Electron wave functions at the nuclear surface

Volume 48B, number 2

PHYSICS LETTERS

21 January 1974

ELECTRON WAVE FUNCTIONS AT THE NUCLEAR SURFACE*

H. OGATA and J. ASA1

Department of Physlcs, Umverstty of Windsor, Windsor, Ontario, Canada Received 1 October 1973 Finite size corrections of BhaUa and Rose are examined m the midway to the point-nucleus hm!t. A possible reason of poor convergence of their positron wave functions is suggested '~t has been known for many years that analysis of beta-decay experiments requires an accurate knowledge of the electron radial wave functions near the nuclear surface [ 1]. Many of the most reliable calculations included the finite deBroglle wavelength effect, the finite nuclear size corrections [2, 3], and the screening effect due to the atomic electronst ~ It has been believed that the most accurate calculation should be obtained by directly solving the Dlrac equation, solutions of which are usually expanded in powers of the radial variable up to a certain order of magnitude without truncation [5], or with a reasonable truncation [6, 7]. Most of such calculations assume that the nuclear charge distribution is uniform throughout the inside of a nucleust 2 . It is the purpose of this paper to examine the suitability of the umform charge distribution and to see if the power expansion method would fail at some particular transition energy and atomic number The radial part of the Dlrac equation is

a is the fine structure constant, Z is the atomic number, p as the nuclear radius and ~ is eZ/2p We use the rationalized relativistic units, 1/= m = c = 1 The radial wave functions with the uniform charge potential (2) have been tabulated by Bhalla and Rose (BR) [5] It has been noted,however, that these wave functions do not approach the values for the point nucleus as the nuclear radius ]s made very small, that these authors phase convention for the positron wave functions is believed to have been explained inconsistently [9], and their values of the positron wave functions seem to behave very strangely at higher energy and at low atomic number [6]. To understand these puzzhng features of the wave functions, we have Introduced a diffused nuclear charge distribution, which reduces to the uniform charge distribution when a metric constant a approaches zero.

dF I d r = ( K I r ) F - [ W

Pe(r) _- Peo

1 VIG

,

V(r) = - ~ Z / r

(r > p ) ,

V(r) = - ~(3 r2/p 2)

(r < p).

(2)

(O
t)

(1)

dG Idr = - ( K / r ) G +[W+I

V]F

,

where F K and G~ are radial wave functions (f~ and g~ ) multiphed by the radial variable r. ~¢ is the angular m o m e n t u m q u a n t u m number and the potential V is usually the one derived from the uniform nuclear charge dtstrlbutlon

* Supported in part by the National Research Council of .Canada t 1 Thxs effect is not considered m this work [e.g 4] -}-2The effect of the non-uniform charge distribution Is proved to be negligibly small [8].

Pe(r) =

(p-- t < r < p +t),

P e ° [ l + e x p { - - g d l{2gd(p

Oe(r) = 0

r/ap)}}]-i (p +t < r)

(3)

gd is the Gudermannxan function and gd-1 is its inverse function. The potential due to this charge distribution can be calculated analytically

V(r)=-~[3-r2/p2]+O(a 2) V(r)=

(0
t),

l~[3-r2/p2+2p/r] +O(a) ( p - t < r < p + t ) ,

V(r) = - ouZ/r

(p +t < r).

(4) 91

Volume 48B, number 2

PHYSICS LETTERS

Lo

21 January 1974

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Fig. 1 Combmations of the electron radial wave functions for a) Z = 84, A = 210, and b) Z = - 9 0 , A = 228, as functions of momentum p 92

Volume 48B, number 2

PHYSICS LETTERS

Using this potential, we have calculated regular and irregular solutions in each region in powers o f r and then connected these wave functions smoothly at two boundaries for various values of a which is related to the thickness parameter t through the relation t = ap sinh- 11. When a is made vanlshingly small, the solutions are essentially the same as those of BR. Even if a is finite, which makes t finite, the solutions are almost unaltered~ 3 . When the potential in the intermediate region is extended to the whole Internal region, the solutions change considerably and when the nuclear radius is made smaller they approach the point-nucleus approximation. Let us call the latter solutions "weak coagulation limit", say, through which we hope BR's solutions are connected to the point-nucleus solutions. To see this situation more clearly, we have calculated combinations o f these wave functions L o, N 0, M 0, L1, L12 and N12, which are often used in the study of the beta decay [ 1 0 - 1 2 ] and the results are shown in fig. la and lb. The broken line shows BR's calculation and our wiggling trapezoidal charge distribution calculation whereas the heavy solid line indicates our calculation of the weak coagulation hmlt. The two other curves are the one calculated from the Konopinskl-Uhlenbeck formula [10] (dash-dot line) and that o f the point-nucleus approximation (thin solid line), added for comparison. When the nuclear radius p is made smaller, together with Z, it is observed that the last three curves become almost completely overlapping with each other. The broken line, however, does not approach the classical limit at all, but stays almost at the original place no matter how large is the thickness of the intermediate region. This shows that even if we make the nuclear surface diffuse, the broken line does neither converge to the classical limit nor diverge from it as long as there is a finite central region of the uniform charge distribution adjacent to the diffused surface. On the other hand, if the radius is kep unchanged but Z is made smaller, the broken line moves toward the other curves. This suggests that the ratio (F~c/G~¢) at the inner boundary may be strongly dependent on Z and that the ratio is significantly different from other cases when there is the uniformly charged central region,

21 January 1974

even though each of the functions Fg and G K are not drastically different from other cases. By inspecting these figures, it may be expected that the use of the wave functions with the asusmptlon of the central uniform charge distribution would lead to results which are quite different from those derived from the weak coagulation hmit. In the case of negatron decays, however, this may not show up if one examines, e.g., the beta spectrum shape, since the broken line and the solid heavy line for L 0 are almost complet¢ly parallel to each other and if one normalizes the value o f L 0 at a particular value of momentum, there may not be observed any significant discrepancy within the range of the value of the momentum of one's Interest. A different situation may be expected if an analysis requires a combination of more than one of these quantities. To find this, we have analysed beta-gamma angular correlation data of 186,188Re [13], but both wave functions, with or without the central uniform charge density, give rise to almost equivalent values of goodness o f fit. The situation is much different for the case of the positron decays. Making the nuclear radius/9 or Z, or both, smaller, we find a general trend similar to the negatron case, except that these curves themselves change very much as Z alone approaches zero. Also, as mentioned earlier [9], BR's wave functions are believed to behave erroneously, and this makes it difficult to see these two curves are related to each other. In order to see what could be wrong with the BR's calculation, we rewrite the eq. (1) in another form.

dF / d r - ( K / r ) F K = - G

/P 1 , (1 ')

dG /dr+(K/r) G = F / e 2 ,

where

PI 1 = W- 1 +~(3-r2/p2),

(5) p ] l = 14I+ 1 + ~ ( 3 - r 2 / p 2 ) . The coupled first order differential equations (1') can readily be separated into two uncoupled second order differential equations for F~ and GK,

t 3 We have examined even a very small value of p - t as far as the model calculation is permitted. 93

PHYSICS LETTERS

Volume 48B, number 2 d2F~

dr

[ +[ d2G --+ dr 2

+t

fimte size corrections prowde much better agreement compared to the weak coagulation limit as expected, and that a finite but very small diffused surface regmn is definitely favoured if any central uniform charge &strlbutlon is assumed [ 15 ]. It is also hkely, m additmn, that if there is any malor clustering taking place reside of a nucleus, it would affect even the low energy behaviour o f the electron wave functions.

d ( l o g P 1) dFTr

dr 2

K(~-I) r~

dr K d(l°gP1) &] r dr +

F = 0,

d(logP2) d G dr

dr

[ - - K ( K + I ) K d ( l ° g P 2 ) P-~P2] ~ +r dr + G~ = 0 .

(6)

If F~ and G K are expanded m powers of r, and subsntuted back into eq. (1'), we get the solutions o f BR. It should be mentioned, however, that eq. (6) has a regular singular point at the origan and that a general solution of the eq. (6) must have another loganthmm term which has been discarded to avoid the infinite probabdlty at the origin. It should also be mentioned that the power expansion is only convergent within a circle of convergence, whose radius as P [3 +(W ¥ 1)/~] l/2 and that this radms of convergence could be less than p for a negatwe Z value and a large value o f W Therefore, the power expanston around the origin may fad for a p a m c u l a r set of values of W and Z in the case of the positron d e c a y t 4 . This situation does not xmprove even if the wave functmns are expanded in powers of r 2. Finally, should the charge distributions studied m this work be closer to reality, one o f these distributions would fit electron scattering data better, even though such a crucial test would not be necessary to examine the low energy behaviour o f the electron wave functions. In this connection, we have analysed tugh energy electron scattering data o f 88Sr [ t 4 ] . It is ln&cated that the electron wave functions with the t 4 To our best knowledge, this fact has not been pomted out

94

21 January 1974

References [1] M Monta, Prog Theoret Phys (Kyoto), Suppl 26 (1963) 1. This review artmle provides a concise description of the story References not listed here may be found in this article, H A Wmdemtilller, Rev Mod Phys. 33 (1961) 574 [2] M.E Rose andC.L Perry, Phys Rev 90 (1953) 479 [3] M E Rose and D.K Holmes, Phys Rev 83 (1951) 190 ]4] W Buhrmg, Nucl Phys 40 (1963) 472, 49 (1963) 190, 61 (1965) 110;65 (1965) 369 [5] C P Bhalla and M E Rose, Oak Ridge National Laboratory report, ORNL-2954 (1960), (unpubhshed), ORNL-3207 (1961), (unpubhshed) [6] J N Huffaker and C E Laird, Nucl Phys. A92 (1967) 584 [ 7] D H. Wilkinson, Nucl. Phys. A158 (1970) 476. [8] H. Behrens and W Buhnng, Nucl Phys A150 (1970) 481,A162 (1971) 111;A179 (1972) 297 [9] W. Bubrmg, Report, Kernforschungszentrum Karlsruhe, KFK 559 (April 1967). [10] E J Konopmska and G E. Uhlenbeck, Phys Rev 60 (1941) 308 [11] E Greuhng, Phys. Rev 61 (1942) 568; A M Smith, Phys Rev 82 (1951) 959 [12] M Morlta and R.S Monta, Phys Rev 109 (1958) 2048; M Monta, Phys Rev. 113 (1959) 1584. [13] M Trudel, E E. ttablb and H Ogata, Phys Rev. C1 (1970) 643 [14] J Alster, B F Gibson, J S McCarthy, M S. Weiss and R M Wright, Phys Rev C7 (1973) 1089 [ 15] J Asal and H Ogata, Physics m Canada 29 (1973) 19, CC4