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A symmetry based mass formula for Λ hypernuclei
NUCLEAR PHYSICS ELSEVIER
A symmetry
Nuclear
A
Physics A639 (1998) 161c-164~
based mass formula for A hypernuclei*
G. Lkvaia, J. Cseha, P. Van Isackerb, 0. Juilletb aInstitute of Nuclear Research of the Hungarian Academy of Sciences, Debrecen, P.O. Box 51, Hungary 4001 bGrand Accelerateur National d’Ions Lourds, B.P. 5027, F-14076 Caen Cedex 5, France In the past couple of years a considerable amount of experimental spectroscopic information has accumulated on A hypernuclei; in particular, the A separation energy has been determined for the ground states of about 40 hypernuclei, including 3 double-A hypernuclei [1,2]. This data set allows the systematic analysis of hypernuclear binding energies in terms of mass formulae obtained as extensions of the Bethe-Weizsacker formula for normal nuclei. Until now, similar extensions have concentrated mainly on the question of the stability of multiply-strange hadronic matter, and the corresponding mass formulae have been derived either from relativistic mean-field calculations [3,4] or from hyperon-nucleus and hyperon-hyperon interactions [5]. Here, we present an alternative route by replacing the pairing term with a symmetry-inspired Majorana term, which allows a simultaneous treatment of normal and A hypernuclei. The idea of applying the Majorana operator in a mass formula was first proposed for nuclei composed of only protons and neutrons. Its justification [6] is that, due to the short-range attractive nature of the residual nuclear interaction, nucleons in the nucleus attempt to maximize their spatial (orbital) symmetry and choose a specific, favored SU(4) representation, in which the eigenvalue of the SU(4) Casimir invariant is minimal. This favored representation is uniquely determined in the ground state of nuclei (see [6] and references), and can be labelled with a four-lined U(4) Young tableau [fr, fs, fs, fd] accounting for the four possible spin-isospin single-particle states available to the nucleons. This scheme can be readily extended to A hypernuclei, when the allowed single-particle states also include the spin-up and spin-down states for A hyperons, and the group structure is expanded to that proposed by Giirsey and Radicati [7]:
U(6)3 (SU43)> b(l)@ SU,(2))C9SUs(2). The proton, neutron, and A hyperon are assigned to the (X, II) = (1,O) fundamental representation of the Sum flavor group, which corresponds to the SU(3) scheme proposed by Sakata [8]. The six-lined Young tableau of the favored U(6) representation is readily obtained from the four-lined one assigned to the core nucleus: simply fs = 1 and fs = 0 or 1 has to be added, for single- and double-A hypernuclei, respectively. *This work is stipported by the OTKA grant No. T22187, the collaboration grant No. 4400 between the Hungarian Academy of Sciences and the French Centre National de Recherche Scientifique, and grant No. .JF345/93 of the US-Hungarian Science and Technology Joint Fund. 03759474198619.00
0 1998 Elsevier Science B.V. All rights
PI1 SO375-9474(98)00265-6
reserved.
162~
G. LPvai et al./Nuclear
The mass formula describing
Physics A639 (1998)
161c-164~
both normal and A hypernuclei
B(N, Z, A) = a,A - a,A213 - ac$
- a,
(N-2)2 A
on an equal footing is
s +a,~
+a,=,
(W
where A = N + Z + A and the strangeness S is S = Y - B with B the baryon number and Y the hypercharge. The effect of the strangeness is represented by a simple linear term, but. it is also implicitly contained in the Majorana term. To reduce the number of parameters an equal mass dependence ry = “im is assumed. The Majorana operator is intimately connected with the Casimir operator relevant to the given algebra (see [6] and references), and for U(6) its eigenyalues can be expressed in terms of the fi (i = 1,. . ,6) labels of the Young tableau as (M) = -i C,“=, fi(fi + 1 - 2i). When applied to normal nuclei, mass formula (2) gives a better rms deviation than the Bethe-Weizsacker formula: considering all 1909 known nuclear masses [9], for example, the rms deviation is 2.68 MeV as opposed to 3.46 MeV. Similar results were found in fits restricted to selected mass regions, as well as in simultaneous fits of normal and A hypernuclei [6]. Furthermore, despite the parametrization differing from the standard one, mass formula (2) reproduces a number of characteristics of nuclear binding energies as well for normal nuclei, such as the location of the valley of stability, that of the maximal binding energy per nucleon (in the Fe-Ni region) and nucleon separation energies. In Table 1, we present the parameter set and rms deviations obtained from the fit of normal and A hypernuclei with 2 5 N, Z. The reason for excluding the lightest nuclei is that their significant contribution to the rms deviation may falsify the fit; furthermore the mass formula is not expected to work for nuclei consisting only of a few nucleons. However, excluding further light nuclei would also mean losing a sizeable portion of the A hypernuclei (see [6], where fits in other mass regions were also considered). We performed fits for normal nuclei (N), hypernuclei (H) and for normal and hypernuclei together (N+H). It is seen in Table 1 that the N and N+H fits yield approximately the same parameter set (which is the consequence of the overwhelming majority of normal nuclei considered in the fit), while the result is somewhat different in the H fit. In particular, the largest differences occur for a, and a,, the parameters of the Coulomb and the asymmetry terms, respectively. This can be explained by the fact that the majority of hypernuclei are from the low-A region. Here the difference of N and Z is relatively small, therefore the asymmetry term does not have too large a contribution to the mass formula. The relative importance of the Coulomb term is also larger for heavy nuclei due to the 2’ factor. The parameters are more balanced in a fit restricted to light nuclei with 2 5 N, Z < 20 domain [6]. Inspecting the difference of the experiment.al and calculated binding energies of the individual A hypernuclei, we found that the latter one falls behind the former one for “f;Fe and 208,Pb with 5 and 12 MeV, respectively, in the N+H fit. Similarly large deviations occur for the neighbouring normal nuclei, which are located near shell closures, so this effect clearly originates from the nuclear shell structure, which, of course, is not handled by mass formula (2). We note that in the H fit these large deviations do not occur, which may also show up in the somewhat different parameter set. The apparent difference of a, in the H and N fits can be explained as the consequence of the combined effect of the Majorana term and the linear strangeness term in (2) for hypernuclei. The linear dependence on S seems justified, since the binding energies of
163~
G. Ltvai et al. /Nuclear Physics A639 (1998) 161c-164~
Table 1 Coefficients in the mass formula (2) and associated rms deviations calculated for 1902 normal and 37 A hypernuclei with 2 5 N, 2. The parameters and rms deviations are in MeV, except y,, = 7m which is dimensionless. Fit
a,
a,
a,
a,
aY
a,,,
yY = 7m
rms
rms
rms
N H N+H
22.73 25.56 22.91
34.93 43.00 35.38
0.60 0.41 0.60
13.36 5.29 13.13
78.81 69.51
13.74 17.24 14.08
0.79 0.75 0.79
(N) 2.62 2.62
(H) 2.07 3.51
(N+H) 2.64
13B [21) are reasonably reproduced, the three double-A hypernuclei (*iHe, i!Be [l], and ,,,, even if they are not included in the fit. A remarkable result is that the A dependence of the Majorana term is close to that of the pairing term (Am3/“) of the usual Bethe-WeizsLcker mass formula: 7m is between 0.7 and 0.8 in most fits. In fact, closer inspection reveals that the Majorana term introduces a splitting between even-even, odd-even and odd-odd (normal) nuclei similarly to the pairing term. This confirms our finding that the Majorana term can be considered as a sophisticated replacement of the pairing term, the generalization of which to A hypernuclei would be somewhat problematic anyway. Formula (2) can also be used to determine the separation energy of the A particle through BA = B(N, 2, A) - B(N, 2). This quantity is only weakly influenced by shell effects, since they approximately cancel out in the difference B(N, 2, A) - B(N, 2). Fig. 1 shows the experimental B* values and the ones calculated using the N+H parameter set in Table 1. The calculated values show the same A dependence as is found experimentally, that is, a gradual increase to Bn = 26 MeV at A = 208. Except for the lightest hypernuclei (up to He), the relative difference rarely exceeds 10%. Inspecting the composition of BP, we find that the leading term is a,, the parameter of the volume term, but it also has an important contribution from the Majorana term, which varies slowly with A and can be approximated as a,A1-Tm[2 +rm(l +x2)1/8, w h ere z = (N- 2)/A. The contribution of the Majorana term is around 5 MeV for the known A hypernuclei with 20 5 A, which is comparable with the magnitude of the surface term for the heaviest hypernuclei. Another important quantity characterizing the AA interaction in the nuclear medium is ABAA = B(N, Z,2) + B(N, 2) - 2B(N, Z,l). This is known only for the three double-A hypernuclei, *iHe, i:Be, and AA 13B The experimental values are 4.66~tO.84, 4.28kO.48, and 4.76f-0.82 MeV, while the mass formula (2) yields 6.70, 3.40, and 2.43 MeV, respectively with the N+H parameter set in Table 1. AB 12~is not known experimentally for heavier nuclei. However, there are various predictions for it. Applying a zero-range AA interaction, for example, it varies with A as ABAA N A-” with Q = 1, while for a finite-range interaction Q 5 1 is expected [lo]. Our formula predicts asymptotically ABAA N a,A-rm in leading order approximation, although there are significant higher-order corrections even for the heaviest hypernuclei. We plan to extend these studies by incorporating shell effects into the mass formula, and by including further types of (strange) hyperons. Both extensions can resort to symmetry based arguments similar to those presented in this contribution.
G. LZoai et al. /Nuclear
164~
I L
,
,
Physics A639 (I998)
”
I
”
.
I
161c-164~
’
“.
,
.
. 25 -
0
. 20 -
.
BA
6
(ivIev;5
:
. 0
.
: e
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00
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10 -$ .B 5 -d ..
oto. 0
’ 50
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150
200
A Figure 1. Bn as the function of A for single-A hypernuclei with 2 5 hi, 2. Dots denote the experimental data, while open circles stand for theoretical values calculated using the parameters of the N+H fit in Table 1. The error in the experimental Bn values is of the order of 0.1 MeV for A < 30, and typically ranges between 0.5 and 1.5 MeV for A > 30.
REFERENCES 1. 2.
H. Bando, T. Motoba and J. iofka, Int. J. Mod. Phys. A 5 (1990) 4021. R. Bertini et al., Phys. Lett. B 83 (1979) 306; G.A. Lalazissis, M.E. Grypeos and SE. Massen, Phys. Rev. C 37 (1988) 2098; P.H. Pile et al., Phys. Rev. Lett. 66 (1991) 2585; S. Aoki et al., Prog. Theor. Phys. 85 (1991) 1287; C.B. Dover, D.J. Millener, A. Gal and D.H. Davis, Phys. Rev. C 44 (1991) 1905; G.A. Lalazissis, Phys. Rev. C 49 (1994) 1412; T. Hasegawa et al., Phys. Rev. C 53 (1996) 1210. 3. J. Schaffner, C.B. Dover, A. Gal, C. Greiner, D. J. Millener and H. Stocker, Ann. Phys. (N.Y.) 235 (1994) 35. 4. W. Greiner, Int. J. Mod. Phys. E 5 (1995) 1. 5. C.B. Dover and A. Gal, Nucl. Phys. A 560 (1993) 559. 6. G. LCvai, J. Cseh, P. Van Isacker and 0. Juillet, to be published. 7. F. Giirsey and L.A. Radicati, Phys. Rev. Lett. 13 (1964) 173. 8. S. Sakata, Prog. Theor. Phys. 16 (1956) 686. 9. G. Audi and A.H. Wapstra, Nucl. Phys. A 565 (1993) 1. 10. A. Gal, private communication.
A symmetry
Nuclear
A
Physics A639 (1998) 161c-164~
based mass formula for A hypernuclei*
G. Lkvaia, J. Cseha, P. Van Isackerb, 0. Juilletb aInstitute of Nuclear Research of the Hungarian Academy of Sciences, Debrecen, P.O. Box 51, Hungary 4001 bGrand Accelerateur National d’Ions Lourds, B.P. 5027, F-14076 Caen Cedex 5, France In the past couple of years a considerable amount of experimental spectroscopic information has accumulated on A hypernuclei; in particular, the A separation energy has been determined for the ground states of about 40 hypernuclei, including 3 double-A hypernuclei [1,2]. This data set allows the systematic analysis of hypernuclear binding energies in terms of mass formulae obtained as extensions of the Bethe-Weizsacker formula for normal nuclei. Until now, similar extensions have concentrated mainly on the question of the stability of multiply-strange hadronic matter, and the corresponding mass formulae have been derived either from relativistic mean-field calculations [3,4] or from hyperon-nucleus and hyperon-hyperon interactions [5]. Here, we present an alternative route by replacing the pairing term with a symmetry-inspired Majorana term, which allows a simultaneous treatment of normal and A hypernuclei. The idea of applying the Majorana operator in a mass formula was first proposed for nuclei composed of only protons and neutrons. Its justification [6] is that, due to the short-range attractive nature of the residual nuclear interaction, nucleons in the nucleus attempt to maximize their spatial (orbital) symmetry and choose a specific, favored SU(4) representation, in which the eigenvalue of the SU(4) Casimir invariant is minimal. This favored representation is uniquely determined in the ground state of nuclei (see [6] and references), and can be labelled with a four-lined U(4) Young tableau [fr, fs, fs, fd] accounting for the four possible spin-isospin single-particle states available to the nucleons. This scheme can be readily extended to A hypernuclei, when the allowed single-particle states also include the spin-up and spin-down states for A hyperons, and the group structure is expanded to that proposed by Giirsey and Radicati [7]:
U(6)3 (SU43)> b(l)@ SU,(2))C9SUs(2). The proton, neutron, and A hyperon are assigned to the (X, II) = (1,O) fundamental representation of the Sum flavor group, which corresponds to the SU(3) scheme proposed by Sakata [8]. The six-lined Young tableau of the favored U(6) representation is readily obtained from the four-lined one assigned to the core nucleus: simply fs = 1 and fs = 0 or 1 has to be added, for single- and double-A hypernuclei, respectively. *This work is stipported by the OTKA grant No. T22187, the collaboration grant No. 4400 between the Hungarian Academy of Sciences and the French Centre National de Recherche Scientifique, and grant No. .JF345/93 of the US-Hungarian Science and Technology Joint Fund. 03759474198619.00
0 1998 Elsevier Science B.V. All rights
PI1 SO375-9474(98)00265-6
reserved.
162~
G. LPvai et al./Nuclear
The mass formula describing
Physics A639 (1998)
161c-164~
both normal and A hypernuclei
B(N, Z, A) = a,A - a,A213 - ac$
- a,
(N-2)2 A
on an equal footing is
s +a,~
+a,=,
(W
where A = N + Z + A and the strangeness S is S = Y - B with B the baryon number and Y the hypercharge. The effect of the strangeness is represented by a simple linear term, but. it is also implicitly contained in the Majorana term. To reduce the number of parameters an equal mass dependence ry = “im is assumed. The Majorana operator is intimately connected with the Casimir operator relevant to the given algebra (see [6] and references), and for U(6) its eigenyalues can be expressed in terms of the fi (i = 1,. . ,6) labels of the Young tableau as (M) = -i C,“=, fi(fi + 1 - 2i). When applied to normal nuclei, mass formula (2) gives a better rms deviation than the Bethe-Weizsacker formula: considering all 1909 known nuclear masses [9], for example, the rms deviation is 2.68 MeV as opposed to 3.46 MeV. Similar results were found in fits restricted to selected mass regions, as well as in simultaneous fits of normal and A hypernuclei [6]. Furthermore, despite the parametrization differing from the standard one, mass formula (2) reproduces a number of characteristics of nuclear binding energies as well for normal nuclei, such as the location of the valley of stability, that of the maximal binding energy per nucleon (in the Fe-Ni region) and nucleon separation energies. In Table 1, we present the parameter set and rms deviations obtained from the fit of normal and A hypernuclei with 2 5 N, Z. The reason for excluding the lightest nuclei is that their significant contribution to the rms deviation may falsify the fit; furthermore the mass formula is not expected to work for nuclei consisting only of a few nucleons. However, excluding further light nuclei would also mean losing a sizeable portion of the A hypernuclei (see [6], where fits in other mass regions were also considered). We performed fits for normal nuclei (N), hypernuclei (H) and for normal and hypernuclei together (N+H). It is seen in Table 1 that the N and N+H fits yield approximately the same parameter set (which is the consequence of the overwhelming majority of normal nuclei considered in the fit), while the result is somewhat different in the H fit. In particular, the largest differences occur for a, and a,, the parameters of the Coulomb and the asymmetry terms, respectively. This can be explained by the fact that the majority of hypernuclei are from the low-A region. Here the difference of N and Z is relatively small, therefore the asymmetry term does not have too large a contribution to the mass formula. The relative importance of the Coulomb term is also larger for heavy nuclei due to the 2’ factor. The parameters are more balanced in a fit restricted to light nuclei with 2 5 N, Z < 20 domain [6]. Inspecting the difference of the experiment.al and calculated binding energies of the individual A hypernuclei, we found that the latter one falls behind the former one for “f;Fe and 208,Pb with 5 and 12 MeV, respectively, in the N+H fit. Similarly large deviations occur for the neighbouring normal nuclei, which are located near shell closures, so this effect clearly originates from the nuclear shell structure, which, of course, is not handled by mass formula (2). We note that in the H fit these large deviations do not occur, which may also show up in the somewhat different parameter set. The apparent difference of a, in the H and N fits can be explained as the consequence of the combined effect of the Majorana term and the linear strangeness term in (2) for hypernuclei. The linear dependence on S seems justified, since the binding energies of
163~
G. Ltvai et al. /Nuclear Physics A639 (1998) 161c-164~
Table 1 Coefficients in the mass formula (2) and associated rms deviations calculated for 1902 normal and 37 A hypernuclei with 2 5 N, 2. The parameters and rms deviations are in MeV, except y,, = 7m which is dimensionless. Fit
a,
a,
a,
a,
aY
a,,,
yY = 7m
rms
rms
rms
N H N+H
22.73 25.56 22.91
34.93 43.00 35.38
0.60 0.41 0.60
13.36 5.29 13.13
78.81 69.51
13.74 17.24 14.08
0.79 0.75 0.79
(N) 2.62 2.62
(H) 2.07 3.51
(N+H) 2.64
13B [21) are reasonably reproduced, the three double-A hypernuclei (*iHe, i!Be [l], and ,,,, even if they are not included in the fit. A remarkable result is that the A dependence of the Majorana term is close to that of the pairing term (Am3/“) of the usual Bethe-WeizsLcker mass formula: 7m is between 0.7 and 0.8 in most fits. In fact, closer inspection reveals that the Majorana term introduces a splitting between even-even, odd-even and odd-odd (normal) nuclei similarly to the pairing term. This confirms our finding that the Majorana term can be considered as a sophisticated replacement of the pairing term, the generalization of which to A hypernuclei would be somewhat problematic anyway. Formula (2) can also be used to determine the separation energy of the A particle through BA = B(N, 2, A) - B(N, 2). This quantity is only weakly influenced by shell effects, since they approximately cancel out in the difference B(N, 2, A) - B(N, 2). Fig. 1 shows the experimental B* values and the ones calculated using the N+H parameter set in Table 1. The calculated values show the same A dependence as is found experimentally, that is, a gradual increase to Bn = 26 MeV at A = 208. Except for the lightest hypernuclei (up to He), the relative difference rarely exceeds 10%. Inspecting the composition of BP, we find that the leading term is a,, the parameter of the volume term, but it also has an important contribution from the Majorana term, which varies slowly with A and can be approximated as a,A1-Tm[2 +rm(l +x2)1/8, w h ere z = (N- 2)/A. The contribution of the Majorana term is around 5 MeV for the known A hypernuclei with 20 5 A, which is comparable with the magnitude of the surface term for the heaviest hypernuclei. Another important quantity characterizing the AA interaction in the nuclear medium is ABAA = B(N, Z,2) + B(N, 2) - 2B(N, Z,l). This is known only for the three double-A hypernuclei, *iHe, i:Be, and AA 13B The experimental values are 4.66~tO.84, 4.28kO.48, and 4.76f-0.82 MeV, while the mass formula (2) yields 6.70, 3.40, and 2.43 MeV, respectively with the N+H parameter set in Table 1. AB 12~is not known experimentally for heavier nuclei. However, there are various predictions for it. Applying a zero-range AA interaction, for example, it varies with A as ABAA N A-” with Q = 1, while for a finite-range interaction Q 5 1 is expected [lo]. Our formula predicts asymptotically ABAA N a,A-rm in leading order approximation, although there are significant higher-order corrections even for the heaviest hypernuclei. We plan to extend these studies by incorporating shell effects into the mass formula, and by including further types of (strange) hyperons. Both extensions can resort to symmetry based arguments similar to those presented in this contribution.
G. LZoai et al. /Nuclear
164~
I L
,
,
Physics A639 (I998)
”
I
”
.
I
161c-164~
’
“.
,
.
. 25 -
0
. 20 -
.
BA
6
(ivIev;5
:
. 0
.
: e
. c
00
j
10 -$ .B 5 -d ..
oto. 0
’ 50
’ 100
I
I
150
200
A Figure 1. Bn as the function of A for single-A hypernuclei with 2 5 hi, 2. Dots denote the experimental data, while open circles stand for theoretical values calculated using the parameters of the N+H fit in Table 1. The error in the experimental Bn values is of the order of 0.1 MeV for A < 30, and typically ranges between 0.5 and 1.5 MeV for A > 30.
REFERENCES 1. 2.
H. Bando, T. Motoba and J. iofka, Int. J. Mod. Phys. A 5 (1990) 4021. R. Bertini et al., Phys. Lett. B 83 (1979) 306; G.A. Lalazissis, M.E. Grypeos and SE. Massen, Phys. Rev. C 37 (1988) 2098; P.H. Pile et al., Phys. Rev. Lett. 66 (1991) 2585; S. Aoki et al., Prog. Theor. Phys. 85 (1991) 1287; C.B. Dover, D.J. Millener, A. Gal and D.H. Davis, Phys. Rev. C 44 (1991) 1905; G.A. Lalazissis, Phys. Rev. C 49 (1994) 1412; T. Hasegawa et al., Phys. Rev. C 53 (1996) 1210. 3. J. Schaffner, C.B. Dover, A. Gal, C. Greiner, D. J. Millener and H. Stocker, Ann. Phys. (N.Y.) 235 (1994) 35. 4. W. Greiner, Int. J. Mod. Phys. E 5 (1995) 1. 5. C.B. Dover and A. Gal, Nucl. Phys. A 560 (1993) 559. 6. G. LCvai, J. Cseh, P. Van Isacker and 0. Juillet, to be published. 7. F. Giirsey and L.A. Radicati, Phys. Rev. Lett. 13 (1964) 173. 8. S. Sakata, Prog. Theor. Phys. 16 (1956) 686. 9. G. Audi and A.H. Wapstra, Nucl. Phys. A 565 (1993) 1. 10. A. Gal, private communication.