The quantity Y identifies the additional quantum numbers, e.g. parity v and ...... Hannachi, R. Chapman, J.C. Lisle, J.N. MO, E. Paul, D.J.G. Love, P.J. Nolan, A.H..
Nuclear Physics A453 (1986) 1044126 ONorth-Holland Publishing Company
THE ROUTHIAN
IN “GAUGE”
SPACE
J.-Y. ZHANG” and J.D. GARRETT The Niels Bohr Institute,
University of Copenhagen, Copenhagen, Denmark and The‘Joint Institute for Heavy Ion Research, Holifield Heavy Ion Research Facility, Oak Ridge, Tennessee 37831, USA J.C. BACELARb The Niels Bohr Institute,
and S. FRAUENDORF
University of Copenhagen, Copenhagen, Denmurk
Received 29 August 1985 (Revised 25 November 1985) Abstract: Formulae are presented for the construction of “double” r-outhians fr-om which both the configuration- and “gauge’‘-space collective energies have been removed. Such values are constructed for a series of ytterbium isotopes varying in mass from 159 to 170 and are presented as a function of X, for constant hw, and as a function of hw for constant h,,. This analysis indicates the prcscnce of sizable correlations for the lowest (q, a) = (+ ,O) configuration (and to a lesser extent for the lowest (+ , i) configuration) at the largest rotational frequencies for which such an analysis can be accomplished. An empirical spectrum of single-neutron states is constructed from values of X,, and is compared both to a similar spectrum of states constructed from excitation energy differences and to cranking-model calculations.
1. Introduction The parallel roles of the deformed and pair fields in the cranking hamiltonian1*2) are well established, see e.g. refs. 3-8). In the presence of a quadrupole deformation and a static pair field the hamiltonian can be written as
[Here the convention2g3) of designating quantities in the spherical and rotating deformed systems by subscript zeros and primes has been adopted.] The Coriolis is included in the hamiltonian to guarantee the plus centrifugal term, -ofx, conservation of angular momentum broken by the addition of a nonspherically-symmetric field, - ~0. Another lagrangian term, --Xi?, also is included to restore the conservation of particle number broken by the pair field, -A(?‘+ 8-). Thus the a Permanent address: Institute of Modern Physics, Lanzhou, Peoples Republic of China. b Present address: Nuclear Science Division, Lawrence Berkeley Laboratory, Berkeley, California, USA. 0375.9474/86/$03,50oElsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
105
J.-Y. Zhang et al. / Routhian in guuge spare
term motion
-hfi
may be considered
in “gauge”
The analogy
space*,
between
as a Coriolis
interaction
associated with rotational
and both A and N as “gauge’‘-space
“gauge”
canonical
space and the normal configuration and particle numbers “gauge’‘-space routhians””
variables.
space can be as a function
exhibited by plotting of the “gauge” lagrangian multiplier A ( = d E/d N ). Such plots are analogues of the routhian and aligned angular momentum plots which have been so important in spectroscopic studies at large angular momentum, see e.g. refs. 2712-15). “Gauge’‘-space alignment (or particle-number) plots, which were introduced 6,7.16) to illustrate the analogy between the two spaces, have been utilized to study the angular-momentum
and particle-number
dependence
of the shape changes
between
spherical and deformed nuclei 6-8) . Recently this technique has been applied 8J7-20) to nonyrast configurations and at very large angular momenta, providing evidence for the quenching of static neutron pair correlations at large angular momenta. Until now the complete analogy between “gauge” and configuration spaces has not been exploited by producing “gauge’‘-space routhians. [See, however, refs. 6,13,21) where comparisons of excitation energies at a constant h are described.] The present paper reports such an analysis for a series of ytterbium isotopes. The normal routhians, e’, are converted to “gauge” space by adding the ground-state binding energies and subtracting the “gauge’‘-space rotational energy, - X N. The resulting “gauge’‘-space (or “double”) routhians, e”, are referred to liquid-drop energies, the appropriate “gauge’‘-space reference. Such an analysis indicates that even at the largest observed rotational frequencies correlations exist for the (7~, a) = (+ , 0) configuration, and to a lesser extent for the ( + ,i) configuration. These correlations probably result from “dynamic” neutron pair correlations 22). An empirical spectrum of single-neutron states is also constructed from the extracted values of the Fermi level A as a function of configuration and rotational frequency. This technique, based on the energy difference for the same configuration in AN = 2 neighbors (i.e. derivatives of the “double” routhian, e “, with respect to neutron number) produces a rather different spectrum of states from that ‘*,19) based on energy differences as a function of configuration within the same isotope. The different spectra of states obtained from these two analysis techniques
are attributed
to the sizeable correlations
observed
for the (r, a) = ( + ,0)
and (+, $) configurations, which to lowest order are independent of neutron number. Finally, the single-neutron spectrum of states obtained from AN = 2 energy differences is compared with unpaired cranking-model calculations. * The term “gauge” space is borrowed from classical electrodynamics9) where the electrical and magnetic field are invariant with respect to a “gauge” transformation that adds a gradient of an arbitrary function to the potential. ** The term routhian is derived from classical mechanics, where a routhian is a function that is a hamiltonian with respect to certain variables and a lagrangian with respect to otherslOJ1). Indeed, the cranking hamiltoman, eq. (l), which explicitly contains the lagrangian terms -WI, and -AN, fits this definition. In the present context, this definition has been extended to the expectation values of this hamiltonian.
106
J.-Y. Zhang et al. / Routhian in gauge space
2. The construction of “gauge”-space routhiaus
2.1. TOTAL
ROUTHIANS
In analogy with the construction of the excitation energy in a rotating system*), or the configuration-space routhian, it also is possible to construct from experimental quantities an excitation energy in which the “gauge’‘-space rotational energy, X,N (or h,Z for isotonic chains), has been removed. Such an energy, when referred to the appropriate reference, the rotating liquid drop, becomes independent of the neutron (or proton) binding energies. The “gauge’‘-space routhian*, E;(Z, N, I, v>, for the configuration of AN = 2 neighboring isotopes N - 1 and N + 1 with proton number 2 and angular momentum I, can be defined as E;(Z,N,I,v)=~[E(Z,N+l,I,v)+E(Z,N-l,I,v)] -X,(Z,
N, I, v)N.
(2)
The quantity Y identifies the additional quantum numbers, e.g. parity v and signature (Y, labelling the configuration of interest. Similarly the “gauge’‘-space routhian for isotones, Ei(Z, N, I, Y), is given by E;(Z,
N, I, V) =+[E(Z+ -h,(Z,
1, N, I, V) +E(Z-
1, N, I, v)]
N, I, Y)Z.
(3)
These expressions are analogous to that of the standard routhian, given by E’(Z, N, I, V) =$[E(Z, -w(Z,
N, I+ 1, V) +E(Z, N, I, Y)I,.
N, I-
E’(Z, N, I, v),
1, v)] (4
Actually the desired quantity is the “double” routhian in which both the rotational and isotopic energies have been subtracted. Such a double routhian for an isotopic chain, El( Z, N, I, v), reduced both in normal (or configuration) space and in * In this and the ensuing discussion we conform to the established practice [see e.g. refs. 2.6-8,13,16*39*41)] in which the appropriate quantity, constructed from experimental values, approximating the intrinsicframe excitation energy is termed the experimental routhian. Likewise, when the equivalent quantity is constructed in “gauge”, or particle-number, space, it is called an experimental “gauge’‘-space routhian. Because these various labels become so complex, and since no theoretical routhians are presented, the term “experimental” is usually omitted. It must, however, be realized that the quantities given are derived from experimental values. When the various formulae are expressed in terms of experimental quantities, that quantity, e.g. angular momentum I, neutron number N, or proton number Z, is given explicitly as an argument in the appropriate equation. After the experimental routhian has been reduced to the theoretically more natural arguments, w, A,, and h,, and has been referred to the appropriate reference configuration, these “natural” quantities are then grven as arguments, see e.g. eqs. (14) (20)-(25). This quantity then is appropriate for comparison with the theoretical routhian calculated with the hamiltonian given in eq. (1).
J.-Y. Zhang et al. / Routhian in gauge space
107
“gauge” space is given by EZ(Z, N, I, V) =i[E(Z,
N+ 1, I+ 1, JJ) fE(Z,
N+ 1, I-
1, V)
+E(Z,N-l,I+l,v)+E(Z,N-l,I-l,v)] -w(Z,
N,I, v)I/&(Z,
A “double” routhian for an isotonic chain, E:(Z,
N,I, v)N.
(5)
N, I, v), can also be defined:
E~(Z,N,I,v)=~[E(Z+l,N,1+1,v)+E(Z+l,N,I-1,~) +E(Z-l,N,I+l,v)+E(Z-l,N,I-l,v)] -o(Z,
N&)1,-$,(Z,
N,I,v)Z.
(6)
Since definitions (2), (3), (5), and (6) contain energy differences for AN or AZ = 2 isotopes and isotones, the difference in binding energies must be considered. Therefore, the energies, E( Z, N, I, Y) are taken to be total energies defined as E(Z, N, 1, v) = -E,(Z,
N) + E,(Z,
N, 1, v>,
(7)
where E,(Z,N)=[ZM(1H)+NM(1n)-M(Z,N)]c2.
(8)
M(lH), M(‘n), and M(Z, N) are the masses of the proton, neutron and the ground state of the nucleus with Z protons and N neutrons. The ground-state binding energies are taken from the most recent compilation23), and the sources 15,19,24-38) of the experimental excitation energies are summarized in table 1. The lagrangian multipliers, XJ Z, N, I, v), Xr( Z, N, I, v), and o (Z, N, I, v), are obtained from the derivative of the total energy with respect to the appropriate TABLE 1
Sources of data used in analysis
Nucleus
159yb
16’Yb 161y,,
16’Yb 163Yb l”Yb 165yb
166Yb 167yb 168fi
169Yb “‘Yb
Neutron number 89 90 91 92 93 94 95 96 97 98 99 100
Refs.
24 1 24-26 27-29 27.28.30 1 31.32 1 32.33 1 15.34 1 35 1 19.34.36 1 19.37 ) 19.36 1 38 1
J.-Y. Zhang et al. / Routhian in gauge space
108
parameter,
N, 2, or I,:
=+[E(z,N+I,I,Y)-E(Z,N-I,I,~)],
=:[E(Z+l,N,I,Y)--E(Z-l,N,~,v)], 4Z,JLW=
dE(Z,
N, 1, y) ar X
E(Z,
N, I+
1, Y) - E(Z,
(9)
(IO)
N, I-
1, V)
(11)
Physically, X, and X, correspond to half the two-neutron and two-proton, separation energies respectively between AN, AZ = 2 neighbors in the same configuration Y and at the same angular momentum I. That is, they are the binding energies of the least bound pair of neutron and protons, respectively, which are coupled to positive parity and angular momentum zero. The rotational frequency o corresponds to half the inband E2 y-ray transition energy in the limit of K = 0.
2.2. REFERENCE CONFIGURATIONS AND REDUCED ROUTHIANS
The total routhians, defined by eqs. (2)-(6), include the energy associated with both intrinsic and rotational excitation for the specific degree (or degrees) of freedom involved (angular momentum, neutron number, or proton number). It is desirable to isolate the energy associated with the intrinsic degrees of freedom. The procedure 2,39) for accomplishing this is to refer the routhians defined by eqs. (2)-(6) to a configuration
in which there is no intrinsic excitation.
For rotations in normal space the reference usually 2, is taken to be a Harris parametrization4’) of the ground-state configuration of even-even systems. [See, ).] In this formulation the moment of inertia is parameterized however, refs. 34,37*41*42 as Y=JJo
+.Q?,
(12)
and the reference energy EL is given by
(13) Indeed the even-even ground-state configuration has no intrinsic excitation; however, the presence of the 9i term indicates frequency-dependent pair and/or shape correlations.
J.-Y. Zhang et nl. / Routhian in gauge space
109
An equally valid, and perhaps even more attractive, procedure is to choose as reference a parametrization of the least-correlated experimental configuration. When the routhians are referred to such a reference configuration, it may be possible to be quantitative about the energy associated with the sum of the various correlations. For light rare-earth nuclei the least-correlated 22) configurations, (r, a) = ( - , l), ( - , 0), ( - , + $), are observed 43) to have nearly frequency-independent moments of inertia* of about 66 (and 76) ti2. MeV-’ below (and above) the proton alignment frequency. These values of the reference moments of inertia correspond to about 85% (and 100%) of a deformed rigid rotor with e2 = 0.25. Since only a small amount of data has been established above the proton alignments for the lightest ytterbium isotopes, .YO= 66 A2. MeV-’ (and $i = 0) has been chosen for the present analysis. The normal (or configuration-space) reduced routhian, e’(Z, N, o, v), is then given by e’(Z, N,w, v) =E’(Z,
N,w, v) -EL(w).
(14)
It is also desirable to isolate the energy associated with the “gauge’‘-space degrees of freedom (N and Z). This is accomplished by referring the data to a “gauge’‘-space routhian constructed from the liquid drop: G%(Z> L) G&
= E&Z,
N,,)
-
A,Nrr, ,
N) = &D(ZLD, N) - X,Z,,.
05) 06)
The static liquid-drop energy, E,,( Z, N), is parameterized 44) as
E,,(Z,N)=(-a1A+a2A2”)
[
l-
K(Yi’]
+ c~Z~/AI’~ - c,Z2/A
(17)
with parameters from ref. 45): a, = 15.4941 MeV , a2 = 17.9437 MeV,
c3 = 0.7053 (equivalent to r0 = 1.2249 fm) , c4 = 1.1530 MeV,
K=1.7826. Since either the isotonic.or isotopic rotational energies, X,N or h,Z, are subtracted in the construction of the “gauge’‘-space and “double” routhians, eqs. (3)-(6), a AIlNLD or ApZLD term also is included in the “gauge’‘-space reference given in eqs. (15) and (16). Such “gauge’‘-space references are strong functions of N and Z; therefore, they must be evaluated at noninteger values of the neutron and proton number (Nr,, and * In principle, the moment of inertia is expected to contain an A5j3 dependence. In the present work where masses only vary from 159-170, this has been ignored even though there is preliminary experimental evidence”) for such a mass dependence.
J.-Y. Zhang et al. / Routhiun in guuge space
110
which are consistent with the positions of the respective neutron or proton Fermi surface (A, or Xr). The values of NLD and Z,, are obtained from the conditions Z,,),
(18) (19) respectively.
A, and A, are taken to be the experimental quantities from eqs. (9) and
(10). The “gauge”-space then given by
reduced routhians,
eL(Z, A,, I, V) and ek(A,, N, I, v), are
e:( Z, A,, 1, V) = E;( Z, A,, 1, v) - E;fb( Z, A,>,
(20)
$(A,,
(21)
N, 1, v) = E;( A,, N, 1, v) - EG( A,, N).
“Double” reduced routhians for isotopic (and isotonic) chains, ez( Z, A,, w, V) (and eF(X,, N, o, v)), from which the rotational energy in both configuration and “gauge” spaces is subtracted, can also be defined: ez( Z, A,, o, v) = El(Z,
A,, w, v> - E;(m)
- E&(Z,
w, v) -E;(w)
e~(X,,N,w,v)=E~(A,,N,
A,),
- E&(X,,
(22)
N).
(23)
TABLE 2
Analogy
between configurationConfiguration
and “gauge’‘-space
space
quantities
“Gauge”
space
hamiltonian deformation lagrangian definitions lagrangian
term term
multiplier
-A(fp++k-)
-eO_ -WI;
-h,fi
w - aE/aIx
X, = aE/c?N
=$(&+I
-J-L,)
=~(J%+I
=iE, routhian single-particle routhian reference
E’ = ;(E,+, + Er_,) e’ = E’ - E’ s
-EN-I)
= $S2” - WI,
EL = even-even g.s.b. or yrast; most uncorrelated state.
“double”
routhian:
“double”
~~=~(E,+,,N+~+EI+~.N-~+~I-~,N+~+EI-~,N-~)-~~~-~~~ single-particle routhian: elf = E” _ E; _ ,Fro ” n
E:, = $(%+I + %-I) o’_= E:, - En’ LD Era = liquid-drop term
-V”
111
J.-Y. Zhung et al. / Routhian in gauge space
The analogy between configurationfor quick reference in table 2.
and “gauge”-space quantities is summarized
2.3. HOW “DOUBLE” ROUTHIANS FOR ISOTOPIC CHAINS ARE ACTUALLY CONSTRUCTED
To clarify the procedure for producing the quantities utilized in the remainder of this paper, a few short rules summarizing their construction are given. (The label 2 will be dropped in the ensuing discussion, since only isotopic chains are considered.) (i) Normal or configuration-space routhians, e’(N, w, v), are extracted for each isotope as a function of rotational frequency Aw and configuration v, and referred to an appropriate reference configuration, see fig. 1. (In the present work Y0 = 66 A2 . MeV-1 and Yi = 0.) The standard procedure described in e.g. refs 2,39,41)and reiterated in the preceding subsections is used. (ii) The rotating-frame total energy, e;(N, o, v), is calculated by adding the binding energy of each isotope to the normal routhian, e’(N, w, v): e&(N, w, v) = -E,(N)
+ e’(N, 0, v).
(24
(This step, of course, is not needed for the construction of the normal routhian, as quantities are only defined for a single nucleus. Therefore, the binding energy is just a constant additive quantity for the normal total routhian, E’(N, w, v), which is removed in the construction of the normal reduced routhian, e’( N, w, v) by referring the system to a reference configuration.) (iii) The neutron Fermi level, A,( N, w, v), is calculated at each rotational frequency of each configuration for AN = 2 pairs of neighbouring isotopes. At this step the “gauge’‘-space particle-number plots (N versus X,), the equivalent of the total alignment plots (1, versus w) in configuration space, can be made. Such plots, shown in fig. 2 are, of course, the bases of the previous “gauge’‘-space analyses, see e.g. refs. 6-8*16-19). (iv) The “double” total routhian is constructed. Since the reduction has already been accomplished in normal space this quantity is calculated from the rotatingframe total energies, ek(N, w, v), given in eq. (24): E~(N,w,v)=$[e;(N+l,w,v)+e’,(N-l,w,v)]
-h,(N,w,v)N.
(25)
For each frequency of each configuration there now exists a value for the total “double” routhians and a value for the Fermi level X,. Therefore, a “double” total routhian plot, (Ez(N, o, v) versus X,(N, w, v)) can be made. However, the rapid change of binding energy as a function of neutron number (= 8 MeV/neutron) masks the interesting effects corresponding to the intrinsic single-particle excitations which are = 1 MeV. (v) The reduced “double” routhian, ef(X,, w, v), is calculated by subtracting the rotating liquid-drop energy (the “gauge’‘-space reference) from the total “double”
112
J.-Y. Zhang et al. / Routhian in gauge space
1 .2
I 4 by
I .6
(MeV)
Fig. 1. Normal configuration-space reduced routhians, e’( Z, N, w, v), for 167,168Yb plotted as a function of Aw. These routhians are referred to a reference configuration having a constant moment of inertia of 66 AZ. MeV-‘. No neutron-pair gap has been added for the odd-N isotope i6’Yb. These data are taken from ref. 19).
routhian. This subtraction is made at constant A, by calculating the rotating liquid-drop reference energy at noninteger neutron number, NLD, obtained from the condition given in eq. (18). Since A, is frequency and configuration dependent, the rotating liquid-drop reference must be calculated for each data point. (vi) The reduced “double” routhian, ei(X,, w, v), is plotted versus A, for constant w and versus w for constant A, (see figs. 3 and 4).
3. What new information is learned from a “gauge’‘-space analysis? 3.1. THE PARTICLE-NUMBER
PLOT REVISITED
The particle-number, or “gauge”-space total alignment plot* can be considered an equation of state relating the “gauge’‘-space canonical variables, N and A,. For a canonical pair the intensive variable (in the present case A,) is defined as the derivative of the total energy with respect to the extensive variable (in the present * Presently defined, such plots were introduced ‘,‘,16) to quantify the particle-number dependence of the change in shape between nearly spherical and information for ground-state configurations is contained in the atomic mass two-neutron (or two-proton) separation energies, are plotted as a function of N ref. 47) for the most recent compilation of such plots.
angular-momentum and deformed nuclei. Similar evaluations46,47), where (or Z). See figs. l-10 of
113
J.-Y. Zhang et al. / Routhian in gauge space TABLE 3 Examples
of canonical
Thermodynamics of extended media extensive variable
“)
intensive variable
“)
equation state
volume,
pressure,
of
V
P
P(V)
variable
Rotating
in some physical
problems Metal in magnetic field
“Gauge’‘-space rotation
nucleus
Fermi level, X
product of volume and internal mag. field, VB external mag.,
(&)=-dE(I) dl
A = dE(N)
_A
T(w)
N(A)
angular
momentum,
rotational
freq., w
I
particle
number,
N
dN
dE(B) V
dB
B(H)
a ) A thermodynamic variable is labelled extensive if its value in a homogeneous is proportional to mass and intensive if it is independent of mass.
system in equilibrium
case N). Phase transitions appear as singularities in such an equation of state. Some analogous examples of pairs of canonical variables are given in table 3, see also ref. 48). Two contrasting examples of particle-number plots (for the ground states of even-mass dysprosium isotopes and for the (+, $) configuration of ytterbium isotopes at ftw = 0.38 MeV) are shown in the right-hand portions of fig. 2. For comparison, the corresponding configuration-space total-alignment plots also are shown in the left-hand portions of this figure. Two patterns are observed in both types of plots. In the top portions of this figure, the intensive variables, X, and o, vary regularly as a function of the experimentally
accessible extensive variables, N and I. This “smooth” behavior is
indicative
of the correlated nature of the nuclear field with respect to the corre-
sponding
degree
of freedom.
Specifically,
it implies
the existence
of a static
deformation of the nuclear pair field of order A, (or a deformation of the field describing the nuclear potential as a function of shape). That is, there is a minimum in the potential energy surface for a nonzero value of A,, (or deformation). The discontinuities in these curves correspond to phase transitions in the equation of state. The singularity in the dysprosium particle-number plot at N = 89 (see the upper right-hand portion of fig. 2) indicates that at this neutron number the ground states of neighbouring even-N dysprosium nuclei no longer have the same correlations. This discontinuity has been identified 6-8) with the transition from nearly spherical to deformed ground-state shapes. Similarly discontinuities, or “ backbends” 4g), in total-align ment plots (see upper left-hand portion of fig. 2) have been observed in many decay sequences of rotational nuclei. They are associated 2,50) with a shift of the yrast sequence to a different intrinsic configuration at the rotational frequency where the Coriolis plus centrifugal forces on a pair of highlyaligned particles are exactly counterbalanced by the pairing.
114
J.-Y. Zhang et al. / Routhian in gauge space I
I
’
I
I l
f
‘-Hf,,
l
yrast
”
/
f t i.
J
f
J” , .2
/
P’
-9
.4
6
”
’
-8
-7
‘s0DY6, yrast
15 -
\ \ /
IO /
5-
-
/ \e \
o0
I
I
.2
.4 i-~w (MeV)
s
I .6
go-
l, -9
I -8
I -7
X, (MeV)
Fig. 2. Comparison of configuration-space alignment plots (left-hand side) and “gauge’‘-space alignment, or particle-number, plots (right-hand side). The plots shown in the top (bottom) portion indicate collective and the (single-particle) behavior. The 16*Hf and lsoDy data are from ref. 14) and ref. 62), respectively, ground-state binding energies of the dysposium isotopes are from ref.23). The data sources for the ytterbium isotopes are summarized in table 1.
In contrast, the irregular patterns of the plots of intensive versus extensive variables (equations of state), shown in the lower portions of fig. 2, are indicative of a noncorrelated, or single-particle, behaviour of these systems with respect to the neutron-pair and shape degrees of freedom. At large rotational frequencies the static neutron pair correlations have been sufficiently quenched 19), so that the lowest ( + , i) configuration in the neighbouring odd-mass ytterbium isotopes no longer have large overlaps. Similarly, the two valence shell neutrons in 150Dys, are not sufficient to produce a static deformed nuclear shape with the resulting rotationally-correlated spectra. Instead, the energy associated with the addition of two neutrons (or two
J.-Y. Zhang et al. / Routhian in gauge space
115
units of angular momentum) depends on the details of the single-particle spectrum of states as a function of N (or 1). Because of the finite number of constituent particles, significant fluctuations occur in the spacings of the resulting single-particle states.
3.2. “DOUBLE” ROUTHIANS AND CORRELATION ENERGIES
Complete information is not contained in “canonical” plots of intensive variables as a function of the extensive variables, as discussed in the preceding subsection. In particular, such analyses lack information on the magnitude of the energy associated with the various correlations. It is often convenient to plot the energy of the system as a function of the intensive variables. The technique to remove the energies associated with the collective pair and rotational degrees of freedom was described in the preceding section. The product of such an analysis is the reduced “double” routhian, ez(X,, w), i.e. the energy associated with the intrinsic particle degrees of freedom in correlated systems, as a function of the two intensive variables, X, and (J. The resulting “double” routhians plotted as a function of X, at constant o, and as a function of w at constant X,, are contained in figs. 3 and 4. Perhaps the most striking feature shown in these figures is the systematic depressing of the “double” routhians for the ( + ,O),and to a lesser extent of the ( + ,$), configuration relative to that of the other negative-parity configurations*. At the lowest rotational frequencies where neutron-pair correlations are expected to be large, this preference is a maximum, > 1 MeV, for the completely paired lowest ( + ,0) configuration. This preference persists, but is greatly reduced in magnitude and appears to still be decreasing, at the highest rotational frequencies. For Aw = 0.40 MeV the (+ , 0) and (+ , $) configuration is preferred by about 250 (and 100) keV relative to the lowest negative-parity configuration. At larger rotational frequencies, insufficient systematic data exists for the construction of “double routhians”. However, routhians for specific isotopes which extend to the highest frequencies indicate 43) that the preference for the ( + , 0) and ( + ,$) configurations continues to decrease with increasing rotational frequency. The X, and o dependence observed for the “double” routhians corresponding to the negative-parity configurations is somewhat different than that of the positiveparity states. At large rotational frequencies the “double” routhians are nearly frequency independent, and there is no systematic preference for a specific configuration to occur lowest in energy. (The observed frequency dependence for Aw > 0.35 MeV and X, = - 8.9 MeV, fig. 4, is attributed to the proton crossing in the light * This is not true for values corresponding simultaneously to the highest rotational frequencies and the most negative value of X,(N = 90), where the complications of both strong isotopic- and configurationdependent shape variations and quasiproton band-crossing must be considered.
116
J.-Y. Zhang et al. / Routhian in gauge space
DOUBLE ROUTHIANS Versus
* Q
-2
I
Xn
Even-N
Odd -N
/*, f-, I-,
/*, kz/ /r,-//, / f-,&I f-,-//2/
01 + 01 -*II-Z-
+-c-L --h-
I hw= 0 MeV
-\ -I \-
/-,hl
0
I -9
-8
-9 A,
I
I
I
-8
-9
-8
(MeV)
Fig. 3. “Double” routhians for a series of ytterbium isotopes plotted as a function of A, for constant fiw. The configurations are labeled by the corresponding conserved quantum numbers (?r, a). The data sources are contained in table 1.
ytterbium systems.) Such a pattern is that expected for uncorrelated configurations; the intrinsic single-particle energies, on the average, should be independent of rotational frequency and configuration. The configuration of the lowest “double” routhian would then vary as a function of the neutron Fermi level, A,, with a relative spacing of up to about half the average single-neutron level spacing (about 75 keV for nondegenerate unpaired states or about 150 keV for twofold degenerate paired states in this mass region). The observation of the expected features of a completely uncorrelated configuration for just those negative-parity configurations, expected to be the least correlated 22), gives confidence, not only in the ability of the chosen references (subsect. 2.2) to remove the effects of rotation and neutron binding energy, but also in the possibility of obtaining a quantitative estimate of the configuration, rotational frequency, and Fermi level dependence of correlation
117
J.-Y. Zhang et al. / Routhian in gauge space
I
t
I
‘P
ROUTHIANS
EXF? DOUBLE
’
Versus
z I
0)
2
-
I
I
)1 W
Even - N
Odd -N
l*,Ol + f-, 01 ---o--f-,1/ -
/f,1/2/ /t,-&, t-,+2 j /-,+I
--m--+ -+ -
I
An=-8.1 MeV
I ;
0 i 0
.2
.4 hw
(MeV)
Fig. 4. “Double” routhians for a series of ytterbium isotopes plotted as a function of Aw for constant A,. The points indicate the frequencies at which the data extrapolations are made. The configurations are labelled by the corresponding conserved quantum number (r, a). The data sources are contained in table 1.
energies*. Indeed at lower rotational frequencies the “double” routhians become more negative, indicating the presence of correlations, as expected. The “double” routhians, shown as a function of Ao and A, in figs. 3 and 4, are in quantitative agreement with current ideas 2,3,22) about correlations in rotating nuclei. Short-range correlations should be present for all low-seniority configurations at l Indeed the configuration-space reference was chosen to have a moment of inertia of about the average value of the highest angular-momentum portion of the negative-parity bands in these nuclei. Therefore, the data really indicate that if strong correlations exist for these negative-parity configurations at large rotational frequencies, they are nearly independent of configuration, rotational frequency, or neutron Fermi level. The measured correlation energies of the positive-parity levels, of course, are relative to any correlations which might have been included in the reference obtained from the negative-parity configurations, e.g. those associated with the proton degrees of freedom.
J.-Y. Zhang et al. / Routhian in gauge space
118
small rotational frequencies. The largest correlations, however, are expected for the “completely paired” ( + , 0) configuration. The magnitude of the total correlation energy of the “completely paired” configuration, gccorr,measured as the energy difference bewen the “double” routhian for the (+ ,O) configuration at fiw = 0 and the lowest uncorrelated configuration in the frequency-independent region at large rotational frequencies, is about 20% smaller than that expected from simple estimates 3, based on uniform level spacing and Ay, the odd-even mass difference, see table 4. With increasing rotational frequency, correlations decrease for all configurations, primarily as a result of the breaking of time-reversal symmetry by the Coriolis and centrifugal forces in a rotating system5r). They survive, however, to the highest frequencies shown in figs. 3 and 4 in that configuration, (+, 0), where they were largest. In the higher-seniority configurations, the contributions to pair correlations are “blocked” for one or more of the pairs of single-neutron configurations. Thus the correlations are expected to be less than, and not to extend to as large rotational frequencies as, that of the (+, 0) configuration. Indeed, this is observed. Sizeable differences, however, are observed between the correlations for the negative- and positive-parity configurations in odd-N isotopes and must be understood. Except at the smallest rotational frequencies, the opposite signature positive-parity levels in this mass region do not correspond to time-reversed single-neutron orbitals. [The large, constant signature splitting predicted, see fig. 5, for these states indicates 52)
TABLE 4 Comparison
of “experimental” total correlation energies and estimates on equally-spaced levels
based
Estimate
“Experimental” isotope
“)
AYC) WeVl
- 8.1 - 8.5 -8.9
-1.50 - 1.75 - 1.85
16*Yb,, 164Ybgq 16’Ybg0
1.040 l.lle) 1.15e)
- 1.80 - 2.05 - 2.20
“) The association between isotope and A, is only approximate. X, is derived from two isotopes (see eq. (9)) and is configuration dependent. b, Experimental estimate of the total correlation energy taken as the energy difference between ez(X,, w=O, v=(+,O)) and ei(X,, w ~0.4, v = lowest, 71= -). ‘) Odd-even mass difference obtained from latest experimental masses 23). d, Estimate of total correlation energy (= -A2,/2d) obtained [see pp. 652-653 of ref. 3)] in the limit of equally-spaced levels. A, = A”ie. and d, the average spacing of two-fold degenerate Nilsson levels, is taken to be 300 keV [refs. 54.55)], ‘) Some masses used to obtain this value are from systematics.
J.-Y. Zhang et al. / Routhian in gauge space
I
6.7 _..----=.-_ es..=.
0
K
- -_=’ -s=_-...=.L~_----_ c c-
.04 fW (fiw,)
119
-
.08
Fig. 5. Calculated spectrum of single-neutron states in the absence of pair correlations. Deformations of c2 = 0.23, e4 = 0.018, and y = - 7” [ref. 19)] were used together with the modified-oscillator parameters of ref. 58). However, for the N = 6 shell, p has been increased to 0.38 in order to reproduce the observed relative spacing of the negative- and positive-parity levels, see fig. 6. The predicted gaps at N = 94 and 97 are indicated. (c, o) = (t , i), (+ , - f), (- , i) and (- , - f) levels are denoted by solid, short-dashed, dot-dashed and long-dashed curves, respectively.
that the limit has been reached in which the rotationally-induced Coriolis and centrifugal forces are dominant.] Therefore, these positive-parity single-neutron configurations do not contribute as much to pair correlations as the nearby negative-parity, single-neutron configurations. Occupying, or “blocking”, a positive-parity configuration will then not have as much of an effect on the neutron-pair correlations as occupying a negative-parity configuration. Thus larger correlations, which survive to larger rotational frequencies, are both expected and observed for the (+ , t) configuration. Recent calculations and analyses indicate [refs. 14,15,19,36,42,48,52,53)] that static neutron-pair correlations are effectively quenched for nuclei in this mass region at Aw 2 0.35-0.40 MeV. The present analysis indicates that the correlation energy is changing for the (+, 0) and (+, :) configuration up to tzw = 0.45 MeV”. This apparent paradox is clarified by recent calculations 22) in the random-phase approximation (RPA) based on single-neutron basis states in a deformed rotor. Such calculations predict sizeable correlations resulting from dynamic pair effects (pan * Although there is insufficient high-spin data above tiw = 0.45 MeV for the systematic construction of “double” routhians, some data extend to larger rotational frequencies. A similar analysis of the larger-frequency individual routhians 43,64) indicates that the correlations for the (+ ,O) and (i , f) configurations extend to about fro = 0.55 and 0.45 MeV, respectively. The estimate for the (+ ,0) configuration, however, is only based on a few nuclei.
120
J.-Y. Zhang et al. / Routhiun in gauge space
vibrations) above the critical frequency for the disappearance of static pair correlations. Indeed the (+ ,O), and to a lesser extent the (+, $), configurations are precisely the configurations for which the dynamic pair effects are expected to be largest at these rotational frequencies. It is also noted that number-projected Hartree-Fock-Bogoliubov calculations for a neighboring isotone 63), 16*Hfg6, indicates that pair correlations persist to arbitrarily large angular momentum for the (+, 0) configuration*. Unfortunately the experimental “double” routhian cannot be constructed to sufficiently large rotational frequencies for the ( + , 0) configuration to test the different predictions of this approach and the RPA calculations discussed in the preceding paragraph - see, however, the footnote on p. 119. Neither have number-projected calculations been made for higher-seniority configurations. More definitive data exist for these configurations, since pair correlations are expected to disappear at lower angular momentum.
3.3. THE REDUCED
“ALIGNMENT”
IN “GAUGE”
SPACE
The “gauge” space analogy of one important configuration-space quantity remains to be discussed: relative “alignment”, or in “gauge’‘-space relative particle number. In configuration space the relative alignment i is the alignment of the intrinsic nucleon relative to the reference configuration chosen. It can be defined 2, as the negative of the slope in the relative routhian plot, fig. 1: i = - de'/do
.
(26)
A similar “gauge’‘-space quantity, the increase in the particle number at a “gauge’‘-space backbend (such as that shown in the upper right-hand portion of fig. 2), has been used ‘) to quantify the number of transitional nuclei. The reduced particle number n (A n, o, Y), however, has never been defined relative to a liquid-drop reference: n(X,, w, V) = -de”(X,, = N(L)
o, v)/dX,(w,
- NI.D(G
V) (27)
It is simply the negative slope of the “double” routhian plotted as a function of the Fermi surface at constant rotational frequency, shown in fig. 3. The reduced particle number is a measure of the number of neutrons (or protons) in the nucleus with respect to the average number given by equally spaced levels. The effects of shell gaps, deformation, local fluctuations in the level spacings, etc. are not included. The positive value of this quantity, given by the negative slope of e” with respect to h,, indicates an abundance of neutrons for the N = 90-100 * The conclusions of the present paper disagree with the statement of Mutz and Ring 63) that “. the knowledge of experimental level schemes alone does not yield information about an eventual pairing collapse”.
J.-Y
ytterbium
Zhang et al. / Routhian in gauge space
121
isotopes with respect to the average. This, of course, is a result of the
prolate deformation
of these isotopes which depresses a large number of single-neu-
tron levels. The increase in N with increasing X, for the most negative values of X, indicates
a larger than average density of levels for the lighter ytterbium isotopes.
For less negative values of h,, the reduced neutron number saturates (a constant slope is observed) indicating a reduced midshell level density approximately that of the average value for the complete shell. The less dramatic slope changes at lower rotational frequencies and for less correlated configurations reflects the averaging over several orbitals for configurations with strong pair correlations.
4. Empirical spectrum of single-neutron states extracted from X n In the absence of static pair correlations, it should be possible to obtain more specific information for other spectroscopic quantities, e.g. nuclear shapes, details of the nuclear potential, and other correlations. A source of such information is the spectrum of the single-particle states, which are the basic excitations of the independent-particle models. To accomplish this a variety of analysis techniques must be developed and utilized to isolate a specific type of information from the experimental spectrum of states, which in general is a complicated mixture of single-particle states “dressed” by the various correlations. An initial attempt to derive information from an empirical spectrum of states constructed from relative excitation energies of single-neutron routhians for rapidly-rotating ytterbium isotopes is reported in refs. 19,52). The spectrum of states resulting from such an analysis is “dressed” by both configuration-dependent deformations and configuration-dependent correlations. Indeed some of the features observed in this empirical spectrum of states appear 19) to be a result of sizeable dynamic neutron pair correlations 22) - see subsect. 3.2.
4.1. THE CONSTRUCTION VALUES
OF THE EMPIRICAL
SPECTRUM
OF STATES FROM
OF X,
It also is possible to derive an empirical spectrum of single-neutron states from the Fermi levels X, extracted from the experimental data. The energy of the single-neutron state is taken to be the value of h, obtained from eq. (9). The quantum numbers corresponding to the configuration with lowest values of X, for neighbouring A, = 2 pairs of isotopes determine the quantum numbers of the level occupied at X,. For example, at Aw = 0.40 MeV the configuration with the lowest values of h, obtained from 163*165Yb has (r, CX)= (-, i), and that obtained from 164,166Yb has ( + , 0). The configuration of the 96th single-neutron level that is added at A, = - 8.520 MeV must then be (-, - $). The sum of such information that exists for
122
J.-Y. Zhung et al. / Routhian in gauge space
-8
-9
-
I-. &I
___ _.-.-.-
.4
.5 fiw
(MeV)
Fig. 6. Empirical spectrum of single-neutron states constructed from values of A, as described in subsect. 4.1. The missing points for the highest-energy (- , $) and ( - , - f) configurations at large values of h w is points for the lowest energy a result of insufficient high-spin data for 166Yb . The missing low-frequency levels is a result of proton band-crossings at these frequencies in the light ytterbium isotopes.
several configurations as a function of rotational frequency defines a spectrum of single-neutron states. An empirical spectrum of states constructed in this manner for rapidly-rotating ytterbium isotopes is shown in fig. 6.
4.2. COMPARISON A, AND
FROM
OF EMPIRICAL EXCITATION
SPECTRA ENERGY
OF STATES
EXTRACTED
FROM
VALUES
OF
DIFFERENCES
At least two differences are observed in this empirically constructed spectrum of single-neutron states when compared to that constructed from excitation energy differences within specific nuclei [see fig. 7 of ref. 52) or fig. 17 of ref. 19)]: (i) The positive-parity states systematically are lower with respect to the neighboring negative-parity states in the spectrum constructed from X, than. they are in that constructed from excitation energy differences. (ii) The alignments for the various configurations vary more for that constructed from excitation energy differences. The variations between the spectra obtained by the two methods can be explained by the correlations which are present for the lowest positive-parity states even at the highest rotational frequencies - see subsect. 3.2. If such correlations for a specific configuration are similar in neighboring even-N or odd-N isotopes, then to a first approximation the spectrum of states constructed from values of h, will be
J.-Y. Zhang et al. / Routhian in gauge space
123
independent of these correlations. (h, is defined as the difference between excitation energies in AN = 2 neighbors - see eq. (9).) Therefore correlations, which remain even at the largest rotational frequencies for the positive-parity levels (see subsect 3.2 and figs. 3 and 4) and are included in the spectrum obtained from relative excitation energies, are to a first approximation, removed from that shown in fig. 6. Similarly the largest slope variations observed in the spectrum obtained from relative excitation energies are thought 19) to be the result of the frequency dependence of these correlations. Even though the empirical spectrum of single-particle states constructed from values of A, is relatively free of correlations, they are sensitive to deformation differences in neighboring even-N and odd-N isotopes. This effect is most important for the single-particle levels of those configurations which are strongly deformation dependent (e.g. the positive-parity single-neutron states in these light ytterbium isotopes). The increase in quadrupole deformation with increasing neutron number expected 56) for these isotopes will lower the empirical levels of the more stronglydeformation dependent positive-parity configurations relative to that of the negative-parity configurations. Such an effect must be more important for the lighter ytterbium isotopes where both the deformation change with mass and the slope of the single-neutron energy level as a function of deformation are larger.
4.3. COMPARISON
OF THE EMPIRICAL
SPECTRUM
OF STATES WITH THEORETICAL
PREDICTIONS
There are significant deviations between the empirically constructed spectrum of single-neutron states, shown in fig. 6 and that calculated from the cranking model with no pair correlations, frequency-independent deformations, and “standard” 57) modified-oscillator parameters [see e.g. fig. 8 of ref. 52)]. In particular, the positiveparity states are predicted too low relative to the negative-parity states. In contrast, the shell-dependent modified-oscillator parameters ‘*) predict the positive-parity states too high relative to the negative-parity states. The agreement with experiment is improved* (compare figs. 5 and 6) by a small increase in p (from 0.34 to 0.38) for the N = 6 neutron shell. Similarly calculations based on Woods-Saxon nuclear potentials 59,60) reproduce the observed level spacing. However, none of these calculations predict the near-equal alignments (slopes) observed for the lowest ( -k , i) and neighboring negative-parity single-neutron levels. This feature, which has also been observed for N = 89 and 91 dysprosium and erbium isotopes using the excitation-energy-difference method 61), is not imderstood. * The slightly larger p( N = 6) used for fig. 20 of ref. 19) (= 0.40) compensates for the correlations which remain for the lowest positive-parity states in odd-N isotopes at the largest rotational frequencies studied.
124
J.-Y. Zhang et al. / Routhian in gauge space 5.
Summary
Formulae are presented for the construction of “double” routhians from which both the configuration- and “gauge’‘-space rotational energies have been removed. Such an analysis, which exploits the complete analogy between configuration and “gauge” spaces, has been applied to a series of ytterbium isotopes varying in mass number from 159 to 170. Various features of “gauge’‘-space total-alignment, or particle-number, plots are illustrated and compared to configuration-space totalalignment plots with similar features. “Double” routhians are presented as a function of X, for constant tzo and as a function of hw for constant h,. Evidence is obtained that the lowest (+, 0) configuration (and to a lesser extent the lowest ( + , i) configuration) is correlated even at the largest values of rotational frequencies (0.40-0.45 MeV) for which such a systematic analysis can be accomplished. These features agree with recent predictions 22) of dynamic pair correlations using the random-phase approximation (RPA) and single-neutron states in a rotating deformed potential. An empirical spectrum of single-neutron states is constructed from values of h,. Such a procedure is based on total energy differences for the same configuration in neighbouring even-N or odd-N isotopes. To a first approximation this prescription should remove those correlations which are dependent on the configuration, but are nearly equal for identical configurations in neighbouring even-N and odd-N isotopes. The differences observed between this empirical spectrum of states and that constructed from excitation energy differences within a specific nucleus can be understood in terms of such configuration-dependent correlations, which are removed when the construction is based on values of h,. Finally it is demonstrated that many of the features of this “uncorrelated” empirical spectrum of states can be described by unpaired cranking calculations based on either modified-oscillator or Woods-Saxon potentials with an appropriate choice of parameters. However, the small alignment of the low-lying positive-parity configuration relative to that of neighbouring negative-parity configurations remains unexplained. Discussions with R. Bengtsson, R.A. Broglia, T. Dossing, G.B. Hagemann, B. Herskind and W. Nazarewicz are acknowledged as is support from the Danish Natural Science Research Council, the Danish Ministry, of Education, the US Department of Energy (contract no. DE-AS05-76ERO-4936) and the US National Science Foundation (contract no. PHY-8406676). One of us (J.-Y. Z.) wishes to thank the Niels Bohr Institute for its kind hospitality during his recent visit. References 1) 2) 3) 4)
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