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Isotope shift constant and nuclear charge model
Z. Fang, O. Redi, H. Stroke
To cite this version:
Z. Fang, O. Redi, H. Stroke. Isotope shift constant and nuclear charge model. Journal de Physique II, EDP Sciences, 1992, 2 (4), pp.877-893. �10.1051/jp2:1992173�. �jpa-00247679�
Classification Physics Abstracts
31.30G-21,10F-35,10B
Isotope shift constant and nuclear charge model *'**
Z. Fang(~,~), O. Redi(~), and H-H- Stroke(~,~)
(~) Department of Physics, New York University, 4 Washington Place, New York, NY-10003, U,S.A.
(~) Present address: Department of Physics, University of Nevada, 4505 Maryland Parkway, Las
Vegas, NV-89154, U-S-A-
(~) On leave from NYU 1991-92 as Scientific Associate at CERN, CH-1211 Geneva 23, Switzer- land
Rdsumd. Nous employons la m6thode de Zimmermann [Z. Phys. A 321 (1985) 23-30], qu'il avait util1s4e dans un calcul de la constante du d4placement isotopique pour une distribution de
charge uniforme, pour l'obtenir avec un mod+le de charge nucl4aire avec forme quasi-trap4zoidale.
Les deux mod+les donnent des r6sultats dont la difference excbde de peu la prdcision des mesures actuelles. Les m6mes param+tres sont utilis4s pour comparer la ddpendance aux deux mod+les de la contribution au d6placement isotopique des moments plus dlev6s de la distribution de la charge nucldaire dans la formulation de Seltzer [Phys. Rev. 188 (1969) 1916-1919]. On trouve que ces
contributions sont essentiellement inddpendantes du mod+le. Des tables de calculs numdriques
sont pr4sent4es.
Abstract. We use the method of Zimmermann [Z. Phys. A 321 (1985) 23-30], which he used to calculate the isotope shift constant for a uniform nuclear charge distribution, to obtain it for a diffuse nuclear charge model. The two models give results that differ slightly on the level of precision of current experiments. The same parameters are used to calculate the model sensitivity of the contributions to the isotope shifts of higher moments of the nuclear charge
distribution as formulated by Seltzer [Phys. Rev. 188 (1969) 1916-1919]. These are found to be essentially model independent. Tables are given of the numerical calculations.
* Work supported at NYU in part by NSF grants PHY-8so3971, INT-8815310, and ECS-8819352. Submitted by Z. Fang in partial fulfillment of PhD degree at New York University.
* *We dedicate this work tc Professor P. Jacquinot for his leadership in the world of science, atomic spec- troscopy and instrumentation over a period of six decades! His warm friendship and cooperation extending
across national borders has been valued by ~1s for many years.
1. Introduction,
The atomic electron has served for many years as a probe of nuclear structure. The nuclear
charge, spin, and magnetic moment were early properties that were thus measured by atomic Spectroscopy. But more detailed knowledge on the nucleus iS also gained by the effect that the Spatially extended nuclear charge and magnetization have on the hyperfine spectrum.
The nuclear charge distribution thus leads to the isotope shift while the distributed nuclear
magnetization gives rise to the 'fhyperfine structure anomaly" or Bohr-Wehskopf effect ill.
We do not discuss the latter here, but only give some useful references [2, 3]. Also, although
we begin by calculating the structure elsects
on the spectrum of a single isotope, for electron atoms (in contradistinction to muonic atoms) we cannot compare the results with sufficient
accuracy with those of a hypothetical point nucleus. A comparison of isotopic vaHations with
experimental data is possible. Here we will ignore two mass dependent effects, which are in the first case trivial (reduced mass) and in the second very difficult to calculate as it depends
on electron correlations (specific mass skiff [4]). The volume dependent isotope shifts (field skiffs), which are most sensitive in transitions that involve s electrons, increase with increasing
mass number A, while the mass dependent shifts decrease with increasing A. Also it has been found IS] that in s - p atomic transitions the specific mass shift is small. Thus for heavier elements the field shift predominates and there is some rationale for ignoring it in the present
dhcussion.
The isotope shifts in the atomic spectrum thus reflect essentially the isotopic variations b of the mean square radius < R~ > of the nuclear charge distribution [6]. Electrons in s and pi
(its small component of the relativistic wave function) orbits, which have a non-zero probabilityj~
at the position of the nucleus, (~b(0)(~, can probe this variation of the charge distribution. With these spherically symmetric electron orbits one is sensitive to b < R~ > caused by a change
in nuclear radius and in nuclear deformation. We restrict the discussion to the dependence
of the spherical isotope shift on the nuclear model. Effects on the isotope shift caused by deformations and by nuclear polarizability [7] are neglected.
Early isotope shift measurements led to a confirmation of the general size of the nuclear radius [8]. These were made
on stable isotopes. While
a few systematic experiments were per-
formed with radioactive isotopes [9] "ols-line" (I.e. preparing spectroscopic lamps or cells with radioisotopes produced at a remote cyclotron or nuclear reactor site, and, in
some instances, followed by mass separation), the bulk of the recent progress has its origin in the "on-line"
work of Otten with the CERN synchrocyclotron and ISOLDE (the isotope separator with on line detection). The work of Otten and his collaborators at the University of Mainz and the contributions from the collaboration of the Laboratoire Aimd Cotton, led by P. Jacquinot (and
later by Liberman) and the Orsay mass separator group of Bernas and Klapisch are included in an early review [10]._ In the dozen years that followed, a large number of experiments were done, in several cases extending over more than twenty isotopes and nuclear isomers. A review of the work, which has led to considerable detailed knowledge on nuclear radius variations has
been made recently by Otten ill].
Not only the quantity, but the quality of the isotope shift data have increased dramatically:
a gain of some three orders of magnitude in precision has been obtained with the use of laser spectroscopy by the elimination of Doppler broadening of the spectral lines. This higher preci- sion also warranted the reixamination of isotope shift calculations [12-IS]. In this contribution
we study further the sensitivity of the calculations of the isotope shift to the model of the nu- clear charge distribution. Although a number of earlier works have considered several nuclear models, recently improved calculations by Zimmermann [13] were made only for a uniform
nuclear charge distribution. We calculate for comparison the isotope shifts for
a "realistic"
dilsuse charge distribution based on hi formalism. We account furthermore for the isotopic variations of the electron wave functions inside the nucleus.
2. Sununary of isotope shift calculations.
The theoretical studies of the isotope field shifts date back to perturbation theory work by Rosenthal and Breit at New York University, and by Racah [16]. The Dirac equations were
solved with certain approximations, e-g- neglecting the binding of the electron compared to its rest mass energy, ignoring core electrons, and in the determination of the normalization of the wave functions. The consequence of lifting the above restrictions was examined by Broch.
ii?] Non-perturbative approaches were also used 11?, 18] and compared to the perturbation
calculations. Bodmer, then at CEItN, analyzed further the dilserent approaches and approxi-
mations [19] and pointed out the application of his formalism to non-uniform nuclear charge distributions [20]. Fradkin [21] considered additionally the potential of the electron shells and nuclear compressibility in accounting for the isotope shifts. Further work along these lines, though differing somewhat in the results from the preceding work [13], was done by Babushkin [22, 23]. Seltzer [24] calculated contributions to the isotope shift constant (definedin Sect. 3.2)
from higher (even) moments of the radial nuclear charge distribution, and Torbhom, Fricke and Rosin [25] made ab-initio isotope shift calculations with the use of a multiconfiguration Dirac- Fock method.
Zimmermann [12] pointed out an error in the normalization used by Babushkin [23]. A re-evaluation of Babushkin's numerical results was given by Blundell et. al. [15]. The most recent calculations of Zimmermann [13], which include higher order contributions in obtaining
the isotope shift constant, G [Eq. (Ii)], were done in order to eliminate errors in past work and to improve the calculations so as to obtain an accuracy on the level of I il provided by
experiment. We use his method to calculate the isotope shift constant for a diffuse nuclear
charge distribution.
3. Zimmermann isotope shift for a difEuse nuclear charge distribution.
3.I THE NUCLEAR POTENTIAL. Here we follow exactly the formalism of Zimmermann (13],
much of which
we reproduce for completeness, but for a dilsuse (quasi-trapezoidal) nuclear charge distribution. For simplicity, we approximate it as earlier [2] by the polynomial in
~ = r/RN (RN, nuclear radius),
P " PO + P2Z~ + p3Z~+P4Z~. (I)
The coefficients p; are determined by a fit to the Hofstadter trapezoidal charge distribution
[26] and of course the condition that f~° pdr = Ze. From the charge distribution we obtain
the potential
V(z) =
)(It
a~z~ a4z~ asz~ a6z~), (2)N
where K + 1+ a2 + a4 + as + a6 and the coefficients aj are determined by the p;. We gave explicit expressions in [2] in terms of the quantities cl " 1.07A~'~F (the radial distance at
which p = po/2), the skin thickness t = 2z3
" 2A0F, and RN " cl + z3, which characterize
the trapezoidal charge distribution [26].
3.2 ZIMMERMANN FORMULATION OF THE ISOTOPE SHIFT. We solve the Dirac equations
as before [27] with the same approximations of neglecting the electron binding energy E~
compared to mc~ (m : electron mass, c : velocity of light) and the contribution of the core electrons to the potential in the vicinity of the nucleus [14, IS]:
27(
~
~ " (l 6~ + ")G~ (3)
27(~(~ + n~~ = (I + e~ u)F~,
where u is the potential energy, 7 " aZ, s = 2Zrlao (ao, Bohr radius), and all energies are expressed in units of mc~ (I.e. e~ = E~/mc~); n
= -(£ + I) for j = £ + ) and £ for j = £ )
(£, electron orbital angular momentum). Zimmermann [13] solves directly for the level shift in
a given isotope Ae~ + e~ ES resulting from Au = u u°, the superscripts 0 referring to the point nuclear case. The result is 11?]
Ae~ = 27(F~G( F$G~)(,~, (4)
where sN is the value of s at r = RN With use of the small argument expansion of the Bessel functions (with the second term of the expansion included by Zimrnermann), matching interior and exterior solutions at sN,
~
K~ +
)
=
jl,~,
(5)~ ~
one obtains the field shift Ae~ in the level [13].
The resulting isotope shift is
b(Ae~)
= )(Ae~)bsN. (6)
sN
After somewhat lengthy arithmetic, following Zimmermann,
~ ~ 27 7 K(~ P)s~2P~~~(1+a~sN))>
b(A6~) " Ci~ ( ~~~
r2(2p) 7 K(K + P) ~N where
~~
~~ ~~~ ~~ ~ (7 It (~ ~~'~ K(~ + P)1' ~~~
and ~~~~~ ~~~~ ~~~~~ ~ ~~
~~~ 2p+1 7-K(n-p) ~l-2p 7-K(n+p) ~~~
We have onfitted the subscript n in the It's in the above equations. r is the gamma function,
p =
@t, and C(~ is the coefficient associated with the Bessel function J2p and J2p+1 solutions of the Dirac equation for s > sN. For s > sN it is in fact the coefficient of the leading
term of the large component of the Dirac wave function.
For an s electron we can then use the Breit-ltosenthal connection to the non-relativistic wave function [16] to obtain
~0 2 ~CO
~3~2(~) (~~)
~~ 2Z~e~ °
Rm is the ltydberg constant in energy units. The Breit- Rosenthal normalization is 4x f/°~b~r~
dr = I. The end result of the above calculations h in agreement with Zimmermann's noting
that to obtain his equation (17) the Rydberg is to be expressed in units of wave numbers. For
an s electron (n
= -I), one usually expresses [28] the isotope shift, equation (6) in terms of an isotope skiff constant, Cs:
b(A~) " )~
l1fi3(°) l~cs> (~~)
where, with the use of equation (8), we obtain Zimmermann's equation (17):
~ + K(I + P)
~2pbRN
(I + assN)> ~~~~
c~ = 2Ryco
~2(2p) ~ + K(I P) ~ ~N
where Ryco is expressed in wave numbers, and as is the value of a evaluated for n
= -I. This
result is independent of the nuclear model.
3.3 SPECIALIzATION To THE DIFFUSE NUCLEAR CHARGE DISTRIBUTION. It is seen that the isotope shift constant depends directly on the variation of RN and through that of K. For the latter, we need to determine the interior functions F~ and G~. Before doing so, we observe
that the first order correction terms assN in (12) give [29] a few percent contributions to C~
even for high Z. Higher order terms are estimated to contribute much less than I il. With an accuracy of ~-10~~ in calculating K~ and a resulting uncertainty of
~- 0.3 to 0A il in Cs, the
hotope shift constant calculated from (12) should have an accuracy of m I il.
The interior solutions (r < RN) of the Dirac equations in the potential of equation (2),
U = -eV(z) are now obtained as done explicitly in earlier work [2, 27]. A series solution for the functions F and G is assumed (first for n
= I, and then for a pi/2 electron, n = + I),
and the recursion formulae for the coefficients determined. K can then be calculated from an evaluation of the ratio FIG at RN (z
= I). As in Zimmermann [13], K is expanded in powers of 7, and evaluated to sufficient accuracy retaining terms up to 7~. As an example, for Z
= 80,
mass number A
= 200, for s and pi/2 electrons, respectively, we find K~ = -0.42987 0.05937~ 0.01237~
Kp = [2.2085 0.30037~ 0.02087~].
7
If we wish to determine the model dependence of the isotope shift constants for pi/2 elec- trons, for which non-relativistically ~b~(0) = 0, we simply define, in a manner analogous to
equation (12), but with n=1,
7 + Kp(p 1)
~2P b~N
(I + apsN). ~~~~
Cp = 2RY~X~ r2(2p) 7 Kp(p+ 1) ~ ~N
With the use of equations (12) and (13) we can thus calculate Cs and Cp for the polynomial approximation of the trapezoidal nuclear charge dhtribution. The results are given in table I.
We also calculate Cs and Cp for the uniform charge distribution with RN " 1.20A~'~F. These
agree with values given for s electrons in table I of [IS].
In equation (S) of [14], we find the "customary" repreqentation for extracting b < r~ > of the nuclear charge distribution from the experimental value of C for an isotopic pair
b < r2 >~xp=
~~~jP
~
b < r2 >man~ar~, (14)
Stan ar
where C~tandard is usually based on the uniform dhtribution above. Hence we would expect
b < ~~ >trape
~
Cirape
$ < r~ >unif Cunif ~~~~
Table I. Isotope skiff constants C for uniform and trapezoidal charge distHbutions calculated by Zimmermann's method with RN " 1.20A~'~F for the former and the parameters given in
the text for the latter.
S Pi12
Z Ai~A2 Ctrape Cunif ) Ctrape Cunif )
10 20- 22 2.19 2.28 0.003 0.004 0.968
II 23- 25 2.48 2.66 0.934 0.005 0.005 0.942
12 24- 26 2.91 3.lS 0.926 0.006 0.007 0.933
13 27- 29 3.25 3.59 0.906 0.008 0.009 0.913
14 28- 30 3.75 4.16 0.901 0.010 0.011 0.906
15 31- 33 4.14 4.67 0.886 0.013 0.01S 0.891
16 32- 34 4.69 S.32 0.882 0.017 0.019 0.886
17 35- 37 5.14 5.91 0.870 0.021 0.023 0.875
18 38- 40 5.62 6.53 0.861 0.025 0.029 0.865
19 39- 41 6.27 7.31 0.858 0.031 0.036 0.861
20 42- 44 6.81 8.02 0.850 0.037 0.044 0.853
21 45- 47 7.39 8.77 0.843 0.045 0.053 0.846
22 48- 50 8.00 9.57 0.837 0.053 0.063 0.840
23 50- 52 8.73 10.48 0.833 0.063 0.076 0.836
24 52- 54 9.49 11.45 0.830 0.075 0.090 0.832
25 53- 55 10.40 12.56 0.828 0.089 0.107 0.830
26 56- 58 II-1? 13.57 0.824 0.104 0.125 0.826
27 57- 59 12.17 14.81 0.822 0.122 0.148 0.824
28 60- 62 13.05 15.96 0.818 0.141 0.171 0.820
29 63- 65 13.98 17.17 0.815 0.162 0.198 0.816
30 64- 66 15.15 18.64 0.813 0.188 0.231 0.815
31 69- 71 15.99 19.80 0.808 0.212 0.262 0.810
32 72- 74 17.08 21.22 0.805 0.242 0.300 0.807
33 75- 77 18.24 22.73 0.803 0.275 0.342 0.804
34 78- 80 19.46 24.34 0.800 0.312 0.390 0.802
35 79- 81 20.99 26.27 0.799 0.375 0.447 0.801
36 82- 84 22.37 28.70 0.797 0.404 0.506 0.799
37 85- 87 23.83 29.98 0.795 0.4SS 0.572 0.797
38 86- 88 25.63 32.27 0.795 0.S18 0.6Sl 0.796
39 89- 91 27.27 34.42 0.793 0.582 0.733 0.794
40 90- 92 29.29 36.98 0.792 0.659 0.830 0.792
41 93- 95 31.14 39.41 0.791 0.738 0.932 0.791
42 95- 97 33.26 42.15 0.790 0.829 1.049 0.790
43 97- 99 35.50 45.04 0.789 0.930 1.178 0.789
44 100-102 37.72 47.94 0.787 1.037 1.316 0.787
45 103-105 40.06 51.01 0.786 1.156 1.469 0.786
46 106-108 42.54 54.26 0.785 1.286 1.638 0.786
47 107-109 45.56 58.12 0.784 1.442 1.837 0.784
48 l10-l12 48.37 61.79 0.783 1.602 2.043 0.783
49 l13-lls Sl.35 65.70 0.782 1.778 2.271 0.782
50 l16-l18 54.52 69.84 0.781 1.972 2.522 0.781
Table I. (continued
s pi12
Z Ai~A2 Ctrape Cunif ~f Ctrape Cunif )
51 121-123 2.781
52 124-126 61.01 78.41 0.779 2.403 3.083 0.779
53 127-129 64.79 83.34 0.778 2.660 3A16 0.778
54 132-134 68.33 88.05 0.776 2.922 3.761 0.778
55 133-135 73.06 94.16 0.776 3.253 4.187 0.777
56 136-138 77.58 100.1 0.776 3.594 4.630 0.777
57 138-140 82.68 106.7 0.776 3.983 5.132 0.776
58 140-142 88.10 l13.7 0.775 4A12 5.686 0.776
59 141-143 94.18 121.5 0.775 4.899 6.314 0.776
60 144-146 100.1 129.2 0.775 5.403 6.968 0.776
61 146-148 106.6 137.7 0.775 5.976 7.708 0.775
62 150-152 l12.9 146.0 0.774 6.567 8.478 0.775
63 lsl-153 120.8 156.0 0.774 7.280 9.398 0.774
64 156-158 127.5 165.0 0.774 7.972 10.30 0.774
65 159-161 135.6 175A 0.773 8.781 11.35 0.774
66 161-163 144.6 187.0 0.773 9.697 12.54 0.774
67 163-165 154.2 199A 0.774 10.71 13.85 0.774
68 166-168 163.9 212.1 0.773 11.79 )5.24 0.774
69 169-171 174A 225.6 0.773 12.98 16.78 0.774
70 172-174 185.5 240.0 0.773 14.28 18.47 0.774
71 175-177 197A 255.5 0.773 15.72 20.33 0.774
72 177-179 210.8 272.6 0.774 17.35 22.43 0.774
73 179-181 225.0 291.0 0.774 19.14 24.74 0.774
74 182-184 239.6 309.8 0.774 21.05 27.22 0.774
75 185-187 255.2 329.9 0.774 23.16 29.94 0.774
76 188-190 271.9 351A 0.774 25.50 32.95 0.774
77 191-193 289.8 374.4 0.774 28.05 36.25 0.774
78 194-196 308.9 399.0 0.775 30.87 39.89 0.774
79 197-199 329.4 425.3 0.775 33.98 43.89 0.775
80 200-202 351.3 453.4 0.775 37.41 48.30 0.775
81 203-205 374.9 483.6 0.776 41.19 53.16 0.775
82 206-208 400.0 515.7 0.776 45.34 58.50 0.776
83 207-209 429.1 552.8 0.777 50.18 64.69 0.776
84 210-212 458.3 590.0 0.777 55.27 71.22 0.777
85 215-217 487.1 626.8 0.778 60.59 78.06 0.777
86 218-220 520.5 669.3 0.778 66.77 85.97 0.777
87 221-223 556.4 714.8 0.779 73.59 94.70 0.778
88 224-226 594.9 763.6 0.780 81.14 104.3 0.778
89 227-229 636.1 815.8 0.780 89.45 l14.9 0.779
90 229-231 682.1 873.8 0.781 98.90 127.0 0.779
91 231-233 731.6 936.1 0.782 109A 140.3 0.780
92 235-237 781.3 998.7 0.783 120.5 154.4 0.781
93 237-239 838.5 1071 0.784 133.3 170.6 0.782
94 239-241 900.1 l148 0.785 147.5 188.7 0.782
95 241-243 966.4 1230 0.786 163.3 208.6 0.783
