- Email: [email protected]
Improved description of M1 transitions by special hamiltonians of the proton-neutron interacting boson model
Volume 215, number 3
PHYSICS LETTERS B
22 December 1988
I M P R O V E D D E S C R I P T I O N O F M1 T R A N S I T I O N S BY S P E C I A L H A M I L T O N I A N S OF THE P R O T O N - N E U T R O N INTERACTING BOSON M O D E L
W. FRANK, P. VON BRENTANO, A. G E L B E R G lnstitul fiir Kernphysik der Universildt zu K6ln, Ziilpicherstrafie 77, D-5000 Cologne 41, Fed. Rep. Germany and
H. H A R T E R lnstitut f~r Theoretische Physil,"der Universitiit Tiibingen, Auf der Morgenstelle I4, D-7400 Tiibingen, Fed. Rep. Germany
Received 8 August 1988; revised manuscript received 28 September 1988
The proton-neutron interacting boson model (IBM-2) describes energies, B(E2) and B(M1 ) values of nuclei. In order to reduce the great number of free IBM-2 parameters two special IBM-2 hamiltonians are proposed which allow a decoupling of the energy and B(E2 ) fit from the determination of the B(MI ) values and the energy of the lowest mixed symmetry 1+ state. This property allows a simple fit procedure of the IBM-2 parameters in both cases.
1. Introduction The interacting boson m o d e l ( I B M - l ) [1,2] describes energies a n d electric q u a d r u p o l e transitions. The p r o t o n - n e u t r o n interacting boson m o d e l (IBM2) [ 1,3,4 ] generalizes the IBM- 1 to include both proton a n d neutron bosons. This generalization makes the m o d e l m o r e realistic and allows a description o f a d d i t i o n a l observables (magnetic dipole transitions, etc.) arising from m i x e d s y m m e t r y states i.e., states which are not fully s y m m e t r i c a l in p r o t o n a n d neutron bosons. The price for this extension is, however, a considerable increase o f the n u m b e r o f free p a r a m eters. The IBM-1 h a m i l t o n i a n has 6 free p a r a m e t e r s for the excitation energies; the IBM-2 h a m i l t o n i a n 21 free parameters. It turns out that the n u m b e r o f free p a r a m e t e r s in the most general IBM-2 h a m i l t o n i a n is in m a n y cases larger than the n u m b e r o f p a r a m e t e r s which are d e t e r m i n e d by the data. Usually this problem is tackled by keeping some IBM-2 p a r a m e t e r s fixed ( m a n y o f them zero). In this note we want to proceed in a slightly different way. Here we reduce the n u m b e r o f free p a r a m e t e r s o f a rather general IBM-2 h a m i l t o n i a n by imposing certain desirable constraints on the IBM-2 h a m i l t o n i a n and E2 opera0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
tor. In particular we want an IBM-2 h a m i l t o n i a n and E2 o p e r a t o r whose energies and E2 transition rates o f the F = Fmax states are i n d e p e n d e n t o f the p a r a m e t e r Zv = ( Z , - Z v ) / 2 if the M a j o r a n a p a r a m e t e r ~ is infinite. This d e m a n d can actually be fulfilled and leads to the specific IBM-2 h a m i l t o n i a n s and E2 operators to be discussed below. First we give an IBM-2 h a m i l t o n i a n [ 1,5,6 ] which contains the most general rank-two part ( a , f i t { n , u)={1,2}): H = e 2 ( n a , +nd,) -bo)L '2 + Z
o~.<.fl
x,~Q.(zo4J)Q~(z~,~)
+ h , . , T ~ 2~ .T~ 2~ + d ~ I ,
T ( E 2 ) = ~ q.Q,~(X,~), oz
Q~,(Z) = d t . s . +cTc[,~s*,+ x( d~cT~) (2) , M=½[F~a×(Fma~+I)-P2I iil
-}-
( d J". s *. - d ~-. s ~-. ) "( cTcT.as . - aV.s . ) E
([d~xd¢~](K)'[~.Xdu](K)),
K ~ 1,3
435
Volume 215, number 3
T(M1 )raM-2 = ( 3 X / ~
PHYSICS LETTERS B
(g,~L~+g~L,,)~N
= ,,/(3/41r) (g~L' +g~Lv)/~N, g~=½(g,+g,,),
g,,,=½(g~--g,,),
where L'
+,/T6 { d; x & ]
)
is the total angular momentum, L v = L , ~ - L , and T~ 2) = [d~*×aTo] (2) for 0 = n, v. The Majorana operator M [7] separates states of different F-spin. As it is known from the Darmstadt experiments the energy of the collective 1 + state, a mixed symmetry ( F = F m a × - 1 ) IBM-2 state, lies in the range of approximately 3 MeV [8 ]. Thus the large energy separation from the low lying F=Fma~ states implies that their corresponding wavefunctions are almost F-spin pure. They become fully symmetric (i.e. totally F-spin pure with F = f m a x ) if the Majorana parameter ~ is set to infinity. These fully symmetrical states in proton and neutron degrees of freedom can be mapped onto IBM-1 states [9-11 ] using the projection method [7]. For ~ - ~ the projection connects the IBM-2 hamiltonian and E2 operator with an IBM-1 hamiltonian and E2 operator which have equal energy and B (E2) values. We now want the IBM-2 hamiltonian to satisfy the following four conditions: (1) It fulfills the consistent Q IBM-2 condition [ 1,12 ] i.e. Z~=Z~ =Z~, and Z, =X,, =Z,~. (2) The projected IBM-1 hamiltonian should have the following form:
Adopting the four conditions, the hamiltonian and E2 operator given above and the projection formulas given in ref. [ 13 ] we obtain only two possible solutions, one hamiltonian which contains a [ Q , ( z , ) / N ~ r + Q v ( z . ) / N v ] 2 t e r m operator (conditions (1), (2), ( 3 ) and (4a)) and another one which contains the Q,'Q~ term (conditions ( 1 ), (2), (3) and (4b)). Both solutions fulfil the consistent Q IBM-1 condition, which that the quadrupole operator in the hamiltonian and the E2 operator have the same Z parameter.
2. Q" Q-hamiltonian Projected IBM- 1. Hamiltonian H = a , rid+ ½~¢'L2+ ½~cQ2(Zs) . Quadrupole operator T(E2 ) = qQ (Zs) • IBM-2, Hamiltonian H = e 2 ( nd~ + na~) +coL '2 + IG~,(2 N ~ N J N Z) QZ + ~dl/[ +G~,{NJ[ZN~(N,~-I)]Q
2
+ N J [ 2 N , ( N ~ - I ) ]Q2} . Quadrupole operator
H,,roj =grlj + coL 2 + lcQ2 ( z ) •
T(E2) = ( 2 q ~ N J N ) O .
(3) The projected IBM-1 hamiltonian and E2 operator are independent of the parameter X,,. (4a) The IBM-2 hamiltonian contains no T~ 2). T ~2) term. Alternatively: (4b) The scalar products of rank-two tensors in the IBM-2 hamiltonian are only products of a proton and a neutron operator. These postulates are rather natural. Many IBM-2 hamiltonians used in fitting fulfil them at least partly [ 1,12 ]. Condition (4b) is the usual assumption of proton-neutron rank-two interaction, whereas (4a) is customary because the T ~: )- T ~2~ term is not often applied.
In both cases
436
22 December 1988
Q:=I N[Q,~(Z,~)/N~: +Q,,(X,,)/N,,] . The following IBM-2 parameters (e2, co, x~,,,, q,~, q,,,
Z., Z.) are related to the IBM-1 parameters (el, x', ~c, Zs, q) and to the IBM-2 parameter Zv:
e2 =e, - IK{ ( 4 - Z 2 +;(2) + [NJ2(N,~ - 1 ) l [ (Zs +Zv) 2 - 4 ] + [ N d 2 ( N ~ - 1 )] [(Zs - Z v ) 2 - 4]}, co=K'/2, x=, = [ N ( N - 1 ) / 4 N = N , IK,
Volume 2 l 5, number 3 z~ =z~ +Zv,
PHYSICS LETTERS B
z, =z~ - z , ,
q,~=qN /2N,~,
q,=qN/2N,
.
3. Q~" Q~-hamiltonian Projected I B M - 1. Hamiltonian
H=e~ n a + 1/¢'L2+ ½/cQ=(Zs) • Quadrupole operator T(E2) =qQ(z~) • IBM-2. Hamiltonian
H = e= (n~/~ +nj,,) +coL '= +K~,,Q.Ob~)'Q.OG) + t G . z .=T ~(=) . T .(2 ) + ~ M .
Quadrupole operator T ( E 2 ) =q,~Q,~(z,~) + q . Q . ( z . )
•
The following IBM-2 parameters (e2, co, x.., q., q.,
Z~, Z~) are related to the IBM-1 parameters (e~, x', x, X~, q) and to the IBM-2 parameter Z~: ez=e,-
½x(4-X~) ,
co=x' /2 , 1c,~, = [ N ( N -
1 ) /2N~N,]x,
K~=Z~+Xv,
X~=X~-Z, . ,
q,~=qN /2N,~,
q~=qN/2N..
4. Discussion Now we want to talk in more detail about these two simple IBM-2 hamiltonians and their corresponding quadrupole operator which is the same for both of them. The proton and neutron quadrupole charges in the E2 operator are no longer independent but linked by the simple relation q J q , = N ~ / N ~ . This result reflects the fact that B ( E 2 ) values between F ~ states deliver the proton and neutron quadrupole charges independently. A second remark concerns the numbers o f free pa-
22 December 1988
rameters in the IBM-2 hamiltonians/E2 operator and the number of parameters in the consistent Q IBM- 1 operators on which they project. The IBM-2 hamiltonian is found to have two extra parameters, the parameter Zv and the Majorana parameter ~. The Majorana parameter ¢ can be determined by the energy of the 1 + state [ 14 ] whereas the parameter X, is fixed by the M I transitions. By construction fitting these two parameters only weakly affects the energies and B ( E 2 ) values of the F = / ' ~ a x states. With the IBM-2 hamiltonian fulfilling the consistent Q IBM-2 condition by definition ( 1 ) in section 2, the fact that the corresponding projections fulfil the consistent Q IBM-1 condition (i.e. the ,~ parameter in the hamiltonian is the same as the X parameter in the quadrupole operator [2,12]) is not a trivial one, since in general this is not correct for the projection of an arbitrary IBM-2 hamiltonian ~l. But, since Casten and Warner have demonstrated that the data are really supporting the consistent Q IBM- 1 formalism [ 2,12 ], it is reasonable to study IBM-2 hamiltonians projecting on consistent Q IBM-I ones. We find two hamiltonians which have a different dependence on the quadrupole operator. The first hamiltonian has a quadrupole interaction between all bosons whereas the second one has a quadrupole interaction between protons and neutrons only. As both hamiltonians project onto the same IBM-I hamiltonian and E2 operator, energies and B ( E 2 ) values of F = Fmax states cannot distinguish between them. Thus one needs to consider additional data to distinguish between the two hamiltonians, as for instance accurately measured B (M 1 ) values for interband transitions or global fits with constant parameters. For data fitting one can start with the consistent Q 1 , IBM-1 hamiltonian H = e ~ n d + ~1c L -~+ u1 c Q 2 (Z) and the E2 operator T ( E 2 ) = q Q ( x ) for energies and B (E2) values. If one of the two IBM-2 hamiltonians is chosen, nearly all IBM-2 parameters are determined by the formulas given above. The remaining parameters Z,, and ¢ can be fixed by the B(M1 ) values respectively the 1 + state energy. For an application of this procedure see ref. [ 15 ]. ~ In particular note that the projection of the usual IBM-2 hamiltonian H=e( nj. + nd.) +coL' 2+ x,~,,Q,~(Z~) "Q~(x, ) + ~ neither fulfills the consistent Q IBM-1 condition nor is it independent ofz,437
Volume 215, number 3
PHYSICS LETTERS B
In most cases the above m e n t i o n e d IBM-1 hamilt o n i a n is sufficient. Otherwise, if one wants to use the most general IBM-1 h a m i l t o n i a n
H=e~ nd+ ½x' L 2+ ½tcQ2(z) d-yZ~3).T(3)+ ~ T ( a ) . T (4) ( T t k ) = [ d * × d ] ~k) for k=3, 4), one has to add the following term:
22 December 1988
Acknowledgement We would like to thank D.D. Warner and R.F. Casten for stimulating discussions, and M.W. Kirson a n d W. Krips for reading the manuscript. This work has been funded by the G e r m a n Federal Minister for Research and Technology ( B M F T ) u n d e r contract n u m b e r 060 K 272.
[ N ( N - 1 )/N~N,,] ( y T ~ 3)' T~v3) + ---TROT(4)--vT(4)] +3~(7y+96)(na.+na~) to the IBM-2 h a m i l t o n i a n (To~k)= [d~×aVo]
5. Summary By imposing a few natural constraints on a very general IBM-2 h a m i l t o n i a n we have obtained two special forms of the IBM-2 h a m i l t o n i a n and a u n i q u e E2 operator with a special relation between the quadrupole charges of protons and neutrons. These special h a m i l t o n i a n s project on the consistent Q IBM-1 h a m i l t o n i a n of Casten and W a r n e r [2,12] and can be considered to be a natural generalization of this h a m i l t o n i a n to IBM-2. Their m a i n feature is the independence of energies and B ( E 2 ) values from the parameter Zv which determines the B(M1 ) values. Furthermore, most of the parameters in these special IBM-2 h a m i l t o n i a n s can be d e t e r m i n e d from a fit clone with the projected IBM-1 hamiltonian. Thus the use of these special IBM-2 operators on the one h a n d saves a lot of computing time for nuclei with large boson n u m b e r s while on the other h a n d it leads to a very transparent d e t e r m i n a t i o n of the IBM-2 parameters from the data.
438
References [ 1] F. Iachello and A. Arima, The Interacting boson model (Cambridge U.P., Cambridge, 1987). [2] R.F. Casten and D.D. Warner, Rev. Mod. Phys. 60 (1988) 389. [ 3 ] T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. B 76 (1978) 139. [4] T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A 309 (1978) 1. [5] R. Bijker, A.E.L. Dieperink, O. Scholten and R. Spanhoff, Nucl. Phys. A 344 (1980) 207. [6] A.E.L. Dieperink, Nucl. Phys. A 421 (1984) 189c. [7 ] H. Harter, A. Gelberg and P. von Brentano, Phys. Lett. B 157 (1985) 1. [8]D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Loludice, F. Palumbo and O. Scholten, Phys. Lett. B 137 (1984) 27. [9] H. Harter, Ph.D. thesis, University of Cologne (1987), unpublished. [ 10] W. Frank, Diplom thesis, University of Cologne (1987), unpublished. [ 11 ] O. Scholten,Ph.D. Thesis, Universityof Groningen (1980), unpublished. [ 12 ] D.D. Warnerand R.F. Casten,Phys. Rev. C 28 ( 1983) 1798. [ 13 ] W. Frank, H. Harter, P. von Brentano and A. Gelberg, Phys. Rev. C., to be published. [ 14] U. Hartmann, D. Bohle, T. Guhr, K.D. Hummel, G. Kilgus, U. Milkau and A. Richter, Nucl. Phys. A 465 (1987) 25. [ 15 ] P. von Brentano,W. Frank, A. Gelberg,H. Harter, W. Krips, R.F. Casten, H.G. B6rnerand B. Krusche, 6th Intern. Symp. on Capture gamma-ray spectroscopy (Leuven, 1987), J. Phys. G 14 Suppl. (1988) S129.
PHYSICS LETTERS B
22 December 1988
I M P R O V E D D E S C R I P T I O N O F M1 T R A N S I T I O N S BY S P E C I A L H A M I L T O N I A N S OF THE P R O T O N - N E U T R O N INTERACTING BOSON M O D E L
W. FRANK, P. VON BRENTANO, A. G E L B E R G lnstitul fiir Kernphysik der Universildt zu K6ln, Ziilpicherstrafie 77, D-5000 Cologne 41, Fed. Rep. Germany and
H. H A R T E R lnstitut f~r Theoretische Physil,"der Universitiit Tiibingen, Auf der Morgenstelle I4, D-7400 Tiibingen, Fed. Rep. Germany
Received 8 August 1988; revised manuscript received 28 September 1988
The proton-neutron interacting boson model (IBM-2) describes energies, B(E2) and B(M1 ) values of nuclei. In order to reduce the great number of free IBM-2 parameters two special IBM-2 hamiltonians are proposed which allow a decoupling of the energy and B(E2 ) fit from the determination of the B(MI ) values and the energy of the lowest mixed symmetry 1+ state. This property allows a simple fit procedure of the IBM-2 parameters in both cases.
1. Introduction The interacting boson m o d e l ( I B M - l ) [1,2] describes energies a n d electric q u a d r u p o l e transitions. The p r o t o n - n e u t r o n interacting boson m o d e l (IBM2) [ 1,3,4 ] generalizes the IBM- 1 to include both proton a n d neutron bosons. This generalization makes the m o d e l m o r e realistic and allows a description o f a d d i t i o n a l observables (magnetic dipole transitions, etc.) arising from m i x e d s y m m e t r y states i.e., states which are not fully s y m m e t r i c a l in p r o t o n a n d neutron bosons. The price for this extension is, however, a considerable increase o f the n u m b e r o f free p a r a m eters. The IBM-1 h a m i l t o n i a n has 6 free p a r a m e t e r s for the excitation energies; the IBM-2 h a m i l t o n i a n 21 free parameters. It turns out that the n u m b e r o f free p a r a m e t e r s in the most general IBM-2 h a m i l t o n i a n is in m a n y cases larger than the n u m b e r o f p a r a m e t e r s which are d e t e r m i n e d by the data. Usually this problem is tackled by keeping some IBM-2 p a r a m e t e r s fixed ( m a n y o f them zero). In this note we want to proceed in a slightly different way. Here we reduce the n u m b e r o f free p a r a m e t e r s o f a rather general IBM-2 h a m i l t o n i a n by imposing certain desirable constraints on the IBM-2 h a m i l t o n i a n and E2 opera0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
tor. In particular we want an IBM-2 h a m i l t o n i a n and E2 o p e r a t o r whose energies and E2 transition rates o f the F = Fmax states are i n d e p e n d e n t o f the p a r a m e t e r Zv = ( Z , - Z v ) / 2 if the M a j o r a n a p a r a m e t e r ~ is infinite. This d e m a n d can actually be fulfilled and leads to the specific IBM-2 h a m i l t o n i a n s and E2 operators to be discussed below. First we give an IBM-2 h a m i l t o n i a n [ 1,5,6 ] which contains the most general rank-two part ( a , f i t { n , u)={1,2}): H = e 2 ( n a , +nd,) -bo)L '2 + Z
o~.<.fl
x,~Q.(zo4J)Q~(z~,~)
+ h , . , T ~ 2~ .T~ 2~ + d ~ I ,
T ( E 2 ) = ~ q.Q,~(X,~), oz
Q~,(Z) = d t . s . +cTc[,~s*,+ x( d~cT~) (2) , M=½[F~a×(Fma~+I)-P2I iil
-}-
( d J". s *. - d ~-. s ~-. ) "( cTcT.as . - aV.s . ) E
([d~xd¢~](K)'[~.Xdu](K)),
K ~ 1,3
435
Volume 215, number 3
T(M1 )raM-2 = ( 3 X / ~
PHYSICS LETTERS B
(g,~L~+g~L,,)~N
= ,,/(3/41r) (g~L' +g~Lv)/~N, g~=½(g,+g,,),
g,,,=½(g~--g,,),
where L'
+,/T6 { d; x & ]
)
is the total angular momentum, L v = L , ~ - L , and T~ 2) = [d~*×aTo] (2) for 0 = n, v. The Majorana operator M [7] separates states of different F-spin. As it is known from the Darmstadt experiments the energy of the collective 1 + state, a mixed symmetry ( F = F m a × - 1 ) IBM-2 state, lies in the range of approximately 3 MeV [8 ]. Thus the large energy separation from the low lying F=Fma~ states implies that their corresponding wavefunctions are almost F-spin pure. They become fully symmetric (i.e. totally F-spin pure with F = f m a x ) if the Majorana parameter ~ is set to infinity. These fully symmetrical states in proton and neutron degrees of freedom can be mapped onto IBM-1 states [9-11 ] using the projection method [7]. For ~ - ~ the projection connects the IBM-2 hamiltonian and E2 operator with an IBM-1 hamiltonian and E2 operator which have equal energy and B (E2) values. We now want the IBM-2 hamiltonian to satisfy the following four conditions: (1) It fulfills the consistent Q IBM-2 condition [ 1,12 ] i.e. Z~=Z~ =Z~, and Z, =X,, =Z,~. (2) The projected IBM-1 hamiltonian should have the following form:
Adopting the four conditions, the hamiltonian and E2 operator given above and the projection formulas given in ref. [ 13 ] we obtain only two possible solutions, one hamiltonian which contains a [ Q , ( z , ) / N ~ r + Q v ( z . ) / N v ] 2 t e r m operator (conditions (1), (2), ( 3 ) and (4a)) and another one which contains the Q,'Q~ term (conditions ( 1 ), (2), (3) and (4b)). Both solutions fulfil the consistent Q IBM-1 condition, which that the quadrupole operator in the hamiltonian and the E2 operator have the same Z parameter.
2. Q" Q-hamiltonian Projected IBM- 1. Hamiltonian H = a , rid+ ½~¢'L2+ ½~cQ2(Zs) . Quadrupole operator T(E2 ) = qQ (Zs) • IBM-2, Hamiltonian H = e 2 ( nd~ + na~) +coL '2 + IG~,(2 N ~ N J N Z) QZ + ~dl/[ +G~,{NJ[ZN~(N,~-I)]Q
2
+ N J [ 2 N , ( N ~ - I ) ]Q2} . Quadrupole operator
H,,roj =grlj + coL 2 + lcQ2 ( z ) •
T(E2) = ( 2 q ~ N J N ) O .
(3) The projected IBM-1 hamiltonian and E2 operator are independent of the parameter X,,. (4a) The IBM-2 hamiltonian contains no T~ 2). T ~2) term. Alternatively: (4b) The scalar products of rank-two tensors in the IBM-2 hamiltonian are only products of a proton and a neutron operator. These postulates are rather natural. Many IBM-2 hamiltonians used in fitting fulfil them at least partly [ 1,12 ]. Condition (4b) is the usual assumption of proton-neutron rank-two interaction, whereas (4a) is customary because the T ~: )- T ~2~ term is not often applied.
In both cases
436
22 December 1988
Q:=I N[Q,~(Z,~)/N~: +Q,,(X,,)/N,,] . The following IBM-2 parameters (e2, co, x~,,,, q,~, q,,,
Z., Z.) are related to the IBM-1 parameters (el, x', ~c, Zs, q) and to the IBM-2 parameter Zv:
e2 =e, - IK{ ( 4 - Z 2 +;(2) + [NJ2(N,~ - 1 ) l [ (Zs +Zv) 2 - 4 ] + [ N d 2 ( N ~ - 1 )] [(Zs - Z v ) 2 - 4]}, co=K'/2, x=, = [ N ( N - 1 ) / 4 N = N , IK,
Volume 2 l 5, number 3 z~ =z~ +Zv,
PHYSICS LETTERS B
z, =z~ - z , ,
q,~=qN /2N,~,
q,=qN/2N,
.
3. Q~" Q~-hamiltonian Projected I B M - 1. Hamiltonian
H=e~ n a + 1/¢'L2+ ½/cQ=(Zs) • Quadrupole operator T(E2) =qQ(z~) • IBM-2. Hamiltonian
H = e= (n~/~ +nj,,) +coL '= +K~,,Q.Ob~)'Q.OG) + t G . z .=T ~(=) . T .(2 ) + ~ M .
Quadrupole operator T ( E 2 ) =q,~Q,~(z,~) + q . Q . ( z . )
•
The following IBM-2 parameters (e2, co, x.., q., q.,
Z~, Z~) are related to the IBM-1 parameters (e~, x', x, X~, q) and to the IBM-2 parameter Z~: ez=e,-
½x(4-X~) ,
co=x' /2 , 1c,~, = [ N ( N -
1 ) /2N~N,]x,
K~=Z~+Xv,
X~=X~-Z, . ,
q,~=qN /2N,~,
q~=qN/2N..
4. Discussion Now we want to talk in more detail about these two simple IBM-2 hamiltonians and their corresponding quadrupole operator which is the same for both of them. The proton and neutron quadrupole charges in the E2 operator are no longer independent but linked by the simple relation q J q , = N ~ / N ~ . This result reflects the fact that B ( E 2 ) values between F ~ states deliver the proton and neutron quadrupole charges independently. A second remark concerns the numbers o f free pa-
22 December 1988
rameters in the IBM-2 hamiltonians/E2 operator and the number of parameters in the consistent Q IBM- 1 operators on which they project. The IBM-2 hamiltonian is found to have two extra parameters, the parameter Zv and the Majorana parameter ~. The Majorana parameter ¢ can be determined by the energy of the 1 + state [ 14 ] whereas the parameter X, is fixed by the M I transitions. By construction fitting these two parameters only weakly affects the energies and B ( E 2 ) values of the F = / ' ~ a x states. With the IBM-2 hamiltonian fulfilling the consistent Q IBM-2 condition by definition ( 1 ) in section 2, the fact that the corresponding projections fulfil the consistent Q IBM-1 condition (i.e. the ,~ parameter in the hamiltonian is the same as the X parameter in the quadrupole operator [2,12]) is not a trivial one, since in general this is not correct for the projection of an arbitrary IBM-2 hamiltonian ~l. But, since Casten and Warner have demonstrated that the data are really supporting the consistent Q IBM- 1 formalism [ 2,12 ], it is reasonable to study IBM-2 hamiltonians projecting on consistent Q IBM-I ones. We find two hamiltonians which have a different dependence on the quadrupole operator. The first hamiltonian has a quadrupole interaction between all bosons whereas the second one has a quadrupole interaction between protons and neutrons only. As both hamiltonians project onto the same IBM-I hamiltonian and E2 operator, energies and B ( E 2 ) values of F = Fmax states cannot distinguish between them. Thus one needs to consider additional data to distinguish between the two hamiltonians, as for instance accurately measured B (M 1 ) values for interband transitions or global fits with constant parameters. For data fitting one can start with the consistent Q 1 , IBM-1 hamiltonian H = e ~ n d + ~1c L -~+ u1 c Q 2 (Z) and the E2 operator T ( E 2 ) = q Q ( x ) for energies and B (E2) values. If one of the two IBM-2 hamiltonians is chosen, nearly all IBM-2 parameters are determined by the formulas given above. The remaining parameters Z,, and ¢ can be fixed by the B(M1 ) values respectively the 1 + state energy. For an application of this procedure see ref. [ 15 ]. ~ In particular note that the projection of the usual IBM-2 hamiltonian H=e( nj. + nd.) +coL' 2+ x,~,,Q,~(Z~) "Q~(x, ) + ~ neither fulfills the consistent Q IBM-1 condition nor is it independent ofz,437
Volume 215, number 3
PHYSICS LETTERS B
In most cases the above m e n t i o n e d IBM-1 hamilt o n i a n is sufficient. Otherwise, if one wants to use the most general IBM-1 h a m i l t o n i a n
H=e~ nd+ ½x' L 2+ ½tcQ2(z) d-yZ~3).T(3)+ ~ T ( a ) . T (4) ( T t k ) = [ d * × d ] ~k) for k=3, 4), one has to add the following term:
22 December 1988
Acknowledgement We would like to thank D.D. Warner and R.F. Casten for stimulating discussions, and M.W. Kirson a n d W. Krips for reading the manuscript. This work has been funded by the G e r m a n Federal Minister for Research and Technology ( B M F T ) u n d e r contract n u m b e r 060 K 272.
[ N ( N - 1 )/N~N,,] ( y T ~ 3)' T~v3) + ---TROT(4)--vT(4)] +3~(7y+96)(na.+na~) to the IBM-2 h a m i l t o n i a n (To~k)= [d~×aVo]
5. Summary By imposing a few natural constraints on a very general IBM-2 h a m i l t o n i a n we have obtained two special forms of the IBM-2 h a m i l t o n i a n and a u n i q u e E2 operator with a special relation between the quadrupole charges of protons and neutrons. These special h a m i l t o n i a n s project on the consistent Q IBM-1 h a m i l t o n i a n of Casten and W a r n e r [2,12] and can be considered to be a natural generalization of this h a m i l t o n i a n to IBM-2. Their m a i n feature is the independence of energies and B ( E 2 ) values from the parameter Zv which determines the B(M1 ) values. Furthermore, most of the parameters in these special IBM-2 h a m i l t o n i a n s can be d e t e r m i n e d from a fit clone with the projected IBM-1 hamiltonian. Thus the use of these special IBM-2 operators on the one h a n d saves a lot of computing time for nuclei with large boson n u m b e r s while on the other h a n d it leads to a very transparent d e t e r m i n a t i o n of the IBM-2 parameters from the data.
438
References [ 1] F. Iachello and A. Arima, The Interacting boson model (Cambridge U.P., Cambridge, 1987). [2] R.F. Casten and D.D. Warner, Rev. Mod. Phys. 60 (1988) 389. [ 3 ] T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. B 76 (1978) 139. [4] T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A 309 (1978) 1. [5] R. Bijker, A.E.L. Dieperink, O. Scholten and R. Spanhoff, Nucl. Phys. A 344 (1980) 207. [6] A.E.L. Dieperink, Nucl. Phys. A 421 (1984) 189c. [7 ] H. Harter, A. Gelberg and P. von Brentano, Phys. Lett. B 157 (1985) 1. [8]D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Loludice, F. Palumbo and O. Scholten, Phys. Lett. B 137 (1984) 27. [9] H. Harter, Ph.D. thesis, University of Cologne (1987), unpublished. [ 10] W. Frank, Diplom thesis, University of Cologne (1987), unpublished. [ 11 ] O. Scholten,Ph.D. Thesis, Universityof Groningen (1980), unpublished. [ 12 ] D.D. Warnerand R.F. Casten,Phys. Rev. C 28 ( 1983) 1798. [ 13 ] W. Frank, H. Harter, P. von Brentano and A. Gelberg, Phys. Rev. C., to be published. [ 14] U. Hartmann, D. Bohle, T. Guhr, K.D. Hummel, G. Kilgus, U. Milkau and A. Richter, Nucl. Phys. A 465 (1987) 25. [ 15 ] P. von Brentano,W. Frank, A. Gelberg,H. Harter, W. Krips, R.F. Casten, H.G. B6rnerand B. Krusche, 6th Intern. Symp. on Capture gamma-ray spectroscopy (Leuven, 1987), J. Phys. G 14 Suppl. (1988) S129.