Calculated properties of μ-mesonic atoms

Nuclear Physics 35 (1962) 2 9 5 - - 3 0 2 ; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

CALCULATED P R O P E R T I E S OF p - M E S O N I C A T O M S KENNETH

W . F O R D t a n d J O H N G. W I L L S ct

University o] Cali/ornia, Los Alamos Scientilic Laboratory, Los Alamos, New Mexico ¢¢t R e c e i v e d 5 F e b r u a r y 1962 R e s u l t s of c a l c u l a t i o n s o f / , - m e s o n i c e n e r g y levels a n d s o m e a u x i l i a r y q u a n t i t i e s der i v e d f r o m t h e b o u n d m u o n w a v e f u n c t i o n s a r e p r e s e n t e d . T h e r e s u l t s a r e e x a c t insofar as t h e n u c l e u s m a y be r e g a r d e d as a s t a t i c s p h e r i c a l l y s y m m e t r i c c h a r g e d i s t r i b u t i o n . D y n a m i c a l corrections are omitted.

Abstract:

1. Introduction The main purpose of this paper is to present the theoretical 2p -+ Is energies of #-mesonic atoms, calculated as accurately as is possible with present knowledge of nuclear charge distributions. Comparison with an older experiment 1) is made, b u t the comparison of the results with recent more precise experiments 2-4) is left to the experimental papers. Some 3d -+ 2p energies have also been calculated and appear in ref. 3). The particularly important case of the 3d --~ 2p transition in phosphorus has been accurately treated b y others 5, e). In the process of obtaining energies, we obtain relativistic muon wave functions, which are useful for other purposes. We present here the quantity Zen relevant to muon capture 7), and the binding correction to the muon magnetic moment s). Much more elaborate calculations of the decay rate of bound muons 9), and of the capture rate in carbon 10), have also made use of these wave functions. Parts of the results presented here have appeared in previous publications 7,s) and in an unpublished report n), and this paper is intended primarily to summarize conveniently in one place all of the numerically calculated quantities of interest, except for the wave functions. Some numerical tables of muon wave functions were given in ref. n). Others may be obtained from the authors.

2. Calculations Details of the numerical problem solved "exactly" with the help of a IBM 704 computer have been given in ref s). In the radial Dirac equations, the * P e r m a n e n t A d d r e s s : P h y s i c s D e p a r t m e n t , B r a n d e i s U n i v e r s i t y , W a l t h a m 54, M a s s a c h u s e t t s . *t P r e s e n t A d d r e s s : P h y s i c s D e p a r t m e n t , U n i v e r s i t y of W a s h i n g t o n , Seattle 5, W a s h i n g t o n . *it W o r k s u p p o r t e d b y t h e U.S. A t o m i c E n e r g y C o m m i s s i o n . 295

296

KENNETH W. FORD AND JOHN G. WILLS

m u o n r e d u c e d m a s s is u s e d 22), a n d t h e n u m e r i c a l results p r e s e n t e d here are b a s e d on the following choice of t h e m u o n m a s s 13): m s = 206.76 m e = 105.65 MeV. F o r helium, a G a u s s i a n nuclear charge d i s t r i b u t i o n is used 24). F o r all h e a v i e r e l e m e n t s a t w o - p a r a m e t e r c h a r g e d i s t r i b u t i o n is used, c h a r a c t e r i z e d b y a TABLE 1 Elements and nuclear parameters Z

Element

A

2 4 6 7 8 9 11 12 13 14 15 16 17 19 20 22 23 24 25 26 28 29 30 37 42 47 48 50 51 56 57 64 73 74 80 81 82 83 92

He Be C N O F Na Mg A1 Si P S C1 K Ca Ti V Cr Mn Fe Ni Cu Zn Rb Mo Ag Cd Sn Sb Ba La Gd Ta W Hg T1 Pb Bi U

4.00 9.02 12.01 14.01 16.00 19.0 23.0 24.3 27.0 28.1 31.0 32.1 35.5 39.1 40.1 47.9 51.0 52.0 54.9 55.9 58.7 63.6 65.4 85.5 96.0 107.9 112.4 118.7 121.8 137.4 138.9 157.3 181.5 183.9 200.6 204.4 207.2 209.0 238.1

rl/A t 0.825 a) 0.83 t 0 . 0 4 D) 1.009 ± 0.015 1.02 1.0454-0.015 1.03 1.03 1.03 ± 0 . 0 2 1.03 1.03 ± 0 . 0 2 1.04 1.04 + 0 . 0 2 1.05 1.06 1.063=i=0.015 1.06 1.057=t= 0.015 1.07 1.07 1.07 1.06 -4-0.02 1.07 1.07 1.075 1.08 1.08 1.08 1.08 1.085 1.09 1.09 1.095 1.10 1.10 ± 0 . 0 2 1.10 1.10 1.11 i 0 . 0 1 1 . 0 9 4 ± 0.01 1.10

n

Req/A t

a)

1.30 1.74 1.69 1.64 1.59 1.56 1.51 1.56 1.49 1.49 1.48 1.46 1.47 1.47 1.47 1.39 1.34 1.38 1.36 1.36 1.34 1.32 1.31 1.27 1.24 1.22 1.22 1.22 1.23 1.21 1.21 1.23 1.23 1.23 1.21 1.18 1.19 1.22 1.21

2.25 3.00 3.2 3.5 3.5 3.7 3.5 3.8 3.8 3.9 4.0 4.0 4.1 4.1 4.6 5.0 4.8 5.0 5.0 5.0 5.4 5.5 6.1 6.8 7.2 7.4 7.3 7.25 7.9 7.8 7.5 7.5 7.5 8.5 9.9 I0.0 7.7 8.5

s) A G a u s s i a n c h a r g e d i s t r i b u t i o n w a s used for h e l i u m : p = Z4r1-8 exp (--ri]r12). b) U n c e r t a i n t i e s in rl]AJ a r e i n d i c a t e d for those e l e m e n t s w h e r e fi t s t o e l e c t r o n s c a t t e r i n g data have been made.

CALCULATED PROPERTIES OF /J-MESONIC ATOMS

297

radial parameter rl and a surface thickness parameter n; this has been used previously to fit high energy electron scattering data 15), and is defined in refs. s, 15). The electric potential derived from this charge distribution can be written down explicitly le). The elements studied and the nuclear parameters assumed are given in TABLE 2 E n e r g y Differences 2p doublet splitting Element

4 Be 6 C 7 N 8 O 9 F 11Na 12 Mg 13 A1 14 Si 15 P 16 S 17 C1 19 K 20 Ca 22 Ti 23 V 24 Cr 25 Mn 26 Fe 28 Ni 29 Cu 30 Zn 37 R b 42 Mo 47 Ag 48 Cd 50 Sn 51Sb 56 Ba 57 La 64 Gd 73 Ta 74 W 80 H g 81 T1 82 P b 83 Bi 92 U

Point nucleus

Finite nucleus s)

0.0000024 0.0000120 0.0000223 0.000038 0.000061 0.000136 0.00019 0.00028 0.00036 0.00048 0.00062 0.00060 0.00079 0.00078 0.00123 0.00120 0.00152 0.00148 0.00222 0.00215 0.00266 0.00254 0.0032 0.0030 0.0037 0.0036 0.0044 0.0041 0.0059 0.0055 0.0068 0.0063 0.0078 0.0073 0.0184 0.0156 0.0310 0.0250 0.0493 0.0372 0.0538 0.0400 0.0638 0.0457 0.0693 0.0485 0.1028 0.0657 0.1109 0.0695 0.182 0.096 0.325 0.136 0.344 0.142 0.489 0.175 0.518 0.184 0.549 0.188 0.580 0.188 0.944 0.234

2pt --~ l s transition energy Point nucleus 0.033344 0.075282 0.10262 0.13420 0.17010 0.25453 0.30311 0.35605 0.41319 0.4747 0.5405 0.6107 0.7640 0.8471 1.0270 1.1235 1.2244 1.3298 1.4396 1.6728 1.7965 1.9246 2.953 3.832 4.838 5.055 5.505 5.738 6.990 7.257 9.302 12.414 12.797 15.263 15.704 16.153 16.613 21.209

Finite nucleus ~)

E x p e r i m e n t b)

0.033332 0.075031 0.10204 0.13309 0.16795 0.24905 0.29437 0.34430 0.39700 0.4524 0.5120 0.5720 0.7027 0.7718 0.9202 1.0011 1.0745 1.1563 1.2391 1.4113 1.4981 1.5883 2.224 2.715 3.201 3.293 3.466 3.543 3.996 4.084 4.631 5.288 5.360 5.850 5.999 6.023 5.993 6.536

a) Finite nucleus values include v a c u u m polarization correction. b) F i t c h a n d R a i n w a t e r 1). F o r m o r e recent e x p e r i m e n t a l results, see refs. s-4).

0.35 0.410

0.955

1.55 1.60

3.50

6.02 6.02

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KENNETH W. FORD AND JOHN G. WILLS

table 1. These parameters are either those which have provided good fits to electron scattering data, or are interpolated between such values. Table 1 also lists the "equivalent-uniform radii", Req (ref. 14)). The unit of length in this and subsequent tables is the femtometer (1 fm = 10-16 m). The radial Dirac equations were solved b y an iteration scheme to yield the energy eigenvalues and the radial wave functions F and G. These wave functions were then used to obtain the vacuum polarization correction to the energy, the binding correction to the magnetic moment, and the capture quantity, ZetfTable 2 gives the calculated 2p doublet splitting, and the 2pt -+ ls transition energy in MeV, the latter being compared with the early experiments of Fitch and Rainwater x). The small discrepancies no longer require discussion and are undoubtedly without significance. It is of some interest to note, however, that because of the higher mass previously assumed for the muon, the energy values for light nuclei appeared to be in better agreement with the theoretical values for nuclear radii of about 1.2A ½than t h e y now are. If taken completely seriously, the transition energies for the light elements (up to Z = 30) would imply nuclear radii as:much as a factor of two too small. These comments do not apply to lead and bismuth energies, whose analyses have been little affected by the revised muon mass. 2.1. V A C U U M P O L A R I Z A T I O N

The vacuum polarization correction to/x-mesonic energy levels, considered for example by Rainwater 17) and Glauber et al. is), is not significantly larger than a much less certain correction arising from nuclear polarization 19), at least for the Is states ~. However, it can be calculated accurately, and accordingly we include this correction in the tabulated 2p -+ ls transition energies. (The nuclear polarization correction is of the same sign, increasing slightly the Is binding energy). Except for the n ----- 2 levels for Z ~ 9, the muon is bound within a distance of the nucleus small compared to the electron Compton wave-length. Therefore Schwinger's small-distance approximation for the vacuum polarization potential 20), suitably modified for a distributed nuclear charge le), m a y be used. (For the 2p levels up to Z = 9, the correction is exceedingly small and is ignored.) The energy shift is, using a wave function normalization f (F2+G2)dr = 1,

AEp = f (F2+G2)Vp(r)dr < O,

(1)

where Vp(r) is the effective potential due to vacuum polarization, given approximately by

Vp(r) : (2~/3:z)[VL(r )-5V(r)],

(2)

? F o r h i g h e r s t a t e s , t h e r e q u i r e d c o n s i s t e n c y of m e s o n m a s s v a l u e s o b t a i n e d f r o m d i f f e r e n t t r a n s i t i o n s h a s b e e n used to d e m o n s t r a t e e x p e r i m e n t a l l y t h e e x i s t e n c e of t h e v a c u u m p o l a r i z a t i o n c o r r e c t i o n 1~).

CALCULATED PROPERTIES OF v-MESOmC ATOMS

in which

V(r)

VL(r)

--lzc(el/r) f p(r')r'{}r--r'l [ln(C[r--r'[/t~e)--

299

is the electrostatic potential, and VL(r ) is 1]

--(r+r')[In(C(r+r')/~e)--l])dr'.

(3)

Here p(r) is the nuclear charge density in protons per unit volume, he is the reduced electron Compton wave length, and C = 1.781. The correction, given b y eq. (1), was evaluated numerically in each case, and is shown for the Is states in table 3. This table includes also the calculated binding energy and the mean value of the potential energy ,

(V> = f (Fl+Gl)V(r)dr.

(4)

2.2. C A P T U R E

The mean nuclear charge density, or muon-nucleus overlap integral, defined by

=

f (FI+GI)P(') dr,

(5)

is of interest to total muon capture rates 21, 22), and to the question of a possible anomalous muon-nucleon interaction. The equivalent quantity l e t f is defined by le4ff =

Y~(gO3

,

(6)

where a o is the muon Bohr radius. Values of l e t f a r e listed in table 3. For light nuclei, we have Z~ff m Z4[1--ll/ao]. (7) For very heavy nuclei l e t t would approach a constant if the central density of charge is constant. For a central charge density of 0.080 protons per fm s one has l e t t --> 4 5 . 2.3. M A G N E T I C M O M E N T

Magnetic moments of #-mesonic atoms have recently been measured to very high precision 13). The theory of the magnetic moment of a bound magnetic muon has been given elsewhere 8), along with some calculated results. Table 3 includes a tabulation of the binding corrections to the moment for Is states, given by

, g/g =

f

(8)

For comparison, the point-nucleus binding correction is 1, Z,2 (Ag/g)potnt =---,(~ ).

(9)

Since all of the calculated quantities depend upon the assumed values of the nuclear parameters, it is important to know how a change of nuclear

300

KENNETH W. FORD AND JOHN G. WILLS

radius or surface thickness would affect the results. The sensitivity of results to variation of the parameters is given in table 4 for four elements. Since these numbers do not need to be known to high accuracy, curves drawn through these four points provide a satisfactory basis for estimating the sensitivity TABLE 3 Ground state properties

Element

E n e r g y a)

2 He 4 Be 6 C 7 N 80 9 F 11Na 12 M g 13 A1 14 Si 15 P 16 S 17 C1 19 K 20 C a 22 T i 23 V 24 Cr 25 M n 26 F e 28 N i 29 Cu 30 Z n 37 R b 42 M o 47 A g 48 Cd 50 S n 51Sb 56 B a 57 L a 64 G d 73 T a 74 W 80 H g 81 T1 82 P b 83 B i 92 U

-- 0.010940 -- 0.044354 -- 0.09979 -- 0.13573 -- 0.17707 -- 0.22360 -- 0.3322 -- 0.3933 -- 0.4604 -- 0.5316 -- 0.6069 -- 0.6878 -- 0.7706 -- 0.9508 - - 1.0468 - - 1.2531 - - 1.3651 - - 1.4710 - - 1.5865 - - 1.7046 - - 1.9515 -- 2.0778 -- 2.2087 -- 3.170 - - 3.936 - - 4.731 -- 4.889 -- 5.198 -- 5.345 - - 6.167 - - 6.332 -- 7.456 -- 8.933 -- 9.102 - - 10.201 --10.464 --10.590 --10.648 --12.175

Vacuum polarization correction

--0.000090 --0.000320 --0.00050 --0.00072 --0.00099 --0.00164 --0.00202 --0.00246 --0.00293 --0.00343 --0.00399 --0.00455 --0.0058 --0.0065 --0.0080 --0.0088 --0.0095 --0.0104 --0.0112 --0.0130 --0.0139 --0.0148 --0.0217 --0.0270 --0.0324 --0.0333 --0.0352 --0.0361 --0.0415 --0.0425 --0.049 --0.057 --0.058 --0.064 --0.067 --0.067 --0.067 --0.075

s) T h i s e n e r g y d o e s n o t i n c l u d e t h e v a c u u m

-- ( V )

0.02187 0.08859 0.19861 0.2696 0.3511 0.4420 0.6530 0.7697 0.8988 1.0343 1.1760 1.3282 1.4794 1.8089 1.9831 2.355 2.560 2.740 2.944 3.149 3.577 3.790 4.013 5.572 6.787 7.995 8.226 8.666 8.865 10.034 10.265 11.768 13.688 13.906 15.361 15.748 15.875 15.885 17.763

Zef f

1.98 3.89 5.72 6.61 7.49 8.32 9.95 10.69 I1.48 12.22 12.91 13.64 14.24 15.53 16.15 17.38 18.04 18.49 19.06 19.59 20.66 21.12 21.61 24.47 26.37 27.95 28.20 28.64 28.79 29.99 30.20 31.34 32.61 32.76 33.81 34.21 34.18 34.00 34.94

polarization correction.

(Ag/g) --0.0000710 --0.000282 --0.000629 --0.000851 --0.001104 --0.001384 --0.002029 --0.002379 --0.00278 --0.00317 --0.00359 --0.00404 --0.00446 --0.00540 --0.00588 --0.00692 --0.00749 --0.00795 --0.00850 --0.00904 --0.01015 --0.01069 --0.01126 --0.01494 --0.01769 --0.02022 --0.02066 --0.02146 --0.02178 --0.02389 --0.02429 --0.02659 --0.02928 --0.02958 --0.03173 --0.03248 --0.03248 --0.03220 --0.03432

CALCULATED PROPERTIES OF /*-MESONIC ATOMS

301

for all elements. The exchange ratios listed are defined b y Xa (x) ---- d(lnA)/d(lnx),

(10)

and give the percent change in A for a 1 % change in x. TABLE 4 Sensitivity to p a r a m e t e r variation 2 p | --~ Is energy

lAg/g!

(p)

Element

6 C 22 Ti 51Sb 83 Bi

X~(Vx)

XE(")

Xp(r~)

Xp(n)

Zo(V~)

Xo(~)

--0.0140 --0.177 --0.525 --0.819

0.00879 0.0657 0.0828 0.109

--0.175 --0.946 --1.61 --2.04

0.112 0.331 0.248 0.260

--0.0277 --0.302 --0.718 --0.953

0.0181 0.113 0.109 0.121

3. C o n c l u s i o n Some of the results presented here have already been applied to the interpretation of experiments on transition energies, capture rates, and magnetic moments. Conceivably some will find a use in the analysis of future experiments on #-mesonic atoms. There is already overwhelming evidence that the/,-meson, in all low energy phenomena, differs from the electron only in mass. The/~-mesonic atom therefore becomes primarily a tool for studying nuclear, atomic and solid state effects, rather than a tool for studying the muon. Most of the remaining effects of interest are dynamical effects, which can be calculated (or estimated) as perturbations to the results of the static calculations presented here. Most of the results reported here were obtained at the Los Alamos Scientific Laboratory. A few were obtained subsequently at the computation centers of the Massachusetts Institute of Technology, of New York University, and of the University of Washington. We are grateful to the authorities of these institutions for the opportunity to use their computing machines. One of us (K.W.F.) is indebted to Professor A. Salam for the hospitality of Imperial College, where this paper was written. References 1) 2) 3) 4) 5) 6)

V. Fitch and J. Rainwater, Phys. Rev. 92 (1953) 789 P. Brix et al., to be published W. Frati et al., to be published Johnson, Hincks a n d Anderson, Phys. Rev., to be published A. P e t e r m a n n and Y. Yamaguchi, Phys. Rev. Lett. 2 (1959) 359 Devons, Gidal, L e d e r m a n and Shapiro, Phys. Rev. Lett. 5 (1960) 330

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W.

FORD

AND

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G. W I L L S

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