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Derivation of microscopic unified Bohr–Mottelson rotational model
Nuclear Physics A 852 (2011) 109–126 www.elsevier.com/locate/nuclphysa
Derivation of microscopic unified Bohr–Mottelson rotational model P. Gulshani ∗ NUTECH Services, 3313 Fenwick Cres., Mississauga, Ontario, L5L 5N1, Canada Received 22 October 2010; accepted 13 January 2011 Available online 18 January 2011
Abstract In a previous article, we derived a microscopic version of the phenomenological Bohr–Mottelson unified rotational model for rotation about a single axis. In this article, we generalize the derivation to that for rotation about all the three axes. As in the previous derivation, we apply the nuclear Hamiltonian directly to the rotational-model wavefunction instead of using the usual canonical transformation. In this way, we avoid using redundant coordinates or imposing any constraints on the rotationally-invariant rotational-model intrinsic wavefunction. We show that, in the transformed nuclear Schrödinger equation, the Coriolis coupling term vanishes exactly only for a choice of the rotational-model Euler angles that is consistent with angleangular momentum commutation relation and rotational invariance of the intrinsic wavefunction. For this choice of the Euler angles, the kinematic moment-of-inertia tensor, collective-rotation velocity field, and flow vorticity have the rigid-flow characteristics. This quantum rigid flow reduces to irrotational free-vortex flow in the limit of a single particle. We derive a microscopic effective rotation-intrinsic unified Schrödinger equation for the states of a rotational band that reduces to the phenomenological, unified, tri-axial quantum rigid-rotor model in the limit that the off-diagonal elements of the kinematic inertia tensor operator can be neglected. The model derivation shows that a multi-fermion system with unpaired or paired (quasi) particles rotates rigidly and a single-particle system rotates irrotationally if the intrinsic system is rotationally invariant. © 2011 Elsevier B.V. All rights reserved. Keywords: Rotational model; Coriolis coupling; Rigid flow; Moment of inertia; Hydrodynamics; Velocity field; Vorticity; Rotational bands; Band mixing
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1. Introduction In a previous article [1], we derived a microscopic version of the Bohr–Mottelson rotational model [2–23] for rotation about a single axis. This was achieved in [1] by applying the nuclear Hamiltonian to the rotational-model wavefunction1 : Ψ = eilθ Φ(ξ ),
LΦ = 0,
Leilθ = h¯ leilθ ,
LΨ = h¯ lΨ
(1)
where the Euler angle θ defines the orientation (with respect to the rotation axis) the intrinsic reference-frame axes relative to the space-fixed frame, l = angular momentum quantum number along the rotation axis, and Φ is a rotationally invariant intrinsic wavefunction (i.e., LΦ = 0, refer to Eq. (1)). In this way, we avoided using redundant coordinates or imposing any constraints on the rotationally-invariant rotational-model intrinsic wavefunction. In Ref. [1], the Euler angle θ was chosen to be consistent with angle-angular momentum commutation relation, and with the rotational-model requirement that Φ be rotationally invariant. The result was the Schrödinger equation: N,3 h¯ 2 l 2 h¯ 2 ∂ 2 · +V + − Φ = EΦ (2) 2 2M 2Irig ∂xnj n,j =1
Because of above choice of θ , the kinematic moment of inertia Irig had the rigid-flow value: Irig ≡ M
A
rn2
(3)
n=1
and there are no terms in Eq. (2) that correspond to a coupling between the intrinsic and rotational motions (i.e., Coriolis coupling terms), which could modify Irig , although Irig is still subject to centrifugal stretching. As a consequence, Ref. [1] showed that the collective-rotation flow n had the rigid-flow characteristics: velocity Vn and vorticity Ω r2 n × Vnrot = 2ω n ≡ ∇ Vn = ω (4) × xn , Ω · 1− n Irig For a system of multi-fermions, the kinematic moment-of-inertia is Irig , with a rigid-flow expectation value if the particles were unpaired and a less-than rigid-flow value if the particles were paired due to a pairing force and fluctuations in Irig . For a single particle (i.e., A = 1) or for a Bose–Einstein condensate, Vn is clearly irrotational, and the above results reduce to those from the usual analysis of the motion of a single particle. The above results show that, for a uni-axial rotation, a multi-fermion system such as a deformed nucleus rotates rigidly if the intrinsic state is invariant under rotation. This uni-axial rigid-flow model is the microscopic counterpart of the Bohr–Mottelson unified uni-axial rotational model. The rigid flow is the only type of flow that is fully consistent with and within the scope of the Bohr–Mottelson rotational model [1]. Any other type of flow would impose additional constraints on the intrinsic state, and may introduce Coriolis coupling terms in Eq. (2), and hence would not be within the scope of the rotational model [1]. 1 This approach differs from the usual method of canonically transforming the particle coordinates to a set of collective and intrinsic coordinates.
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In this article, we generalize the above results (presented in more detail in [1]) to the case of rotation about all the three axes, and derive a microscopic version of the general tri-axial Bohr–Mottelson rotational model. In Section 2, we review briefly the usual method of canonical transformation of the particle coordinates to a set of Euler angles and intrinsic coordinates, and the resulting Coriolis coupling and moment of inertia (more detail and references to the related work by other authors are presented in Ref. [1] and will not be repeated in this article). Then we express the results of the canonical transformation in terms of the space-fixed particle coordinates. Section 3 generalizes the microscopic derivation the uni-axial unified rotational model given in [1] to that of a tri-axial rotational model. In Section 3.1, we use a rotation-intrinsic product wavefunction (implied by the canonical decomposition in Section 2) and apply to it the spacefixed nuclear Hamiltonian to derive an effective rotation-intrinsic Schrödinger equation. This result is compared with that obtained from the canonical transformation method in Section 2. In Section 3.2, we derive a condition for which the Coriolis-coupling term vanishes, and which is also consistent with the Euler angle–angular momentum commutation relations, and obtain the resulting rotation-intrinsic Schrödinger equation and moment of inertia. In Section 3.3, we derive a microscopic unified rotational model for a tri-axial nucleus and show that it reduces to the phenomenological unified rotor model if the inertia tensor is assumed to be diagonal. In Section 3.4, we examine the properties and solution of the microscopic unified tri-axial-nucleus Schrödinger equation and derive an expression for the inverse of rigid-flow inertia tensor in terms of the quadrupole moment tensor for use in a future application of the model. In Section 3.5, we examine the properties of the quantum flow obtained in Section 3.2. Section 4 concludes this article. 2. Canonical transformation of Schrödinger equation, Coriolis coupling and moment of inertia This section reviews briefly the usual method of canonical transformation used by a number of authors to describe microscopically nuclear collective rotational motion. This review provides the background material needed for the analysis presented in Section 3. More detail on the canonical transformation, references to related work by other authors, and difficulties encountered in this approach are presented in detail in Ref. [1] and will not be repeated in this article. 2.1. Review of usual canonical transformation The usual method of canonical transformation is to start from the transformation of particles coordinates xnj (n = 1, . . . , N , j = 1, 2, 3, where N = total number of particles) to the coordi (ξ ) (A = 1, 2, 3, σ = 1, . . . , 3N − 3) in a reference frame rotated through the three nates xnA σ Euler angles θs (s = 1, 2, 3) given by: (ξσ ) xnj = RAj (θs )xnA
(5)
where RAj is the Aj ’s component of the orthogonal 3 × 3 matrix R (in this article centre of mass motion is ignored for simplicity but can be easily included). Both θs and ξσ are functions are not all independent (i.e., some of them are of xnj to be defined below. The coordinates xnA redundant) because they must be independent of θs and hence must satisfy certain conditions. This redundancy introduces significant difficulties in calculations involving the intrinsic system. Applying the transformation in Eq. (5) to the N -particle Hamiltonian:
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H≡
N,3 1 2 1 pnj + 2M 2 n,j =1
N,3
V (xnj , xmk )
(6)
n,j,m,k=1
(where M is the particle mass, pnj is the j th component of nth particle linear momentum, and V is a rotationally-invariant two-body interaction) and after some lengthy algebraic manipulations, we decompose H into the following three parts [24,25]: H = Hint +
3
F A LA +
A=1
3 1 −1 IAB LA LB 2
(7)
A,B=1
where the intrinsic Hamiltonian Hint , which includes the nuclear potential, depends on the (3N − 3) coordinates ξσ and the derivatives ∂ξ∂σ . The three operators FA are functions of ξσ and ∂ξ∂σ , and LA are the three components of the angular momentum operator along the body-fixed axes (for simplicity, spin angular momentum is ignored in this article but can be readily included, −1 is the AB component of the inverse of the kinematic moment for example refer to [26]). IAB of inertia tensor I, which is a function of ξσ and is in general not diagonal. In Eq. (7), the second term is the Coriolis interaction, and the third term is the rotational kinetic energy. The −1 . The rotational kinetic energy term includes the effect of centrifugal stretching through IAB Coriolis-interaction term couples the intrinsic and rotational motions. This term complicates the computation of the eigenstates of the transformed Hamiltonian in Eq. (7). The functional form and strength of the Coriolis coupling operators FA and I in Eq. (7) depend on the choice of θs . It is therefore desirable to choose θs to eliminate the Coriolis coupling term, as was done in [1]. Other choices of θs are discussed in [1]. 2.2. Derivation of angular momentum–Euler angle relationship and microscopic expressions −1 for FA and IAB In this subsection, we express the angular momentum operators LA in Eq. (7) in terms of θs , −1 in Eq. (7) in terms of space-fixed particle coordinates and derive expressions for FA and IAB xnj . These results are used in Section 3. σ First, we express LA in terms of θs . Using Eq. (5) and noting that ∂ξ ∂θs = 0, we obtain (in this article, summation over repeated indices is assumed unless stated otherwise): ∂xnj ∂RAj ∂ ∂ ∂ = · = RBj xnA ∂θs ∂θs ∂xnj ∂θs ∂xnB −1 ∂RAj ∂ ∂ 1 ∂RAj ≡ = RBj xnA − xnB RBj LAB 2 ∂θs ∂xnB ∂xnA i h¯ ∂θs where we have used the fact that the matrix ∂ ∂ ≡ RBj ∂xnB ∂xnj
∂RAj ∂θs
(8)
RBj is anti-symmetric in indices A and B, and (9)
From Eq. (8) we readily derive the relation: −i h¯
∂ = λAs LA ∂θs
and its inverse relation:
(10)
P. Gulshani / Nuclear Physics A 852 (2011) 109–126 −1 LA = −i hλ ¯ sA
∂ ∂θs
113
(11)
where LA ≡ 12 εABC LAB , and the 3 × 3 matrix λ is defined by: λAs ≡
3 ∂RBj 1 εABC RCj 2 ∂θs
(12)
A,B=1
where εABC is the third rank antisymmetric Levi-Civita tensor. The inverse λ−1 of λ is defined by: δts = λ−1 tA λAs ,
δAB = λAs λ−1 sB
(13) ∂ ∂ξσ
Eq. (11) shows that LA is independent of ξσ and . From Eq. (11), we conclude that [Lj, ξσ ] = 0, i.e., ξσ is invariant under rotation and hence ξσ are functions of the invariants of the xn · xm |, | xn − xm |, etc. Therefore, group SO(3), such as | xn |, | ∂ ∂ Lj k ≡ −i h¯ xnj − xnk ∂xnk ∂xnj ∂ ∂ ∂θs ∂θs ∂ξσ ∂ξσ = −i h¯ xnj − xnk − i h¯ xnj − xnk ∂xnk ∂xnj ∂θs ∂xnk ∂xnj ∂ξσ ∂ ∂ ∂ = [Lj k , θs ] + [Lj k , ξσ ] = [Lj k , θs ] (14) ∂θs ∂ξσ ∂θs Therefore, we conclude that the intrinsic angular momentum operator [Lj k , ξσ ] ∂ξ∂σ vanishes naturally for any choice of θs due to the invariance of ξσ under rotation similar to that of the radial distance in a single-particle system [1]. It is informative to construct the matrix λ in Eq. (12). Using the orthogonal matrix R given in [27], we obtain: cos θ1 cos θ2 cos θ3 − sin θ1 sin θ3 − cos θ1 cos θ2 sin θ3 − sin θ1 cos θ3 cos θ1 sin θ2
R= ⎛
sin θ1 cos θ2 cos θ3 + cos θ1 sin θ3 − sin θ1 cos θ2 sin θ3 + cos θ1 cos θ3 sin θ1 sin θ2
(15)
⎞
− sin θ2 cos θ3 sin θ3 0 cos θ3 0 ⎠ λ = ⎝ sin θ2 sin θ3 0 1 cos θ2 ⎛ − cos θ3 / sin θ2 sin θ3 / sin θ2 cos θ3 sin θ3 λ−1 = ⎝ cot θ2 sin θ3 cot θ2 cos θ3
− sin θ2 cos θ3 sin θ2 sin θ3 cos θ2
⎞ 0 0⎠ 1
(16)
−1 in Eq. (7) in terms of space-fixed particle coordinates xnj . Next, we express FA and IAB Taking commutator of θs with the Hamiltonian H in Eq. (7) and using Eq. (11), we obtain:
1 −1 1 −1 LA [LB , θs ] + IAB [LA , θs ]LB [H, θs ] = FA [LA , θs ] + IAB 2 2 h¯ 2 −1 −1 ∂λ−1 −1 −1 −1 sB I λ = −i hλ F − i hI λ L − ¯ sA A ¯ AB sA B 2 AB tA ∂θt Multiplying Eq. (17) on the left by λAs and summing over s, we obtain:
(17)
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FA =
i i h¯ −1 −1 −1 ∂λAs −1 LB − ICB λtC λsB λAs [H, θs ] − IAB 2 ∂θt h¯
(18)
Taking commutator of the θt with the first line in Eq. (17), we obtain: −1 [LA , θt ][LB , θs ] [H, θs ], θt = IAB
(19)
Substituting Eq. (11) into Eq. (19), we obtain: −1 = IAB
1
[H, θs ], θt
(20) ¯ Substituting H in Eq. (6) into Eqs. (18) and (20), and assuming momentum independent interaction V , we obtain:
∂ ∂θs ∂θs ∂ i h¯ +2 λAs FA = − 2M ∂xnj ∂xnj ∂xnj ∂xnj ∂θs ∂θs ∂θt i h¯ ∂θt ∂λAt 1 λAs λBt · · LB − · · − M ∂xnj ∂xnj 2M ∂xnj ∂xnj ∂θs
∂θs ∂ ∂θs ∂θs i h¯ ∂ ∂θt 1 λAs λBt =− λAs + 2λAs · · LB (21) − 2M ∂xnj ∂xnj ∂xnj ∂xnj M ∂xnj ∂xnj λAs λBt h2
where, in the second line on the right-hand side of Eq. (21), we have used the chain rule: ∂λAt ∂θs
=
∂λAt ∂xnj
∂θs ∂xnj
·
. Eq. (21) is used in Section 3.
3. Derivation of microscopic unified rotational model and nature of rotational motion In this section, we derive a microscopic effective intrinsic-rotation multi-particle Schrödinger equation resembling that of the unified rotational model for a tri-axial nucleus. We show that the Coriolis-coupling terms in this equation are identical to that for Eqs. (7) and (21) when acting on the model wavefunction. The Coriolis-coupling terms are shown to vanish only for a rigid flow. We obtain the corresponding kinematic moment of inertia, and collective-rotation velocity field. We also show that the eigenfunctions of a tri-axial nucleus can be naturally expanded in terms of the eigenfunctions that resemble those for an axially symmetric nucleus. 3.1. Decomposition of Schrödinger equation into rotational and intrinsic parts For the rotation–intrinsic motion implied by the decomposition of the Hamiltonian in Eq. (7), we use a superposition of product wavefunction in the form: ΨLM =
L
L DMK (θs )ΦK (ξσ )
(22)
K=−L
The intrinsic wavefunction ΦK in Eq. (22) has the property: LA ΦK (ξσ ) = 0
(23)
Unlike the rotational-model wavefunction [2–23], the wavefunction ΨLM in Eq. (22) is not symmetrized because the approach that we are taking in this article does not in general admit the rotational-model symmetry as will be discussed in more detail below. Consistent with the decomposition of the angular momentum operator into rotational and intrinsic parts given in Section 2.2,
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we interpret the rotational-model condition in Eq. (23) to mean that (i) ΦK is a function of varixn · xm |, | xn − xm |, etc., and (ii) ΦK is a zero ables that are invariant under rotation such as | xn |, | angular momentum eigenstate. In deriving the microscopic unified intrinsic–rotation Schrödinger equation presented, we avoid using the unknown intrinsic coordinates ξσ , and express all quantities in terms of the space-fixed coordinates. To achieve this, we apply the original Hamiltonian in Eq. (6) to the wavefunction ΨLM in Eq. (22)2 and obtain: L DMK ΦK (E − V ) K
L L 2 ∂ 2 DMK h¯ 2 ∂ΦK ∂DMK L ∂ ΦK DMK =− +2 · + ΦK 2 2 2M ∂x nj ∂x nj ∂xnj ∂xnj
(24)
n,j,K
L Because DMK is a function of θs only, we obtain, using chain rule and Eq. (10), the result: L ∂DMK ∂θs ∂ L i ∂θs L = · DMK = λAs LA DMK ∂xnj ∂xnj ∂θs h¯ ∂xnj
(25)
Using Eq. (25), chain rule, and Eq. (10), we obtain: L ∂ 2 DMK ∂θs ∂θt i i ∂θs ∂ ∂ L L LA DMK = · λAs LA DMK + · λAs · 2 ∂x ∂x ∂x ∂x ∂θ h h ∂xnj ¯ ¯ nj nj nj nj t ∂θs 1 ∂θs ∂θt ∂ i L L λAs LA DMK − 2· · λAs λBt LB LA DMK (26) = · h¯ ∂xnj ∂xnj h¯ ∂xnj ∂xnj Substituting Eqs. (25) and (26) into Eq. (24), we obtain the effective intrinsic-rotation Schrödinger equation: i h¯ ∂ΦK ∂θs h¯ 2 ∂ 2 L L DMK − + V − E Φ − · λAs LA DMK 0= K 2 2M M ∂x ∂x ∂x nj nj nj K nj K ∂θs ∂ i h¯ L − ΦK λAs LA DMK 2M ∂xnj ∂xnj K ∂θs 1 ∂θt L + ΦK · λAs λBt LB LA DMK (27) 2M ∂xnj ∂xnj K
We now show that the second and third terms on the right-hand side of Eq. (27) are the same as the Coriolis interaction term FA LA in Eq. (7) acting on the wavefunction ΨLM in Eq. (22). Using [LA , FA ] = 0 (refer to Section 2.1) and Eqs. (21) and (23), we have (sum over A is assumed): L FA LA ΨLM = (FA ΦK ) LA DMK
∂θs i h¯ ∂ ∂θs ∂ΦK L LA DMK ΦK =− λAs + 2λAs · (28) 2M ∂xnj ∂xnj ∂xnj ∂xnj 2 This approach is reminiscent of the generator coordinate method of Peierls and Yoccoz [16,17,19,20,22,28–32]. However, in the Peierls–Yoccoz’s method the Euler angels are parameters and not dynamical variables and the intrinsic wavefunction is not rotationally invariant and is a function of the Euler angles. For this reason, projection to states of good angular momentum is necessary in Peierls–Yoccoz’s method.
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The right-hand side of Eq. (28) is seen to be identical to the sum of the second and third terms on the right-hand side of Eq. (27). Therefore, we conclude that the second and third terms on the right-hand side of Eq. (27) arise from the action of the Coriolis interaction term FA LA in Eq. (7), which couples the intrinsic and rotational motions. 3.2. Condition for vanishing of Coriolis interaction As noted in Section 2.1, it is desirable (as a step in solving the rotation–intrinsic Schrödinger Eq. (27) to choose the θs such that the Coriolis-coupling term FA ΦK in Eq. (27) or (28) vanishes, and hence to account completely for the effect of the Coriolis interaction on the rotational (i.e., I) and intrinsic motions. In the resulting equation, the intrinsic and rotational motions would be decoupled except for the centrifugal stretching effect in I, as in the single-particle motion [1]. Therefore, we want to choose θs such that the quantity FA ΦK in Eq. (28) vanishes, i.e., to satisfy3 : ∂θs ∂ ∂θs ∂ΦK λAs + 2λAs · =0 (29) ΦK ∂xnj ∂xnj ∂xnj ∂xnj Furthermore, we require that the condition in Eq. (29) imposes no constraints on ΦK beyond that required by the rotational model condition in Eq. (23). This requirement facilitates the solution of Eq. (27) as it avoids supplementary constraints (and hence additional complexities) that ΦK must otherwise satisfy. Eq. (29) is satisfied by requiring each of the two terms in the square brackets on the left-hand side of Eq. (29) to vanish separately. The second term on the left-hand side of Eq. (29) vanishes if: ∂θs ∂ΦK · =0 ∂xnj ∂xnj
(30)
In view of the rotational model condition on ΦK in Eq. (23), we fulfill Eq. (30) by choosing θs ∂θs such that ∂x · ∂x∂nj is proportional to an angular momentum componentLkj ≡ −ih(xnk ∂x∂nj − nj xnj ∂x∂nk ). Therefore, we choose
∂θs ∂xnj
to be proportional to xnk , i.e.,
∂θs = χjsk x nk ∂xnj 3
(31)
k=1
for each s, and each of the three matrices χ s to be anti-symmetric, i.e., s χjsk = −χkj ,
for j = k,
s and χjj = 0 for all j
(32)
Substituting Eqs. (31) and (32) into the left-hand side of Eq. (30), we then obtain: 3 Multiplying Eq. (29) by Φ ∗ and its complex conjugate by Φ and adding the resulting two equations, we obtain: ∇ n · K K 2 (|ΦK | λAs ∇n θs ) = 0. This is the equation of continuity (or conservation) equation for the intrinsic-collective probability n θs is the collective-rotation velocity current density, where |ΦK |2 is the intrinsic probability density and V nrot A ≡ λAs ∇
field (refer to Section 3.5). Therefore, Eq. (29) expresses the vanishing of the net outflow, from a unit volume, of the intrinsic probability density at the collective velocity, and hence the decoupling of the intrinsic and collective motions
λ−1 C ∂θs = sA nj as in the case of rotation about a single axis [1]. We also note that the solution ∂x 2 of Eq. (29) for arbitrary nj
constants Cnj does not appear to be useful as noted in the uni-axial rotation case in [1].
|ΦK |
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∂θs ∂ΦK ∂ΦK ∂ ∂ 1 ΦK · = χjsk xnk = − χjsk xnk − xnj ∂xnj ∂xnj ∂xnj 2 ∂xnj ∂xnk 1 s 1 s = (33) χj k Lkj ΦK = χ εkj l Ll ΦK = 0 2i h¯ 2i h¯ j k where the last equality follows from Eq. (23). Thus, for the choice of θs in Eqs. (31) and (32), the second term on the left-hand side of Eq. (29) vanishes without constraining ΦK beyond that in Eq. (23). We note that a choice of θs different from that in Eq. (31) would necessarily constrain ΦK beyond Eq. (23), and hence would take us outside the scope of the rotational model. Substituting Eqs. (31) and (32) into the first term on the left-hand side of Eq. (29), we obtain: ∂θs ∂ ∂ ∂ λAs χjsk xnk = xnk λAs χjsk λAs = ∂xnj ∂xnj ∂xnj ∂xnj −1 ∂ ∂ 1 λAs χjsk = xnk Lkj λAs χjsk = (34) − xnj 2 ∂xnj ∂xnk 2i h¯ where in Eq. (34) the second equality is obtained because j = k due to anti-symmetry of χjsk (refer to Eq. (32)) and hence xnk and ∂x∂nj commute. Eq. (34) shows that, for the Coriolis term to vanish, the quantity λAs χjsk must be invariant under rotation. To show that Ll (λAs χjsk ) in Eq. (34) vanishes, we need to determine χjsk explicitly as follow. Using Eq. (11), we have the “conjugate” variable commutation relation: −1 −1 [Lj, θs ] = RAj [LA, θs ] = −i hR ¯ Aj λsA = −i hλ ¯ sj
(35)
Using Eqs. (31) and (35), we have:
∂θs ∂θs 1 −i h¯ −1 x ε ε = [L θ ] = [L θ ] = − x −i hλ ¯ sj j, s j kl kl, s j kl nk nj 2 2 ∂xnl ∂xnl −i h¯ i h ¯ s rig εj kl xnk χlls xnl − xnl χkls xnl = χ · IAB j = 2 M
(36)
where the pseudo-vector χ s and the rigid-flow moment of inertia I rig are defined in terms of the mass quadrupole moment tensor Q by: 1 χjs ≡ εj kl χkls , 2
I rig ≡ Tr(Q) − Q,
and Qj k ≡ Mxnj xnk
(37)
Multiplying Eq. (36) by I rig −1 , the inverse of I rig , we obtain: rig −1
χjs = −Mλ−1 sk Ikj
(38)
Substituting Eq. (38) into the right-hand side of Eq. (34) and using the first of Eqs. (37), we obtain: rig −1 Lkj λAs χjsk = Lkj λAs εj kl χls = −2Ll λAs Mλ−1 sk Ik l rig −1 rig −1 rig −1 = −2MLl IAl = −2MLl RBl IAB = −2MIAB Ll (RBl ) (39) rig −1
rig −1
The last equality in Eq. (39) follows from the fact that Ll and IAB commute because IAB is an intrinsic quantity as stated in Section 2.1 in relation to Eq. (7) (refer to [24,25]). Using the property: LCD (RBl ) = −i h(R ¯ Dl δBC − RCl δBD )
(40)
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which is derived using Eq. (5) (refer also to [25] Appendix A), we have: 1 Ll (RBl ) = RAl LA (RBl ) = RAl εACD LCD (RBl ) 2 i h¯ = − RAl εACD (RDl δBC − RCl δBD ) 2 i h¯ = − εACD (δAD δBC − δAC δBD ) = 0 (41) 2 where the last equality follows from the anti-symmetry of the tensor εACD . From Eqs. (39) and (41) it follows that the right-hand side of Eq. (34) vanishes and hence: ∂θs ∂ (42) λAs = 0 ∂xnj ∂xnj for the choice of θs in Eq. (31). From Eqs. (33) and (42), we conclude that the Coriolis interaction terms in Eqs. (29), and (28), and (27) vanish for the choice of θs in Eqs. (31) and (32). Furthermore, the choice of θs in Eqs. (31) and (32) does not constrain the intrinsic wavefunction ΦK beyond Eq. (23). 3.3. Microscopic unified rotational model For the choice of θs in Eqs. (31) and (32), Eq. (27) reduces to: h¯ 2 ∂ 2 L 0= DMK − + V − E ΦK 2 2M ∂xnj K nj +
∂θs 1 ∂θt L ΦK · λAs λBt LB LA DMK 2M ∂xnj ∂xnj
(43)
K
L∗ and summing over M and using the closure Multiplying Eq. (43) from the left by DMK relation [27]: L∗ L DMK (44) DMK = δKK M
we obtain the effective unified intrinsic-rotation Schrödinger equation: h¯ 2 ∂ 2 0= − + V − E ΦK 2 2M ∂xnj nj +
∂θs 1 ∂θt L∗ L ΦK · λAs λBt DMK LB LA DMK 2M ∂xnj ∂xnj
(45)
M,K
To determine the kinematic moment of inertia in Eq. (45), we use Eqs. (31) and (38), and the property: εj kk εj ll = δkl δk l − δkl δk l to obtain:
(46)
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∂θs ∂θt 1 εj kk χks εj ll χlt Qkl λAs λBt · λAs λBt = χjsk xnk χjt l xnl λAs λBt = ∂xnj ∂xnj M 1 s t 1 s t rig χl χl Qkk − χls χkt Qkl λAs λBt = χ χ I λAs λBt = M M l k kl rig −1 −1 rig −1 rig λtl Il l Ikl λAs λBt
= Mλ−1 sk Ik l
rig −1
= MIAB
(47)
Substituting Eq. (47) into Eq. (45), we obtain the effective unified intrinsic-rotation Schrödinger equation: h¯ 2 ∂ 2 1 rig −1 L∗ L + 0= − + V − E Φ ΦK IAB DMK (48) LB LA DMK K 2 2M 2 ∂xnj nj
M,K
Eq. (48) is a microscopic counterpart of the phenomenological Bohr–Mottelson unified rigidrig rotor Schrödinger equation except that tensor IAB is not diagonal.4 We can express the rotational rig −1 kinetic energy term in Eq. (48) in terms of the principle-axis components of IAB and the rig −1 component of the angular momentum along the principal axes of IAB , but then the action of rig −1 ∂ L and the action of the principle-axis component of the angular momentum on DMK ∂xnj on IAB would be complicated (although the final answer should be the same for both representations). In Eq. (48), we evaluate the action of the angular momentum operators on the Wigner rotation matrix using the well-known properties [16,17,19,22,27,36]: L L L L L3 DMK = h¯ KDMK , L± DMK = h¯ (L ± K)(L ∓ K + 1)DMK∓1 , L± = L1 ± iL2
(49)
and obtain the effective unified rotation–intrinsic asymmetric rotor Schrödinger equation: h¯ 2 ∂ 2 h¯ 2 rig −1 rig −1 I11 L(L + 1) − K 2 + V + + I22 EΦK = − 2 2M 4 ∂xnj nj h¯ 2 K 2 rig −1 h¯ 2 rig −1 rig −1 rig −1 ΦK + I33 I11 + − I22 − 2iI12 2 8 × (L + K + 2)(L − K − 1)(L + K + 1)(L − K)ΦK+2 h¯ 2 rig −1 rig −1 rig −1 I11 − I22 + 2iI12 8 × (L − K + 2)(L + K − 1)(L − K + 1)(L + K)ΦK−2 rig −1 h¯ 2 rig −1 + (2K − 1) I13 − iI23 (L + K + 1)(L − K)ΦK+1 4 rig −1 h¯ 2 rig −1 + (2K + 1) I13 + iI23 (L − K + 1)(L + K)ΦK−1 4 +
(50)
4 It is noted that we obtain the rigid-flow moment of inertia in Eq. (48) for any inter-particle interaction V whereas the cranking model [4,16,17,19,20,22,33–35] predicts the rigid-body value for the moment of inertia when one uses a self-consistency condition between the potential and particle density. However, Eq. (48) assumes that the interaction V is rotationally invariant and hence it implicitly assumes consistency between the potential and density.
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3.4. Properties and solution of microscopic unified rotational model If in Eq. (50) we ignore the operator
∂2 2 ∂xnj
+ V and assume the inertia tensor I rig −1 to be
diagonal and an adjustable parameter, then Eq. (50) would be identical to the phenomenological Bohr–Mottelson asymmetric (i.e., tri-axial) rotor Schrödinger equation [17]. Because the operrig −1 rig −1 ator I rig −1 in not diagonal and is not generally axially symmetric (i.e., I11 = I22 ), the intrinsic wavefunction ΦK is not invariant under rotation of π about the body-fixed 3 axis (in contrast to the usual symmetry used in the phenomenological rotational model). One can show that the angular momentum component LA (A = 3) does not commute with the Hamiltonian in Eq. (7) unless I rig −1 is assumed to be diagonal and axially symmetric. It is also not possible to obtain an intrinsic–rotation Schrödinger equation for the axially symmetric condition by considering, as is done in the phenomenological rotor model [16,19,20,22], the third Euler angle as an intrinsic variable and include it in the intrinsic wavefunction. If we do that, then the intrinsic wavefunction would not vanish when acted on by the components of the angular momentum operator because these components are functions of the third Euler angle. However, Eq. (50) is invariant under time reversal (i.e., complex conjugation and reversal of angular-momentum ∗ = (−1)K Φ direction) if we require that ΦK −K , i.e., ΦK transforms under time reversal as a spherical harmonic. We note that for a single-particle system, the inertia tensors I rig and I rig −1 are diagonal and rig symmetric and I33 = 0 and we must require K = 0. Therefore, for a single particle, Eq. (50) reduces to: h¯ 2 ∂ 2 h¯ 2 + V + rig L(L + 1) ΦK (51) EΦK = − 2 2M 2I ∂xnj nj
This equation generalizes the Schrödinger equation for a single particle in two spatial dimensions given in an earlier article [1]. We may obtain a solution of Eq. (50) by writing: o ΦK = AK ΦK
where
(52)
o ΦK
is a solution of the following equation: h¯ 2 ∂ 2 h¯ 2 rig −1 rig −1 o I11 L(L + 1) − K 2 = − +V + + I22 E o ΦK 2 2M 4 ∂xnj nj h¯ 2 K 2 rig −1 o I33 + ΦK 2
Eq. (50) then becomes: o E − E o AK ΦK =
h¯ 2 rig −1 rig −1 rig −1 I11 − I22 − 2iI12 8 o × (L + K + 2)(L − k − 1)(L + K + 1)(L − K)AK+2 ΦK+2 +
h¯ 2 rig −1 rig −1 rig −1 I − I22 + 2iI12 8 11
(53)
P. Gulshani / Nuclear Physics A 852 (2011) 109–126
×
121
o (L − K + 2)(L + K − 1)(L − K + 1)(L + K)AK−2 ΦK−2
rig −1 h¯ 2 rig −1 o (2K − 1) I13 − iI23 (L + K + 1)(L − K)AK+1 ΦK+1 4 rig −1 h¯ 2 rig −1 o + (2K + 1) I13 + iI23 (L − K + 1)(L + K)AK−1 ΦK−1 4 +
(54)
∗ and integrating over the spatial coordinates, we obtain an algeMultiplying Eq. (54) by ΦK braic equation relating the coefficients AK , AK±1 , and AK±2 as a function of E − E o and the matrix elements of the components of the rigid-flow inertia tensor I rig −1 appearing in Eq. (54). ∗ ∗ Multiplying Eq. (54) separately by each of the wavefunctions ΦK±1 and ΦK±2 and integrating over the spatial coordinates, we obtain in total five algebraic equations for the five unknowns AK , AK±1 , and AK±2 , as a function of E − E o . For non-trivial solutions to these equations, the determinant of the coefficient matrix making up these five equations must vanish. This condition provides a fifth order polynomial equation in E − E o that can be solved numerically for the eigenvalue E − E o for each value of K and L. In solving Eq. (54), we must ensure that the rotational model condition LA ΦK (ξ ) = 0 in Eq. (23) is satisfied. This condition can be fulfilled by filling each of the lowest-lying single-particle states in pairs of particles (giving zero total angular momentum) in a variational or with perturbative Slater-determinant solution of Eq. (54). We may also supplement Eq. (54) with Eq. (23) and use a Lagrange multiplier method as in the cranking model. Eq. (50) or (54) does not imply that the energy spectrum E in Eq. (50) or (54) would resemble the quantum-rotor spectrum (i.e., L(L + 1) type) or the dynamic moment of inertia obtained from E would be I rig . This would be the case if the centrifugal stretching in the rigid-flow moment of inertia I rig is small, i.e., fluctuations in I rig are small (in relation to the intrinsic energy spacing) as may be the case in deformed nuclei.5 We note that Eq. (53) would be identical to the phenomenological rotor-model Schrödinger equation for an axially symmetric nucleus if we assume that I rig −1 is axially symmetric. In fact, Eq. (53) is obtained if we drop the summation in Eq. (22), i.e., use the wavefunction for a member of a single rotational band, and repeat the steps that result in Eq. (50). To facilitate the determination of the matrix elements ΦK |I rig −1 |ΦK , we need an expression for I rig −1 . We can express I rig −1 in terms of the mass quadrupole tensor Q in Eq. (37) as follows. Suppose that the orthogonal matrix R diagonalizes Q, and hence I rig and I rig −1 (using Eq. (37) for I rig ) have diagonal elements Iα , Iα , and I1α (α = 1, 2, 3) respectively. Then:
QAB = RαA RαB Iα , rig −1
IAB
= RαA RαB
rig IAB = RαA RαB Iα = RαA RαB Tr(Q) − Iα ,
1 Iα
(55)
Then, we have (with α, β, γ in cyclic order): rig −1
IAB
= RαA RαB
1 1 = RαA RαB Iα Iβ + Iγ
5 The result that the kinematic moment of inertia in Eq. (50) or (54) has the rigid-flow value and that the dynamic moment of inertia may also have the rigid-flow value in absence of short-range pairing forces may be consistent with the results in [35,37,38], which show that the moment of inertia has approximately the rigid-flow value for a wavefunction that is a superposition of the states of a number of major harmonic oscillator shells.
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= RAα RBα = RAα RBα
(Iα + Iβ )(Iα + Iγ ) I 2 + Iα (Iγ + Iβ ) + Iβ Iγ = RAα RBα α (Iβ + Iγ )(Iα + Iβ )(Iα + Iγ ) det(I rig ) Iα2 + Iα (Iγ + Iβ ) + 12 (Iγ + Iβ )2 − 12 (Iγ2 + Iβ2 )
det(I rig ) I 2 + Iα Iα (Q) + 12 Iα2 (Q) − 12 Iα (Q2 ) ≡ RAα RBα α det(I rig ) 1 1 2 2 Q + [Tr(Q)] − 2 Tr(Q2 ) = AB 2 det(I rig )
(56)
where we have used the definitions: Iα Q2 ≡ Tr Q2 − Iα2
Iα (Q) ≡ Tr(Q) − Iα ,
(57)
Now: det I rig = Tr(Q) − I1 Tr(Q) − I2 Tr(Q) − I3 = (I1 I2 + I1 I3 + I2 I3 ) Tr(Q) − det(Q) 2 1 Tr(Q) − Tr Q2 · Tr(Q) − det(Q) = 2 Substituting Eq. (58) into Eq. (56), we obtain:
(58)
2
I
rig −1
Tr(Q ) 2Q2 + [Tr(Q)]2 {1 − [Tr(Q)] 2} 1 · = 2 [Tr(Q)]3 det(Q) {1 − Tr(Q ) } − 2 det(Q)
(59)
[Tr(Q)]2
To simplify the many-body expression in Eq. (59), we assume, as is normally done, incompressibility of nuclear matter and set to a constant the nuclear mass quadrupole volume det(Q). In the terms involving the ratio of the trace and determinant functions, we assume that Tr(Q) ≈ 3I , Tr(Q2 ) ≈ 3I 2 , and det(Q) ≈ I 3 (where I is a diagonal element of Q), which may be appropriate for small nuclear quadrupole deformation. These approximations reduce Eq. (59) to the following manageable two-body operator: I rig −1 ≈
3Q2 + [Tr(Q)]2 6 det(Q)
(60)
One may also use other appropriate approximations to simplify Eq. (59). For example, for a large nuclear quadrupole shape deformation, one may use the approximate expression: I rig −1 ≈
4Q2 + [Tr(Q)]2 8 det(Q)
(61)
3.5. Rigid-flow nature of rotation implied by rotational model In this section we show that the choice of θs given in Eqs. (31), (32), and (38) implies a rigidflow nuclear rotation. To define a velocity field corresponding to the collective rotational motion given by Eqs. (31), (32), and (38), we note that, in the rotational kinetic energy term in Eq. (45), L∗ L L D L the quantity M DMK B A MK is a c-number, and is generally a complex number unless rig −1 is diagonal or approximately so (as Eq. (50) shows) as, for example, for a single-particle I motion. The remaining quantities in the rotational kinetic energy term in Eq. (45) are variables.
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Therefore, we define (as is done in the footnote 3 in connection with the vanishing of Coriolisinteraction terms) a collective-rotation velocity field per unit angular momentum component Ll as follows: 1 s λsl ∇θ Vnrot l ≡ (62) M Substituting Eqs (31) and (38) into Eq. (62), we obtain: 1 s 1 χj k xnk λls = εj kk χks xnk λls M M l rig −1 rig −1 = −εj kk λ−1 xnk ≡ −εj kk xnk ωkl = ω × xn j sl Il k xnk λls = −εj kk Ilk
rot l Vnj ≡
(63)
Eq. (63) resembles the well-known rigid-flow velocity field, where the rigid-flow angular velocity ω l per unit angular momentum component Ll is defined as follows: rig −1
ωjl ≡ Ij l
(64)
Substituting Eq. (62) into Eq. (45), we obtain: h¯ 2 ∂ 2 M rot A rot B L∗ L V Vn 0= − + V − E ΦK + ΦK DMK LB LA DMK 2 2M 2 n ∂xnj nj
(65)
M,K
The second term in Eq. (65) is a generalization of the usual the rotational kinetic energy in terms of the collective-rotation velocity field for an asymmetric quantum rotor. The divergence of the collective-rotation velocity field Vnrot l in Eq. (62) is: l 1 l 1 1 rig −1 B n · ω n · Vnrot B = ∇ × xn = L = LA IAB = 0 (66) ∇ ·ω = LA ω A i h i h i h ¯ ¯ ¯ n rig −1
The last equality in Eq. (66) is a consequence of the fact that IAB is an intrinsic quantity (refer Section 2.1). Eq. (66) shows that the collective-rotation velocity field in the body-fixed reference frame is solenoidal (i.e., has zero divergence). (This result is analogous to that in a rigid-body rotation where the angular velocity is a constant.) However, the quantum-fluid flow n |ΦK |2 = 0, unlike the situation in the usual continuumimplied by Vnrot l is compressible, i.e., ∇ = 0 where ∇ · (ρ V) · V = 0 implies that ∇ρ = 0 and hence the density fluid continuity equation ∇ ρ is a constant. Eq. (64) shows that, except in the motion of a single-particle system treated in the previous article [1], the multi-particle angular velocity is not singular at the coordinate-system origin (unless all the particles are located at the origin, which is a very unlikely situation particularly for a fermion system with an anti-symmetrized wavefunction). That is, the rigid flow given by Eqs. (31), (32), (38), (63), and (64) does not possess any vortices. Specifically, we compute the n of Vnrot l in Eq. (63) (no summation over n): vorticity Ω l n × Vnrot l = ∇ n × ω nl ≡ ∇ Ω × xn l n · ω n )ω n xn + ω n · xn ) − xn ∇ (67) l − ω ·∇ l (∇ l = ( xn · ∇ and using the identity (refer to Eqs. (37) and (64)): ∂ ∂ rig −1 rig −1 rig −1 ∂ rig ωkl = Ikl = −Ikl Ilk I ∂xnj ∂xnj ∂xnj k l rig −1 rig −1 ∂ δk l Tr(Q) − Qk l = −Ikl Ilk ∂xnj
(68)
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we obtain (no summation over n): l n × Vnrot l ≡ ∇ Ωnk k l l k 2 = 2ωk − 2Mωl ωk δl k xnj − xnk xnl j j + Mωll ωk (2δl k xnj xnk
− δj l xnk xnk − δj k xnl xnk )
(69)
For a single particle (i.e., for N = 1), the first two terms on the right-hand side of Eq. (69) cancel one another and the last term vanishes (because rotation can occur only about an axis per nl vanishes pendicular to the radius vector, refer to [1]). Therefore, for a single-particle system, Ω l rot l everywhere except at the origin, where ωk and Vn are singular and a free-line vortex exists (refer to [1]). The single-particle velocity field has no other singularities and is not affected by the nodal points of the wavefunction, as shown in [1]. For two or more particles, the vorticity in Eq. (69) does not vanish anywhere (except at infinity, i.e., xn → ∞, where it is very unlikely to find the particles since the wavefunction vanishes there, and except when all the particles are l , av located simultaneously at the origin, which is also very unlikely). The average vorticity Ω l n in Eq. (69) over all the particles, is: obtained by averaging Ω 1 1 l j 1 l j lk − ωl ωl − ωll ωl Qj k Ωav ≡ Ωnk = 2ωkl 1 − N n N N ωll ωl ωk ωll Tr(Q) 3 l + = 2ωk 1 − − δkl + l l Tr(Q) (70) 2N 2N N 2N lk in Eq. (70) vanishes for N = 1 and rapidly approaches the rigid-flow value 2ωl as N inΩav k av is about 95% of the rigid-flow value. creases. For example, for light nuclei (N ∼ 20), Ω We, therefore, conclude that the rigid-flow velocity field given in Eqs. (31), (32), (38), (63), and (64) approaches the rigid-flow velocity field for large values of N and is irrational and freevortex for a single particle. These results indicate that a multi-particle fermion system collectively rotates rigidly as unpaired or paired (quasi-) particles, and a Bose–Einstein condensate system rotates irrotationally as does a single-particle system. However, the energy spectrum would depend on the interaction (compare single-particle motion discussed in [1]).
4. Concluding remarks In this article, we have derived a microscopic version of the phenomenological unified Bohr– Mottelson nuclear rotational model without imposing any constraints on the rotational-model intrinsic wavefunction, and without using redundant or intrinsic coordinates, working entirely with the space-fixed particle coordinates. The microscopic, unified rotational model is derived from a direct action of the multi-particle Hamiltonian operator on the rotational-model wavefunction. This approach circumvents the difficulties in the usual canonical transformation to intrinsic and collective Euler-angle coordinates, where the unknown intrinsic coordinates are generally not readily tractable. (The approach in this article is consistent with and generalizes the transformation of the Hamiltonian of a single-particle system to radial and angular components.) We derive an effective unified intrinsicrotation Schrödinger equation, expressed in terms of the space-fixed particle coordinates and momenta, that includes a general rotational-kinetic-energy term and Coriolis-coupling term.
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We show that the Coriolis-coupling term in the Schrödinger equation vanishes exactly only for a choice of the Euler angles that is consistent with the rotational-model requirement, namely that the intrinsic wavefunction be invariant under rotation. This choice of the Euler angles does not impose any constraints on the rotationally-invariant intrinsic wavefunction. It is shown that any other choice of the Euler angles would require the velocity field to depend on the intrinsic wavefunction, and the intrinsic wavefunction would be rotationally non-invariant (i.e., non-vanishing under the action of the angular momentum) violating the rotational model requirement. For the choice of the Euler angles consistent with the rotational model as stated above and satisfying the angle-angular momentum commutation relations, the kinetic-energy term in the Schrödinger equation becomes identical to that for a rigid-flow tri-axial quantum rotor with nondiagonal rigid-flow moment-of-inertia tensor. (Note that a rigid flow is similar to a rigid-body motion except that the particles are not frozen at their positions and the angular velocity is not a constant.) The corresponding collective-rotation velocity field and its vorticity are shown to have the rigid-flow characteristics. For a single particle, the velocity field is irrotational and is singular, and hence has a free-line vortex at the origin. The single-particle velocity field has no other singularities and is not affected by the nodal points of the wavefunction. For two and more particles, the velocity field has no singularity anywhere and has a vorticity that approaches in form to that for a rigid flow as the number of particles increases. For the rigid-flow choice of the Euler angles, the effective microscopic unified intrinsicrotation Schrödinger equation reduces to that of the phenomenological Bohr–Mottelson unified rotational model for a tri-axial rigid-flow quantum rotor if the kinematic moment-of-inertia tensor is assumed to be diagonal. We propose a method for solving the Schrödinger equation in terms of the eigenfunctions of the axially-symmetric-rotor Schrödinger equation. (For a single particle, the effective microscopic unified intrinsic-rotation Schrödinger equation reduces naturally to the well-known expression for the single-particle Schrödinger equation in the radial coordinate.) The model derivation presented in this article clarifies the assumptions and approximations that underlie the phenomenological adiabatic unified nuclear rotational model. Specifically, the impact of the redundant coordinates is minimized by requiring the intrinsic wavefunction to be a zero angular momentum eigenstate, and the phenomenological model should be using the rigidflow kinematic moment of inertia tensor. The model derivation and results presented in this and previous articles elucidate simply the kinematics and dynamics of the rotational motion of a multi-particle system such a deformed nucleus, and its connection to the rotational motion of a single particle, and provide a simple transparent microscopic means of analyzing nuclear rotational motion and determining the energy spectrum. In particular, the results show that a multi-fermion system with unpaired or paired (quasi) particles rotates rigidly and a single-particle system rotates irrotationally if the intrinsic system is rotationally invariant. Residual short-range interaction and fluctuations in the kinematic moment of inertia reduce the rigid-flow value of the kinematic moment of inertia to agree with the experimental results, but the quasi-particles still rotate rigidly. In a future article, we will investigate the solution of Eq. (54) for deformed rotational nuclei and compare the results with those of the phenomenological rotational models. References [1] [2] [3] [4]
P. Gulshani, Nucl. Phys. A 832 (2010) 18. A. Bohr, Kgl. Dan. Mat. Fys. Medd. 26 (14) (1952). A. Bohr, B.R. Mottelson, Dan. Mat. Fys. Medd. 27 (16) (1953). A. Bohr, B.R. Mottelson, Dan. Mat. Fys. Medd. 30 (1) (1955).
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G. Alaga, K. Alder, A. Bohr, B.R. Mottelson, Dan. Mat. Fys. Medd. 29 (9) (1955). K. Alder, A. Bohr, T. Huus, B.R. Mottelson, A. Winther, Rev. Mod. Phys. 28 (1956) 432. A. Faessler, W. Greiner, Z. Physik 168 (1962) 425. A. Faessler, W. Greiner, Z. Physik 170 (1962) 105. A. Faessler, W. Greiner, Z. Physik 177 (1962) 190. A. Faessler, Nucl. Phys. 59 (1964) 177. A. Faessler, W. Greiner, R.K. Sheline, Nucl. Phys. 62 (1965) 241. A. Faessler, W. Greiner, R.K. Sheline, Nucl. Phys. 70 (1965) 33. A. Faessler, W. Greiner, R.K. Sheline, Nucl. Phys. 80 (1965) 417. J.D. Rogers, Annu. Rev. Nucl. Sci. 15 (1965) 241. J.P. Davidson, Rev. Mod. Phys. 37 (1965) 105. D.J. Rowe, Nuclear Collective Motion, Methuen, London, 1970. J.M. Eisenberg, W. Greiner, Nuclear Models, vol. 1, North-Holland, Amsterdam, 1970. G. Gnesuss, W. Greiner, Nucl. Phys. A 171 (1971) 449. A. de Shalit, H. Feshbach, Theoretical Nuclear Physics, vol. I, John Wiley and Sons, New York, 1974. A. Bohr, B.R. Mottelson, Nuclear Structure, vol. II, Benjamin, New York, 1975. A. Bohr, Rev. Mod. Phys. 48 (1976) 365. P. Ring, P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag, New York, 1980. T. Gustafson, Dan. Mat. Fys. Medd. 30 (5) (1955). P. Gulshani, in: Proc. of VII Latin American Symposium on Nuclear Physics and Applications, Cusco, Peru, June 11–16, 2007, in: AIP Conference Proceedings, vol. 947, 2007. P. Gulshani, Can. J. Phys. 86 (2008) 1. F.M. Villars, G. Cooper, Ann. Phys. 56 (1970) 224. M.E. Rose, Elementary Theory of Angular Momentum, John Wiley and Sons, New York, 1957. R.E. Peierls, J. Yoccoz, Proc. Roy. Soc. (London) 70 (1957) 381. A.D. Zeh, Z. Physik 188 (1965) 361. A.D. Zeh, Z. Physik 188 (1965) 467. A.D. Zeh, Z. Physik 188 (1965) 470. H. Rouhaninezad, J. Yoccoz, Nucl. Phys. 78 (1966) 353. D.R. Inglis, Phys. Rev. 103 (1956) 1786. J.G. Valatin, Proc. Roy. Soc. (London) 238 (1956) 132. T. Troudet, R. Arvieu, Ann. Phys. 134 (1981) 1. A. Bohr, B.R. Mottelson, Nuclear Structure, vol. I, Benjamin, New York, 1975. J.P. Draayer, K.J. Weeks, Nucl. Phys. A 413 (1984) 215. M.G. Vassanji, D.J. Rowe, Nucl. Phys. A 426 (1984) 205.
Derivation of microscopic unified Bohr–Mottelson rotational model P. Gulshani ∗ NUTECH Services, 3313 Fenwick Cres., Mississauga, Ontario, L5L 5N1, Canada Received 22 October 2010; accepted 13 January 2011 Available online 18 January 2011
Abstract In a previous article, we derived a microscopic version of the phenomenological Bohr–Mottelson unified rotational model for rotation about a single axis. In this article, we generalize the derivation to that for rotation about all the three axes. As in the previous derivation, we apply the nuclear Hamiltonian directly to the rotational-model wavefunction instead of using the usual canonical transformation. In this way, we avoid using redundant coordinates or imposing any constraints on the rotationally-invariant rotational-model intrinsic wavefunction. We show that, in the transformed nuclear Schrödinger equation, the Coriolis coupling term vanishes exactly only for a choice of the rotational-model Euler angles that is consistent with angleangular momentum commutation relation and rotational invariance of the intrinsic wavefunction. For this choice of the Euler angles, the kinematic moment-of-inertia tensor, collective-rotation velocity field, and flow vorticity have the rigid-flow characteristics. This quantum rigid flow reduces to irrotational free-vortex flow in the limit of a single particle. We derive a microscopic effective rotation-intrinsic unified Schrödinger equation for the states of a rotational band that reduces to the phenomenological, unified, tri-axial quantum rigid-rotor model in the limit that the off-diagonal elements of the kinematic inertia tensor operator can be neglected. The model derivation shows that a multi-fermion system with unpaired or paired (quasi) particles rotates rigidly and a single-particle system rotates irrotationally if the intrinsic system is rotationally invariant. © 2011 Elsevier B.V. All rights reserved. Keywords: Rotational model; Coriolis coupling; Rigid flow; Moment of inertia; Hydrodynamics; Velocity field; Vorticity; Rotational bands; Band mixing
* Tel.: +1 905 569 8233.
E-mail address: [email protected]. 0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.01.013
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1. Introduction In a previous article [1], we derived a microscopic version of the Bohr–Mottelson rotational model [2–23] for rotation about a single axis. This was achieved in [1] by applying the nuclear Hamiltonian to the rotational-model wavefunction1 : Ψ = eilθ Φ(ξ ),
LΦ = 0,
Leilθ = h¯ leilθ ,
LΨ = h¯ lΨ
(1)
where the Euler angle θ defines the orientation (with respect to the rotation axis) the intrinsic reference-frame axes relative to the space-fixed frame, l = angular momentum quantum number along the rotation axis, and Φ is a rotationally invariant intrinsic wavefunction (i.e., LΦ = 0, refer to Eq. (1)). In this way, we avoided using redundant coordinates or imposing any constraints on the rotationally-invariant rotational-model intrinsic wavefunction. In Ref. [1], the Euler angle θ was chosen to be consistent with angle-angular momentum commutation relation, and with the rotational-model requirement that Φ be rotationally invariant. The result was the Schrödinger equation: N,3 h¯ 2 l 2 h¯ 2 ∂ 2 · +V + − Φ = EΦ (2) 2 2M 2Irig ∂xnj n,j =1
Because of above choice of θ , the kinematic moment of inertia Irig had the rigid-flow value: Irig ≡ M
A
rn2
(3)
n=1
and there are no terms in Eq. (2) that correspond to a coupling between the intrinsic and rotational motions (i.e., Coriolis coupling terms), which could modify Irig , although Irig is still subject to centrifugal stretching. As a consequence, Ref. [1] showed that the collective-rotation flow n had the rigid-flow characteristics: velocity Vn and vorticity Ω r2 n × Vnrot = 2ω n ≡ ∇ Vn = ω (4) × xn , Ω · 1− n Irig For a system of multi-fermions, the kinematic moment-of-inertia is Irig , with a rigid-flow expectation value if the particles were unpaired and a less-than rigid-flow value if the particles were paired due to a pairing force and fluctuations in Irig . For a single particle (i.e., A = 1) or for a Bose–Einstein condensate, Vn is clearly irrotational, and the above results reduce to those from the usual analysis of the motion of a single particle. The above results show that, for a uni-axial rotation, a multi-fermion system such as a deformed nucleus rotates rigidly if the intrinsic state is invariant under rotation. This uni-axial rigid-flow model is the microscopic counterpart of the Bohr–Mottelson unified uni-axial rotational model. The rigid flow is the only type of flow that is fully consistent with and within the scope of the Bohr–Mottelson rotational model [1]. Any other type of flow would impose additional constraints on the intrinsic state, and may introduce Coriolis coupling terms in Eq. (2), and hence would not be within the scope of the rotational model [1]. 1 This approach differs from the usual method of canonically transforming the particle coordinates to a set of collective and intrinsic coordinates.
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In this article, we generalize the above results (presented in more detail in [1]) to the case of rotation about all the three axes, and derive a microscopic version of the general tri-axial Bohr–Mottelson rotational model. In Section 2, we review briefly the usual method of canonical transformation of the particle coordinates to a set of Euler angles and intrinsic coordinates, and the resulting Coriolis coupling and moment of inertia (more detail and references to the related work by other authors are presented in Ref. [1] and will not be repeated in this article). Then we express the results of the canonical transformation in terms of the space-fixed particle coordinates. Section 3 generalizes the microscopic derivation the uni-axial unified rotational model given in [1] to that of a tri-axial rotational model. In Section 3.1, we use a rotation-intrinsic product wavefunction (implied by the canonical decomposition in Section 2) and apply to it the spacefixed nuclear Hamiltonian to derive an effective rotation-intrinsic Schrödinger equation. This result is compared with that obtained from the canonical transformation method in Section 2. In Section 3.2, we derive a condition for which the Coriolis-coupling term vanishes, and which is also consistent with the Euler angle–angular momentum commutation relations, and obtain the resulting rotation-intrinsic Schrödinger equation and moment of inertia. In Section 3.3, we derive a microscopic unified rotational model for a tri-axial nucleus and show that it reduces to the phenomenological unified rotor model if the inertia tensor is assumed to be diagonal. In Section 3.4, we examine the properties and solution of the microscopic unified tri-axial-nucleus Schrödinger equation and derive an expression for the inverse of rigid-flow inertia tensor in terms of the quadrupole moment tensor for use in a future application of the model. In Section 3.5, we examine the properties of the quantum flow obtained in Section 3.2. Section 4 concludes this article. 2. Canonical transformation of Schrödinger equation, Coriolis coupling and moment of inertia This section reviews briefly the usual method of canonical transformation used by a number of authors to describe microscopically nuclear collective rotational motion. This review provides the background material needed for the analysis presented in Section 3. More detail on the canonical transformation, references to related work by other authors, and difficulties encountered in this approach are presented in detail in Ref. [1] and will not be repeated in this article. 2.1. Review of usual canonical transformation The usual method of canonical transformation is to start from the transformation of particles coordinates xnj (n = 1, . . . , N , j = 1, 2, 3, where N = total number of particles) to the coordi (ξ ) (A = 1, 2, 3, σ = 1, . . . , 3N − 3) in a reference frame rotated through the three nates xnA σ Euler angles θs (s = 1, 2, 3) given by: (ξσ ) xnj = RAj (θs )xnA
(5)
where RAj is the Aj ’s component of the orthogonal 3 × 3 matrix R (in this article centre of mass motion is ignored for simplicity but can be easily included). Both θs and ξσ are functions are not all independent (i.e., some of them are of xnj to be defined below. The coordinates xnA redundant) because they must be independent of θs and hence must satisfy certain conditions. This redundancy introduces significant difficulties in calculations involving the intrinsic system. Applying the transformation in Eq. (5) to the N -particle Hamiltonian:
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H≡
N,3 1 2 1 pnj + 2M 2 n,j =1
N,3
V (xnj , xmk )
(6)
n,j,m,k=1
(where M is the particle mass, pnj is the j th component of nth particle linear momentum, and V is a rotationally-invariant two-body interaction) and after some lengthy algebraic manipulations, we decompose H into the following three parts [24,25]: H = Hint +
3
F A LA +
A=1
3 1 −1 IAB LA LB 2
(7)
A,B=1
where the intrinsic Hamiltonian Hint , which includes the nuclear potential, depends on the (3N − 3) coordinates ξσ and the derivatives ∂ξ∂σ . The three operators FA are functions of ξσ and ∂ξ∂σ , and LA are the three components of the angular momentum operator along the body-fixed axes (for simplicity, spin angular momentum is ignored in this article but can be readily included, −1 is the AB component of the inverse of the kinematic moment for example refer to [26]). IAB of inertia tensor I, which is a function of ξσ and is in general not diagonal. In Eq. (7), the second term is the Coriolis interaction, and the third term is the rotational kinetic energy. The −1 . The rotational kinetic energy term includes the effect of centrifugal stretching through IAB Coriolis-interaction term couples the intrinsic and rotational motions. This term complicates the computation of the eigenstates of the transformed Hamiltonian in Eq. (7). The functional form and strength of the Coriolis coupling operators FA and I in Eq. (7) depend on the choice of θs . It is therefore desirable to choose θs to eliminate the Coriolis coupling term, as was done in [1]. Other choices of θs are discussed in [1]. 2.2. Derivation of angular momentum–Euler angle relationship and microscopic expressions −1 for FA and IAB In this subsection, we express the angular momentum operators LA in Eq. (7) in terms of θs , −1 in Eq. (7) in terms of space-fixed particle coordinates and derive expressions for FA and IAB xnj . These results are used in Section 3. σ First, we express LA in terms of θs . Using Eq. (5) and noting that ∂ξ ∂θs = 0, we obtain (in this article, summation over repeated indices is assumed unless stated otherwise): ∂xnj ∂RAj ∂ ∂ ∂ = · = RBj xnA ∂θs ∂θs ∂xnj ∂θs ∂xnB −1 ∂RAj ∂ ∂ 1 ∂RAj ≡ = RBj xnA − xnB RBj LAB 2 ∂θs ∂xnB ∂xnA i h¯ ∂θs where we have used the fact that the matrix ∂ ∂ ≡ RBj ∂xnB ∂xnj
∂RAj ∂θs
(8)
RBj is anti-symmetric in indices A and B, and (9)
From Eq. (8) we readily derive the relation: −i h¯
∂ = λAs LA ∂θs
and its inverse relation:
(10)
P. Gulshani / Nuclear Physics A 852 (2011) 109–126 −1 LA = −i hλ ¯ sA
∂ ∂θs
113
(11)
where LA ≡ 12 εABC LAB , and the 3 × 3 matrix λ is defined by: λAs ≡
3 ∂RBj 1 εABC RCj 2 ∂θs
(12)
A,B=1
where εABC is the third rank antisymmetric Levi-Civita tensor. The inverse λ−1 of λ is defined by: δts = λ−1 tA λAs ,
δAB = λAs λ−1 sB
(13) ∂ ∂ξσ
Eq. (11) shows that LA is independent of ξσ and . From Eq. (11), we conclude that [Lj, ξσ ] = 0, i.e., ξσ is invariant under rotation and hence ξσ are functions of the invariants of the xn · xm |, | xn − xm |, etc. Therefore, group SO(3), such as | xn |, | ∂ ∂ Lj k ≡ −i h¯ xnj − xnk ∂xnk ∂xnj ∂ ∂ ∂θs ∂θs ∂ξσ ∂ξσ = −i h¯ xnj − xnk − i h¯ xnj − xnk ∂xnk ∂xnj ∂θs ∂xnk ∂xnj ∂ξσ ∂ ∂ ∂ = [Lj k , θs ] + [Lj k , ξσ ] = [Lj k , θs ] (14) ∂θs ∂ξσ ∂θs Therefore, we conclude that the intrinsic angular momentum operator [Lj k , ξσ ] ∂ξ∂σ vanishes naturally for any choice of θs due to the invariance of ξσ under rotation similar to that of the radial distance in a single-particle system [1]. It is informative to construct the matrix λ in Eq. (12). Using the orthogonal matrix R given in [27], we obtain: cos θ1 cos θ2 cos θ3 − sin θ1 sin θ3 − cos θ1 cos θ2 sin θ3 − sin θ1 cos θ3 cos θ1 sin θ2
R= ⎛
sin θ1 cos θ2 cos θ3 + cos θ1 sin θ3 − sin θ1 cos θ2 sin θ3 + cos θ1 cos θ3 sin θ1 sin θ2
(15)
⎞
− sin θ2 cos θ3 sin θ3 0 cos θ3 0 ⎠ λ = ⎝ sin θ2 sin θ3 0 1 cos θ2 ⎛ − cos θ3 / sin θ2 sin θ3 / sin θ2 cos θ3 sin θ3 λ−1 = ⎝ cot θ2 sin θ3 cot θ2 cos θ3
− sin θ2 cos θ3 sin θ2 sin θ3 cos θ2
⎞ 0 0⎠ 1
(16)
−1 in Eq. (7) in terms of space-fixed particle coordinates xnj . Next, we express FA and IAB Taking commutator of θs with the Hamiltonian H in Eq. (7) and using Eq. (11), we obtain:
1 −1 1 −1 LA [LB , θs ] + IAB [LA , θs ]LB [H, θs ] = FA [LA , θs ] + IAB 2 2 h¯ 2 −1 −1 ∂λ−1 −1 −1 −1 sB I λ = −i hλ F − i hI λ L − ¯ sA A ¯ AB sA B 2 AB tA ∂θt Multiplying Eq. (17) on the left by λAs and summing over s, we obtain:
(17)
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FA =
i i h¯ −1 −1 −1 ∂λAs −1 LB − ICB λtC λsB λAs [H, θs ] − IAB 2 ∂θt h¯
(18)
Taking commutator of the θt with the first line in Eq. (17), we obtain: −1 [LA , θt ][LB , θs ] [H, θs ], θt = IAB
(19)
Substituting Eq. (11) into Eq. (19), we obtain: −1 = IAB
1
[H, θs ], θt
(20) ¯ Substituting H in Eq. (6) into Eqs. (18) and (20), and assuming momentum independent interaction V , we obtain:
∂ ∂θs ∂θs ∂ i h¯ +2 λAs FA = − 2M ∂xnj ∂xnj ∂xnj ∂xnj ∂θs ∂θs ∂θt i h¯ ∂θt ∂λAt 1 λAs λBt · · LB − · · − M ∂xnj ∂xnj 2M ∂xnj ∂xnj ∂θs
∂θs ∂ ∂θs ∂θs i h¯ ∂ ∂θt 1 λAs λBt =− λAs + 2λAs · · LB (21) − 2M ∂xnj ∂xnj ∂xnj ∂xnj M ∂xnj ∂xnj λAs λBt h2
where, in the second line on the right-hand side of Eq. (21), we have used the chain rule: ∂λAt ∂θs
=
∂λAt ∂xnj
∂θs ∂xnj
·
. Eq. (21) is used in Section 3.
3. Derivation of microscopic unified rotational model and nature of rotational motion In this section, we derive a microscopic effective intrinsic-rotation multi-particle Schrödinger equation resembling that of the unified rotational model for a tri-axial nucleus. We show that the Coriolis-coupling terms in this equation are identical to that for Eqs. (7) and (21) when acting on the model wavefunction. The Coriolis-coupling terms are shown to vanish only for a rigid flow. We obtain the corresponding kinematic moment of inertia, and collective-rotation velocity field. We also show that the eigenfunctions of a tri-axial nucleus can be naturally expanded in terms of the eigenfunctions that resemble those for an axially symmetric nucleus. 3.1. Decomposition of Schrödinger equation into rotational and intrinsic parts For the rotation–intrinsic motion implied by the decomposition of the Hamiltonian in Eq. (7), we use a superposition of product wavefunction in the form: ΨLM =
L
L DMK (θs )ΦK (ξσ )
(22)
K=−L
The intrinsic wavefunction ΦK in Eq. (22) has the property: LA ΦK (ξσ ) = 0
(23)
Unlike the rotational-model wavefunction [2–23], the wavefunction ΨLM in Eq. (22) is not symmetrized because the approach that we are taking in this article does not in general admit the rotational-model symmetry as will be discussed in more detail below. Consistent with the decomposition of the angular momentum operator into rotational and intrinsic parts given in Section 2.2,
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we interpret the rotational-model condition in Eq. (23) to mean that (i) ΦK is a function of varixn · xm |, | xn − xm |, etc., and (ii) ΦK is a zero ables that are invariant under rotation such as | xn |, | angular momentum eigenstate. In deriving the microscopic unified intrinsic–rotation Schrödinger equation presented, we avoid using the unknown intrinsic coordinates ξσ , and express all quantities in terms of the space-fixed coordinates. To achieve this, we apply the original Hamiltonian in Eq. (6) to the wavefunction ΨLM in Eq. (22)2 and obtain: L DMK ΦK (E − V ) K
L L 2 ∂ 2 DMK h¯ 2 ∂ΦK ∂DMK L ∂ ΦK DMK =− +2 · + ΦK 2 2 2M ∂x nj ∂x nj ∂xnj ∂xnj
(24)
n,j,K
L Because DMK is a function of θs only, we obtain, using chain rule and Eq. (10), the result: L ∂DMK ∂θs ∂ L i ∂θs L = · DMK = λAs LA DMK ∂xnj ∂xnj ∂θs h¯ ∂xnj
(25)
Using Eq. (25), chain rule, and Eq. (10), we obtain: L ∂ 2 DMK ∂θs ∂θt i i ∂θs ∂ ∂ L L LA DMK = · λAs LA DMK + · λAs · 2 ∂x ∂x ∂x ∂x ∂θ h h ∂xnj ¯ ¯ nj nj nj nj t ∂θs 1 ∂θs ∂θt ∂ i L L λAs LA DMK − 2· · λAs λBt LB LA DMK (26) = · h¯ ∂xnj ∂xnj h¯ ∂xnj ∂xnj Substituting Eqs. (25) and (26) into Eq. (24), we obtain the effective intrinsic-rotation Schrödinger equation: i h¯ ∂ΦK ∂θs h¯ 2 ∂ 2 L L DMK − + V − E Φ − · λAs LA DMK 0= K 2 2M M ∂x ∂x ∂x nj nj nj K nj K ∂θs ∂ i h¯ L − ΦK λAs LA DMK 2M ∂xnj ∂xnj K ∂θs 1 ∂θt L + ΦK · λAs λBt LB LA DMK (27) 2M ∂xnj ∂xnj K
We now show that the second and third terms on the right-hand side of Eq. (27) are the same as the Coriolis interaction term FA LA in Eq. (7) acting on the wavefunction ΨLM in Eq. (22). Using [LA , FA ] = 0 (refer to Section 2.1) and Eqs. (21) and (23), we have (sum over A is assumed): L FA LA ΨLM = (FA ΦK ) LA DMK
∂θs i h¯ ∂ ∂θs ∂ΦK L LA DMK ΦK =− λAs + 2λAs · (28) 2M ∂xnj ∂xnj ∂xnj ∂xnj 2 This approach is reminiscent of the generator coordinate method of Peierls and Yoccoz [16,17,19,20,22,28–32]. However, in the Peierls–Yoccoz’s method the Euler angels are parameters and not dynamical variables and the intrinsic wavefunction is not rotationally invariant and is a function of the Euler angles. For this reason, projection to states of good angular momentum is necessary in Peierls–Yoccoz’s method.
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The right-hand side of Eq. (28) is seen to be identical to the sum of the second and third terms on the right-hand side of Eq. (27). Therefore, we conclude that the second and third terms on the right-hand side of Eq. (27) arise from the action of the Coriolis interaction term FA LA in Eq. (7), which couples the intrinsic and rotational motions. 3.2. Condition for vanishing of Coriolis interaction As noted in Section 2.1, it is desirable (as a step in solving the rotation–intrinsic Schrödinger Eq. (27) to choose the θs such that the Coriolis-coupling term FA ΦK in Eq. (27) or (28) vanishes, and hence to account completely for the effect of the Coriolis interaction on the rotational (i.e., I) and intrinsic motions. In the resulting equation, the intrinsic and rotational motions would be decoupled except for the centrifugal stretching effect in I, as in the single-particle motion [1]. Therefore, we want to choose θs such that the quantity FA ΦK in Eq. (28) vanishes, i.e., to satisfy3 : ∂θs ∂ ∂θs ∂ΦK λAs + 2λAs · =0 (29) ΦK ∂xnj ∂xnj ∂xnj ∂xnj Furthermore, we require that the condition in Eq. (29) imposes no constraints on ΦK beyond that required by the rotational model condition in Eq. (23). This requirement facilitates the solution of Eq. (27) as it avoids supplementary constraints (and hence additional complexities) that ΦK must otherwise satisfy. Eq. (29) is satisfied by requiring each of the two terms in the square brackets on the left-hand side of Eq. (29) to vanish separately. The second term on the left-hand side of Eq. (29) vanishes if: ∂θs ∂ΦK · =0 ∂xnj ∂xnj
(30)
In view of the rotational model condition on ΦK in Eq. (23), we fulfill Eq. (30) by choosing θs ∂θs such that ∂x · ∂x∂nj is proportional to an angular momentum componentLkj ≡ −ih(xnk ∂x∂nj − nj xnj ∂x∂nk ). Therefore, we choose
∂θs ∂xnj
to be proportional to xnk , i.e.,
∂θs = χjsk x nk ∂xnj 3
(31)
k=1
for each s, and each of the three matrices χ s to be anti-symmetric, i.e., s χjsk = −χkj ,
for j = k,
s and χjj = 0 for all j
(32)
Substituting Eqs. (31) and (32) into the left-hand side of Eq. (30), we then obtain: 3 Multiplying Eq. (29) by Φ ∗ and its complex conjugate by Φ and adding the resulting two equations, we obtain: ∇ n · K K 2 (|ΦK | λAs ∇n θs ) = 0. This is the equation of continuity (or conservation) equation for the intrinsic-collective probability n θs is the collective-rotation velocity current density, where |ΦK |2 is the intrinsic probability density and V nrot A ≡ λAs ∇
field (refer to Section 3.5). Therefore, Eq. (29) expresses the vanishing of the net outflow, from a unit volume, of the intrinsic probability density at the collective velocity, and hence the decoupling of the intrinsic and collective motions
λ−1 C ∂θs = sA nj as in the case of rotation about a single axis [1]. We also note that the solution ∂x 2 of Eq. (29) for arbitrary nj
constants Cnj does not appear to be useful as noted in the uni-axial rotation case in [1].
|ΦK |
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∂θs ∂ΦK ∂ΦK ∂ ∂ 1 ΦK · = χjsk xnk = − χjsk xnk − xnj ∂xnj ∂xnj ∂xnj 2 ∂xnj ∂xnk 1 s 1 s = (33) χj k Lkj ΦK = χ εkj l Ll ΦK = 0 2i h¯ 2i h¯ j k where the last equality follows from Eq. (23). Thus, for the choice of θs in Eqs. (31) and (32), the second term on the left-hand side of Eq. (29) vanishes without constraining ΦK beyond that in Eq. (23). We note that a choice of θs different from that in Eq. (31) would necessarily constrain ΦK beyond Eq. (23), and hence would take us outside the scope of the rotational model. Substituting Eqs. (31) and (32) into the first term on the left-hand side of Eq. (29), we obtain: ∂θs ∂ ∂ ∂ λAs χjsk xnk = xnk λAs χjsk λAs = ∂xnj ∂xnj ∂xnj ∂xnj −1 ∂ ∂ 1 λAs χjsk = xnk Lkj λAs χjsk = (34) − xnj 2 ∂xnj ∂xnk 2i h¯ where in Eq. (34) the second equality is obtained because j = k due to anti-symmetry of χjsk (refer to Eq. (32)) and hence xnk and ∂x∂nj commute. Eq. (34) shows that, for the Coriolis term to vanish, the quantity λAs χjsk must be invariant under rotation. To show that Ll (λAs χjsk ) in Eq. (34) vanishes, we need to determine χjsk explicitly as follow. Using Eq. (11), we have the “conjugate” variable commutation relation: −1 −1 [Lj, θs ] = RAj [LA, θs ] = −i hR ¯ Aj λsA = −i hλ ¯ sj
(35)
Using Eqs. (31) and (35), we have:
∂θs ∂θs 1 −i h¯ −1 x ε ε = [L θ ] = [L θ ] = − x −i hλ ¯ sj j, s j kl kl, s j kl nk nj 2 2 ∂xnl ∂xnl −i h¯ i h ¯ s rig εj kl xnk χlls xnl − xnl χkls xnl = χ · IAB j = 2 M
(36)
where the pseudo-vector χ s and the rigid-flow moment of inertia I rig are defined in terms of the mass quadrupole moment tensor Q by: 1 χjs ≡ εj kl χkls , 2
I rig ≡ Tr(Q) − Q,
and Qj k ≡ Mxnj xnk
(37)
Multiplying Eq. (36) by I rig −1 , the inverse of I rig , we obtain: rig −1
χjs = −Mλ−1 sk Ikj
(38)
Substituting Eq. (38) into the right-hand side of Eq. (34) and using the first of Eqs. (37), we obtain: rig −1 Lkj λAs χjsk = Lkj λAs εj kl χls = −2Ll λAs Mλ−1 sk Ik l rig −1 rig −1 rig −1 = −2MLl IAl = −2MLl RBl IAB = −2MIAB Ll (RBl ) (39) rig −1
rig −1
The last equality in Eq. (39) follows from the fact that Ll and IAB commute because IAB is an intrinsic quantity as stated in Section 2.1 in relation to Eq. (7) (refer to [24,25]). Using the property: LCD (RBl ) = −i h(R ¯ Dl δBC − RCl δBD )
(40)
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which is derived using Eq. (5) (refer also to [25] Appendix A), we have: 1 Ll (RBl ) = RAl LA (RBl ) = RAl εACD LCD (RBl ) 2 i h¯ = − RAl εACD (RDl δBC − RCl δBD ) 2 i h¯ = − εACD (δAD δBC − δAC δBD ) = 0 (41) 2 where the last equality follows from the anti-symmetry of the tensor εACD . From Eqs. (39) and (41) it follows that the right-hand side of Eq. (34) vanishes and hence: ∂θs ∂ (42) λAs = 0 ∂xnj ∂xnj for the choice of θs in Eq. (31). From Eqs. (33) and (42), we conclude that the Coriolis interaction terms in Eqs. (29), and (28), and (27) vanish for the choice of θs in Eqs. (31) and (32). Furthermore, the choice of θs in Eqs. (31) and (32) does not constrain the intrinsic wavefunction ΦK beyond Eq. (23). 3.3. Microscopic unified rotational model For the choice of θs in Eqs. (31) and (32), Eq. (27) reduces to: h¯ 2 ∂ 2 L 0= DMK − + V − E ΦK 2 2M ∂xnj K nj +
∂θs 1 ∂θt L ΦK · λAs λBt LB LA DMK 2M ∂xnj ∂xnj
(43)
K
L∗ and summing over M and using the closure Multiplying Eq. (43) from the left by DMK relation [27]: L∗ L DMK (44) DMK = δKK M
we obtain the effective unified intrinsic-rotation Schrödinger equation: h¯ 2 ∂ 2 0= − + V − E ΦK 2 2M ∂xnj nj +
∂θs 1 ∂θt L∗ L ΦK · λAs λBt DMK LB LA DMK 2M ∂xnj ∂xnj
(45)
M,K
To determine the kinematic moment of inertia in Eq. (45), we use Eqs. (31) and (38), and the property: εj kk εj ll = δkl δk l − δkl δk l to obtain:
(46)
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∂θs ∂θt 1 εj kk χks εj ll χlt Qkl λAs λBt · λAs λBt = χjsk xnk χjt l xnl λAs λBt = ∂xnj ∂xnj M 1 s t 1 s t rig χl χl Qkk − χls χkt Qkl λAs λBt = χ χ I λAs λBt = M M l k kl rig −1 −1 rig −1 rig λtl Il l Ikl λAs λBt
= Mλ−1 sk Ik l
rig −1
= MIAB
(47)
Substituting Eq. (47) into Eq. (45), we obtain the effective unified intrinsic-rotation Schrödinger equation: h¯ 2 ∂ 2 1 rig −1 L∗ L + 0= − + V − E Φ ΦK IAB DMK (48) LB LA DMK K 2 2M 2 ∂xnj nj
M,K
Eq. (48) is a microscopic counterpart of the phenomenological Bohr–Mottelson unified rigidrig rotor Schrödinger equation except that tensor IAB is not diagonal.4 We can express the rotational rig −1 kinetic energy term in Eq. (48) in terms of the principle-axis components of IAB and the rig −1 component of the angular momentum along the principal axes of IAB , but then the action of rig −1 ∂ L and the action of the principle-axis component of the angular momentum on DMK ∂xnj on IAB would be complicated (although the final answer should be the same for both representations). In Eq. (48), we evaluate the action of the angular momentum operators on the Wigner rotation matrix using the well-known properties [16,17,19,22,27,36]: L L L L L3 DMK = h¯ KDMK , L± DMK = h¯ (L ± K)(L ∓ K + 1)DMK∓1 , L± = L1 ± iL2
(49)
and obtain the effective unified rotation–intrinsic asymmetric rotor Schrödinger equation: h¯ 2 ∂ 2 h¯ 2 rig −1 rig −1 I11 L(L + 1) − K 2 + V + + I22 EΦK = − 2 2M 4 ∂xnj nj h¯ 2 K 2 rig −1 h¯ 2 rig −1 rig −1 rig −1 ΦK + I33 I11 + − I22 − 2iI12 2 8 × (L + K + 2)(L − K − 1)(L + K + 1)(L − K)ΦK+2 h¯ 2 rig −1 rig −1 rig −1 I11 − I22 + 2iI12 8 × (L − K + 2)(L + K − 1)(L − K + 1)(L + K)ΦK−2 rig −1 h¯ 2 rig −1 + (2K − 1) I13 − iI23 (L + K + 1)(L − K)ΦK+1 4 rig −1 h¯ 2 rig −1 + (2K + 1) I13 + iI23 (L − K + 1)(L + K)ΦK−1 4 +
(50)
4 It is noted that we obtain the rigid-flow moment of inertia in Eq. (48) for any inter-particle interaction V whereas the cranking model [4,16,17,19,20,22,33–35] predicts the rigid-body value for the moment of inertia when one uses a self-consistency condition between the potential and particle density. However, Eq. (48) assumes that the interaction V is rotationally invariant and hence it implicitly assumes consistency between the potential and density.
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3.4. Properties and solution of microscopic unified rotational model If in Eq. (50) we ignore the operator
∂2 2 ∂xnj
+ V and assume the inertia tensor I rig −1 to be
diagonal and an adjustable parameter, then Eq. (50) would be identical to the phenomenological Bohr–Mottelson asymmetric (i.e., tri-axial) rotor Schrödinger equation [17]. Because the operrig −1 rig −1 ator I rig −1 in not diagonal and is not generally axially symmetric (i.e., I11 = I22 ), the intrinsic wavefunction ΦK is not invariant under rotation of π about the body-fixed 3 axis (in contrast to the usual symmetry used in the phenomenological rotational model). One can show that the angular momentum component LA (A = 3) does not commute with the Hamiltonian in Eq. (7) unless I rig −1 is assumed to be diagonal and axially symmetric. It is also not possible to obtain an intrinsic–rotation Schrödinger equation for the axially symmetric condition by considering, as is done in the phenomenological rotor model [16,19,20,22], the third Euler angle as an intrinsic variable and include it in the intrinsic wavefunction. If we do that, then the intrinsic wavefunction would not vanish when acted on by the components of the angular momentum operator because these components are functions of the third Euler angle. However, Eq. (50) is invariant under time reversal (i.e., complex conjugation and reversal of angular-momentum ∗ = (−1)K Φ direction) if we require that ΦK −K , i.e., ΦK transforms under time reversal as a spherical harmonic. We note that for a single-particle system, the inertia tensors I rig and I rig −1 are diagonal and rig symmetric and I33 = 0 and we must require K = 0. Therefore, for a single particle, Eq. (50) reduces to: h¯ 2 ∂ 2 h¯ 2 + V + rig L(L + 1) ΦK (51) EΦK = − 2 2M 2I ∂xnj nj
This equation generalizes the Schrödinger equation for a single particle in two spatial dimensions given in an earlier article [1]. We may obtain a solution of Eq. (50) by writing: o ΦK = AK ΦK
where
(52)
o ΦK
is a solution of the following equation: h¯ 2 ∂ 2 h¯ 2 rig −1 rig −1 o I11 L(L + 1) − K 2 = − +V + + I22 E o ΦK 2 2M 4 ∂xnj nj h¯ 2 K 2 rig −1 o I33 + ΦK 2
Eq. (50) then becomes: o E − E o AK ΦK =
h¯ 2 rig −1 rig −1 rig −1 I11 − I22 − 2iI12 8 o × (L + K + 2)(L − k − 1)(L + K + 1)(L − K)AK+2 ΦK+2 +
h¯ 2 rig −1 rig −1 rig −1 I − I22 + 2iI12 8 11
(53)
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×
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o (L − K + 2)(L + K − 1)(L − K + 1)(L + K)AK−2 ΦK−2
rig −1 h¯ 2 rig −1 o (2K − 1) I13 − iI23 (L + K + 1)(L − K)AK+1 ΦK+1 4 rig −1 h¯ 2 rig −1 o + (2K + 1) I13 + iI23 (L − K + 1)(L + K)AK−1 ΦK−1 4 +
(54)
∗ and integrating over the spatial coordinates, we obtain an algeMultiplying Eq. (54) by ΦK braic equation relating the coefficients AK , AK±1 , and AK±2 as a function of E − E o and the matrix elements of the components of the rigid-flow inertia tensor I rig −1 appearing in Eq. (54). ∗ ∗ Multiplying Eq. (54) separately by each of the wavefunctions ΦK±1 and ΦK±2 and integrating over the spatial coordinates, we obtain in total five algebraic equations for the five unknowns AK , AK±1 , and AK±2 , as a function of E − E o . For non-trivial solutions to these equations, the determinant of the coefficient matrix making up these five equations must vanish. This condition provides a fifth order polynomial equation in E − E o that can be solved numerically for the eigenvalue E − E o for each value of K and L. In solving Eq. (54), we must ensure that the rotational model condition LA ΦK (ξ ) = 0 in Eq. (23) is satisfied. This condition can be fulfilled by filling each of the lowest-lying single-particle states in pairs of particles (giving zero total angular momentum) in a variational or with perturbative Slater-determinant solution of Eq. (54). We may also supplement Eq. (54) with Eq. (23) and use a Lagrange multiplier method as in the cranking model. Eq. (50) or (54) does not imply that the energy spectrum E in Eq. (50) or (54) would resemble the quantum-rotor spectrum (i.e., L(L + 1) type) or the dynamic moment of inertia obtained from E would be I rig . This would be the case if the centrifugal stretching in the rigid-flow moment of inertia I rig is small, i.e., fluctuations in I rig are small (in relation to the intrinsic energy spacing) as may be the case in deformed nuclei.5 We note that Eq. (53) would be identical to the phenomenological rotor-model Schrödinger equation for an axially symmetric nucleus if we assume that I rig −1 is axially symmetric. In fact, Eq. (53) is obtained if we drop the summation in Eq. (22), i.e., use the wavefunction for a member of a single rotational band, and repeat the steps that result in Eq. (50). To facilitate the determination of the matrix elements ΦK |I rig −1 |ΦK , we need an expression for I rig −1 . We can express I rig −1 in terms of the mass quadrupole tensor Q in Eq. (37) as follows. Suppose that the orthogonal matrix R diagonalizes Q, and hence I rig and I rig −1 (using Eq. (37) for I rig ) have diagonal elements Iα , Iα , and I1α (α = 1, 2, 3) respectively. Then:
QAB = RαA RαB Iα , rig −1
IAB
= RαA RαB
rig IAB = RαA RαB Iα = RαA RαB Tr(Q) − Iα ,
1 Iα
(55)
Then, we have (with α, β, γ in cyclic order): rig −1
IAB
= RαA RαB
1 1 = RαA RαB Iα Iβ + Iγ
5 The result that the kinematic moment of inertia in Eq. (50) or (54) has the rigid-flow value and that the dynamic moment of inertia may also have the rigid-flow value in absence of short-range pairing forces may be consistent with the results in [35,37,38], which show that the moment of inertia has approximately the rigid-flow value for a wavefunction that is a superposition of the states of a number of major harmonic oscillator shells.
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= RAα RBα = RAα RBα
(Iα + Iβ )(Iα + Iγ ) I 2 + Iα (Iγ + Iβ ) + Iβ Iγ = RAα RBα α (Iβ + Iγ )(Iα + Iβ )(Iα + Iγ ) det(I rig ) Iα2 + Iα (Iγ + Iβ ) + 12 (Iγ + Iβ )2 − 12 (Iγ2 + Iβ2 )
det(I rig ) I 2 + Iα Iα (Q) + 12 Iα2 (Q) − 12 Iα (Q2 ) ≡ RAα RBα α det(I rig ) 1 1 2 2 Q + [Tr(Q)] − 2 Tr(Q2 ) = AB 2 det(I rig )
(56)
where we have used the definitions: Iα Q2 ≡ Tr Q2 − Iα2
Iα (Q) ≡ Tr(Q) − Iα ,
(57)
Now: det I rig = Tr(Q) − I1 Tr(Q) − I2 Tr(Q) − I3 = (I1 I2 + I1 I3 + I2 I3 ) Tr(Q) − det(Q) 2 1 Tr(Q) − Tr Q2 · Tr(Q) − det(Q) = 2 Substituting Eq. (58) into Eq. (56), we obtain:
(58)
2
I
rig −1
Tr(Q ) 2Q2 + [Tr(Q)]2 {1 − [Tr(Q)] 2} 1 · = 2 [Tr(Q)]3 det(Q) {1 − Tr(Q ) } − 2 det(Q)
(59)
[Tr(Q)]2
To simplify the many-body expression in Eq. (59), we assume, as is normally done, incompressibility of nuclear matter and set to a constant the nuclear mass quadrupole volume det(Q). In the terms involving the ratio of the trace and determinant functions, we assume that Tr(Q) ≈ 3I , Tr(Q2 ) ≈ 3I 2 , and det(Q) ≈ I 3 (where I is a diagonal element of Q), which may be appropriate for small nuclear quadrupole deformation. These approximations reduce Eq. (59) to the following manageable two-body operator: I rig −1 ≈
3Q2 + [Tr(Q)]2 6 det(Q)
(60)
One may also use other appropriate approximations to simplify Eq. (59). For example, for a large nuclear quadrupole shape deformation, one may use the approximate expression: I rig −1 ≈
4Q2 + [Tr(Q)]2 8 det(Q)
(61)
3.5. Rigid-flow nature of rotation implied by rotational model In this section we show that the choice of θs given in Eqs. (31), (32), and (38) implies a rigidflow nuclear rotation. To define a velocity field corresponding to the collective rotational motion given by Eqs. (31), (32), and (38), we note that, in the rotational kinetic energy term in Eq. (45), L∗ L L D L the quantity M DMK B A MK is a c-number, and is generally a complex number unless rig −1 is diagonal or approximately so (as Eq. (50) shows) as, for example, for a single-particle I motion. The remaining quantities in the rotational kinetic energy term in Eq. (45) are variables.
P. Gulshani / Nuclear Physics A 852 (2011) 109–126
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Therefore, we define (as is done in the footnote 3 in connection with the vanishing of Coriolisinteraction terms) a collective-rotation velocity field per unit angular momentum component Ll as follows: 1 s λsl ∇θ Vnrot l ≡ (62) M Substituting Eqs (31) and (38) into Eq. (62), we obtain: 1 s 1 χj k xnk λls = εj kk χks xnk λls M M l rig −1 rig −1 = −εj kk λ−1 xnk ≡ −εj kk xnk ωkl = ω × xn j sl Il k xnk λls = −εj kk Ilk
rot l Vnj ≡
(63)
Eq. (63) resembles the well-known rigid-flow velocity field, where the rigid-flow angular velocity ω l per unit angular momentum component Ll is defined as follows: rig −1
ωjl ≡ Ij l
(64)
Substituting Eq. (62) into Eq. (45), we obtain: h¯ 2 ∂ 2 M rot A rot B L∗ L V Vn 0= − + V − E ΦK + ΦK DMK LB LA DMK 2 2M 2 n ∂xnj nj
(65)
M,K
The second term in Eq. (65) is a generalization of the usual the rotational kinetic energy in terms of the collective-rotation velocity field for an asymmetric quantum rotor. The divergence of the collective-rotation velocity field Vnrot l in Eq. (62) is: l 1 l 1 1 rig −1 B n · ω n · Vnrot B = ∇ × xn = L = LA IAB = 0 (66) ∇ ·ω = LA ω A i h i h i h ¯ ¯ ¯ n rig −1
The last equality in Eq. (66) is a consequence of the fact that IAB is an intrinsic quantity (refer Section 2.1). Eq. (66) shows that the collective-rotation velocity field in the body-fixed reference frame is solenoidal (i.e., has zero divergence). (This result is analogous to that in a rigid-body rotation where the angular velocity is a constant.) However, the quantum-fluid flow n |ΦK |2 = 0, unlike the situation in the usual continuumimplied by Vnrot l is compressible, i.e., ∇ = 0 where ∇ · (ρ V) · V = 0 implies that ∇ρ = 0 and hence the density fluid continuity equation ∇ ρ is a constant. Eq. (64) shows that, except in the motion of a single-particle system treated in the previous article [1], the multi-particle angular velocity is not singular at the coordinate-system origin (unless all the particles are located at the origin, which is a very unlikely situation particularly for a fermion system with an anti-symmetrized wavefunction). That is, the rigid flow given by Eqs. (31), (32), (38), (63), and (64) does not possess any vortices. Specifically, we compute the n of Vnrot l in Eq. (63) (no summation over n): vorticity Ω l n × Vnrot l = ∇ n × ω nl ≡ ∇ Ω × xn l n · ω n )ω n xn + ω n · xn ) − xn ∇ (67) l − ω ·∇ l (∇ l = ( xn · ∇ and using the identity (refer to Eqs. (37) and (64)): ∂ ∂ rig −1 rig −1 rig −1 ∂ rig ωkl = Ikl = −Ikl Ilk I ∂xnj ∂xnj ∂xnj k l rig −1 rig −1 ∂ δk l Tr(Q) − Qk l = −Ikl Ilk ∂xnj
(68)
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we obtain (no summation over n): l n × Vnrot l ≡ ∇ Ωnk k l l k 2 = 2ωk − 2Mωl ωk δl k xnj − xnk xnl j j + Mωll ωk (2δl k xnj xnk
− δj l xnk xnk − δj k xnl xnk )
(69)
For a single particle (i.e., for N = 1), the first two terms on the right-hand side of Eq. (69) cancel one another and the last term vanishes (because rotation can occur only about an axis per nl vanishes pendicular to the radius vector, refer to [1]). Therefore, for a single-particle system, Ω l rot l everywhere except at the origin, where ωk and Vn are singular and a free-line vortex exists (refer to [1]). The single-particle velocity field has no other singularities and is not affected by the nodal points of the wavefunction, as shown in [1]. For two or more particles, the vorticity in Eq. (69) does not vanish anywhere (except at infinity, i.e., xn → ∞, where it is very unlikely to find the particles since the wavefunction vanishes there, and except when all the particles are l , av located simultaneously at the origin, which is also very unlikely). The average vorticity Ω l n in Eq. (69) over all the particles, is: obtained by averaging Ω 1 1 l j 1 l j lk − ωl ωl − ωll ωl Qj k Ωav ≡ Ωnk = 2ωkl 1 − N n N N ωll ωl ωk ωll Tr(Q) 3 l + = 2ωk 1 − − δkl + l l Tr(Q) (70) 2N 2N N 2N lk in Eq. (70) vanishes for N = 1 and rapidly approaches the rigid-flow value 2ωl as N inΩav k av is about 95% of the rigid-flow value. creases. For example, for light nuclei (N ∼ 20), Ω We, therefore, conclude that the rigid-flow velocity field given in Eqs. (31), (32), (38), (63), and (64) approaches the rigid-flow velocity field for large values of N and is irrational and freevortex for a single particle. These results indicate that a multi-particle fermion system collectively rotates rigidly as unpaired or paired (quasi-) particles, and a Bose–Einstein condensate system rotates irrotationally as does a single-particle system. However, the energy spectrum would depend on the interaction (compare single-particle motion discussed in [1]).
4. Concluding remarks In this article, we have derived a microscopic version of the phenomenological unified Bohr– Mottelson nuclear rotational model without imposing any constraints on the rotational-model intrinsic wavefunction, and without using redundant or intrinsic coordinates, working entirely with the space-fixed particle coordinates. The microscopic, unified rotational model is derived from a direct action of the multi-particle Hamiltonian operator on the rotational-model wavefunction. This approach circumvents the difficulties in the usual canonical transformation to intrinsic and collective Euler-angle coordinates, where the unknown intrinsic coordinates are generally not readily tractable. (The approach in this article is consistent with and generalizes the transformation of the Hamiltonian of a single-particle system to radial and angular components.) We derive an effective unified intrinsicrotation Schrödinger equation, expressed in terms of the space-fixed particle coordinates and momenta, that includes a general rotational-kinetic-energy term and Coriolis-coupling term.
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We show that the Coriolis-coupling term in the Schrödinger equation vanishes exactly only for a choice of the Euler angles that is consistent with the rotational-model requirement, namely that the intrinsic wavefunction be invariant under rotation. This choice of the Euler angles does not impose any constraints on the rotationally-invariant intrinsic wavefunction. It is shown that any other choice of the Euler angles would require the velocity field to depend on the intrinsic wavefunction, and the intrinsic wavefunction would be rotationally non-invariant (i.e., non-vanishing under the action of the angular momentum) violating the rotational model requirement. For the choice of the Euler angles consistent with the rotational model as stated above and satisfying the angle-angular momentum commutation relations, the kinetic-energy term in the Schrödinger equation becomes identical to that for a rigid-flow tri-axial quantum rotor with nondiagonal rigid-flow moment-of-inertia tensor. (Note that a rigid flow is similar to a rigid-body motion except that the particles are not frozen at their positions and the angular velocity is not a constant.) The corresponding collective-rotation velocity field and its vorticity are shown to have the rigid-flow characteristics. For a single particle, the velocity field is irrotational and is singular, and hence has a free-line vortex at the origin. The single-particle velocity field has no other singularities and is not affected by the nodal points of the wavefunction. For two and more particles, the velocity field has no singularity anywhere and has a vorticity that approaches in form to that for a rigid flow as the number of particles increases. For the rigid-flow choice of the Euler angles, the effective microscopic unified intrinsicrotation Schrödinger equation reduces to that of the phenomenological Bohr–Mottelson unified rotational model for a tri-axial rigid-flow quantum rotor if the kinematic moment-of-inertia tensor is assumed to be diagonal. We propose a method for solving the Schrödinger equation in terms of the eigenfunctions of the axially-symmetric-rotor Schrödinger equation. (For a single particle, the effective microscopic unified intrinsic-rotation Schrödinger equation reduces naturally to the well-known expression for the single-particle Schrödinger equation in the radial coordinate.) The model derivation presented in this article clarifies the assumptions and approximations that underlie the phenomenological adiabatic unified nuclear rotational model. Specifically, the impact of the redundant coordinates is minimized by requiring the intrinsic wavefunction to be a zero angular momentum eigenstate, and the phenomenological model should be using the rigidflow kinematic moment of inertia tensor. The model derivation and results presented in this and previous articles elucidate simply the kinematics and dynamics of the rotational motion of a multi-particle system such a deformed nucleus, and its connection to the rotational motion of a single particle, and provide a simple transparent microscopic means of analyzing nuclear rotational motion and determining the energy spectrum. In particular, the results show that a multi-fermion system with unpaired or paired (quasi) particles rotates rigidly and a single-particle system rotates irrotationally if the intrinsic system is rotationally invariant. Residual short-range interaction and fluctuations in the kinematic moment of inertia reduce the rigid-flow value of the kinematic moment of inertia to agree with the experimental results, but the quasi-particles still rotate rigidly. In a future article, we will investigate the solution of Eq. (54) for deformed rotational nuclei and compare the results with those of the phenomenological rotational models. References [1] [2] [3] [4]
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