Buehring, W.

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April 1967Institut für Experimentelle KernphysikE1eetran Radial Wave Functian far Pasitan Beta DecaysW. BühringKFK 559B. H.KERNFORSCHUNGSZENTRUM KARLSRUHEApril 1967 KFK 559II. Physikalisches Institut der Universität Heidelbergund Institut für Experimentelle Kernphysikauf dem Kernforschungszentrum KarlsruneElectron Radial Wave Function for Positon Beta DecaysWolfgang BühringGesellschaft für Kernforschung m.b.H., KarlsruheAbstract:It is pointed out that the tables of electron radialwave functions by Bhalla and Rose are incQrrect asfar as positons are concerned.- 2 -We infer from recent papers 1,2) that there is considerableinterest in accurate electron radial wave functions (ERWFs)for positon beta decays. Ln particular, allowed Fermi transitionsgive, if they are superallowed, important information on theuniversal Fermi interaction 1) or, if they are isospin forbidden,on the isospin impurity of the nuclear states 2). In both ofthese applications one needs rather accurateERWFs in orderto arrive at reliable conclusions. For these reasons it seems nownecessary to point out that the tables of ERWFs by Bhalla andRose 3-5), which are widely used, do not agree with our (unpublis~Jresults as far as positons are concerned. In this note we givean example of our results in order to demonstrate the disagreementin a quantitative way. Also we suggest the reason why the resultsof Bhalla and Rose might be incorrect, and discuss the evidencewe have for the reliability of our results.The ERWFs under consideration correspond to a uniformly extendednuclear charge distribution of radius ~ , which is not screenedby atomic electrons. Thus we are concerned with a weIl definedproblem, and results obtained by different authors should beidentical. In this connection it is necessary to know thatBhalla and Rose 3-5), although they give the value 0.4285 for theradius constant (which relates the nuclear radius ~ to the massnumber A), actually 6) used a slightly smaller value ofapproximately 0.4276. Ii this fact is allowed for, our resultsagree with the tables of Bhalla and Rose in the case of negatons.For positons, however, the wave functions fand g deviate as isdemonstrated in table 1, whereas the ratios f/g and the phaseshifts Li. still agree e· Therefore i t is the normalization which isincorrect.- 3 -It might be surprising that the normalization can be wrong forpositons while it is correct for negatons, since formally theonly difference between these two cases is the sign of thenuclear charge Z. However, Bhalla and Rose need sign conventionsin order to derive their 3) eqs. (18b) and (19). Equation (18b)implies the choice~ri{~-~)1::: Sir (1) (~+ ii) 11 Vj.Aj1.1which is in accordance with their expliclt statements concerningthe phases of ~ and ~. But eq .(2 9 ) implies the choicee.1-[il,-,n ~ ('4+i<t)/IVd'2+~1.twhich is contradictory in the case of positons correspondingto Z < O. This might give a plausible explanation for theobserved discrepancy.In our calculation the normalization is accomplished in adifferent way which does not depend on sign conventions norexplizitlyon the sign of Z. Thus, since we obtain correctresults in the case of negatons, our results for positons arealso expected to be correct. But we have even stronger evidencethat our results are reliable. Using a drasticly di~ferent method,namely the general method developed for the screened field 7),we obtain identical results both in the case of negatons and ofpositons.ERWFs for specific cases can be supplied on request.The calculations were performed on the Siemens 2002 computerat Heidelberg.Z(A)-26(56)-90(228)p1 .06.01 .. 06.0Ref. f 1B+R 380783Bü 380765B+R 429430Bü 428518B+R 275241Bü 274496B+R 340055Bü 323908g1253264253248205286204834712205710278766965730532f 1/g1150350150353209186209203386463386463443377443386tan L1 1100844100844272140272132588269588278279122279126f_ 1668681668518282437281592138055136820942027883843g-1897678897453506407504914507689503143394814370433f_ 1/g_ 1744901744906557727557703271928271931238600238597tan6"' 1114650114647228038227931202384202392607019607008Table 1Comparison of some results fram Bhalla and Rose (B+R) and from the present work (Bü). The notationof B+R is used, signs and powere of ten are omitted. The very small deviations of f/g and tan~ areprobably due to the fact that our radius constant 0.4276 is not quite exactly the same as that ofB+R. (Also, since some parts of the caloulation by B+R are performed to an aoouraoy of 10-6 only,their final results, in particular tanLl, are not always expected to be accurate to six digits).The values of fand of g, however, show a significant discrepancy.I~I- 5 -References1) J.M. Freeman, J.G. Jenkin, G. lVIurray and W.E. Burcham,Phys. Rev. Letters ~ (1966) 9592) L. van Neste, R. Coussement and J.P. Deutsch, PhysicsLetters 23 (1966) 1223) C.P. Bhalla and lVI.E. Rose, Oak Ridge National LaboratoryReport ORNL-2954, (1960)4) C.P. Bhalla and lVI.E. Rose, Oak Ridge National LaboratoryReport ORNL-3207, (1962)5) C.P. Bhalla and lVI.E. Rose, Phys. Rev. 128 (1962) 7746) C.P. Bhalla, private communication7) w. Bühring, Nuclear Physics 61 (1965) 1-10

Electron radial wave function for positon beta decays

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Electron radial wave function for positon beta decays

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