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Simple relation between radiative pion absorption and muon-capture in nuclei
Volume 28B. number 2
SIMPLE
PHYSICS
RELATION AND
LETTERS
11 November
BETWEEN RADIATIVE PION MUON-CAPTURE IN NUCLEI
M. RHO and A. SHERWOOD Centre d’Etudes Nuclkaires de Saclay, Gif-sur-Yvette, Received
1 October
1968
ABSORPTION
France
1968
It is shown numerically that for light nuclei with J = 0 ‘. T = 0 ground state, the ratio of the total radiative 1s pion capture rate r(n-) to the total muon capture rate r(p-) is very nearly independent of nuclear structure, depending only on the ratio of squares of pionic and muonic atomic wave functions averaged over the nuclear dimension.
The radiative r- absorption in nuclei Ai’, ?I- + Ai - + y + Af, provides three means of studying nuclear structure [l]. Firstly, the process being theoretically well understood can provide ways of investigating pion-nucleus interaction through the S-orbit v- wave function. Secondly, the process goes predominantly through the axial vector current with momentum transfer equivalent to pion mass (mr) and therefore can provide informations on the axial vector matrix elements of the p-capture and in particular on the corrections to the usual impulse approximation. Thirdly and perhaps the most interesting is the possibility of studying the spin-isospin giant resonances which are members of giant resonance supermultiplets [2]. In this letter, we discuss the first piont. Using the formal similarity [3] of the pion-induced process to the muon process with the same initial state and summed over final states, /L- + Ai - V+ + Af, we demonstrate that the ratio of the capture rates for S-orbit muons and pions is very nearly independent of the nuclear structure. The ratio thus depends only on the pionic 1s wave function, since the muonic wave function is well known. This is shown for the J = O+, T = 0 nuclei 4He, I2C, 160 and %a with experimental photo-absorption cross-section and nuclear models for correction terms only. Consider the capture of S-orbit muons and pions t in the initial nucleus Ai to go to all final states in Af_ The case of muon capture is wellknown [2] and so we discuss below only the pion -? For A ) ‘7. the P-orbit absorption becomes significant, and it would be difficult to measure directly the capture from lS-orbit for higher mass nuclei. Thus 46Ca we include in the list may be academic. but it helps clarify the theoretical aspect we are connected with.
102
process. By the PCAC hypothesis with the softpion theorem [2], one obtains the transition operator to order m,/M and in impulse approximation T
=a
6 ‘Tam)&* exp(-ik Ui
i=l
ri) + R ,
(1)
where R contains terms proportional to the pion momentum q. Here E is the photon polarization vector, k the photon momentum, and r(-) changes proton into neutron. The operator becomes simple when it operates on an 1s pionic wave function as R gives terms propertional to ]Vcp1S(‘)12 and this can be neglected, as cPlS(r) varies very slowly within the nucleus [l]. The usual algebra (polarization sum etc.) leads to the simple formula for the total transition rate l?(n-) (ti = c = 1) Const. I;Tvl2 J $
x
C(g)
f
X
Vl~72!-)cr(i)exp(-ik,xi)li)
(2) 2,
where $v is the S-orbit pionic wave function averaged over nuclear dimension *, .k i the energy carried away by the photon equa f to (m, - B.E. of s-mesic atom - AEfi between nuclear states). We point out the difference between the matrix element in eq. (2) and the axial vector matrix element in n capture [2]: in eq. (2), the summand is linear in photon energy k, whereas in the n capture it is quadratic in neutrino energy v. The crucial point of our calculation is that we can treat eq. (2) in the same way as for the muon axial vector matrix elements. So using the idea of Foldy and Walecka [2], we relate the major * For 12C and 160, the variation is *40/L over the nuclear
dimension
[4].
Volume 28B.
number 2
PHYSICS
Contributions
to the capture
ratio
LETTERS Table 1 ( r(?r-)/I’@-))
11 November
from each factor
Input
1968
in eq. (3).
Results
Nucleus
b osc
EY
EU
EY-Eres
EV-Eres
(F)
(MeV)
(MeV)
(MeV)
(MeV)
1.38
141
107
113
79
0.80 -t 0.09
1.64
1.30 f 0.20
3.57 -t 0.57
12C
1.60
141
107
117
83
0.93 + 0.15
1.06
0.99 * 0.18
2.72 i 0.51
160
1.80
142
108
119
85
1.20 * 0.15
0.92
1.10 = 0.19
3.02 f 0.53
40Ca
2.30
144
110
125
91
1.70 * 0.34
0.55
0.94 f 0.22
2.59 f 0.61
4
He
1cCn l+C/J
Ii7 l’/.l
Tabulated are osc. parameters (b osc) for shell model and the “maximum” photon (I+) and neutrino (E,) energies. Average photon and neutrino energies used in the closure approximation are Ey - Eres and Ev - Eres where Eres = = peak energy in photon-absorption cross-section. We use the symbol IT/II* for the integral ratio in eq. (3), and the ratio of matrix elements corresponds to (&/I~) (l+Cr)/(l +Cp). Errors indicate the uncertainty in the theoretical estimate, treated symmetrically. An estimate of 10% error (based on 160) is assigned to I~/IIJ for all nuclei.
part of eq. (2) to an energy weighted integral over photo-absorption cross-section oy(E). To do so, we confine the discussion to the J = O+, T = 0 ground state and define the proportionality constant n by
F(*) l(_fl~ d-)0(i)exp(-ik. = 9[37
($j(fl~
ri)jO)12 =
rim)exp(-ik.
f-i)jO)12]
If the SU(4) supermultiplet symmetry holds, then 77is exactly unity [2]. In general, however, this may not be true [5], the deviation from unity being as much as 30%. This uncertainty of n is not serious if one takes the ratio of eq. (2) to the corresponding muon catpure rate which is also dominated by an axial vector term. The ratio takes the form 2 r(v_) _A 7, x -q’r-l r(n-)
I1
corrections defined by C = (OM + RT)/D where “OM” stands for contributions from multipoles other than dipole, ‘D” for dipole, and “RT” for recoil correction terms. For the pion capture, the recoil term appears in R and hence its contribution is neglected. Ey and Ev are the energies that the photon and neutrino would carry away if the transition were to occur between members of an isobaric multiplet [2]. The constant X is
x G; + 317G; +nG2p - 2nGpG& (
Here FA is the axial vector coupling constant, aTI is the pion-decay constant divided by rnr, Gv . . . are effective n-capture constants in units of M4. Numerically lo-12 X = 2.60, 2.75 and 2.85 respectively for-g = 0.7, 1, 1.5. The weak dependence on q allows us then to take X = = (2.75 f 50/O)x 1012. To compute the integrals in eq. (3), we have taken the photo-absorption cross-section data [6] from Dolbilkin for 12C and 160~ from Nikolaev for 40Ca and from Gorbunov and Spiridonov for 4He. Evaluating the elastic form factors F(q) = (l/A)@
where Fn and Fp are elastic form factors introduced to correct for the effect of momentum transfer neglected in using oy(E) [a], C, and Cp
.
F exp(iq-
ri IlO)
in shell model [2], we obtain the ratios of integrals Iv/In given in table 1. Here, the source of large uncertainty is that the experimentally observed ay(E) does not saturate the dipole sum rule, some strength being found in higher energy. 103
Volume 28B. number 2
PHYSICS
We believe, however, that the rapid decrease of the integrands at high energy and taking the ratio in eq. (3) compensate for the possible error t. As for the lower limit, it has been checked that adE) decreases fast enough as E - 0 that the error is small. For instance, in l60, the error is found to be less than 2% if one takes 20 MeV instead of 15 MeV for the lower limit. The calculation of the correction factors “C” depends on nuclear model. But since C involves ratio, one may take any model one wishes. Any
erYOY incurred in Cclwill be reflected same direction. Therefore the ratio
in C ,, in the
(1 +Cx)/(l +CcI) should be much less model-dependent than C is. With this in mind, we first make the closure approximation [‘I], and then calculate the corrections in the shell model with the oscillator parameters given in the table. Also wherever available, we use the sum over partial transitions as calculated for 160 and 40Ca by Luyten et al. [?‘I for muon capture. We obtain error estimates from the difference between the two models. For 4He and I2C, we have assumed the same error as in 160 and 40Ca. For I2C, the allowed transition O+ - l+ needs also be taken Into account. This is due to the fact [2] that the ground state is not a good SU(4) singlet in l2C. Of course, allowing n + 1 would in part take into account such deviation. In order to keep n close to unity, we prefer to include its contribution explicitly. The relevant matrix element can be extracted for desired momentum transfer from inelastic electron scattering 12C(O+) + 12C *(l+; 15.1 MeV) at 1800. Such a matrix element for k = 92 MeV which dominates the p-capture process has been evaluated to within 5% by Foldy and Wale&a [8]. We extend their analysis to k = 126 MeV to obtain the necessary matrix element for v- capture. These matrix elements divided by the dipole matrix elements evaluated above give the correction C(l+). The quantity 1,(1 + Cx)/Zp(l + CCL)in the table corresponds exactly to the ratio of nuclear matrix elements. Despite the kinematic differences which are reflected in (1 + C n )/( 1 + Cp ) and I,/$, the resultant is, except for 4He, surprisingly close to unity. It is tempting to ascribe this deviation in 4He to the failure in the p-capture $ The contribution
from higher energy region was tested for 160 with a following model: we took a model lvhere o,,(E) mm 7 mb for 27 C E C 50 MeV which adds enough strength to saturate the dipole sum rule. This contribution raised the integral ratio only by 6’5. Since much of the strength must lie higher up. we believe the error in terminating integral at 27 MeV can be safely neglected.
104
fLPI----nn I I p1no
II Novemoer
r~txi
theory as it was found by Foldy and Wale&a [2] that the procedure used here underestimates the rate by about 30%. More detailed analysis of p-capture rate is needed before one decides that the 4He discrepancy is real. Finally the capture rate ratio is given solely in terms of I;Z;s/7gj2 on the last column. The 7r~.12 is known and therefore the knowledge of ]&,I i would enable us to calculate I?(a-) in terms of r(,u-). Or conversely the measurement of I?(s-) would determine /&]2, thus enabling us to study the n--nucleus interaction supplementing informations from n-mesic atoms. No direct data for r(a-) are available for any of the nuclei considered. However the wave functions qr(‘) for 12C and I60 have been calculated by Seki and Cromer [4] by fitting the a-mesic data of Jen ins et al. [9]. Using them we evaluated l?n/?p 1$ = 1.29 and 1.17 respectively for I2C and 160. This gives the ratio r(n-)/r(p-) = 3.51 x 1012 for l2C and = 3.54 x x I012 for 160. In summary, we found that for J = O+, T = 0 nuclei,
r(n-)/r(p-1 x Const
x IT,/?,
I2 ,
(4)
independently of nuclear matrix elements. This simple relation holds only for the S-orbit pion. The question remains whether this relation can be checked experimentally. We are grateful to Dr. J. Le Tourneux and Dr. H. Stern for helpful discussions. We thank Miss C. Chevereau for her help in the numerical computation.
References 1. J.Delorne and T.E.O.Ericson, Phys. Letters 21 (1966) 98; D. K. Anderson and J. M. Eisenberg, Phys. Letters 22 (1966) 164. 2. L. L. Foldy and J. D. Walecka, Nuovo Cimento 34 (1964) 1026. 3. Low-energy theorem with PCAC has been used to show the formal analogy; M. Ericson and A. Figureau, Nucl. Phys. B3 (1967) 609;
H. Pietschmann et al.. Phys.Rev.Letters 19 (1967) 1259; D. Griffiths and C. W. Kim, Phys. Rev.. to be published. Phys.Rev.156 (1967) 93. 4. R.Seki and A.H.Cromer, 5. M.Rho. Canadian Summer Institute of Nuclear and particle physics, August 1967. to be published; Phys. Rev. Letters 19 (1967) 248. 6. D. S. Dolbilkin. Proc. P. N. Lebedev Physics Institute, Vol.36 (1967) 17; F.A.Nikolaev. ibid, Vol.36 (1967) 77; A. N. Gorbunov and V. M. Spiridonov. Soviet Phys. JETP 7 (1958) 600.
Volume 28B, number 2
PHYSICS
7. J. P.Luyten et al., Nucl.Phys.41 (1963) 236. 8. L. L. Foldy and J.D. Walecka, Phys.Rev. 140 (1965) B1339. 9. D.A.Jenkins et al., Phys.Rev.Letters 17 (1966). more recent measurements give different numbers
LETTERS
11 November
1968
for the width of 1s level. This would certainly change the wave functions and hence the ratio. Furhter experiments are needed before one can get “accurate” wave functions; see R. J. Harris et al . . Phys.Rev. Letters 20 (1968) 505.
105
SIMPLE
PHYSICS
RELATION AND
LETTERS
11 November
BETWEEN RADIATIVE PION MUON-CAPTURE IN NUCLEI
M. RHO and A. SHERWOOD Centre d’Etudes Nuclkaires de Saclay, Gif-sur-Yvette, Received
1 October
1968
ABSORPTION
France
1968
It is shown numerically that for light nuclei with J = 0 ‘. T = 0 ground state, the ratio of the total radiative 1s pion capture rate r(n-) to the total muon capture rate r(p-) is very nearly independent of nuclear structure, depending only on the ratio of squares of pionic and muonic atomic wave functions averaged over the nuclear dimension.
The radiative r- absorption in nuclei Ai’, ?I- + Ai - + y + Af, provides three means of studying nuclear structure [l]. Firstly, the process being theoretically well understood can provide ways of investigating pion-nucleus interaction through the S-orbit v- wave function. Secondly, the process goes predominantly through the axial vector current with momentum transfer equivalent to pion mass (mr) and therefore can provide informations on the axial vector matrix elements of the p-capture and in particular on the corrections to the usual impulse approximation. Thirdly and perhaps the most interesting is the possibility of studying the spin-isospin giant resonances which are members of giant resonance supermultiplets [2]. In this letter, we discuss the first piont. Using the formal similarity [3] of the pion-induced process to the muon process with the same initial state and summed over final states, /L- + Ai - V+ + Af, we demonstrate that the ratio of the capture rates for S-orbit muons and pions is very nearly independent of the nuclear structure. The ratio thus depends only on the pionic 1s wave function, since the muonic wave function is well known. This is shown for the J = O+, T = 0 nuclei 4He, I2C, 160 and %a with experimental photo-absorption cross-section and nuclear models for correction terms only. Consider the capture of S-orbit muons and pions t in the initial nucleus Ai to go to all final states in Af_ The case of muon capture is wellknown [2] and so we discuss below only the pion -? For A ) ‘7. the P-orbit absorption becomes significant, and it would be difficult to measure directly the capture from lS-orbit for higher mass nuclei. Thus 46Ca we include in the list may be academic. but it helps clarify the theoretical aspect we are connected with.
102
process. By the PCAC hypothesis with the softpion theorem [2], one obtains the transition operator to order m,/M and in impulse approximation T
=a
6 ‘Tam)&* exp(-ik Ui
i=l
ri) + R ,
(1)
where R contains terms proportional to the pion momentum q. Here E is the photon polarization vector, k the photon momentum, and r(-) changes proton into neutron. The operator becomes simple when it operates on an 1s pionic wave function as R gives terms propertional to ]Vcp1S(‘)12 and this can be neglected, as cPlS(r) varies very slowly within the nucleus [l]. The usual algebra (polarization sum etc.) leads to the simple formula for the total transition rate l?(n-) (ti = c = 1) Const. I;Tvl2 J $
x
C(g)
f
X
Vl~72!-)cr(i)exp(-ik,xi)li)
(2) 2,
where $v is the S-orbit pionic wave function averaged over nuclear dimension *, .k i the energy carried away by the photon equa f to (m, - B.E. of s-mesic atom - AEfi between nuclear states). We point out the difference between the matrix element in eq. (2) and the axial vector matrix element in n capture [2]: in eq. (2), the summand is linear in photon energy k, whereas in the n capture it is quadratic in neutrino energy v. The crucial point of our calculation is that we can treat eq. (2) in the same way as for the muon axial vector matrix elements. So using the idea of Foldy and Walecka [2], we relate the major * For 12C and 160, the variation is *40/L over the nuclear
dimension
[4].
Volume 28B.
number 2
PHYSICS
Contributions
to the capture
ratio
LETTERS Table 1 ( r(?r-)/I’@-))
11 November
from each factor
Input
1968
in eq. (3).
Results
Nucleus
b osc
EY
EU
EY-Eres
EV-Eres
(F)
(MeV)
(MeV)
(MeV)
(MeV)
1.38
141
107
113
79
0.80 -t 0.09
1.64
1.30 f 0.20
3.57 -t 0.57
12C
1.60
141
107
117
83
0.93 + 0.15
1.06
0.99 * 0.18
2.72 i 0.51
160
1.80
142
108
119
85
1.20 * 0.15
0.92
1.10 = 0.19
3.02 f 0.53
40Ca
2.30
144
110
125
91
1.70 * 0.34
0.55
0.94 f 0.22
2.59 f 0.61
4
He
1cCn l+C/J
Ii7 l’/.l
Tabulated are osc. parameters (b osc) for shell model and the “maximum” photon (I+) and neutrino (E,) energies. Average photon and neutrino energies used in the closure approximation are Ey - Eres and Ev - Eres where Eres = = peak energy in photon-absorption cross-section. We use the symbol IT/II* for the integral ratio in eq. (3), and the ratio of matrix elements corresponds to (&/I~) (l+Cr)/(l +Cp). Errors indicate the uncertainty in the theoretical estimate, treated symmetrically. An estimate of 10% error (based on 160) is assigned to I~/IIJ for all nuclei.
part of eq. (2) to an energy weighted integral over photo-absorption cross-section oy(E). To do so, we confine the discussion to the J = O+, T = 0 ground state and define the proportionality constant n by
F(*) l(_fl~ d-)0(i)exp(-ik. = 9[37
($j(fl~
ri)jO)12 =
rim)exp(-ik.
f-i)jO)12]
If the SU(4) supermultiplet symmetry holds, then 77is exactly unity [2]. In general, however, this may not be true [5], the deviation from unity being as much as 30%. This uncertainty of n is not serious if one takes the ratio of eq. (2) to the corresponding muon catpure rate which is also dominated by an axial vector term. The ratio takes the form 2 r(v_) _A 7, x -q’r-l r(n-)
I1
corrections defined by C = (OM + RT)/D where “OM” stands for contributions from multipoles other than dipole, ‘D” for dipole, and “RT” for recoil correction terms. For the pion capture, the recoil term appears in R and hence its contribution is neglected. Ey and Ev are the energies that the photon and neutrino would carry away if the transition were to occur between members of an isobaric multiplet [2]. The constant X is
x G; + 317G; +nG2p - 2nGpG& (
Here FA is the axial vector coupling constant, aTI is the pion-decay constant divided by rnr, Gv . . . are effective n-capture constants in units of M4. Numerically lo-12 X = 2.60, 2.75 and 2.85 respectively for-g = 0.7, 1, 1.5. The weak dependence on q allows us then to take X = = (2.75 f 50/O)x 1012. To compute the integrals in eq. (3), we have taken the photo-absorption cross-section data [6] from Dolbilkin for 12C and 160~ from Nikolaev for 40Ca and from Gorbunov and Spiridonov for 4He. Evaluating the elastic form factors F(q) = (l/A)@
where Fn and Fp are elastic form factors introduced to correct for the effect of momentum transfer neglected in using oy(E) [a], C, and Cp
.
F exp(iq-
ri IlO)
in shell model [2], we obtain the ratios of integrals Iv/In given in table 1. Here, the source of large uncertainty is that the experimentally observed ay(E) does not saturate the dipole sum rule, some strength being found in higher energy. 103
Volume 28B. number 2
PHYSICS
We believe, however, that the rapid decrease of the integrands at high energy and taking the ratio in eq. (3) compensate for the possible error t. As for the lower limit, it has been checked that adE) decreases fast enough as E - 0 that the error is small. For instance, in l60, the error is found to be less than 2% if one takes 20 MeV instead of 15 MeV for the lower limit. The calculation of the correction factors “C” depends on nuclear model. But since C involves ratio, one may take any model one wishes. Any
erYOY incurred in Cclwill be reflected same direction. Therefore the ratio
in C ,, in the
(1 +Cx)/(l +CcI) should be much less model-dependent than C is. With this in mind, we first make the closure approximation [‘I], and then calculate the corrections in the shell model with the oscillator parameters given in the table. Also wherever available, we use the sum over partial transitions as calculated for 160 and 40Ca by Luyten et al. [?‘I for muon capture. We obtain error estimates from the difference between the two models. For 4He and I2C, we have assumed the same error as in 160 and 40Ca. For I2C, the allowed transition O+ - l+ needs also be taken Into account. This is due to the fact [2] that the ground state is not a good SU(4) singlet in l2C. Of course, allowing n + 1 would in part take into account such deviation. In order to keep n close to unity, we prefer to include its contribution explicitly. The relevant matrix element can be extracted for desired momentum transfer from inelastic electron scattering 12C(O+) + 12C *(l+; 15.1 MeV) at 1800. Such a matrix element for k = 92 MeV which dominates the p-capture process has been evaluated to within 5% by Foldy and Wale&a [8]. We extend their analysis to k = 126 MeV to obtain the necessary matrix element for v- capture. These matrix elements divided by the dipole matrix elements evaluated above give the correction C(l+). The quantity 1,(1 + Cx)/Zp(l + CCL)in the table corresponds exactly to the ratio of nuclear matrix elements. Despite the kinematic differences which are reflected in (1 + C n )/( 1 + Cp ) and I,/$, the resultant is, except for 4He, surprisingly close to unity. It is tempting to ascribe this deviation in 4He to the failure in the p-capture $ The contribution
from higher energy region was tested for 160 with a following model: we took a model lvhere o,,(E) mm 7 mb for 27 C E C 50 MeV which adds enough strength to saturate the dipole sum rule. This contribution raised the integral ratio only by 6’5. Since much of the strength must lie higher up. we believe the error in terminating integral at 27 MeV can be safely neglected.
104
fLPI----nn I I p1no
II Novemoer
r~txi
theory as it was found by Foldy and Wale&a [2] that the procedure used here underestimates the rate by about 30%. More detailed analysis of p-capture rate is needed before one decides that the 4He discrepancy is real. Finally the capture rate ratio is given solely in terms of I;Z;s/7gj2 on the last column. The 7r~.12 is known and therefore the knowledge of ]&,I i would enable us to calculate I?(a-) in terms of r(,u-). Or conversely the measurement of I?(s-) would determine /&]2, thus enabling us to study the n--nucleus interaction supplementing informations from n-mesic atoms. No direct data for r(a-) are available for any of the nuclei considered. However the wave functions qr(‘) for 12C and I60 have been calculated by Seki and Cromer [4] by fitting the a-mesic data of Jen ins et al. [9]. Using them we evaluated l?n/?p 1$ = 1.29 and 1.17 respectively for I2C and 160. This gives the ratio r(n-)/r(p-) = 3.51 x 1012 for l2C and = 3.54 x x I012 for 160. In summary, we found that for J = O+, T = 0 nuclei,
r(n-)/r(p-1 x Const
x IT,/?,
I2 ,
(4)
independently of nuclear matrix elements. This simple relation holds only for the S-orbit pion. The question remains whether this relation can be checked experimentally. We are grateful to Dr. J. Le Tourneux and Dr. H. Stern for helpful discussions. We thank Miss C. Chevereau for her help in the numerical computation.
References 1. J.Delorne and T.E.O.Ericson, Phys. Letters 21 (1966) 98; D. K. Anderson and J. M. Eisenberg, Phys. Letters 22 (1966) 164. 2. L. L. Foldy and J. D. Walecka, Nuovo Cimento 34 (1964) 1026. 3. Low-energy theorem with PCAC has been used to show the formal analogy; M. Ericson and A. Figureau, Nucl. Phys. B3 (1967) 609;
H. Pietschmann et al.. Phys.Rev.Letters 19 (1967) 1259; D. Griffiths and C. W. Kim, Phys. Rev.. to be published. Phys.Rev.156 (1967) 93. 4. R.Seki and A.H.Cromer, 5. M.Rho. Canadian Summer Institute of Nuclear and particle physics, August 1967. to be published; Phys. Rev. Letters 19 (1967) 248. 6. D. S. Dolbilkin. Proc. P. N. Lebedev Physics Institute, Vol.36 (1967) 17; F.A.Nikolaev. ibid, Vol.36 (1967) 77; A. N. Gorbunov and V. M. Spiridonov. Soviet Phys. JETP 7 (1958) 600.
Volume 28B, number 2
PHYSICS
7. J. P.Luyten et al., Nucl.Phys.41 (1963) 236. 8. L. L. Foldy and J.D. Walecka, Phys.Rev. 140 (1965) B1339. 9. D.A.Jenkins et al., Phys.Rev.Letters 17 (1966). more recent measurements give different numbers
LETTERS
11 November
1968
for the width of 1s level. This would certainly change the wave functions and hence the ratio. Furhter experiments are needed before one can get “accurate” wave functions; see R. J. Harris et al . . Phys.Rev. Letters 20 (1968) 505.
105