Empirical proton-neutron interaction energies. Linearity and saturation phenomena

Empirical proton-neutron interaction energies. Linearity and saturation phenomena

Volume 227, number 1 PHYSICS LETTERS B 17 August 1989 EMPIRICAL PROTON-NEUTRON INTERACTION ENERGIES. LINEARITY AND SATURATION PHENOMENA J.-Y. Z H A...

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Volume 227, number 1

PHYSICS LETTERS B

17 August 1989

EMPIRICAL PROTON-NEUTRON INTERACTION ENERGIES. LINEARITY AND SATURATION PHENOMENA J.-Y. Z H A N G ,.b,d, R.F. C A S T E N a,d a n d D.S. B R E N N E R b ~' Brookhaven National Laboratory, Upton, N Y 11973, USA h Clark University, Worcester, MA 01610, USA ~ Joint Institute for Heavy Ion Research, ORNL, Oak Ridge, TN 37830, USA Institut fiir Kernphysik, Universitiit K6ln, D-5000 Cologne, FRG

Received 23 March 1989; revised manuscript received 6 June 1989

A method to extract empirical proton-neutron interaction energies for individual protons and neutrons as well as integrated valence p-n interaction strengths is discussed. The results give direct experimental support to the NpNn scheme, verifying the linearity of the p-n interaction strength against NpN~ early in a shell and its subsequent saturation. They also reflect the behavior of recently calculated quadrupole p-n strengths which account for the empirical saturation in B(E2:0 + ~ 2 + ) values, and point to the importance of the monopole p-n interaction as well.

It is by n o w c o m m o n l y accepted that the T = 0 c o m p o n e n t o f the p r o t o n - n e u t r o n ( p - n ) i n t e r a c t i o n is p r i m a r i l y r e s p o n s i b l e for c o n f i g u r a t i o n m i x i n g a n d the onset o f d e f o r m a t i o n a n d collectivity in heavy nuclei [ 1 - 6 ] . It was recently suggested [7] that a f i r s t o r d e r e s t i m a t e o f the integrated p - n strength is the p r o d u c t N o N n of the n u m b e r o f v a l e n c e p r o t o n s a n d v a l e n c e n e u t r o n s . In effect, the N o N , s c h e m e a s s u m e s that the p - n i n t e r a c t i o n is orbit i n d e p e n d e n t . However, this is a n o v e r s i m p l i f i c a t i o n a n d , especially in a d e f o r m e d field, the o v e r l a p o f p r o t o n a n d n e u t r o n wave f u n c t i o n s d e p e n d s o n the relative i n c l i n a t i o n s of their orbits to the n u c l e a r e q u a t o r i a l plane. In ref. [8] a r e f i n e m e n t ( d e n o t e d I&,,I ) to the N p N , , s c h e m e was c o n s t r u c t e d by associating NON,, with the q u a d r u p o l e p - n i n t e r a c t i o n a n d i n t e g r a t i n g the latter over all occupied Nilsson orbits. ISon I helps linearize the b e h a v i o r o f such sensitive i n d i c a t o r s o f collectivity as B ( E 2 : 0 + ~ 2 + ) values. Calculated ]Son I values for rare earth nuclei are s h o w n in fig. 1. T h e b e h a v i o r o f ISp, I is s i m p l e to u n d e r s t a n d . In def o r m e d nuclei well before mid-shell, both p r o t o n s a n d n e u t r o n s enter d o w n s l o p i n g e q u a t o r i a l ( N i l s s o n ) orJ Permanent address: Institute of Modern Physics, Lanzhou, P.R. China. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )

i

I [

[Spn[ 3o0

i

~ 88

~ [ I 2oo I : ~ Loo-

. o • • • ,

90 92 96 98 ,oo io4

~'NpNn/

x 94 ~ i

o

+ + +÷

~~

i

o

"~

°*

~ 0 ,0- s..... j

%5

~-

'i °

i

rp:,

:

~ ~ ,~,

~/, o

~ ~oo

~ 2oo

...........

NpNr,

~. . . .

300

Fig. 1. Calculated values of the integrated quadrupole p-n interaction ISp,[ for N=88-104. So° is normalized to NpN, for the N=92 isotones. (From ref. [8].) The inset shows the analytic behavior, eq. (5), of [Spn(./) ] for two values of Fp as a function ofF, in units of K/4.

Volume 227, number 1

PHYSICS LETTERS B

bits with high mutual spatial overlaps. Thus, the p - n interaction strength should scale roughly with NpNn. near mid-shell, however, some protons and neutrons will be in level or upsloping orbits inclined at relatively large angles to the nuclear equatorial plane. Their spatial overlaps with nucleons in downsloping orbits will be significantly less. Increments to Sp, will thus be less, leading to an asymptotically flat behavior. It would be interesting to have an experimental basis for these theoretical ideas. It is the purpose of this letter to show that it is indeed possible to extract empirical p - n interactions for many nuclei, that they reflect both the the linearity and saturation features just described, and that they also point to the key role of the monopole p - n interaction. First, however, it is interesting to note that the behavior of Sp, is actually a rather general consequence of the Pauli principle. It is easy and informative to show this analytically in the limit of a single j shell. It has been shown in ref. [9] that the quadrupole p - n interaction in the ground state can be written, after recoupling, as a sum over all filled orbits of products of expectation values of the quadrupole moment operators qp and qn in each single particle orbitj. These operators have even tensor character: if we assume good seniority, v(2 + ) = 2 , these expectation values can be calculated [9] from the matrix elements (j"2 + I Q2 Ij"2 + > for a j" configuration in terms of those in t h e j 2 configuration by

(j,,2+lqlj,,2+>_ 2 j + l - 2 n 2j-3

(j22+lqlj22+>.

(1)

For a nucleus with Np (iV,) valence protons (neutrons),

IS, nO) l Np Nn

Z

=kpk.

(2jp+l-2np)(2j,+l-2n.),

(2)

Ilp,lln -- O

where kp, kn are constants independent of Np and Nn. We define the running indices f , a n d f , and the quantities Fp and Fn, which are twice the fractional occupancy of the shell, as 2rip fP=2jp+l' 2Np F p - 2jp+ 1 '

2n, f"- 2j.+l' r,-

2N. 2j, +~

'

(3)

17 August 1989

0 4 f p , f , , Fp, F. < 1 .

(3 cont'd)

Assuming for simplicity that jp=jn =j, we have ISpn(j) I = g

~ [ (1 --fp) ( 1 --fn) ] fp ,fn = 0

Fp Fn

0 0 or

I&.0) I~

¼gfpF.(2-Fp) ( 2 - F o ) ,

(5)

where K contains all constant factors. [ Past mid-shell, the integration is from Fp (or Fn) to 0 resulting in a similar expression with factors ( F p - 2 ) and (Fo - 2 ) . 1 ISonU) I is sketched in fig. 1 and displays the same initial linearity and subsequent saturation as IS~nl. This is evident directly from eqs. (4) and (5) since, at the beginning of a shell, where Fp, Fn << 1, I Spn [ ~ FpFn~ NpNn while near mid-shell (Fp ~ Fn -~ 1 ) the integrand in eq. (4) nearly vanishes causing the integral to saturate as Fp, Fn increase. This derivation in a spherical basis is shown to illustrate the origin of the behavior of Spn and is primarily of pedagogic interest. In a realistic case, many j shells are filling and a deformed Nilsson basis is more appropriate. Nevertheless, qualitatively, the contribution of each Sp,(j) to Sp, is as in eq. (5) and the same overall pattern persists. To see if such theoretical behavior is reflected empirically, we now exploit a recent technique [ 10 ] to isolate the residual p - n interaction of the last two protons with the last two neutrons. This interaction energy, denoted a V2p_2,, is defined [ 10 ] by the double difference of binding energies

8V2p 2n(Z-I- 1, N-I- 1 ) = [B(Z+2, N + 2 ) - B ( Z + 2 , -

[B(Z,

N+2)-B(Z, N)].

N)]

(6)

The meaning of eq. (6) is schematically illustrated in fig. 2. Each term in eq. (6) is composed of p-p, n n, and p - n contributions as well as contributions to the average spherical field where, by the latter term, we mean that part of the interaction that has nothing to do with the deformation driving part of the p - n interaction. The first difference isolates the interaction of the last two neutrons with all Z + 2 protons

Volume 227, number 1

/~

Y

PHYSICS LETTERS B

V

) (Z+2)

/~

'~. @ [(N+2)

V

j --

(N)]

,,Interaction of last 2 neutrons with (Z+2)~ [ protons and with each other and their } contribution to sphericat average field /

./ --

(Z)

@ [(N+2) - -

(N)]

tlnteraction of last 2 neutrons with (Z) {protons and with each other and their} ~contribution to sphericaP average fieldt

Net Interaction: last 2 neutrons with last 2 protons

Fig. 2. Schematic illustration of binding energy differences embodied in eq. (6).

and with each other and contains their contributions to the spherical average field. The second term is the interaction of the last two neutrons with Z protons and with each other and their contribution to the average field. The difference of these two quantities cancels the p - p and n - n interactions as well as the mean field contributions, giving the net interaction of the last two neutrons with the last two protons. For an attractive p - n interaction 6Vpn is negative. The average interaction of each of the last two protons with each of the last two neutrons, denoted 6Vpn, is just ~6 ~p-2n. Note also, since 6 Vpn is empirically obtained, it is not just the quadrupole p - n interaction component but contains all relevant multipoles. The approach here is reminiscent of closed loop difference equation techniques [ 11 ] used to predict nuclear masses but the present construction is designed to isolate the residual p - n interaction. It also has links to that used by Sakai [ 12] for specificjp-jn multiplets near closed shells. Finally, a similar philosophy to eq. (6) was used, in an inverse sense, in ref. [6 ] to calculate intruder state energies in singly magic or nearly magic nuclei, with Vpn estimated in a semi-empirical manner in a monopole approximation. We stress that 6Vp,, is not the integrated p - n interaction but that of the last nucleons. To obtain an integratedstrength, denoted Vpn,we must sum 6 Vpnover all valence protons and neutrons. Manipulation of eq. (6) gives the general expression, valid for either particles or holes, Vpn( Z + 6p, X + a n ) = JapSn { [ B ( Z + 2c~p, N + 2dn) - B ( Z + 2C~p,No) ] - [B(Zo, N+ 2~.)

-B(Zo, No) ]},

(7)

17 August 1989

where (Z o, No) specify the nearest magic numbers and &p (&n) is ( + 1 ) if the valence protons (neutrons) are particle-like and ( - 1 ) if they are hole-like. Although this p - n interaction is labelled by the intermediate odd-odd nucleus the calculation only involves even-even nuclei. Eq. (7) is a generalization of a semi-empirical expression used near singly magic nuclei in ref. [6]. Note that eq. (7) often entails binding energies from highly proton or neutron rich nuclei [i.e., B ( Z + 2 , No) or B(Zo, N + 2 ) ] , especially when Z or N is near mid-shell. Such data are not always available. In two mass regions, the proton hole-neutron particle nuclei near A = 100 and the proton particle-neutron hole nuclei nearA = 130, the data do exist [ 13 ] to extract Vvn values to compare with the linear portions of [Spn [ curves. This procedure of course involves the approximation that the interaction of specific protons and neutrons is unchanged when subsequent nucleons are added. For the near-spherical nuclei involved here, this should be a reasonable approximation. For the A = 100 region, the binding energies B(50, 50) and B(48, 50) were obtained by extrapolation but they are constant for all nuclei in this region, and all Cd nuclei, respectively, and therefore have little or no effect on the trends in Vpn. The empirical Vp,,are plotted in fig. 3. As with Sp,, in fig. 1, they display a remarkable linearity against NpNn. The many doublets of overlapping circles are particularly striking confirmation of the correlation between Vp,,and NpNn since they represent nuclei with very different numbers of valence protons and neutrons yet similar NpN, products. [The small spreads in Vpn at NpNn= 15 are an extreme situation involving nuclei with (NpNn)= (1, 15) compared to (Np, Nn) = (3, 5), (5, 3 ). ] These results give strong support to, and justification for the rationale behind, the NpNn scheme which, heretofore, was motivated mainly by phenomenological utility. We note that successful semi-empirical mass formulas will also contain an implicit linearity against NpN,, arising, for example, from the ( N - Z ) 2 symmetry term or from deformation energy terms. Here, however, the emphasis is on explicitly isolating and empirically testing for this effect. The data do not exist to extend these plots into potential saturation regions. We can, however, still test for saturation by determining if 16 Vp,,I decreases near

Volume 227, number 1

PHYSICS LETTERS B

I

40

I

I

I

17 August 1989

I

I

40

A~100

I

I

I

I

I

I

I

A~130

O

3

32

0 >~ 24

0

8

@

O

--= >~- - 16

"--c >~

oo

O

O

0 O°

O O

oo

0 0 0

g o@

o8 o 0 :~

I

10

I

I

20 30

I [ 40 50 NpNn

I I 60 70

I 80

0

L I0

I

I

20 30

I

I

40 50 NpNn

1 I 60 70

[ 80

Fig. 3. Integrated empirical p-n interaction Vp, plotted against NpNnfor two mass regions. Errors are always less than the size of the circles.

mi d shell. To this end, a n u m b e r o f 8 Vp, values are given in table 1 for rare-earth nuclei. While there are numerous small-scale fluctuations, there are also quite clear trends that reflect the calculated b e h a v i o r o f Spn and its physical interpretation. F o r example, the 83rd and 84th neutrons occupy equatorial, downsloping Nilsson orbits. So do the protons just after Z = 50 but, as Z increases, the protons enter successively m o r e polar orbits and interact less with the neutrons: hence 8Vp, should, and does, drop with increasing Z for N = 8 3 . In contrast, when Z is near mid-shell, [Vpnl should increase with increasing N as the neutrons oc-

cupy orbits having greater overlap with mid-shell proton orbits. Th e data (e.g., for Z = 65 ) support this although the fluctuations are greater. Finally, if neutrons and protons fill their respective shells " i n step", I ~ V p n I should be rather stable as observed along diagonals in table 1. Most importantly, there is clearly a general decrease from upper left toward lower middle o f table 1 from values near ~ 4 0 0 keV to values averaging ~ 2 7 5 keV. F o r example, 6Vpn averages - 3 8 5 keV for Z = 53, 55 and N = 83-87, compared to - 270 keV for Z = 65, 67, N = 89-93. Thus, the latter region will

Table 1 fiVpovalues (in MeV ), for the rare-earth region. (Note that, although the ~ Vp. are labelled for convenience by the intermediate odd Z, N values, the binding energies involved refer to even-even nuclei.) Errors are nearly always < 10 keV (usually < 4 keV) except for the most neutron rich isotopes of a few elements (Z = 53-59, 65 ) where they are near 30 keV. Z

N 83

53 55 57 59 61 63 6~ 67

-0.415 -0.329 -0.363 -0.336 -0.307 -0.289 -0.286

85

87

89

91

-0.419 -0.338 -0.320 -0.292 -0.286 -0.328 -0.295

-0.378 -0.317 -0.300 -0.308 -0.308 -0.332

-0.370 -0.395 -0.359 -0.322 -0.284

-0.475 -0.306 -0.284 -0.263

93

95

97

99

-0.312 -0.278 -0.255

-0.314 -0.275

-0.376 -0.299

-0.362

Volume 227, number 1

PHYSICS LETTERS B

contribute less to the I vp, I integrals which will therefore increase more slowly near mid-shell. This saturation reflects that seen in calculated ISp,I values. One important and revealing difference, however, is that, while ISpnl becomes asymptotically flat near mid-shell, 6Vpn never vanishes and I Vp, I increases throughout the shell. The reason is that the empirical Vpn values automatically include monopole as well as quadrupole interactions. The former is always attractive (it is independent of orbit inclinations) and provides a "base" to the 8Vpn values [5,6,14]. The data suggest that this essential component has the strength of a couple hundred keV per p - n pair. This is broadly consistent with other estimates obtained near closed shells [6,14]. The quadrupole component then varies from ~ - 2 0 0 keV early in the shell to near zero at mid-shell. To summarize, we have discussed a method for empirically extracting the p - n interaction, 6 Vpn, of the last neutron and proton in a given nucleus, and the total valence p - n interaction, Vpn, over extensive sequences of nuclei. Vpn is initially linear in NpNnbut subsequently increases more slowly, thus giving empirical support to the NpN,, scheme and its refinements, providing an empirical basis for model calculations of p - n interactions, and highlighting the importance of the monopole p - n interaction.

17 August 1989

We are grateful to J.D. Garrett and K. Heyde for discussions. Research was performed under USDOE contracts DE-AC02-76CH00016 and DE-FG0288ER40417. References [ 1 ] A. de Shalit and M. Goldhaber, Phys. Rev. 92 ( 1953 ) 1211. [2] I. Talmi, Rev. Mod. Phys. 34 (1962) 704. [3] P. Federman and S. Pinel, Phys. Lett. B 69 (1977) 385. [4] R.F. Casten et al., Phys. Rev. Lett. 47 (1981 ) 1433. [ 5 ] K. Heyde et al., Phys. Lett. B 155 ( 1985 ) 303. [ 6 ] G.E. Arenas Peris and P. Federman, Phys. Rev, C 38 ( 1988 ) 493; Phys. Len. B 173 (1986) 359. [7] R.F. Casten, Phys. Rev. Lett. 54 (1985) 1991. [ 8 ] R.F. Casten, K. Heyde and A. Wolf, Phys. Lett. B 208 ( 1988 ) 33. [9] K. Heyde, in: Nuclear structure in the Zr region, eds. R.A. Meyer and K. Sistemich (Springer, Berlin, 1988) p. 3; and private communication. [10]J.-Y. Zhang, C.-S. Wu, C.-H. Yu and J.D. Garrett, Contemporary topics in nuclear structure physics (Cocoyoc, 1988), Abslract Volume, p. 109; J.-Y. Zhang, C.-S. Wu, C.-H. Yu, M. Carpenter and J.D. Garrett, to be published. [ I 1 ] G.T. Garvey et al., Rev. Mod. Phys. 41 ( 1969 ) 51 ; G.T. Garvey and I. Kelson, Phys. Rev. Lett. 16 ( 1966 ) 197. [ 12] M. Sakai, Nucl. Phys. A 345 (1980) 232. [ 13 ] A.H. Wapstra, G. Audi and R. Hoekstra, At. Data Nucl. Data Tables 39 (1988) 281. [14] J.P. Schiffer and W.W. True, Rev. Mod. Phys. 48 (1976) I91: A. Molinari et al., Nucl. Phys. A 239 (1975) 45.