Observation of electric quadrupole X-ray transitions in muonic thallium, lead and bismuth

(l.E.4:3.C]

Nuclear Physics Al96 (1972) 452-464; Not to be reproduced

OBSERVATION

by photoprint

OF ELECTRIC

IN MUONIC H.

SCHNEUWLY

E.

KANKELEIT

H.

QUADRUPOLE

THALLIUM,

+, L. SCHELLENBERG ++, K.

LEAD

1,

++, H. K. WALTER

R.

X-RAY

TRANSITIONS

AND BISMUTH

+, H. BACKE

LINDENBERGER

W. U. SCHRC)DER

@ North-Holland Publishing Co., Amsterdam

or microfilm without written permission from the publisher

++, R. ENGFER+++, M.

+++ and

PEARCE

t:,

A. ZEHNDER

U. JAHNKE

C.

PETITJEAN

:, +++,

ttt

CERN, Geneva, Switzerland Received

16 June

1972

Abstract: Electric quadrupole X-ray transitions (5g --f 3d, 4f j 2p, and 3d j 1s) have been observed in muonic Tl, Pb and Bi. From the 3 --f 1 transitions, energy splittings of the n = 3 levels were deduced. From a comparison of the relative intensities of El and E2 transitions, the population ratios 5g/Sf, 4f/4d, and 3d/3p were deduced. These ratios are well reproduced by a cascade calculation assuming a statistical initial population at n = 20, including K, L and M shell conversion. In the case of *05T1 discrepancies between the experimental and the calculated 3d-is/3p-1s intensity ratio can be explained by nuclear excitation. From the 3~3 + Is% intensity in *09Bi we can deduce the ratio of the radiationless to the X-ray transition width and give limits for prompt neutron emission from the 3d level.

E

ATOMIC PHYSICS, E2 transitions,

MESIC ATOMS Tl, Pb, Bi; measured p-mesic p-cascade, radiationless transitions, population

E,, I.; deduced ratios.

1. Introduction Up to now electric quadrupole transitions in muonic atoms have received little attention. This is partly due to the fact that E2 transitions are very weak. Early calculations by Wiik ‘) and Htifner ‘) predicted that only in the case of nd + Is transitions in medium and heavy muonic atoms would the E2 intensity exceed 1 y0 of the competing El transition intensities. In “‘Bi, for instance, the theoretical ratio of the transition probabilities is:

P(3d -+ 1s) -- = 0.046. P(3d + 2p) In electronic atoms, E2 transitions and other weak higher multipole transitions have already been observed “). But due to the lower energy of such transitions the E2/El intensity ratios are smaller by a factor of lo3 than in muonic atoms. At present, t Visitor ++ Visitor +++ Visitor t Visitor t: Visitor

from Institut de Physique, Universite de Fribourg, Switzerland. from Institut fiir Technische Kernphysik der TH Darmstadt, Germany. from Laboratorium fiir Hochenergiephysik der ETH, Zurich, Switzerland. from Hahn-Meitner Institut fiir Kernforschung, Berlin, Germany. from University of Victoria, BC, Canada. 452

X-RAY

TRANSITIONS

453

muonic E2 transitions can be observed in heavy atoms only. Here the finite size of the nucleus and the fine structure of the atomic levels provide an energy separation of the E2 transitions from the closest lying El transitions that is large enough to be resolved by Ge(Li) detectors. In the thallium isotopes for instance, the finite size of the nucleus “) leads to an energy separation of the 3d, -+ Is, E2 transition from the 3p+ -+ Is, El transition of about 76 keV at 8.3 MeV, whereas for a point nucleus these transition energies would be degenerate. Also, the fine structure splits the 3d levels and 3p levels amounts to more than 40 keV. If there is no accidental coincidence of the energy levels - in “‘Bi the 3d, and 3p, levels have nearly the same energy (fig. 2) - all four 3 + 1 transitions can be observed. Muonic E2 transitions in 208Pb were observed by Martin et al. ‘) and Jenkins et al. “). This paper reports on measurements and calculations of E2 transitions in muonic thallium, lead and bismuth. Some preliminary results on bismuth have already been reported ‘). The experimental method and the analysis of the data are described in sect. 2. Some details of the cascade calculations are given in sect. 3. The measured and calculated energies and intensities of the E2 transitions are discussed in sect. 4. 2. Experiment The measurements were performed at the CERN muon channel. The experimental arrangement is described in detail elsewhere “). A coaxial Ge(Li) detector was used, having an active volume of 25 cm3 and a resolution of 6.5 keV at 8.3 MeV. Energy spectra, prompt and delayed with respect to a muon stop, and calibration spectra were taken simultaneously with the aid of an on-line computer. In this way delayed y-rays resulting from muon capture by the nucleus can be separated from the prompt muonic X-rays. Metallic targets of natural isotopic abundance of thallium, lead and bismuth were used. The target thickness was typically 1.5 g/cm2. The interesting parts of the experimental muon spectra are shown in figs. I,3 and 4 for thallium and fig. 2 for bismuth. They were analysed with a fit program “), approximating the experimental line shape by a Gaussian function and up to two exponential tails on both sides of the peak, folded with the same Gaussian. The line position was defined as the centre of gravity of the fitted function cut at 20 % of its height. The intensity of a line was given by the integrated area in the range - 1.5 to + 1.5 half-widths from the middle of the line at half height. The energy calibration was obtained from the simultaneously measured spectra of different radioactive sources ( 5‘Co, 16N and 228Th). As the only high-energy calibration line available was the 6129.96kO.46 keV line from I60 in neutron-irradiated water, an extrapolation of the calibration up to 8 MeV was necessary for the identification of the 3d -+ 1s transition lines. The absolute energies can be given to an accuracy of only about 2 keV. Therefore, we prefer to quote energy differences, which can be obtained with a higher accuracy, using the photopeak, single- and doubleescape peak energy separations.

et al.

H. SCHNEUWLY

454 N

3-l

I

7.30

.

(DE )

I

I

I

I

7.25

Transitions

7.40

7.35

E[MeV]

Fig. 1. Double-escape peaks of the muonic 3p + Is El and 3d --f 1s E2 transitions in natural thallium. The arrows with superscripts refer to the 20sT1 isotope, the other arrows to the corresponding transitions in z03T1, which lie about 10 keV higher.

il

160

_

140

_

100

_

p-

209

61

3 - 1

TRANSITIONS

(D.E.)

z’ z

L P z ;

60-

I

I

I

I

I

7.50

7.55

7.60

7.65

I

Fig. 2. Double-escape

peaks of the 3 + 1 transitions in muonic bismuth.

The intensities of transitions up to 3 MeV were obtained from a calibration of the photopeak efficiency of the Ge(Li) detector with a set of calibrated radioactive sources taking into account the extension of the target and the self-absorption in the target. No intensity-calibration lines were available for transitions around 8 MeV. Therefore, the absolute intensities were determined in the following way. The intensity ratio

X-RAY TRANSITIONS

209 2d3h-2ph

P-

Bi

x2

4dhf2ph

\

4% -2ph

4f?‘r2~‘1

I

50

100

150

700

800

coo

CHANNEL

Fig. 3. Photopeaks

and double-escape

counts channel

peaks of the 4d + 2p El and 4f + 2p E2 transitions muonic bismuth. /L -TI

3dsh-

OL

in

2p+

DE

1

I

1.3

1.4

E [f&V]

Fig. 4. Photopeaks of the 5f -+ 3d El and 5g + 3d E2 transitions in muonic thallium. transition lines give rise to the high-energy tail of the El transition lines.

The E2

456

H. SCHNEUWLY

et al.

Z(3d, --r ls+)/Z(3d, --+2p,) is not affected by nuclear excitation, direct neutron emission or any details of the cascade. Therefore, taking the theoretical value of this ratio and the measured 3d, --) 2p, intensity per muon stop [from ref. “) for thallium and from Backe a) and Zehnder ‘) for bismuth], we can calculate the absolute 3d, -+ Is+ intensity as a calibration point close in energy to the 3p --) 1s transitions. The relative intensities of the closely lying 3p -+ 1s El and 3d -+ Is E2 transitions have been obtained from the intensities of the double-escape peaks, which are strong in this energy region. Using the double-escape peak efficiencies measured by Cline lo) for similar detector sizes, a correction of less than 1 % was attached to the measured intensity ratio. The E2 transitions were then identified by their energies and approximate intensities relative to the neighbouring El transitions. Sum coincidences have been estimated from known solid angle and efficiency of the detector to contribute less than 0.1 % to the measured intensities; this is confirmed by the fact that sum coincidence events such as (2p --) Is) + (4f + 3d) were not found in the energy spectrum.

3. Cascade calculations In order to calculate the populations of the states n = 5 fed by transitions from higher states, a non-relativistic cascade program based on a code developed by Hiifner [ref. ii)] was used. The muons were assumed to be captured into the levels with main quantum number n = 14 or n = 20, with the captures being statistically distributed over all fine structure states, i.e. proportional to (2Z+ 1). Only dipole transitions with Aj = 0, + 1 and An 1 1 were considered for the higher muonic levels. In the cascade starting at n = 14 the only non-radiative processes were taken to be Auger K and L electron emission. The electron emission probabilities were calculated according to Eisenberg and Kessler I’). For the calculation starting at the n = 20 levels, we have included the conversion of the M shell for low-energy transitions. As can be seen from the paper of Backe etaZ.4), the intensities of the observed transitions calculated in this way for thallium agree much better with the measured ones for all observed transitions than the traditional cascade calculation starting at n = 14 without M shell conversion. For the calculation of the M conversion, the L conversion coefficient was multiplied by a factor of 0.2 as a rough average for the aM/crLratio i3). The population of the states with n < 5 was obtained from the intensities of the transitions coming from n > 5 levels, calculated with the non-relativistic cascade program referred to above, and the intensities of the transitions coming from n 5 5 levels were calculated relativistically as described below. For n $ 5 only radiative transitions need be considered, the competing Auger electron emission being less than 0.1 ‘A. The probability of an Lth order muonic electric transition between atomic states ]nZj) and In’l’j’) is given by the formula i4)

457

X-RAY TRANSITIONS

P(L) = ac

2(L+ 1) L[(2L+1)!!]2

E

2L+1

( Kc)

in the long-wavelength approximation. Here a is the fine structure constant and E is the transition energy. The radial matrix element reads

where f and g are the radial components of the muonic Dirac-wave function. The functionsfand g were calculated with a computer code originally developed by Acker [ref. ‘“)I, assuming a two-parameter Fermi-type charge distribution of the nucleus. The long-wavelength approximation holds only if kR < 1, where k is the wave number of the X-ray and R the linear dimension of the system i6). For the transition probabilities, under consideration, kR is always less than 0.2. The errors in the calculated probabilities from this approximation were therefore estimated to be less than 4 %.

4. Discussion 4.1. ENERGY

SPLITTINGS OF THE 3d -t Is LEVELS

Martin et al. ‘) reported on a discrepancy between the measured 3p, + 3d, energy splitting and the calculated value in “*Pb . In a later paper “) this discrepancy was interpreted as a statistical fluctuation. For this reason we present in table 1 the measured and calculated energy differences between the it = 3 fine structure levels obtained from the transitions 3p, + Is+, 3p+ -+ Is,, 3d, -+ Is, and 3d, -+ Is+, for thallium and bismuth. Since the fine structure components of the three lead isotopes were not resolved experimentally, the energy splittings could not be determined accurately in the case of lead. For comparison, we therefore list in table 1 the values obtained by Jenkins et al. “) in the case of 208Pb. For each of the two thallium isotopes the theoretical energies have been calculated using the charge distribution parameters c and t and corrections given in ref. “). Because of the large isotope shift of about 10 keV (see table 2) between 203T1 and 205T1, the positions of all the 3 + 1 transitions of both isotopes could be determined, except that of the 3p+ + Is+ transition in ‘03T1 because its energy agrees within 1 keV with that of the five times stronger 3d, --, Is+ transition in “‘Tl. In the fitting procedure the energy splitting of the 3p+ + ls+ and 3p+ --f Is, transitions in 203T1 and their intensity ratio were therefore held fixed and set equal to the theoretical values. As can be seen from table 1, there is a good agreement between the experimental and the theoretical energy differences.

458

H. SCHNEUWLY

et al.

TABLE 1

Comparison

of theoretical

Levels

and experimental energy differences (keV) of the 3d and 3p levels in muonic thallium, lead and bismuth

Z03T1 exp.

2O5Tl

theor.

exp.

209Bi

z”sPb “)

theor.

exp.

theor.

exp.

theor.

theor.

3d3-3d+

41.14&0.50

40.83 40.89hO.30

40.81 42.74kO.32

42.79 44.07AO.54

44.80 44.85

3pq-3dq

34.3610.60

34.61 34.61kO.30

35.03 37.05kO.53

38.22 40.3550.71

41.33 40.22

3p+-3d3

75.51 f0.40

75.44 75.51 f0.30

75.84

84.43 f0.73

86.13 85.07

3p+--3d+

29.04 30.10&0.50

29.51

3P+-3P)

46.40 45.4OkO.50

46.27 47.56kO.82

37.36 35.93 47.49

48.77 49.14

c (fm)

6.620 “)

6.629 b,

6.668

6.682 ‘) 6.660

I (fm)

2.261 “)

2.282 “)

2.230

2.248 ‘) 2.248

The charge distribution parameters are given in the last two lines. “) All values from ref. 6). “) From ref. ‘), ‘) From ref. Is). TABLE 2

Isotope shift between 203Tl and 205T1 Transition

A&& (keV)

%I,(~~~T~) (kev)

~%r,(~~~Tl) (keV)

3d+ + lsi

10.64&0.40

10.49

8354.69

8344.20

3di. + Is+

10.39+0.50

10.47

8313.86

8303.39

3P+ + 1st

10.39+0.60

10.07

8389.30

8379.23

3P+ + ls+

“)

9.94

8342.90

8332.96

The energies are calculated using the c, t parameters of ref. 4, (compare table 2). “) Experimentally this transition in *03T1 could not be separated from the stronger 3d+ + transition in 205Tl.

IS+ E2

In the case of muonic ‘09Bi it is known ‘vi7 ) that in addition to the Ml and E2 interaction with the nucleus, there are two different E3 resonance processes involving the 3d, level. One involves the 3d, -+ 2p, transition, the other one the 3d, + 2pS transition, which are in resonance with the J$’ and 3’ nuclear states, respectively. The effect was estimated by two fitting procedures, taking into account or neglecting the quadrupole splitting of the 3d hyperfine structure components and the nuclear mixing. It was largest for the 3p+ --* 3d, energy splitting, where it amounts to 0.6 keV, i.e. just within the error. Therefore this resonance has been neglected in the theoretical values of table 1. As can be seen from table 1 the agreement between the theoretical and experimental values is good. The small discrepancy of energy differences involving the 3p+ level can be reduced by changing the parameters of the charge distribution. If we take instead of c = 6.682 fm given by Bardin et al. Ia) its lower limit, i.e. c = 6.660 fm, we obtain a much better agreement between the theoretical and experimental level energy splittings.

X-RAY TRANSITIONS 4.2. INTENSITIES

459

OF THE 3 -+ 1 TRANSITIONS

4.2.1. Thallium. The experimental transition intensity ratios in muonic thallium are compared to predictions in table 3. The 3d, -+ Is, intensity for 205Tl has been TABLE 3

Comparison

of experimental

and theoretical intensity ratios in muonic thallium

*OsTl

=Tl

203.205Tl

exp.

exp.

theor. n = 14

1.21 hO.20

1.20&0.15

0.80

1.22

1.22

3dt + Is+ (3p++3d+) -+ Is+

0.49hO.10

0.65 kO.07

0.45

0.50

0.58

(3p++3d+) + Is+ 3P3 + 1st

2.48 kO.52

1.86 kO.22

1.77

2.45

2.09

0.62 10.07

0.89f0.10

0.61

0.61

0.75

0.50f0.06

0.46

0.46

0.46

1.74f0.19

1.44

2.20

1.95

Intensity ratio

3d + 1s 3p -+ Is

2.2OhO.32

“) With nuclear mixing: (8.8*0.9)

% excitation

203.205Tl 2OSTI ____ ___ theor. theor. mix.“) n = 20 n = 20

of the Z = %- nuclear level [ref. 4)].

corrected for the disturbing 3p+ -+ Is+ transition in *03T1, using the theoretical 3p fine structure intensity ratio. The theoretical intensity ratios obtained from the two different cascade calculations referred to in sect. 3 differ significantly from each other, see for instance, the (3d + ls)/(3p + 1s) ratios. In ‘03T1 a hypothetical nuclear mixing with IZ 1 3 muonic states would not change the results significantly. Prompt neutron emission is less probable than in “‘Tl because of the higher neutron binding energy. Table 3 shows that in ‘03T1 the agreement is excellent between the measured intensity ratios and the theoretical intensity ratios calculated assuming a statistical initial distribution at the n = 20 level with K, L and M shell conversion. For 205T1 the resonance excitation of (8.8 kO.9) ‘A of the nuclear 3- level “) was taken into account in the calculations. This excitation reduces the 3d, --) Is+ transition intensity by a factor of about 0.83. The experimental and theoretical intensity ratios are in good agreement if we take this nuclear mixing into account. It may be that the agreement would be even better if there were radiationless decay from the n = 3 levels as is the case of, for instance, “‘Bi which is explained below. But so far no quantitative results exist about prompt neutron emission in thallium.

460

H. SCHNEUWLY

et al.

4.2.2. Bismuth. In the case of bismuth (table 4) we have both effects on the muonic cascade: prompt-neutron emission and nuclear mixing. From experiments of Hargrove et al. ’ 9*‘O), it is known that (7f2) ‘A prompt neutrons are emitted per muon stop in *09Bi. Their measured prompt neutron spectrum shows a series limit and a

Comparison

of experimental

TABLE4 and theoretical intensity ratios in muonic lead and bismuth *OSgi

Pb

theor. b, n = 20

1.272

1.908

exp. zosPb “)

theor. n = 20

1.13*0.17

1.70+0.39

1.245

1.61 ho.23

3d+ + 1s) _______.- + ls+ (3pp+3d+)

0.55 +0.08

0.48 +0.07

0.500

0.62&0.07

0.501

0.609

(3Pt+3d+) + 1st 3P) --f ls*

2.07+0.31

3.54kO.75

2.492

2.60&0.50

2.538

3.133

0.6410.08

0.612

0.75kO.20

0.612

0.720

“) Values taken from ref. 6). “) With nuclear mixing [ref. I’)] and radiationless

Relative intensities

exp.

theor. ?I = 20

exp.

decay from 3p levels (r,.,./r’x

TABLE5 of 3p and 3d transitions in muonic thallium and bismuth

exp.

exp.

theor.

exp.

theor. 0.2861

3d+ + 2pt_

0.252 kO.051

0.2864

0.245 kO.020

3dt -+ Is+

0.0125f0.0025

0.0142

0.0125~0.0010

3p* + IS) 3P+ -+ 1s) 3dt -+ Is+

= 0.5).

0.0147

0.0052&0.0012

0.0053

0.0053

0.0102&0.0024

0.0105*0.0022

0.0117

0.0078~0.0012

0.0116

0.0206kO.0047

0.0146+0.0031 “)

0.0232 “)

0.0166&0.0025

0.0204 ‘)

The intensities are given per muon stop and normalized to the 5 --f 4 transitions. [For thallium see ref. 4), and for bismuth refs. **9).] “) Corrected for the unresolved 3p+ + 1s) transitions in 203T1. b, With nuclear mixing for ‘05T1: 0.0195. ‘) With nuclear mixing [see ref. I’)]. Without nuclear mixing a value of 0.0241 was calculated.

large peak corresponding to radiationless 3 -+ 1 transitions in the muonic atom. The 3 3 1 transitions have energies larger than the neutron binding energy. Whether the prompt neutrons arise from the np + 1s (with it 2 3) or from the 3d + 1s transitions can be deduced through the following arguments. Assuming an initial population distribution of the form (21+ 1)e”’ in the II = 14 level, the calculations of Srinivasan

X-RAY

TRANSITIONS

461

and Sundaresan “) predict intensities of np --, 1s radiationless transitions in “‘Bi leading to prompt neutron emission that agree with the experimental value of Hargrove et al. ’ ‘, ’ “) . But this cascade calculation needs a population of all np levels with n 2- 3 of 13 y0 to 18 %, assuming a ratio of the radiationless to X-ray transition widths for all np + 1s transitions of Z,.,,/Zx = 0.6. However, a cascade calculation starting with statistically distributed population at n = 20 level, which is in better agreement with all experimental data (subsects. 4.3 and 4.4), yields a population of the 3p level of 2.7 % and the sum over all np levels with n 2 3 gives only a total population of 4.7 % per stopped muon. This is too small to explain 7 % prompt neutrons. The ratio of the measured and calculated 3p., + ls+ intensities (table 5) is about 3, which would lead to a ratio of radiationless transition width to X-ray transition width of Z,.,./Zx (np -+ 1s) = 0.5kO.2 in agreement with the calculations of Srinivasan and Sundaresan yielding Z,.i./Zx(np + 1s) = 0.6 to 1.0. The 3p, -+ 2s+ and 3p+ + 2~ intensities cannot be used to check the Z,.,./Zx ratio because the corresponding photopeaks are masked by the double-escape peaks of the 3d, + 2p, X-ray and the nuclear y-transition (4’ + s-), respectively. Assuming that our Zr.i./Zx ratio is the same for all np + 1s radiationless transitions, a rate of 1 Y0prompt neutrons per muon stop is expected to originate from 3p + 1s radiationless transitions and about 0.7 o/ofrom higher np -+ 1s transitions. The larger remaining fraction of prompt neutrons must be ejected in other transitions. The 3d level is about 30 times more populated by the cascade than the 3p level. Even if a much smaller fraction of the muons which pass via the 3d level would contribute to the radiationless transitions, since there is no collective E2 state known which acts as a doorway for the compound decay 22), it can be assumed that the remaining 3 ‘A to 5 ‘A prompt neutrons originate from the 3d level. This assumption is supported by the observed 3d, + Is+ intensity given in table 5. It is known that two different resonance processes involving the 3d, level *, 1‘) reduce the 3d, + Is, transition intensity by a factor of 0.85. Taking into account the nuclear mixing [(7.3 L-0.9) % per muon stop] according to Backe et al. “) for the 3d, + Is+ intensity, we can determine the ratio of the radiationless and X-ray transition widths for the 3d+ level. If we assume that the Z,.l./Zx(3p + 1s) is exactly 0.5, we obtain from the comparison of the measured and calculated Z(3d, -+ ls&Z(3p+ -+ Is+) intensity ratio a value for Z,.l./Zx(3d + 1s). Since the errors on the measured intensity and intensity ratio are relatively large, it is only possible to give limits which are nearly the same from the 3d, and 3d, levels. The ratio of the radiationless transition width to the X-ray transition width for the 3d level is therefore given in the limits: 0.06 c Z,.,./Zx(3d + 1s) < 0.4. 4.2.3. Lead. The fine structure components of the three lead isotopes were not resolved experimentally. An analysis done in the same way as we have for bismuth is therefore not possible. Reference should be made to the papers of Anderson et al. “) and Jenkins et al. 6), who performed experiments on separate isotopes. Experimental evidence for prompt neutron emission has been found in 207Pb by Kessler et al. 20). It seems evident that prompt neutrons are also emitted from 208Pb, which would per-

H. SCHNEUWLY

462

et al.

haps explain the discrepancy between the calculated intensity ratios and the given data for “‘Pb. But so far no quantitative results exist about prompt neutron emission in “‘Pb. 4.3. INTENSITIES

OF HIGHER

E2 TRANSITIONS

The 4d + 2p El transitions and the 4f + 2p E2 transitions have nearly the same energy. Therefore, for the comparison of their intensity ratios, the errors arising from the efficiency calibration of the detector and from the self-absorption in the target can be neglected. However, the intensities are weak compared to the background in this energy region (see the bismuth spectrum in fig. 3) and a small change in the background fitting parameters strongly affects the fitted intensities. The same argument holds for the comparison of the 5f --) 3d with the 5g --) 3d transitions (see thallium spectrum in fig. 4). In this case, the E2 transitions are only 3-4 keV (at 1.4 MeV) higher in energy than the El transitions[ref. “)I, and their presence leads only to an asymmetry in the 5 -+ 3 peak. In the fitting procedure the energy difference of theE1 and E2 transitions was held fixed and set equal to the theoretical one. Table 6 gives a comparison between several El/E2 and E2/E2 intensity ratios and Comparison

of experimental

TABLE 6 and theoretical intensity ratios for muonic thallium,

Tl theor. n=14

Tl theor. n=20

Tl exp.

Pb theor. n = 20

Pb exp.

0.586

0.586

0.48&-0.13

0.586

0.401

0.544

0.4550.13

0.400

0.533

0.418

lead and bismuth

209Bi

*OgBi

theor. n = 20

exp.

0.43hO.15

0.586

0.8710.40

0.572

0.41 *to.14

0.596

0.47rto.21

0.64kO.12

0.556

0.5210.08

0.579

0.81 zto.22

0.559

0.65kO.15

0.584

0.491to.14

0.609

0.79iO.23

0.719

0.719

0.50&0.18

0.719

0.719

0.39*0.14

0.142

0.179

0.19~0.05

0.186

0.193

0.1510.05

5gp --f 3ds 54 + 3dt

0.134

0.175

0.21 hO.04

0.182

0.189

0.35io.10

5g + 3d 5f +3d

0.140

0.178

0.19+0.05

0.187

0.196

0.26f0.09

4f+ -+2P* 4dt -j 213-5 4f$ -+ 4de + 4f -+ ___ 4d +

2P+ “) 2~+ 2p “) 2p

5gt 5gp 5g; 54

3d+ 3dq 3d+ 3dt

--f -+ + +

“) The intensity transition using the “) For the total 4fs + 2p+ and 4d+

of the 4d+ + 2~3 transition has been corrected for the underlying 44 + 2p+ calculated (4f+ --f 2pt)/(4fs -+ 2p*) branching ratio. 4f + 2p and 4d + 2p transition intensities the intensities of the transitions + 2p+ respectively, were taken into account using calculated branching ratios.

X-RAY TRANSITIONS

463

the theoretical calculations in thallium, lead and bismuth. The errors quoted for the measured intensity ratios are the statistical errors only. In particular, in the 5 + 3 case they do not include the error from fixing the energy difference between E2 and El. For thallium, the agreement with the cascade calculation starting with a statistical population distribution at the n = 20 level (see sect. 3) is significantly better than with the usual one starting at II = 14. For lead, the experimental intensity ratios agree very well with those of Jenkins et al. “) for “‘Pb. In the case of bismuth, the theoretical 4 + 2 transitions agree with the experimental ones, but for the 5 + 3 transitions, where the statistics are very low, the agreement is not so satisfactory. 4.4. POPULATION

RATIOS

The population ratios 3d/3p, 4f/4d and 5g/5f can be deduced from the measured relative intensities of El and E2 transitions with the same main quantum numbers n. For this purpose the branching ratios for the competing transitions have to be known, Comparison

3d/3p

TABLE7 of the population ratios 3d/3p, 4f/4d and 5g/5f

*‘-Tl exp.

z05T1 exp.

32.114.8

25.4zk2.8

Tl

ZOSgi

Pb

theor. n = 14

theor. n = 20

20.02

32.08 “)

4f/4d

15214.1

9.79

13.12

5gl5f

6.9*1.7

5.09

6.56

exp.

theor. n = 20

exp.

31.96 11.1*3.0

theor. n = 20 31.85

13.32

15.8k4.6

13.51

6.67

8.9+3.0

6.77

“) With nuclear mixing for 20dT1 [ref. ‘)I: 28.62.

which can be reliably calculated. Table 7 shows the comparison of the deduced population ratios and the calculated ones. For the 3d/3p ratio in “‘Tl the nuclear mixing in the 3d+ level “) improves the agreement with the experimental value. The agreement with the cascade starting at n = 20 is seen to be much better than with n = 14. Although the n = 20 calculation represents a useful improvement in the procedure for identifying weak lines, it is not possible to draw any conclusion about which atomic states are initially formed. The initial population is distributed over both Hand Z,and variations in the initial distribution affect the population of low-lying states; i.e. a large number of level populations would need to be accurately known in order for the initial distribution to be determined. The increase of the ratios in table 7, which results if a statistical population of n = 20 instead of n = 14 is assumed, can be understood as a consequence of the gradual enrichment of high angular momentum states in the course of cascade transitions. For the 3d/3p population ratio this effect is particularly pronounced, because the 3p level is fed from pure ns states (3 6 s 2 20), whose populations decrease more than those for the higher I-values, when one starts the cascade at n = 20 instead of n = 14.

464

H. SCHNEUWLY

et al.

We thank J. Hiifner for many clarifying discussions. This work was supported by the Bundesministerium fiir Bildung und Wissenschaft and the Schweizerischer Nationalfonds. References 1) B. Wiik, Z. Phys. 178 (1964) 231 2) J. Hilfner, Z. Phys. 190 (1966) 81 3) R. H. Garstang, Advances in atomic and molecular processes, ed. D. R. Bates (Academic Press, New York, 1962) ch. I; D. Layzer and R. H. Garstang, Ann. Rev. Astron. Astrophys. 6 (1968) 449 4) H. Backe, R. Engfer, U. Jahnke, E. Kankeleit, R. M. Pearce, C. Petitjean, L. Schellenberg, H. Schneuwly, W. U. Schriider, H. K. Walter and A. Zehnder, Nucl. Phys. Al89 (1972) 472 5) P. Martin, G. H. Miller, R. E. Welsh, D. A. Jenkins and R. J. Powers, Phys. Rev. Lett. 25 (1970) 1406 6) D.A. Jenkins, R. J. Powers, P. Martin, G. H. Miller and R. E. Welsh, Nucl. Phys. A175 (1971) 73 7) H. Backe, R. Engfer, U. Jahnke, E. Kankeleit, K. H. Lindenberger, C. Petitjean, H. Schneuwly, W. U. Schriider and H. K. Walter, Hyperfine interactions in excited nuclei, ed. G. Goldring and R. Kalish (Gordon and Breach, London, 1971) p. 729 8) H. Backe, Z. Phys. 241 (1971) 435 9) A. Zehnder, Diplomarbeit, ETH Zurich, 1970 10) J. E. Cline, IEEE Trans. Nucl. Sci. 15 (1968) 198 11) J. Hiifner, Z. Phys. 195 (1966) 365 12) Y. Eisenberg and D. Kessler, Nuovo Cim. 19 (1961) 1195; Y. Eisenberg and D. Kessler, Phys. Rev. 130 (1963) 2349 13) R. S. Hager and E. C. Seltzer, Nuclear Data Tables A4 (1968) l-235 14) A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) 15) H. L. Acker, G. Backenstoss, C. Daum, J. C. Sens and S. A. de Wit, Nucl. Phys. 81 (1966) 1 16) M. E. Rose, Multipole fields (Wiley, New York, 1955) 17) H. Backe, R. Engfer, E. Kankeleit, W. U. Schroder, H. K. Walter and K. Wien, High-energy physics and nuclear structure, ed. S. Devons (Plenum Press, New York, 1970) p. 166; H. Backe, R. Engfer, E. Kankeleit, R. M. Pearce, C. Petitjean, L. Schellenberg, H. Schneuwly, W. U. Schriider, H. K. Walter and A. Zehnder, Nuclear polarization and nuclear excitation in muonic Tl and Bi, presented at Fourth Int. Conf. on high-energy physics and nuclear structure, Dubna, USSR, 7-l 1 Sept. 1971; W. Y. Lee, M. Y. Chen, S. C. Cheng, E. R. Macagno, A. M. Rushton and C. S. Wu, Nucl. Phys. A181 (1972) 14 18) T. T. Bardin, R. C. Cohen, S. Devons, D. Hitlin, E. Macagno, J. Rainwater, K. Runge and C. S. Wu, Phys. Rev. 160 (1967) 1043 19) C. K. Hargrove, E. P. Hincks, G. R. Mason, R. J. McKee, D. Kessler and S. Ricci, Phys. Rev. Lett. 23 (1969) 215 20) D. Kessler, V. Chan, C. K. Hargrove, E. P. Hincks, G. R. Mason, R. J. McKee and S. Ricci, High-energy physics and nuclear structure, ed. S. Devons (Plenum Press, New York, 1970) p. 144 21) V. Srinivasan and M. K. Sundaresan, Can. J. Phys. 49 (1971) 621 22) J. Hiifner and F. Scheck, Muonic atoms, in Muon physics, ed. V. W. Hughes and C. S. Wu, to be published 23) H. L. Anderson, C. K. Hargrove, E. P. Hincks, J. D. McAndrew, R. J. McKee, R. D. Barton and D. Kessler, Phys. Rev. 187 (1969) 1565