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A rotational model for Fe57
i
"
:
i
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Nuclear Physics 24 (1961) 18~27; 1~1 North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission front the publisher
A ROTATIONAL
M O D E L F O R Fe s¢
R. D. LAWSON and 5I. H. MACFARLANE * .4 rgonne Natioual Laboratory, .4 rgot~w, Illinois t*
Received 2 J a n u a r y 1961 A b s t r a c t : A model of Fe ~'~ is examined, in which the odd neutron is considered to move in the field of m~ axially symmetric rotor, The effect of band mixing is included, For a prelate deformation with ~5from (1.15 to 0.2, reasonable agreement with experiment is obtaiued for m a n y of tim properties of the states below 1 MeV in Fe ~7. However, properties which depend on the intrinsic wave function of the ½- ground state are in serious disagreement with experiment. The predictions of the model are insensitive to reasonable changes in the m o m e n t of inertia of the rotor and in the parameters (other than the deformation) characterizing the shape of the Nilsson potential.
1. I n t r o d u c t i o n
Although recent experimentai studies have determined many of the properties of low-lying levels in Fe 57 with considerable accuracy, interpretation of these data has been mainly qualitative. A conventional shell-model calculation would at the very least demand explicit consideration of three neutrons and two proton holes - a calculation which is prohibitively tedious. Similar difficulties, encountered in the (ds) shell, have been avoided by the use of rotational models 1-4). Tile qualitative success of such calculations prompted us to examine an extreme single-particle rotational model of Fe sT. Tile underlying physical picture is that of a single nucleon moving in the detormed field of an axially symmetric rotor. The parameters of this model are the moment of inertia of the rotor and the quantities characterizing the potentia~ well. We have used a potential of the type discussed by Nilsson s) _ a (leff~rmed harmonic oscillator with spin-orbit coupling. Our calculation differs f' ov~iearlier studies of the rotational properties of light nuclei in that we do not a|~ w ourselves the freedom to fix the positions of the unperturbed bands to giw optimum agreement with experiment. Instead, we fix the parameters of t;,~e well and the rotor, and thereby determine the Hamiltonian of the model completely. All subsoquent comparisons with experiment are made in terms of the eigenvalues arm eigenfunctions of this Hamiltonian. The effects of band mixing introduced by the term (li2/2,f)I • j in the Hamiltonian are, of course, very ,mvort,mt ; ,,, since li~/2.f is now more than ten times as great as in the Now at Department of Physics, University of Rochester, Rochester, New York. • ~ Wo, k performed under the auspices of ~he U. S. Atomic Energy Commission. 18
A ROTATIONAL
MODEL
FOR Fe ~7
] t)
rare-earth region. We find in fact that the only significant parameter of our calculation is the deformation of the well; reasonable variations of the other parameters produce remarkably little change in the predictions of the model. Earlier calculations, wherein the positions of the bands were arbitrarily adjusted, suggest considerable sensitivity to the parameters of the model. This sensitivity is greatly reduced when the model is applied consistently. Satisfactory agreement is obtained with the observed spectrum and with various of the static and dynamic properties of the levels of Fe ~7 below 1 MeV. However, quantities that depend on the intrinsic wave flmction of the -~-ground state are in marked disa~eement with experiment. 2. G a l c u l a t i o n a n d R e s u l t s
The Hamiltonian of our model is G) H
2.~¢ [(I--J)2-- (I,---]~)-] @Hlntrlnsl c 1i °.
t~2
2,f [12~' ]'z-- 2Iz21 + ..... ([+]_ 2J
+I_]~) +H,ntr,ns,e,
(1)
~
where ,J~ is tile moment of inertia of the rotor, I - - j the angular momentum of the core. The Nilsson Hamiltonian, H~nmnstc ires four adjustable parameters: the oscillator strength ]/e~o, the deformation 6 (or ~]) (ref. G)), and the :
I/2-.
ZO
112-
t9
5/2" :-~
1t2 5/2,~
I P,'/;~
/
3/2"
7/23/2 W2-
5/2 t5
5/2
5/2J/2-
2 ~ _ s ~ - - - - - -/ 12 5t2...j.~J'JJ !3
5/2-
tO
7/2 " t / j -0,5
-0,2
-0.!
-6
-4
-2
1I $ 0.t *
0
2
0.~'
#
4
0.5 t / a -
-4
6
Fig. 1. E n e r g y levels in the (If, 2p) shell as a flmction of the d e f o r m a t i o n ,1. Tlfis d i a g r a m is p a r t of fig. 5 in t e l s),
In a single-particle rotational model, all nucleons but the lm:~t are placcd in pairs in the lowest available Nilsson orbits. The possibility of promotiztg such
20
R.D.
L A W S O N A N D M. It. M A C F A R L A N E
pMrs to unfilled levels is neglected, which is sensible only when there is a sizeable energy separation between the last core orbit and the first odd-particle orbit, in other words, it is not meaningful to apply a single-particle rotational m~lel close to a crossing in the Nilsson level diagram. In the case of Fc 5~, it will be seen from fig. 1 that for --2 < ~/<: 4 no such crossings occur. Thus we may reasonably hope that our single-particle calculation will provide a good first approximation to a many-particle calculation with residual forces. Difficulties with crossings in the Nilsson level diagram may already have been encountered in the (ds) shell. It is believed that beyond SW we are dealing with negative deformations 7). In Si ~9 we then have a large energy separation between the last core orbit (Nilsson level no. 7) and the first single-particle orbits (no. 8 and no. 9). Accordingly one would expect, and indeed finds *, that the positive-parity levels in Si 29 are well described by the model under consideration. On the other hand, in Sz3 the last core orbit can be either no. 8 or no. 9, which lie very close in energy and actually intersect for deformations in the region of interest. The expectations of the above "crossing" arguraents are agMn borne out in practice 8); the single-particle rotational model cannot give adequate agreelnent with experiment for Sza. The wave function of the rth level of spin i in Fe 57 is 9) ~llr ._=: * M
~ K,S
A Irc AIr
" Ko"PMKS
=,<,s ~'~ "l~s {V ~ -16z - - ~~. (1-t-R,)D~fKZKs} ,
(2)
where S distinguishes different intrinsic states of the same K and the operator R~ produces a rotation ~f 180 ° about an ~xis perpendicular to the symmetry (z) axis of the rotor. The mixing amplitudes A~i~[sare obtained by diagonalizing the Hamiltonian (l) with r~spect to the basic states ¢~s. For positive deformations, the single-particle eigenfunctions ZKs are those of Nilsson's orbits no. 20 and no. 2,6 (K -~=g 1_ ), no. 16and no. 19 (K = ~ - ) a n d n o . 15 (K------~-) (ref. s)). Si,ce we are ignoring core excitation, we do not attach any significance to the predictions of our model for levels above 1 MeV in FeSL Orbits no. 15 and no. 26 ha.,',~ only slight influence on the energies and wave functions below 1 MeV; thei~ omi~,sion would riot significantly alter the calculation. The energy matrices were diagonalized for Nilsson's choice of single-particle parameters and several wdues of ~. Two values of ~2[2J were used: 0.14 MeV (-~ the excitation energy of the first excited state in Fe 5~) and 0.11 Men. The fact that this rather sizeable variation in ~2/2~6 makes little difference to the predictions of the model indicates that the precise value chosen for the moment of inertia is not of critical importance to the calculation. The resulting spectra are strewn in fig. 2. See pp+ 665+666 of ref. ~7).
A
ROTATIONAL
MODEL
~512 ---7/2
'~" 1.0
Yc 57
21
~5/2 ~5/27/2 ~7/2
5/2~T/2
----712
¢D
----5/2
s: z
....
FOR
5/2
0.8
:t,-
--312 := ta
i t
--312
- - - - 112
~5/2
__312--312
~312
--5/2 --312
_~5/2 ~5/2
5/2 - - -
312--3t2
--112
....
3/2
-
- 5/a
:
: 1/2
3/2
1/2
0 t-
--512--512----112 +1
THEORY
+2
?;z
+5
..... 5/2
....
112 ...... 112
+2
-.3
+4
1z
THEORY
~.~:0.14
~-,~=o,It
EXPE~?iME~,|T
Fig. 2. S p e c t r u m of Fe'% The first three c o h m m s show the s p e c t r a o b t a i n e d for po~dti~ c defc~rma-. tion (r] = 1, 2 and 3) a~c..,~-/2,¢* z<~, " ,= 0.14 MeV. Tile n e x t three columns arc the resultina .,p~(.~,~ . . . . ~' when ~] = ~ 9 .~ and 4 with h=/2..¢ = 0.11 MeV. Column 7 shows the e x p e r i m e n t a l results. TABLE • S t a t i c m u l t i p o l e m o m e n t s and s t r i p p i n g w i d t h s of some of the low-lying states in I:e a7 for various values of the d e f o r m a t i o n a n d m o m e n t of inertia of the rotor. Magnetic i Magnetic i Quadrupole m o m e n t of the m o m e n t of t h e ! m o m e n t of the ........................ 4.~* "/~ *? i ground state first e x c i t e d ~1 first excited 2//~, i'4"/r~ " t in n.m, s t a t e in n.m. i s t a t e in b a r n s .......... 7-- ........................................... 7 ......................................................... Experiment 4-0.0903 a) --0.153 b) ! negz~tive la) i .~ 0.'3 11! ~ O.25 ~} ..........................
>
i I P
-:-0 39
--0.42
+0.04
2 i
+o.58
-!-0.77
+0.05
3
+0.70
+ 0.06
~- 0.02
2
+0.59
3
+0.71
-r 0.02
+0.78
--0.53
1
i
+0.05
.............
-~0.(!4
i
0.5l 45O
]
'= o . 0 0 2
}
0 o~2
120
[
O.~d2
3.32 i
I I , 1
~ 0.0o5
1 I
!-' .............................. i.................... 1. 1
3.08
;
0
t
Jl
]
4
-o.12
i
1.os
i o.o07 .....
L
In c a l c u l a t i n g matrLx elements of tile m a g n e t i c dipole o p e r a t o r s , a value of Z ] A == 6.464 was t a k e n for g~. The q u a n t i t y listed in c o l u m n 5 is a m e a s u r e of the relative s t r e n g t h of the p r o t o n groups to the t h i r d excited s t a t e (~-)* a n d t o t h e g r o u n d s t a t e d o u b l e t in Fe56(d, p)Fe~L ,f~ is the spectroscopic factor for the g r o u n d state. 8) G. W. L u d w i g and H. H. VVoodbury, Phys. Rev. 117 (19601 1286 b) S. S. H a n n a , J. Heberle, C. Littlejohn, G. J, Perlow, R, S. P r e s t o n and D. H. Vincent, P h y s . Rev. L e t t e r s 4 (1960) 177
~2
R. D. LAWSON AND M. tI. MACFARLANE TABLE 2 Reduced tra~,sition-probabilities for low-lying states in Fe 5~ B(E2; { - -* ~--)! B(E2; [[- -,. ~.~-) B(M1; { - -.- ~-) xl04Se ~:cm4 ! × 1 0 *se a c m 4 x l 0 a°e 2cm ~
B(M1;,~- -~ ½-)] × 1 0 a°e 2cm ~
Experimex t XVeisskopf estimate >
0.0013
200
l
1
0.0005
0.0003
0.0044
5.6
2
0.0020
0.0028
1.23
7.3
3
0.0047
0.0057
i.i7
19.9
0.0023
0.0026
1.41
9.0
3
0.0054
0.0052
1.22
24.1
4
0.010
0.0074
0,83
47.8
"9
zd iei
"
ti ,,~
a) A. T. G, Ferguson, M. A. Grace and J. O. Newton, Nuclear Physics 17 (1960) 9 ~'i G R, t, lshop, M. A. Grace, C. E. Johnson, A. C. Knipper, I4. R. Lemmer, J. perez y J o r b a and R. G. Scurlock, Phil. Mag. 46 (1955) 951 ~:) H. R. Lemmer, O. J. A. Segaert amd M. A. Grace, Proc. Phys. Soc. (London) A 68 (1955) 701
I',y use of standard techniques 9), the energy eigenfunctions (2), were then used to calculate various moments and transition rates. The electromagnetic properties of the core enter the calculation through the gyromagnetic ratio gR and the intrinsic quadrupole moment Q0. The results are summarized in tables 1 and 2.
3. C o m p a r i s o n w i t h E x p e r i m e n t 3,L SPECTRUM
I,~ Fc 57 :there are five known levels below 1 MeV (ref. lo)). The spins of :he three Iow~:~:~tare well established, whereas that of the 365-keV state is fixed by st~ii~t)it~,~~,, .~ data li) a.s either ~-- or ½-. Nothing is known about the spin or parity -~ d:e state at 705 keV. From fig. 2 it is evident that the calculated and observed s4~<:~ctra are in moderate agreement for either value of the moment-of-inertia *r,tram,~t~x-~ ." and ~/ ~" 3-4, A spin of ~- rather than ½- is predicted for the third excited state. The 705-keV level is not observed in stripping ¢ from Fe 56 and is not prominent in pickup r.,) from Fe 5s. These characteristics are consistent with its ¢ We wish to thank Dr, P a r r y for sending us his results prior to publication.
A ROTATIONAL
MODEL
FOR
F e b7
~}~
identification as the ~ - state predicted by our model at about 1 MeV, On the other hand, we may, in fact, be dealing with a complicated excitation of *:hecore. Our calculated energy levels tend to be more widely spread than :is found experimentally. This is reasonable, since one of the effects of the neglected core excit~tions would be to compress the lower part of the predicted spectra. In view of the additional error involved in simulating the nuclear potential by a harmonic oscillator, we conclude that agreement between the theoretical and experimental energy levels is adequate. 3.2. E2 T R A N S I T I O N R A T E S A N D Q U A D R U P O L E MOMENTS
Since Fe 57 is an odd-neutron nucleus and the electric multipole operators depend only on the proton coordinates, the static quadrupole moments and E2 matrix elements are proportional to the intrinsic quadrupole moment Q0 of the core. The multiplicative factors are simple overlap integrals involving ++le intrinsic wave function of the odd neutron, The quadrupole moment of the first excited state is given by 9)
For I = {, the factor in brackets has a value of + ~ when K = ~ and - 1 when K = ½. Thus, although Qo is positive, a negative value can be obtained for Q provided that the internal wave function is predominately K =-- ½. hi this way we obtain agreement with the negative sign deduced by Bersohn ,3) for the quadrupole moment of the { - first excited state (See table 1). For a uniformly charged spheroid, Qo is given 14) ~, in terms of the deformation parameter 6 by Qo = } Ze R0~d(l+0,17O+ . . . ) , (4) where R 0, the radius of the charge distribution, is taken in this calculation to be 1.2At × 10-la cm. Using this expression and the eigenfunctions (2), w, obtain rather good agreement with the experimental values of B (E2) for the transition from the -~- level at 136 keV to the ground and first excited stat( s (se.e table 2). Since the matrix elements of the quadrupole operator depend directly on the distortion of the core, this satisfactory description of quadrupole moments and E2 transition rates sugge:sts that a rotational model of Fe 57 is not unreasonable. 3.3. M1 T R A N S I T I O N R A T E S A N D M A G N E T I C MO~IENTS
The calculated magnetic moment of the first excited -~- state is a sensitive function of the deformation, passing through the experimental vahm for r~ between 1 and 2 and again between 3 and 4. As we have seen, the larger value of deformation gives reasonable agreement with experiment both for the spectrum and for the electric quadrupole properties of Fe~L The matrix element of the M1 transition between the second ({-) and first t See in particular eq. (V12) of ref. 1~) where in fl =-~ ( ~ / ~ t ~ ' : d ~ 1.06 ~.
~4
R, D, LAWSON AND M, H. MACFARLANE
({-) excited states of Fe 57 is overestimated by a factor of two (for ~/m 4). In rotational-model calculations in the (ds) shell, Paul has obtained about the same quMity of agreement between theory and experiment for M 1 transitions in F vl (see table 1 of ref. x), while Paul and Montague find larger discrepancies tor M1 transitions in Na °-3 (see tame 1 of ref. 4). In analyzing M1 data in the rare-earth region, the standard procedure has been to treat the intrinsic matrix dements of the magnetic-dipole operator as empirical parameters. The values of these parameters required to fit the M1 transition rates frequently differ markedly from calculated values obtained from Nilsson's wave functions. It is, ~herefore, rather hard to compare our results with those obtained for M1 transitions in the rare-earth nuclei. Kerman, in his semiempirical analysis o) of b;md mixing in W lss (a nucleus similar to Fe 5~ in that it has an odd neutron, a ground-state spin of {-, and an equilibrium deformation ~ ~ 0.2), obtains agreement to within 50 O//ofor nine M1 matrix elements (see table 4 of ref. 9)), which is rather better than we have done. We conclude that the agreement between theory and experiment for the 122-keV M1 transition in Fe 57is marginal. On the other hand, the calmflated values of the ground-state magnetic moment and the M1 transition rate between the first excited state and ground state are quite clearly in disagreement with experiment. The discrepancy in die ground-state magnetic moment is very similar to that found in W ~s3 and Os *~" (see table 7b of ref. 15)). In each case we are dealing with a ~}- oddneutron mmleus with a magnetic moment close to 0.1 nuclear magnetons. For :inch mmh:i both the rotational model ? and the shell model 1~) (even with configuration mixing) predict magnetic moments very close to the Schmidt value (0.638 nm). This disagreement indicates serious deficiencies in either the magnetic-moment operator or the ground-state nuclear wave function, or both. The fact that agreement with experiment for the 14-keV M 1 matrix element is cor~siderably worse than for the 122-keV matrix element sugge.~ts that the main trouble lies in the ground-state nuclear wave function. 3~4. S T R I P P I N G AND P I C K U P
The reaction Fe56(d, p)Fe 5~ has recently been studied~l). Strong I = 1 tr~msitions are observed to the unresolved ground-state doublet and to the third excited state at 365 keV; the excited state at 136 keV has l = 3. ;n comparing these experimental data with the predictions of our model, the :relevant quantities are the reduced widths 02. Since the spin of the target nucleus is zero, 02 for the rth lew~'l of spin 1 in Fe 67 is given by 17) ?t
: ( L r) = (:2z-',-l){Z K,S
See eq. (26) of ref. s). *t See in particular eq. (lII. 203) and p. 611 of t e l 1~).
P. R O T A T I O N A L M O D E L
FOR
Fe 57
~
where 0o2(1) is the appropriate single-particle reduced width and cre:'sis the amplitude of angular m o m e n t u m I in l,us. We do not give quantitative estimates of the relative strength of l = 1 and l = 3 transitions because too little is known about the single-particle reduced widths. However, the large calculated spectroscopic factor o5¢ of the ~-- ~.~.ate ( > 0.5 for all cases considered)is consistent with the fact that a strong 1 = 3 transition is ob~;erved. In comparing the reduced widths of the two 1 = 1 transitions, only the spectroscopic factors 5 ° need be considered since the single-particle reduced width cancels. From table 1, we see t h a t the calculated value of [4-~*/(2,~'!,-!-4..~)ii for r] ~ 4, is close to 1, in fair agreement with the observed wdue of 0.6. The caletdated contribution of the gromld state to the combined 1 =: 1 reduced width of the ground-state doublet is negligible, amounting to 1,.~;~than 1 °/o of the single-particle reduced width. This is in direct contradiction to tile fact that an l = 1 transition to the Fe ~Gground state is seen quite cle,arly ~8) in Fe57(d, t)Fe ~G. The strength of the transition in question indicates, very rougly, that ttle reduced width is about a quarter of the single-particle value, This discrepancy in the ground-state reduced width, involving a factor of between 20 and ,'30, is further clear indication that the grour~d-sta~:e wave function of our model is seriously defective. 4. D i s c u s s i o n Within an oscillator shell, the Nilsson parameters. For the' /If, 2p) shell we m a y and the single-particle level separations namely, E(2p,})--E(2p~) and E(lf~)--E(2p~)
potential is characterized by thr,:.e take these to b,- the &'formation d in the limit of zero deform, atio:~, = A(p) --:- A(fp).
So far we have followed Nilsson's original calculation, taking A(p) == 1.6 MeV and el (fp) = 0.8 MeV. To test the sensitivity of the model to changes in the parameters characterizing the single-particle potential, we repeated the calculation using A (p) = 1.,5 MeV, ./__1(fp) :-- 0..5 MeV and ]i~/2J = 0, 14. ]"he spectra for r/-= 2 and 71 == 3 were not significantly altered from those shown in fig. 2. Furthermore, the overlap between correspond.ing eigenvalues (same value of ,rj) from the two sets of single-particle parameters was in each case greater t h a n 99 %. We conclude that the predictions of the model are insensitive to reasonable changes in the single-particle parameters, ahvays remaining within the harmonic-oscillator approximation. It does not seem possible to obtain agreement with experiment for a singleparticle model with ne~,~;ttive deformation. For ~1 . . . . 2, a ~ ground state is
~f~
R. D, LAWSON AND M. H. MACFARLANE
obtained, the next level lying at about 500 keV excitation. At , / = --4 (see fig. 1) we encounter a crossing in the Nilsson diagram involving orbits no. 15 and no. 16 so that, as discussed earlier, an extreme single-particle model cannot be used. Finally, for ~/-- -- 6, we obtain a { ground state with tee lowest { state lying about 500 keV too high. Further, the second } state is predicted to show no stripping from Fe se, in disagreement with experiment. Vervier and Bartholomew xg) have discussed the levels of Fe byin a qualitative fashion on the basis of a rotational model. Assigning K -----} to the first excited state, they are led by the negative sign of the observed quadrupole moment to select an oblate deformation, z/~-, --3. As mentioned above, orbits no. 1,5 and no, 16 are then close to an intersection. Although Vervier and Bartholom,.~w consider the promot,.'on of pairs of particles among these levels (and level no. 17 which is dose by), they neglect any residual pairing forces. In addition, the effects of band mixing, which we have seen to be quite large, are not considered. Since the negative sign of the measured quadrupole moment can be obtained for v , sitive deformation, the above argument for negative deformation is not conclu.dve. i n principle, the equilibrium deformation of a nucleus can be determined by minimizing the total energy as a function of deformation. Calculations in the (ds) shell 7) predict a sharp change from prolate to oblate deformations around A = 26; such a change seems to be observed, but at A = 28-29. Thus we expect a change of sign in the deformation around the middle of the (fp) shell, probably somewhere between Fe and Ni. However, calculations of the type under consideration are not sufficiently reliable to locate it exactly. The present model might also be expected to be applicable to Ni 59, which has the same (odd) number of neutrons as FeSL However, the ground-state spin of Ni s9 is {, which we cannot obtain on the single-particle rotational model without proceeding to rather large negative deformations (l*/[ ~ 6). Since a change of sign of ~ between Fe 5' and Ni s9 would not be unexpected, it is possible that a reasonable description of Ni 59 could be obtained at z / ~ --4 with a threebody rotational model including residual two-body ~nteractions.
5. Summary and Conclusions As a model of Fc a~, we consider a neutron in the deformed field of an axially ~symmetric rotor. Full account is taken of the mixing of rotational bands. For prolate shapes ~corresponding to ~ between 0.15 and 0.2, a moderately accurate description is thereby obtained of many of the known properties of levels below 1 MeV in Fe 5~. Serious discrepancies are, however, encountered for quantities which depend on the intrinsic wave function of the { - ground state. The predictions of the model are remarkably insensitive to reasonable variatim,s in the mament of inertia of the rotor and in the parameters (other than
A ROTATIONAL
MODEL
FOR
F e 57
27
the deformation) characterizing the deformed (harmonic-oscillator) potential. This significant fact is obscured if, instead of using the full rotational Hamiltonian, the relative positions of the pure bands are arbitrarily adju~;t~;d to fit the observed spectrum. Since agreement with experiment for Fe 57 is not fully satisfactory and, moreover, cannot be significantly improved by reasonable adjustments of the parameters of the model, the single-particle rotational picture must be refined. We might consider a many-particle rotational model, including residual two-body forces between the extra-core nucleons. Such a calculation wouht bc similar in spirit and in degree of complication to conventional shell-~[iodel studies. Alternatively, we might carry the single-particle approach one step further, replac,_'ng the axially symmetric rotor by something better able to describe the Fe 6~ core. One such model is that of a neutrort in the field of an asymmetric ~'otor. References 1) E. B. Paul, Phil. Mag. 15 (1957) 311 2) G. R~kavy, Nuclear Physics 4 (1957) 375 3) A. E. Litherland, H. McManus, E. B. Paul, D. A. Bromley and H. E. Gore, C~m. J. Ph3~;. 36 (1958) 378 4) E. ]3. Paul and J. H. Montague, Nuclear Physics 8 (1958) 61 5) S. G. Nilsso:~, Mat. Fys. Medd. Dan. Vid. Sclsk. 29, No. 16 f1955) 6) A. K. Kerman, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 15 (1956} 7) Proc. Intern. Conf. on Nuclear Structure, Kingston, Sept. i960 (North-tloll~md Publishing Co., Amsterdam, 1960) p. 441 8) R. Bhatt, to be published 9) A. K. Kerman, in Nuclear Reactions, ed. by P. M. Endt and M. Derncur (North.-Hollz~.,.~d Publishing Co., Amste.'dam, 1959) vol. I, Ch. X) 10) Nuclear Data Sheets 11) A. W. Dalton, G. Parry, H. D. Scott and S. Swierszczewski, to be published 12) B. Zeidman, private communication 13) R. ]3ershon, Phys. Rev. Letters 4 (1960) 609 14) Alder, Bohr, Huus, Mottelsou and \Vinther, Revs. Mod. Phys. 28, (1956) 432 15) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1, No. 8 (1959) 16) H. Noya, A. Arima and H. Horie, Prog. Theor. Phys. (Japa.n) Supplement 8 (1958) 33 17) M. H. Macfarlane and J. B. French, Revs. Mod. Phys. 32 (1960) 567 18) P,. Zeidman, J. I,. Yntema and B. J. R~Lz, Phys. Rev. 120 (19Ii0) 1723 19) J. F. Vervier and G. A. Bartholomew, Proc. ln*ern. Conf. on Nuclc~,r Structure, Kingston, Sept. 1960 (North-Holland Publishing Co., Amsterdam, 1969) p. 650
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i
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Nuclear Physics 24 (1961) 18~27; 1~1 North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission front the publisher
A ROTATIONAL
M O D E L F O R Fe s¢
R. D. LAWSON and 5I. H. MACFARLANE * .4 rgonne Natioual Laboratory, .4 rgot~w, Illinois t*
Received 2 J a n u a r y 1961 A b s t r a c t : A model of Fe ~'~ is examined, in which the odd neutron is considered to move in the field of m~ axially symmetric rotor, The effect of band mixing is included, For a prelate deformation with ~5from (1.15 to 0.2, reasonable agreement with experiment is obtaiued for m a n y of tim properties of the states below 1 MeV in Fe ~7. However, properties which depend on the intrinsic wave function of the ½- ground state are in serious disagreement with experiment. The predictions of the model are insensitive to reasonable changes in the m o m e n t of inertia of the rotor and in the parameters (other than the deformation) characterizing the shape of the Nilsson potential.
1. I n t r o d u c t i o n
Although recent experimentai studies have determined many of the properties of low-lying levels in Fe 57 with considerable accuracy, interpretation of these data has been mainly qualitative. A conventional shell-model calculation would at the very least demand explicit consideration of three neutrons and two proton holes - a calculation which is prohibitively tedious. Similar difficulties, encountered in the (ds) shell, have been avoided by the use of rotational models 1-4). Tile qualitative success of such calculations prompted us to examine an extreme single-particle rotational model of Fe sT. Tile underlying physical picture is that of a single nucleon moving in the detormed field of an axially symmetric rotor. The parameters of this model are the moment of inertia of the rotor and the quantities characterizing the potentia~ well. We have used a potential of the type discussed by Nilsson s) _ a (leff~rmed harmonic oscillator with spin-orbit coupling. Our calculation differs f' ov~iearlier studies of the rotational properties of light nuclei in that we do not a|~ w ourselves the freedom to fix the positions of the unperturbed bands to giw optimum agreement with experiment. Instead, we fix the parameters of t;,~e well and the rotor, and thereby determine the Hamiltonian of the model completely. All subsoquent comparisons with experiment are made in terms of the eigenvalues arm eigenfunctions of this Hamiltonian. The effects of band mixing introduced by the term (li2/2,f)I • j in the Hamiltonian are, of course, very ,mvort,mt ; ,,, since li~/2.f is now more than ten times as great as in the Now at Department of Physics, University of Rochester, Rochester, New York. • ~ Wo, k performed under the auspices of ~he U. S. Atomic Energy Commission. 18
A ROTATIONAL
MODEL
FOR Fe ~7
] t)
rare-earth region. We find in fact that the only significant parameter of our calculation is the deformation of the well; reasonable variations of the other parameters produce remarkably little change in the predictions of the model. Earlier calculations, wherein the positions of the bands were arbitrarily adjusted, suggest considerable sensitivity to the parameters of the model. This sensitivity is greatly reduced when the model is applied consistently. Satisfactory agreement is obtained with the observed spectrum and with various of the static and dynamic properties of the levels of Fe ~7 below 1 MeV. However, quantities that depend on the intrinsic wave flmction of the -~-ground state are in marked disa~eement with experiment. 2. G a l c u l a t i o n a n d R e s u l t s
The Hamiltonian of our model is G) H
2.~¢ [(I--J)2-- (I,---]~)-] @Hlntrlnsl c 1i °.
t~2
2,f [12~' ]'z-- 2Iz21 + ..... ([+]_ 2J
+I_]~) +H,ntr,ns,e,
(1)
~
where ,J~ is tile moment of inertia of the rotor, I - - j the angular momentum of the core. The Nilsson Hamiltonian, H~nmnstc ires four adjustable parameters: the oscillator strength ]/e~o, the deformation 6 (or ~]) (ref. G)), and the :
I/2-.
ZO
112-
t9
5/2" :-~
1t2 5/2,~
I P,'/;~
/
3/2"
7/23/2 W2-
5/2 t5
5/2
5/2J/2-
2 ~ _ s ~ - - - - - -/ 12 5t2...j.~J'JJ !3
5/2-
tO
7/2 " t / j -0,5
-0,2
-0.!
-6
-4
-2
1I $ 0.t *
0
2
0.~'
#
4
0.5 t / a -
-4
6
Fig. 1. E n e r g y levels in the (If, 2p) shell as a flmction of the d e f o r m a t i o n ,1. Tlfis d i a g r a m is p a r t of fig. 5 in t e l s),
In a single-particle rotational model, all nucleons but the lm:~t are placcd in pairs in the lowest available Nilsson orbits. The possibility of promotiztg such
20
R.D.
L A W S O N A N D M. It. M A C F A R L A N E
pMrs to unfilled levels is neglected, which is sensible only when there is a sizeable energy separation between the last core orbit and the first odd-particle orbit, in other words, it is not meaningful to apply a single-particle rotational m~lel close to a crossing in the Nilsson level diagram. In the case of Fc 5~, it will be seen from fig. 1 that for --2 < ~/<: 4 no such crossings occur. Thus we may reasonably hope that our single-particle calculation will provide a good first approximation to a many-particle calculation with residual forces. Difficulties with crossings in the Nilsson level diagram may already have been encountered in the (ds) shell. It is believed that beyond SW we are dealing with negative deformations 7). In Si ~9 we then have a large energy separation between the last core orbit (Nilsson level no. 7) and the first single-particle orbits (no. 8 and no. 9). Accordingly one would expect, and indeed finds *, that the positive-parity levels in Si 29 are well described by the model under consideration. On the other hand, in Sz3 the last core orbit can be either no. 8 or no. 9, which lie very close in energy and actually intersect for deformations in the region of interest. The expectations of the above "crossing" arguraents are agMn borne out in practice 8); the single-particle rotational model cannot give adequate agreelnent with experiment for Sza. The wave function of the rth level of spin i in Fe 57 is 9) ~llr ._=: * M
~ K,S
A Irc AIr
" Ko"PMKS
=,<,s ~'~ "l~s {V ~ -16z - - ~~. (1-t-R,)D~fKZKs} ,
(2)
where S distinguishes different intrinsic states of the same K and the operator R~ produces a rotation ~f 180 ° about an ~xis perpendicular to the symmetry (z) axis of the rotor. The mixing amplitudes A~i~[sare obtained by diagonalizing the Hamiltonian (l) with r~spect to the basic states ¢~s. For positive deformations, the single-particle eigenfunctions ZKs are those of Nilsson's orbits no. 20 and no. 2,6 (K -~=g 1_ ), no. 16and no. 19 (K = ~ - ) a n d n o . 15 (K------~-) (ref. s)). Si,ce we are ignoring core excitation, we do not attach any significance to the predictions of our model for levels above 1 MeV in FeSL Orbits no. 15 and no. 26 ha.,',~ only slight influence on the energies and wave functions below 1 MeV; thei~ omi~,sion would riot significantly alter the calculation. The energy matrices were diagonalized for Nilsson's choice of single-particle parameters and several wdues of ~. Two values of ~2[2J were used: 0.14 MeV (-~ the excitation energy of the first excited state in Fe 5~) and 0.11 Men. The fact that this rather sizeable variation in ~2/2~6 makes little difference to the predictions of the model indicates that the precise value chosen for the moment of inertia is not of critical importance to the calculation. The resulting spectra are strewn in fig. 2. See pp+ 665+666 of ref. ~7).
A
ROTATIONAL
MODEL
~512 ---7/2
'~" 1.0
Yc 57
21
~5/2 ~5/27/2 ~7/2
5/2~T/2
----712
¢D
----5/2
s: z
....
FOR
5/2
0.8
:t,-
--312 := ta
i t
--312
- - - - 112
~5/2
__312--312
~312
--5/2 --312
_~5/2 ~5/2
5/2 - - -
312--3t2
--112
....
3/2
-
- 5/a
:
: 1/2
3/2
1/2
0 t-
--512--512----112 +1
THEORY
+2
?;z
+5
..... 5/2
....
112 ...... 112
+2
-.3
+4
1z
THEORY
~.~:0.14
~-,~=o,It
EXPE~?iME~,|T
Fig. 2. S p e c t r u m of Fe'% The first three c o h m m s show the s p e c t r a o b t a i n e d for po~dti~ c defc~rma-. tion (r] = 1, 2 and 3) a~c..,~-/2,¢* z<~, " ,= 0.14 MeV. Tile n e x t three columns arc the resultina .,p~(.~,~ . . . . ~' when ~] = ~ 9 .~ and 4 with h=/2..¢ = 0.11 MeV. Column 7 shows the e x p e r i m e n t a l results. TABLE • S t a t i c m u l t i p o l e m o m e n t s and s t r i p p i n g w i d t h s of some of the low-lying states in I:e a7 for various values of the d e f o r m a t i o n a n d m o m e n t of inertia of the rotor. Magnetic i Magnetic i Quadrupole m o m e n t of the m o m e n t of t h e ! m o m e n t of the ........................ 4.~* "/~ *? i ground state first e x c i t e d ~1 first excited 2//~, i'4"/r~ " t in n.m, s t a t e in n.m. i s t a t e in b a r n s .......... 7-- ........................................... 7 ......................................................... Experiment 4-0.0903 a) --0.153 b) ! negz~tive la) i .~ 0.'3 11! ~ O.25 ~} ..........................
>
i I P
-:-0 39
--0.42
+0.04
2 i
+o.58
-!-0.77
+0.05
3
+0.70
+ 0.06
~- 0.02
2
+0.59
3
+0.71
-r 0.02
+0.78
--0.53
1
i
+0.05
.............
-~0.(!4
i
0.5l 45O
]
'= o . 0 0 2
}
0 o~2
120
[
O.~d2
3.32 i
I I , 1
~ 0.0o5
1 I
!-' .............................. i.................... 1. 1
3.08
;
0
t
Jl
]
4
-o.12
i
1.os
i o.o07 .....
L
In c a l c u l a t i n g matrLx elements of tile m a g n e t i c dipole o p e r a t o r s , a value of Z ] A == 6.464 was t a k e n for g~. The q u a n t i t y listed in c o l u m n 5 is a m e a s u r e of the relative s t r e n g t h of the p r o t o n groups to the t h i r d excited s t a t e (~-)* a n d t o t h e g r o u n d s t a t e d o u b l e t in Fe56(d, p)Fe~L ,f~ is the spectroscopic factor for the g r o u n d state. 8) G. W. L u d w i g and H. H. VVoodbury, Phys. Rev. 117 (19601 1286 b) S. S. H a n n a , J. Heberle, C. Littlejohn, G. J, Perlow, R, S. P r e s t o n and D. H. Vincent, P h y s . Rev. L e t t e r s 4 (1960) 177
~2
R. D. LAWSON AND M. tI. MACFARLANE TABLE 2 Reduced tra~,sition-probabilities for low-lying states in Fe 5~ B(E2; { - -* ~--)! B(E2; [[- -,. ~.~-) B(M1; { - -.- ~-) xl04Se ~:cm4 ! × 1 0 *se a c m 4 x l 0 a°e 2cm ~
B(M1;,~- -~ ½-)] × 1 0 a°e 2cm ~
Experimex t XVeisskopf estimate >
0.0013
200
l
1
0.0005
0.0003
0.0044
5.6
2
0.0020
0.0028
1.23
7.3
3
0.0047
0.0057
i.i7
19.9
0.0023
0.0026
1.41
9.0
3
0.0054
0.0052
1.22
24.1
4
0.010
0.0074
0,83
47.8
"9
zd iei
"
ti ,,~
a) A. T. G, Ferguson, M. A. Grace and J. O. Newton, Nuclear Physics 17 (1960) 9 ~'i G R, t, lshop, M. A. Grace, C. E. Johnson, A. C. Knipper, I4. R. Lemmer, J. perez y J o r b a and R. G. Scurlock, Phil. Mag. 46 (1955) 951 ~:) H. R. Lemmer, O. J. A. Segaert amd M. A. Grace, Proc. Phys. Soc. (London) A 68 (1955) 701
I',y use of standard techniques 9), the energy eigenfunctions (2), were then used to calculate various moments and transition rates. The electromagnetic properties of the core enter the calculation through the gyromagnetic ratio gR and the intrinsic quadrupole moment Q0. The results are summarized in tables 1 and 2.
3. C o m p a r i s o n w i t h E x p e r i m e n t 3,L SPECTRUM
I,~ Fc 57 :there are five known levels below 1 MeV (ref. lo)). The spins of :he three Iow~:~:~tare well established, whereas that of the 365-keV state is fixed by st~ii~t)it~,~~,, .~ data li) a.s either ~-- or ½-. Nothing is known about the spin or parity -~ d:e state at 705 keV. From fig. 2 it is evident that the calculated and observed s4~<:~ctra are in moderate agreement for either value of the moment-of-inertia *r,tram,~t~x-~ ." and ~/ ~" 3-4, A spin of ~- rather than ½- is predicted for the third excited state. The 705-keV level is not observed in stripping ¢ from Fe 56 and is not prominent in pickup r.,) from Fe 5s. These characteristics are consistent with its ¢ We wish to thank Dr, P a r r y for sending us his results prior to publication.
A ROTATIONAL
MODEL
FOR
F e b7
~}~
identification as the ~ - state predicted by our model at about 1 MeV, On the other hand, we may, in fact, be dealing with a complicated excitation of *:hecore. Our calculated energy levels tend to be more widely spread than :is found experimentally. This is reasonable, since one of the effects of the neglected core excit~tions would be to compress the lower part of the predicted spectra. In view of the additional error involved in simulating the nuclear potential by a harmonic oscillator, we conclude that agreement between the theoretical and experimental energy levels is adequate. 3.2. E2 T R A N S I T I O N R A T E S A N D Q U A D R U P O L E MOMENTS
Since Fe 57 is an odd-neutron nucleus and the electric multipole operators depend only on the proton coordinates, the static quadrupole moments and E2 matrix elements are proportional to the intrinsic quadrupole moment Q0 of the core. The multiplicative factors are simple overlap integrals involving ++le intrinsic wave function of the odd neutron, The quadrupole moment of the first excited state is given by 9)
For I = {, the factor in brackets has a value of + ~ when K = ~ and - 1 when K = ½. Thus, although Qo is positive, a negative value can be obtained for Q provided that the internal wave function is predominately K =-- ½. hi this way we obtain agreement with the negative sign deduced by Bersohn ,3) for the quadrupole moment of the { - first excited state (See table 1). For a uniformly charged spheroid, Qo is given 14) ~, in terms of the deformation parameter 6 by Qo = } Ze R0~d(l+0,17O+ . . . ) , (4) where R 0, the radius of the charge distribution, is taken in this calculation to be 1.2At × 10-la cm. Using this expression and the eigenfunctions (2), w, obtain rather good agreement with the experimental values of B (E2) for the transition from the -~- level at 136 keV to the ground and first excited stat( s (se.e table 2). Since the matrix elements of the quadrupole operator depend directly on the distortion of the core, this satisfactory description of quadrupole moments and E2 transition rates sugge:sts that a rotational model of Fe 57 is not unreasonable. 3.3. M1 T R A N S I T I O N R A T E S A N D M A G N E T I C MO~IENTS
The calculated magnetic moment of the first excited -~- state is a sensitive function of the deformation, passing through the experimental vahm for r~ between 1 and 2 and again between 3 and 4. As we have seen, the larger value of deformation gives reasonable agreement with experiment both for the spectrum and for the electric quadrupole properties of Fe~L The matrix element of the M1 transition between the second ({-) and first t See in particular eq. (V12) of ref. 1~) where in fl =-~ ( ~ / ~ t ~ ' : d ~ 1.06 ~.
~4
R, D, LAWSON AND M, H. MACFARLANE
({-) excited states of Fe 57 is overestimated by a factor of two (for ~/m 4). In rotational-model calculations in the (ds) shell, Paul has obtained about the same quMity of agreement between theory and experiment for M 1 transitions in F vl (see table 1 of ref. x), while Paul and Montague find larger discrepancies tor M1 transitions in Na °-3 (see tame 1 of ref. 4). In analyzing M1 data in the rare-earth region, the standard procedure has been to treat the intrinsic matrix dements of the magnetic-dipole operator as empirical parameters. The values of these parameters required to fit the M1 transition rates frequently differ markedly from calculated values obtained from Nilsson's wave functions. It is, ~herefore, rather hard to compare our results with those obtained for M1 transitions in the rare-earth nuclei. Kerman, in his semiempirical analysis o) of b;md mixing in W lss (a nucleus similar to Fe 5~ in that it has an odd neutron, a ground-state spin of {-, and an equilibrium deformation ~ ~ 0.2), obtains agreement to within 50 O//ofor nine M1 matrix elements (see table 4 of ref. 9)), which is rather better than we have done. We conclude that the agreement between theory and experiment for the 122-keV M1 transition in Fe 57is marginal. On the other hand, the calmflated values of the ground-state magnetic moment and the M1 transition rate between the first excited state and ground state are quite clearly in disagreement with experiment. The discrepancy in die ground-state magnetic moment is very similar to that found in W ~s3 and Os *~" (see table 7b of ref. 15)). In each case we are dealing with a ~}- oddneutron mmleus with a magnetic moment close to 0.1 nuclear magnetons. For :inch mmh:i both the rotational model ? and the shell model 1~) (even with configuration mixing) predict magnetic moments very close to the Schmidt value (0.638 nm). This disagreement indicates serious deficiencies in either the magnetic-moment operator or the ground-state nuclear wave function, or both. The fact that agreement with experiment for the 14-keV M 1 matrix element is cor~siderably worse than for the 122-keV matrix element sugge.~ts that the main trouble lies in the ground-state nuclear wave function. 3~4. S T R I P P I N G AND P I C K U P
The reaction Fe56(d, p)Fe 5~ has recently been studied~l). Strong I = 1 tr~msitions are observed to the unresolved ground-state doublet and to the third excited state at 365 keV; the excited state at 136 keV has l = 3. ;n comparing these experimental data with the predictions of our model, the :relevant quantities are the reduced widths 02. Since the spin of the target nucleus is zero, 02 for the rth lew~'l of spin 1 in Fe 67 is given by 17) ?t
: ( L r) = (:2z-',-l){Z K,S
See eq. (26) of ref. s). *t See in particular eq. (lII. 203) and p. 611 of t e l 1~).
P. R O T A T I O N A L M O D E L
FOR
Fe 57
~
where 0o2(1) is the appropriate single-particle reduced width and cre:'sis the amplitude of angular m o m e n t u m I in l,us. We do not give quantitative estimates of the relative strength of l = 1 and l = 3 transitions because too little is known about the single-particle reduced widths. However, the large calculated spectroscopic factor o5¢ of the ~-- ~.~.ate ( > 0.5 for all cases considered)is consistent with the fact that a strong 1 = 3 transition is ob~;erved. In comparing the reduced widths of the two 1 = 1 transitions, only the spectroscopic factors 5 ° need be considered since the single-particle reduced width cancels. From table 1, we see t h a t the calculated value of [4-~*/(2,~'!,-!-4..~)ii for r] ~ 4, is close to 1, in fair agreement with the observed wdue of 0.6. The caletdated contribution of the gromld state to the combined 1 =: 1 reduced width of the ground-state doublet is negligible, amounting to 1,.~;~than 1 °/o of the single-particle reduced width. This is in direct contradiction to tile fact that an l = 1 transition to the Fe ~Gground state is seen quite cle,arly ~8) in Fe57(d, t)Fe ~G. The strength of the transition in question indicates, very rougly, that ttle reduced width is about a quarter of the single-particle value, This discrepancy in the ground-state reduced width, involving a factor of between 20 and ,'30, is further clear indication that the grour~d-sta~:e wave function of our model is seriously defective. 4. D i s c u s s i o n Within an oscillator shell, the Nilsson parameters. For the' /If, 2p) shell we m a y and the single-particle level separations namely, E(2p,})--E(2p~) and E(lf~)--E(2p~)
potential is characterized by thr,:.e take these to b,- the &'formation d in the limit of zero deform, atio:~, = A(p) --:- A(fp).
So far we have followed Nilsson's original calculation, taking A(p) == 1.6 MeV and el (fp) = 0.8 MeV. To test the sensitivity of the model to changes in the parameters characterizing the single-particle potential, we repeated the calculation using A (p) = 1.,5 MeV, ./__1(fp) :-- 0..5 MeV and ]i~/2J = 0, 14. ]"he spectra for r/-= 2 and 71 == 3 were not significantly altered from those shown in fig. 2. Furthermore, the overlap between correspond.ing eigenvalues (same value of ,rj) from the two sets of single-particle parameters was in each case greater t h a n 99 %. We conclude that the predictions of the model are insensitive to reasonable changes in the single-particle parameters, ahvays remaining within the harmonic-oscillator approximation. It does not seem possible to obtain agreement with experiment for a singleparticle model with ne~,~;ttive deformation. For ~1 . . . . 2, a ~ ground state is
~f~
R. D, LAWSON AND M. H. MACFARLANE
obtained, the next level lying at about 500 keV excitation. At , / = --4 (see fig. 1) we encounter a crossing in the Nilsson diagram involving orbits no. 15 and no. 16 so that, as discussed earlier, an extreme single-particle model cannot be used. Finally, for ~/-- -- 6, we obtain a { ground state with tee lowest { state lying about 500 keV too high. Further, the second } state is predicted to show no stripping from Fe se, in disagreement with experiment. Vervier and Bartholomew xg) have discussed the levels of Fe byin a qualitative fashion on the basis of a rotational model. Assigning K -----} to the first excited state, they are led by the negative sign of the observed quadrupole moment to select an oblate deformation, z/~-, --3. As mentioned above, orbits no. 1,5 and no, 16 are then close to an intersection. Although Vervier and Bartholom,.~w consider the promot,.'on of pairs of particles among these levels (and level no. 17 which is dose by), they neglect any residual pairing forces. In addition, the effects of band mixing, which we have seen to be quite large, are not considered. Since the negative sign of the measured quadrupole moment can be obtained for v , sitive deformation, the above argument for negative deformation is not conclu.dve. i n principle, the equilibrium deformation of a nucleus can be determined by minimizing the total energy as a function of deformation. Calculations in the (ds) shell 7) predict a sharp change from prolate to oblate deformations around A = 26; such a change seems to be observed, but at A = 28-29. Thus we expect a change of sign in the deformation around the middle of the (fp) shell, probably somewhere between Fe and Ni. However, calculations of the type under consideration are not sufficiently reliable to locate it exactly. The present model might also be expected to be applicable to Ni 59, which has the same (odd) number of neutrons as FeSL However, the ground-state spin of Ni s9 is {, which we cannot obtain on the single-particle rotational model without proceeding to rather large negative deformations (l*/[ ~ 6). Since a change of sign of ~ between Fe 5' and Ni s9 would not be unexpected, it is possible that a reasonable description of Ni 59 could be obtained at z / ~ --4 with a threebody rotational model including residual two-body ~nteractions.
5. Summary and Conclusions As a model of Fc a~, we consider a neutron in the deformed field of an axially ~symmetric rotor. Full account is taken of the mixing of rotational bands. For prolate shapes ~corresponding to ~ between 0.15 and 0.2, a moderately accurate description is thereby obtained of many of the known properties of levels below 1 MeV in Fe 5~. Serious discrepancies are, however, encountered for quantities which depend on the intrinsic wave function of the { - ground state. The predictions of the model are remarkably insensitive to reasonable variatim,s in the mament of inertia of the rotor and in the parameters (other than
A ROTATIONAL
MODEL
FOR
F e 57
27
the deformation) characterizing the deformed (harmonic-oscillator) potential. This significant fact is obscured if, instead of using the full rotational Hamiltonian, the relative positions of the pure bands are arbitrarily adju~;t~;d to fit the observed spectrum. Since agreement with experiment for Fe 57 is not fully satisfactory and, moreover, cannot be significantly improved by reasonable adjustments of the parameters of the model, the single-particle rotational picture must be refined. We might consider a many-particle rotational model, including residual two-body forces between the extra-core nucleons. Such a calculation wouht bc similar in spirit and in degree of complication to conventional shell-~[iodel studies. Alternatively, we might carry the single-particle approach one step further, replac,_'ng the axially symmetric rotor by something better able to describe the Fe 6~ core. One such model is that of a neutrort in the field of an asymmetric ~'otor. References 1) E. B. Paul, Phil. Mag. 15 (1957) 311 2) G. R~kavy, Nuclear Physics 4 (1957) 375 3) A. E. Litherland, H. McManus, E. B. Paul, D. A. Bromley and H. E. Gore, C~m. J. Ph3~;. 36 (1958) 378 4) E. ]3. Paul and J. H. Montague, Nuclear Physics 8 (1958) 61 5) S. G. Nilsso:~, Mat. Fys. Medd. Dan. Vid. Sclsk. 29, No. 16 f1955) 6) A. K. Kerman, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 15 (1956} 7) Proc. Intern. Conf. on Nuclear Structure, Kingston, Sept. i960 (North-tloll~md Publishing Co., Amsterdam, 1960) p. 441 8) R. Bhatt, to be published 9) A. K. Kerman, in Nuclear Reactions, ed. by P. M. Endt and M. Derncur (North.-Hollz~.,.~d Publishing Co., Amste.'dam, 1959) vol. I, Ch. X) 10) Nuclear Data Sheets 11) A. W. Dalton, G. Parry, H. D. Scott and S. Swierszczewski, to be published 12) B. Zeidman, private communication 13) R. ]3ershon, Phys. Rev. Letters 4 (1960) 609 14) Alder, Bohr, Huus, Mottelsou and \Vinther, Revs. Mod. Phys. 28, (1956) 432 15) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1, No. 8 (1959) 16) H. Noya, A. Arima and H. Horie, Prog. Theor. Phys. (Japa.n) Supplement 8 (1958) 33 17) M. H. Macfarlane and J. B. French, Revs. Mod. Phys. 32 (1960) 567 18) P,. Zeidman, J. I,. Yntema and B. J. R~Lz, Phys. Rev. 120 (19Ii0) 1723 19) J. F. Vervier and G. A. Bartholomew, Proc. ln*ern. Conf. on Nuclc~,r Structure, Kingston, Sept. 1960 (North-Holland Publishing Co., Amsterdam, 1969) p. 650