ISSN 1062�8738, Bulletin of the Russian Academy of Sciences. Physics, 2012, Vol. 76, No. 8, pp. 849–853. © Allerton Press, Inc., 2012.Original Russian Text © V.O. Sergeev, 2012, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2012, Vol. 76, No. 8, pp. 949–953.

849

Gamma transitions of M4 multipolarity are charac�terized by the ban on orbital angular momentum l notactually affecting them, in contrast to transitions oflower multipolarity. In addition, collective effects arequite unlikely to occur here. We may assume that thesetransitions are not affected by these factors, and it istherefore interesting to compare the reduced probabili�ties of such transitions with the values calculated usingthe single�particle model, of course considering thelimited character of this model application.

Our calculations were performed using Moshk�ovsky’s formulas in [1], which take into account forthe magnetic properties of nucleons. The formulas fornuclei with odd numbers of protons and neutrons aredifferent due to differences in their magnetic moments(+2.79 nucl. magn. for a free proton and –1.91 nucl.magn. for a free neutron) and factors gl (gl = 1.0 for aproton and gl = 0 for a neutron). Our calculations con�sidered statistical factors whose values were generallyS = 1.5 – 1.8 for M4 transitions. For the р1/2 → g9/2transition, the value is S = 5; and for the reverse tran�sition, it is S = 1. M4 transitions are usually stronglyconverted, and the most advanced BRICC code wasused to calculate the theoretical values for the coeffi�cients of internal conversion [2].

Databases [2] and [3] were used to determineexperimental values of the reduced probabilities of M4transitions, and in each case these data were checked.

Hindrance factors of M4 transitions Fhind weredetermined as the ratio of experimental and theoreti�cal values for the reduced probability of hindrance:

Fhind = Т1/2(М4)exp/Т1/2 (М4)th.The Table presents data on 95 known M4 transi�

tions. The data on M4 transitions are assigned to sec�tions (a) through (e) of the Table, in accordance withthe quantum single�particle configurations ascribed tothe levels between which the investigated transitionsoccur (i.e., the main components). These configura�

tions are indicated in the section subheadings. Section(f) presents data on M4 transitions in odd�odd nuclei;section (g) presents data on 131Sb, 135Cs, and 179Hfnuclides. The second column gives the energies ofM4 transitions. Quantum characteristics of the statesbetween which transitions occur are given in thetable’s third column. Fractions (% of the number ofdecays) of the M4 transitions for corresponding iso�meric states are given in the table’s next to the last col�umn. The last column gives the values for theM4�transition hindrance factors, calculated in themanner described above.

With regard to section (a), which presents data ontransitions of the 2р1/2 ↔ 1g9/2 –type in nuclideswith an odd number of protons, note that the experi�mental estimate of the intensity of the 64.3 keVM4 transition in a 99Rh nucleus (<0.16%) given in [4]correspond to the hindrance factor F > 40. This valuefalls out from the graph shown in Fig. 1 for the depen�dence of the hindrance factors of this type of M4 tran�sition on Z. In this case, the rather smooth depen�dence F(Z) allows us to estimate F ≈ 4 by means ofinterpolation (the asterisk in Fig. 1). It corresponds toa possible intensity of ≈2% for an isomeric M4 transi�tion in a 99Rh nucleus.

Figures 1–4 show the dependences of Moshkovskyhindrance factor Fhind on the number of protons orneutrons for M4 transitions of various types. Charac�teristics of the levels between which transitions occurare noted in the figure captions, signifying the compo�nents of the single�particle wave functions that makethe main contribution to the probability of a consid�ered M4 transition. The figures show that values of thehindrance factors calculated using Moshkovsky’s for�mulas tend to diminish as the number of protons orneutrons approaches the magic numbers N = 50(Fig. 1), N = 82 (Fig. 2), N = 126 (Fig. 3), and Z = 50(Fig. 4). In odd isotopes of Pb that have the magic

Systematics of M4 TransitionsV. O. Sergeev

St. Petersburg State University, St. Petersburg, 198504 Russiae�mail: vserg2009@bk.ru

Abstract—Reduced probabilities of known γ transitions of M4 multipolarity are analyzed. Hindrance factorsof 95 M4 transitions are calculated and presented. It is shown that the investigated M4 transitions are inhib�ited in regard to estimates using Moszkovsky formulas, and their mean hindrance factor is 5.5. The factorsare close to unity in two cases only, but the intensity of a weak M4 transition could be overestimated in anexperiment due to the summation effect. Dependences of the hindrance factors on the number of protonsand neutrons are shown for M4 transitions between states with dominant single�particle componentsπ1g9/2 → π2p1/2, ν1h11/2 → ν2d3/2, and ν1i13/2 → ν2f5/2.

DOI: 10.3103/S1062873812080242

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SERGEEV

Values for the hindrance factors of M4 transitions

(a) 2p1/2 ↔ 1g9/2–type transitions in nuclides with an odd number of protons

Nuclide Energy of M4 transition, keV → Fraction

of isomeric M4 transition, % Hindrance factor

87Y48 380.82 9/2+ → 1/2– 98.4(3) 5.44(15)89Y50 908.96 9/2+ → 1/2– 100 3.77(12)91Y52 555.58 9/2+ → 1/2– 100 9.05(30)91Nb50 104.60 1/2–

→ 9/2+ 93 (4) 4.07(18)93Nb52 30.77 1/2–

→ 9/2+ 100 6.21(27)95Nb54 235.69 1/2–

→ 9/2+ 94.4(24) 8.74(26)97Nb56 743.35 1/2–

→ 9/2+ 100 12.1(5)93Tc50 391.83 1/2–

→ 9/2+ 76.6 (11) 2.95 (6)95Tc52 38.89 1/2–

→ 9/2+ 3.9 (5) 4.5 (6)97Tc54 96.57 1/2–

→ 9/2+ 100 5.71(12)99Tc56 142.68 1/2–

→ 9/2+ 0.79 (8) 9.3 (10)95Rh50 543.3 1/2–

→ 9/2+ 88 (6) 1.88(12)97Rh52 258.7 1/2–

→ 9/2+ 5.6 (7) 2.9 (4)99Rh54 64.3 9/2+ → 1/2– [1.7] syst. [4.0] syst.101Rh56 157.41 1/2–

→ 9/2+ 7.2 (3) 6.35(26)107In58 678.5 1/2–

→ 9/2+ 100 6.43(10)109In60 649.90 1/2–

→ 9/2+ 100 7.3 (4)111In62 537.22 1/2–

→ 9/2+ 100 8.31(25)113In64 391.69 1/2–

→ 9/2+ 100 8.87(9)115In66 336.24 1/2–

→ 9/2+ 95.0 (7) 8.84(12)117In68 315.30 1/2–

→ 9/2+ 47 (2) 5.26 (21)119In70 311.37 1/2–

→ 9/2+ 5.6 (15) 6.6 (18)121In72 313.69 1/2–

→ 9/2+ 1.2 (2) 7.2 (12)(b) h11/2 ↔ 2d3/2–type transitions in nuclides with an odd number of neutrons117Sn67 156.02 11/2–

→ 3/2+ 100 2.19( 13)

119Sn69 65.66 11/2– → 3/2+ 100 2.05 (3)

121Sn71 6.29 11/2– → 3/2+ 100 1.42 (5)

12Te69 81.788 11/2– → 3/2+ 88.6 (11) 3.41 (8)

123Te71 88.46 11/2– → 3/2+ 100 2.94 (17)

125Te73 109.27 11/2– → 3/2+ 100 3.11 (21)

127Te75 88.26 11/2– → 3/2+ 97.6 2.85 (8)

129Te77 105.50 11/2–

→ 3/2+ 63 (17) 2.7 (3)131Te79 182.26 11/2–

→ 3/2+ 22.2 (16) 2.67 (21)133Te81 334.26 11/2–

→ 3/2+ 17.5 (30) 2.1 (3)129Te75 196.56 11/2–

→ 3/2+ 100 5.19 (15)131Te77 163.93 11/2–

→ 3/2+ 100 3.37 (12)133Te79 233.22 11/2–

→ 3/2+ 100 3.39 (11)135Te81 526.56 11/2–

→ 3/2+ 100 3.25 (3)133Ba77 277.93 11/2–

→ 3/2+ 100 6.57 (28)135Ba79 268.22 11/2–

→ 3/2+ 100 4.32 (12)137Ba81 661.657 11/2–

→ 3/2+ 100 3.93 (4)137Ce79 254.29 11/2–

→ 3/2+ 99.22 4.72 (15)139Ce81 754.24 11/2–

→ 3/2+ 100 4.57 (10)139Nd79 231.15 11/2–

→ 3/2+ 11.8 (4) 4.8 (4)141Nd81 756.51 11/2–

→ 3/2+ 100 5.55 (14)141Sm79 175.8 11/2–

→ 3/2+ 0.31 (3) 4.5 (5)143Sm81 754.0 11/2–

→ 3/2+ 99.76 (6) 5.96 (20)145Gd81 721.8 11/2–

→ 3/2+ 94.3 (5) 5.84 (23)147Dy81 678.4 11/2–

→ 3/2+ 35 (4) 6.2 (7)149Er81 630.5 11/2–

→ 3/2+ 3.5 (7) 5.6 (13)(c) 2p1/2 ↔ 1g9/2–type transitions in nuclides with an odd number of neutrons69Zn39 438.64 9/2+

→ 1/2– 99.967 5.61 (11)85Kr49 304.87 1/2–

→ 9/2+ 21.4 (4) 3.50 (14)85Sr47 238.66 1/2–

→ 9/2+ 86.6 (6) 4.98 (17)87Sr49 388.53 1/2–

→ 9/2+ 99.70 (8) 3.53 (11)89Zr49 587.4 1/2–

→ 9/2+ 93.77 (22) 3.45 (11)91Mo49 653.0 1/2–

→ 9/2+ 50.1 (12) 4.48 (25)93Ru49 734.4 1/2–

→ 9/2+ 22.0 (23) 5.1 (6)

iI π

fI π

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 76 No. 8 2012

SYSTEMATICS OF M4 TRANSITIONS 851

Table. (Contd.)

Nuclide Energy of M4 transition, keV → Fraction

of isomeric M4 transition, % Hindrance factor

(d) li13/2 ↔ 2f5/2–type transitions in nuclides with an odd number of neutrons193Pt115 135.50 13/2+

→ 5/2– 100 10.9 (3)195Pt117 129.5 13/2+

→ 5/2– 99.76 (2) 9.1 (9)197Pt119 346.5 13/2+

→ 5/2– 96.7 (4) 8.2 (3)193Hg113 101.25 13/2+

→ 5/2– 7.2 (5) 8.9 (7)195Hg115 122.78 13/2+

→ 5/2– 54.2 (20) 7.2 (6)197Hg117 164.97 13/2+

→ 5/2– 91.4 (1) 6.54 (20)199Hg119 374.1 13/2+

→ 5/2– 100 6.03 (15)197Pb115 234.1 13/2+

→ 5/2– 14 (2) 5.6 (8)199Pb117 424.85 13/2+

→ 5/2– 93 (3) 4.35 (13)201Pb119 629.1 13/2+

→ 5/2– 100 4.08 (20)203Pb121 825.2 13/2+

→ 5/2– 100 3.46 (10)205Pb123 1013.4 13/2+

→ 5/2– 0.64 (6) 2.8 (3)207Pb125 1063.66 13/2+

→ 5/2– 100 3.97 (15)199Pî115 238 (1) 13/2+

→ 5/2– 2.1 (10) 4.5 (23)201Pî117 417.9 13/2+

→ 5/2– 56 (14) 5.2 (13)203Pî119 641.7 13/2+

→ 5/2– 100 3.73 (29)

(e) h11/2 ↔ 2d3/2–type transitions in nuclides with an odd number of protons193Ir 116 80.24 11/2–

→ 3/2+ 100 9.3 (3)195Ir 118 100 11/2–

→ 3/2+ ~5 5 (5) but 0193Au 114 290.2 11/2–

→ 3/2+ [0.46(3)] [0.72 (5)]*,195Au 116 318.58 11/2–

→ 3/2+ 100 8.7 (4)197Au118 409.15 11/2–

→ 3/2+ 0.53 (2) 8.2 (3)207Tl126 997.1 11/2–

→ 3/2+ 96.5 (2) 6.3 (6)* The intensity of g290 was overstimated due to the summation (see [3]).

(f) transitions in odd–odd nuclei68Cu 636.9 6– → 2+ 10.8 (18) 11.3 (19)102Rh 98.8 6+ → 2– 5.0 (6) 510 (60)108Ag 30.33 6+

→ 2– 8.7 (9) 1280 (130)*

110Ag 116.48 6+ → 2– 1.36 (6) 860 (40)*

134Cs 138.74 8– → 4+ 0.51 (5) 6.1 (6)**

182Ta 356.47 10– → 6+ 1.64 (12) 72 (5)***

184Re 83.31 8+ → 4– 99.93 (3) 89 (6)***

190Ir 148.7 11– → 7+ 5.6 (9) 8.3( 13)196Au 174.91 12– → 8+ 100 3.14 (10)198Au 115.2 12– → 8+ 100 4.7 (8)196Tl 120.1 7+ → 3– 4.5 (2) 3.60 (16)198Tl 260.9 7+ → 3– 46 (2) 7.5 (5)206Tl 564.2 12– → 8+ 11.8 (20) 70 (12)206Tl 316.8 12– → 8+ 10.6 (30) 2.9 (1)

* The γ transition between states and

** The γ transition between states and

*** Transition in a deformed nucleus.

(g) transitions in 131Sb, 135Cs, and 179Hf nuclides131Sb 1676.1 15/2– → 7/2+ 2.5 (5) [0.7]*

135Cs 846.1 19/2– → 11/2+ 100 940 (20)179Hf 375.0 1/2– → 9/2+ 0.026 54 (10)**

* The intensity of 1676.1 keV γ rays in experiment [5] could be overestimated due to the summation effect.** The γ transition in a deformed nucleus.

iI π

fI π

( ) 37 2 6

9 2 5 2g d+

−⎡ ⎤π ν⎣ ⎦

1

21 2 5 2 .p d

−

−⎡ ⎤π ν⎣ ⎦

( )[ ]or d 58

7 2 2 11 2g h−

π ν [ ]4

11 2, 11 2 .h h+

ν

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SERGEEV

number of protons Z = 82, the values for the hindrancefactors of M4 transitions are relatively low: 3 < F <6(Fig. 4). This is consistent with the shell model and thesingle�particle character of Moszkovsky’s formulas.As the number of nucleons beyond a closed shellincreases, the single�particle nature of γ transitionsbecomes inappropriate, the probability of a transitiondeclines, and the values of the hindrance factors rise.The single�particle model is not applicable if weincrease the number of nucleons beyond the closed

shell, and we have to consider other models of thenucleus, e.g., the deformed nuclei model.

Figure 5 shows a statistical diagram of the values ofhindrance factors for known M4 transitions, implyingthat the average hindrance factor of M4 transitions isF = 5.5. It should be noted that most (75%) of the hin�drance factor values lie in the relatively narrow rangeof F = 3–9, allowing one to use M4 transitions to esti�mate the probability of isomeric state decay throughcompeting channels.

10

5

165605550 N

F

Z = 41

Z = 43

Z = 49

Z = 45

Fig. 1. Dependence of the hindrance factors of М4 transi�tions of the 1g9/2 → 2p1/2 type in nuclei with an oddnumber of protons on the number of neutrons. The stardenotes hindrance factor F = 4.0 of the 64.3 keV M4 tran�sition in a 99Rh nucleus, obtained by interpolation.

5

2

80757065 N

F

Z = 56

Z = 52

Z = 54

Z = 50

Z = 58

Fig. 2. Dependence of the hindrance factors of M4 transi�tions of the 1h11/2 → 2d3/2 type in nuclei with an evennumber of protons on the number of neutrons.

10

5

125120115 N

F

Z = 78

Z = 80

Z = 82

Fig. 3. Same as in Fig. 2, but for transitions of 1i13/2 →

2f5/2 in nuclei with an even number of protons.

10

5

4540 Z

FN = 56

N = 54

N = 50

N = 52

Fig. 4. Same as in Fig. 2, but for transitions of 1g9/2 →

2p1/2 in nuclei with an even number of neutrons on thenumber of protons.

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 76 No. 8 2012

SYSTEMATICS OF M4 TRANSITIONS 853

In [2] and in a number of other publications fromthe Brookhaven Nuclear Data Center, reduced proba�bilities of γ transitions (including M4 transitions) weregiven in units calculated using the Weisskopf formula.These units are associated with γ transitions of electric

multipolarity. As a result, all M4 transitions are—according to [2]—accelerated, which is inexplicable.As was mentioned above, Moszkowsky’s formulas takeinto account for the magnetic properties of nucleonsand reflect the magnetic properties of γ transitionsmore naturally. We showed above that all known M4transitions are hindered to one degree or another withrespect to Moszkowsky’s single�particle estimate (seethe diagram in Fig. 5), and we can explain their hin�drance, at least qualitatively.

ACKNOWLEDGMENTS

The author is grateful to E.P. Grigoryev for hishelpful discussions on this work.

REFERENCES

1. Moszkowski, S.A., Phys. Rev., 1953, vol. 89, p. 474;Moszkowski, S.A., Alpha, Beta and Gamma–Ray Spec�troscopy, Siegbahn, K., Ed., Amsterdam: North�Hol�land, 1966; Moscow: Fizmatgiz, 1969, chap. 15.

2. National Nuclear Data Centre. http://www.nndc.bnl.gov

3. Firestone, R.B., Tables of Isotopes, New York: JohnWiley and Sons, 1998.

4. Gasior, M., et al., Acta Phys. Pol. B, 1972, vol. 3, p. 153.5. Genevey, J., et al., Eur. J. Phys. A, 2000, vol. 9, p. 191.

12

4

Fhind

N

14

8

10

0

2

5

2 3 4 5 6 7 8 9 10 12

Fig. 5. Dependence of the number of М4 transitions N onthe value of the corresponding hindrance factor F, calcu�lated using Moshkovsky’s formulas.