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Lifetime of the 32+ , 1.013 MeV state and the excited-core model of 27Al
1.E.I:3.A]
NuclearPhysics A91 ( 1 9 6 7 ) 4 7 2 - - 4 8 0 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from lhe publisher
L I F E T I M E O F T H E ½+, 1.013 M e V S T A T E AND
THE EXCITED-CORE MODEL
O F 27A1
D. EVERS, J. HERTEL, T. W. RETZ-SCHMIDT and S. J. SKORKA II. Institut fiir Experimentalphysik, Universitiit Hamburg
Received 11 July 1966 Abstract: Nuclear resonance fluorescence scattering was observed to measure the lifetime of the
second excited state in 2~A1at 1.013 MeV. The 37Al(p, p'y)-reaction in a solid aluminium target served as a Doppler-broadened gamma source. The scattering experiment yielded a mean life of tm ~ 2.2 ±0.3 ps, corresponding to a magnetic dipole transition strength of (11.5 ± 1.6) • 10-3 Weisskopf units. The low-lying levels of 37A1are discussed in detail in terms of the excited-core model, taking a mixing of the two ~+ excited states into account. Agreement between the model predictions and the experimental data are obtained for a 15 % quasi-particle admixture to the ~+ state at 1.013 MeV. To fit the energies of the multiplet states the core-hole interaction must contain multipoles of orders up to four. E [
l
NUCLEAR REACTIONS ~rAI(7, y), E = 1.013 MeV; measured a. ZTA1level deduced/~.
[
J
1. I n t r o d u c t i o n
The excited-core m o d e l 1, 2) has recently been a p p l i e d by T h a n k a p p a n 3) to 27A1. The low-lying levels o f 27A1 are interpreted as resulting f r o m the c o u p l i n g o f the ld~ p r o t o n hole to the first excited (2 +) state in 28Si. This gives rise to a multiplet with spins ½+, 3+ . . . . ~+ (see fig. 1). A w e a k coupling in 27A1 is indicated by the fact t h a t the d e f o r m a t i o n s o f the sdshell nuclei change sign between 26Mg a n d 28Si (see ref. 4)). F u r t h e r m o r e a discussion o f 27A1 in the strong-coupling collective m o d e l 5) yields a n u m b e r o f serious difficulties. The t r a n s i t i o n f r o m a strong-coupling to a w e a k - c o u p l i n g description going f r o m 25Mg to 27AI has been d e m o n s t r a t e d conclusively by Crawley et al. 6), who c o m p a r e d the inelastic scattering o f 17.5 M e V p r o t o n s by these two nuclei. These a u t h o r s also showed t h a t the ( 2 J + 1) rule for the d i s t r i b u t i o n o f the excitation cross section o f the different m e m b e r s o f the m u l t i p l e t is excellently fulfilled if a certain mixing o f the ~-+ g r o u n d state a n d the 5+ m e m b e r o f the multiplet is t a k e n into account. Similar results were o b t a i n e d b y other a u t h o r s 7 - 1 o ) by inelastic scattering o f deuterons, s-particles a n d neutrons. R e p r e s e n t i n g the core-hole interaction b y scalar p r o d u c t s o f tensors o f orders up to four, T h a n k a p p a n 3) was able to r e p r o d u c e the energies o f the multiplet states as well as m a n y M 1 a n d E2 t r a n s i t i o n p r o b a b i l i t i e s when adjusting the m o d e l p a r a m e t e r s 472
EXCITED-CORE MODEL
473
to fit the experimental excitation energies. The second ~+ 2 level at 2.98 MeV is unaccounted for in his calculations. Furthermore there is a serious disagreement with respect to the magnetic dipole component of the ground state transition of the a+ z state at 1.013 MeV. This transition probability disagrees by more than a factor of eight from the model predictions, while the branching ratio to the ½+ state at 0.842 MeV is reproduced closely. We therefore felt that a remeasurement of the absolute lifetime of the ~+ state was necessary, particularly since the four existing measurements of the lifetime contradict each other considerably; Metzger et al. 11) got t m = .t . .7+1.2 . . o.5 ps while Booth et al. 12) found t m . . . . n+3"s . 1 . 8 ps. An earlier measurement of Vanhuyse et al. 13) yielded +2.9 t m = (4.1_1.6) • 10 -2 ps, while two further experiments is) set a lower limit at MeV
J~
4.05,i
I/2 +
3.674
m
3.000 2.97~
I12 +
2.73'
912 + 3•2 + 5•2 +
2.212
712 +
1.013 Q842
3•2 + I12 +
5•2 +
27At Fig. 1. The lower energy levels in 27A1. tm > 1 ps. A new self-absorption experiment was recently published by Hough et al. 14) who found t m = .n. .~+o.s . 0.2 ps. In our experiment the nuclear resonance fluorescence method was used, combined with a pulsed beam background discrimination technique 16) and employing recent studies 17) of the slowing down of recoil nuclei in solid targets. This experimental method allows the achievement of high statistical accuracy by performing simple scattering experiments. These measurements are described in the following sect. 2. Sect. 3 contains results of new calculations of the transition probabilities in 27A1 within the framework of the excited-core model. In these calculations the ~+ 2 state at 2.98 MeV has also been included. This level is interpreted as quasi-particle excitation
D. EVERS et al.
474
arising from the ld k single-particle shell-model state. Mixing of the two -~+ states is allowed for. In the usual notation [JcJ, J ) (Jc is the angular m o m e n t u m of the core, j that of the hole and J the total angular m o m e n t u m of the system), the model wave functions employed in the calculations are g.s°
~9(s) = AI0 s, s > + x / 1 _A212 ~, 5>,
0.842 MeV
~k(1) = 125,½>,
1.013 MeV
~k(3) = C [ 2 ~ , 2 a > + x / 1 - C 210~,3>,
2.21
~(~)
MeV
= 12~,g),
2.731 MeV
5 ~ 5 )-x/1-A ~9(~*) = A]2 ~,
2
2.98
MeV
0 ( 3 . ) = CI0 3, 3 ) - x / 1 - C 2 1 2 ~, 3),
3.00
MeV
0(~)
[0 ~-, s),
= 12~,9).
Higher admixtures like [2 3, j ) were neglected, as well as possible admixtures to the ~1 + state at 0.842 MeV resulting from higher ~1 + states. The parameters A and C are expected to be close to one if the core-excitation model in its simple version is valid. In sect. 4 the results of the calculations and the model parameters are discussed. It turns out that it is possible to remove the above-mentioned discrepancy concerning the 3+ state by a small admixture (C 2 < 1) to this state and to find a small number of parameters in terms of which a great number of transition probabilities, mixing and branching ratios, as well as moments and excitation energies are interrelated.
2. Measurement of the lifetime of the 1.013 MeV state
The experiment was performed in a standard ring geometry similar to previous experiments 18). The pulsed beam of 3 MeV protons had a pulse frequency of 1 Mc and an average beam current of 0.7 #A at 1 nsec pulse length. The target consisted of 99.98 % metallic aluminium. The NaI(TI)-XP1040 detector was gated with a 5 nsec time window centered at the arrival time of the scattered gamma quanta. An additional circuit, gated in anticoincidence to this time window, was connected to a digital energy stabilization system working on the 661 keV peak of a weak x37Cs radioactive source mounted close to the detector. The time-of-flight system not only reduced the uncorrelated background by a factor of 200, it also discriminated against effects of neutrons from target and beam impurities. The total number of resonant gamma rays from the target was calculated according to the method explained in detail in ref. 17). Energy-loss data of Lindhard et al. 19) were used. Because of the long lifetime of the level in question the angular distribution of the resonant gamma rays could be assumed isotropic, due to the angular scattering
EXCITED-CORM EODEL
475
of the excited recoils. For a small correction of the angular distribution of the resonance scattering, the E2/M1 mixing ratio of the 1.013 MeV radiation was taken to be 6 = - 0 . 3 2 (ref. 22)). The straightforward evaluation of the scattering experiment (fig. 2) yielded for the mean life of the 1.013 MeV state t m = 2.2_+0.3 psec, taking a ground state branching of 97.7 ~ (ref. s)). Our result agrees well with that of Metzger et al. 11), but disagrees with all others except for the two lower limits 15) (see sect. 1).
(refs.12-14)) 50C
~
/
N ( Er,t m)
,r'~140 ( .$
g3oo J
E >
E
J
o
100 ~
Q5
tin=
I
1
I
1.5 Mean life ~s~
2.2
I
2
0.3
ps
25 I
Fig. 2. Evaluation o f the experiment. T h e solid curve is the theoretical r e s o n a n c e intensity as a function o f the m e a n life tm for a n incident p r o t o n energy o f 3 MeV. T h e straight line represents the result o f the resonance scattering c o u n t i n g rate, including the experimental errors (one s t a n d a r d deviation). T h e intersection defines the m e a n life. T h e uncertainty o f the theoretical curve is due to the uncertainty o f the used energy loss data (see ref. 17) for m o r e details).
The E2 partial transition probability of the 1.013 MeV state was measured by Alklaazov et al. 2o) and Alexander et al. 21) by Coulomb excitation (tin(E2) = (1.7+0.3)" 10 -v sec). Together with our lifetime measurements one gets [3[ = 0.40_+0.05 in fair agreement with an independent result of McCallum 22) (6 = - 0 . 3 2 + 0 . 1 4 ) . The reduced transition probabilities can be calculated to be B(M1, ] ~ 5) = 0.020+0.003[p2], B(E2, ~ ~ ~) = 50_12[e 2 • fm*]. With the known branching to the 0.842 MeV state (2.4 ~ ) , one can further deduce B(M1, ~2 ~ ½) = 0.12_+ 0.02 [#2 ] assuming an upper limit of [ M ] 2 2 ~ 50 for the E2 transition strength of this branch. The corresponding mixing ratio is [8[(3 ~ ½) < 0.07.
476
D . EVERS
et aL
3. The core-excitation model of 27A1 Starting with the wave functions introduced in sect. 1 and using well-known relations of tensor algebra 23), it is possible, within the framework of the excited-core model, to express the relevant electromagnetic matrix elements as a function of a few quantities. (i) the gyromagnetic ratios gc of the core and gp of the particle; assuming a collective nature of the core excitation, gc = Z/A. (ii) the core transition matrix elements (0[Idg'~2)[[2) and (2[[dt'~2)112); the former is found to be 17.3 fm 2 from the 2 + --* 0 + transition probability in 2asi (ref. 25)). The latter one and the single hole matrix element (d:-l[Idg(p2)rld~l) were adjusted to fit the experimental data for the electric quadrupole moment of the ground state 24) and for the E2 transition probability of the 7+ member to the ground state 2v). Both of these quantities are fairly well known. (iii) The particle transition matrix elements M(M1) = (3[[dt,(pl)[]~_) and M(E2) = 3 (2) 5 • (~[[~'p [l~-), these were calculated according to the single-particle estimates. The fact that the d+ shell is partially filled was taken into account (ref. 26)). (iv) Finally, the mixing parameters A and C which were adjusted empirically to yield the correct M1 ground state transition probabilities of the -72+ and ~÷ 2 states respectively. Only four adjustable parameters were introduced: A, C, (d211lJt~(p2)lld~ 1) and (211~2)112). It should be noted that these parameters are expected to fall within rather limited ranges if the model is valid. The following values for the parameters were found: A = 0.885, C = - 0.922, (d~lll._~'(p2)lld~l) = 5.1 fm 2, (211,~'~2)112) ~ 1.3 fm 2. While A and C are defined rather sharply by the experimental data, the two matrix elements, particularly the latter one, are not. It may be noted that (2[1~'~2)[12)<< (011~'~e)ll2), (even (2[IJt~2)[12) = 0 is consistent with the E2 transition probability of the ~+ state within its rather small experimental error implying the intrinsic quadrupole moment of the 2 + state in 28Si to be zero). It should be mentioned, moreover, that the single-particle magnetic dipole quantities gp = 1.916 and M(M1) = 2.01 /~o both had to be reduced empirically by 20 ~o to obtain agreement with the magnetic moment of the ground state and the magnetic dipole ground state transition of the second 2 + state at 2.98 MeV. The reduced values gp = 1.533 and M ' ( M 1 ) = 1.62 Po were used throughout to calculate all M1 transition probabilities. The results of the calculations based on the above set of parameters are compared with the available experimental information (refs. 24, 27)) in table 1.
477
EXCITED-CORE MODEL
4. D i s c u s s i o n of the theoretical results T h e d a t a in t a b l e 1 s h o w a g r e e m e n t w i t h i n t h e s t a n d a r d e r r o r s o f the e x p e r i m e n t a l results in p r a c t i c a l l y all cases. P a r t i c u l a r l y , c o m p l e t e a g r e e m e n t is o b t a i n e d f o r t h e M 1 t r a n s i t i o n s o f t h e t w o 3+ states to t h e g r o u n d state by t h e s m a l l s i n g l e - p a r t i c l e adm i x t u r e o f 1 - C 2 = 0.15 to t h e 1.013 M e V state. TABLE 1 Comparison of the calculated and experimental values of various quantities related to the electromagnetic properties of the low-lying levels in 27A1 Transition
-+ ~ -+ ½ ---~ ~* _~. ~
~* --~ ~ ~* -+ ~ ~-+~
B( M 1) (#02)
B(E2) (e~ • fm 4)
exp.
theor,
0.020±0.003 0.12 ±0.02 0.11 ±0.01 0.042 +0.017 o.o,4 0.81 +o.2~ --O.54 0.35 ±0.04
0.020 0.528 0.110 0.010 0.591 0.342
exp.
theor.
48±20 50~ 12
59.0 28.4
53± 5 19-4- 9.5
53.0 12.9
< 53 52± 6
33.5 48.6
Mixing ratio 6(E2/M1) exp. ,~ -+ -+ -+ ~*~
~ ½ ~ ~
6" -+ ~ ]* -+ ~ ~- -+ -}
theor.
--0.32 ~0.14 ~ 0.07(16!) 0.43 ±0.04 0.80 ±0.45([6[) (--0.038 ±0.055) (--0.226_+_0.064)
--0.320 0.0025 0.406 --0.821
--0.03 ±0.03
0.069 --0.248 0.0096
Branching ratio exp.
F@*) F@)
theor.
0.80±0.05
0.822
0.20_0.10
0.148
magn. moment (g.s.) el. quadrupole moment (g.s.)
3.6385 15.2
3.6385 (#o) 15.2
(fm 2)
M o r e o v e r , t h e ~5 + m i x i n g p a r a m e t e r A w h i c h was o b t a i n e d by a d j u s t m e n t to the M 1 t r a n s i t i o n o f t h e 5 + state is in g o o d a g r e e m e n t w i t h t h e results o f inelastic scatteri n g e x p e r i m e n t s (see t a b l e 2) a n d w i t h t h e t h e o r e t i c a l result o f T h a n k a p p a n 3), w h o o b t a i n e d A by fitting t h e c o r e - h o l e i n t e r a c t i o n p a r a m e t e r s to the e x c i t a t i o n energies
478
D. EVERS et al.
of the multiplet states. The parameter A has a considerable influence on practically all transitions. There are only two marked disagreements displayed in table 1. The first is the M1 transition probability of the 170 keV transition between the )+ and the ½+ members of the multiplet. The experimental value for this transition depends on the measurement 28) of the rather small branching ratio of the 1.013 MeV state ((2.4±0.3) ~ , see sect. 2). Agreement of the theoretical value of B(M1, ~ ~ ½) can only be obtained by a rather large quasi particle (2s~) admixture to the ½+ state (approximately 60 ~ ) since the single-particle transition between the two admixtures is/-forbidden. A small ]2 3, j ) admixture to both involved states could, on the other hand, further decrease the discrepancy. The branching ratio of the ~+ 2 state is rather difficult to measure. A remeasurement is desirable, particularly since the striking disagreement between the two recent nuclear resonance fluorescence experiments on the lifetime of this state seems to indicate a larger branching to the ½+ state: The measurement of Hough 14) (see sect. 1) was achieved in self-absorption (which is proportional to F(Fo/F)), while our result was TABLE 2 C o m p a r i s o n o f different values for the mixing p a r a m e t e r 1 - - A s Source
Ref.
1--A 2
(n, n') (d, d') (p, p,) Theory This work
9) 7) 6) a)
0.18±0.07 0.28 4-0.03 0.23 0.174 0.217
obtained by resonance scatter&g (which is proportional to F(Fo/F)2). Assuming an increase of the branching ratio to approx. 10 ~ would bring the ~+ ~ ½+ M1 transition in agreement with the theory and would, at the same time, decrease the disagreement of the two measurements by about one upper standard deviation of Hough's experiment. All other results would be practically unaffected (C = - 0 . 9 2 8 instead of C = -0.922). The second disagreement shown in table 1 is that of the mixing ratio of the 5* ~ transition. This mixing ratio was obtained by Ophel et al. 5) by triple correlation measurements together with the corresponding mixing ratio of the ground state transition of the -52* state (6 = - 0 . 0 3 8 ± 0 . 0 5 5 ) . The latter one is in disagreement with a recent measurement of the E2 ground state transition rate of the 25. state by Lombard 29), which would give ]61 = 0.80±0.45, assuming a total width of this a~+ao meV. This situation seems to indicate that the (first mentioned) level 27) o f F = vv_4o mixing ratio of the ~* ~ ~ branch might be in error. A measurement of this mixing ratio is being planned in our laboratory.
EXCITED-CORE MODEL
479
The calculation of the energy spacings of the multiplet involves the assumption of an interaction Hamiltonian between the hole and the core. It is of interest to check if agreement between theory and experiment can be gained with the obtained parameters. Following Thankappan 3), we assume ~int
~--" - - E r f r ( T(r)"
t(pr)),
where T~") is a tensor operator of rank r operating only on the degrees of freedom of the core, etc. It turns out that close agreement can - in correspondence to Thankappan - only be obtained by admitting tensor interactions of ranks up to four. Since the parameters A, (d~ Xl I.-~2)lld~- x) and (211~'~2)112) were already fixed by the experimental transition rates (it is jt,~2) = 1c"'<2),etc.), the number of free parameters is one less than TABLE 3 Experimental and calculated energies of levels in 27A1 State
Eexp
Etheor
½ 5 ~52* 5"
0.842 1.01 2.21 2.73 2.98 3.00
0.862 1.03 2.15 2.73 2.98 3.07
necessary to obtain a trivial fit, even if a term r = 0 is added. With the interaction parameters (see ref. 3) for notations) = -0.161 MeV,
133 = - 2 . 5 5 MeV,
r/ = 0.0694 M e V . fm -4,
e4 = - 1 . 6 7 MeV,
and a core excitation energy of E c = 1.657 MeV (instead of 1.772 MeV in 28Si) we obtain the excitation energies accumulated in table 3. The small difference of the core excitation energy corresponds to a rank r = 0 interaction component. The agreement of the ~* energy (2.98 MeV) was obtained by adjusting the ~+ quasiparticle excitation to be 2.62 MeV. All calculated energies differ by less than 2.5 ~o from the experimental values. It should be noted that q is already fixed by the parameter A and the above mentioned rank 2 matrix elements. References 1) 2) 3) 4) 5)
R. D. Lawson and J. L. Uretsky, Phys. Rev. 108 (1957) 1300 A. de-Shalit, Phys. Rev. 122 (1960) 1530 V. K. Thankappan, Phys. Rev. 141 (1966) 957 J. Bar-Touv and I. Kelson, Phys. Rev. 158 (1965) 1035 T. R. Ophel and B. T. Lawergren, Nuclear Physics 52 (1964) 417
480 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29)
D. EVERS et al. G. M. Crawley and G. T. Garvey, Phys. Lett. 19 (1965) 228 H. Niewodnizanski et al., Nuclear Physics 55 (1964) 386 J. Kokame, K. Fukunaya and H. Nakamura, Phys. Lett. 14 (1965) 234 G. C. Bonazzola et aL, Phys. Rev. 140 (1965) B835 P. H. Stelson et al., Nuclear Physics 68 (1965) 97 F. R. Metzger, C. P. Swann and V. K. Rasmussen, Nuclear Physics 16 (1960) 568 E. C. Booth and K. A. Wright, Nuclear Physics 35 (1962) 472 V. J. Vanhuyse and G. J. Vanpraet, J. Phys. Rad. 21 (1960) 290 J. H. Hough and W. L. Mouton, Nuclear Physics 76 (1966) 248 L. A. Schaller and W. C. Miller, Bull. Am. Phys. Soc. 9 (1964) 666; L. J. Lidofsky, L. Mo, Y. K. Lee and C. S. Wu, Bull. Am. Phys. Soc. 10 (1965) 119 S. J. Skorka, Nucl. Instr. 28 (1964) 192 S. J. Skorka et al., Nuclear Physics 81 (1966) 370 S. J. Skorka et al., Nuclear Physics 68 (1965) 177 J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128 D. G. Alkhazov et al., ZhETF (USSR) 35 (1959) 736 K. F. Alexander and V. Bredel, Nuclear Physics 17 (1960) 153 G. C. McCallum, Phys. Rev. 123 (1961) 568 A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) K. Way et al., Nuclear Data Sheets (Nat. Research Council, Washington, D.C.) S. J. Skorka and T. W. Retz-Schmidt, Nuclear Physics 46 (1963) 225 S. A. Moszkowski, Alpha-, beta- and gamma-ray spectroscopy, Vol. 2, ed. by K. Siegbahn (North-Holland Publ. Co., Amsterdam, 1965)p. 884 S. J. Skorka, Nuclear Data (1967) to be published; references quoted in ref. 3) E. Almquist et al., Nuclear Physics 19 (1960) 1 R. Lombard, P h . D . thesis, University of Paris (1964) unpublished, quoted in ref. z)
NuclearPhysics A91 ( 1 9 6 7 ) 4 7 2 - - 4 8 0 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from lhe publisher
L I F E T I M E O F T H E ½+, 1.013 M e V S T A T E AND
THE EXCITED-CORE MODEL
O F 27A1
D. EVERS, J. HERTEL, T. W. RETZ-SCHMIDT and S. J. SKORKA II. Institut fiir Experimentalphysik, Universitiit Hamburg
Received 11 July 1966 Abstract: Nuclear resonance fluorescence scattering was observed to measure the lifetime of the
second excited state in 2~A1at 1.013 MeV. The 37Al(p, p'y)-reaction in a solid aluminium target served as a Doppler-broadened gamma source. The scattering experiment yielded a mean life of tm ~ 2.2 ±0.3 ps, corresponding to a magnetic dipole transition strength of (11.5 ± 1.6) • 10-3 Weisskopf units. The low-lying levels of 37A1are discussed in detail in terms of the excited-core model, taking a mixing of the two ~+ excited states into account. Agreement between the model predictions and the experimental data are obtained for a 15 % quasi-particle admixture to the ~+ state at 1.013 MeV. To fit the energies of the multiplet states the core-hole interaction must contain multipoles of orders up to four. E [
l
NUCLEAR REACTIONS ~rAI(7, y), E = 1.013 MeV; measured a. ZTA1level deduced/~.
[
J
1. I n t r o d u c t i o n
The excited-core m o d e l 1, 2) has recently been a p p l i e d by T h a n k a p p a n 3) to 27A1. The low-lying levels o f 27A1 are interpreted as resulting f r o m the c o u p l i n g o f the ld~ p r o t o n hole to the first excited (2 +) state in 28Si. This gives rise to a multiplet with spins ½+, 3+ . . . . ~+ (see fig. 1). A w e a k coupling in 27A1 is indicated by the fact t h a t the d e f o r m a t i o n s o f the sdshell nuclei change sign between 26Mg a n d 28Si (see ref. 4)). F u r t h e r m o r e a discussion o f 27A1 in the strong-coupling collective m o d e l 5) yields a n u m b e r o f serious difficulties. The t r a n s i t i o n f r o m a strong-coupling to a w e a k - c o u p l i n g description going f r o m 25Mg to 27AI has been d e m o n s t r a t e d conclusively by Crawley et al. 6), who c o m p a r e d the inelastic scattering o f 17.5 M e V p r o t o n s by these two nuclei. These a u t h o r s also showed t h a t the ( 2 J + 1) rule for the d i s t r i b u t i o n o f the excitation cross section o f the different m e m b e r s o f the m u l t i p l e t is excellently fulfilled if a certain mixing o f the ~-+ g r o u n d state a n d the 5+ m e m b e r o f the multiplet is t a k e n into account. Similar results were o b t a i n e d b y other a u t h o r s 7 - 1 o ) by inelastic scattering o f deuterons, s-particles a n d neutrons. R e p r e s e n t i n g the core-hole interaction b y scalar p r o d u c t s o f tensors o f orders up to four, T h a n k a p p a n 3) was able to r e p r o d u c e the energies o f the multiplet states as well as m a n y M 1 a n d E2 t r a n s i t i o n p r o b a b i l i t i e s when adjusting the m o d e l p a r a m e t e r s 472
EXCITED-CORE MODEL
473
to fit the experimental excitation energies. The second ~+ 2 level at 2.98 MeV is unaccounted for in his calculations. Furthermore there is a serious disagreement with respect to the magnetic dipole component of the ground state transition of the a+ z state at 1.013 MeV. This transition probability disagrees by more than a factor of eight from the model predictions, while the branching ratio to the ½+ state at 0.842 MeV is reproduced closely. We therefore felt that a remeasurement of the absolute lifetime of the ~+ state was necessary, particularly since the four existing measurements of the lifetime contradict each other considerably; Metzger et al. 11) got t m = .t . .7+1.2 . . o.5 ps while Booth et al. 12) found t m . . . . n+3"s . 1 . 8 ps. An earlier measurement of Vanhuyse et al. 13) yielded +2.9 t m = (4.1_1.6) • 10 -2 ps, while two further experiments is) set a lower limit at MeV
J~
4.05,i
I/2 +
3.674
m
3.000 2.97~
I12 +
2.73'
912 + 3•2 + 5•2 +
2.212
712 +
1.013 Q842
3•2 + I12 +
5•2 +
27At Fig. 1. The lower energy levels in 27A1. tm > 1 ps. A new self-absorption experiment was recently published by Hough et al. 14) who found t m = .n. .~+o.s . 0.2 ps. In our experiment the nuclear resonance fluorescence method was used, combined with a pulsed beam background discrimination technique 16) and employing recent studies 17) of the slowing down of recoil nuclei in solid targets. This experimental method allows the achievement of high statistical accuracy by performing simple scattering experiments. These measurements are described in the following sect. 2. Sect. 3 contains results of new calculations of the transition probabilities in 27A1 within the framework of the excited-core model. In these calculations the ~+ 2 state at 2.98 MeV has also been included. This level is interpreted as quasi-particle excitation
D. EVERS et al.
474
arising from the ld k single-particle shell-model state. Mixing of the two -~+ states is allowed for. In the usual notation [JcJ, J ) (Jc is the angular m o m e n t u m of the core, j that of the hole and J the total angular m o m e n t u m of the system), the model wave functions employed in the calculations are g.s°
~9(s) = AI0 s, s > + x / 1 _A212 ~, 5>,
0.842 MeV
~k(1) = 125,½>,
1.013 MeV
~k(3) = C [ 2 ~ , 2 a > + x / 1 - C 210~,3>,
2.21
~(~)
MeV
= 12~,g),
2.731 MeV
5 ~ 5 )-x/1-A ~9(~*) = A]2 ~,
2
2.98
MeV
0 ( 3 . ) = CI0 3, 3 ) - x / 1 - C 2 1 2 ~, 3),
3.00
MeV
0(~)
[0 ~-, s),
= 12~,9).
Higher admixtures like [2 3, j ) were neglected, as well as possible admixtures to the ~1 + state at 0.842 MeV resulting from higher ~1 + states. The parameters A and C are expected to be close to one if the core-excitation model in its simple version is valid. In sect. 4 the results of the calculations and the model parameters are discussed. It turns out that it is possible to remove the above-mentioned discrepancy concerning the 3+ state by a small admixture (C 2 < 1) to this state and to find a small number of parameters in terms of which a great number of transition probabilities, mixing and branching ratios, as well as moments and excitation energies are interrelated.
2. Measurement of the lifetime of the 1.013 MeV state
The experiment was performed in a standard ring geometry similar to previous experiments 18). The pulsed beam of 3 MeV protons had a pulse frequency of 1 Mc and an average beam current of 0.7 #A at 1 nsec pulse length. The target consisted of 99.98 % metallic aluminium. The NaI(TI)-XP1040 detector was gated with a 5 nsec time window centered at the arrival time of the scattered gamma quanta. An additional circuit, gated in anticoincidence to this time window, was connected to a digital energy stabilization system working on the 661 keV peak of a weak x37Cs radioactive source mounted close to the detector. The time-of-flight system not only reduced the uncorrelated background by a factor of 200, it also discriminated against effects of neutrons from target and beam impurities. The total number of resonant gamma rays from the target was calculated according to the method explained in detail in ref. 17). Energy-loss data of Lindhard et al. 19) were used. Because of the long lifetime of the level in question the angular distribution of the resonant gamma rays could be assumed isotropic, due to the angular scattering
EXCITED-CORM EODEL
475
of the excited recoils. For a small correction of the angular distribution of the resonance scattering, the E2/M1 mixing ratio of the 1.013 MeV radiation was taken to be 6 = - 0 . 3 2 (ref. 22)). The straightforward evaluation of the scattering experiment (fig. 2) yielded for the mean life of the 1.013 MeV state t m = 2.2_+0.3 psec, taking a ground state branching of 97.7 ~ (ref. s)). Our result agrees well with that of Metzger et al. 11), but disagrees with all others except for the two lower limits 15) (see sect. 1).
(refs.12-14)) 50C
~
/
N ( Er,t m)
,r'~140 ( .$
g3oo J
E >
E
J
o
100 ~
Q5
tin=
I
1
I
1.5 Mean life ~s~
2.2
I
2
0.3
ps
25 I
Fig. 2. Evaluation o f the experiment. T h e solid curve is the theoretical r e s o n a n c e intensity as a function o f the m e a n life tm for a n incident p r o t o n energy o f 3 MeV. T h e straight line represents the result o f the resonance scattering c o u n t i n g rate, including the experimental errors (one s t a n d a r d deviation). T h e intersection defines the m e a n life. T h e uncertainty o f the theoretical curve is due to the uncertainty o f the used energy loss data (see ref. 17) for m o r e details).
The E2 partial transition probability of the 1.013 MeV state was measured by Alklaazov et al. 2o) and Alexander et al. 21) by Coulomb excitation (tin(E2) = (1.7+0.3)" 10 -v sec). Together with our lifetime measurements one gets [3[ = 0.40_+0.05 in fair agreement with an independent result of McCallum 22) (6 = - 0 . 3 2 + 0 . 1 4 ) . The reduced transition probabilities can be calculated to be B(M1, ] ~ 5) = 0.020+0.003[p2], B(E2, ~ ~ ~) = 50_12[e 2 • fm*]. With the known branching to the 0.842 MeV state (2.4 ~ ) , one can further deduce B(M1, ~2 ~ ½) = 0.12_+ 0.02 [#2 ] assuming an upper limit of [ M ] 2 2 ~ 50 for the E2 transition strength of this branch. The corresponding mixing ratio is [8[(3 ~ ½) < 0.07.
476
D . EVERS
et aL
3. The core-excitation model of 27A1 Starting with the wave functions introduced in sect. 1 and using well-known relations of tensor algebra 23), it is possible, within the framework of the excited-core model, to express the relevant electromagnetic matrix elements as a function of a few quantities. (i) the gyromagnetic ratios gc of the core and gp of the particle; assuming a collective nature of the core excitation, gc = Z/A. (ii) the core transition matrix elements (0[Idg'~2)[[2) and (2[[dt'~2)112); the former is found to be 17.3 fm 2 from the 2 + --* 0 + transition probability in 2asi (ref. 25)). The latter one and the single hole matrix element (d:-l[Idg(p2)rld~l) were adjusted to fit the experimental data for the electric quadrupole moment of the ground state 24) and for the E2 transition probability of the 7+ member to the ground state 2v). Both of these quantities are fairly well known. (iii) The particle transition matrix elements M(M1) = (3[[dt,(pl)[]~_) and M(E2) = 3 (2) 5 • (~[[~'p [l~-), these were calculated according to the single-particle estimates. The fact that the d+ shell is partially filled was taken into account (ref. 26)). (iv) Finally, the mixing parameters A and C which were adjusted empirically to yield the correct M1 ground state transition probabilities of the -72+ and ~÷ 2 states respectively. Only four adjustable parameters were introduced: A, C, (d211lJt~(p2)lld~ 1) and (211~2)112). It should be noted that these parameters are expected to fall within rather limited ranges if the model is valid. The following values for the parameters were found: A = 0.885, C = - 0.922, (d~lll._~'(p2)lld~l) = 5.1 fm 2, (211,~'~2)112) ~ 1.3 fm 2. While A and C are defined rather sharply by the experimental data, the two matrix elements, particularly the latter one, are not. It may be noted that (2[1~'~2)[12)<< (011~'~e)ll2), (even (2[IJt~2)[12) = 0 is consistent with the E2 transition probability of the ~+ state within its rather small experimental error implying the intrinsic quadrupole moment of the 2 + state in 28Si to be zero). It should be mentioned, moreover, that the single-particle magnetic dipole quantities gp = 1.916 and M(M1) = 2.01 /~o both had to be reduced empirically by 20 ~o to obtain agreement with the magnetic moment of the ground state and the magnetic dipole ground state transition of the second 2 + state at 2.98 MeV. The reduced values gp = 1.533 and M ' ( M 1 ) = 1.62 Po were used throughout to calculate all M1 transition probabilities. The results of the calculations based on the above set of parameters are compared with the available experimental information (refs. 24, 27)) in table 1.
477
EXCITED-CORE MODEL
4. D i s c u s s i o n of the theoretical results T h e d a t a in t a b l e 1 s h o w a g r e e m e n t w i t h i n t h e s t a n d a r d e r r o r s o f the e x p e r i m e n t a l results in p r a c t i c a l l y all cases. P a r t i c u l a r l y , c o m p l e t e a g r e e m e n t is o b t a i n e d f o r t h e M 1 t r a n s i t i o n s o f t h e t w o 3+ states to t h e g r o u n d state by t h e s m a l l s i n g l e - p a r t i c l e adm i x t u r e o f 1 - C 2 = 0.15 to t h e 1.013 M e V state. TABLE 1 Comparison of the calculated and experimental values of various quantities related to the electromagnetic properties of the low-lying levels in 27A1 Transition
-+ ~ -+ ½ ---~ ~* _~. ~
~* --~ ~ ~* -+ ~ ~-+~
B( M 1) (#02)
B(E2) (e~ • fm 4)
exp.
theor,
0.020±0.003 0.12 ±0.02 0.11 ±0.01 0.042 +0.017 o.o,4 0.81 +o.2~ --O.54 0.35 ±0.04
0.020 0.528 0.110 0.010 0.591 0.342
exp.
theor.
48±20 50~ 12
59.0 28.4
53± 5 19-4- 9.5
53.0 12.9
< 53 52± 6
33.5 48.6
Mixing ratio 6(E2/M1) exp. ,~ -+ -+ -+ ~*~
~ ½ ~ ~
6" -+ ~ ]* -+ ~ ~- -+ -}
theor.
--0.32 ~0.14 ~ 0.07(16!) 0.43 ±0.04 0.80 ±0.45([6[) (--0.038 ±0.055) (--0.226_+_0.064)
--0.320 0.0025 0.406 --0.821
--0.03 ±0.03
0.069 --0.248 0.0096
Branching ratio exp.
F@*) F@)
theor.
0.80±0.05
0.822
0.20_0.10
0.148
magn. moment (g.s.) el. quadrupole moment (g.s.)
3.6385 15.2
3.6385 (#o) 15.2
(fm 2)
M o r e o v e r , t h e ~5 + m i x i n g p a r a m e t e r A w h i c h was o b t a i n e d by a d j u s t m e n t to the M 1 t r a n s i t i o n o f t h e 5 + state is in g o o d a g r e e m e n t w i t h t h e results o f inelastic scatteri n g e x p e r i m e n t s (see t a b l e 2) a n d w i t h t h e t h e o r e t i c a l result o f T h a n k a p p a n 3), w h o o b t a i n e d A by fitting t h e c o r e - h o l e i n t e r a c t i o n p a r a m e t e r s to the e x c i t a t i o n energies
478
D. EVERS et al.
of the multiplet states. The parameter A has a considerable influence on practically all transitions. There are only two marked disagreements displayed in table 1. The first is the M1 transition probability of the 170 keV transition between the )+ and the ½+ members of the multiplet. The experimental value for this transition depends on the measurement 28) of the rather small branching ratio of the 1.013 MeV state ((2.4±0.3) ~ , see sect. 2). Agreement of the theoretical value of B(M1, ~ ~ ½) can only be obtained by a rather large quasi particle (2s~) admixture to the ½+ state (approximately 60 ~ ) since the single-particle transition between the two admixtures is/-forbidden. A small ]2 3, j ) admixture to both involved states could, on the other hand, further decrease the discrepancy. The branching ratio of the ~+ 2 state is rather difficult to measure. A remeasurement is desirable, particularly since the striking disagreement between the two recent nuclear resonance fluorescence experiments on the lifetime of this state seems to indicate a larger branching to the ½+ state: The measurement of Hough 14) (see sect. 1) was achieved in self-absorption (which is proportional to F(Fo/F)), while our result was TABLE 2 C o m p a r i s o n o f different values for the mixing p a r a m e t e r 1 - - A s Source
Ref.
1--A 2
(n, n') (d, d') (p, p,) Theory This work
9) 7) 6) a)
0.18±0.07 0.28 4-0.03 0.23 0.174 0.217
obtained by resonance scatter&g (which is proportional to F(Fo/F)2). Assuming an increase of the branching ratio to approx. 10 ~ would bring the ~+ ~ ½+ M1 transition in agreement with the theory and would, at the same time, decrease the disagreement of the two measurements by about one upper standard deviation of Hough's experiment. All other results would be practically unaffected (C = - 0 . 9 2 8 instead of C = -0.922). The second disagreement shown in table 1 is that of the mixing ratio of the 5* ~ transition. This mixing ratio was obtained by Ophel et al. 5) by triple correlation measurements together with the corresponding mixing ratio of the ground state transition of the -52* state (6 = - 0 . 0 3 8 ± 0 . 0 5 5 ) . The latter one is in disagreement with a recent measurement of the E2 ground state transition rate of the 25. state by Lombard 29), which would give ]61 = 0.80±0.45, assuming a total width of this a~+ao meV. This situation seems to indicate that the (first mentioned) level 27) o f F = vv_4o mixing ratio of the ~* ~ ~ branch might be in error. A measurement of this mixing ratio is being planned in our laboratory.
EXCITED-CORE MODEL
479
The calculation of the energy spacings of the multiplet involves the assumption of an interaction Hamiltonian between the hole and the core. It is of interest to check if agreement between theory and experiment can be gained with the obtained parameters. Following Thankappan 3), we assume ~int
~--" - - E r f r ( T(r)"
t(pr)),
where T~") is a tensor operator of rank r operating only on the degrees of freedom of the core, etc. It turns out that close agreement can - in correspondence to Thankappan - only be obtained by admitting tensor interactions of ranks up to four. Since the parameters A, (d~ Xl I.-~2)lld~- x) and (211~'~2)112) were already fixed by the experimental transition rates (it is jt,~2) = 1c"'<2),etc.), the number of free parameters is one less than TABLE 3 Experimental and calculated energies of levels in 27A1 State
Eexp
Etheor
½ 5 ~52* 5"
0.842 1.01 2.21 2.73 2.98 3.00
0.862 1.03 2.15 2.73 2.98 3.07
necessary to obtain a trivial fit, even if a term r = 0 is added. With the interaction parameters (see ref. 3) for notations) = -0.161 MeV,
133 = - 2 . 5 5 MeV,
r/ = 0.0694 M e V . fm -4,
e4 = - 1 . 6 7 MeV,
and a core excitation energy of E c = 1.657 MeV (instead of 1.772 MeV in 28Si) we obtain the excitation energies accumulated in table 3. The small difference of the core excitation energy corresponds to a rank r = 0 interaction component. The agreement of the ~* energy (2.98 MeV) was obtained by adjusting the ~+ quasiparticle excitation to be 2.62 MeV. All calculated energies differ by less than 2.5 ~o from the experimental values. It should be noted that q is already fixed by the parameter A and the above mentioned rank 2 matrix elements. References 1) 2) 3) 4) 5)
R. D. Lawson and J. L. Uretsky, Phys. Rev. 108 (1957) 1300 A. de-Shalit, Phys. Rev. 122 (1960) 1530 V. K. Thankappan, Phys. Rev. 141 (1966) 957 J. Bar-Touv and I. Kelson, Phys. Rev. 158 (1965) 1035 T. R. Ophel and B. T. Lawergren, Nuclear Physics 52 (1964) 417
480 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29)
D. EVERS et al. G. M. Crawley and G. T. Garvey, Phys. Lett. 19 (1965) 228 H. Niewodnizanski et al., Nuclear Physics 55 (1964) 386 J. Kokame, K. Fukunaya and H. Nakamura, Phys. Lett. 14 (1965) 234 G. C. Bonazzola et aL, Phys. Rev. 140 (1965) B835 P. H. Stelson et al., Nuclear Physics 68 (1965) 97 F. R. Metzger, C. P. Swann and V. K. Rasmussen, Nuclear Physics 16 (1960) 568 E. C. Booth and K. A. Wright, Nuclear Physics 35 (1962) 472 V. J. Vanhuyse and G. J. Vanpraet, J. Phys. Rad. 21 (1960) 290 J. H. Hough and W. L. Mouton, Nuclear Physics 76 (1966) 248 L. A. Schaller and W. C. Miller, Bull. Am. Phys. Soc. 9 (1964) 666; L. J. Lidofsky, L. Mo, Y. K. Lee and C. S. Wu, Bull. Am. Phys. Soc. 10 (1965) 119 S. J. Skorka, Nucl. Instr. 28 (1964) 192 S. J. Skorka et al., Nuclear Physics 81 (1966) 370 S. J. Skorka et al., Nuclear Physics 68 (1965) 177 J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128 D. G. Alkhazov et al., ZhETF (USSR) 35 (1959) 736 K. F. Alexander and V. Bredel, Nuclear Physics 17 (1960) 153 G. C. McCallum, Phys. Rev. 123 (1961) 568 A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) K. Way et al., Nuclear Data Sheets (Nat. Research Council, Washington, D.C.) S. J. Skorka and T. W. Retz-Schmidt, Nuclear Physics 46 (1963) 225 S. A. Moszkowski, Alpha-, beta- and gamma-ray spectroscopy, Vol. 2, ed. by K. Siegbahn (North-Holland Publ. Co., Amsterdam, 1965)p. 884 S. J. Skorka, Nuclear Data (1967) to be published; references quoted in ref. 3) E. Almquist et al., Nuclear Physics 19 (1960) 1 R. Lombard, P h . D . thesis, University of Paris (1964) unpublished, quoted in ref. z)