), 2328-2332.PHYSICAL REVIE% A VOLUME 33, NUMBER 4Shell correction for the stopping power of K electronsAPRIL 1986P. T. Leung' and M. L. RustgiPhysics Department, State Uniuersity ofNew Fork at Buffalo, Buffalo, New York 14260S. A. T. LongLangley Research Center, National Aeronautics and Space Administration, Hampton, Virginia 23865(Received 25 February 1985; revised manuscript received 12 August 1985)In view of the inapplicability of the asymptotic expressions for the stopping number available inthe literature at high energies, an alternative approach is taken to compute the shell correction to thestopping number of E electrons. Anholt's formula for the E-shell ionization has been used to calcu-late the excitation function for longitudinal interaction and numerical integration over energy hasbeen carried out to evaluate the shell correction. Comparison with other theoretical calculations ismade. It is proposed that, with the inclusion of relativistic effects, an asymptotic expansion of thestopping number with a leading-term logarithmic in the energy of the incident particle would bemore meaningful and might enable one to extract the relativistic contribution to the shell correctionfrom it.INTRODUCTIONThe evaluation of the shell corrections to the Bethe for-mula for the stopping power of matter has remained anactive area of research ever since the pioneering work ofBethe and collaborators in the early 1950's. ' By consid-ering the optical oscillator strength and the mean excita-tion energy of the target atom, they were able to obtainsome asymptotic formulas for the shellwise stoppingnumber (8;) for various target elements using screenedhydrogenic orbitals. The shell correction (C;) could thenbe extracted from a comparison made between the exact8; obtained from numerical integration and the asymptot-ic 8; so obtained. The results of this approach have beenreviewed in the literature by Uehling and Fano. '"Developments along this line have been carried on byKhandelwal to cover the entire Periodic Table for the Kand I. shells which have the most important contributionsto shell corrections and, quite recently, improved asymp-totic formulas have been obtained using Hartree-Slater os-cillator strengths by Khandelwal. 9 Aside from this line ofdevelopment, recently there have been different ap-proaches attempted by others such as the extraction of Cx.and CL from explicit Born-approximation calculations onsubshell stopping powers by McGuire' and the kinetictheoretical description by Sabin and Oddershede" basedon Sigmund's formalism. ' In spite of all these, Walske'sresults ' have many times been regarded as references forcomparison between "hydrogenic calculations" with ex-perimental or other theoretical results. ' ' It is the pur-pose of this paper to point out that due to the nonrela-tivistic nature, the asymptotic formulas given in the litera-ture are not apphcable for high-incident energies andhence the calculation of the shell correction by exploitingthese formulas may not have general validity. Further-more, it will be desirable to include the relativistic natureof the K electrons for heavy elements. %e have thereforein this work followed an alternative approach to integratethe shell correction numerically and the results are com-pared with those from other theoretical investigations.%e shall limit ourselves only to the E shell contribution(Cx) in this paper and always take proton as the incidentparticle.THE STOPPING NUMBERSThe asymptotic formulas of Walske and Khandelwal 'can be summarized as follows. The stopping number 8xis expanded in inverse power of the incident particle'svelocity as follows8K(~K 9ir ) SK(~K )ln9K+ TK(~K ) CK(~K gx )CK(~K~ 7K) UK('gir)tlirt + ~K(le)'tlK + ' (2)In (1) and (2), (9» is the screening parameter which in-creases with the atomic number of the target atom andsix=u /Ztt with u being the incident particle's velocityand Zx —Z —0.3 the effective charge of the target atom.The coefficients SK, Tx, Utt, and Vtt are listed in Refs. 8and 9 and we have used everything in atomic units in thispaper. Because of the nonrelativistic nature of both theincident particle and the E electrons assumed in derivingthese formulas, the results are not applicable in the limitof very large pe's corresponding to high incident energiesor in the case with heavy target elements. For high-energy incident protons, it is more suitable to express g~in the form137 'ZK0.938Epas explained later on with Ep being the energy of the pro-ton in GeV.33 2328 1986 The American Physical Society33 SHELL CORRECTION FOR THE STOPPING POWER OF. . . 2329While Bx can be described by the asymptotic form us-ing Eq. (1), it can also be obtained by exact integration.In the first Born approximation and by considering onlythe direct Coulomb interaction (i.e., only the longitudinalphoton effect), Bx can be written asBx(8x re)= Jii, dW Jg dQ P—g(Q),with8 iFp(Q) jNw(Q) = (4)where Fii (Q) is the inelastic Coulomb form factor for theexcitations and ionization of the target atom. W and Qare the energy and momentum transfers in atomic units.From kinematics and conservation of energy and momen-tum, one can set W;„=Ox and Q;„=8' /4gx. ' Forhydrogenlike wave functions, E~(Q)~has the follow-ing form according to Anholtexp ——tan '[2k/(Q+1 —k )]k=2'd'~~w(Q)~' (3Q+ ~)Q3 [1—exp( —2n. /k)][(Q —k +1) +4k ]' (5)whered= 1+ ZgQ2'2 —11+(W —1)2'2 —1(6)heavy target elements. Furthermore, it is also worthwhileto point out that the expansion formulas for Bx given inthe literature lose their asymptotic nature completely oncethe relativistic nature of the K ele:trons is taken into tak-en into account (the 1 factor) as shown in Fig. 2. In thefollowing, therefore, we attempt to recompute Cx includ-ing the d term by following an alternative method origi-nally suggested in Walske's paper.(7)arises from the normalization constants since we haveused semirelativistic Darwin wave functions to describethe atomic K electron. Similar but somewhat differentexpressions have also been obtained by Davidovic et al. 'but since Anholt's results agree better with experiment, wehave here adopted his formalism. Previous formulationsusing Schrodinger wave functions would be equivalent tothe above results by setting d =1.' a in Eq. (7) standsfor the fine-structure constant. Numerical integration forEq. (3) has been carried out for various elements using a256&&256 points Gaussian quadrature for all energies. Atlow energies, our results reproduce the previous results ifwe set d =1. At higher energies, where the asymptoticformulas of Walske and Khandelwal are not valid, our ex-actly integrated results for Bx with and without the effectof d are shown in Fig. l. It is clear from this figure thatthe use of the Darwin wave function gives very differentresults from those using the Schrodinger wave functionfor target elements with a large Z, since the K electronsof the heavy atoms are quite relativistic in nature. In fact,Anholt had pointed out earlier' that the tables publishedpreviously ' have to be modified due to the inclusion ofthe factor d, although for light elements such as carbonthere is essentially no difference with or without the dterm. However, for light elements, i)x becomes very largeat large incident energies and hence Eqs. (1) and (2) can-not be applied. Taking into consideration this fact andthe effect of the d term, we conclude that the previousCx's calculated from Eqs. (1) and (2) are of questionableaccuracy for high-incident energies for both light andt 5.0-I4.0-i20-I l.o-(O.O-7O-2.0—l}I 0-I(0 i O2Ep (GeV)FIG. 1. Stopping number 8~ and the effect of d on it. Thesolid line represents the exact results of this work and the dash-dotted line represents the integrated result without the factor d .The curve labeled a is for C (8J(.—0.64) with the range of q~ be-ing 60~ gz ~600; the integrated results with or without d al-most overlap with each other. The curve b is for Ni (8~ ——0.8)with the range of q~ being 5 ~ gJ(- ~ 25 and the curve c is for Pm(8~ ——0.9) with the range of q~ being 1 ~ q~ & S.5.2330 P. T. LEUNG, M. L. RUSTGI, AND S. A. T. LONG 33CALCULATION OF CgBethe, Brown, and Walske have considered the following function:W2/4X(e~,q~)= f dW f, dg —Pw(g) —f dW f, dQ P—w(0)W QI Wg 0 0—lim f dW f dg —P (0)—»m f dW J dg —Pw(Q)i /4g» Q g 0 0 Qi Qand have shown that X A-In' +8 for all rls and fixed ez. Hence by comparing (1) and (8), one can find C» as—Cg ——Bg —X .(8)In what follows, we shall first rederive Eq. (8) of Ref. 4 in a bit simpler way than the original one and then apply numer-ical integration techniques to compute Cz from Eq. (9).The first integral of Eq. (8) can be rewritten asJ dW f dQ —Pw(g)= fw dW fi~4 dQ Nw(g)W 0 W 4g~+f dW f, dg —P (Q)+f dW f dQ P—(Q). (10)Substitute (10) into (8), recombine terms and we can writeW2/4X(e„p,)= f" dW J„,' dg —Pw(g) —f dW f, dg —Pw(0)+ f dW J„, dQ —[Pw(g) —P (0)]—f dw f„dg—Pw(g) .Substituting (3) and (11) into (9), we getW~/4—Cz= f dW f, dQ Pw(g) —f dW f, dQ Pw(g)+ f dW f, „dQ Pw(0)~KW W ], /4g~+ J, dW f,„dQ Pw(g)+—J, dW f, dQ —[(tw(Q) —Pw(0)l (12)Let us denote the last integral by C' and rewrite it as follows:00 ]/4"& 1 00 1/4q~C'= f dW f dQ [4'w(g) —Pw(0)l —f dw f, dg —[O (Q) —0 (0)] . (13)Interchanging the order of integration in the first term in (13) and applying the Bethe suin rule' in the formw =, d8' wo (14)W2/4dg —Pw(g) —f dW f, dg —Pw(g)+ f dW f, dW—Pw(0)—C = f" dWJ",~K00 1/4g~—fw dW f dQ —[kw(g) —0'w(0)1.we get only the second term left for C. Substituting back in (12) and neglecting the fourth term in (12) because it be-comes less than 10 for g& & 0.5, one obtainsIf we rewrite the last term in (15) asW~/4—fw "W f, dQ —[Aw(g) —Nw(0)1 —fw dW f,„dQ—4'w(g)+ fw dW J,„dQ—Nw(0»JS iK W IEC(16)33 SHELL CORRECTION FOR THE STOPPING PO%ER OF. . . 23310.4-8,7-Ni (6)„=0.8)0.2-Il.80.2 OA 0.6 0.80 (0 )2l /q„FIG. 3. Comparison between this work and that of %alske{Ref. 4) for Ck. The solid line represents the results of thepresent work and the dashed line represents %'alske's results.0 2 4I I I I6 Q 10 l2 (4 l6 Ie ZG~Ksubstitute (16) back in (15), simplify and regroup terms,we arrive finally at Eq. (8) of Ref. (4),(17)FIG. 2, Comparison of the integrated results for 8~ with theasymptotic formulas given in Ref. 8. The solid line representsthe exact result with the factor d . The dash-dotted linerepresents the integrated result without the factor d and thedotted line represents the results of the asymptotic formulas.cerned. In fact, it will be very interesting if one can arguephysically what the trend of the C» should be withrespect to the variation of atomic number of the targetelement. It should also be remarked that the curves inFig. 3 are trustworthy only for not too high values of i)»since the expansion of C» in O(l/i)») will never vanishsince i)» approaches a maximum value as the velocity ofthe incident particle approaches that of light, whereas weexpect that no shell correction is necessary for incidentparticles with infinitely high energies. We suggest thelower limit 0.2&1/i)» for the C» curves to be trust-worthy. Furthermore, since the Bethe sum rule [Eq. (14)]holds exactly only for nonrelativistic atomic wave func-tions while we have been using Darwin wave functions inour formalism, the results for heavy target element (e.g.,for 8» —0.9) will not be very accurate. However, sincethe term d does not play an important role for lightatoms (see Fig. 1), we expect that our results for Ni(8» —0.g) and Al (8» —0.7) would still be quite accurateThe comparison of C» for Al from different calcula-tions is shown in Fig. 4. It is interesting to note thatNote that we have only applied the sum rule once in ourderivation. With the Pii(Q) being given above [Eqs.(4)—(7)], we have computed C» via numerical integrationfor various elements. Incidentally, we found that it ismore convenient to manipulate with the four integrals inEq. (15) than the two in Eq. (17) because of faster conver-gence and above all, the first term in Eq. (15) is just the8» which we have computed before and the third term isindependent of i)». In Fig. 3 we show a comparison ofthe results obtained in this work and %alske s original re-sults for two elements: Ni (8» ——0.8) and Pm (8» —0.9).Figure 4 shows a comparison between the ~present resultwith the results from other calculations. ' ').6-l.5-(4-i.0-0.9-0.7-DISCUSSION AND CONCLUSION 040 0.5 l. 5 2.0It is seen in Fig. 3 that the results obtained in this workare in general smaller than those of Walske's original cal-culations. Furthermore, the results here seem to show areverse trend as far as the screening parameter (8») is con-FIG. 4. Comparison of Cz for Al from different theoreticalcalculations. a is the result of this work, b is %'alske's results(Ref. 4), c is the result of McGuire (Ref. 10), and d is the resultof Sabin and Oddershede (Ref. 11).2332 P. T. LEUNG, M. L. RUSTGI, AND S. A. T. LONG 33Bkll,'5.0"0.0-lO,O-1.0-O. i l.oI2E, (GeV)I10l.OO. lII,O 2 3(G v) l O4FIG. 5. Stopping number for Ni. a includes only longitudi-nal effect and b includes the sum of longitudinal, transverse,and spin-flip effect.FIG. 6. Stopping number for U. a includes only longitudinaleffect, b includes the sum of longitudinal and transverse effects,and c includes the sum of longitudinal, transverse, and spin-flipeffects.while the present result is quite close to %alske's originalresult, both results are consistently smaller than the resultof McGuire. '0 It may be more interesting to note that theresult of Sabin and Oddershede" lies just between the tworesults mentioned above, being very close to the "hydro-genic results" for high-incident energies and very close toMcGuire's result for low-incident energies. It would be ofgreat interest if we can understand more physically thebehaviors of these various results. Finally, it might alsobe of interest if one can flnd new asymptotic formulas forB» which better flt the exact integrated results in Fig. 1and reproduce the C» curves in Fig. 4 at the same time.However, from previous investigations ' the "saturationcharacteristics" of the B»'s will disappear if we also in-clude the relativistic effects, namely, the transverse andspin-flip effects. ' ' Figures 5 and 6 show the results forNi and U from which we can see the importance of thespin-flip effect for heavy elements. Furthermore, it isfound that with the inclusion of relativistic effects, B» be-comes almost linear in ln(E&) and hence we propose thatit might be more meaningful to seek asymptotic expansionof B» with a leading logarithmic term in energy (ratherthan in velocity) of the incident particle. In other words,should be defined to be proportional to energy only ifall these other relativistic effects are included. In thatcase, we might be able to extract the shell correction accu-rately for all energies with the relativistic effects (mainlythe spin-flip effect for heavy target atoms) on C» beingincluded. Of course, one is always left with the possibilitythat other processes (e.g., pair creation in the atomic fieldof the target) may come in which suppress the B» curveagain towards saturation at high-incident energies so thatan expansion in velocity (-rl») with modified coefficientsmay still be possible. All these will be left as a futurestudy in the relativistic effects on the shell corrections ofBethe's theory of stopping power of matter.ACKNOWLEDGMENTThis work was partially supported by the U.S. NationalAeronautics and Space Administration under Grant No.NAG1442. The authors are grateful to Dr. L. N. Pandeyfor his help with the numerical work.'Permanent address: Physics Department, Tamkang University,Tansui, Taiwan 251, Republic of China.'L. M. Brown, Phys. Rev. 79, 297 (1950).M. A. Bethe, L. M. Brown, and M. C. Walski, Phys. Rev. 79,413 (1950).M. C. Walske and H. A. Bethe, Phys. Rev. 83, 457 (1951).4M. C. Walske, Phys. Rev. 88, 1283 (1952).5M. C. Walske, Phys. Rev. 101, 940 (1956).6E. A. Uehling, Ann. Rev. Nucl. Sci. 4, 315 (1954).7U. Fano, Ann. Rev. Nucl. Sci. 13, 1 (1963).G. S. Khandelwal, Nucl. Phys. A116, 97 {1968).G. S. Khandelwal, Phys. Rev. A 26, 2983 (1982).E. J. McGuire, Phys. Rev. A 28, 49 (1983); 28, 53 {1983);28,57 (1983).J. R. Sabin and J. Oddershede, Phys. Rev. A 26, 3209 {1982);29, 1757 (1984).P. Sigmund, Phys. Rev. A 26, 2497 (1982).'3H. Bichsel and L. E. Porter, Phys. Rev. A 25, 2499 (1982).H. H. Andersen et a/. , Phys. Rev. A 16, 1929 (1977).&5E. Merzbacher and H. W. Lewis, Encyclopedia of Physics,edited by S. Fliigge (Springer-Verlag, Berlin, 1958), Vol. 34, p.166.R. Anholt, Phys. Rev. A 19, 1004 (1979).7D. M. Davidovic, B. L. Moiseiwitsch, and P. H. Norrington,J. Phys. B 11, 947 {1978).G. S. Khandelwal, B. H. Choi, and E. Merzbacher, At. Data1, 103 (1969).'9See, e.g. , H. A. Bethe and R. W. Jackiw, Intermediate Quanturn Mechanics (Benjamin/Cummings, Reading, 1968), pp.303—306.~ See the remark by Fano, Ref. 7, Sec. 4.4.2'P. T. Leung and M. L. Rustgi, Phys. Rev. A 27, 540 {1983).

Shell correction for the stopping power of K electrons

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Shell correction for the stopping power of K electrons

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