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Bifurcations in the cranking model
cm *.--
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Physics Reports 264 (1996) 279-295
Bifurcations in the cranking model E.R. Marshalek Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
Abstract An informal presentation is given of a new theorem associated with the self-consistent cranking method, dubbed the Cranking Bifurcation Theorem, which extends the range of the cranking technique to generate certain types of collective multiphonon anharmonic solutions of time-dependent mean-field equations. Applications to the cranked volume-conserving oscillator are offered as simple examples.
1. Introduction Before launching into my main topic, I would like to say that it is a pleasure and honor to participate in the Belyaev fest, especially since work of Professor Belyaev played an important role in my early career. While a graduate student, I avidly studied his famous paper on pairing correlations that was published in a Danish journal [ 11. The effort proved rewarding, enabling me to do my thesis calculations treating collective vibrations in deformed nuclei microscopically for the first time [ 21. Not long afterwards as a postdoc, I got the idea that mean-field theories should be treated as classical field theories subject to quantization. Since such field theories are non-linear, quantization appeared difficult. The key to the solution was provided by the seminal paper of Belyaev and Zelevinsky [ 31, which introduced the concept of boson mappings into nuclear physics and spawned the cascade of developments reviewed in a recent article [4]. The topic that will be discussed now is closely related to these developments. Recently, there has been a revival of interest in multiphonon excitations. For example, Aprahamian believes she has found evidence for at least 17 cases of two-phonon y-vibrations in rare-earth nuclei, many of which would have to be highly anharmonic, as judged by the ratio of energy spacings [ 51. In addition, there is growing experimental evidence for two-phonon octupole levels, and two-phonon giant dipole and quadrupole resonances in various spherical nuclei [ 61. Although many methods are available for describing anharmonic multiphonon excitations, including, for example, boson mappings [4], the approach that I will now describe, which is a technique for obtaining periodic solutions of time-dependent mean-field equations, is arguably the simplest and allows the summation of anharmonicities to all orders, albeit in a semiclassical approximation. The 0370-1573/96/$9.50 @ 1996 Elsevier Science B.V. All rights reserved SSDIO370-1573(95)00043-7
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main limitation of the method is that it can only be applied to nuclei having equilibrium mean-field solutions with an axis of rotational symmetry (spherical or axially symmetric deformed shapes)although these can correspond to excited as well as ground-state configurations-and the vibrations built on these configurations must project an angular momentum K # 0 on the equilibrium symmetry axis. This means that the method cannot describe monopole and /? quadrupole vibrations, for which K = 0, but it can describe, for example, y quadrupole (K = 2)) octupole (K = 1,2,3) and giant dipole (K = 1) vibrations. The physical picture of such modes as a surface wave-an instantaneous distortion rotating about the symmetry axis-suggests the possibility of a stationary description in a frame rotating with the wave. Such considerations then lead to the cranking model, but with cranking about the (equilibrium) symmetry axis. The normal cranking method is applied to nuclei having a broken rotational symmetry in the ground-state mean field. The cranking axis is chosen such that the system is deformed in any plane perpendicular to this axis. The technique then generates the usual low-lying rotational bands found in permanently deformed nuclei. On the other hand, the common wisdom is that cranking a mean field about a symmetry axis just carries it into itself, leading to nothing new, except possibly a redefinition of the Fermi surface (equivalent to introducing a “tilted Fermi surface”). However, I have found that this is not necessarily so-at certain critical cranking frequencies new solutions that break the symmetry bifurcate from the symmetric one. This has led me to formulate what I call the Cranking Bifurcation Theorem (CBT) , which may be stated as follows: - If a system of coupled anharmonic oscillators has an axially symmetric equilibrium configuration (vacuum), then self-consistent cranking about this axis yields families of symmetry-breaking solutions that bifurcate from the vacuum solution at the critical rotational frequencies L2, = wJK~, where We is the small-oscillation (RPA) frequency of any mode carrying Kp # 0 units of angular momentum along the symmetry axis. Each family describes the dynamics of the system on submanifolds of phase space characterized by a single non-vanishing action Nfi = (I - IO) / Kp, where I is the total angular momentum of the mode and I0 that of the vacuum. A heuristic proof will be given shortly. This theorem has a broad range of applications, encompassing time-dependent mean-field approximations such as the time-dependent Hartree-Fock (TDHF) and Hat-tree-Fock-Bogoliubov (TDHFB) approximations, phenomenological mean-field models such as the Nilsson-Strutinsky model, empirical collective models (Bohr-Mottelson, IBM, etc.) and liquiddrop models. The TDHF approximation, for example, is replaced by the static cranked Hartree-Fock (CHF) approximation. In lowest-order perturbation theory, one obtains the familiar RPA equations from a purely static cranking argument, as we shall see. In a non-perturbative treatment, one has to crank about a symmetry axis and search for bifurcations at the critical frequencies given by the theorem. This requires an RPA program to locate these frequencies and a CHF program to follow a given broken symmetry solution for cranking frequencies away from the critical value. In this way, one generates a solution that is a continuous function of the cranking frequency in some neighborhood of a bifurcation point. Of course, the physical solutions correspond to suitably quantized values of the angular momentum and describe multiphonon excitations of a mode with K # 0. The excitation phonon is no longer the RPA phonon away from the bifurcation point, but rather a non-linear phonon whose structure takes into account renormalizations accruing from the anharmonic mixing with other modes, including those with K = 0. If the CHF solution is obtained by exact diagonalization rather than perturbation theory in the neighborhood of a critical point, then large-amplitude vibrations can be described with no difficulty and without the need for an adiabatic approximation.
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The TDHF (or other time-dependent mean-field) solution can be generated afterwards from the CHF solution by transforming from the rotating to the laboratory frame. In most cases, however, this step is unnecessary.
2. A heuristic
proof of the CBT
Consider an even-even nucleus whose equilibrium shape is either spherical or axially deformed in some mean-field configuration (the “vacuum”) which is normally the ground state but may be an excited state as well. The physical picture of a collective vibration that projects K # 0 units of angular momentum along a symmetry axis corresponds to a rotating distortion, a surface wave traveling around this axis. Since such a mode is degenerate, it is possible to choose linear combinations of normal coordinates that describe a uniformly rotating wave. From a quanta1 viewpoint, these modes may be described in terms of boson creation operators BL carrying Kfi units of angular momentum along the symmetry axis. Since the cranking model is basically a classical approximation, these boson creation operators will be replaced by the corresponding c-number complex coordinates p; , (p, and i& then become canonically conjugate variables). These coordinates are sometimes called phasor variables. If H is the quanta1 Hamiltonian, then (H) ’ , will denote the classical Hamiltonian function of &, p,. The equilibrium points of the system relative to a frame rotating uniformly with angular velocity 0 about the 3-axis are solutions of the set of equations
(1) where
(ff’) = (fg - fi(J3), is the so-called Routhian, (_Z3)being the classical angular-momentum axis (3-axis) given by *
(2) component
along the symmetry
(3) The diagonal quadratic form for (.Z3) is, of course, a consequence of designating the 3-axis as the symmetry axis with the appropriate choice of phasor variables. The constant IO is the spin of the axially symmetric vacuum. For the ground state configuration lo = 0, but for an excited state vacuum, non-zero values of IO may occur, as, for example, in the case of an oblate rotational-band terminus configuration. This is illustrated in Fig. ( 1). If IO = 0, then the excitations occur in time-reversal conjugate pairs corresponding to & and p?, , with the sum in Eq. (3) understood to run over both time-reversal partners and K_, = -Kp. If Z, # 0, then the total spin of an excited mode is given by
’ One may regard (H) as the expectation value with respect to a coherent state. * For convenience, ti = 1 will be used throughout this paper.
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Fig. 1. The rotational motion of the bifurcating solution (solid ellipse) The rotations may be in the same (a, c) or opposite senses (b, d).
compared with that of the vacuum (dashed circle).
10 + Kp IP, 1’7 where Kp can be positive or negative. In this case, time-reversal symmetry is broken for a fixed choice of vacuum. ’ The difference between the two cases is illustrated in Fig. 1. For simplicity, let us first consider small oscillations (RPA) about the vacuum configuration. In that case, the classical Hamiltonian is
(H) = Wo) = Eo(Z0) + c
@/&P/l
7
(4)
/I where E. ( IO) is a constant, the vacuum energy, and /?t( has been chosen as a normal-mode with corresponding frequency wP. From Eq. (3)) the Routhian (2) becomes (H’) = Eo(Zo) - fiZo + c(wfi P
- &M;P,.
coordinate
(5)
Then, Eqs. ( 1) imply P,(%
-
OK,) =O.
(6)
In general, this set of equations has two kinds of solutions: the trivial solution in which all p, = 0, and non-trivial solutions in which a single coordinate p, # 0, while p, = 0 for ,u # ,uo. 4 However, the non-trivial solutions are possible only for a discrete set of cranking frequencies given by L!=L& G
wcL/Kp,
Kp # 0.
(7)
should be noted that there are no non-trivial solutions for those modes with Kp = 0.The trivial solution just corresponds to the axially symmetric vacuum (unproductive cranking), while the nontrivial solutions are deformed about the 3-axis, the magnitude of the deformation depending on p,,. It
s However, one can also study excitations built on the time-reversed vacuum having spin -Ia to obtain the time-reversed states, as shown in Fig. 1. Thus, while (a) and (b) do not correspond to time-reversed states, (c) and (d), which are excitations built on the time reversed vacuum, are the time reverses of (a) and (b), respectively. 4 For simplicity, we do not discuss the resonant case in which some of the frequencies wfi are commensurate.
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Now, Eq. (6) does not determine pm, which reflects the independence of the frequency and amplitude for a harmonic oscillator. However, \p, 1 can be fixed via Eq. (3) by prescribing a value (.Z3) = I, where Z takes on suitably quantized values, which leads to N,
= &@I2 = (I - lo)&.
(8)
Of course, the sign of Z must be chosen taking into regard that of Km so that the r.h.s. of (8) is positive. The total energy (lab frame!) for this solution is then given by E = (H) = Eo(Zo) + ww IP,l* = Eo(Zo) + wti (Z - Zo) /K,.
(9)
For example, for quadmpole vibrations about a spherical ground state, one has Z. = 0 and the cranking procedure engages only those modes with 1Km 1= 1,2. For 1KM 1= 2, the excitation spectrum is given by AE = iwmZ = wMn, where n is the number of phonons. Therefore, the cranking solution selects the sequence of aligned multiphonon excitations with (quantized) spins Z = 2n = 0,2,4, . . ., which, for the lowest frequency We, are often yrast states, sometimes referred to as quasirotational states. For IKMI = 1, the excitation spectrum is given by AE = w,Z = oMn, corresponding to multiphonon states with spins Z = n = 2,3,4,. . . As shown explicitly in Ref. [7] for the U(5) limit of the interacting boson model (IBM) [ 81, such solutions correspond to rotation about a tilted axis rather than a principal axis of the density distribution, thereby breaking signature symmetry. Next, consider the more general situation in which (H) contains anharmonic corrections to the small-oscillation part in the form of an arbitrary, possibly infinite polynomial in the coordinates which may be the result of a Taylor expansion about the equilibrium point (vacuum). Ply PP The argument is greatly simplified if one chooses the coordinates such that (H) is given by the Birkhoff-Gustavson (BG) normal form [ 9,101. In this connection, one should distinguish between the non-resonant and the resonant cases. In the non-resonant case, the small-oscillation frequencies are either incommensurate or else the commensurability is the consequence of a global symmetry that forbids resonant coupling terms. In that case, as proven by Birkhoff [ 91, the Hamiltonian can be transmuted by a series of canonical transformations into an infinite polynomial in the action variables N, defined by Np = L’;Pp 3
(10)
with the conjugate angle variables being cyclic 5 . With the assumption that the cooordinates p;, ,8, have already been chosen as the transformed variables, the most general Birkhoff normal form for (H) is given by
where the numerical coefficients ZX(*~) ( ,LL~,cL*, . . . , ,x,J are completely symmetric in the indices and the NP constitute a formal set of constants of motion that are in involution. The resonant case, which 5 The obvious quantum analog is the perturbative diagonalization of a boson Hamiltonian by successive unitary transformations to provide a function of the boson number operators. This has been called the diagonal boson representation.
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is analogous to degenerate perturbation theory in quantum mechanics [ Ill, is too complicated to deal with here. From Eqs. ( 1) -( 3), ( 11)) the stationary points in the rotating frame are obtained as solutions of the set of equations
Sin_
1
I)! ~ ,,,,plh(2~)(ll,~i,.‘.~,-l)Ni(l.‘~NIL.I c
+...
1 =*.
In this case, as in the harmonic limit, one still has the trivial solution in which every p, = 0 and, in addition, there are the non-trivial solutions given by p, = 6,,,p,. Thus, equating to zero the corresponding term in brackets in Eqs. (12) with NP = NMSP,+,, gives the following series in N,: 0 - 0, = hC4’(,u~,..Q,) Ki’&
1 + . . . + cn _ Ij!h”“‘(~~.
. . +uo>K~‘(~,,)“-
+ ... 7
(13)
which can be reverted to give a series solution for NM in powers of L’- L$. 6 Hence, in the presence of anharmonicity, a non-trivial solution can exist for a continuous range of rotational frequencies 0 in some neighborhood of each critical frequency L& given by Eq. (7), and these critical frequencies are points at which an anharmonic branch bifurcates from the vacuum solution. It is also interesting to note that if 0, > 0, which is often the case, and hc4) (,LQJ.L~) < 0 while the higher-orders are small, then the solution “backbends” in some neighborhood of L?,, i.e. it exists only for 0 < 0,. The energy corresponding to such an anharmonic branch may be obtained trivially by substituting in Eq. ( 1 I), which merely limits the dynamics to a particular submanifold the solution NP = N,6,,, in phase space, ipso facto proving that the cranking technique gives the correct classical energy. When the action variable is expressed in terms of the prescribed angular momentum through Eq. (8), the energy takes the form of an expansion in powers of I - lo:
E= (H) = Eo(Zo) fw,
+...
(14) Then, Eq. (13) just becomes equivalent to the familiar Hamilton relation
This completes the heuristic argument. It should be emphasized that the BG normal form was introduced here solely to facilitate the justification of the cranking technique, not as an end in itself. It should be obvious from the foregoing that if the normal form couId be calculated in nuclei, then cranking would become superfluous. ‘This implies for pw, an expansion
in powers of (a - 0,)“‘.
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To see the relevance of the foregoing argument to nuclei, one should first consider how canonical coordinates can be introduced into nuclear mean fields. The most straightforward route is to first perform a generalized Holstein-Primakoff (GHP) boson mapping of the nuclear many-body system [4]. Then, one may appeal to a theorem [ 12-141 according to which the replacement of the boson operators by classical c-numbers 7 results in a canonical parametrization of the one-body density matrix. Depending on the system and the type of GHP mapping chosen, one can obtain in this way any of the standard TDMF approximations, with the Hamiltonian (energy) being a functional of the c-number canonical variables (which may be identified with the original /?t, p,). While this can be the starting point for calculating a BG normal form, it is generally preferable to work with the standard forms of cranked mean-field equations, using, for example, the non-canonical density matrix elements themselves in the variational procedure.
3. Static cranking
derivation
of the RPA
The foregoing discussion is sufficiently general to encompass a wide variety of Hamiltonian dynamical systems, including nuclei in a mean-field approximation. A simple microscopic example of the bifurcation theorem is afforded by the derivation of the RPA eigenfrequency equation for a separable multipole-multipole interaction of rank L using the static cranking method (a slightly less general derivation has been given in [ 1.51) . This can also be regarded as an independent method for proving the theorem for mean-field approximations, which is easily generalized to arbitrary interactions. The many-body cranked Hartree mean-field Hamiltonian ZZMFis then given by
HMF
= Ho
-
0J3 - x c
sQ;&
(16)
,
K=-L
where Ho is the uncranked axially symmetric mean-field Hamiltonian, the operator QK is a component of an L-pole tensor, SQ; is the change in the corresponding deformation parameter, and x is the strength of the interaction. Since HO and the angular-momentum component J3 are assumed to commute, HO - 0J3 may be chosen as the zeroth-order Hamiltonian and the rest of Eq. ( 16) treated as a perturbation. With IO) denoting the unperturbed axially symmetric configuration (not necessarily the ground state) and In) the orthogonal excited (or de-excited) configurations, we have that HoJO) = &IO), HoIn) = &In). In addition, one has J310) = ZolO) 3
J3b)
= &b)>
(17)
where it is assumed for generality that the vacuum IO) carries angular momentum eigenvector of H MF to first order in the 8Q; is given by
IO. The perturbed
(18)
’ Of course, one must choose the proper ordering of operators prior to the c-number
replacement.
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The first-order changes_ S&K in the deformation parameters are determined conditions SQK = (PcJQF#~) - SK,aQO,where Q0 E (O]&,]O),* as follows:
by the self-consistency
(19) It is immediately seen that there is a trivial solution with all the 6QK = 0, for which /PO) = IO), and non-trivial solutions with one pair SQK = (- l)“SQT, # 0, for which the corresponding term in brackets vanishes, thereby giving rise to the eigenfrequency formula
+c n+O
I<4&-KlO> I2 1 E,,-Eo+w
(20)
=;’
provided that the cranking frequency is given by L2 = L$ = W/K and K # 0. A second equation is obtained by letting 0 ---f --a and thus w --f --w. This pair of equations correspond to the what was called the RPA for “non-Hermitian fields” [ 161. Equations of this type were first derived for odd-A nuclei by Bes and Chung [ 171, which shows that the present cranking approach applies to odd nuclei as well as even-even ones. If IO) is an excited configuration, one must consider frequencies of both signs, with negative values of o corresponding to de-excitations. Moreover, as long as 1, # 0, there is no symmetry for modes with dl = +K and dl = -K. If, however, Z. = 0, the pair of equations collapses to the single well-known RPA dispersion formula c
l(n]&(Q,
+ (-l)KQ-~)]O)12~E, (En - Eo)2 - io2
n#O
- Eo)
=-
1
for which the modes with fK are degenerate. To determine the lowest-order for the mode in question, one must prescribe the spin using the condition VOJ
33 ]Po)
(21)
2x’ deformation
change 8QK
= 1.
(22)
Actually, this fixes lsQK] as follows: (23) where BK( w) is an RF’A mass parameter, which will not be given here explicitly. What this result shows is that the deformation associated with the symmetry breaking is proportional to the vibrational amplitude. The perturbative approach can be carried to higher orders for values of 0 # L$, by using an expansion in powers of ID - L&I ‘I’, where the square root may be regarded as a critical exponent. In this way, one generates the Taylor expansion equivalent to the BG expansion discussed earlier. However, as already emphasized, the primary aim here is not to use perturbation theory but to diagonalize the cranked mean field directly, thereby summing the anharmonicities. It should also be clear that the method just outlined applies not only to spherical nuclei, but also to axially symmetric deformed nuclei, an application never considered before. A second point to be emphasized is that * The condition
of axial symmetry does not require that Qo vanish, although it may do so for a spherical nucleus.
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since the 8QK do not vanish for a non-trivial solution, not only is the axial symmetry broken, but the cranking axis may not coincide with a principal axis of the density distribution. Therefore, some of the solutions can describe tilted rotation, which involves broken signature symmetry as well. This is particularly obvious for the case of quadrupole deformation (L = 2), where the ]K] = 1 mode corresponds to a tilted ellipsoid since it implies the existence of Y2i and Y2_, components in the mean field. Unfortunately, time constraints do not permit us to consider the topic of tilted rotation, which is a subject of growing interest [ 181.
4. Example:
The anisotropic
oscillator shell model
As a soluble example, one may consider the volume-conserving anisotropic harmonic oscillator (VCHO) model. This is just the model of nucleons in a pure deformed harmonic oscillator whose equipotential volumes are independent of deformation. The cranked version of this model as applied to conventional rotational bands has been discussed by numerous authors, but perhaps most exhaustively by Troudet and Arvieu [ 191. It has been shown that the VCHO can be interpreted as the Hartree approximation applied to a special internucleon interaction that includes separable many-body components in addition to the quadrupole-quadrupole interaction [ 20-221. With the the aid of the CBT, new solutions of the cranked VCHO have recently been found [ 15,23,24]. We first consider solutions describing multiphonon giant isoscalar y (K = 2) vibrations built on the IO = 0 ground state of an axially symmetric system. In Fig. 2, the angular momentum (denoted there by (L3)) is plotted as a function of the cranking frequency, with scaling chosen to give a universal plot [ ifI E _Z(nl + i) involving a sum over quanta perpendicular to the symmetry axis for the orbitals occupied in the ground state]. One sees that the solution, which happens to backbend, bifurcates at the critical frequency 0, = wRPA/2, where wRPAE awl is precisely the well known RPA frequency for this mode [25], WI G w, = wy being the common frequency of the deformed oscillator potential perpendicular to the symmetry axis in the ground state. This result is in accord with the CBT. It is also interesting to note that the moment of inertia for this kind of rotation is always less than the rigid-body value, although it approaches this value for high angular momentum. In the case of normal rotation in this model, the moment of inertia has exactly the rigid-body value in the limit of low angular momentum. In Fig. 3, the excitation energy is plotted as a function of the angular momentum; the deviation from a straight line (which represents the RPA) is a measure of the anharmonicity. This may seem large at first sight, but taking into account the scaling, one realizes that, except for very light nuclei, the anharmonicity is miniscule for states with a.few phonons, which is in line with recent data on giant resonances. For an even-even nucleus, the physical states correspond to aligned y-vibrational bandheads having (L3) = K = 0,2,4, . . . In Fig. (4), one sees a typical total energy surface as a function of the deformation parameters .szOand ~22 [ 261 for a non-zero angular momentum (I = 10). The twin degenerate minima just correspond to two possible elliptic deformations at an angle of 7r/2; i.e. the semimajor axis can lie along the l-axis or the a-axis. For Z = 0, there is only one minimum with E22 = 0, corresponding to axial symmetry. For any I # 0, the twin minima with broken axial symmetry appear, while the axial shape corresponds to a point on the crest. This is an example of a pitchfork bifurcation. Finally, the evolution of the shape for this solution as a function of L?/L$ is shown in Fig. 5, assuming that the ground state is a superdeformed prolate spheroid with an axis ratio of 2. It should be noted that for the VCHO
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0
20
40
80
100
C,,/(h?J Fig. 2. A scaled universal plot of the angular momentum about an equilibrium symmetry axis. Fig. 3. A scaled universal plot of the excitation about an equilibrium symmetry axis.
vs the rotational
frequency for the anisotropic
energy vs. the angular momentum
I
=
Fig. 4. The total energy surface as a function of the deformation about an equilibrium symmetry axis.
for the anisotropic
oscillator
cranked
oscillator
cranked
10
parameters
for the anisotropic
oscillator potential cranked
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8
2 0
-0.0
0.2
0.4
0.6
0.8
1.0
Q/R, Fig. 5. The evolution of the nuclear shape as a function of the rotational frequency cranked about an equilibrium symmetry axis.
potential,
wi/wi
for the anisotropic
oscillator
potential
= (Xi) /R2 which defines the square of a semiaxis of the density distribution, with 1/3. One sees that along the backbending curve the initial cigar shape with the R2 = ((X:)(X;)(X:>) 3-axis of symmetry evolves through triaxial shapes into an infinitely long needle oriented along the 2-axis as R -+ 0 and the angular momentum (L3) + 03, in the interim passing through an oblate shape with the l-axis of symmetry. Of course, this corresponds to the minimum on the right-hand side of Fig. 4; the twin solution ends up as a needle aligned along the 2-axis. An example of multiphonon vibrations built on an axially symmetric excited configuration with I0 # 0 is presented in Figs. 6 and (7) in which the angular momentum is labeled by (.I,) = I. The vacuum configuration is the well-known ground-band termination state of *ONe with spin IO = 8. The RPA excitations built on this state were studied in detail some time ago within the framework of the VCHO by Kurasawa [27], who found that there are three modes with AZ = K = f2. As shown in Fig. 6, which is a spin vs. rotational frequency plot, cranking of the IO = 8 configuration about the symmetry axis gives rise to three bifurcations (solid lines) at the cranking frequencies w/2, where the values of o agree exactly in magnitude with the RPA frequencies. Also shown are the degenerate time-reversed solutions (dashed lines) obtained by cranking the time-reversed configuration with Z. = -8, with all rotational frequencies reversed in sign. The segment marked a represents a multiphonon anharmonic excitation branch with AZ = K = 2, which corresponds to the mode type marked (a) in Fig. 1. The segment marked b represents a multiphonon anharmonic excitation branch with AZ = K = -2, which corresponds to the mode type marked (b) in Fig. 1. The segment designated by g.b. is formally a AZ = K = -2 de-excitation (negative-frequency) branch, but this mode, which is a “vibration” seen from the perspective of the IO = 8 state, is really the ground rotational band, as already noted by Kurasawa in his RPA analysis. The time-reversed solutions are physically equivalent and give rise to the same energy vs. angular momentum plot Fig. 7, as long as one uses 111, which is the observed spin. Equivalently, one may always follow only the solid trajectories as long as the b trajectory satisfies (.Zx) = Z > 0, but when (Jx) turns negative, switching to the corresponding portion of the time-reversed trajectory that has (Jx) > 0, which gives rise to
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\
__,~__.____________!,____.
I,=-8
5
10
15
20
25
I Fig. 6. Bifurcating solutions based on the band termination state. The solid lines emanate from the configuration units of spin, while the dashed lines are the time-reverse solutions. Fig. 7. The excitation energy vs. spin corresponding
with 8
to the solutions in Fig. 6.
the abrupt change in Fig. 7. The forking structure is also quite interesting and may be a signature for band termination. The reader may be curious about the relation of the signs of the bifurcation frequencies in Fig. 6 to whether a mode is an excitation or de-excitation. This may be answered as follows. Let w,, wb, and -tigb denote the actual RPA frequencies, where the first two are positive, being excitation modes and the last negative, being a de-excitation mode built on the ZO= 8 state. In the neighborhood of a bifurcation point, it is sufficient to use the following classical cranked RPA Hamiltonian corresponding to Eq. (5)
(ff) = Eo(Z0) -
f2zo+ ( @a-
2a)s%h +
(wb
+ 2fi>p,'/-?b + (-@gb+ 2fi)&&,
(24)
which takes into account whether AZ = K = &2. The condition (6) then gives the bifurcation points occurring at 0 = wa/2, -@b/2, and @@/2, in agreement with Fig. (6). It should be noted that Troudet and Arvieu [ 191 had found the trajectory a (but not b), but had no way of realizing that it could be interpreted as a vibrational band built on the band termination state. There also exist AZ = K = f 1 modes that can be found with the aid of the CBT, but these will be discussed in a more comprehensive paper. It is worthwhile mentioning again that the CBT is applicable not only to microscopic mean fields, but also to phenomenological collective models, such as the Bohr-Mottelson model [28] and the IBM [ 81. Since these models are bosonized from the outset, the classical approximation is obtained by simply replacing the bosons by c-numbers in the Hamiltonian, or, what is essentially equivalent, averaging in a coherent state, and then invoking Eq. (1). Examples of such applications to the IBM are given in Refs. [7,23] _
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5. Some problems to which the CBT can be applied 5.1. How repeatable is a phonon? If an excitation mode has a pure particle-hole or two-quasiparticle nature, a double excitation is ruled out by the Pauli principle. In order for a “phonon” to be repeatable, it is obviously necessary that it be a collective mode. A fundamental question that this line of inquiry might be able to answer is: how many times can a given phonon excitation be repeated? Of course, the answer depends in part on the definition of the phonon, which may explain the conflicting answers given in the literature. It appears that the best and ultimate definition is the non-linear phonon of the diagonal boson representation used in the BG representation. An equivalent definition can also be obtained from the self-consistent coordinate method (SCCM), which is very popular in Japan [ 291. From the viewpoint of the BG representation, there is no clear upper limit on the number N, of times that a given phonon may be repeated. This is because the treatment there is based on a Taylor expansion, which only guarantees the existence of a solution in a fuzzy neighborhood of the critical point. A clear answer may possibly emerge from an exact, i.e. non-perturbative solution of the non-linear cranking equations. It may then be possible, for example, for a solution to disappear for some N, (or Z) , thereby signaling a cutoff on repeatability. 5.2.
Development
of realistic codes
While most of the applications of the CBT thus far have concentrated on simple models in order to gain insights into the nature of the solutions, realistic codes can certainly be developed. These can be either fully self-consistent HFB or CHB codes or they can be based on phenomenological deformed potential models, such as the Nilsson or Saxon-Woods potentials, with the inclusion of pairing correlations. While standard codes designed for normal cranking can be trivially modified for cranking about a symmetry axis, this is only the first step. At least an RPA code is also required to locate the bifurcation points. Even better would be a code that includes the lowest-order perturbation corrections to the RPA, equivalent to the fourth-order BG approximation, which would provide a good starting guess for the main code. In the case of phenomenological deformed potential models, I have been able to show that the RPA limit of the cranking equations exactly corresponds to the result of the vibrating potential model (VPM) [ 30,311. Once a bifurcation associated with a particular RPA root (usually the lowest) is located, the corresponding solution must be followed as a function of the rotational frequency, or, preferably, the spin. Since for a given spin there may be a large number of self-consistent solutions, say, one for each RPA root, there is the problem of ascertaining that the desired solution is actually being followed. Some of the problems that occur in the cranking model for normal rotational bands may also come to haunt the application to vibrations in some nuclei. This includes well-known difficulties associated with level crossings [ 321 and with the phase-transitional collapse of pairing. While many of the interesting applications, such as to multiphonon y-vibrations, involve low spins, it should be noted that the cranking frequencies are relatively high, making pairing collapse more likely. While on the one hand the same prescriptions used for normal rotation can be applied in such situations, for example, using particle-number projection to deal with a pairing collapse, on the other hand, the anomalies themselves may represent valid physical mechanisms for terminating a vibrational band. A
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pairing collapse in such a situation might be interpreted as the dissolution of a vibrational mode by virtue of mixing with a large number of quasiparticle excitations that block the pairing correlations. Should this happen for a boson number NP < 2, one could conclude that the mode in question is non-repeatable. 5.3. Octupole excitations
in superdeformed
nuclei
Since the supershells in superdeformed nuclei are composed of approximately equal numbers of single-particle levels of opposite parity, octupole correlations are expected to be very important [ 331. Numerous calculations have been carried out for these nuclei, using the stretched octupole-octupole interaction [ 211 and either a pure deformed harmonic-oscillator potential [ 34-361 or a more realistic Nilsson potential with pairing [37]. In all cases, the RPA was used. However, since some of the octupole modes are expected to be quite soft, it is very important that anharmonic corrections be studied. The cranking technique described here seems ideally suited for this purpose as long as one is content to study the ]K( = 1,2,3 octupole modes. Unfortunately, the K = 0 mode is not directly accessible in this way. 5.4. Is there a universal anharrnonic vibrator? Recently, Casten, Zamfir and Brenner “universal” quadrupole vibrator formula E(Z) = nE(2,+)
on the basis of energy
systematics
proposed
+ +z(lz - 1)&d
for the aligned n phonon Ejiri [39] formula E(I)
[38]
levels with spins I = 2n, which is phenomenologically
=aZ+bl(Z+l).
the
(25) equivalent
to the
(26)
What is remarkable is not the expressions themselves, which have been known for a long time, but rather that the anharmonicity parameter c4 or b is nearly constant in each of three broad regions of the nuclear periodic table covering all even-even nuclei with 2 2 38 that are not well-deformed. With I0 = 0 for the ground state and KM = 2, this is essentially a truncation of Eq. (14) with E4 = h c4)( ,uO, ,uO). This indicates that the present formalism applied perturbatively to nuclei with spherical ground states can be used to calculate the parameter ~~ or b and test whether a microscopic theory can predict a universal value. In fact, I effectively obtained expressions for this parameter within the framework of the pairing plus quadrupole-quadrupole force model 22 years ago using an early precursor of the present formalism [40]. The only numerical calculations were in the vicinity of single-closed shell nuclei, then yielding values with the opposite sign of the universal ~4. It certainly would be worthwhile doing such calculations again over a broader region of nuclei and with improved forces. 5.5. Vibrations in odd-A nuclei The CBT may provide a new approach to the old subject of the coupling of odd nucleons to collective vibrations. The (uncranked) vacuum state should be chosen as the equilibrium solution
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293
including the odd particle so that It, # 0. If this is axially symmetric then the CBT can be applied. One may then anticipate two types of bifurcations: those that correspond to the scattering of the odd nucleon and those that correspond to excitations from the even-even core. The former take into account the effect on the single-particle (quasiparticle) energies of the coupling to the collective vibrations, while the latter includes the collective vibrations themselves. Since the vacuum state is at least two-fold degenerate, one must, of course, make a particular choice. For example, in a deformed nucleus one may choose the vacuum ]!Po) = LZ;IO), where 10) is the mean-field state of the even-even core and LZ;creates the odd nucleon, with angular momentum I0 = Kp. Corresponding to a collective excitation projecting K units of angular momentum on the symmetry axis in the even-even nucleus, one expects two modes in the odd nucleus that project K f Kp units of angular momentum for a onephonon excitation. These modes are not expected to be degenerate since in one case the nucleon is rotating in the same sense as the collective wave, while in the other case it is rotating in the opposite sense. Of course, if the vacuum is chosen instead as the time-reversed state IPi) = TuL]O), then I0 = -Kp, but the physics is the same, only the sense of rotations is reversed. The physical picture is basically that shown in Fig. 1. As mentioned earlier, in the RPA limit, one recovers equations of the type used long ago by B&s and Chung [ 171 for describing y-vibrations in odd-A deformed nuclei, which certainly deserved more study. The proposal here goes much further since the effects of anharmonicity to all orders can be included in the cranking approach, which is not limited to deformed nuclei, but can also be applied to spherical nuclei. 5.6. Electromagnetic
transitions and moments
The discussion thus far has made no mention of the calculation of electromagnetic transitions and moments in the cranking approach. In fact, transition matrix elements can be calculated between levels lying along the same cranked trajectory, for example, in the case of y-vibrations, between the K = 4 two-phonon and the K = 2 one-phonon bandheads. The best derivation for the prescription, which uses only mean-field wave functions, is provided by the self-consistent cranking + RPA (SCC+RPA) method [ 411 briefly discussed below. The calculation of static moments has never presented a problem for the cranking model-one need only calculate the expectation values with respect to the cranked wave function. Of course, there are many possible applications, not the least of which is the old problem of accounting theoretically for the sizeable quadrupole moments of spherical nuclei measured by the reorientation effect, which, to my knowledge, has never been satisfactorily resolved in a microscopic theory. 5.7. Pairing vibrations with non-zero spin Although the CBT applies to all spherical nuclei, one does not expect satisfactory results using the cranked HFB approximation for a nucleus with two particles or two holes outside of a closed shell for the simple reason that the BCS or HFB approximations are not very accurate in this case. However, the levels in such a nucleus may be regarded as pairing vibrations, i.e. pair-transfer modes built on the adjacent closed-shell system. Thus, for example, the levels in Te or Cd isotopes can be regarded as pairing vibrations built on a suitable Sn isotope. If the pairing vibration has non-zero angular momentum, such as the first 2+ level of, for example l14Cd, it can be obtained by cranking ‘14Sn. This may provide a new approach for computing properties of nuclei with two particles or
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holes added to a closed shell. Of course, one can continue the cranking trajectory with f2 pairs added to the closed shell, etc.
to reach 4+ states
5.8. Further rejinements The cranking description of vibrations has two major drawbacks. First, while anharmonicities of all orders can be summed, the approximation is basically a classical one with a superimposed BohrSommerfeld quantization condition on the angular momentum. Lacking are certain quanta1 fluctutation corrections that cannot be incorporated in the quantization condition. Second, the cranking formalism only describes levels that are harmonics of the same fundamental mode. For example, to return to the paradigm of the y-vibration, the cranking trajectory covers the one-phonon states B12 IO),9 with I = (KI = 2, and the aligned two-phonon states(Bi,)2]0), with I = JKI = 4, but does not cover the mixed two-phonon state BiBI, IO), with I = ]KJ = 0, nor, for that matter, totally unrelated modes such as P-vibrations. The possibility exists to compensate for both shortcomings by going beyond the cranked mean field approximation to include at least the quantized RPA fluctuations about the steady rotation. This is the SCC+RPA mentioned before [41]. While the discussion there centers about normal high-spin states, there is no reason why the formalism should not also apply to cranked vibrations. It has the following advantages. First, it gives rise to zero-point correlation corrections along the cranked trajectory. Second, states that do not lie on the cranked trajectory may possibly be reached by applying the SCC+RPA phonon operators to the correlated vacuum state; i.e. the cranked trajectory with correlations, which acts as the reference. Third, the SCCtRPA provides a well-defined formalism for deriving transition matrix elements that goes beyond the cranked meanfield approximation. By using boson expansions about the cranked mean-field solution, it is possible in principle to include higher-order quanta1 correlation effects not included in the SCC+RPA. So far, this idea has been tested only in the U(5) limit of the IBM [ 71.
Acknowledgements The author thanks A. Aprahamian and R. Nazmitdinov was supported in part by DOE grant DE-FG02-91ER40640.
for stimulating
conversations.
This work
References 111 S.T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1959) 131. 121 E.R. Marshalek, Lawrence Berkeley Laboratory report, UCRL-10046 ( 1962). [31 S.T. Belyaev and V.G. Zelevinsky, Nucl. Phys. 39 ( 1962) 582. 141 A. Klein and E.R. Marshalek, Rev. Mod. Phys.63 (1991) 375. [51 X. Wu, A. Aprahamian, S.M. Fischer, W. Reviol, G. Liu and J.X. Saladin, Phys. Rev. C 49 ( 1994) 1837; A. Aprahamian, X. Wu, S. Fischer, W. Reviol and J.X. Saladin, Rev. Mex. Fis. 39, suppl. 2 ( 1993) 1; H.G. Bomer et al., Phys. Rev. Lett. 66 (1991) 691. [61 M. Piiparinen et al., Z. Phys. A 337 (1990) 387; H.J. Wollersheim et al., Z. Phys. A341 (1992) 137; ’ Of course, signature symmetry
requires that the physical state be a linear combination
of the states with K = f2.
E.R. Marshalek/Physics
[7]
[S] [9]
[lo] [ 1l] [ 121 [ 131 [ 141 [ 151
[ 161 [ 171 [ 181
[ 191 [20] [21] [22] [23] [24] [25] [26] [ 271 [28] [29] [30] [31] [32] [33] [34] [ 351 [36] [37] [38] [ 391 [40] [41]
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R.F. Casten et al., Phys. Lett. B 297 (1992) 19; M. Deleze et al., Nucl. Phys. A 554 (1993) 1; M. Piiparinen et al., Phys. Rev. Lett. 70 ( 1993) 150; J. Ritman et al., Phys. Rev. Lett. 70 (1993) 533; J.A. Scarpaci et al., Phys. Rev. Lett. 71 (1993) 3766; C. Siems et al., Phys. Lett. B 320 (1994) 1. E.R. Marshalek, Tipsy rotators, in: M. Vallieres and D.H. Feng, eds., Symposium on Contemporary Physics, Drexel University, 1991 (World Scientific, Singapore, 1993) 191. A. Arima and F. lachello, The Interacting Boson Model (Cambridge Univ. Press, Cambridge, 1987). G.D. Birkhoff, Dynamical Systems, Vol. 10 (Am. Math. Sot., New York, 1927). F.G. Gustavson, Astron. J. 71 (1966) 670. F. Sakata and T. Marumori, in: B.-L. Hao, ed., Directions in Chaos, Vol. 4 (World Scientific, Singapore, 1992) 459. E.R. Marshalek and J. Weneser, Phys. Rev. C 2 ( 1970) 1682. E.R. Marshalek and G. Holzwarth, Nucl. Phys. A 191 (1972) 438. J.P. Blaizot and E.R. Marshalek, Nucl. Phys. A 309 (1978) 422, 453. E.R. Marshalek, Nucl. Phys. A 559 (1993) 291~. C.G. Andersson and J. Krumlinde, Nucl. Phys. A 291 ( 1977) 21. D.R. B&s and C.Y. Chung, Nucl. Phys. 86 ( 1966) 581. S. Frauendorf and T. Bengtsson, in: Future Directions in Nuclear Physics with 4~ Gamma Detection Systems of the New Generation, Strasbourg, 1991, AIP Conference Proceedings 259, 223; S. Frauendorf and F.R. May, in: International Conference on Nuclear Structure at High Angular Momentum, Ottawa, May 1992, Vol. 2, (Chalk River Report AECL-10613, 1992) 177; S. Frauendorf, Nucl. Phys. A 557 (1993) 259~. T. Troudet and R. Arvieu, Ann. of Phys. (N.Y.) 134 (1981) 1. E.R. Marshalek, Phys. Rev. Lett. 51 (1983) 1534; Phys. Rev. C 29 (1984) 640; Phys. Lett. B 244 (1990) 1. T. Kishimoto et al., Prog. Theor. Phys., Suppl. No. 74-75 (1983) 170. H. Sakamoto and T. Kishimoto, Nucl. Phys. A 501 (1989) 205,242; H. Sakamoto, Nucl. Phys. A 557 (1993) 583~. E.R. Marshalek and R.G. Nazmitdinov, Phys. Lett. B 300 (1993) 199. E.R. Marshalek, R.G. Nazmitdinov and I. Ragnarsson, Izv. Akad. Nauk, Ser. Fiz. 57 ( 1993) 54. T. Suzuki and D.J. Rowe, Nucl. Phys. A 289 (1977) 461. S.E. Larsson, Phys. Scrip. 8 (1973) 17. H. Kurasawa, Prog. Theor. Phys. 64 ( 1980) 2055. A. Bohr and B. Mottelson, Nuclear Structure, Vol. 2 (W.A. Benjamin, Reading, 1975). T. Marumori, T. Maskawa, F. Sakata and A. Kuriyama, Prog. Theor. Phys. 64 ( 1980) 1294. E.R. Marshalek and J.O. Rasmussen, Nucl. Phys. 43 (1963) 438. D.J. Rowe, Nuclear Collective Motion (Methuen, London, 1970) Chs. 4, 16. I. Hamamoto, Nucl. Phys. A 271 (1976) 15. W. Nazarewicz and J. Dobaczewski, Phys. Rev. Lett. 68 (1992) 154. S. Mizutori, Y.R. Shimizu and K. Matsuyanagi, Prog. Theor. Phys. 83 ( 1990) 666; 85 ( 1991) 559; 86 ( 1991) 13 1. T. Nakatsukasa, S. Mizutori and K. Matsuyanagi, Prog. Theor. Phys. 87 (1992) 607. R. Nazmitdinov and S. Aberg, Phys. Lett. B 289 (1992) 238. S. Mizutori, T. Nakatsukasa, K. Arita, Y.R. Shimizu and K. Matsuyanagi, Nucl. Phys. A 557 (1993) 125~. R.F. Casten, N.V. Zamfir and D.S. Brenner, Phys. Rev. Lett. 71 (1993) 227. H. Ejiri et. al., J. Phys. Sot. Jpn. 24 ( 1968) 1189. E.R. Marshalek, Nuovo Cimento Lett. 4 ( 1972) 631. E.R. Marshalek, Nucl. Phys. A 275 (1977) 416.
__ tB
ELSEVIER
PHYSICS
REPORTS
Physics Reports 264 (1996) 279-295
Bifurcations in the cranking model E.R. Marshalek Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
Abstract An informal presentation is given of a new theorem associated with the self-consistent cranking method, dubbed the Cranking Bifurcation Theorem, which extends the range of the cranking technique to generate certain types of collective multiphonon anharmonic solutions of time-dependent mean-field equations. Applications to the cranked volume-conserving oscillator are offered as simple examples.
1. Introduction Before launching into my main topic, I would like to say that it is a pleasure and honor to participate in the Belyaev fest, especially since work of Professor Belyaev played an important role in my early career. While a graduate student, I avidly studied his famous paper on pairing correlations that was published in a Danish journal [ 11. The effort proved rewarding, enabling me to do my thesis calculations treating collective vibrations in deformed nuclei microscopically for the first time [ 21. Not long afterwards as a postdoc, I got the idea that mean-field theories should be treated as classical field theories subject to quantization. Since such field theories are non-linear, quantization appeared difficult. The key to the solution was provided by the seminal paper of Belyaev and Zelevinsky [ 31, which introduced the concept of boson mappings into nuclear physics and spawned the cascade of developments reviewed in a recent article [4]. The topic that will be discussed now is closely related to these developments. Recently, there has been a revival of interest in multiphonon excitations. For example, Aprahamian believes she has found evidence for at least 17 cases of two-phonon y-vibrations in rare-earth nuclei, many of which would have to be highly anharmonic, as judged by the ratio of energy spacings [ 51. In addition, there is growing experimental evidence for two-phonon octupole levels, and two-phonon giant dipole and quadrupole resonances in various spherical nuclei [ 61. Although many methods are available for describing anharmonic multiphonon excitations, including, for example, boson mappings [4], the approach that I will now describe, which is a technique for obtaining periodic solutions of time-dependent mean-field equations, is arguably the simplest and allows the summation of anharmonicities to all orders, albeit in a semiclassical approximation. The 0370-1573/96/$9.50 @ 1996 Elsevier Science B.V. All rights reserved SSDIO370-1573(95)00043-7
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main limitation of the method is that it can only be applied to nuclei having equilibrium mean-field solutions with an axis of rotational symmetry (spherical or axially symmetric deformed shapes)although these can correspond to excited as well as ground-state configurations-and the vibrations built on these configurations must project an angular momentum K # 0 on the equilibrium symmetry axis. This means that the method cannot describe monopole and /? quadrupole vibrations, for which K = 0, but it can describe, for example, y quadrupole (K = 2)) octupole (K = 1,2,3) and giant dipole (K = 1) vibrations. The physical picture of such modes as a surface wave-an instantaneous distortion rotating about the symmetry axis-suggests the possibility of a stationary description in a frame rotating with the wave. Such considerations then lead to the cranking model, but with cranking about the (equilibrium) symmetry axis. The normal cranking method is applied to nuclei having a broken rotational symmetry in the ground-state mean field. The cranking axis is chosen such that the system is deformed in any plane perpendicular to this axis. The technique then generates the usual low-lying rotational bands found in permanently deformed nuclei. On the other hand, the common wisdom is that cranking a mean field about a symmetry axis just carries it into itself, leading to nothing new, except possibly a redefinition of the Fermi surface (equivalent to introducing a “tilted Fermi surface”). However, I have found that this is not necessarily so-at certain critical cranking frequencies new solutions that break the symmetry bifurcate from the symmetric one. This has led me to formulate what I call the Cranking Bifurcation Theorem (CBT) , which may be stated as follows: - If a system of coupled anharmonic oscillators has an axially symmetric equilibrium configuration (vacuum), then self-consistent cranking about this axis yields families of symmetry-breaking solutions that bifurcate from the vacuum solution at the critical rotational frequencies L2, = wJK~, where We is the small-oscillation (RPA) frequency of any mode carrying Kp # 0 units of angular momentum along the symmetry axis. Each family describes the dynamics of the system on submanifolds of phase space characterized by a single non-vanishing action Nfi = (I - IO) / Kp, where I is the total angular momentum of the mode and I0 that of the vacuum. A heuristic proof will be given shortly. This theorem has a broad range of applications, encompassing time-dependent mean-field approximations such as the time-dependent Hartree-Fock (TDHF) and Hat-tree-Fock-Bogoliubov (TDHFB) approximations, phenomenological mean-field models such as the Nilsson-Strutinsky model, empirical collective models (Bohr-Mottelson, IBM, etc.) and liquiddrop models. The TDHF approximation, for example, is replaced by the static cranked Hartree-Fock (CHF) approximation. In lowest-order perturbation theory, one obtains the familiar RPA equations from a purely static cranking argument, as we shall see. In a non-perturbative treatment, one has to crank about a symmetry axis and search for bifurcations at the critical frequencies given by the theorem. This requires an RPA program to locate these frequencies and a CHF program to follow a given broken symmetry solution for cranking frequencies away from the critical value. In this way, one generates a solution that is a continuous function of the cranking frequency in some neighborhood of a bifurcation point. Of course, the physical solutions correspond to suitably quantized values of the angular momentum and describe multiphonon excitations of a mode with K # 0. The excitation phonon is no longer the RPA phonon away from the bifurcation point, but rather a non-linear phonon whose structure takes into account renormalizations accruing from the anharmonic mixing with other modes, including those with K = 0. If the CHF solution is obtained by exact diagonalization rather than perturbation theory in the neighborhood of a critical point, then large-amplitude vibrations can be described with no difficulty and without the need for an adiabatic approximation.
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The TDHF (or other time-dependent mean-field) solution can be generated afterwards from the CHF solution by transforming from the rotating to the laboratory frame. In most cases, however, this step is unnecessary.
2. A heuristic
proof of the CBT
Consider an even-even nucleus whose equilibrium shape is either spherical or axially deformed in some mean-field configuration (the “vacuum”) which is normally the ground state but may be an excited state as well. The physical picture of a collective vibration that projects K # 0 units of angular momentum along a symmetry axis corresponds to a rotating distortion, a surface wave traveling around this axis. Since such a mode is degenerate, it is possible to choose linear combinations of normal coordinates that describe a uniformly rotating wave. From a quanta1 viewpoint, these modes may be described in terms of boson creation operators BL carrying Kfi units of angular momentum along the symmetry axis. Since the cranking model is basically a classical approximation, these boson creation operators will be replaced by the corresponding c-number complex coordinates p; , (p, and i& then become canonically conjugate variables). These coordinates are sometimes called phasor variables. If H is the quanta1 Hamiltonian, then (H) ’ , will denote the classical Hamiltonian function of &, p,. The equilibrium points of the system relative to a frame rotating uniformly with angular velocity 0 about the 3-axis are solutions of the set of equations
(1) where
(ff’) = (fg - fi(J3), is the so-called Routhian, (_Z3)being the classical angular-momentum axis (3-axis) given by *
(2) component
along the symmetry
(3) The diagonal quadratic form for (.Z3) is, of course, a consequence of designating the 3-axis as the symmetry axis with the appropriate choice of phasor variables. The constant IO is the spin of the axially symmetric vacuum. For the ground state configuration lo = 0, but for an excited state vacuum, non-zero values of IO may occur, as, for example, in the case of an oblate rotational-band terminus configuration. This is illustrated in Fig. ( 1). If IO = 0, then the excitations occur in time-reversal conjugate pairs corresponding to & and p?, , with the sum in Eq. (3) understood to run over both time-reversal partners and K_, = -Kp. If Z, # 0, then the total spin of an excited mode is given by
’ One may regard (H) as the expectation value with respect to a coherent state. * For convenience, ti = 1 will be used throughout this paper.
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Fig. 1. The rotational motion of the bifurcating solution (solid ellipse) The rotations may be in the same (a, c) or opposite senses (b, d).
compared with that of the vacuum (dashed circle).
10 + Kp IP, 1’7 where Kp can be positive or negative. In this case, time-reversal symmetry is broken for a fixed choice of vacuum. ’ The difference between the two cases is illustrated in Fig. 1. For simplicity, let us first consider small oscillations (RPA) about the vacuum configuration. In that case, the classical Hamiltonian is
(H) = Wo) = Eo(Z0) + c
@/&P/l
7
(4)
/I where E. ( IO) is a constant, the vacuum energy, and /?t( has been chosen as a normal-mode with corresponding frequency wP. From Eq. (3)) the Routhian (2) becomes (H’) = Eo(Zo) - fiZo + c(wfi P
- &M;P,.
coordinate
(5)
Then, Eqs. ( 1) imply P,(%
-
OK,) =O.
(6)
In general, this set of equations has two kinds of solutions: the trivial solution in which all p, = 0, and non-trivial solutions in which a single coordinate p, # 0, while p, = 0 for ,u # ,uo. 4 However, the non-trivial solutions are possible only for a discrete set of cranking frequencies given by L!=L& G
wcL/Kp,
Kp # 0.
(7)
should be noted that there are no non-trivial solutions for those modes with Kp = 0.The trivial solution just corresponds to the axially symmetric vacuum (unproductive cranking), while the nontrivial solutions are deformed about the 3-axis, the magnitude of the deformation depending on p,,. It
s However, one can also study excitations built on the time-reversed vacuum having spin -Ia to obtain the time-reversed states, as shown in Fig. 1. Thus, while (a) and (b) do not correspond to time-reversed states, (c) and (d), which are excitations built on the time reversed vacuum, are the time reverses of (a) and (b), respectively. 4 For simplicity, we do not discuss the resonant case in which some of the frequencies wfi are commensurate.
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283
Now, Eq. (6) does not determine pm, which reflects the independence of the frequency and amplitude for a harmonic oscillator. However, \p, 1 can be fixed via Eq. (3) by prescribing a value (.Z3) = I, where Z takes on suitably quantized values, which leads to N,
= &@I2 = (I - lo)&.
(8)
Of course, the sign of Z must be chosen taking into regard that of Km so that the r.h.s. of (8) is positive. The total energy (lab frame!) for this solution is then given by E = (H) = Eo(Zo) + ww IP,l* = Eo(Zo) + wti (Z - Zo) /K,.
(9)
For example, for quadmpole vibrations about a spherical ground state, one has Z. = 0 and the cranking procedure engages only those modes with 1Km 1= 1,2. For 1KM 1= 2, the excitation spectrum is given by AE = iwmZ = wMn, where n is the number of phonons. Therefore, the cranking solution selects the sequence of aligned multiphonon excitations with (quantized) spins Z = 2n = 0,2,4, . . ., which, for the lowest frequency We, are often yrast states, sometimes referred to as quasirotational states. For IKMI = 1, the excitation spectrum is given by AE = w,Z = oMn, corresponding to multiphonon states with spins Z = n = 2,3,4,. . . As shown explicitly in Ref. [7] for the U(5) limit of the interacting boson model (IBM) [ 81, such solutions correspond to rotation about a tilted axis rather than a principal axis of the density distribution, thereby breaking signature symmetry. Next, consider the more general situation in which (H) contains anharmonic corrections to the small-oscillation part in the form of an arbitrary, possibly infinite polynomial in the coordinates which may be the result of a Taylor expansion about the equilibrium point (vacuum). Ply PP The argument is greatly simplified if one chooses the coordinates such that (H) is given by the Birkhoff-Gustavson (BG) normal form [ 9,101. In this connection, one should distinguish between the non-resonant and the resonant cases. In the non-resonant case, the small-oscillation frequencies are either incommensurate or else the commensurability is the consequence of a global symmetry that forbids resonant coupling terms. In that case, as proven by Birkhoff [ 91, the Hamiltonian can be transmuted by a series of canonical transformations into an infinite polynomial in the action variables N, defined by Np = L’;Pp 3
(10)
with the conjugate angle variables being cyclic 5 . With the assumption that the cooordinates p;, ,8, have already been chosen as the transformed variables, the most general Birkhoff normal form for (H) is given by
where the numerical coefficients ZX(*~) ( ,LL~,cL*, . . . , ,x,J are completely symmetric in the indices and the NP constitute a formal set of constants of motion that are in involution. The resonant case, which 5 The obvious quantum analog is the perturbative diagonalization of a boson Hamiltonian by successive unitary transformations to provide a function of the boson number operators. This has been called the diagonal boson representation.
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is analogous to degenerate perturbation theory in quantum mechanics [ Ill, is too complicated to deal with here. From Eqs. ( 1) -( 3), ( 11)) the stationary points in the rotating frame are obtained as solutions of the set of equations
Sin_
1
I)! ~ ,,,,plh(2~)(ll,~i,.‘.~,-l)Ni(l.‘~NIL.I c
+...
1 =*.
In this case, as in the harmonic limit, one still has the trivial solution in which every p, = 0 and, in addition, there are the non-trivial solutions given by p, = 6,,,p,. Thus, equating to zero the corresponding term in brackets in Eqs. (12) with NP = NMSP,+,, gives the following series in N,: 0 - 0, = hC4’(,u~,..Q,) Ki’&
1 + . . . + cn _ Ij!h”“‘(~~.
. . +uo>K~‘(~,,)“-
+ ... 7
(13)
which can be reverted to give a series solution for NM in powers of L’- L$. 6 Hence, in the presence of anharmonicity, a non-trivial solution can exist for a continuous range of rotational frequencies 0 in some neighborhood of each critical frequency L& given by Eq. (7), and these critical frequencies are points at which an anharmonic branch bifurcates from the vacuum solution. It is also interesting to note that if 0, > 0, which is often the case, and hc4) (,LQJ.L~) < 0 while the higher-orders are small, then the solution “backbends” in some neighborhood of L?,, i.e. it exists only for 0 < 0,. The energy corresponding to such an anharmonic branch may be obtained trivially by substituting in Eq. ( 1 I), which merely limits the dynamics to a particular submanifold the solution NP = N,6,,, in phase space, ipso facto proving that the cranking technique gives the correct classical energy. When the action variable is expressed in terms of the prescribed angular momentum through Eq. (8), the energy takes the form of an expansion in powers of I - lo:
E= (H) = Eo(Zo) fw,
+...
(14) Then, Eq. (13) just becomes equivalent to the familiar Hamilton relation
This completes the heuristic argument. It should be emphasized that the BG normal form was introduced here solely to facilitate the justification of the cranking technique, not as an end in itself. It should be obvious from the foregoing that if the normal form couId be calculated in nuclei, then cranking would become superfluous. ‘This implies for pw, an expansion
in powers of (a - 0,)“‘.
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To see the relevance of the foregoing argument to nuclei, one should first consider how canonical coordinates can be introduced into nuclear mean fields. The most straightforward route is to first perform a generalized Holstein-Primakoff (GHP) boson mapping of the nuclear many-body system [4]. Then, one may appeal to a theorem [ 12-141 according to which the replacement of the boson operators by classical c-numbers 7 results in a canonical parametrization of the one-body density matrix. Depending on the system and the type of GHP mapping chosen, one can obtain in this way any of the standard TDMF approximations, with the Hamiltonian (energy) being a functional of the c-number canonical variables (which may be identified with the original /?t, p,). While this can be the starting point for calculating a BG normal form, it is generally preferable to work with the standard forms of cranked mean-field equations, using, for example, the non-canonical density matrix elements themselves in the variational procedure.
3. Static cranking
derivation
of the RPA
The foregoing discussion is sufficiently general to encompass a wide variety of Hamiltonian dynamical systems, including nuclei in a mean-field approximation. A simple microscopic example of the bifurcation theorem is afforded by the derivation of the RPA eigenfrequency equation for a separable multipole-multipole interaction of rank L using the static cranking method (a slightly less general derivation has been given in [ 1.51) . This can also be regarded as an independent method for proving the theorem for mean-field approximations, which is easily generalized to arbitrary interactions. The many-body cranked Hartree mean-field Hamiltonian ZZMFis then given by
HMF
= Ho
-
0J3 - x c
sQ;&
(16)
,
K=-L
where Ho is the uncranked axially symmetric mean-field Hamiltonian, the operator QK is a component of an L-pole tensor, SQ; is the change in the corresponding deformation parameter, and x is the strength of the interaction. Since HO and the angular-momentum component J3 are assumed to commute, HO - 0J3 may be chosen as the zeroth-order Hamiltonian and the rest of Eq. ( 16) treated as a perturbation. With IO) denoting the unperturbed axially symmetric configuration (not necessarily the ground state) and In) the orthogonal excited (or de-excited) configurations, we have that HoJO) = &IO), HoIn) = &In). In addition, one has J310) = ZolO) 3
J3b)
= &b)>
(17)
where it is assumed for generality that the vacuum IO) carries angular momentum eigenvector of H MF to first order in the 8Q; is given by
IO. The perturbed
(18)
’ Of course, one must choose the proper ordering of operators prior to the c-number
replacement.
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The first-order changes_ S&K in the deformation parameters are determined conditions SQK = (PcJQF#~) - SK,aQO,where Q0 E (O]&,]O),* as follows:
by the self-consistency
(19) It is immediately seen that there is a trivial solution with all the 6QK = 0, for which /PO) = IO), and non-trivial solutions with one pair SQK = (- l)“SQT, # 0, for which the corresponding term in brackets vanishes, thereby giving rise to the eigenfrequency formula
+c n+O
I<4&-KlO> I2 1 E,,-Eo+w
(20)
=;’
provided that the cranking frequency is given by L2 = L$ = W/K and K # 0. A second equation is obtained by letting 0 ---f --a and thus w --f --w. This pair of equations correspond to the what was called the RPA for “non-Hermitian fields” [ 161. Equations of this type were first derived for odd-A nuclei by Bes and Chung [ 171, which shows that the present cranking approach applies to odd nuclei as well as even-even ones. If IO) is an excited configuration, one must consider frequencies of both signs, with negative values of o corresponding to de-excitations. Moreover, as long as 1, # 0, there is no symmetry for modes with dl = +K and dl = -K. If, however, Z. = 0, the pair of equations collapses to the single well-known RPA dispersion formula c
l(n]&(Q,
+ (-l)KQ-~)]O)12~E, (En - Eo)2 - io2
n#O
- Eo)
=-
1
for which the modes with fK are degenerate. To determine the lowest-order for the mode in question, one must prescribe the spin using the condition VOJ
33 ]Po)
(21)
2x’ deformation
change 8QK
= 1.
(22)
Actually, this fixes lsQK] as follows: (23) where BK( w) is an RF’A mass parameter, which will not be given here explicitly. What this result shows is that the deformation associated with the symmetry breaking is proportional to the vibrational amplitude. The perturbative approach can be carried to higher orders for values of 0 # L$, by using an expansion in powers of ID - L&I ‘I’, where the square root may be regarded as a critical exponent. In this way, one generates the Taylor expansion equivalent to the BG expansion discussed earlier. However, as already emphasized, the primary aim here is not to use perturbation theory but to diagonalize the cranked mean field directly, thereby summing the anharmonicities. It should also be clear that the method just outlined applies not only to spherical nuclei, but also to axially symmetric deformed nuclei, an application never considered before. A second point to be emphasized is that * The condition
of axial symmetry does not require that Qo vanish, although it may do so for a spherical nucleus.
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287
since the 8QK do not vanish for a non-trivial solution, not only is the axial symmetry broken, but the cranking axis may not coincide with a principal axis of the density distribution. Therefore, some of the solutions can describe tilted rotation, which involves broken signature symmetry as well. This is particularly obvious for the case of quadrupole deformation (L = 2), where the ]K] = 1 mode corresponds to a tilted ellipsoid since it implies the existence of Y2i and Y2_, components in the mean field. Unfortunately, time constraints do not permit us to consider the topic of tilted rotation, which is a subject of growing interest [ 181.
4. Example:
The anisotropic
oscillator shell model
As a soluble example, one may consider the volume-conserving anisotropic harmonic oscillator (VCHO) model. This is just the model of nucleons in a pure deformed harmonic oscillator whose equipotential volumes are independent of deformation. The cranked version of this model as applied to conventional rotational bands has been discussed by numerous authors, but perhaps most exhaustively by Troudet and Arvieu [ 191. It has been shown that the VCHO can be interpreted as the Hartree approximation applied to a special internucleon interaction that includes separable many-body components in addition to the quadrupole-quadrupole interaction [ 20-221. With the the aid of the CBT, new solutions of the cranked VCHO have recently been found [ 15,23,24]. We first consider solutions describing multiphonon giant isoscalar y (K = 2) vibrations built on the IO = 0 ground state of an axially symmetric system. In Fig. 2, the angular momentum (denoted there by (L3)) is plotted as a function of the cranking frequency, with scaling chosen to give a universal plot [ ifI E _Z(nl + i) involving a sum over quanta perpendicular to the symmetry axis for the orbitals occupied in the ground state]. One sees that the solution, which happens to backbend, bifurcates at the critical frequency 0, = wRPA/2, where wRPAE awl is precisely the well known RPA frequency for this mode [25], WI G w, = wy being the common frequency of the deformed oscillator potential perpendicular to the symmetry axis in the ground state. This result is in accord with the CBT. It is also interesting to note that the moment of inertia for this kind of rotation is always less than the rigid-body value, although it approaches this value for high angular momentum. In the case of normal rotation in this model, the moment of inertia has exactly the rigid-body value in the limit of low angular momentum. In Fig. 3, the excitation energy is plotted as a function of the angular momentum; the deviation from a straight line (which represents the RPA) is a measure of the anharmonicity. This may seem large at first sight, but taking into account the scaling, one realizes that, except for very light nuclei, the anharmonicity is miniscule for states with a.few phonons, which is in line with recent data on giant resonances. For an even-even nucleus, the physical states correspond to aligned y-vibrational bandheads having (L3) = K = 0,2,4, . . . In Fig. (4), one sees a typical total energy surface as a function of the deformation parameters .szOand ~22 [ 261 for a non-zero angular momentum (I = 10). The twin degenerate minima just correspond to two possible elliptic deformations at an angle of 7r/2; i.e. the semimajor axis can lie along the l-axis or the a-axis. For Z = 0, there is only one minimum with E22 = 0, corresponding to axial symmetry. For any I # 0, the twin minima with broken axial symmetry appear, while the axial shape corresponds to a point on the crest. This is an example of a pitchfork bifurcation. Finally, the evolution of the shape for this solution as a function of L?/L$ is shown in Fig. 5, assuming that the ground state is a superdeformed prolate spheroid with an axis ratio of 2. It should be noted that for the VCHO
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0
20
40
80
100
C,,/(h?J Fig. 2. A scaled universal plot of the angular momentum about an equilibrium symmetry axis. Fig. 3. A scaled universal plot of the excitation about an equilibrium symmetry axis.
vs the rotational
frequency for the anisotropic
energy vs. the angular momentum
I
=
Fig. 4. The total energy surface as a function of the deformation about an equilibrium symmetry axis.
for the anisotropic
oscillator
cranked
oscillator
cranked
10
parameters
for the anisotropic
oscillator potential cranked
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289
8
2 0
-0.0
0.2
0.4
0.6
0.8
1.0
Q/R, Fig. 5. The evolution of the nuclear shape as a function of the rotational frequency cranked about an equilibrium symmetry axis.
potential,
wi/wi
for the anisotropic
oscillator
potential
= (Xi) /R2 which defines the square of a semiaxis of the density distribution, with 1/3. One sees that along the backbending curve the initial cigar shape with the R2 = ((X:)(X;)(X:>) 3-axis of symmetry evolves through triaxial shapes into an infinitely long needle oriented along the 2-axis as R -+ 0 and the angular momentum (L3) + 03, in the interim passing through an oblate shape with the l-axis of symmetry. Of course, this corresponds to the minimum on the right-hand side of Fig. 4; the twin solution ends up as a needle aligned along the 2-axis. An example of multiphonon vibrations built on an axially symmetric excited configuration with I0 # 0 is presented in Figs. 6 and (7) in which the angular momentum is labeled by (.I,) = I. The vacuum configuration is the well-known ground-band termination state of *ONe with spin IO = 8. The RPA excitations built on this state were studied in detail some time ago within the framework of the VCHO by Kurasawa [27], who found that there are three modes with AZ = K = f2. As shown in Fig. 6, which is a spin vs. rotational frequency plot, cranking of the IO = 8 configuration about the symmetry axis gives rise to three bifurcations (solid lines) at the cranking frequencies w/2, where the values of o agree exactly in magnitude with the RPA frequencies. Also shown are the degenerate time-reversed solutions (dashed lines) obtained by cranking the time-reversed configuration with Z. = -8, with all rotational frequencies reversed in sign. The segment marked a represents a multiphonon anharmonic excitation branch with AZ = K = 2, which corresponds to the mode type marked (a) in Fig. 1. The segment marked b represents a multiphonon anharmonic excitation branch with AZ = K = -2, which corresponds to the mode type marked (b) in Fig. 1. The segment designated by g.b. is formally a AZ = K = -2 de-excitation (negative-frequency) branch, but this mode, which is a “vibration” seen from the perspective of the IO = 8 state, is really the ground rotational band, as already noted by Kurasawa in his RPA analysis. The time-reversed solutions are physically equivalent and give rise to the same energy vs. angular momentum plot Fig. 7, as long as one uses 111, which is the observed spin. Equivalently, one may always follow only the solid trajectories as long as the b trajectory satisfies (.Zx) = Z > 0, but when (Jx) turns negative, switching to the corresponding portion of the time-reversed trajectory that has (Jx) > 0, which gives rise to
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x
\
Reports 264 (1996) 279-295
\
__,~__.____________!,____.
I,=-8
5
10
15
20
25
I Fig. 6. Bifurcating solutions based on the band termination state. The solid lines emanate from the configuration units of spin, while the dashed lines are the time-reverse solutions. Fig. 7. The excitation energy vs. spin corresponding
with 8
to the solutions in Fig. 6.
the abrupt change in Fig. 7. The forking structure is also quite interesting and may be a signature for band termination. The reader may be curious about the relation of the signs of the bifurcation frequencies in Fig. 6 to whether a mode is an excitation or de-excitation. This may be answered as follows. Let w,, wb, and -tigb denote the actual RPA frequencies, where the first two are positive, being excitation modes and the last negative, being a de-excitation mode built on the ZO= 8 state. In the neighborhood of a bifurcation point, it is sufficient to use the following classical cranked RPA Hamiltonian corresponding to Eq. (5)
(ff) = Eo(Z0) -
f2zo+ ( @a-
2a)s%h +
(wb
+ 2fi>p,'/-?b + (-@gb+ 2fi)&&,
(24)
which takes into account whether AZ = K = &2. The condition (6) then gives the bifurcation points occurring at 0 = wa/2, -@b/2, and @@/2, in agreement with Fig. (6). It should be noted that Troudet and Arvieu [ 191 had found the trajectory a (but not b), but had no way of realizing that it could be interpreted as a vibrational band built on the band termination state. There also exist AZ = K = f 1 modes that can be found with the aid of the CBT, but these will be discussed in a more comprehensive paper. It is worthwhile mentioning again that the CBT is applicable not only to microscopic mean fields, but also to phenomenological collective models, such as the Bohr-Mottelson model [28] and the IBM [ 81. Since these models are bosonized from the outset, the classical approximation is obtained by simply replacing the bosons by c-numbers in the Hamiltonian, or, what is essentially equivalent, averaging in a coherent state, and then invoking Eq. (1). Examples of such applications to the IBM are given in Refs. [7,23] _
E. R. Marshalek/ Physics Reports 264 (1996) 279-295
291
5. Some problems to which the CBT can be applied 5.1. How repeatable is a phonon? If an excitation mode has a pure particle-hole or two-quasiparticle nature, a double excitation is ruled out by the Pauli principle. In order for a “phonon” to be repeatable, it is obviously necessary that it be a collective mode. A fundamental question that this line of inquiry might be able to answer is: how many times can a given phonon excitation be repeated? Of course, the answer depends in part on the definition of the phonon, which may explain the conflicting answers given in the literature. It appears that the best and ultimate definition is the non-linear phonon of the diagonal boson representation used in the BG representation. An equivalent definition can also be obtained from the self-consistent coordinate method (SCCM), which is very popular in Japan [ 291. From the viewpoint of the BG representation, there is no clear upper limit on the number N, of times that a given phonon may be repeated. This is because the treatment there is based on a Taylor expansion, which only guarantees the existence of a solution in a fuzzy neighborhood of the critical point. A clear answer may possibly emerge from an exact, i.e. non-perturbative solution of the non-linear cranking equations. It may then be possible, for example, for a solution to disappear for some N, (or Z) , thereby signaling a cutoff on repeatability. 5.2.
Development
of realistic codes
While most of the applications of the CBT thus far have concentrated on simple models in order to gain insights into the nature of the solutions, realistic codes can certainly be developed. These can be either fully self-consistent HFB or CHB codes or they can be based on phenomenological deformed potential models, such as the Nilsson or Saxon-Woods potentials, with the inclusion of pairing correlations. While standard codes designed for normal cranking can be trivially modified for cranking about a symmetry axis, this is only the first step. At least an RPA code is also required to locate the bifurcation points. Even better would be a code that includes the lowest-order perturbation corrections to the RPA, equivalent to the fourth-order BG approximation, which would provide a good starting guess for the main code. In the case of phenomenological deformed potential models, I have been able to show that the RPA limit of the cranking equations exactly corresponds to the result of the vibrating potential model (VPM) [ 30,311. Once a bifurcation associated with a particular RPA root (usually the lowest) is located, the corresponding solution must be followed as a function of the rotational frequency, or, preferably, the spin. Since for a given spin there may be a large number of self-consistent solutions, say, one for each RPA root, there is the problem of ascertaining that the desired solution is actually being followed. Some of the problems that occur in the cranking model for normal rotational bands may also come to haunt the application to vibrations in some nuclei. This includes well-known difficulties associated with level crossings [ 321 and with the phase-transitional collapse of pairing. While many of the interesting applications, such as to multiphonon y-vibrations, involve low spins, it should be noted that the cranking frequencies are relatively high, making pairing collapse more likely. While on the one hand the same prescriptions used for normal rotation can be applied in such situations, for example, using particle-number projection to deal with a pairing collapse, on the other hand, the anomalies themselves may represent valid physical mechanisms for terminating a vibrational band. A
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pairing collapse in such a situation might be interpreted as the dissolution of a vibrational mode by virtue of mixing with a large number of quasiparticle excitations that block the pairing correlations. Should this happen for a boson number NP < 2, one could conclude that the mode in question is non-repeatable. 5.3. Octupole excitations
in superdeformed
nuclei
Since the supershells in superdeformed nuclei are composed of approximately equal numbers of single-particle levels of opposite parity, octupole correlations are expected to be very important [ 331. Numerous calculations have been carried out for these nuclei, using the stretched octupole-octupole interaction [ 211 and either a pure deformed harmonic-oscillator potential [ 34-361 or a more realistic Nilsson potential with pairing [37]. In all cases, the RPA was used. However, since some of the octupole modes are expected to be quite soft, it is very important that anharmonic corrections be studied. The cranking technique described here seems ideally suited for this purpose as long as one is content to study the ]K( = 1,2,3 octupole modes. Unfortunately, the K = 0 mode is not directly accessible in this way. 5.4. Is there a universal anharrnonic vibrator? Recently, Casten, Zamfir and Brenner “universal” quadrupole vibrator formula E(Z) = nE(2,+)
on the basis of energy
systematics
proposed
+ +z(lz - 1)&d
for the aligned n phonon Ejiri [39] formula E(I)
[38]
levels with spins I = 2n, which is phenomenologically
=aZ+bl(Z+l).
the
(25) equivalent
to the
(26)
What is remarkable is not the expressions themselves, which have been known for a long time, but rather that the anharmonicity parameter c4 or b is nearly constant in each of three broad regions of the nuclear periodic table covering all even-even nuclei with 2 2 38 that are not well-deformed. With I0 = 0 for the ground state and KM = 2, this is essentially a truncation of Eq. (14) with E4 = h c4)( ,uO, ,uO). This indicates that the present formalism applied perturbatively to nuclei with spherical ground states can be used to calculate the parameter ~~ or b and test whether a microscopic theory can predict a universal value. In fact, I effectively obtained expressions for this parameter within the framework of the pairing plus quadrupole-quadrupole force model 22 years ago using an early precursor of the present formalism [40]. The only numerical calculations were in the vicinity of single-closed shell nuclei, then yielding values with the opposite sign of the universal ~4. It certainly would be worthwhile doing such calculations again over a broader region of nuclei and with improved forces. 5.5. Vibrations in odd-A nuclei The CBT may provide a new approach to the old subject of the coupling of odd nucleons to collective vibrations. The (uncranked) vacuum state should be chosen as the equilibrium solution
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including the odd particle so that It, # 0. If this is axially symmetric then the CBT can be applied. One may then anticipate two types of bifurcations: those that correspond to the scattering of the odd nucleon and those that correspond to excitations from the even-even core. The former take into account the effect on the single-particle (quasiparticle) energies of the coupling to the collective vibrations, while the latter includes the collective vibrations themselves. Since the vacuum state is at least two-fold degenerate, one must, of course, make a particular choice. For example, in a deformed nucleus one may choose the vacuum ]!Po) = LZ;IO), where 10) is the mean-field state of the even-even core and LZ;creates the odd nucleon, with angular momentum I0 = Kp. Corresponding to a collective excitation projecting K units of angular momentum on the symmetry axis in the even-even nucleus, one expects two modes in the odd nucleus that project K f Kp units of angular momentum for a onephonon excitation. These modes are not expected to be degenerate since in one case the nucleon is rotating in the same sense as the collective wave, while in the other case it is rotating in the opposite sense. Of course, if the vacuum is chosen instead as the time-reversed state IPi) = TuL]O), then I0 = -Kp, but the physics is the same, only the sense of rotations is reversed. The physical picture is basically that shown in Fig. 1. As mentioned earlier, in the RPA limit, one recovers equations of the type used long ago by B&s and Chung [ 171 for describing y-vibrations in odd-A deformed nuclei, which certainly deserved more study. The proposal here goes much further since the effects of anharmonicity to all orders can be included in the cranking approach, which is not limited to deformed nuclei, but can also be applied to spherical nuclei. 5.6. Electromagnetic
transitions and moments
The discussion thus far has made no mention of the calculation of electromagnetic transitions and moments in the cranking approach. In fact, transition matrix elements can be calculated between levels lying along the same cranked trajectory, for example, in the case of y-vibrations, between the K = 4 two-phonon and the K = 2 one-phonon bandheads. The best derivation for the prescription, which uses only mean-field wave functions, is provided by the self-consistent cranking + RPA (SCC+RPA) method [ 411 briefly discussed below. The calculation of static moments has never presented a problem for the cranking model-one need only calculate the expectation values with respect to the cranked wave function. Of course, there are many possible applications, not the least of which is the old problem of accounting theoretically for the sizeable quadrupole moments of spherical nuclei measured by the reorientation effect, which, to my knowledge, has never been satisfactorily resolved in a microscopic theory. 5.7. Pairing vibrations with non-zero spin Although the CBT applies to all spherical nuclei, one does not expect satisfactory results using the cranked HFB approximation for a nucleus with two particles or two holes outside of a closed shell for the simple reason that the BCS or HFB approximations are not very accurate in this case. However, the levels in such a nucleus may be regarded as pairing vibrations, i.e. pair-transfer modes built on the adjacent closed-shell system. Thus, for example, the levels in Te or Cd isotopes can be regarded as pairing vibrations built on a suitable Sn isotope. If the pairing vibration has non-zero angular momentum, such as the first 2+ level of, for example l14Cd, it can be obtained by cranking ‘14Sn. This may provide a new approach for computing properties of nuclei with two particles or
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holes added to a closed shell. Of course, one can continue the cranking trajectory with f2 pairs added to the closed shell, etc.
to reach 4+ states
5.8. Further rejinements The cranking description of vibrations has two major drawbacks. First, while anharmonicities of all orders can be summed, the approximation is basically a classical one with a superimposed BohrSommerfeld quantization condition on the angular momentum. Lacking are certain quanta1 fluctutation corrections that cannot be incorporated in the quantization condition. Second, the cranking formalism only describes levels that are harmonics of the same fundamental mode. For example, to return to the paradigm of the y-vibration, the cranking trajectory covers the one-phonon states B12 IO),9 with I = (KI = 2, and the aligned two-phonon states(Bi,)2]0), with I = JKI = 4, but does not cover the mixed two-phonon state BiBI, IO), with I = ]KJ = 0, nor, for that matter, totally unrelated modes such as P-vibrations. The possibility exists to compensate for both shortcomings by going beyond the cranked mean field approximation to include at least the quantized RPA fluctuations about the steady rotation. This is the SCC+RPA mentioned before [41]. While the discussion there centers about normal high-spin states, there is no reason why the formalism should not also apply to cranked vibrations. It has the following advantages. First, it gives rise to zero-point correlation corrections along the cranked trajectory. Second, states that do not lie on the cranked trajectory may possibly be reached by applying the SCC+RPA phonon operators to the correlated vacuum state; i.e. the cranked trajectory with correlations, which acts as the reference. Third, the SCCtRPA provides a well-defined formalism for deriving transition matrix elements that goes beyond the cranked meanfield approximation. By using boson expansions about the cranked mean-field solution, it is possible in principle to include higher-order quanta1 correlation effects not included in the SCC+RPA. So far, this idea has been tested only in the U(5) limit of the IBM [ 71.
Acknowledgements The author thanks A. Aprahamian and R. Nazmitdinov was supported in part by DOE grant DE-FG02-91ER40640.
for stimulating
conversations.
This work
References 111 S.T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1959) 131. 121 E.R. Marshalek, Lawrence Berkeley Laboratory report, UCRL-10046 ( 1962). [31 S.T. Belyaev and V.G. Zelevinsky, Nucl. Phys. 39 ( 1962) 582. 141 A. Klein and E.R. Marshalek, Rev. Mod. Phys.63 (1991) 375. [51 X. Wu, A. Aprahamian, S.M. Fischer, W. Reviol, G. Liu and J.X. Saladin, Phys. Rev. C 49 ( 1994) 1837; A. Aprahamian, X. Wu, S. Fischer, W. Reviol and J.X. Saladin, Rev. Mex. Fis. 39, suppl. 2 ( 1993) 1; H.G. Bomer et al., Phys. Rev. Lett. 66 (1991) 691. [61 M. Piiparinen et al., Z. Phys. A 337 (1990) 387; H.J. Wollersheim et al., Z. Phys. A341 (1992) 137; ’ Of course, signature symmetry
requires that the physical state be a linear combination
of the states with K = f2.
E.R. Marshalek/Physics
[7]
[S] [9]
[lo] [ 1l] [ 121 [ 131 [ 141 [ 151
[ 161 [ 171 [ 181
[ 191 [20] [21] [22] [23] [24] [25] [26] [ 271 [28] [29] [30] [31] [32] [33] [34] [ 351 [36] [37] [38] [ 391 [40] [41]
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