Nuclear compressibility and its effect on nuclear binding energies from the study of muonic atoms
K V SUBBA RAO and A A KAMAL
Physics Department, University College of Science, Osmania University, Hyderabad 500 007, India
MS received 20 June 1983; revised 14 October 1983
Abstract. The variation of nuclear parameter with mass number elicits information about nuclear compressibility. Analysis of muonic x-ray transitions provides an elegant method to investigate the behaviour of the nuclear parameter ro. It is observed from the behaviour of r o that nuclei in the region A ~< 70 are highly compressible while ~hose in the region A ~ 210 arc almost incompressible. The behaviour of ro is incorporated into the semi-empirical mass formula through the Coulomb energy term. From the modified mass formula thus obtained binding energies of about 440 spherical nuclei have been calculated. The results suggest that
nuclear compressibility imposes certain relationship between excess binding energies (Eex p - Ecal) and neutron, proton number. The present study also points out that shell effects exhibited by nuclear binding energies cannot be accounted for by simply varying the coefficients of the mass formula: on the other hand extra terms are necessary to explain them.
Keywords. Muonic atom: nuclear compressibility; nuclear binding energies; mass tbrmula.
1. Introduction
It is well-known that the semi-empirical mass formula (equation (1)) reproduces only the main trend in the variation of nuclear binding energies with mass number A and nuclear charge Z.
E = a,,A - a s A 2/3 - a r Z Z A -1/3 - a s y m ( N - Z ) 2 A t +6apairA . . (1)
In the above mass formula a~., as, ac, asy m and apart are the coefficients of the volume energy, surface energy, coulomb energy, symmetry energy and pairing energy respectively. The sign of the pairing energy term is determined by the 6 value, which is + 1 for even-even nuclei, 0 for odd nuclei and - 1 for odd-odd nuclei. The mass formula is discussed in detail by Wapstra (1958). The quantity (E~x p - Eta l), called the excess binding energy, is known to exhibit particularly large deviations in the vicinity of magic numbers (Wapstra 1958). The observed discrepancies between the experimental and calculated binding energies (Eex v - Eta I) are attributed mainly to the inadequacies of the assumptions involved in the liquid drop model employed in developing (1).
Important assumptions mainly suspected are: (i) the statistical assumptions, especially in connection with light nuclei, (ii) the assumption of uniform density; (iii) the assumption that nuclei are incompressible; and (iv) absence of shell structure of nuclei.
It is well established that because of assumption (i), the mass formula is not strictly valid for light nuclei. Assumptions (ii) and (iii) are too stringent and must be regarded with scepticism. It is also well-known that shell effects are too important to be ignored.
Besides, the liquid drop model is inadequate to explain nuclear deformations. The 71
individual contributions from the above mentioned physical effects may be in the same or opposite direction and they may not be independent. Hence, the entire problem of apportioning the binding energies of nuclei to the various above mentioned physical effects is expected to be very complicated. In the present study a method of investigating the effect of nuclear compressibility on the nuclear binding energies is described.
2. Analysis of muonic x-rays---nuclear compressibility
In the nucleus, if the charge is assumed to be distributed uniformly within a sphere having a radius R = ro A1/3, the parameter ro would be directly proportional to the cube root of volume per nucleon or inversely to the cube root of average nucleon density. Although R represents the charge radius, it is reasonable to assume that ro represents average nucleon density, because the charge and mass distributions in a nucleus are nearly identical (Hill 1957). The A x/3 dependence of the radius R is well known and is established by a variety of experiments. However, in many investigations like scattering experiments, study of isotope and isotone shifts in optical spectra and in the evaluation of nuclear binding energies the ro value is assumed to be constant---each experiment assuming a different value. Since it is proportional to the average nucleon density, the constancy of ro invariably implies the acceptance of incompressibility of nuclear matter--a fact which is not strictly valid. In fact, many discrepancies in the interpretation of various experimental results are attributed to this assumption. For example, the isotope shifts predicted b y t h e relation R = ro A1/3 are nearly twice the experimentally observed values (Devons and Duerdoth 1969). But, if the parameter ro is allowed to vary, better results could be obtained (Subba Rao and Kamal 1983).
Analysis of muonic x-ray transitions is an established technique to study nuclear structure and provides an elegant method to investigate the compressibility of nuclear matter. Although uniform model is not very realistic, under the assumption of uniform charge density analysis of muonic x-rays gets simplified and yields reasonably good results (Subba Rao and Kamal 1980). It is observed that from such an analysis the variation of parameter ro with mass number A can be obtained (Subba Rao and Kamal
1983).
The radial part of the Dirac equation for a two-component wave function with a large component g and a small componentfis solved to obtain the energy levels of the muon in the electrostatic field of the nucleus.
d f _ k f 1
dr r ~C { W - V(r) - laC 2 } g, (2a)
dg 1 U V(r) + l~C 2 } f - kg (2b)
Tr=V6 T"
In (2) V(r) is the potential generated by the uniform charge density and has the form of an harmonic oscillator potential within the radius R:
V(r) = - (3Ze2/2R) + ( Z e 2 / 2 R 3 ) r 2, r <~ R
= - Ze2/r, r >1 R. (3)
The energy levels obtained are then corrected for vacuum polarisation effect. The method of solving the Dirac equation and calculating the parameter r 0 is given in detail
fr
1-4;
1.3~
1"3~
ro
1.3C
1.2E
1.22
I I I I I I~0
o o
tt') t30
II II II II I| II II II
U Z U Z N Z N Z
z"
1.18 I I I I I I I I I
20 6 0 1 0 0 140 180 22(
A
Figure 1. V a r i a t i o n o f p a r a m e t e r r o w i t h m a s s n u m b e r A.
elsewhere (Barrett 1977; Subba Rao and Kamal 1980, 1983). The r 0 values calculated for various spherical nuclei (Subba Rao and Kamal 1983) are depicted in figure 1. It is observed that ro decreases from 1.44 fm at A = 27 to 1"19 fm at A = 209. But in the region A ~< 70 the decrease is very rapid. Thereafter the ro value decreases rather slowly and in the region of heavy nuclei (A ~ 210), its variation is very small. From this behaviour of ro (figure 1) it can be inferred that average nucleon density increases rapidly up to A ,,- 70, gradually thereafter and becomes almost constant in the region of heavy nuclei. This phenomenon indicates clearly that light nuclei (A ~< 70) are highly compressible while the heavy nuclei (A ~ 210) are compressible to a very small extent.
The rapid decrease of ro with A in light nuclei is partly due to surface thickness and the associated surface energy term, which are relatively large for light nuclei. It is also observed that the general behaviour of ro is punctuated with strong shell effects (figure 1). The behaviour of r o is approximately represented by the equation (Subba
Rao and Kamal 1983)
ro = a exp (b/A), (4)
where a and b are constants. Due to the exponential term in it (4) is valid only in the region A > 25. The least square fit to the r o values yields a = 1-1553 fm and b = 5.837.
The behaviour of ro can be incorporated into the mass formula (1) through the coefficient ac of the coulomb energy term. This procedure permits the study of the effect o f nuclear compressibility on nuclear binding energies.
3. Evaluation of nuclear binding energies
In view of the assumption that nuclear charge is uniformly distributed with radius R = ro Ax/3, the coulomb constant ac in (1) is given as
a~ = 3e2/5ro . (5)
Because of (4), coefficient a~ may be rewritten as:
3e 2
a~ = -~-a exp ( - b / A ) (6)
or a~ = a~ exp ( - b / A ) (7)
where a'c = 3e2/5a. Accordingly, (1) gets modified as E = a ~ A - a s A2/3 - a ' ~ Z 2 A - l / 3 e x p ( - b / A )
_ asym( N _ Z ) 2 A - x + t~apairA - 1/2 (8) Thus the nuclear compressibility effect on the binding energies of nuclei is to make the coefficient o f coulomb term ac a function o f mass number A. It is to be noted that surface energy term a, also involves the parameter r o. But in deriving the surface energy from liquid drop model, the nuclear matter distribution is considered, whereas the parameter ro in the present study is obtained from nuclear charge distribution.
Although, as mentioned earlier, the behaviour o f ro can be treated as representing the behaviour of average nucleon density, it would not be appropriate to use the parameter r 0 to estimate the surface energy.
Selecting a few nuclei from the various spherical nuclei studied, the coefficients in (8) are determined by making a least-square fit to the experimental binding energies (Wapstra 1958). The values obtained are
a~ = 15-9914 MeV (15-68 MeV), a, = 19.7624 MeV (18-56 MeV), a'~ = 0.6706 MeV (0.717 MeV), asy m --- 28.05 MeV (28.06 MeV), apair = 11"00 MeV (11.00 MeV).
For comparison the corresponding values obtained by Myers and Swiatecki (1966) are given in the brackets. With the above determined coefficients, binding energies o f 441
M eV
16
8
i
ILl I0.
~ 0 u.I
8
-16"-'-
l - - - - r r
20 28 50
_ _ ~ _ _ I , _ _ I t
0 20 4 0
1 T---"
82 126
f
,'
t,//I;//,
' 't/ ;lfl : i!l//f /
'//
1 ~ _ _ _ . t ~ . _ , ~
60 80 120 14(
N
Figure 2. Variation of excess binding energy with neutron number N.
spherical nuclei in the region 13 ~< Z ~< 85 are evaluated. The term (Eex p - Ecal) giving the difference between the experimental binding energies and those calculated from (8) is plotted against neutron number N (figure 2) and proton number Z (figure 3).
4. Results and discussion
The deviations of excess binding energies obtained by Wapstra (1958) and Myers and Swiatecki (1966) display the shell effects very strongly. The isotopic and isotonic deviations are characterised by a change in the sign of slopes, thereby exhibiting peaks (Wapstra 1958) or valleys (Myers and Swiatecki 1966) in the vicinity of magic numbers.
The magnitude of deviations is particularly large at the magic numbers. It is interesting to observe that deviations of excess binding energies obtained in the present work (figures 2 and 3) do not display the shell effects as clearly as mentioned above. It is noteworthy that in many nuclei the excess binding energy varies directly with neutron number N (figure 2) and inversely with proton number Z (figure 3). This variation is approximately linear in many nuclei. However, the general pattern of the variations still indicates shell effects--the deviations are generally much larger in the vicinity of magic numbers. Moreover, the slope of the curves exhibits the tendency to change suddenly around the magic numbers N = 20, 50, 82 and 126 (figure 2). It is also observed that the
MeV r !
2 0 28
T
50 82
' 1
16
8
u t.d
~ 0
h i
-16[-
10 20
Figure 3.
\ ,
o
\
30 4 0 50 60 75
Z
V a r i a t i o n o f excess b i n d i n g e n e r g y w i t h p r o t o n n u m b e r Z.
80 9 0
deviations are equally distributed on both sides of the line (Eex p - Eeal) = 0. Another noteworthy feature is that the curves are approximately parallel. In these respects, the curves have strong resemblance to those obtained by Myers and Swiatecki (1966).
Recently, Myers and Schmidt (1981) calculated the root mean square (RMS) radius of various nuclei as predicted by the liquid drop model. They plotted the differences between the experimentally measured RMS radii and those calculated from liquid drop model. In almost every isotopic sequence, the differences plotted slope steeply downward to the right. This isotopic behaviour is similar to the one observed in the present Work (figure 2) except that the curves slope downward to the left. Myers and Schmidt attribute this behaviour to the effect of neutron skin. These features indicate that nuclear compressibility imposes certain relationship between excess binding energies and neutron, proton numbers.
5. Conclusions
As only the coulomb term gets altered by the nuclear compressibility, it is not surprising that excess binding energies still display large deviations. Moreover, it is now realised that shell effects displayed by excess binding energies cannot be accounted for by just varying the coefficients in the mass formula; additional terms are necessary to explain them. Nilsson et al (1969) showed that such a procedure can yield fruitful results and
reduce the discrepancy between the experimental and calculated binding energies. In their work, the mass formula was modified to account for nuclear deformations and diffuseness of nuclear surface, apart from adding extra terms to include shell effects.
Calculations performed in the rare-earth and actinide regions agree well with the experimental results. The shell effects around the magic number Z = 82 are successfully explained and excess binding energies are within __+ 2 MeV. The results of Nilsson et al (1969) indicate that in the high-Z region a major portion of the excess binding energies is due to the shell structure of the nuclei. But, as already pointed out, in heavy nuclei the degree of compressibility is very small and hence, its contribution to excess binding energies can be expected to be small. Therefore, the real test for any modified mass formula lies in the medium and low-Z regions wherein nuclei exhibit remarkable compressibility.
The present study demonstrates that the analysis of muonic x-ray transitions provides an elegant method to study the nuclear compressibility and its effect on nuclear binding energies. The experimental data on muonic transition energies currently available is very limited and when the experimental data for more nuclei become available, it is expected that the effect of nuclear compressibility on r.uclea~:
binding energies can be investigated in greater detail.
Acknowledgements
One of the authors (KVSR) would like to thank uGc for the award of a research fellowship.
References
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