The incompressibility of nuclear matter and the breathing mode

Physics Letters B 271 ( 1991 ) 12-16 North-Holland

P H YSIC S k ETT ER S B

The incompressibility of nuclear matter and the breathing mode J.M. Pearson Laboratoire de Physique NuclOaire, UniversitOde MontrPal, Montreal, Quebec, Canada H3C 3J7 Received 18 April 1991; revised manuscript received 30 April 1991

It is shown that a unique value of the nuclear-matter incompressibility Kv cannot be extracted from the breathing-mode data. There is thus no basis for the often quoted figure of 300 MeV. However, the data establish a correlation between K, and the thirdorder derivative of the nuclear-matter saturation curve.

The incompressibility o f infinite nuclear m a t t e r ( I N M ) is defined as (1) °

where e ( p ) is the energy per nucleon at d e n s i t y p , and Po is the saturation density (actually, Kv is always defined for the s y m m e t r i c case, p° =pp = ½Po). P r i m a r i l y because of the interest in stellar collapse, a considerable effort has been m o u n t e d over the last few years in an a t t e m p t to d e t e r m i n e this quantity from laboratory nuclear physics, since it is felt that this is one of the few ways in which one can impose experimental constraints on the equation o f state o f nuclear m a t t e r at super- and sub-nuclear densities (for recent reviews on the relevance to supernova explosions see, for example, Cooperstein and Baron [ 1 ], and Bethe [2]). The most prolific source of experimental information on K, has been the giant isoscalar m o n o p o l e resonance ( G I M R ) , the so-called breathing mode. Two different procedures have been used for extracting values of Kv from the measured b r e a t h i n g - m o d e energies, Ebr. The first consists o f m a k i n g RPA calculations [3], or semi-classical a p p r o x i m a t i o n s thereto [4], o f the G I M R in several finite nuclei for various effective forces (or relativistic mean-field theories), characterized by different values of K,. The force that has the best agreement with e x p e r i m e n t Supported in part by NSERC of Canada. 12

then d e t e r m i n e s the best value o f Kv. This approach is usually quoted [ 3 ] as giving Kv= 210 _+30 MeV. The second and more direct way that has been used to extract the value of Kv from the measured values o f Ebr is to define a finite-nucleus incompressibility according to K(A, Z) = (M/h2)R2E2r,

(2)

where R is the R M S matter radius of the nucleus in question. One then fits the K ( A , Z ) measured in as m a n y nuclei as possible to the l e p t o d e r m o u s expansion [ 5 ]

K(A, Z)=Kv+KsfA t/3.q_Kvsi2q_KcoulZ2A - 4 / 3

.t_Kssi2A-i/3.t_KcvA 2/3,

(3)

where 1= 1 - 2 Z / A . In this way one determines in principle not only the required Kv but also the other parameters, Ksf (surface t e r m ) , Kvs ( v o l u m e - s y m metry t e r m ) , etc. The last two terms, the surfaces y m m e t r y and curvature terms, respectively, are o f higher order, and are often neglected. Actually, making this expansion, and in particular taking the leading term as being Kv, the incompressibility of INM, presupposes that the breathing m o d e can be described by the scaling model. However, to a good approximation this has been well justified [4,6] for all real nuclei with A > 100 (see below). Fitting, then, in this way a n u m b e r of precision measurements on various Sn and Sm isotopes and on 2°Spb, Sharma et al. [7,8] find Kv=300_+25 MeV, in ap-

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parent contradiction with the results of the first approach. The contradiction is rendered all the more acute by the observation of Stocker and Sharma [ 9 ] that to fit the measured Eb,- within the framework of a a-o) model a value of K, considerably smaller than 300 MeV is required. Also, using a highly generalized Skyrme-type force with a very flexible saturating mechanism, Farine et al. [ 10] have been unable to fit the measured Eb, ifKv is larger than 210 MeV. The object of this note is to re-examine the analysis of Sharma et al. [7,8] in an attempt to resolve this paradox. It is well known that because so few data points are available the least-squares fit of the parameters of the expansion (3) is subject to serious stability problems. For this reason, Sharma et al. [7,8] first set K~ = K~.~= 0, having shown that their results are quite insensitive to K,.v. Next, in order to have only three free parameters in the fit, they express the Coulomb coefficient Kc,,~ in terms of Kv, using the relation given by Treiner et al. [ 11 ] Kco~J = 3 e 2 ( 1215 ) 5 r ~ \ K, - 1 2 . 5 m e V ,

(4)

4 3 where utrb = 1/Po. Now this relation is justified on the grounds that it holds for a wide class of forces, given in table 5 of ref. [ 11 ]. However, very few of these forces have K, close to 300 MeV, and of those that do, none is known to fit the breathing-mode energies. Moreover, Skyrme forces that deviate significantly from eq. (4) have now been found [ 10 ]. Thus the conclusion of Sharma et al. [7,8] may not be consistent with the assumptions that they make at the outset.

14 N o v e m b e r 1991

Let us therefore re-examine the fits of Sharma et al. [ 7,8 ], relaxing the contentious constraint (4). We take just seven data points, as shown in the first two columns of table 1; the first six come from ref. [7] and the last from ref. [ 12 ]. The precision measurements [7] of ~4*Sm is discarded, on account of the questionable sphericity of this nucleus, and likewise all measurements on lighter nuclei, because of the possible failure of the scaling model. For the corresponding experimental values of K(A, Z), as given by eq. (2) and denoted by Ke,p(A, Z ) in the fourth column of table 1, we need the RMS matter radii R, which we obtain by performing HF calculations with an effective force that gives a precision fit to the measured charge radii (third column of table 1 ). Sharma et al. [7,8] have essentially the same radii as we do

[131. We note that since eq. (3) is linear in the coefficients [as long as the constraint (4) is not applied], the least-squares fit to the data reduces to a set of linear algebraic equations. The procedure is altogether standard (see, for example, ret: [ 14] ), but we recall the essentials in order to make it quite clear what exactly we are doing. Setting first Kss=Kcv=0, we rewrite eq. (3) in the form 4

y, = ~ g,~,K~ ,

(5)

~t=l

where Yi (i = 1, ..., 7 ) represent the data points Kexp(A,, Zi), and the K~ are the components of the vector K = (K,., Ksr, Kvs, Kcoul). We then define the matrix S, l, =

g, vg,l, t--1

,

(6)

0"7

Table 1 See text for explanations. All q u a n t i t i e s except R in MeV.

I I2Sn nf4Sn ~16Sn 12°Sn

124Sn '44Sm 2°8pb Z2

]']br

R (fro)

/kcxp(A, Z )

K, (A, Z )

K2(A, Z )

K3(A, Z )

15.88 _+0.14 15.80_+ 0.14 15.69_+0.16 15.52 _+0.15 15.35 _+0.16 15.13_+0.14 13.9 _+0.3

4.56 4.59 4.62 4.67 4.72 4.93 5.54

126.4 + 2.2 126.8 _+2.2 126.7_+2.6 126.7 _+2.4 126.6 _+2.6 134.2+2.5 143.0_+6.2

126.5 126.6 126.7 126.7 126.6 134.3 142.8

126.3 126.8 127.0 127.0 126.3 134.0 139.7

126.6 126.5 126,4 126.6 126,9 134,5 145,2

0.0116

0.0609

0.0914

13

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Table 2 Parameter sets, as explained in text (all quantities in MeV ).

K,. K~r K~,~ h'c-o~

Set 1

Set 2

Set 3

218.9 _+ 411.8 -470.2 _+1336.3 -90.00 _+ 942.5 1.334+_ 27.13

351.3 -879.9 -421.0 -8.124

119.6 -163.0 158.3 8.429

where 05 is the error in the data point y,, and

<= i_t i

O'7

Then

K:S-IV

(8)

and the error in K~, is given by

O'~ ,=, i °~(0K//~-=(S-I)//; 1, ].~=l ..... 4. \ ay, ]

(9)

(The errors of the different K~, are strongly correlated, in the sense that a change in one parameter within its error bars requires a change in the other parameters if the fit is to be maintained. This is particularly evident in table 2 - see below. ) The values of the four coefficients, K,,, K(.ou1, K~and K~ resulting from this fit are shown in the second column of table 2 (set 1 ). The fifth column of table 1, K, (A, Z ) , shows the corresponding values of K(A, Z). Comparing with the values of K~p(A, Z) given in the preceding column, we see the excellence of our fit. Likewise, the estimate for K, of 218 MeV might be noted with some interest, but the large error bar of + 4 1 2 MeV makes this result meaningless. To understand the origin and significance of these large error bars, we show in the last two columns of

14 November 1991

table 2 two completely different parameter sets, set 2 ( K , = 351 MeV) and set 3 ( K , = 120 MeV), for which the corresponding values of K(A, Z) are given in the last two columns of table 1, Ka(A, Z) and K3(A, Z ) , respectively. The agreement with experiment for all three sets is excellent, the predicted values of K(A, Z ) always falling within the experimental error bars. (Sets 2 and 3 were obtained, of course, by refitting to data that had been slightly modified within the original error bars. However, the provenance of these new parameter sets is immaterial for our purpose, and in principle they could have been guessed. For this reason we show no error bars for these sets, although they are the same as for set 1, insofar as they are obtained from a fit to the modified data. The only nucleus that varies significantly between the various sets is 2°spb, which has much larger experimental error bars. It will be seen that as K, decreases K(208, 82) somewhat counter-intuitively increases. ) All three parameter sets are seen to be almost equally plausible. Admittedly, the lowest X2 is indeed found for set 1, i.e., K~ = 21 8 MeV, but the Xe for the other two sets are also very small, and there is absolutely no basis for rejecting these sets. We conclude that with four free parameters the seven G I M R data points allow all values of K~ over the range 120-351 MeV (and probably over a wider range). This, of course, simply reflects the instability of the fit, and is the basis of the long-standing practice [ 7,8,1 1,15 ] of imposing physical constraints, such as eq. (4), on the parameters in order to reduce the n u m b e r of degrees of freedom in the least-squares fit. But no matter how statistically stable the final value for K,, it will be no more reliable than the physics that has gone into the constraints acting between the coefficients. To understand the significance of the

Table 3 3-parameter minimization for different fixed values of K,. All quantities in MeV.

14

K,

Ksf

K,,~

K(-ouL

Z2

100.0 150.0 200.0 250.0 300.0 350.0 400.0

-84.60+47.9 --246.7 +47.9 -408.9 _ + 4 7 . 9 -571.0 +47.9 -733.2 +47.9 -895.3 +47.9 -1057.5 +47.9

180.6 +101 66.83+101 -46.94_+101 -160.7 _ + 1 0 1 -274.5 +101 -388.3 _+101 -502.1 +101

9.145 +2.06 5.861 "+2.06 2.577 +2.06 -0.7065"+2.06 -3.990 _+2.06 -7.274 +_2.06 -10.56 +2.06

0.095 0.040 0.014 0.017 0.050 0.113 0.205

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constraint (4) we minimize 2'2 with respect to the three coefficients Kcou~, K~f and K,.~ for different fixed values of K,. (table 3 ). The variation of Kco~. with K,. is shown by the shaded band in fig. 1, the finite width representing the computed errors. (The three parameter sets of tables 1 and 2 fall in this band. ) Now the correlation between Kco~ and K, represented by eq. (4) is shown by the curve in fig. 1. It is seen to intersect with the experimentally allowed zone in the vicinity of 300 MeV, which explains the unique result obtained in refs. [7, 8]. With only a modest modification o f e q . (4) it would have been possible to obtain an equally unambivalent result of, say, 200 MeV for K,. Given the counter-examples to eq. (4) that are already known [10], and in the absence of any fundamental reason for believing in this or any other constraint between the parameters, we conclude that it is impossible to extract from the G I M R s a unique value for K,, or indeed for any other of the coefficients of the expansion (3). However, this does not mean that the measured breathing-mode energies are irrelevant to the deterruination of the parameters of the underlying effective force or relativistic mean-field theory. But to see whether a given force (or mean-field theory) is compatible with the measurements it will be necessary to calculate the actual values of the breathing-mode energies for the force (or mean-field theory) in question: it will not be enough simply to determine the incompressibility K,,, since, as we have seen, no

IO

Kcoul. (MeV) v?

-IO

14 November 1991

unique value of this is determined by experiment, i.e., the approach adopted by the Blaizot group [3,5,6 ] is the only legitimate one. At the present time there is no published force that gives a completely acceptable fit to the G I M R data. The situation with regard to the Kv-Kcou~ correlation for three well known effective forces is shown in fig. 2: these are the SkM* and $3 Skyrme forces (see ref. [ 16] for a convenient summary o f the parameters), and the D 1 finite-range force used in refs. [ 3,5 ]. In this figure we have re-drawn the band showing the experimental K,.-K~.ou~ correlation to take account of the small errors introduced by the scaling approximation, as indicated by table 1 of teE [6]: in fitting to eq. (3) one should increase K(A, Z) by approximately 10%. Of these three forces only $3 lies close to the experimental zone, but this of course is a necessaw rather than sufficient condition to fit the G I M R data, and in the case of $3 we know that it does not [3], Ksr being insufficiently negative for the large value o f & . Finally, we note that within the framework of the scaling model the relation

Kcou,_ 3e2( K'

5.0 ~ - 8

)

(10)

always holds [ 5 ], K' being the INM parameter K ' = - 2 7 (p3 d-~e(P)'~

dP~ J,,o

(l l)

Thus our analysis shows that even if the data do not determine a unique value of K,, they do imply a unique relation between the second and third derivatives of the INM saturation curve at the equilibrium density. We see that Kcou~acts as a rather sensitive probe of the equation of state, of obvious interest for the extrapolation to super- and sub-nuclear densities. Let us examine in the light of this last remark the semiphenomenological BCK equation of state [ 17 ], which has been widely used for supernova simulation. For symmetric nuclear matter this equation reads

e(p)=e(po)+ ~

+

,

(12)

-20 Fig. 1. Correlation between K~ and Kcou~; shaded band corresponds to breathing-mode data, curve to eq. (4).

where u=p/po, and 7 is an adjustable parameter ( > 1 ). Usually the value K,.= 180 MeV (symmetric 15

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nuclear matter) is taken, which is generally considered to be "soft". However, since this equation of state is intended to be used only for uniform nuclear matter no surface properties are specified, so it is impossible to assert at the present time that this value of K, really is incompatible with the G I M R data. On the other hand, in view of eqs. (10) and ( 11 ) it is a matter of trivial algebra to show to what extent this equation of state is compatible with the experimental correlation that we have established between Kv and KcouL. We find that eq. (12) implies

14 November 1991

teed that a calculation of the breathing-mode energies would result in an agreement with their experimental values, any more than in the case of the $3 force. I thank M. Brack, M.M. Sharma, and W. Stocker for helpful communications. G. Beaudoin, L.-A. Hamel, and H. Jeremie patiently instructed me in the statistical analysis of experiments. S. Das Gupta kindly read the manuscript.

3e 2

Kcou~= 5to ( 7 - 3 ? , ) .

(13)

Referring now to fig. 2 we find that with an acceptable value ofpo we must have ; , < 0 . 0 3 K , - 3.3, where K,. is expressed in MeV. Thus for K , = 180 MeV we must have ),<2.2, while for K , < 145 MeV we must have ?,< 1, which is impossible. For completeness we also consider the much harder FP equation of state [ 18] ( K v = 2 4 0 MeV). Estimating K' from table 1 of ref. [18] to be 650 MeV, we see from fig. 2 that this equation of state is very close to the experimental K c o u l - K v band. However, a force underlies this equation of state, and it is not guaran-

KCoul I0 /MeV) ioo

- I0

~K////"~.

200
I _

300

I

400

V)

Gogny D 1

~, SkM E] $ 3 (D FP -20

Fig. 2. Experimental correlation between K~ and ]~'L',ml, corrected for errors in scaling model. The four points refer to various forces, as explained in the text.

16

References [I] J. Cooperstein and E.A. Baron, in: Supernovae, ed. A. Petschek (Springer, Berlin, 1990) p. 213. [2] H.A. Bethe. Rev. Mod. Phys. 62 (1990) 801. [3] J.P. Blaizot, D. Gogny and B. Grammaticos, Nucl. Phys, A 265 (1976) 315. [4] P. Gleissl, M. Brack, J. Meyer and P. Quentin, Ann. Phys. (NY) 1 9 7 ( 1 9 9 0 ) 2 0 5 . [5] J.P. Blaizot, Phys. Rep. 64 (1980) 171. [6] J.P. Blaizot and B. Grammaticos, Nucl. Phys. A 355 ( 1981 ) 115. [7] M.M. Sharma, W.T.A. Borghols, S. Brandenburg, S. Crona, A. van der Woude and M.N. Harakeb, Phys. Rev. C 38 (1988) 2562. [8] M.M. Sharma, W. Stocker, P. Gleissl and M. Brack, Nucl. Phys. A 504 (1989) 337. [ 9 ] W. Stocker and M.M. Sharma, Z. Phys., to be published, [ 10] M. Farine, J.M. Pearson and F. Tondeur, unpublished. [ 11 ] J. Treiner, H. Krivine, O. Bohigas and J. Martorell, Nucl. Pbys. A371 (1981) 253. [12]S. Brandenburg, W.T.A. Borghols, A.G. Drentje, L.P. Ekstr6m, M.N. Harekeh, A. van der Woude, A. H/lkanson, L. Nilsson, N. Olsson, M. Pignanelli and R. De Leo, Nucl. Phys. A 466 (1987) 29. [ 13] M.M. Sharma, private communication. [14] J. Mathews and R.L. Walker, Mathematical methods of physics ( B e n j a m i n / C u m m i n g s , Menlo Park, CA, 1964), section 14.7. [15]J.P. Blaizot, in: The nuclear equation of state, eds. W. Greiner and H. St6cker ( Plenum, New York, 1989 ) p. 679. [16] M. Brack, C. Guet and H.-B. Hfikansson, Phys. Rep. 123 (1985) 275. [ 17] E. Baron, J. Cooperstein and S. Kahana, Nucl. Phys. A 440 (1985) 744. [18] B. Friedman and V.R. Pandharipande, Nucl. Phys. A 361 (1981) 502.