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The scissors mode in the presence of a neutron skin
cm *.__
__ El
ELSEVIER
13
March 1997
PHYSICS
LElTERS
6
Physics Letters B 395 (1997) 145-150
The scissors mode in the presence of a neutron skin D.D. Warner a, F? Van Isacker b a CCLRC Daresbury Laboratory Daresbury Warrington WA4 4AD, UK b GANIL, BP 5027, F-14021 Caen Cedex, France
Received 12 November 1996; revised manuscript received 13 January 1997 Editor: C. Mahaux
Abstract The effect of a neutron skin on the collective scissors mode in very neutron-rich nuclei is investigated. The algebraic approach is used to show that the development of a neutron skin can give rise to an additional low-lying mode involving out-of-phase motion of the skin neutrons against the remaining nucleons of the core. This “soft” scissors mode represents the analogue of the soft dipole mode already postulated to occur in such nuclei. @ 1997 Elsevier Science B.V.
In recent years, experiments with radioactive beams from projectile fragmentation facilities have revealed [l] the presence of a neutron halo in several of the lightest nuclei on the neutron drip line. This is now understood to arise when the last one or two neutrons are in low angular momentum orbits very near the top of the well so that their wave functions have a very extended distribution which is manifest empirically in an anomalously large matter radius. There is, however, a distinctly different phenomenon which is predicted in some Hartree-Fock calculations [2,3] to occur in heavier nuclei in which an excess of several neutrons builds up so that the neutron density actually extends out significantly further than that of the protons, resulting in a mantle of dominantly neutron matter. The presence of this neutron “skin” may affect collective modes of nuclear excitation which involve the outof-phase motion of neutrons against protons, such as the giant dipole resonance (GDR) [4] and the scissors mode [ 51. There is also then the possibility of a “soft” dipole mode [ 61 in which the core nucleons move against the more weakly bound skin neutrons. Recently, the effect of an increasing skin thickness on the energy of these three modes was investigated [7] with a simple approach based on classical density oscillations, in which the change in the potential energy in each case was estimated from the density overlaps as a function of displacement. Not surprisingly, the effect on the first two modes was found to be minimal, while the soft dipole mode drops rapidly in energy, relative to the GDR. It is the purpose of this Letter to investigate further the behaviour of the scissors mode in the presence of a neutron skin. The basic algebraic approach associated with the interacting boson model (IBM) [ 81 which has previously proved particularly enlightening in characterising the normal scissors mode [ 9,101 will be employed. It will be shown that, as in the dipole case, the existence of a soft scissors mode may also be postulated. 0370-2693/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO370-2693(97)00086-5
146
D.D. Warnel; P Van Isacker/Physics Letters B 395 (1997) 145-150
In the following protons are denoted with n- and neutrons with V. The incorporation of both neutrons and protons in the IBM involves the algebra of the product U,( 6) @ U, (6). In the normal application of the IBM to nuclei nearer to stability, the nucleon pairs can be described microscopically in terms of the known shell structure. In the problem addressed here, the shell structure at the extremes of stability is not known and no such link can be established quantitatively at this stage. Nevertheless, the algebra of U(6) can still be used to provide an elegant and simple way of looking at the quadrupole degrees of freedom available to the three component system which results from the development a neutron skin sufficient in extent to be at least partially decoupled from the core nucleons. Moreover, it will become apparent later that some features of the results are actually independent of the particular unitary algebra involved. The starting point for the quadrupole modes of a nucleus with an additional neutron skin is taken as a triple product involving an additional algebra U, (6) with the remaining core neutrons being described by U, (6). The dynamical algebra of the system is then U,(6)
&ml
@ U,,(6) L
EJ U,(6) 1
3
(1)
[Nv,l
[%I
where each U( 6) algebra is characterised by a number of bosons Ni that are coupled symmetrically to [Nil. The fact that the skin neutrons are assumed to interact weakly with the core neutrons and protons, which interact strongly with each other, is represented in the reduction of ( 1) by coupling the corresponding U(6) algebra of the neutron skin after those describing the core nucleons. The reduction thus proceeds as U,(6)
@U,,(6)
@&s(6)
x&,(6)
@U,,(6)
1 U,,,,(6).
(2)
(6) has a subalgebra structure familiar from IBM-I, and, specifically, the three The triple-sum algebra UrrVCVs usual limits [ 81, U(5), SU(3) and O(6), can be obtained as subchains of (2). We stress that the results derived below are not only valid in the dynamical symmetries but also for intermediate situations. By virtue (6) it is assumed that an appropriate identical mixture of U( 5)) SU( 3) and 0( 6) is of the presence of U,,, valid for all three subsystems r, vC and Y,. More general situations, where the deformations of the subsystems are different, can be envisaged in an algebraic treatment but are not considered here. It is the coupled nature of the algebra U,,(6) in IBM-2 that permits states with mixed symmetry [ 111, the lowest of which in deformed nuclei represent the normal scissors mode. In the reduction (2), U,,., (6) is characterised by irreducible representations [NC - f, f] where NC is the number of nucleon pairs in the core, NC = N, + NVC. The lowest states are then contained in the representation [NC, 01, which denotes the totally symmetric coupling. The lowest states of mixed symmetry are in the next representation, [NC - 1, l] . The triple-sum algebra UrVCy,(6) is characterised by up to three rows, with the lowest couplings arising from [NC, 0] x [NV,] being [N, 0, 0] and [N - 1, 1, 0] , N denoting the total number of bosons. Hence the first nonsymmetric representation resulting from the triple-sum algebra describes symmetric coupling of the core nucleons and non-symmetric coupling of the skin neutrons. However, the non-symmetric representation [N - 1,1,0] of U ryCu,(6) may also arise from the product [NC - 1, l] x [NV,]. In this case, it is the core nucleons which are coupled non-symmetrically. The result is that there are now two scissors modes, one representing out-of-phase motion between the neutrons and protons in the core and the other denoting an oscillation between the core and the skin where, in this case, as in the soft dipole mode, the core protons carry the core neutrons with them. Denoting these two classes of states as 1%~) and ]SSa) respectively, where CYdescribes all further sublabels of U(6), we have ISa), = I[&1 x [&I
[!=a), = I[N,rl x [&I where the subscript
x [N,,l; [N,+Nv,-I,11 x [N,,l; [N,r+N,,,Ol
a is used to distinguish
x [Ksl;
[N-l,
x [N,,l; [N-l,
1,014,
1,01~$,
(3) from other scissors states discussed
(3) below.
D.D. Warnel; F! Van Isacker/Physics Letters B 395 (1997) 145-150
147
While the intuitive approach to the relative interaction strengths of the constituents leads to the two modes described above, it is also possible to propose a scissors mode involving an oscillation of the protons against all the neutrons rather than just those in the core. Similarly, the soft scissors mode always involves the skin neutrons but these can oscillate against just the protons rather than all the nucleons in the core. These additional possibilities can be represented algebraically and emerge from the remaining two possible reductions of ( 1) ,
U,(6) @U,(6) @b,(6) 1
(b) U,v,(6) ~3U,(6) Cc> U,(6) @U,,(6)
>
u
m,,(6).
(4)
In what follows, the classifications or limits (2) and (4) are denoted as a, b and c, respectively. In the same way as discussed above for limit a, each of the reductions b and c gives rise to a “normal” and “soft” scissors mode, the latter being distinguished by’ non-symmetric coupling of the skin neutrons with one of the other constituents. The scissors states in limits b and c are
I~~~o-I~~~1~~~,,l~~~,,l;~~,+~,,,~l~~~,,l;~~-l,1,0la), IS& = IINrl x [&I x [%$I; [N,I x [N,,+N,,,01; [N-l,
l,Ola),
(5)
where the first corresponds to an oscillation of the core neutrons against the protons and the skin neutrons while in the second the protons oscillate against all neutrons. The latter assumption is usually taken for the scissors state but it is as yet unclear whether such will be the case in nuclei with a large neutron excess. The former assumption is, a priori, unreasonable. The soft-scissors states in limits b and c are
ISWC = I[N,l x r&l x [~vsl; [N,l x [N++N+-L
11; [N-l,
Lola),
(6)
where the first corresponds to an oscillation of the skin neutrons against the protons while in the second they oscillate against the core neutrons. Again, one of the modes, the second in this case, appears unphysical. Thus the basis a which stemmed from the physically intuitive choice of coupling remains the most attractive. Nevertheless, the fact that the three U( 6) algebras are coupled to the same final quantum numbers [ N - 1, 1, 0] implies that the three bases are connected by a unitary transformation and thus any one state can be expressed as a suitable mixture in one of the other bases. Moreover, any of the bases (2) or (4) can be generated with the hamiltonian
fi=&W,v,UN
+P&Wm,Wl
+~~22[Uv,v,(6)1 +&W,rvcvs(6)l,
(7)
where & [ U,( 6) ] denotes the quadratic Casimir operator of the algebra U,( 6) obtained by adding the generators of subsystems i and j. This Casimir operator is related to a Majorana operator which gives zero when acting on states symmetric in U
is the Majorana
ez[Uij(6)]
operator associated
= (Ni+Nj)(Ni+Nj+5)-2&ij.
with Uij (6)) (9)
For firV,u, the same result holds with Ni + Nj = N, + NYC+ NV,. The three limits introduced above are recovered for specific choices of the parameters in the hamiltonian (8)) that is, B = C = 0 in limit a, A = C = 0 in limit b, and A = B = 0 in limit c.
D.D. Warnel; P. Van Isacker/Physics Letters B 395 (1997) 145-150
148
The energy contribution of (8) to the total hamiltonian can be obtained analytically for states with Urrv,v, (6) symmetry [N - 1, 1,0] , to which normal and soft scissors states belong. Its matrix elements in basis a are given by
Note that the energy matrix is independent of a, the sublabels of U?ru,v, (6). As a result, the energy eigenvalues of [N - 1, 1,0] states are obtained by diagonalising a 2 x 2 matrix, which can be carried out analytically. As a side result of the diagonalisation of (lo), the transformation between the various normal and soft scissors states as defined in (3)) (5)) and (6) can be obtained. In particular, the following overlaps involving the physically reasonable states are found:
(11) Clearly, the scissors and soft-scissors states in any one limit are orthogonal; e.g. .(SalSScu), = 0. Note, however, that the modes are not orthogonal if different limits are considered, albeit involving reasonable assumptions concerning the nature of the modes. For instance, if one considers an oscillation of the protons against all neutrons for the scissors state (limit c) and an oscillation of the skin neutrons against all other nucleons for the soft scissors state (limit a), the overlap is then 112
.(S4=4a =
[ N;:)N] . (N
77
UC
(12)
v
The overlaps ( 11) and ( 12) can be interpreted as recoupling (or Racah) coefficients in U( 6). It is seen that they come out as square roots of ratios of the various boson numbers involved. They have a general structure which is independent of the particular unitary algebra [U(6) in this case] but depends solely on the character of the representations (symmetric [ Ni] and mixed-symmetric [ Ni - 1 , 1] ) and on the order of the coupling. Returning now to our initial postulate, represented by limit a, the energy of the scissors mode, from (10) with B = C = 0, is A (N, + NV, ) + D N while for the soft scissors it is DN. The centre of gravity of the orbital Ml strength distribution in rare earth nuclei indicates an excitation energy for the normal mode of E, w 3 MeV. The signs of A and D are determined to be positive by the requirement that the fully symmetric representation of U(6) should lie lowest in energy. The soft mode will thus be lower than the normal mode, the extent depending on the relative magnitudes of the two coefficients. The only difference in the constituents involved in the interactions represented by the two Casimir operators is the skin neutrons. Thus it may be reasonable to assume that the two constants have roughly equal magnitude, if the number of neutrons in the skin is small compared to the total, in which case the soft mode should appear at approximately half the energy of the normal one, resulting in a predicted energy for the soft mode of E, - 1.5 MeV. The other two terms in the hamiltonian (8) can now be used to perturb this pure limit; for example, with B # 0 the dependence of the ratio of energies on B can be estimated by expanding in Nvs/N,
ESS -N Es -
A(&
+N,r) +BN,r
2A(Nz+ + Nn-)
2A2 - AB - 2B2 Nvs -3 + 4A(2A-B) N
(13)
D.D. Warner; R Van Isacker/Physics Letters B 395 (1997) 145-150
149
which was found assuming A = D, C = 0, and B < 2A, and neglecting the difference between NV, and N,, in the second term. It can be seen that the energy ratio increases for non-zero B. The characteristic excitation of these mixed-symmetry modes is via magnetic dipole transitions. In even-even nuclei the existence of l+ scissors states excited in (e, e’) or (y, y’) is by now well established. The IBM-2 prediction for the Ml strength towards the If state corresponding to ]SCX),in the above is
BtMl;OGf-+ 12) = -&
where g, and g, are the boson g factors. The function IBM, (
0
(14)
-gzJ2f(N)N?rNv, f(N)
is known analytically
in the three limits of the
U(5) (15)
The U(5), SU( 3) and O(6) limits correspond to vibrational, deformed and y-soft nuclei, respectively, and in each case, candidates for states of mixed symmetry exist [ 10,12,13]. Eq. (14) is valid for the scissors state of limit c in which all the protons oscillate against all neutrons. We stress that, with an appropriate choice of interpolating function f(N) (e.g., the one of Ref. [ 141)) ( 14) is also valid for a mixture of the three limits and corresponds to the sum rule Ml strength, which is concentrated in one state in the three symmetry limits but is fragmented over several states in general. Likewise, all results reported below for the soft scissors mode are valid for intermediate situations. A similar expression can be derived for the dipole strength to the soft scissors state of limit a by considering the separate contributions to the Ml operator from the core and the skin neutrons, p(M1)
= g,r&
+ gp& = g,&r + g&
+ g,%,,
(16)
and this yields B(M1; 0; --+ l&J
= $(gr
- sp)‘f(N)
NNN+N; ?T UC
This result can also be obtained using the sum rule ( 14) and the transformation The soft scissors mode a can be written in basis c as
(17) between
the different
bases.
(18) The sum rule (14) implies that the Ml strength to the state ]SSCX), is zero. This is entirely consistent with the fact that, as deduced earlier, here the oscillation is one involving core neutrons against skin neutrons. The result ( 17) then follows immediately from ( 14) and ( 18). The corresponding expressions in other limits can be derived in a similar fashion. From ( 14) and ( 17) one finds the following simple result for the ratio of B(M1) ‘s in the soft and normal scissors modes: B(Ml;O&
--f l&J
B(MI;O&
-+ l;J
N,N,, = (NV + N,,)N,’
(19)
This value is identical to that found previously for the El sum rule ratio of the soft and giant dipole resonances [7] so that again, the result depends only on the relative number of constituents in the subsystems, rather than on the specific algebra chosen.
D.D. Wamel; I? Van Isacker/Physics Letters B 395 (1997) 145-150
150
In summary, it has been shown that the algebraic approach can be extended to describe a three-component system, as arises in the presence of a neutron skin in very neutron-rich nuclei. The relative coupling strengths between the components can be represented by the choice of coupling scheme and the method reveals the possibility of a soft scissors mode in which the skin neutrons undergo out-of-phase oscillations against the remaining nucleons. The analogy with the soft dipole mode is clear and the probability of finding an empirical realisation of either depends critically on the extent to which the neutron skin develops in heavier nuclei as the neutron drip line is approached. In fact, recent results from Hartree-Fock RPA calculations [ 151 suggest that the separation between the degrees of freedom of the core and skin in quadrupole collective states may be sufficient to allow new collective modes involving the skin neutrons alone. In the current example, if the extent of the skin is at least comparable to the range of the proton-neutron interaction, the notion of a partial decoupling of it from the core becomes valid and the soft dipole modes may manifest themselves, albeit in a possibly fragmented form. The corollary is, of course, that evidence for such modes in the form of low-lying l+ states with enhanced Ml decays could serve as an empirical signature of the development of a neutron skin. We wish to thank M.A. Nagarajan and F. Iachello for useful discussions. This work is supported France-British Alliance grant PN 96-058 and by the EU grant CHGE-CT-94-00-56.
References [I] [2] [3] [4] [5] [6] [7] [ 81 [9]
[lo] [ 1 l] [ 121 [ 131 [ 141 [ 151
I. Tanihata et al., Phys. Rev. Lett. 55 (1985) 2676; Phys. Lett. B 206 (1988) 592. N. Fukunishi, T. Otsuka and I. Tanihata, Phys. Rev. C 48 (1993) 1648. J. Dobazcewski, W. Nazarcwicz and T.R. Werner, Z. Phys. A 354 (1996) 27. M. Goldhaber and E. Teller, Phys. Rev. 74 (1948) 1046. N. Lo Iudice and E Palumbo, Phys. Rev. L&t. 41 (1978) 1532. PG. Hansen and B. Jonson, Europhys. Lett. 4 (1987) 409; K. Ikeda, Nucl. Phys. A 538 (1992) 355~. P Van Isacker, M.A. Nagarajan and D.D. Warner, Phys. Rev. C 45 (1992) R13. E Iachello and A. Arima, The interacting boson model (Cambridge Univ. Press, Cambridge, E Iachello, Phys. Rev. Lett. 53 (1984) 1427. D. Bohle et al., Phys. Len B 137 (1984) 27. P. Van Isacker et al., Ann. Phys. 171 (1986) 253. W.D. Hamilton, A. IrbSick and J.P. Elliott, Phys. Rev. Lett. 53 (1984) 2469. I? von Brentano et al., Phys. Rev. Lett. 76 (1996) 2029. P. von Neumann-Cosel et al., Phys. Rev. Lett. 75 (1995) 4178. M. Yokoyama, T. Otsuka and N. Fukunishi, Phys. Rev. C 52 (1995) 1122.
1987).
by the
__ El
ELSEVIER
13
March 1997
PHYSICS
LElTERS
6
Physics Letters B 395 (1997) 145-150
The scissors mode in the presence of a neutron skin D.D. Warner a, F? Van Isacker b a CCLRC Daresbury Laboratory Daresbury Warrington WA4 4AD, UK b GANIL, BP 5027, F-14021 Caen Cedex, France
Received 12 November 1996; revised manuscript received 13 January 1997 Editor: C. Mahaux
Abstract The effect of a neutron skin on the collective scissors mode in very neutron-rich nuclei is investigated. The algebraic approach is used to show that the development of a neutron skin can give rise to an additional low-lying mode involving out-of-phase motion of the skin neutrons against the remaining nucleons of the core. This “soft” scissors mode represents the analogue of the soft dipole mode already postulated to occur in such nuclei. @ 1997 Elsevier Science B.V.
In recent years, experiments with radioactive beams from projectile fragmentation facilities have revealed [l] the presence of a neutron halo in several of the lightest nuclei on the neutron drip line. This is now understood to arise when the last one or two neutrons are in low angular momentum orbits very near the top of the well so that their wave functions have a very extended distribution which is manifest empirically in an anomalously large matter radius. There is, however, a distinctly different phenomenon which is predicted in some Hartree-Fock calculations [2,3] to occur in heavier nuclei in which an excess of several neutrons builds up so that the neutron density actually extends out significantly further than that of the protons, resulting in a mantle of dominantly neutron matter. The presence of this neutron “skin” may affect collective modes of nuclear excitation which involve the outof-phase motion of neutrons against protons, such as the giant dipole resonance (GDR) [4] and the scissors mode [ 51. There is also then the possibility of a “soft” dipole mode [ 61 in which the core nucleons move against the more weakly bound skin neutrons. Recently, the effect of an increasing skin thickness on the energy of these three modes was investigated [7] with a simple approach based on classical density oscillations, in which the change in the potential energy in each case was estimated from the density overlaps as a function of displacement. Not surprisingly, the effect on the first two modes was found to be minimal, while the soft dipole mode drops rapidly in energy, relative to the GDR. It is the purpose of this Letter to investigate further the behaviour of the scissors mode in the presence of a neutron skin. The basic algebraic approach associated with the interacting boson model (IBM) [ 81 which has previously proved particularly enlightening in characterising the normal scissors mode [ 9,101 will be employed. It will be shown that, as in the dipole case, the existence of a soft scissors mode may also be postulated. 0370-2693/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO370-2693(97)00086-5
146
D.D. Warnel; P Van Isacker/Physics Letters B 395 (1997) 145-150
In the following protons are denoted with n- and neutrons with V. The incorporation of both neutrons and protons in the IBM involves the algebra of the product U,( 6) @ U, (6). In the normal application of the IBM to nuclei nearer to stability, the nucleon pairs can be described microscopically in terms of the known shell structure. In the problem addressed here, the shell structure at the extremes of stability is not known and no such link can be established quantitatively at this stage. Nevertheless, the algebra of U(6) can still be used to provide an elegant and simple way of looking at the quadrupole degrees of freedom available to the three component system which results from the development a neutron skin sufficient in extent to be at least partially decoupled from the core nucleons. Moreover, it will become apparent later that some features of the results are actually independent of the particular unitary algebra involved. The starting point for the quadrupole modes of a nucleus with an additional neutron skin is taken as a triple product involving an additional algebra U, (6) with the remaining core neutrons being described by U, (6). The dynamical algebra of the system is then U,(6)
&ml
@ U,,(6) L
EJ U,(6) 1
3
(1)
[Nv,l
[%I
where each U( 6) algebra is characterised by a number of bosons Ni that are coupled symmetrically to [Nil. The fact that the skin neutrons are assumed to interact weakly with the core neutrons and protons, which interact strongly with each other, is represented in the reduction of ( 1) by coupling the corresponding U(6) algebra of the neutron skin after those describing the core nucleons. The reduction thus proceeds as U,(6)
@U,,(6)
@&s(6)
x&,(6)
@U,,(6)
1 U,,,,(6).
(2)
(6) has a subalgebra structure familiar from IBM-I, and, specifically, the three The triple-sum algebra UrrVCVs usual limits [ 81, U(5), SU(3) and O(6), can be obtained as subchains of (2). We stress that the results derived below are not only valid in the dynamical symmetries but also for intermediate situations. By virtue (6) it is assumed that an appropriate identical mixture of U( 5)) SU( 3) and 0( 6) is of the presence of U,,, valid for all three subsystems r, vC and Y,. More general situations, where the deformations of the subsystems are different, can be envisaged in an algebraic treatment but are not considered here. It is the coupled nature of the algebra U,,(6) in IBM-2 that permits states with mixed symmetry [ 111, the lowest of which in deformed nuclei represent the normal scissors mode. In the reduction (2), U,,., (6) is characterised by irreducible representations [NC - f, f] where NC is the number of nucleon pairs in the core, NC = N, + NVC. The lowest states are then contained in the representation [NC, 01, which denotes the totally symmetric coupling. The lowest states of mixed symmetry are in the next representation, [NC - 1, l] . The triple-sum algebra UrVCy,(6) is characterised by up to three rows, with the lowest couplings arising from [NC, 0] x [NV,] being [N, 0, 0] and [N - 1, 1, 0] , N denoting the total number of bosons. Hence the first nonsymmetric representation resulting from the triple-sum algebra describes symmetric coupling of the core nucleons and non-symmetric coupling of the skin neutrons. However, the non-symmetric representation [N - 1,1,0] of U ryCu,(6) may also arise from the product [NC - 1, l] x [NV,]. In this case, it is the core nucleons which are coupled non-symmetrically. The result is that there are now two scissors modes, one representing out-of-phase motion between the neutrons and protons in the core and the other denoting an oscillation between the core and the skin where, in this case, as in the soft dipole mode, the core protons carry the core neutrons with them. Denoting these two classes of states as 1%~) and ]SSa) respectively, where CYdescribes all further sublabels of U(6), we have ISa), = I[&1 x [&I
[!=a), = I[N,rl x [&I where the subscript
x [N,,l; [N,+Nv,-I,11 x [N,,l; [N,r+N,,,Ol
a is used to distinguish
x [Ksl;
[N-l,
x [N,,l; [N-l,
1,014,
1,01~$,
(3) from other scissors states discussed
(3) below.
D.D. Warnel; F! Van Isacker/Physics Letters B 395 (1997) 145-150
147
While the intuitive approach to the relative interaction strengths of the constituents leads to the two modes described above, it is also possible to propose a scissors mode involving an oscillation of the protons against all the neutrons rather than just those in the core. Similarly, the soft scissors mode always involves the skin neutrons but these can oscillate against just the protons rather than all the nucleons in the core. These additional possibilities can be represented algebraically and emerge from the remaining two possible reductions of ( 1) ,
U,(6) @U,(6) @b,(6) 1
(b) U,v,(6) ~3U,(6) Cc> U,(6) @U,,(6)
>
u
m,,(6).
(4)
In what follows, the classifications or limits (2) and (4) are denoted as a, b and c, respectively. In the same way as discussed above for limit a, each of the reductions b and c gives rise to a “normal” and “soft” scissors mode, the latter being distinguished by’ non-symmetric coupling of the skin neutrons with one of the other constituents. The scissors states in limits b and c are
I~~~o-I~~~1~~~,,l~~~,,l;~~,+~,,,~l~~~,,l;~~-l,1,0la), IS& = IINrl x [&I x [%$I; [N,I x [N,,+N,,,01; [N-l,
l,Ola),
(5)
where the first corresponds to an oscillation of the core neutrons against the protons and the skin neutrons while in the second the protons oscillate against all neutrons. The latter assumption is usually taken for the scissors state but it is as yet unclear whether such will be the case in nuclei with a large neutron excess. The former assumption is, a priori, unreasonable. The soft-scissors states in limits b and c are
ISWC = I[N,l x r&l x [~vsl; [N,l x [N++N+-L
11; [N-l,
Lola),
(6)
where the first corresponds to an oscillation of the skin neutrons against the protons while in the second they oscillate against the core neutrons. Again, one of the modes, the second in this case, appears unphysical. Thus the basis a which stemmed from the physically intuitive choice of coupling remains the most attractive. Nevertheless, the fact that the three U( 6) algebras are coupled to the same final quantum numbers [ N - 1, 1, 0] implies that the three bases are connected by a unitary transformation and thus any one state can be expressed as a suitable mixture in one of the other bases. Moreover, any of the bases (2) or (4) can be generated with the hamiltonian
fi=&W,v,UN
+P&Wm,Wl
+~~22[Uv,v,(6)1 +&W,rvcvs(6)l,
(7)
where & [ U,( 6) ] denotes the quadratic Casimir operator of the algebra U,( 6) obtained by adding the generators of subsystems i and j. This Casimir operator is related to a Majorana operator which gives zero when acting on states symmetric in U
is the Majorana
ez[Uij(6)]
operator associated
= (Ni+Nj)(Ni+Nj+5)-2&ij.
with Uij (6)) (9)
For firV,u, the same result holds with Ni + Nj = N, + NYC+ NV,. The three limits introduced above are recovered for specific choices of the parameters in the hamiltonian (8)) that is, B = C = 0 in limit a, A = C = 0 in limit b, and A = B = 0 in limit c.
D.D. Warnel; P. Van Isacker/Physics Letters B 395 (1997) 145-150
148
The energy contribution of (8) to the total hamiltonian can be obtained analytically for states with Urrv,v, (6) symmetry [N - 1, 1,0] , to which normal and soft scissors states belong. Its matrix elements in basis a are given by
Note that the energy matrix is independent of a, the sublabels of U?ru,v, (6). As a result, the energy eigenvalues of [N - 1, 1,0] states are obtained by diagonalising a 2 x 2 matrix, which can be carried out analytically. As a side result of the diagonalisation of (lo), the transformation between the various normal and soft scissors states as defined in (3)) (5)) and (6) can be obtained. In particular, the following overlaps involving the physically reasonable states are found:
(11) Clearly, the scissors and soft-scissors states in any one limit are orthogonal; e.g. .(SalSScu), = 0. Note, however, that the modes are not orthogonal if different limits are considered, albeit involving reasonable assumptions concerning the nature of the modes. For instance, if one considers an oscillation of the protons against all neutrons for the scissors state (limit c) and an oscillation of the skin neutrons against all other nucleons for the soft scissors state (limit a), the overlap is then 112
.(S4=4a =
[ N;:)N] . (N
77
UC
(12)
v
The overlaps ( 11) and ( 12) can be interpreted as recoupling (or Racah) coefficients in U( 6). It is seen that they come out as square roots of ratios of the various boson numbers involved. They have a general structure which is independent of the particular unitary algebra [U(6) in this case] but depends solely on the character of the representations (symmetric [ Ni] and mixed-symmetric [ Ni - 1 , 1] ) and on the order of the coupling. Returning now to our initial postulate, represented by limit a, the energy of the scissors mode, from (10) with B = C = 0, is A (N, + NV, ) + D N while for the soft scissors it is DN. The centre of gravity of the orbital Ml strength distribution in rare earth nuclei indicates an excitation energy for the normal mode of E, w 3 MeV. The signs of A and D are determined to be positive by the requirement that the fully symmetric representation of U(6) should lie lowest in energy. The soft mode will thus be lower than the normal mode, the extent depending on the relative magnitudes of the two coefficients. The only difference in the constituents involved in the interactions represented by the two Casimir operators is the skin neutrons. Thus it may be reasonable to assume that the two constants have roughly equal magnitude, if the number of neutrons in the skin is small compared to the total, in which case the soft mode should appear at approximately half the energy of the normal one, resulting in a predicted energy for the soft mode of E, - 1.5 MeV. The other two terms in the hamiltonian (8) can now be used to perturb this pure limit; for example, with B # 0 the dependence of the ratio of energies on B can be estimated by expanding in Nvs/N,
ESS -N Es -
A(&
+N,r) +BN,r
2A(Nz+ + Nn-)
2A2 - AB - 2B2 Nvs -3 + 4A(2A-B) N
(13)
D.D. Warner; R Van Isacker/Physics Letters B 395 (1997) 145-150
149
which was found assuming A = D, C = 0, and B < 2A, and neglecting the difference between NV, and N,, in the second term. It can be seen that the energy ratio increases for non-zero B. The characteristic excitation of these mixed-symmetry modes is via magnetic dipole transitions. In even-even nuclei the existence of l+ scissors states excited in (e, e’) or (y, y’) is by now well established. The IBM-2 prediction for the Ml strength towards the If state corresponding to ]SCX),in the above is
BtMl;OGf-+ 12) = -&
where g, and g, are the boson g factors. The function IBM, (
0
(14)
-gzJ2f(N)N?rNv, f(N)
is known analytically
in the three limits of the
U(5) (15)
The U(5), SU( 3) and O(6) limits correspond to vibrational, deformed and y-soft nuclei, respectively, and in each case, candidates for states of mixed symmetry exist [ 10,12,13]. Eq. (14) is valid for the scissors state of limit c in which all the protons oscillate against all neutrons. We stress that, with an appropriate choice of interpolating function f(N) (e.g., the one of Ref. [ 141)) ( 14) is also valid for a mixture of the three limits and corresponds to the sum rule Ml strength, which is concentrated in one state in the three symmetry limits but is fragmented over several states in general. Likewise, all results reported below for the soft scissors mode are valid for intermediate situations. A similar expression can be derived for the dipole strength to the soft scissors state of limit a by considering the separate contributions to the Ml operator from the core and the skin neutrons, p(M1)
= g,r&
+ gp& = g,&r + g&
+ g,%,,
(16)
and this yields B(M1; 0; --+ l&J
= $(gr
- sp)‘f(N)
NNN+N; ?T UC
This result can also be obtained using the sum rule ( 14) and the transformation The soft scissors mode a can be written in basis c as
(17) between
the different
bases.
(18) The sum rule (14) implies that the Ml strength to the state ]SSCX), is zero. This is entirely consistent with the fact that, as deduced earlier, here the oscillation is one involving core neutrons against skin neutrons. The result ( 17) then follows immediately from ( 14) and ( 18). The corresponding expressions in other limits can be derived in a similar fashion. From ( 14) and ( 17) one finds the following simple result for the ratio of B(M1) ‘s in the soft and normal scissors modes: B(Ml;O&
--f l&J
B(MI;O&
-+ l;J
N,N,, = (NV + N,,)N,’
(19)
This value is identical to that found previously for the El sum rule ratio of the soft and giant dipole resonances [7] so that again, the result depends only on the relative number of constituents in the subsystems, rather than on the specific algebra chosen.
D.D. Wamel; I? Van Isacker/Physics Letters B 395 (1997) 145-150
150
In summary, it has been shown that the algebraic approach can be extended to describe a three-component system, as arises in the presence of a neutron skin in very neutron-rich nuclei. The relative coupling strengths between the components can be represented by the choice of coupling scheme and the method reveals the possibility of a soft scissors mode in which the skin neutrons undergo out-of-phase oscillations against the remaining nucleons. The analogy with the soft dipole mode is clear and the probability of finding an empirical realisation of either depends critically on the extent to which the neutron skin develops in heavier nuclei as the neutron drip line is approached. In fact, recent results from Hartree-Fock RPA calculations [ 151 suggest that the separation between the degrees of freedom of the core and skin in quadrupole collective states may be sufficient to allow new collective modes involving the skin neutrons alone. In the current example, if the extent of the skin is at least comparable to the range of the proton-neutron interaction, the notion of a partial decoupling of it from the core becomes valid and the soft dipole modes may manifest themselves, albeit in a possibly fragmented form. The corollary is, of course, that evidence for such modes in the form of low-lying l+ states with enhanced Ml decays could serve as an empirical signature of the development of a neutron skin. We wish to thank M.A. Nagarajan and F. Iachello for useful discussions. This work is supported France-British Alliance grant PN 96-058 and by the EU grant CHGE-CT-94-00-56.
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1987).
by the