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On the parity mixing in single-nucleon orbitals
I
3.C
[ I
Nuclear Physics 80 (1966) 353--366; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
O N THE PARITY MIXING I N S I N G L E - N U C L E O N
ORBITALS
J.-P. AMIET Institut fiir Theoretische Physik der Universitiit, Heidelberg, Germany and P. HUGUENIN Institut de Physique de r Universitd, Neuehdtel, Switzerland t Received 15 October 1965 Abstract: A mixture of parity in single-particle orbitals leads to specific properties of the nuclear wave function. These properties are reviewed here. The wave function is obtained by projecting out a part of definite parity from a product function. Formulae are given for some quantities, such as the magnetic dipole moment, the magnetic dipole transitions, etc., which appreciably deviate from the values predicted by the usual shell model. It is shown why a parity mixing is not incompatible with the fact that definite/-values are measured in stripping reactions. The influence of this parity mixing on nuclear spectra is studied. In the case of odd nuclei, it is similar to that of the spin-orbit coupling if the spectrum of the preceding even nucleus satisfies a simple condition.
1. Introduction This p a p e r is i n t e n d e d to s u p p l e m e n t an earlier one dealing with a p a r t i c u l a r extension o f the usual H a r t r e e - F o c k ( H F ) a p p r o x i m a t i o n s in n u c l e a r t h e o r y 1). It is shown in ref. 1) t h a t single-nucleon o r b i t a l s w i t h o u t p a r i t y c o u l d be well suited for solving the p r o b l e m o f self-consistent fields in the presence o f a strong t e n s o r force. I t seems w o r t h w h i l e to r e m a r k at this p o i n t t h a t the p e r t i n e n t a r g u m e n t o f Villars 2) a b o u t the c o n s e r v a t i o n o f p a r i t y in single-particle o r b i t a l s loses its weight if the forces are spin d e p e n d e n t . I n this case, one can b r e a k up the s y m m e t r y o f p a r i t y in the H F solutions w i t h o u t losing the spherical symmetry. A direct calculation o f the p a r i t y mixture in single-nucleon o r b i t a l s b y solving the H F e q u a t i o n s is clearly p r e m a t u r e . O u r l a c k o f k n o w l e d g e c o n c e r n i n g effective shortrange forces w o u l d m a k e the significance o f an a t t e m p t in this direction m o r e o r less d o u b t f u l . This does n o t i m p l y t h a t the a p p r o x i m a t e t r e a t m e n t p r o p o s e d in the last section o f ref. 1) is meaningless; we c a n n o t p r o v e t h a t it is a true a p p r o x i m a t i o n to H F , b u t it is at least a m e a n i n g f u l p a r a m e t r i z a t i o n whose f o r m is d i c t a t e d b y H F . T h e o n l y a d a p t a b l e p a r a m e t e r which is used (except the coefficients o f the spini n d e p e n d e n t c e n t r a l p o t e n t i a l ) is the effective c o u p l i n g c o n s t a n t in f r o n t o f the onep i o n - e x c h a n g e p o t e n t i a l in which the b-function 3) has been included. t This work has been partly supported by the "Fonds National Suisse de la Recherche Scientifique." 353
354
J.-P.
AMIET AND
P.
HUGUENIN
A wave function for the nucleus constructed from orbitals without parity has many properties that we shall discuss here, in relation with the deviations from the usual shell model that they imply for many matrix elements. In the main part of this paper we shall treat the parity-mixing coefficients in the nuclear wave function as free parameters. We first propose to look for those nuclear properties which are easily explainable in this model (without taking the mixing parameters beyond a reasonable range) and for which the usual shell model fails to give satisfactory predictions. It is hoped that in the future one can test the consistency of the model by comparing different quantities which depend on the same mixing parameters. Secondly, we shall show why a parity mixture of the single-particle orbitals cannot be detected in direct reactions at the present stage of the theory of nuclear reactions. Finally, we intend to give various matrix elements which can be easily computed once parity-mixing coefficients have been calculated, as for instance in the framework of the method proposed in ref. x) or in more elaborate variation calculus. We shall only consider nuclei with a large excess of neutrons; light nuclei require more care in the treatment of the isospin. 2. The Wave Function
To construct the trial wave function for a nucleus we first take the Slater determinant ~, = d 17 Zvjmt, (2.1) vjmt
where the single-particle orbitals goj,,t form a complete set of orthonormal functions, they are eigenfunctions of j2, Ja and za (third component of the isospin operator) and can be split into two parts " + -)~vjmt, " Xvjmt = ~vjmt
(2.2)
which are eigenvectors of the parity operator/7. No restrictive assumption is made on the t-dependence, i.e.
~±X,jm~½ # X,j,,±÷,
~± = ½(~4-i~2).
(2.3)
This means that the wave functions of a proton (t = + ½) and of a neutron (t = - ½), both labelled with (v, j, m), are different; therefore, ~k is not an eigenfunction of T 2, the total isospin operator, even for closed shells. This result from the H F calculation whenever the two-body forces depend on the isospin multiplet t. Thus, the function should be projected twice in order to obtain definite parity and isospin. For heavy nuclei exclusively dealt with here, the large excess of neutrons together with the antisymmetrization keep the mean value of T 2 close to an integer. This small deviation t If the forces are of pure z(1) • T(2) character, it is advantageous to take T-Z+~r(x) = )¢-~r(-x).
SINGLE-NUCLEON ORBITALS
355
(not due to Coulomb forces!) will have no sensitive influence on the matrix elements that we study in the sequel. In order to compare the function (2.1) with the usual independent-particle wave function, let us introduce another complete set of orthonormal functions, q~gmt(X), which now satisfy "C~(~0t = (~t, + ~ - - t "
T o simplify, we shall suppose that the expansion of each orbital in this basis can be limited to two terms in a good approximation *; one has then
X,,j,,,t ~" ~,,,~jtq),,,q,.t + [3.,q, q),,_,,j,,,t,
(2.4)
with I~n~j,I2 + Ifl,~,l 2 = 1. The coefficients 0¢ and fl are so defined, that without parity mixing one has ~ = 1, fl = 0. In that case, @ is reduced to the usual form
d
1-I
(2.5)
nnjmt <_F
where F is the Fermi level. Because of the antisymmetry of the wave function parity mixing (fl =6 0) among occupied states does not affect the wave function. Only the states ( n n j m t > L ) which mix outside the occupation contribute. Thus, the function ~ takes the form ~ ~--- ~
1--[ (~n:~jtfPnnjrrlt"~- flnr~jt(Pn--njrat) H ~n'g'j'm't" " L < nnjmt ~ F n'n'j'm't' < L
(2.6)
It must be noted that in practical cases proton and neutron mixed states have different quantum numbers, owing to the Coulomb energy. Therefore, for closed j-shells, ff is eigenvector of T z as long as Coulomb corrections to the orbitals are not taken into account. Moreover, all orbitals in (2.6) are orthogonal to each other because the mixed states have different quantum numbers (j, m). Now, the nuclear states are obtained by projecting ~ onto subspaces of definite parity (see ref. i)),
~
= N~(~+nH~),
N~ --- 2(l+z~(~l/-/l~/)),
(2.7)
where/-/is the parity operator and n = +_ 1 are its eigenvalues. For closed j-shells the function ~ , = + ~ describes the ground state, and ~ , = _ 1 is an excited state of spin zero which automatically appears in the model. For nuclei with one nucleon outside closed j-shells, the two corresponding states are ~lnjmt = d~(O~njt ~Onjmt(X)(2Y)O++ flnjt (12- njmt(X)~o- ), e_
=
+ fl j,
(2.8)
This form has the advantage that it can be tentatively used for other odd nuclei too. * This is n o t always possible, as will be seen in sect. 3.
356
J.-P. AMIET AND P. HUGUENIN
The core functions ~o+ and ~ o - are then obtained from the product function (2.1) by first coupling the spin to zero. The expansion of the function ~ , in powers of ~ and ¢/is a sum of Slater determinants which can be interpreted as a special combination of particle-hole excitations of the initial state (2.5). It is similar to the wave functions of Arima et al. 4, 5), who use perturbation theory for the computation of/-forbidden M1 transitions. On the other hand, it can be used in a larger framework of self-consistent fields involving, for instance, a spatial deformation 6). The ground state wave function for even nuclei 7%+ has an entirely different structure from that of the state ~ o - . Consider the definition (2.7) where ~k has the form (2.6). When the parity mixture vanishes (/~ ---, 0) the state 7', = + 1 tends towards a single Slater determinant, namely that given in (2.5). On the other hand, the vector ~ = - 1 has no definite limit for { ~ j t --" 0} as numerator and denominator (N~) in (2.7) vanish simultaneously. The limit is always a sum of Slater determinants which depend on the order in which the various/~'s are set equal to zero. The same is true for the two vectors defined in (2.8). In other words, the functions of the type of ~o* to some extent retain the structure of an independent particle wave function and the states that they describe (ground state or low-lying excited state) must be taken into account for the filling of the next j-shells. The functions of the second type have the structure of a collective state which cannot be "occupied" in the sense of a shell model. We shall use these properties in sect. 5. We eventually want to establish that the mean value of the parity, ( / 7 > = ( ~1/71~>, is negligible compared to 1. This property greatly simplifies the calculation of many matrix elements. The order of magnitude of , estimated using (2.6), is certainly correct; when only j-shells with different j-values are mixed up, a straightforward calculation gives
= I7 ([~J,tj[ 2 - Iflj,tjl2) 2j+ x. j , tj
The quantities [~l2 - [fll2 are always smaller than 1. Provided that the parity mixing has no negligible effect, one expercts that in at least a few j-shells the upper limit of these differences will be less than 0.9. In which case, the following inequality holds as a good approximation:
<
e -0"lM << 1,
(2.9)
here, M is the number of mixed states.
3. Magnetic Dipole Moments and Electromagnetic Transitions Let A q
= E
/=1
(3.1)
357
SINGLE-NUCLEON ORBITALS
be any single-particle operator of tensor rank k in configuration space, but of any type in isospin space. Further, let * Ifljmt ~" d~)~jrat I-[ )~j'm't' ~- ~Xjmt(X)I~J=O(Xl jJm't"
"'"
XA-1)
(3.2)
be the trial function (2.1) for a nucleus with only one nucleon (j, m, t) outside closed j-shells. The mean value of O~k] in the state ~ i , m (obtained from ~kjmt according to (2.7)) is equal to <
[k]
2
,
(k)
[k]
'
v~jm,lOq IV~jm,> = ~-[+]
when H and -o- qm commute. The ratio of the two contributions has the same order of magnitude as < H >, which is a small number (see (2.9)). I f one neglects the small second term, one obtains (k > 1)
(,e=jmtlO~k:ll ~P~jmt> ~ •
(3.3)
~j'm'," = dZj's',' ~J =o(X . . . . . Xa- 1),
(3.4)
Now, let be the corresponding trial function for a state where the off-shell nucleon is excited. ~}= 0 differs from ~j= o because of changes in the core polarization. The diagonal matrix elements of O~kl also have with (3.4) the form (3.3). In the case of off-diagonal matrix elements, one finds:
<~=jm, lOq[klI~=,j,s,,,> ,~,
(3.5)
where re' = rt, ( - r e ) if O~ka commutes (anti-commutes) w i t h / / . The factor < ~1~' > may appreciably differ from 1, although the individual single-particle functions are not greatly affected by the change in the last nucleon state. We obtain from (3.5) the probability of an electromagnetic transition tt of multipolarity k,
T(k) = 8rcc
e2
(k+l)(~)2k+l
hc
k [ ( 2 k + 1)!!] 2
B(k),
where the reduced transition rate B for electric transitions is nel(k) --
0t ' ~- lak 2ji+l
+ bk+ dk 121<~[~'>l 2, (3.6a) t For the sake of brevity the subscript ~, which completes the labelling of the basis functions Z will be omitted in the sequel. t* For notation see ref. 7).
358
J.-P. AMIET AND P. HUGUENIN
and for magnetic transitions is Bmag(k) =
I
( h ) '2
+----1 2ji ~mc
[ak-l-t-bt~-l
+ck_l+dk_l[2]<~l~'>12,
(3.6b)
..¢t,tkl -- 2x/k(2k+ 1) {6t,¢gL[Y . tk- 11xj[ll]tkl+((k+l)t~t--~t.~gL)[Y[k-1]xStq']tkl}. k+l Here, Pt is the intrinsic magnetic moment in Bohr magneton units of the single nucleon which makes the transition, jtll and stll the operators of total angular momentum, and spin, respectively, and y m the spherical harmonics of order k, all three in the form of irIeductible tensors (standard contra-variant form of Fano and Racah 8)). It is clear that two of the four reduced matrix elements in (3.6) always vanish, according to the multipolarity and the type of the transition; it is convenient to use here the parities n i and nf of the initial and final state instead of the orbital quantum numbers l i and ly (/is fixed by [ l - j l -- ½ and n = ( - 1)l). The four factors ak, bk, Ck, dk are radial integrals defined as follows:
ak(ji , if) =
dr r k +2f*ht(r)f, tfja(r),
bk(Ji, jf) =
d r r k+2f_=,j,t(r)f_,,~ja(r), *
ck(ji,jf) = fO d r r*+2"*'J~,jt,(r )f_,fja(r), dk(Ji, Jf) =
fo
'°drrk+2... z • J-nthd.Zr.~,. )J,fjft~.r)
(3.7)
The functions fnjt(r) and f-,~jt(r) are the radial parts of the two parity components of the orbital gj=t(x) (see (2.2)). In writing (3.7), we have made the convention to set the n of (2.2) equal to the parity of the nucleus state in which X appears. This simplifies the correspondence with the usual shell model; namely, when one allows the parity mixing to tend towards zero the component g -~ of Z vanishes, together with three of the four coefficients (3.7), and one finds again for B ( k ) the prediction of the shell model. The values (3.6) will not differ much from those of a shell-model calculation without parity mixing, except when the latter model introduces additional selection rules. For instance, for M1 transitions between two states with total angular momenta j = l+½ a n d j ' = l ' - ½ , I F - l [ = 2 where l and l' are the orbital angular momenta of the extra-core nucleon, the shell model predi6ts no transition (/-forbidden). On the other hand, we obtain from (3.6) in the same case the following non-zero reduced
359
S I N G L E - N U C L E O N ORBITALS
transition rate: Bmag(1) =
~mc
4~z(2ji+l )
b2(JiJf)l(Ji-~zill$llJf-ztf)lz[(~[~')[2'
(3.8)
where zq = ~, and the magnetic moment operator # = g~l+gsS. Similarly the diagonal matrix elements of p deviate in a characteristic way from the Schmidt values; using (3.3) one finds
1/ •
j
,
Pj = Y (2j+1)(2j +1)
{(1-bo)(J~llpl]j~)+bo(j-zcllplIj-~)},
(3.9)
where bo(j,j) is defined according to (3.7) by setting k = 0, zq -- nf = re, and Ji = Jf = J- It is worthwhile remarking that bo(j,j) is a measure of the parity mixture in the orbital ~(Smt(X); indeed, remembering (2.2) and taking k = 0 i n ( 3 . 7 ) , one has b0(j, j) -- f [Z~-~[2 d3x.
In cases where the EM transition has a mixture of M1 and E2 multipolarities, we find the following interference ratio for the transition of a proton *:
t~= (JnllE2[lj'n)= (JnllMlllj'n)
( - 1 ) s-½ (j~½ -
2 j~)VJi+jf+l 0 10
2mco/c a2-b2, gs-gt bo
(3.10)
where a2, b2 and b 0 are radial integrals defined in (3.7). We are aware that this single-particle estimate for the quadrupole coefficient a 2 - b2 could be very poor, and that pairing energy and quadrupole vibrations should be taken into account 10). But we are more concerned here with the sign of 6 which, unlike the magnetic dipole moments, depends on that of the parity-mixing coefficient bo(ji,jf). In order to illustrate some specific aspects of this section, we shall now review the case of four nuclei 133Cs, 135Cs, 139La and 141Pr, for which many data are available 7+ where the 3 + and ~subsheUs are involved. Shell-model predictions apply best to 139La and 141Pr which have respectively closed j-shells minus, plus one proton; tentatively, we also include Z33Cs and 135Cs for which eq. (2.8) is to be used. The magnetic moment of the ground state and the B(1) value (3.8) of the M1 transition between the ~ + and 3 + states are known 11) for all four nuclei. The magnetic moment of the 3 + excited state (81 keV) of 133Cs has also been measured 12). These data are listed in table 1. The missing values (in brackets in the table) have been estimated by interpolating between known magnetic moments in the 3 + and 7 + shells 13). In table 1, column I, we have listed the coefficients bo which make the predictions (3.8) and (3.9) fit these experimental data. They are slowly varying functions of the f The explicit definition of 6 has unfortunately not been given in the work of Biedenharn and Rose 9). We have established expression (3.10) for 6 by using the same phase conventions as these authors (they also agree with ref. s)).
360
J.-P. AMIET AND P. HUGUENIN
occupation of the shell, even across the point where the ~+ shell is filled; this fact is consistent with the present model where calculated mixing parameters are not expected to change rapidly with increasing A. This aspect is not present in some other models, as, for example, that of quadrupole vibration ~o) (see concluding remarks in ref. 11)). TABLE 1 I Transition
Experimental data
AE jn ---~j'n" (keV) xaaCs
lanes XaOLa lllpr
{+ ~ ~-+ ~+ --* {+ ~+ ~ ½+ 3 + ~ ~+
II
81 250 166 145
a) Ref. 12).
~
B(M1) kt~-+ (cm 2 . 10_ao)
/t~+ 2.58 2.73 2.78 (2.8)
3.4 (3.6) (3.8) 4.0
0.49 0.93 0.52 0.44
(~(E2/M1) --0.16-4-0.01 a) 0.034-0.03 b) 0.07±0.02 b)
"~ ~
'~ ~
0.21 0.25 0.26 (0.26)
~"
~ ~
0.35 0.025 (0.30) 0.035 (0.25) 0.026 0.20 0.029
I
(E21 0.27 0.27 0.25 0.23
--13 --13 --13 +13
b) Ref. 18).
If we assume that the orbitals Z~÷ and Z~÷ can be approximated according to (2.4), one obtains the relations bo(3, 3) = fl~,
bo(~, ~) = fl~,
7 ~)[ 5 = [fllfl~l = x/bo(7, ~)bo(~, 7 s {). (3.11) Ibo(~,
In this approximation the off-diagonal matrix element of/~ is determined by the diagonal ones, i.e. the reduced transition rate B ( M I ) can be calculated up to the factor ( ¢] ¢ ' ) from the magnetic dipole moments of the initial and final states. The values fbo({, 3)1 deduced from the empirical values of bo(~-,-~) 7 7 and bo(}) by use of (3.11) are listed in table 1, column II. Even if one admits that the factor ( ¢1 ¢ ' ) could be as small as ½, the coefficients bo(-~, {) calculated in this way are still five times too large. It is true that (3.10) gives an upper limit, as a consequence of the Schwarz inequality
Ibo(j,j')l= <=
drr2]f j_,(r)] 2
?"
drr2]f j,_,(r)] 2
Io °
= x/bo(j,j)bo(j',j'),
but we do not think that this error could account for more than half of the discrepancy; we are thus left with a reduced transition rate B(M1), proportional to bo(~,~), 7 s which is about six times too large. This inconsistency shows that the present model cannot fully explain the magnetic dipole moments and M1 transitions at the same time.
SINGLE-NUCLEON ORBITALS
361
In particular, the good values obtained by Freed and Kisslinger 14) for the magnetic moments in the ~+ subshell strongly suggest that we have grossly overestimated the parameter b o (}, ½) by assuming that it alone is responsible for the deviation of /~+ from the Schmidt value. A set of coefficients fl,j~ has been calculated for all nuclei with closed j-shells by Mfiller 15), who solved eq. (5.10) of ref. 1); for this calculation the antisymmetry of the nuclear wave function has been neglected and only the first few coefficients fl,iof each orbital have been retained. From the values he computed for A = 140 with a filled ~+ shell, one obtains bo(~, ~r) = 0.030,
bo(~,~) = 0.19,
7 y) 5 = -0.056. bo(~,
(3.12)
These values apply to the case of 139La and 141Pr, but can also be tentatively used for Cs since they do not vary rapidly with the occupation number of the sub-shell ~+. The prediction (3.12) for the off-diagonal element bo(~-,~-) 7 s agrees well with the empirical values (table 1, column I) of all four nuclei if the core polarization (1< ~l~' >1 < 1) is taken into account. The coefficient bo(~,~-) 7 7 is about ] of the value required to explain the magnetic moment #(~ +) of a 39La ' but the other one, b o(~, ~2), fails completely to account for the deviation from the Schmidt value of#(~ +) of 141Pr. These results support the conclusion of the above analysis; the reduced transition rate tends to be too large, and the corrections to the Schmidt values for the magnetic dipole moment are too small. In fact, this situation is rather satisfying as B(M1) will certainly be reduced if one takes the core polarization and the pairing effect seriously into account; moreover, it is expected, as remarked above, that the addition of other types of couplings are going to increase the corrections to the magnetic moments. For instance, the corrections of Freed and Kisslinger 14) complement ours very well 7 + subshells. in the case of the ~s + and ~The coefficient b0(~-,~-) ~ s is predicted in the present model with a specific sign which can be checked experimentally in mixed M1 and E2 transitions. The parameters of interference 6 which have been measured 12, as) for the EM transitions between the + a n d s -+ states are listed in table 1. The theoretical values (column Ill) have been obtained by using (3.10). The factor a 2 - b 2 (defined in (3.7)) which characterises the quadrupole part of the transition has been calculated with the radial dependence of the orbitals computed by MSller as). One obtains a 2 - b 2 = - 16.3 fm 2. The discrepancy between the experimental and theoretical values for 6 is not surprising since, in the present case, the single-particle approximation yields values that are too small for E2 transitions (with or without parity mixture) and too lalge for M1 (see above). Anyway, our interest here lies mostly in the sign of 6 which should not be too sensitive to the method of calculation of the fl values. It is satisfying to find that the predicted signs fit the data in two cases. We have no good explanation for the discrepancy in the case of 139La" In fact, one would rather expect a good result for 141Pr
362
J.-P.
AMIET AND
P.
HUGUEN1N
and ~39La since, if one believes in similar properties for one particle and for one hole, 6 changes sign under the permutation of initial and final states. Moreover, we do not think that the signs of bo(½, ~-) for 139La and 141Pr should be opposite; the regularity of B(M1) practically excludes a change of signs which implies that bo goes through zero as a function of the number of nucleons. The very small value of 6 (compatible with zero) rather suggests that the E2 part of the transition changes its sign unexpectedly, making the single-particle estimate meaningless.
4. Direct Reactions
The Born approximation of the differential cross section for stripping reactions on a closed j-shells nucleus reads da
2re
dO
hv i
pf(E)l(e'~'f"p~bf(Z, N + 1)1V(xv - x.)l~ki(Z, N)e~ik' "t=P+xn~fd(Xp-- X,)>I 2, (4.1)
where the subscripts i and f refer to the initial and final states respectively, and n, p and d the neutron, proton and deuteron. By introducing in (4.1) the trial function ~=j,, defined in (2.8) for the final state one obtains, neglecting the antisymmetrization of the wave function, da_
dQ
2 2~
~"J hv i Pf(E)l('fd(Xp-- Xn)e~il'l "{x"+=*)i V(Xp-- X")[~O"jm(X)e'k" xP)12"
(4.2)
The only difference between the cross section calculated with and without parity mixing (usual shell model) is the appearance of the factor ~ in the former case (~,g = 1 means no parity mixing). The component q~_=j,, of the orbital (2.4), which describes the absorbed neutron, does not enter this stage of approximation. The smallest value of ~=~ that we expect (about ½), still differs from 1 by far less than the difference between the calculated and measured values of da/dI2. This explains why a parity mixture in the single-particle orbitals cannot be checked by fittings the cross sections of the stripping reactions with theoretical predictions made in Born approximation. The theoretical cross section for direct reactions on odd nuclei reveal a special feature of the model. The easiest cases to describe with the present formalism are the nuclei with closed j-shell plus an outside nucleon (for the pick-up) and a hole in a closed j-shell (for the stripping). We shall restrict ourselves to the first case, as the formulae for the second one are very similar. The wave function for the target is given in (2.8). We have to consider two channels t, for when the outside nucleon has been removed the core may be left in t The interest of this property has been first pointed out to us by Dr. H. D. Zeh.
363
SINGLE-NUCLEON ORBITALS
either state, 4 o . o r # o - . On the same assumption as above, one obtains de+ 2 21r dO = ~j~ ~ vi Pf(E)l(e*'kf' (xP+x*~fdlV ( x p - x,)lq~,~jm(X,)e 'k' "xP)l z,
de_
d f f = B2j
Pf(E)l(e~'kfO~"+"n)fdlV(xp--x")ltP-=jm(X')e'k''"")i
'
(4.3)
(4.4)
where ~ j and fl~j are the parity mixing parameters of the orbitals of the last neutron (see (2.4)). The/-values of these two channels differ by one unit, and the ratio o f their single-particle strengths is fl,,j/~z,,j, instead of zero, as the usual shell model predicts. This quotient should in most cases oscillate between 5 to 30 ~o. The fact that there are practically no known 0 - excited states in the spectrum of middle to heavy even nuclei may be deceptive, because many levels are still unassigned in the energy region where we would expect them (see sect. 5), and we hope that some 0 - states could be discovered if one looked specifically for them. It seems to us that in a precise pick-up experiment one should be able to detect these levels by means o f the channel (4.4), if it does in fact exist.
5. Nuclear Levels
It has been shown in ref. 1) how a tensor force can induce a parity mixture in the H F orbitals, provided it is strong enough compared to the increase in kinetic energy. The result is a shift of single-nucleon energy levels which contributes to the splitting o f the levels j = l _ ½ in the same way as a one-body ! • s term. Another question is to decide to what extent each of the spin-dependent forces contribute to this splitting; we believe, along with m a n y other authors 4, 17, 18) that the strong two-nucleon tensor force may play an essential role in this respect. We want to discuss here again a few effects of the parity mixing on nuclear spectra, but from a different point of view. We shall leave aside H F calculations and study directly the relation between the spectra of an even nucleus and the neighbouring odd nucleus by means of the projected wave functions (2.8). One of the reasons for not working here with H F is that the energy-splitting between the two core levels of opposite parity defined in (2.7) turns out completely wrong in this framework and can only be correctly calculated by taking the pairing energy into account (this is the same situation as for the inertial moments of deformed nuclei). Let us provisionally assume that an even nucleus has an excited state with J = 0, = - 1. We denote by ff~o- the corresponding state vector and by ~o+ that of the ground state. Both are supposed to be exact eigenvectors of the total Hamiltonian H. We now construct four wave functions,
(5.1)
364
J.-P.
A M I E T A N D P. H U G U E N I N
related to the next odd nucleus t; the quantum numbers n a n d j are by definition the parity and the total angular m o m e n t u m of the ground state. The two orbitals q~,j and q~_=j are supposed to yield the minima of the mean energy in the states q~*,j~o+ • I f necessary, the two state vectors cp,=i~ o- have to be made orthogonal to the t w o other ones by standard methods. The energy difference
As = (~o + q~-,,jlHl~o÷ e - , q > - ( ' ~ o ÷ ~o,,jlnl¢'o+ m,,j>
(5.2)
corresponds to a single-particle excitation in the sense of the shell model. I f As is larger than the excitation energy AE of the state ~ o - , then the mean energies in the four states defined in (5.1) look as indicated at the left of fig. I, it one makes the~ reasonable assumption:
Without mixin 9
With mixing
Type of s/ate
[2 J+~)
/1~
IAE
~
Corrective
12J -r )
5ingle part.
I1 j-rr 5
Coltect,¢e
II J ÷rr )
S ingLepa~t
Fig. 1. The effect of the parity mixing on nuclear spectra. The distinction between collective and_ single-particle states is made according to the discussion at the end of sect. 2.
Now, we define four new state vectors, [ljrc) = ~1 0o÷ ~P~j+/~l ~ o - q~-~j, I l j - = > = ai ~ o - ~=j+fl't ~o+ q~-=j, 12j~> = ~2 ~ o - ~°-~j+/~2 ~o÷ q~j, !
(5.3)'
t
12j-=> = ~2 ¢o+ q~-~j+/32 ¢ o - q ~ , which reduce to the four initial ones when the/Y parameters vanish. This corresponds to a parity mixture within the orbitals of the extra-core nucleon, as can be esaily seen from (2.8). It is a well-known fact that when one mixes two states having the same quantum numbers in the framework of a variational calculus, they repulse each other. * Throughout this section we drop irrelevant quantum numbers; moreover, the product_ if(x) ~p(x. . . . . xn) is supposed to be antisymmetrized.
SINGLE-NUCLEON ORBITALS
365
Thus, in the case of the vector (5.3), the splitting of the two intermediate levels in fig. 1 becomes larger and the ground-state energy decreases. Similarly, the above procedure can be used for the other states, ~o~,i,~o+ and ~ , j ' ~ o - , where j ' = j + 1. Without parity mixing, the two states qT_~jOo, and tp_,j,~o, are degenerate if the exchange energy (due to antisymmetrization) is neglected t. The two orbitals qT_~iand q~_,j, correspond to the degenerate doublet with l = j + ½ of the shell model without spin-oIbit coupling. When the state ~P-i,~o ÷ is mixed with q~,j, ~o- (according to (5.3)), its energy decreases like that of the groundstate. These preliminary remarks enable us now to discuss the filling of the next j-shell in the present scheme. If the energy difference A E and Ae ((5.2)) satisfy the inequality A E < Ae,
(5.4)
the level shift due to the parity mixing will split up the quasi-degenerate doublet, ~P-~j,~o+, ~o-~j~o- with the largest spin j ' at the bottom. This feature remains unchanged as one iterates the whole above procedure when adding pairs of nucleons to the core; then one finds that the shell ( j ' , - n ) is filled before the (j, - n ) shell as expected. On the other hand, if A E > Ae
holds, then both states ~p_,j, ~o÷ and q~-,j~o+ move downward ( ( ~ _ n j ~ O + is repulsed by tp,j~ o- which lies initially higher). In this case any crossing of levels is possible and a wrong spin will surely often be predicted for the ground state. In conclusion, the actual model cannot be meaningful if the condition (5.4) is not fulfilled. In other words, the (hypothetical) 0 - state of the core should not lie higher than the singleparticle excited state tp_~j. Since the energy difference A~ between the two states q ~ j ~ o ÷ is strongly conditioned by the centrifugal barrier, one may approximate it according to the shell model by h2 Ae ~ .... (2l+ 1), 2mR z
where R is the nucleus radius, m the nucleon mass and l = j + ½ The condition (5.4) becomes then AE <
h2 2mR 2
(2/+ 1).
(5.5)
In conclusion, an important test of the validity of the present model consists in confirming the presence of 0 - excited states in the right energy range (5.5) in the spectrum of even nuclei. We assume here that the effect of two-body L • S forces is negligible.
366
J . =P.
AMIET A N D
P.
HUGUENIN
6. Conclusion We believe to have shown how a mixture of parity in single-nucleon orbitals can easily explain the/-forbidden M1 transition and, at the same time, account for a large part of the deviation of magnetic dipole moments from the Schmidt values. Quantitatively the question remains open as in the case of the "spin-orbit" splitting induced by the tensor force 1). The first numerical results are nevertheless encouraging. We think that an improved version of the present model (with respect to the quadrupole transitions) could be particularly useful for a systematic understanding of the magnitude and the sign of the ratio 6 in mixed ~-transitions. Generally, it seems to us that a definite advantage could be gained in staiting with orbitals which are not eigenstates of the parity operator, rather than trying to reach the special properties that this mixture of parity conveys to the nuclear wave function through a perturbation expansion. To conclude, the decisive test for or against the legitimacy of the model lies less in elaborate calculations than in the discovery or identification of 0 - excited states of even nuclei in the approximate energy range that we have fixed in the previous section. We are aware that this is not a simple experimental task, but it seems worthwhile to decide whether or not the parity must be a good quantum number for singlenucleon orbitals, as is the case for the nuclear wave function. We are grateful to Professor J. H. D. Jensen for his stimulating interest in this subject and to Professor E. Bodenstedt for fruitful discussions. We are indebted to Dr. K. Dietrich who carefully read the manuscript.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
J.-P. Amiet and P. Huguenin, Nuclear Physics 46 (1963) 171 F. Villars, Proc. Int. School of Physics, E. Fermi Varenna (1961) ed. by V. F. Weisskopf (1963) L. Hulth6n and M. Sugawara, Encyclopedia of physics, ed. by S. Fltigge, Vol. 39 (1957) p. 1 A. Arima, Nuclear Physics 18 (1960) 196 A. Arima, H. Horie and M. Sano, Prog. Theor. Phys. 17 (1957) 567 I. Kelson, Phys. Lett. 16 (1965) 143 A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) U. Fano and G. Racah, Irreductible tensorial sets (Academic Press, New York, 1959) L. C. Biedenharn and M. E. Rose, Revs. Mod. Phys. 25 (1953) 729 R. A. Sorensen, Phys. Rev. 133 (1964) B281, 132 (1963) 2270 J. S. Geiger, R. L. Graham, I. Bergstr6m and F. Brown, Nuclear Physics 68 (1965) 352 E. Bodenstedt, H. J. KOrner and E. Matthias, Nuclear Physics U (1959) 584 E. Bodenstedt, Forts. Phys. 10 (1962) 321 N. Freed and L. S. Kisslinger, Nuclear Physics 25 (1961) 611 J. MOiler, Diplomarbeit, Universitiit Bonn (1963) unpublished J. N. Haag, D. A. Shirley and D. H. Templeton, Phys. Rev. 129 (1963) 1601 B. Jancovici, Prog. Theor. Phys. 22 (1959) 585 P. Goldhammer, Phys. Rev. 122 (1961) 207
3.C
[ I
Nuclear Physics 80 (1966) 353--366; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
O N THE PARITY MIXING I N S I N G L E - N U C L E O N
ORBITALS
J.-P. AMIET Institut fiir Theoretische Physik der Universitiit, Heidelberg, Germany and P. HUGUENIN Institut de Physique de r Universitd, Neuehdtel, Switzerland t Received 15 October 1965 Abstract: A mixture of parity in single-particle orbitals leads to specific properties of the nuclear wave function. These properties are reviewed here. The wave function is obtained by projecting out a part of definite parity from a product function. Formulae are given for some quantities, such as the magnetic dipole moment, the magnetic dipole transitions, etc., which appreciably deviate from the values predicted by the usual shell model. It is shown why a parity mixing is not incompatible with the fact that definite/-values are measured in stripping reactions. The influence of this parity mixing on nuclear spectra is studied. In the case of odd nuclei, it is similar to that of the spin-orbit coupling if the spectrum of the preceding even nucleus satisfies a simple condition.
1. Introduction This p a p e r is i n t e n d e d to s u p p l e m e n t an earlier one dealing with a p a r t i c u l a r extension o f the usual H a r t r e e - F o c k ( H F ) a p p r o x i m a t i o n s in n u c l e a r t h e o r y 1). It is shown in ref. 1) t h a t single-nucleon o r b i t a l s w i t h o u t p a r i t y c o u l d be well suited for solving the p r o b l e m o f self-consistent fields in the presence o f a strong t e n s o r force. I t seems w o r t h w h i l e to r e m a r k at this p o i n t t h a t the p e r t i n e n t a r g u m e n t o f Villars 2) a b o u t the c o n s e r v a t i o n o f p a r i t y in single-particle o r b i t a l s loses its weight if the forces are spin d e p e n d e n t . I n this case, one can b r e a k up the s y m m e t r y o f p a r i t y in the H F solutions w i t h o u t losing the spherical symmetry. A direct calculation o f the p a r i t y mixture in single-nucleon o r b i t a l s b y solving the H F e q u a t i o n s is clearly p r e m a t u r e . O u r l a c k o f k n o w l e d g e c o n c e r n i n g effective shortrange forces w o u l d m a k e the significance o f an a t t e m p t in this direction m o r e o r less d o u b t f u l . This does n o t i m p l y t h a t the a p p r o x i m a t e t r e a t m e n t p r o p o s e d in the last section o f ref. 1) is meaningless; we c a n n o t p r o v e t h a t it is a true a p p r o x i m a t i o n to H F , b u t it is at least a m e a n i n g f u l p a r a m e t r i z a t i o n whose f o r m is d i c t a t e d b y H F . T h e o n l y a d a p t a b l e p a r a m e t e r which is used (except the coefficients o f the spini n d e p e n d e n t c e n t r a l p o t e n t i a l ) is the effective c o u p l i n g c o n s t a n t in f r o n t o f the onep i o n - e x c h a n g e p o t e n t i a l in which the b-function 3) has been included. t This work has been partly supported by the "Fonds National Suisse de la Recherche Scientifique." 353
354
J.-P.
AMIET AND
P.
HUGUENIN
A wave function for the nucleus constructed from orbitals without parity has many properties that we shall discuss here, in relation with the deviations from the usual shell model that they imply for many matrix elements. In the main part of this paper we shall treat the parity-mixing coefficients in the nuclear wave function as free parameters. We first propose to look for those nuclear properties which are easily explainable in this model (without taking the mixing parameters beyond a reasonable range) and for which the usual shell model fails to give satisfactory predictions. It is hoped that in the future one can test the consistency of the model by comparing different quantities which depend on the same mixing parameters. Secondly, we shall show why a parity mixture of the single-particle orbitals cannot be detected in direct reactions at the present stage of the theory of nuclear reactions. Finally, we intend to give various matrix elements which can be easily computed once parity-mixing coefficients have been calculated, as for instance in the framework of the method proposed in ref. x) or in more elaborate variation calculus. We shall only consider nuclei with a large excess of neutrons; light nuclei require more care in the treatment of the isospin. 2. The Wave Function
To construct the trial wave function for a nucleus we first take the Slater determinant ~, = d 17 Zvjmt, (2.1) vjmt
where the single-particle orbitals goj,,t form a complete set of orthonormal functions, they are eigenfunctions of j2, Ja and za (third component of the isospin operator) and can be split into two parts " + -)~vjmt, " Xvjmt = ~vjmt
(2.2)
which are eigenvectors of the parity operator/7. No restrictive assumption is made on the t-dependence, i.e.
~±X,jm~½ # X,j,,±÷,
~± = ½(~4-i~2).
(2.3)
This means that the wave functions of a proton (t = + ½) and of a neutron (t = - ½), both labelled with (v, j, m), are different; therefore, ~k is not an eigenfunction of T 2, the total isospin operator, even for closed shells. This result from the H F calculation whenever the two-body forces depend on the isospin multiplet t. Thus, the function should be projected twice in order to obtain definite parity and isospin. For heavy nuclei exclusively dealt with here, the large excess of neutrons together with the antisymmetrization keep the mean value of T 2 close to an integer. This small deviation t If the forces are of pure z(1) • T(2) character, it is advantageous to take T-Z+~r(x) = )¢-~r(-x).
SINGLE-NUCLEON ORBITALS
355
(not due to Coulomb forces!) will have no sensitive influence on the matrix elements that we study in the sequel. In order to compare the function (2.1) with the usual independent-particle wave function, let us introduce another complete set of orthonormal functions, q~gmt(X), which now satisfy "C~(~0t = (~t, + ~ - - t "
T o simplify, we shall suppose that the expansion of each orbital in this basis can be limited to two terms in a good approximation *; one has then
X,,j,,,t ~" ~,,,~jtq),,,q,.t + [3.,q, q),,_,,j,,,t,
(2.4)
with I~n~j,I2 + Ifl,~,l 2 = 1. The coefficients 0¢ and fl are so defined, that without parity mixing one has ~ = 1, fl = 0. In that case, @ is reduced to the usual form
d
1-I
(2.5)
nnjmt <_F
where F is the Fermi level. Because of the antisymmetry of the wave function parity mixing (fl =6 0) among occupied states does not affect the wave function. Only the states ( n n j m t > L ) which mix outside the occupation contribute. Thus, the function ~ takes the form ~ ~--- ~
1--[ (~n:~jtfPnnjrrlt"~- flnr~jt(Pn--njrat) H ~n'g'j'm't" " L < nnjmt ~ F n'n'j'm't' < L
(2.6)
It must be noted that in practical cases proton and neutron mixed states have different quantum numbers, owing to the Coulomb energy. Therefore, for closed j-shells, ff is eigenvector of T z as long as Coulomb corrections to the orbitals are not taken into account. Moreover, all orbitals in (2.6) are orthogonal to each other because the mixed states have different quantum numbers (j, m). Now, the nuclear states are obtained by projecting ~ onto subspaces of definite parity (see ref. i)),
~
= N~(~+nH~),
N~ --- 2(l+z~(~l/-/l~/)),
(2.7)
where/-/is the parity operator and n = +_ 1 are its eigenvalues. For closed j-shells the function ~ , = + ~ describes the ground state, and ~ , = _ 1 is an excited state of spin zero which automatically appears in the model. For nuclei with one nucleon outside closed j-shells, the two corresponding states are ~lnjmt = d~(O~njt ~Onjmt(X)(2Y)O++ flnjt (12- njmt(X)~o- ), e_
=
+ fl j,
(2.8)
This form has the advantage that it can be tentatively used for other odd nuclei too. * This is n o t always possible, as will be seen in sect. 3.
356
J.-P. AMIET AND P. HUGUENIN
The core functions ~o+ and ~ o - are then obtained from the product function (2.1) by first coupling the spin to zero. The expansion of the function ~ , in powers of ~ and ¢/is a sum of Slater determinants which can be interpreted as a special combination of particle-hole excitations of the initial state (2.5). It is similar to the wave functions of Arima et al. 4, 5), who use perturbation theory for the computation of/-forbidden M1 transitions. On the other hand, it can be used in a larger framework of self-consistent fields involving, for instance, a spatial deformation 6). The ground state wave function for even nuclei 7%+ has an entirely different structure from that of the state ~ o - . Consider the definition (2.7) where ~k has the form (2.6). When the parity mixture vanishes (/~ ---, 0) the state 7', = + 1 tends towards a single Slater determinant, namely that given in (2.5). On the other hand, the vector ~ = - 1 has no definite limit for { ~ j t --" 0} as numerator and denominator (N~) in (2.7) vanish simultaneously. The limit is always a sum of Slater determinants which depend on the order in which the various/~'s are set equal to zero. The same is true for the two vectors defined in (2.8). In other words, the functions of the type of ~o* to some extent retain the structure of an independent particle wave function and the states that they describe (ground state or low-lying excited state) must be taken into account for the filling of the next j-shells. The functions of the second type have the structure of a collective state which cannot be "occupied" in the sense of a shell model. We shall use these properties in sect. 5. We eventually want to establish that the mean value of the parity, ( / 7 > = ( ~1/71~>, is negligible compared to 1. This property greatly simplifies the calculation of many matrix elements. The order of magnitude of , estimated using (2.6), is certainly correct; when only j-shells with different j-values are mixed up, a straightforward calculation gives
= I7 ([~J,tj[ 2 - Iflj,tjl2) 2j+ x. j , tj
The quantities [~l2 - [fll2 are always smaller than 1. Provided that the parity mixing has no negligible effect, one expercts that in at least a few j-shells the upper limit of these differences will be less than 0.9. In which case, the following inequality holds as a good approximation:
<
e -0"lM << 1,
(2.9)
here, M is the number of mixed states.
3. Magnetic Dipole Moments and Electromagnetic Transitions Let A q
= E
/=1
(3.1)
357
SINGLE-NUCLEON ORBITALS
be any single-particle operator of tensor rank k in configuration space, but of any type in isospin space. Further, let * Ifljmt ~" d~)~jrat I-[ )~j'm't' ~- ~Xjmt(X)I~J=O(Xl jJm't"
"'"
XA-1)
(3.2)
be the trial function (2.1) for a nucleus with only one nucleon (j, m, t) outside closed j-shells. The mean value of O~k] in the state ~ i , m (obtained from ~kjmt according to (2.7)) is equal to <
[k]
2
,
(k)
[k]
'
v~jm,lOq IV~jm,> = ~-[
when H and -o- qm commute. The ratio of the two contributions has the same order of magnitude as < H >, which is a small number (see (2.9)). I f one neglects the small second term, one obtains (k > 1)
(,e=jmtlO~k:ll ~P~jmt> ~
(3.3)
~j'm'," = dZj's',' ~J =o(X . . . . . Xa- 1),
(3.4)
Now, let be the corresponding trial function for a state where the off-shell nucleon is excited. ~}= 0 differs from ~j= o because of changes in the core polarization. The diagonal matrix elements of O~kl also have with (3.4) the form (3.3). In the case of off-diagonal matrix elements, one finds:
<~=jm, lOq[klI~=,j,s,,,> ,~
(3.5)
where re' = rt, ( - r e ) if O~ka commutes (anti-commutes) w i t h / / . The factor < ~1~' > may appreciably differ from 1, although the individual single-particle functions are not greatly affected by the change in the last nucleon state. We obtain from (3.5) the probability of an electromagnetic transition tt of multipolarity k,
T(k) = 8rcc
e2
(k+l)(~)2k+l
hc
k [ ( 2 k + 1)!!] 2
B(k),
where the reduced transition rate B for electric transitions is nel(k) --
0t ' ~- lak
+ bk
358
J.-P. AMIET AND P. HUGUENIN
and for magnetic transitions is Bmag(k) =
I
( h ) '2
+----1 2ji ~mc
[ak-l
+ck_l
(3.6b)
..¢t,tkl -- 2x/k(2k+ 1) {6t,¢gL[Y . tk- 11xj[ll]tkl+((k+l)t~t--~t.~gL)[Y[k-1]xStq']tkl}. k+l Here, Pt is the intrinsic magnetic moment in Bohr magneton units of the single nucleon which makes the transition, jtll and stll the operators of total angular momentum, and spin, respectively, and y m the spherical harmonics of order k, all three in the form of irIeductible tensors (standard contra-variant form of Fano and Racah 8)). It is clear that two of the four reduced matrix elements in (3.6) always vanish, according to the multipolarity and the type of the transition; it is convenient to use here the parities n i and nf of the initial and final state instead of the orbital quantum numbers l i and ly (/is fixed by [ l - j l -- ½ and n = ( - 1)l). The four factors ak, bk, Ck, dk are radial integrals defined as follows:
ak(ji , if) =
dr r k +2f*ht(r)f, tfja(r),
bk(Ji, jf) =
d r r k+2f_=,j,t(r)f_,,~ja(r), *
ck(ji,jf) = fO d r r*+2"*'J~,jt,(r )f_,fja(r), dk(Ji, Jf) =
fo
'°drrk+2... z • J-nthd.Zr.~,. )J,fjft~.r)
(3.7)
The functions fnjt(r) and f-,~jt(r) are the radial parts of the two parity components of the orbital gj=t(x) (see (2.2)). In writing (3.7), we have made the convention to set the n of (2.2) equal to the parity of the nucleus state in which X appears. This simplifies the correspondence with the usual shell model; namely, when one allows the parity mixing to tend towards zero the component g -~ of Z vanishes, together with three of the four coefficients (3.7), and one finds again for B ( k ) the prediction of the shell model. The values (3.6) will not differ much from those of a shell-model calculation without parity mixing, except when the latter model introduces additional selection rules. For instance, for M1 transitions between two states with total angular momenta j = l+½ a n d j ' = l ' - ½ , I F - l [ = 2 where l and l' are the orbital angular momenta of the extra-core nucleon, the shell model predi6ts no transition (/-forbidden). On the other hand, we obtain from (3.6) in the same case the following non-zero reduced
359
S I N G L E - N U C L E O N ORBITALS
transition rate: Bmag(1) =
~mc
4~z(2ji+l )
b2(JiJf)l(Ji-~zill$llJf-ztf)lz[(~[~')[2'
(3.8)
where zq = ~, and the magnetic moment operator # = g~l+gsS. Similarly the diagonal matrix elements of p deviate in a characteristic way from the Schmidt values; using (3.3) one finds
1/ •
j
,
Pj = Y (2j+1)(2j +1)
{(1-bo)(J~llpl]j~)+bo(j-zcllplIj-~)},
(3.9)
where bo(j,j) is defined according to (3.7) by setting k = 0, zq -- nf = re, and Ji = Jf = J- It is worthwhile remarking that bo(j,j) is a measure of the parity mixture in the orbital ~(Smt(X); indeed, remembering (2.2) and taking k = 0 i n ( 3 . 7 ) , one has b0(j, j) -- f [Z~-~[2 d3x.
In cases where the EM transition has a mixture of M1 and E2 multipolarities, we find the following interference ratio for the transition of a proton *:
t~= (JnllE2[lj'n)= (JnllMlllj'n)
( - 1 ) s-½ (j~½ -
2 j~)VJi+jf+l 0 10
2mco/c a2-b2, gs-gt bo
(3.10)
where a2, b2 and b 0 are radial integrals defined in (3.7). We are aware that this single-particle estimate for the quadrupole coefficient a 2 - b2 could be very poor, and that pairing energy and quadrupole vibrations should be taken into account 10). But we are more concerned here with the sign of 6 which, unlike the magnetic dipole moments, depends on that of the parity-mixing coefficient bo(ji,jf). In order to illustrate some specific aspects of this section, we shall now review the case of four nuclei 133Cs, 135Cs, 139La and 141Pr, for which many data are available 7+ where the 3 + and ~subsheUs are involved. Shell-model predictions apply best to 139La and 141Pr which have respectively closed j-shells minus, plus one proton; tentatively, we also include Z33Cs and 135Cs for which eq. (2.8) is to be used. The magnetic moment of the ground state and the B(1) value (3.8) of the M1 transition between the ~ + and 3 + states are known 11) for all four nuclei. The magnetic moment of the 3 + excited state (81 keV) of 133Cs has also been measured 12). These data are listed in table 1. The missing values (in brackets in the table) have been estimated by interpolating between known magnetic moments in the 3 + and 7 + shells 13). In table 1, column I, we have listed the coefficients bo which make the predictions (3.8) and (3.9) fit these experimental data. They are slowly varying functions of the f The explicit definition of 6 has unfortunately not been given in the work of Biedenharn and Rose 9). We have established expression (3.10) for 6 by using the same phase conventions as these authors (they also agree with ref. s)).
360
J.-P. AMIET AND P. HUGUENIN
occupation of the shell, even across the point where the ~+ shell is filled; this fact is consistent with the present model where calculated mixing parameters are not expected to change rapidly with increasing A. This aspect is not present in some other models, as, for example, that of quadrupole vibration ~o) (see concluding remarks in ref. 11)). TABLE 1 I Transition
Experimental data
AE jn ---~j'n" (keV) xaaCs
lanes XaOLa lllpr
{+ ~ ~-+ ~+ --* {+ ~+ ~ ½+ 3 + ~ ~+
II
81 250 166 145
a) Ref. 12).
~
B(M1) kt~-+ (cm 2 . 10_ao)
/t~+ 2.58 2.73 2.78 (2.8)
3.4 (3.6) (3.8) 4.0
0.49 0.93 0.52 0.44
(~(E2/M1) --0.16-4-0.01 a) 0.034-0.03 b) 0.07±0.02 b)
"~ ~
'~ ~
0.21 0.25 0.26 (0.26)
~"
~ ~
0.35 0.025 (0.30) 0.035 (0.25) 0.026 0.20 0.029
I
(E21 0.27 0.27 0.25 0.23
--13 --13 --13 +13
b) Ref. 18).
If we assume that the orbitals Z~÷ and Z~÷ can be approximated according to (2.4), one obtains the relations bo(3, 3) = fl~,
bo(~, ~) = fl~,
7 ~)[ 5 = [fllfl~l = x/bo(7, ~)bo(~, 7 s {). (3.11) Ibo(~,
In this approximation the off-diagonal matrix element of/~ is determined by the diagonal ones, i.e. the reduced transition rate B ( M I ) can be calculated up to the factor ( ¢] ¢ ' ) from the magnetic dipole moments of the initial and final states. The values fbo({, 3)1 deduced from the empirical values of bo(~-,-~) 7 7 and bo(}) by use of (3.11) are listed in table 1, column II. Even if one admits that the factor ( ¢1 ¢ ' ) could be as small as ½, the coefficients bo(-~, {) calculated in this way are still five times too large. It is true that (3.10) gives an upper limit, as a consequence of the Schwarz inequality
Ibo(j,j')l= <=
drr2]f j_,(r)] 2
?"
drr2]f j,_,(r)] 2
Io °
= x/bo(j,j)bo(j',j'),
but we do not think that this error could account for more than half of the discrepancy; we are thus left with a reduced transition rate B(M1), proportional to bo(~,~), 7 s which is about six times too large. This inconsistency shows that the present model cannot fully explain the magnetic dipole moments and M1 transitions at the same time.
SINGLE-NUCLEON ORBITALS
361
In particular, the good values obtained by Freed and Kisslinger 14) for the magnetic moments in the ~+ subshell strongly suggest that we have grossly overestimated the parameter b o (}, ½) by assuming that it alone is responsible for the deviation of /~+ from the Schmidt value. A set of coefficients fl,j~ has been calculated for all nuclei with closed j-shells by Mfiller 15), who solved eq. (5.10) of ref. 1); for this calculation the antisymmetry of the nuclear wave function has been neglected and only the first few coefficients fl,iof each orbital have been retained. From the values he computed for A = 140 with a filled ~+ shell, one obtains bo(~, ~r) = 0.030,
bo(~,~) = 0.19,
7 y) 5 = -0.056. bo(~,
(3.12)
These values apply to the case of 139La and 141Pr, but can also be tentatively used for Cs since they do not vary rapidly with the occupation number of the sub-shell ~+. The prediction (3.12) for the off-diagonal element bo(~-,~-) 7 s agrees well with the empirical values (table 1, column I) of all four nuclei if the core polarization (1< ~l~' >1 < 1) is taken into account. The coefficient bo(~,~-) 7 7 is about ] of the value required to explain the magnetic moment #(~ +) of a 39La ' but the other one, b o(~, ~2), fails completely to account for the deviation from the Schmidt value of#(~ +) of 141Pr. These results support the conclusion of the above analysis; the reduced transition rate tends to be too large, and the corrections to the Schmidt values for the magnetic dipole moment are too small. In fact, this situation is rather satisfying as B(M1) will certainly be reduced if one takes the core polarization and the pairing effect seriously into account; moreover, it is expected, as remarked above, that the addition of other types of couplings are going to increase the corrections to the magnetic moments. For instance, the corrections of Freed and Kisslinger 14) complement ours very well 7 + subshells. in the case of the ~s + and ~The coefficient b0(~-,~-) ~ s is predicted in the present model with a specific sign which can be checked experimentally in mixed M1 and E2 transitions. The parameters of interference 6 which have been measured 12, as) for the EM transitions between the + a n d s -+ states are listed in table 1. The theoretical values (column Ill) have been obtained by using (3.10). The factor a 2 - b 2 (defined in (3.7)) which characterises the quadrupole part of the transition has been calculated with the radial dependence of the orbitals computed by MSller as). One obtains a 2 - b 2 = - 16.3 fm 2. The discrepancy between the experimental and theoretical values for 6 is not surprising since, in the present case, the single-particle approximation yields values that are too small for E2 transitions (with or without parity mixture) and too lalge for M1 (see above). Anyway, our interest here lies mostly in the sign of 6 which should not be too sensitive to the method of calculation of the fl values. It is satisfying to find that the predicted signs fit the data in two cases. We have no good explanation for the discrepancy in the case of 139La" In fact, one would rather expect a good result for 141Pr
362
J.-P.
AMIET AND
P.
HUGUEN1N
and ~39La since, if one believes in similar properties for one particle and for one hole, 6 changes sign under the permutation of initial and final states. Moreover, we do not think that the signs of bo(½, ~-) for 139La and 141Pr should be opposite; the regularity of B(M1) practically excludes a change of signs which implies that bo goes through zero as a function of the number of nucleons. The very small value of 6 (compatible with zero) rather suggests that the E2 part of the transition changes its sign unexpectedly, making the single-particle estimate meaningless.
4. Direct Reactions
The Born approximation of the differential cross section for stripping reactions on a closed j-shells nucleus reads da
2re
dO
hv i
pf(E)l(e'~'f"p~bf(Z, N + 1)1V(xv - x.)l~ki(Z, N)e~ik' "t=P+xn~fd(Xp-- X,)>I 2, (4.1)
where the subscripts i and f refer to the initial and final states respectively, and n, p and d the neutron, proton and deuteron. By introducing in (4.1) the trial function ~=j,, defined in (2.8) for the final state one obtains, neglecting the antisymmetrization of the wave function, da_
dQ
2 2~
~"J hv i Pf(E)l('fd(Xp-- Xn)e~il'l "{x"+=*)i V(Xp-- X")[~O"jm(X)e'k" xP)12"
(4.2)
The only difference between the cross section calculated with and without parity mixing (usual shell model) is the appearance of the factor ~ in the former case (~,g = 1 means no parity mixing). The component q~_=j,, of the orbital (2.4), which describes the absorbed neutron, does not enter this stage of approximation. The smallest value of ~=~ that we expect (about ½), still differs from 1 by far less than the difference between the calculated and measured values of da/dI2. This explains why a parity mixture in the single-particle orbitals cannot be checked by fittings the cross sections of the stripping reactions with theoretical predictions made in Born approximation. The theoretical cross section for direct reactions on odd nuclei reveal a special feature of the model. The easiest cases to describe with the present formalism are the nuclei with closed j-shell plus an outside nucleon (for the pick-up) and a hole in a closed j-shell (for the stripping). We shall restrict ourselves to the first case, as the formulae for the second one are very similar. The wave function for the target is given in (2.8). We have to consider two channels t, for when the outside nucleon has been removed the core may be left in t The interest of this property has been first pointed out to us by Dr. H. D. Zeh.
363
SINGLE-NUCLEON ORBITALS
either state, 4 o . o r # o - . On the same assumption as above, one obtains de+ 2 21r dO = ~j~ ~ vi Pf(E)l(e*'kf' (xP+x*~fdlV ( x p - x,)lq~,~jm(X,)e 'k' "xP)l z,
de_
d f f = B2j
Pf(E)l(e~'kfO~"+"n)fdlV(xp--x")ltP-=jm(X')e'k''"")i
'
(4.3)
(4.4)
where ~ j and fl~j are the parity mixing parameters of the orbitals of the last neutron (see (2.4)). The/-values of these two channels differ by one unit, and the ratio o f their single-particle strengths is fl,,j/~z,,j, instead of zero, as the usual shell model predicts. This quotient should in most cases oscillate between 5 to 30 ~o. The fact that there are practically no known 0 - excited states in the spectrum of middle to heavy even nuclei may be deceptive, because many levels are still unassigned in the energy region where we would expect them (see sect. 5), and we hope that some 0 - states could be discovered if one looked specifically for them. It seems to us that in a precise pick-up experiment one should be able to detect these levels by means o f the channel (4.4), if it does in fact exist.
5. Nuclear Levels
It has been shown in ref. 1) how a tensor force can induce a parity mixture in the H F orbitals, provided it is strong enough compared to the increase in kinetic energy. The result is a shift of single-nucleon energy levels which contributes to the splitting o f the levels j = l _ ½ in the same way as a one-body ! • s term. Another question is to decide to what extent each of the spin-dependent forces contribute to this splitting; we believe, along with m a n y other authors 4, 17, 18) that the strong two-nucleon tensor force may play an essential role in this respect. We want to discuss here again a few effects of the parity mixing on nuclear spectra, but from a different point of view. We shall leave aside H F calculations and study directly the relation between the spectra of an even nucleus and the neighbouring odd nucleus by means of the projected wave functions (2.8). One of the reasons for not working here with H F is that the energy-splitting between the two core levels of opposite parity defined in (2.7) turns out completely wrong in this framework and can only be correctly calculated by taking the pairing energy into account (this is the same situation as for the inertial moments of deformed nuclei). Let us provisionally assume that an even nucleus has an excited state with J = 0, = - 1. We denote by ff~o- the corresponding state vector and by ~o+ that of the ground state. Both are supposed to be exact eigenvectors of the total Hamiltonian H. We now construct four wave functions,
(5.1)
364
J.-P.
A M I E T A N D P. H U G U E N I N
related to the next odd nucleus t; the quantum numbers n a n d j are by definition the parity and the total angular m o m e n t u m of the ground state. The two orbitals q~,j and q~_=j are supposed to yield the minima of the mean energy in the states q~*,j~o+ • I f necessary, the two state vectors cp,=i~ o- have to be made orthogonal to the t w o other ones by standard methods. The energy difference
As = (~o + q~-,,jlHl~o÷ e - , q > - ( ' ~ o ÷ ~o,,jlnl¢'o+ m,,j>
(5.2)
corresponds to a single-particle excitation in the sense of the shell model. I f As is larger than the excitation energy AE of the state ~ o - , then the mean energies in the four states defined in (5.1) look as indicated at the left of fig. I, it one makes the~ reasonable assumption:
Without mixin 9
With mixing
Type of s/ate
[2 J+~)
/1~
IAE
~
Corrective
12J -r )
5ingle part.
I1 j-rr 5
Coltect,¢e
II J ÷rr )
S ingLepa~t
Fig. 1. The effect of the parity mixing on nuclear spectra. The distinction between collective and_ single-particle states is made according to the discussion at the end of sect. 2.
Now, we define four new state vectors, [ljrc) = ~1 0o÷ ~P~j+/~l ~ o - q~-~j, I l j - = > = ai ~ o - ~=j+fl't ~o+ q~-=j, 12j~> = ~2 ~ o - ~°-~j+/~2 ~o÷ q~j, !
(5.3)'
t
12j-=> = ~2 ¢o+ q~-~j+/32 ¢ o - q ~ , which reduce to the four initial ones when the/Y parameters vanish. This corresponds to a parity mixture within the orbitals of the extra-core nucleon, as can be esaily seen from (2.8). It is a well-known fact that when one mixes two states having the same quantum numbers in the framework of a variational calculus, they repulse each other. * Throughout this section we drop irrelevant quantum numbers; moreover, the product_ if(x) ~p(x. . . . . xn) is supposed to be antisymmetrized.
SINGLE-NUCLEON ORBITALS
365
Thus, in the case of the vector (5.3), the splitting of the two intermediate levels in fig. 1 becomes larger and the ground-state energy decreases. Similarly, the above procedure can be used for the other states, ~o~,i,~o+ and ~ , j ' ~ o - , where j ' = j + 1. Without parity mixing, the two states qT_~jOo, and tp_,j,~o, are degenerate if the exchange energy (due to antisymmetrization) is neglected t. The two orbitals qT_~iand q~_,j, correspond to the degenerate doublet with l = j + ½ of the shell model without spin-oIbit coupling. When the state ~P-i,~o ÷ is mixed with q~,j, ~o- (according to (5.3)), its energy decreases like that of the groundstate. These preliminary remarks enable us now to discuss the filling of the next j-shell in the present scheme. If the energy difference A E and Ae ((5.2)) satisfy the inequality A E < Ae,
(5.4)
the level shift due to the parity mixing will split up the quasi-degenerate doublet, ~P-~j,~o+, ~o-~j~o- with the largest spin j ' at the bottom. This feature remains unchanged as one iterates the whole above procedure when adding pairs of nucleons to the core; then one finds that the shell ( j ' , - n ) is filled before the (j, - n ) shell as expected. On the other hand, if A E > Ae
holds, then both states ~p_,j, ~o÷ and q~-,j~o+ move downward ( ( ~ _ n j ~ O + is repulsed by tp,j~ o- which lies initially higher). In this case any crossing of levels is possible and a wrong spin will surely often be predicted for the ground state. In conclusion, the actual model cannot be meaningful if the condition (5.4) is not fulfilled. In other words, the (hypothetical) 0 - state of the core should not lie higher than the singleparticle excited state tp_~j. Since the energy difference A~ between the two states q ~ j ~ o ÷ is strongly conditioned by the centrifugal barrier, one may approximate it according to the shell model by h2 Ae ~ .... (2l+ 1), 2mR z
where R is the nucleus radius, m the nucleon mass and l = j + ½ The condition (5.4) becomes then AE <
h2 2mR 2
(2/+ 1).
(5.5)
In conclusion, an important test of the validity of the present model consists in confirming the presence of 0 - excited states in the right energy range (5.5) in the spectrum of even nuclei. We assume here that the effect of two-body L • S forces is negligible.
366
J . =P.
AMIET A N D
P.
HUGUENIN
6. Conclusion We believe to have shown how a mixture of parity in single-nucleon orbitals can easily explain the/-forbidden M1 transition and, at the same time, account for a large part of the deviation of magnetic dipole moments from the Schmidt values. Quantitatively the question remains open as in the case of the "spin-orbit" splitting induced by the tensor force 1). The first numerical results are nevertheless encouraging. We think that an improved version of the present model (with respect to the quadrupole transitions) could be particularly useful for a systematic understanding of the magnitude and the sign of the ratio 6 in mixed ~-transitions. Generally, it seems to us that a definite advantage could be gained in staiting with orbitals which are not eigenstates of the parity operator, rather than trying to reach the special properties that this mixture of parity conveys to the nuclear wave function through a perturbation expansion. To conclude, the decisive test for or against the legitimacy of the model lies less in elaborate calculations than in the discovery or identification of 0 - excited states of even nuclei in the approximate energy range that we have fixed in the previous section. We are aware that this is not a simple experimental task, but it seems worthwhile to decide whether or not the parity must be a good quantum number for singlenucleon orbitals, as is the case for the nuclear wave function. We are grateful to Professor J. H. D. Jensen for his stimulating interest in this subject and to Professor E. Bodenstedt for fruitful discussions. We are indebted to Dr. K. Dietrich who carefully read the manuscript.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
J.-P. Amiet and P. Huguenin, Nuclear Physics 46 (1963) 171 F. Villars, Proc. Int. School of Physics, E. Fermi Varenna (1961) ed. by V. F. Weisskopf (1963) L. Hulth6n and M. Sugawara, Encyclopedia of physics, ed. by S. Fltigge, Vol. 39 (1957) p. 1 A. Arima, Nuclear Physics 18 (1960) 196 A. Arima, H. Horie and M. Sano, Prog. Theor. Phys. 17 (1957) 567 I. Kelson, Phys. Lett. 16 (1965) 143 A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) U. Fano and G. Racah, Irreductible tensorial sets (Academic Press, New York, 1959) L. C. Biedenharn and M. E. Rose, Revs. Mod. Phys. 25 (1953) 729 R. A. Sorensen, Phys. Rev. 133 (1964) B281, 132 (1963) 2270 J. S. Geiger, R. L. Graham, I. Bergstr6m and F. Brown, Nuclear Physics 68 (1965) 352 E. Bodenstedt, H. J. KOrner and E. Matthias, Nuclear Physics U (1959) 584 E. Bodenstedt, Forts. Phys. 10 (1962) 321 N. Freed and L. S. Kisslinger, Nuclear Physics 25 (1961) 611 J. MOiler, Diplomarbeit, Universitiit Bonn (1963) unpublished J. N. Haag, D. A. Shirley and D. H. Templeton, Phys. Rev. 129 (1963) 1601 B. Jancovici, Prog. Theor. Phys. 22 (1959) 585 P. Goldhammer, Phys. Rev. 122 (1961) 207