Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

A NEW PRESCRIPTION FOR DETERMINING P A RTICLE-H O LE INTERACTIONS NEAR CLOSED SHELLS

K. HEYDE l, j. JOLIE 2, j. MOREAU 3, j. RYCKEBUSCH 2, M. WAROQUIER 4

Institute for Nuclear Physics, Proeftuinstraat 86, B-9000 Gent, Belgium

and

J.L. WOOD

Instituut voor Kern- en Stralingsfysika, Celestijnenlaan 200 D, B-3030 Heverlee (Leuven), Belgium and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Received 27 March 1986; revised manuscript received 10 June 1986

Starting from one-nucleon separation energies near doubly-closed shells as well as of the exper imental centroid for par- t i c l e -ho le excitations, a new prescript ion for obtaining residual par t ic le -ho le matr ix e lements is presented. It is shown tha t the major difference with the standard Blomqvist prescription is a shift in the energy that can be est imated in a graphical way.

In carrying out a study of particle-hole excitations in nuclei near doubly-closed shells, one needs a knowledge of the spherical single-particle energies e a (a =- n a, la,Ja, ma) and, in addition, one needs to know the residual particle-hole nuclear and Coulomb interactions energies. The Tamm-Dancoff (TDA) or Random-Phase Approximation (RPA) are the appro- priate means for calculating microscopically the low- lying particle-hole excitations in such nuclei. If the particle-hole excitations are rather pure, the energy can be obtained as

Ep_h(J) = e p - e h + ( p h - 1 ; J I V I p h - 1 ; J ) , (1)

where the residual interaction V contains both the nuclear and Coulomb part and the energies ep, e h can be determined in a self-consistent way from Hartree- Fock equations in a given nucleus A(Z, N).

1 Also at Rijksuniversiteit Gent, STVS & LEKF, Krijgslaan $9, B-9000 Gent, Belgium.

2 " I W O N L " fellow. 3 "Aspirant" at the NFWO. 4 ,,Bevoegdverklaard navorser" at the NFWO.

A semi-empirical method for relating the lowest, non-collective particle-hole configurations near doubly-closed shells to the experimental proton (or neutron) separation energies has been used [1-11 ] in recent years. This semi-empirical method, as first outlined by Blomqvist [ 1,2], uses the prescription

7r ff E p _ h = S p ( Z , N ) - S p ( Z + I , N ) + Vp_ h , (2a)

or

E~_ h = S n ( Z , N ) - S n ( Z , N + 1)+ V~_h, (28)

for proton or neutron particle-hole excitations, re- spectively. This enables one to obtain a residual particle-hole interaction from experimental protons (Sp) or neutron (Sn) separation energies and the known experimental energies Ep_ h. Here, Ep_ h is used throughout as the centroidenergy of known members of the particle-hole multiplet and thus Vp_h is an average matrix element (averaged over the possible spin values i.e. (ph- 1 ;JI Vlph- 1 ;j)).

A first conspicous feature in this method can be noted around 208pb, where a residual proton par- ticle-hole interaction Vp_ n for n 1 h9/23sl/12 of

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Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

-0 .324 MeV was obtained [3] (a recent evaluation by Daehnick [11] gives a value o f - 0 . 2 9 0 MeV) whereas for the neutron particle-hole excitation v2g9/23Pl/1 a value o f - 0 . 0 3 0 MeV resulted [2]. Correcting for the attractive Coulomb particle-hole interaction (a value of-~-0.200 MeV results in the Pb region), which is a well-established quantity [ 12- 14], differences between proton and neutron particle- hole matrix elements remain. Moreover, starting from a given two-body residual interaction, slightly re- pulsive (+0.050 to +0.250 MeV) matrix elements re- sult [14-16]. In the light of the above observations, we inspect the prescription (2) more closely.

It becomes clear that the remaining differences for proton and neutron particle-hole interaction energies deduced from eq. (2) and calculated, using different residual interactions, have to be found in the identi- fication of Sp(Z, N) - Sp(Z + 1,N) with the differ- ence of Hartree-Fock energies ep - e h for a given nucleus A(Z, iV). From Koopmans' theorem, we know that in the nucleus A(Z, N), the proton separa- ti~)n energy is an estimate for determining the proton energy i.e.

- -eh(Z , N ) = Sp(Z, N ) , - c p ( Z , N ) = Sp(Z + l , N ) , (3)

Eq. (3) can be obtained from Hartree-Fock theory (using density-independent interactions) under the assumption that single-particle wave functions in the nucleiA(Z,N),A + I (Z+ 1,N) andA - I ( Z - 1,N) do not change, but only the occupation of the orbital changes. This also implies identical single-particle energies in these nuclei. It is clear that in more realis-

tic situations, the above conditions are not fulfilled. K6hler and Lin [17] have therefore carried out a critical analysis of Koopmans' theorem.

When considering particle-hole excitations in a given nucleusA(Z, N), we need, in principle, both proton (or neutron) separation energies of eq. (3) in the same nucleus. Since the single-particle energies in the nucleus A + 1 (Z + 1, N) are slightly shifted from their position in the nucleus A(Z, N) (see fig. 1) by an amount ZSep, Sp(Z + 1,N) is not the correct quantity to consider. Thus, there exists besides the Coulomb matrix element and the nuclear particle- hole matrix element a third term, which takes into account the difference in single-particle energy for a given orbital jp above the Fermi level in a nucleus A(Z, N) compared to the nucleus A + I(Z + 1,N) i.e.

6p(Z + 1 ,N) = c p ( Z , N ) + Aep , (4)

with dXep a positive quantity. The latter effect, being a monopole correction, is depicted diagrammatically in fig. 1. It clearly shows that -Sp(Z + 1 ,N) is giving a too small estimate of the ep value to be used in the actual calculations of particle-hole states in nuclei. We then obtain the p - h excitation energy as

g p _ h ( J ) = Sp(Z, N) - Sp(Z + I ,N) - Acp

+ (ph-1;JlV[ph-1;J), (5)

and similarly for neutron excitations. There now exist different possibilities to deter-

mine Aep : (i) Going beyond the approximations implied by

%

~ ' E : h.

- V 0

l/ Sp ( Z , N ) - - r - - - - - -

I E:p"

E h '

-Vo+6

/ S p ( ;=*I,N ) - - r ~

(i~, N) (7.1, N)

Fig. 1. Schematic representation of the changes induced in the proton single-particle energy (and thus of the proton separation ener- gy) (see also eqs. (3) and (4)) in a Woods-Saxon potential (Z, N) and (Z + 1, N).

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Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

Koopmans ' theorem in H a r t r e e - F o c k theory and/or using density dependent interactions has been studied [ 17]. The shift in single-particle energy Aep when adding one proton to a nucleus A(Z, N) (self- energy correction) leads to the expression

e p ( Z + 1 ,N) = ep(Z, N) + ~ (pp ' lVIpp ' )np, (6) p,

where the summation (p ') goes over the orbitals available to the extra particle and n~, denotes the

Y number of particles in the orbital p . Because of the presence of the larger, attractive Coulomb part in V, one gets

[6p(Z + 1, N)[ < [ep (Z, N ) [ . (7)

In deriving eq. (6), no rearrangement nor changes in the external potent ial determining ep were included. Such effects will bring about an extra variation of ep when going from a nucleus A(Z, N) to a nucleus A + l(z + 1,U).

(ii) One can calculate Aep making use of a W o o d s - Saxon potential [18]. For, respectively, the A = 20, 40, 60, 100, 132 and 208 regions with the W o o d s - Saxon potential itself being dependent on A, Z and N in an explicit way, Aep was calculated. The results are shown in table 1 where the shift in single-particle energy for nuclei differing by one unit in charge is clearly seen. This lat ter energy shift was also pointed out by Bernes et al. [4].

(iii) Within the liquid drop model, the difference in separating a proton from the nucleus A + I (Z + 1, N) and A(Z, N) includes a part that is not deter- mining the par t ic le -ho le interaction in a given nucleus. This quanti ty can be seen to come from the Coulomb and symmetry energy by considering the Bethe-Weisz~icker mass formula

B(A, Z )=alA +azA2/3 +a3Z2A-1/3

+ a 4 ( N - Z)2A - 1 + 6pair , (8)

where the nominal (globally) fi t ted values o f a 3 and a 4 are - 0 . 7 and - 2 8 MeV, respectively. Then, using

Sp(A)= M(A - 1,Z - 1 , N ) + M ( 1 H ) - M ( A , Z, N) , (9)

with M(A, Z, N) = ZM(1 H) + NM(1 n) - B(A, Z, N), eq. (2) reduces to

/T /T E p _ h = _ 2 a 3 A - 1 / 3 _ 2 a 4 A - 1 + Vp_ h

(to lowest o rder ) . (10)

Table 1 The single-particle energy for protons in a Woods-Saxon po- tential (in MeV) are given for different closed-shell regions (A = 20, 40, 60, 100, 132,208). In a given region, the varia- tion in single-particle energy results from the change in vol- ume (A-dependence), Coulomb (Z-dependence) and symmetry terms (N-Z dependence) in the Woods-Saxon potential [18]. The Woods-Saoxn potential used has the form

V o = [-51 + ~'z 33(N-Z)[A[ (MeV),

Vas= [22 - rz 14(N-Z)[A] (MeV),

VW.S. = V o (1 + exp[(r-Ro)]a]) -1

- V 1 s(d/dr) (1 + exp[(r-R'0)/a] )-1 .

We use in all cases r0 = 1.27 fm, rl s = 1.15 fm; a = 0.7 fm with Ro = roA 1/3 ; Rb =- rx s A 1/3. The numbers between brackets for the A = 60 region are calculated using a slightly changed Vo, p = -59 MeV.

A Nucleus

lP3/2

20 200 -17.652 21F -16.167 22Ne -14.657

40

ld5/2 2Sl/2. ld3/2

4°Ca -12.639 -7.381 -3.602 41Sc -11.814 -6.568 -2.915 42Ti -10.984 -5.749 -2.220

60

lf7/2

6°Ni -9.089 (-13.431) 61Cu -8.509 (-12.767) 62Zn -7.926 (-12.300)

100

2Pl/2 2P3/2 lfs/2

l°°Zr -11.752 -13.613 -13.101 l°rNb -11.333 -13.178 -12.711 102Mo -10.913 -12.742 -12.320

132

lg9/2 2pl/2 2P3/2

132Sn -15.265 -15.904 -17.447 laaSb -14.909 -15.611 -17.063 la4Te -14.553 -15.238 -16.680

208

lhl 1/2 2d3/2 3Sl/2 lh9/2

2°7T1 -9.440 -8.474 -7.946 -3.908 2°spb -9.181 -8.197 -7.662 -3.667 2°9Bi -8.910 -7.919 -7.378 -3.424 210po -8.650 -7.643 -7.094 -3.183

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Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

- [ .~--- SpJZ ,N)- SplZ 1,N)

2 - ~ _ ex tr.

D p o r t .

n,- 13_ I o I I I I I

i[ ~÷1 i~÷3 ;z.5 Z.7 - - PROTON NUMBER

Fig. 2. Outline of the new prescription in order to determine the unperturbed particle-hole energy in a given nucleus A(Z, N), where Z, N denotes the shell closure, from empirical proton separation energies. The dashed line denotes the stan- dard Blomqvist prescription; the full line the new prescrip- tion. Similar figures can be drawn for S2p. When considering Sn, S2n, curves as a function of changing neutron number have to be drawn.

The term - 2 a 3 A - 1 / 3 is the Coulomb contribution and - 2 a 4 A - 1 the symmetry energy contribution: for the above values o f a 3 and a4, these contributions would be 240 keV and 270 keV, respectively, for A = 208.

We now have developed the following method in order to obtain the particle-hole interaction from proton separation energies, using eq. (5). When plotting the proton (neutron) separation energy (see fig. 2), instead of taking the differences Sp(Z, N) - Sp(Z + 1, N) (or Sn(Z, iV) - Sn(Z, N + 1)), one ex'tra- polates the curve of Sp (or Sn) values above the closed-shell gap discontinuity down to the value of Z (or N) corresponding to the closed-shell nucleus. Thus, eq. (5) will give the residual particle-hole (nuclear plus Coulomb) interaction matrix elements. A simple prescription for determining the unper- turbed particle-hole energy is now obtained. Values of Aep so obtained, starting from the experimental proton and neutron separation energies are given in

Table 2 Values of Aep (z~en) obtained from proton (neutron) separa- tion energies using the graphical method as outlined in fig. 2 and in the text.

(Z, N) Aep (Z, N) ae n

82, 126 0.407 82, 126 0.062 50, 82 0.515 50, 50 40, 60 0.612 40, 50 0.230 40, 50 0.535 28, 28 0.623 28, 32 0.995 20, 28 0.353 20, 28 0.783 20, 20 0.215

8, 12 1.170

table 2. In comparing these results with the calculated Woods-Saxon energy shifts Aewoods_Saxon, ~i similar trend, as a function of the atomic mass num- ber A, can be observed. For the 208pb nucleus, using the Blomqvist prescription [2] one obtains an aver-

7r age particle-hole interaction of Vp_ h = -0 .324 MeV; the new prescription gives V ~r - = 0 025 MeV,

p - n "

a value which is much smaller and slightly positive. 71" The values of Vp_ h so obtained still contain the

diagonal Coulomb particle-hole interaction. Cor- recting for the latter quantities, the nuclear particle- hole interaction matrix elements can be obtained and turn out to be of the order of the theoretical nuclear particle-hole matrix elements, calculated for different residual interactions [ 14-16] . The matrix elements, obtained in the empirical way give a rather good de- scription o f the average behaviour and can be used to determine the approximate energy for more complex particle-hole excitations (2p -2h ; 4 p - 4 h ; ...) [ 19].

It is essential to remark that, in extracting part icle- hole matrix elements, we make the assumption of a pure particle or hole character for the ground state in the A + 1 or A - 1 nucleus, respectively. For the lowest-lying single-particle ( -hole) states, fragmenta- tion will often result which has consequences for the definition of the single-particle(-hole) energies in odd-mass nuclei. Differences in the Aep values, when comparing the results of tables 1 and 2 can probably be partly due to such fragmentation. In that respect, a detailed discussion has been given by Baranger [20]. We moreover point out that a large literature exists on the precise definition of single-particle(-hole) en- ergies, when starting from the experimental situation in actual nuclei e.g. Baranger [20], Dieperink et al.

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Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

[21], Kol tun [22], Engelbrecht and Weidemiiller [23].

Before concluding, we point out that the deter- ruination of par t i c le -ho le interactions from experi- mental data has received much at tention over the last decades. In light nuclei in particular, studies were carried out by Bansal, French [24] and Zamick [25]. There, it was shown that the par t ic le -ho le interac- tion is largely dependent on the isospin of the states considered i.e. T = 0 states are more bound than T = 1 states. Similar conclusions were obtained for medium heavy nuclei by Jahn et al. [26] and Mairle [27]. The results obtained here (heavy nuclei), where par t i c le -ho le excitations are either of pro ton or neu- tron character are not discussed within the context of isospin, although here too (see fig. 2), simple linear relations for the single-part icle(-hole) energies result [22,28]. We also mention the extensive studies by Schiffer and True [9] and more recently by Daehnick [ 11 ] where a large compilat ion of two-body residual interaction matrix elements is presented.

In the present article, we have shown how residual par t ic le -ho le matrix elements can be obtained start- ing from one-nucleon separation energies and the knowledge of the experimental pa r t i c le -ho le centroid energy in single-closed shell nuclei. We have also shown that the major difference with the Blomqvist prescription results due to contr ibut ions from Coulomb and symmetry energy. We point out that a simple graphical method can be used. Moreover, we indicate that the matr ix elements thus obtained can be used to calculate the approximate energy for n p - nh excitations.

The two methods, discussed above, result since there is no unique way to divide the total energy of a state into a one-body and a two-body part. Depend- ing on the prescription one uses (eq. (2) or eq. (5)), differences in both the one-body and two-body terms result. All derived quantities, however, do not differ when consistently using one or another prescription. Whereas the Blomqvist method is more phenomeno- logical, the present method results into single-particle energies and par t i c le -ho le interaction matr ix elements that can be compared with H a r t r e e - F o c k energies and calculated two-body matr ix elements for given residual interactions.

The authors are grateful to the IIKW and the NFWO for constant support . They are indebted to

J. Blomqvist, R.A. Meyer, R.F. Casten, P. Van Isacker and G. Wenes for interesting discussions. This work has been performed in part under contract DE- AS05-80ER-10599 of the US Department of Energy.

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