With a residual interaction consisting of density-independent two- and three-body parts, we obtain within Hartee-Fock-Bogolyubov theory at the same time good results on the odd-even staggering of charge radii and on trends of the pairing energies. We present results for Sn and some isotopic chains in the Ba region

Regge, U.

Zawischa, Dieter

English

Institutional Repository of Leibniz Universität Hannover

VOLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JULY 1988
Isotopic Dependence of Nuclear Charge Radii and Pairing Energies
U. Regge and D. Zawischa
Institut fu rT'heoretische Physik, Universitct't Hannover, West Germany
(Received 5 February 1988)
With a residual interaction consisting of density-independent two- and three-body parts, we obtain
within Hartree-Fock-Bogolyubov theory at the same time good results on the odd-even staggering of
charge radii and on trends of the pairing energies. We present results for Sn and some isotopic chains in
the Ba region.
PACS numbers: 21.10.Dr, 21.10.Ft, 21.30.+y, 27.60.+j
In a recent paper it has been shown that the staggering of nuclear charge radii can easily be explained if one as-
sumes a suitable three- or four-body part in the effective residual interaction. Here by a more rigorous treatment using
a repulsive three-body contact force we reproduce systematic trends of the pairing energies and obtain in addition im-
proved results on the charge radii.
For the derivation of our equations we start with the Hamiltonian
H=ge;a; a/+ 2 g Vjjt/a( a, a/ak+ 6 g Vi,)(m„a(a, aka„aml,i, 1 .(. ) i i (,() .i i (I)
I tj kl ij kfmn
where V~J())/and V~j~j)/ „are not antisymmetrized. With the employment of the method of Green's functions, in the re-
sulting hierarchy of equations the n-particle function is connected with the (n+ I )-particle function by the two-body in-
teraction, and with the (n+2)-particle function by V/3). Following Gorkov3 we introduce the anomalous Green's func-
tions F,Fi which obey equations of motion analogous to those for G, and factorize G 2) as well as G('). After Fourier
transformation, we finally arrive at equations of standard Hartree-Fock-Bogolyubov form
G;j(((+co) =G;, (/u+co)+gk(Gg(jt+co)(rk(G( (//+co) ak(F/', (it+co—)l,
Fij (/u+ c0) gk/ G l (/tt co) k k(F(j(jt + co) +k(G(j(/t + co) l.
(2a)
(2b)
(4)
Equations (6) represent a self-consistent Hartree-Fock
problem and may be solved by iterative diagonalization
of h;j =e;8j+I j. In the basis fv;j of the corresponding
eigenstates, p and thus G are diagonal, so that it will
be convenient to solve Eqs. (5) just in this basis. This is
done by the standard iteration procedure.
After adding and subtracting the term g;t r,~a; aj, we
transform the Hamiltonian (1) into this basis. Writing
Eqs. (3) and (4) as
(5a)G=G+G BIG —G~,
F =G SI *F 6 j). G-, —- (sb)
The chemical potential jt is fixed by average particle-number conservation. The quantities I and 5 are given by
~
—(2) I ~ —(3) m —(3)
rij ZVi(jkPk(+ 2 Z Vi(njkmPmnPk(+ 4 w Vikljmn(ttmn/k/~
kl kfmn kfmn
I m-(2) I m -(3)
~ij 2 ZV(jk(tt/k(+ 2 Zi Vijnk(mpmntj/k/~
kf klmn
with p;j =(2tci) 'fdcoG;, (co) being the normal and
p;( = (2/ci) ' fdco Fl (co) the anomalous (or pairing)
density, and Vjk( and V;,k/ „denoting fully antisym-
—(2} (3)
metrized matrix elements.
Equations (2) are solved in two steps. We write (in an
obvious shorthand notation) I =I +bI and obtain by
iteration and resummation the equations
where the subscript —indicates the negative-energy ar-
gument. G and I" are determined by
(6a)
p(2)+ p(p)+ p(y)
I p(2)+ I &(p)
(7)
(8)
r'=V"' '+-,' V"' ' ',
,
', =„".G, ( ). (6b)
its ground-state expectation value becomes
g, =pe,p„„+g p„,„,(-,' r„'2),+ —,' r„'&„),—r,', „,)+ —,' gy,*„p„,„,.
VI V2 Vl V2
1988 The American Physical Society
VOLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JULY 1988
The term containin~ I corrects for those contribu-
tions of V and V already contained in the single-
particle levels e„. Working from the beginning in this
representation, we would have obtained directly Eqs. (5).
But the procedure sketched above gives more insight into
the way double countings are to be avoided, and a
definite prescription to calculate I . Expression (9) gives
the correct contributions of the residual interaction to
the energy. Starting with a Hamiltonian containing a
single-particle potential, the absolute value of the
ground-state energy cannot be calculated, because we
cannot tell how much of the interaction defining this po-
tential will be counted twice. But taking the same initial
potential for a whole chain of nuclei, we are able to cal-
culate energy differences, the unknown terms dropping
out. The assumption is, of course, that the residual in-
teraction yields a good approximation to BI, i.e., the
change in the single-particle potential.
Once arrived at Eqs. (5) and (9) we take as single-
particle basis the eigenstates of a Woods-Saxon poten-
tial. '
Deriving Eqs. (5) within the Green's-function formal-
ism we see that in the definitions of I and 6 there should
appear an effective interaction rather than the bare one,
accounting for higher-order contributions as shown by
Migdal. 6 The arguments concerning the difference be-
tween particle-particle (pp) and particle-hole (ph) in-
teraction apply to the three-body force as well. Because
of time-reversal symmetry the pph and phh matrix ele-
ments in the second terms of Eqs. (3) and (4) should be
parametrized in the same way, whereas the third term in
Eq. (3) is of ppp type and may be treated differently.
Extending the argumentation of II'lemt, Moszkowski,
and Speth we antisymmetrize the three-body part in the
pp but not in the ph channels. As a bonus this avoids the
problem of the so-called "nuclear ferromagnetism" in
systems without spin saturation, because all the critical
F "=C(fi5(r) —r2), (lo)
where f is different for like and for unlike particles. In
terms of the more familiar fo,fo we have
ft =fo+fo' f.=fo fo'.
For the pp (pairing) force (acting only between like par-
contributions proportional to the spin densities arise from
exchange terms of the interaction matrix elements. In
this connection we note that, regarding the form of the p
dependence, our prescription is equivalent to that of
Backman, Jackson, and Speth, '0 whereas the parametri-
zation of Waroquier, Heyde, and Wenes" leads to a par-
tial suppression of the exchange terms, i.e., no proper an-
tisymmetrization as well.
It is known that any density dependence in some
sense corresponds to many-body parts in the effective in-
teraction. But there is one important thing to note: In
contrast to the Hartree-Fock case, in the presence of
pairing a three-body force gives rise to contributions
which could only be interpreted in terms of a two-
particle interaction, if one allowed for a generalized den-
sity dependence including the anomalous density
With an interaction like, e.g. , that of Ref. 10, Decharge
and Gogny' or Dobaczewski, Flocard, and Treiner' the
Green's-function formalism would not give the term
—,
' I ~~~ in Eq. (7), which is the essential one producing
the odd-even staggering of the charge radii (and the kink
at magic numbers) as has been explained in detail in
Ref. l.
For the two-body part of the effective interaction we
make an Ansatz like that of Migdal, but without density
interpolation, because we assume all the density depen-
dence to be produced by V 3~. In the ph channel we
write
1.2 0.4
0.8
&r &A, (fm )
122' 124' A
r
/
/
0.4 -0.4
12+ 12+ 12+ 12+ 13+ 132) 134) 13+ 13
E-- y
-z
exper ident
—— pr esent theory (a)
pr esent theory (b)
FIG. 1. Relative mean-square charge radii for the Sn and Ba isotopic chains. Experimental values taken from Ref. 16; present
theory (a) results from Ref. 1; present theory (b) this work.
150
VoLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JUr +1988
1.5'
Qp(Z, N) (Mev) TABLE I. Parameters of the effective interaction. The
units of V and V are cubed femtometers, the other param-
eters are dimensionless. Cp =386 MeV fm .
Lp V (P) V (P) V (y) V (y)
Sn —0.60
Ba —0.50
—2.45 —1.90 —1 ~ 80 10.0 16.0 12.0 31.0
—2.40 —2. 13 —1.81 10.0 16.0 19.0 58.0
0.5
0. 10
La
Sa
Cs
Xe
O. il O. po' (N-z) /~
FIG. 2. Proton pairing energies defined according to Ref. 2.
The lines are drawn to guide the eye. Dashed lines: experi-
mental values calculated from Ref. 17; solid lines: present
work.
ticles) we have"
F~~ =COLS(r~ —r2) —,' [1 —P (1,2)]. (12)
V~', uzi or vvv interaction,
V (a) V„', zzv or xvv interaction, (13)
where a =p, p. In Ref. 1 only the p-dependent contribu-
tions of V '), i.e., the last term in Eq. (3), were taken
into account, whereas the other ones depending on the
normal density p were considered to be already well ap-
proximated by the fixed density interpolation of the two-
body force used. Then density changes enter only linear-
ly into Eqs. (7) and (8).
In the present calculations all terms originating from
L is taken slightly difl'erent for proton-proton and
neutron-neutron interaction, because in heavy nuclei
with NcZ the configuration spaces for protons and neu-
trons are generally not equivalent.
For the three-body part we take a 6 interaction with
different strength factors CoV3 and CoV3' in the pph
and ppp channel, respectively, the superscripts corre-
sponding to those in Eqs. (7) and (8). We again distin-
guish between interaction of like or unlike particles in
each of these channels:
the three-body component are treated self-consistently,
giving rise to an additional quadratic dependence of I
and 5 on variations of p. This yields on the one hand the
essential rearrangement term —,' g„,,p„„,I „~, in Eq. (9),
without which even binding-energy differences cannot
sensibly be calculated. On the other hand, I (~), depend-
ing on p and contributing to Eq. (7) with the opposite
sign as compared to I (2), produces the negative curva-
ture of the radii as a function of neutron number, experi-
mentally observed, ' ' e.g. , in the tin isotopes (Fig. 1).
As mentioned above, the odd-even staggering as well as
the kink at the closed shell exhibited by our Ba results is
produced by I ~ . We want to stress that these are no
deformation effects, since we are working entirely within
a spherical model [all densities entering Eqs. (5) being
explicitly rotationally invariant '2).
The treatment of 5 ~ is of little influence on the
charge radius, but it significantly affects the pairing en-
ergies. When adding neutrons, this contribution to the
pair potential, also originating from the repulsive three-
body component of our interaction, gets stronger, which
is equivalent to a reduction of the effective pairing in-
teraction strength. This, in our opinion, is the cause of
the isotopic efl'ect in the pairing energies observed by
Vogel, Johnson, and Hansen. 2 In particular, we get the
decrease of the proton pairing with increasing neutron
number (Fig. 2) without the invocation of proton-
neutron pairing. '
We calculate the quantity Ap(Z, N) defined according
to Ref. 2 from Eq. (9) evaluated for different proton and
neutron numbers with ' Ba as reference nucleus. The
interaction and potential parameters are given in Tables
I and II, respectively. All single-particle states up to
three major shells above the Fermi energy are included
in the configuration space. In the Ba region the level
spacings around eF are adjusted according to the experi-
TABLE II. Parameters of the Woods-Saxon potential defined according to Ref. 5. Except
for Vp the Ba values are taken from Ref. 19; b is the oscillator length.
Vp
(Mev)
p n
Rp
(fm)
Qp
(fm)
p n p
RLs
(fm)
QLg
(fm) Rc b
(fm) (fm)
Sn 52.9 43.5 6.33
Ba 57.5 45.8 6.63
6.33 0.67 0.67 32 32 6.33 6.33 0.67 0.67 6.33 2.236
6.40 0.79 0.66 27 32 6.40 6.19 0.59 0.64 6.58 2.277
151
VOLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JULY 1988
mental values quoted in Ref. 19, in Sn the —", (in-
truder) state is shifted down by 1.3 MeV to obtain the
correct level ordering.
Figure 2 shows our results for dp(Z, X) for some iso-
topic chains around Ba compared to the experimental
values. ' Apart from shell effects the decreasing trend is
reproduced very well. Note that all these results are ob-
tained without any parameter variation. The slightly
different parameters for Sn can be tolerated in view of
the extremely simple form of our interaction.
Thus we conclude that with a suitable three-body part
in the effective interaction one is easily able to explain at
the same time general properties of nuclear charge radii
and pairing energies.
This work was supported by the Deutsche Forschungs-
gemeinschaft, No. Za-119. We thank Dr. R. Stapel for
many valuable discussions. Numerical calculations have
been performed at the Regionales Rechenzentrum fiir
Niedersachsen (RRZN).
'D. Zawischa, U. Regge, and R. Stapel, Phys. Lett. B 185,
299 (1987).
P. Vogel, B. Johnson, and P. G. Hansen, Phys. Lett. 139B,
227 (1984).
3L. P. Gorkov, Zh. Eksp. Teor. Fiz. 34, 505 (1958) [Sov.
Phys. JETP 7, 347 (1958)].
4P. Ring and P. Schuck, The Nuclear Many-Body Problem
(Springer-Verlag, Berlin, 1980).
~J. Speth, E. Werner, and W. Wild, Phys. Rep. 33C, 127
(1977).
sA. B. Migdal, Theory of Finite Fermi Systems and Appli
cation to Atomic Nuclei (Wiley, New York, 1967).
7V. Klemt, S. A. Moszkowski, and J. Speth, Phys. Rev. C 14,
302 (1976).
SB. D. Chang, Phys. Lett. 56B, 205 (1975).
9D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626 (1972).
' S.-O. Backman, A. D. Jackson, and J. Speth, Phys. Lett.
56B, 209 (1975).
' ' M. Waroquier, K. Heyde, and G. Wenes, Nucl. Phys.
A404, 269 (1983).
'2J. Decharge and D. Gogny, Phys. Rev. C 21, 1568 (1980).
' J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys.
A422, 103 (1984).
14D. Zawischa, Phys. Lett. 155B, 309 (1985).
' P. Aufmuth, K. Heilig, and A. Steudel, At. Data Nucl.
Data Tables 37, 455 (1987), and references therein.
16K. Heilig, Hyperfine Interact. 24-26, 349 (1985).
17A. H. Wapstra and G. Audi, Nucl. Phys. A432, 1 (1985).
' A. S. Jensen and A. Miranda, Nucl. Phys. A449, 331
(1986).
19C. Conci, thesis, Jiilich University, 1984 (unpublished).
152


Isotopic dependence of nuclear charge radii and pairing energies

Institutionelles Repositorium der Leibniz Universität Hannover

VOLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JULY 1988Isotopic Dependence of Nuclear Charge Radii and Pairing EnergiesU. Regge and D. ZawischaInstitut fu rT'heoretische Physik, Universitct't Hannover, West Germany(Received 5 February 1988)With a residual interaction consisting of density-independent two- and three-body parts, we obtainwithin Hartree-Fock-Bogolyubov theory at the same time good results on the odd-even staggering ofcharge radii and on trends of the pairing energies. We present results for Sn and some isotopic chains inthe Ba region.PACS numbers: 21.10.Dr, 21.10.Ft, 21.30.+y, 27.60.+jIn a recent paper it has been shown that the staggering of nuclear charge radii can easily be explained if one as-sumes a suitable three- or four-body part in the effective residual interaction. Here by a more rigorous treatment usinga repulsive three-body contact force we reproduce systematic trends of the pairing energies and obtain in addition im-proved results on the charge radii.For the derivation of our equations we start with the HamiltonianH=ge;a; a/+ 2 g Vjjt/a( a, a/ak+ 6 g Vi,)(m„a(a, aka„aml,i, 1 .(. ) i i (,() .i i (I)I tj kl ij kfmnwhere V~J())/and V~j~j)/ „are not antisymmetrized. With the employment of the method of Green's functions, in the re-sulting hierarchy of equations the n-particle function is connected with the (n+ I )-particle function by the two-body in-teraction, and with the (n+2)-particle function by V/3). Following Gorkov3 we introduce the anomalous Green's func-tions F,Fi which obey equations of motion analogous to those for G, and factorize G 2) as well as G('). After Fouriertransformation, we finally arrive at equations of standard Hartree-Fock-Bogolyubov formG;j(((+co) =G;, (/u+co)+gk(Gg(jt+co)(rk(G( (//+co) ak(F/', (it+co—)l,Fij (/u+ c0) gk/ G l (/tt co) k k(F(j(jt + co) +k(G(j(/t + co) l.(2a)(2b)(4)Equations (6) represent a self-consistent Hartree-Fockproblem and may be solved by iterative diagonalizationof h;j =e;8j+I j. In the basis fv;j of the correspondingeigenstates, p and thus G are diagonal, so that it willbe convenient to solve Eqs. (5) just in this basis. This isdone by the standard iteration procedure.After adding and subtracting the term g;t r,~a; aj, wetransform the Hamiltonian (1) into this basis. WritingEqs. (3) and (4) as(5a)G=G+G BIG —G~,F =G SI *F 6 j). G-, —- (sb)The chemical potential jt is fixed by average particle-number conservation. The quantities I and 5 are given by~—(2) I ~ —(3) m —(3)rij ZVi(jkPk(+ 2 Z Vi(njkmPmnPk(+ 4 w Vikljmn(ttmn/k/~kl kfmn kfmnI m-(2) I m -(3)~ij 2 ZV(jk(tt/k(+ 2 Zi Vijnk(mpmntj/k/~kf klmnwith p;j =(2tci) 'fdcoG;, (co) being the normal andp;( = (2/ci) ' fdco Fl (co) the anomalous (or pairing)density, and Vjk( and V;,k/ „denoting fully antisym-—(2} (3)metrized matrix elements.Equations (2) are solved in two steps. We write (in anobvious shorthand notation) I =I +bI and obtain byiteration and resummation the equationswhere the subscript —indicates the negative-energy ar-gument. G and I" are determined by(6a)p(2)+ p(p)+ p(y)I p(2)+ I &(p)(7)(8)r'=V"' '+-,' V"' ' ',,', =„".G, ( ). (6b)its ground-state expectation value becomesg, =pe,p„„+g p„,„,(-,' r„'2),+ —,' r„'&„),—r,', „,)+ —,' gy,*„p„,„,.VI V2 Vl V21988 The American Physical SocietyVOLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JULY 1988The term containin~ I corrects for those contribu-tions of V and V already contained in the single-particle levels e„. Working from the beginning in thisrepresentation, we would have obtained directly Eqs. (5).But the procedure sketched above gives more insight intothe way double countings are to be avoided, and adefinite prescription to calculate I . Expression (9) givesthe correct contributions of the residual interaction tothe energy. Starting with a Hamiltonian containing asingle-particle potential, the absolute value of theground-state energy cannot be calculated, because wecannot tell how much of the interaction defining this po-tential will be counted twice. But taking the same initialpotential for a whole chain of nuclei, we are able to cal-culate energy differences, the unknown terms droppingout. The assumption is, of course, that the residual in-teraction yields a good approximation to BI, i.e., thechange in the single-particle potential.Once arrived at Eqs. (5) and (9) we take as single-particle basis the eigenstates of a Woods-Saxon poten-tial. 'Deriving Eqs. (5) within the Green's-function formal-ism we see that in the definitions of I and 6 there shouldappear an effective interaction rather than the bare one,accounting for higher-order contributions as shown byMigdal. 6 The arguments concerning the difference be-tween particle-particle (pp) and particle-hole (ph) in-teraction apply to the three-body force as well. Becauseof time-reversal symmetry the pph and phh matrix ele-ments in the second terms of Eqs. (3) and (4) should beparametrized in the same way, whereas the third term inEq. (3) is of ppp type and may be treated differently.Extending the argumentation of II'lemt, Moszkowski,and Speth we antisymmetrize the three-body part in thepp but not in the ph channels. As a bonus this avoids theproblem of the so-called "nuclear ferromagnetism" insystems without spin saturation, because all the criticalF "=C(fi5(r) —r2), (lo)where f is different for like and for unlike particles. Interms of the more familiar fo,fo we haveft =fo+fo' f.=fo fo'.For the pp (pairing) force (acting only between like par-contributions proportional to the spin densities arise fromexchange terms of the interaction matrix elements. Inthis connection we note that, regarding the form of the pdependence, our prescription is equivalent to that ofBackman, Jackson, and Speth, '0 whereas the parametri-zation of Waroquier, Heyde, and Wenes" leads to a par-tial suppression of the exchange terms, i.e., no proper an-tisymmetrization as well.It is known that any density dependence in somesense corresponds to many-body parts in the effective in-teraction. But there is one important thing to note: Incontrast to the Hartree-Fock case, in the presence ofpairing a three-body force gives rise to contributionswhich could only be interpreted in terms of a two-particle interaction, if one allowed for a generalized den-sity dependence including the anomalous densityWith an interaction like, e.g. , that of Ref. 10, Dechargeand Gogny' or Dobaczewski, Flocard, and Treiner' theGreen's-function formalism would not give the term—,' I ~~~ in Eq. (7), which is the essential one producingthe odd-even staggering of the charge radii (and the kinkat magic numbers) as has been explained in detail inRef. l.For the two-body part of the effective interaction wemake an Ansatz like that of Migdal, but without densityinterpolation, because we assume all the density depen-dence to be produced by V 3~. In the ph channel wewrite1.2 0.40.8&r &A, (fm )122' 124' Ar//0.4 -0.412+ 12+ 12+ 12+ 13+ 132) 134) 13+ 13E-- y-zexper ident—— pr esent theory (a)pr esent theory (b)FIG. 1. Relative mean-square charge radii for the Sn and Ba isotopic chains. Experimental values taken from Ref. 16; presenttheory (a) results from Ref. 1; present theory (b) this work.150VoLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JUr +19881.5'Qp(Z, N) (Mev) TABLE I. Parameters of the effective interaction. Theunits of V and V are cubed femtometers, the other param-eters are dimensionless. Cp =386 MeV fm .Lp V (P) V (P) V (y) V (y)Sn —0.60Ba —0.50—2.45 —1.90 —1 ~ 80 10.0 16.0 12.0 31.0—2.40 —2. 13 —1.81 10.0 16.0 19.0 58.00.50. 10LaSaCsXeO. il O. po' (N-z) /~FIG. 2. Proton pairing energies defined according to Ref. 2.The lines are drawn to guide the eye. Dashed lines: experi-mental values calculated from Ref. 17; solid lines: presentwork.ticles) we have"F~~ =COLS(r~ —r2) —,' [1 —P (1,2)]. (12)V~', uzi or vvv interaction,V (a) V„', zzv or xvv interaction, (13)where a =p, p. In Ref. 1 only the p-dependent contribu-tions of V '), i.e., the last term in Eq. (3), were takeninto account, whereas the other ones depending on thenormal density p were considered to be already well ap-proximated by the fixed density interpolation of the two-body force used. Then density changes enter only linear-ly into Eqs. (7) and (8).In the present calculations all terms originating fromL is taken slightly difl'erent for proton-proton andneutron-neutron interaction, because in heavy nucleiwith NcZ the configuration spaces for protons and neu-trons are generally not equivalent.For the three-body part we take a 6 interaction withdifferent strength factors CoV3 and CoV3' in the pphand ppp channel, respectively, the superscripts corre-sponding to those in Eqs. (7) and (8). We again distin-guish between interaction of like or unlike particles ineach of these channels:the three-body component are treated self-consistently,giving rise to an additional quadratic dependence of Iand 5 on variations of p. This yields on the one hand theessential rearrangement term —,' g„,,p„„,I „~, in Eq. (9),without which even binding-energy differences cannotsensibly be calculated. On the other hand, I (~), depend-ing on p and contributing to Eq. (7) with the oppositesign as compared to I (2), produces the negative curva-ture of the radii as a function of neutron number, experi-mentally observed, ' ' e.g. , in the tin isotopes (Fig. 1).As mentioned above, the odd-even staggering as well asthe kink at the closed shell exhibited by our Ba results isproduced by I ~ . We want to stress that these are nodeformation effects, since we are working entirely withina spherical model [all densities entering Eqs. (5) beingexplicitly rotationally invariant '2).The treatment of 5 ~ is of little influence on thecharge radius, but it significantly affects the pairing en-ergies. When adding neutrons, this contribution to thepair potential, also originating from the repulsive three-body component of our interaction, gets stronger, whichis equivalent to a reduction of the effective pairing in-teraction strength. This, in our opinion, is the cause ofthe isotopic efl'ect in the pairing energies observed byVogel, Johnson, and Hansen. 2 In particular, we get thedecrease of the proton pairing with increasing neutronnumber (Fig. 2) without the invocation of proton-neutron pairing. 'We calculate the quantity Ap(Z, N) defined accordingto Ref. 2 from Eq. (9) evaluated for different proton andneutron numbers with ' Ba as reference nucleus. Theinteraction and potential parameters are given in TablesI and II, respectively. All single-particle states up tothree major shells above the Fermi energy are includedin the configuration space. In the Ba region the levelspacings around eF are adjusted according to the experi-TABLE II. Parameters of the Woods-Saxon potential defined according to Ref. 5. Exceptfor Vp the Ba values are taken from Ref. 19; b is the oscillator length.Vp(Mev)p nRp(fm)Qp(fm)p n pRLs(fm)QLg(fm) Rc b(fm) (fm)Sn 52.9 43.5 6.33Ba 57.5 45.8 6.636.33 0.67 0.67 32 32 6.33 6.33 0.67 0.67 6.33 2.2366.40 0.79 0.66 27 32 6.40 6.19 0.59 0.64 6.58 2.277151VOLUME 61, NUMBER 2 PHYSICAL REVIEW LETTERS 11 JULY 1988mental values quoted in Ref. 19, in Sn the —", (in-truder) state is shifted down by 1.3 MeV to obtain thecorrect level ordering.Figure 2 shows our results for dp(Z, X) for some iso-topic chains around Ba compared to the experimentalvalues. ' Apart from shell effects the decreasing trend isreproduced very well. Note that all these results are ob-tained without any parameter variation. The slightlydifferent parameters for Sn can be tolerated in view ofthe extremely simple form of our interaction.Thus we conclude that with a suitable three-body partin the effective interaction one is easily able to explain atthe same time general properties of nuclear charge radiiand pairing energies.This work was supported by the Deutsche Forschungs-gemeinschaft, No. Za-119. We thank Dr. R. Stapel formany valuable discussions. Numerical calculations havebeen performed at the Regionales Rechenzentrum fiirNiedersachsen (RRZN).'D. Zawischa, U. Regge, and R. Stapel, Phys. Lett. B 185,299 (1987).P. Vogel, B. Johnson, and P. G. Hansen, Phys. Lett. 139B,227 (1984).3L. P. Gorkov, Zh. Eksp. Teor. Fiz. 34, 505 (1958) [Sov.Phys. JETP 7, 347 (1958)].4P. Ring and P. Schuck, The Nuclear Many-Body Problem(Springer-Verlag, Berlin, 1980).~J. Speth, E. Werner, and W. Wild, Phys. Rep. 33C, 127(1977).sA. B. Migdal, Theory of Finite Fermi Systems and Application to Atomic Nuclei (Wiley, New York, 1967).7V. Klemt, S. A. Moszkowski, and J. Speth, Phys. Rev. C 14,302 (1976).SB. D. Chang, Phys. Lett. 56B, 205 (1975).9D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626 (1972).' S.-O. Backman, A. D. Jackson, and J. Speth, Phys. Lett.56B, 209 (1975).' ' M. Waroquier, K. Heyde, and G. Wenes, Nucl. Phys.A404, 269 (1983).'2J. Decharge and D. Gogny, Phys. Rev. C 21, 1568 (1980).' J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys.A422, 103 (1984).14D. Zawischa, Phys. Lett. 155B, 309 (1985).' P. Aufmuth, K. Heilig, and A. Steudel, At. Data Nucl.Data Tables 37, 455 (1987), and references therein.16K. Heilig, Hyperfine Interact. 24-26, 349 (1985).17A. H. Wapstra and G. Audi, Nucl. Phys. A432, 1 (1985).' A. S. Jensen and A. Miranda, Nucl. Phys. A449, 331(1986).19C. Conci, thesis, Jiilich University, 1984 (unpublished).152

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Isotopic dependence of nuclear charge radii and pairing energies

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