aa r X i v : . [ nu c l - t h ] D ec The Nuclear Born Oppenheimer Method and NuclearRotations
Nouredine ZettiliDepartment of Physical & Earth SciencesJacksonville State UniversityJacksonville, AL 36265
Abstract
We deal here with the application of the Nuclear Born Oppenheimer(NBO) method to the description of nuclear rotations. As an edifying il-lustration, we apply the NBO formalism to study the rotational motion ofnuclei which are axially-symmetric and even, but whose shells are not closed.We focus, in particular, on the derivation of expressions for the rotationalenergy and for the moment of inertia. Additionally, we examine the con-nection between the NBO method and the self-consistent cranking (SCC)model. Finally, we compare the moment of inertia generated by the NBOmethod with the Thouless-Valantin formula and hence establish a connectionbetween the NBO method and the large body of experimental data.PACS numbers: 21.60.-n, 21.60.Ev, 21.10.Re Introduction
Since nuclear and molecular rotation-vibration spectra present many striking analo-gies, and since the Born-Oppenheimer (BO) approximation[1] of molecular physicswas shown to be very accurate in describing molecular rotations and vibrations[2], itwill be interesting to explore the possibility of using the BO approximation to describenuclear collective rotations.Exploiting the analogy between nuclear and molecular dynamics, Villars introduceda microscopic method[3], [4] to describe nuclear collective motion. This method, tobe called the Nuclear Born-Oppenheimer (NBO) method, was developed along thelines of the molecular BO approximation by constructing a factorable trial functionmodeled after the BO ansatz.Using an analytically solvable model[5], we have shown that the NBO method isvery accurate for adiabatic collective motion[6]. Since the NBO method is a quan-tum mechanical prescription, we have shown that it offers a suitable framework fordescribing the zero-point fluctuations[7]; we have also shown that the method offersan accurate description of small-amplitude collective oscillations[8] and that it yieldsthe random phase approximation (RPA) equations[9]. Additionally, we have appliedthe NBO method to study nuclear collective motion[10] and examined its connectionwith the collective model of Bohr[11].So, having applied the NBO method to the study of small amplitude motion, wehave yet to apply it to nuclear collective rotations. In this work we want to achieve justthat aim; namely, we want to apply the general BO formalism outlined in Ref.[10] tothe description of nuclear rotational states. As an illustration, we will apply the NBOformalism to study the rotations of nuclei that are axially-symmetric and even, butwith non-closed shells. We will focus, in particular, on the derivation of expressionsfor the energy and for the moment of inertia. Additionally, we shall examine theconnection of the NBO method with the successful self-consistent cranking (SCC)model.In Sec. 2, we present a brief outline of the NBO formalism and how it appliesto nuclear nuclear collective motion. We then devote Sec. 3 to the applicationof the NBO method to study the rotational states of axially symmetric nuclei; in Using an elementary solvable model, Moshinksy and Kittel[2] have shown that the BOapproximation is very accurate for both the molecular energy and wave function: E BO E exact = 1 − χ and |h ψ BO | ψ exact i| = 1 − χ , where χ is equal to the ratio of electronic to nuclear masses(i.e., χ = m e /M ≃ ). articular, we will derive an expression for the rotational energy. In Sec. 4, wepresent a discussion on the connection of the NBO method to the self-consistentcranking (SCC) model. To describe nuclear collective motion within the framework of the NBO method, weneed to introduce a tensor operator ˆ Q αβ ; that is, to be able to describe collectiverotations and vibrations of nuclei, we need to introduce a set of operators that are theelements of a symmetric cartesian tensor operator ˆ Q αβ . In the rest of this work, weshall use Greek subscripts to refer to a space-fixed frame of reference; Latin subscriptswill be used later to refer to a body-fixed frame. The operator ˆ Q αβ is assumed todepend on the various nucleonic variables – positions ~x i , momenta, ~p i , and spins, ~s i .In addition, we assume that ˆ Q αβ are one-body operators, symmetric, even under timereversal, have a continuous eigenvalue spectrum ( q αβ ) , and commute with any othercomponent ˆ Q γδ of ˆ Q ( i.e. , [ ˆ Q αβ , ˆ Q γδ ] = 0 ). Let ˆ K αβ be the canonical conjugateof ˆ Q αβ : [ i ˆ K αβ , ˆ Q γδ ] = 12 ( δ αγ δ βδ + δ αδ δ βγ ) . (2.1) The NBO method consists of the following two essential steps[3], [4], [10]: • First, we need to construct a suitable representation for the nucleus’ Hamilto-nian ˆ H by decomposing it into a series ˆ H = ˆ H + X αβ ˆ H αβ ˆ K αβ + 12 X αβ X γδ ˆ H αβγδ ˆ K αβ ˆ K γδ + · · · , (2.2) where all coefficient operators ˆ H , ˆ H αβ , ˆ H αβ,γδ , . . . commute with ˆ Q αβ . • Second, we make use a factorable trial function h x | ψ i = Z Y α ≤ β dq αβ h x | δ ( q αβ − ˆ Q αβ ) | Φ( q ) i g ( q ) , (2.3) where h x | Φ( q ) i is the intrinsic wave function, and g ( q ) is the collective ampli-tude. We will use x to abbreviate for the set of nucleonic variables – position ~x i , momentum ~p i ,and spin, ~s i . fter constructing the Hamiltonian and the wave function, we can calculate themean energy by a simple application of ˆ H to | ψ i : h ψ | ˆ H | ψ i = Z Y α ≤ β dq αβ g ∗ ( q ) h Φ( q ) | δ ( q αβ − ˆ Q αβ ) { ˜ H | Φ i g ( q ) + X αβ ˜ H αβ | Φ i k αβ g ( q )+ 12 X αβ X γδ ˜ H αβ,γδ | Φ i k αβ k γδ g ( q ) + · · ·} , (2.4) where the k αβ are operators that act on g ( q ) ; they obey commutation relations withthe q αβ isomorphic with (2.1) [ ik αβ , q γδ ] = 12 ( δ αγ δ βδ + δ αδ δ βγ ) . (2.5) The few lowest expressions of ˜ H K are given by ˜ H = ˆ H − X αβ h ˆ H, i ˆ Q αβ i ˆ G αβ + 12 X αβ X γδ hh ˆ H, i ˆ Q αβ i , i ˆ Q γδ i ˆ G αβ ˆ G γδ + · · · , ˜ H αβ = h ˆ H, i ˆ Q αβ i − X γδ hh ˆ H, i ˆ Q αβ i , i ˆ Q γδ i ˆ G γδ + · · · , (2.6)˜ H αβ γδ = hh ˆ H, i ˆ Q αβ i , i ˆ Q γδ i + · · · , where ˆ G αβ is a one particle operator that acts on | φ i ; it is defined by the action of k αβ on the parameter q in | φ i k αβ h x | Φ( q ) i = 12 i (1 + δ αβ ) ∂∂q αβ h x | Φ( q ) i = h x | ˆ G αβ | Φ( q ) i . (2.7) We should note that the mean energy expression (2.4) was derived within a space-fixed or lab frame. However, in the description of permanently deformed (non spher-ical) nuclei, it is more convenient to employ a body-fixed frame of reference. Here,we take the axes of the body-fixed frame along the three principal axes of q αβ whichare defined by the unit vectors ˆ e a ( a = 1 , , , and specify their orientation with re-spect to the space-fixed frame by three Euler angles[23] θ s (i.e., θ, ϕ, ψ ) : ˆ e a = ˆ e a ( θ s )with ˆ e a · ˆ e b = δ ab and ˆ e a × ˆ e b = E abc ˆ e c where E abc is the antisymmetric tensor( E = 1 = −E etc.). The collective degrees of freedom can be separated intorotational and vibrational terms by transforming ˆ Q αβ to the body-fixed frame; thatis, by means of the principal axes transformation of the tensor operator ˆ Q αβ : q αβ = X a =1 e αa ( θ ) e βa ( θ ) q a , (2.8)4 here e αa ( ≡ ˆ e α · ˆ e a ) , the α th component of the unit vector ˆ e a , depends on the threeEuler angles θ s . In the transformation to the body-fixed frame, we have essentiallyreplaced the six collective coordinates q αβ by the three q a ’s and the three Eulerangles. The matrices e αβ obey the orthogonality relations: P a e αa e βa = δ αβ and P α e αa e αb = δ ab .We can now introduce rotation operators ˆ L a about the body-fixed axes. TheEuler angles specifying the orientation of the intrinsic frame need to be viewed asdynamical variables; for instance, the unit vector ˆ e a satisfy the commutation rules ofa vector operator h ˆ L [ ab ] , e αc i = ie αa δ bc − ie αb δ ac . (2.9) We can easily verify from (2.9) that these operators obey the commutation relations h ˆ L a , ˆ L b i = − i E abc ˆ L c , (2.10) which differ in sign from the commutation rules of ordinary angular momentum[12]because they refer to the moving axes and hence do not have the same commutationproperties as angular-momentum components along space fixed axes. For instance,we have h ˆ L , ˆ L i = − i ˆ L . The space fixed components ˆ L αβ of ~ L can be obtainedby rotation: ˆ L αβ = P ab e αa e βb ˆ L ab .In conjunction with the replacement of q αβ by the variables q a and θ s , we seekan expression for the operator k αβ in terms of the ˆ L αβ and a set of three operators p a conjugate to q a , with [ p a , q b ] = iδ ab . We can verify[10] that k αβ transforms likean operator that acts on θ s and q a : k αβ = 12 X ab e αa e βb q a − q b ˆ L [ ab ] + X a e αa e βa p a = X ab e αa e βb "
12 (1 + δ αβ ) ˆ L [ ab ] q a − q b + δ ab p a . (2.11) Additionally, we can ascertain that k αβ is Hermitian with regard to the volume element Q α ≤ β dq αβ , which can be shown to transform like: Z dτ = Z Y α ≤ β dq αβ = Z Y a =1 dq a ( q − q )( q − q )( q − q ) d Ω , (2.12) where d Ω is the usual angular element[23] d Ω = sin θdθdϕdψ .Using the relations (2.11) and (2.12), we can now express (2.4) and (2.6) inthe body-fixed frame. For this, note first that under the transformation (2.8) from he Lab frame to the body-fixed system, the quantities g ( q αβ ) , h x iα s iα | Φ( q αβ ) i , and Q α ≤ β δ ( q αβ − ˆ Q αβ ) become f ( q a , θ s ) , h x ′ ia s ′ ia | φ ( q a ) i , and Q a δ ( q a − ˆ Q aa ) Q a ≤ b δ ( Q ab ) ,respectively, which in turn will be abbreviated to f ( q, θ ) , h x ′ i s ′ i | φ ( q ) i , and δ ( q − ˆ Q ) .Next, we can show[10] that the action of the total angular momentum ˆ J αβ on | ψ i can be expressed in terms of ˆ L αβ on the collective amplitude f ( q a , θ s ) : h x | ˆ J αβ | ψ i = Z dτ h x | δ ( q − ˆ Q ) | φ ( q ) i ˆ L αβ f ( q, θ ) . (2.13) In this new representation, the operator ˆ G αβ of (2.6) is rotated into ˆ G ab : h x ′ ia s ′ ia | ˆ G ab | φ ( q a ) i = 12 (1 − δ ab ) h x ′ ia s ′ ia | ˆ J [ ab ] | φ ( q a ) i q a − q b + δ ab h x ′ ia s ′ ia | ˆ G a | φ ( q a ) i . (2.14) Finally, using Eq. (2.11), (2.13) and (2.14), we have shown in Ref.[10] that themean energy (2.4) is given in the body-fixed frame of reference by h ψ | ˆ H | ψ i = Z dτ f ∗ ( q, θ ) h φ ( q ) | δ ( q − ˆ Q ) (cid:26) ˆ H − X a,b =1 ˙ˆ Y [ ab ] ( ˆ J [ ab ] − ˆ L [ ab ] ) − X a =1 ˙ˆ Q a ( ˆ G a − p a ) + 12 X abcd ˆ B [ ab ] , [ cd ] ( ˆ J [ ab ] − ˆ L [ ab ] )( ˆ J [ cd ] − ˆ L [ cd ] )+ X abc ˆ B [ ab ] ,c ( ˆ J [ ab ] − ˆ L [ ab ] )( ˆ G c − p c )+ 12 X ab ˆ B a,b ( ˆ G a − p a )( ˆ G b − p b ) (cid:27) | φ ( q ) i f ( q, θ ) , (2.15) where ˆ Y [ ab ] = ˆ Q ab q a − q b , ˙ˆ Y [ ab ] = h i ˆ H, ˆ Y [ ab ] i , ˙ˆ Q a = h i ˆ H, ˆ Q a i , (2.16)ˆ B [ ab ] , [ cd ] = [ ˙ˆ Y [ ab ] , i ˆ Y [ cd ] ] , ˆ B [ ab ] ,c = [ ˙ˆ Y [ ab ] , i ˆ Q c ] , ˆ B ab = [ ˙ˆ Q a , i ˆ Q b ] . (2.17) Note that, in deriving the mean energy (2.15), we have terminated the series (2.4) atthe quadratic terms in k αβ . This termination is justified by the validity of the adiabaticapproximation in the present case, since we are dealing with nuclear dynamics for Recall that ˆ J [ ab ] and ˆ G a operate on the intrinsic state | φ ( q a ) i , but ˆ L [ ab ] and p a operate onthe collective state f ( q a , θ s ) (i.e., ˆ L [ ab ] acts on θ s and p a on q a ). hich the time evolution of the collective variables is assumed to be slow on the scaleof a single-particle (nucleonic) motion.As we are going to see next, the rotational and vibrational degrees of freedomappear explicitly in the energy expression (2.15); we will also show how to deriveexpressions for the collective rotational energy and for the moment of inertia. Consider a permanently deformed, non spherical nucleus. Since we are interested inrotational motion only, we assume the nucleus to be in its vibrational ground state.In this case, we assume that the collective amplitude f ( q a , Ω) of (2.15) separates intoa vibrational part, g ( q a − ¯ q a ) , and a rotational part, D (Ω) : f ( q a , Ω) = g ( q a − ¯ q a ) D (Ω) . (3.18) The vibrational collective amplitude g ( q a − ¯ q a ) represents here the zero-point oscil-lations about the equilibrium values, ¯ q a , of q a ( a = 1 , , . Hence, the wave function | ψ i of the system becomes ( c.f. Eq. (2.3)): | ψ i = Z dτ d Ω δ ( q − ˆ Q ) | φ ( q a ) i g ( q a − ¯ q a ) D (Ω) , (3.19) with dτ = ( q a − q )( q − q )( q − q ) Q a =1 dq a and d Ω = dϕdψ sin θdθ ( c.f. Eq.(2.12)), and where δ ( q − ˆ Q ) is used to abbreviate Q a =1 δ ( q a − ˆ Q aa ) Q a
Consider the axis 3, of the body-fixed frame, to be theaxis of symmetry for the system. As a consequence of the axial symmetry, we have: ¯ q a = ¯ q = ¯ q and B = B ≡ B = B .Now, since ˆ H , ˆ ~J and ˆ J z mutually commute commute, they possess joint eigen-functions. The structure of our trial function allows it to be an exact eigenfunction of ˆ ~J and ˆ J , but provides only a variational approximation to the energy. In the caseof axial symmetry, this trial function | ψ i can be obtained from (3.19) by expanding | φ ( q a ) i D (Ω) in terms of the Wigner D − functions : | ψ IM i = Z dτ d Ω δ ( q − ˆ Q ) s I + 116 π X K (cid:20) | φ K ( q a ) iD IMK (Ω)+( − I | φ − K ( q a ) D IM − K (Ω) (cid:21) g ( q a − ¯ q a ) , (3.21) where | φ K i is an eigenfunction of ˆ J ( i.e. , ˆ J | φ K i = K | φ K i ) and D IMK is an eigen-function to ˆ L ( i.e. , ˆ L D IMK = K D IMK ) , ˆ ~ L and ˆ L z . It then follows that ˆ ~J | ψ IM i = I ( I + 1) | ψ IM i , J Z | ψ IM i = M | ψ IM i . (3.22) For the simpler case of the K = 0 band, the wave function | ψ IM i is given by: | ψ IM i = Z dτ d Ω δ ( q − ˆ Q ) s I + 18 π | φ ( q a ) iD IM (Ω) g ( q − ¯ q a ) . (3.23) Note that (as a consequence of axial symmetry) the following important relation holdsfor both forms, (3.21) and (3.23), of | φ i D (Ω) : ( ˆ J − ˆ L ) | φ i D (Ω) = 0 . (3.24) In this case of axial symmetry, and after omitting E + E coupl , we can see thatthe mean-energy (3.20) reduces to: h ψ IM | ˆ H | ψ IM i = Z dτ d Ω g ∗ ( q a − ¯ q a ) D ∗ (Ω) h φ (¯ q a ) | δ ( q − ˆ Q ) × " ˆ H + B X a =1 ˆ L a | φ (¯ q a ) i g D (Ω)+ E + E , (3.25) ~J is the total angular momentum and ˆ J z is its Z component with respect to the Lab frame. The definition of D IM − K (Ω) used here is that of Bohr-Mottelson ith ˆ H = ˆ H − X a =1 ˙ˆ Y ˆ J a + 12 B X a =1 ˆ J a , (3.26) E = Z dτ d Ω g ∗ D ∗ h φ | δ ( q − ˆ Q ) X a =1 ( ˙ˆ Y a − B ˆ J a ) ˆ L a | φ i g ( q a − ¯ q a ) D (Ω) , E = ( B + B ) Z dτ d Ω g ∗ D ∗ h φ | δ ( q − ˆ Q )( ˆ J − ˆ L )( ˆ J − ˆ L ) | φ i g D (Ω) , (3.27) where | φ ( q a ) i D (Ω) is given by (3.21) or (3.23), depending on whether one is inter-ested in the K = 0 band or the K = 0 band.We should now specify the description of the intrinsic structure of the system.To this end, we assume that the intrinsic state | φ (¯ q a ) i is given by a mean fieldapproximation such that h φ (¯ q a ) | ˆ Q | φ (¯ q a ) i = ¯ q is equal to h φ | ˆ Q | φ i = ¯ q ( i.e ,such that ¯ q = ¯ q , the axial symmetry condition). This can be achieved by means ofa constrained variational principle.Let us now look at the determination of the collective tensor operator ˆ Q . Wedetermine the particle-hole ( ph ) components of the tensor operator ˆ Q ab such that h φ | ˙ˆ Y a − B ˆ J a | φ i is variationally stable, i.e. , δ h φ | ˙ˆ Y a − B ˆ J a | φ i = 0 ( a = 1 , . (3.28) This variational condition insures that the simple expression (3.21) for | ψ i is adequateto describe the rotational energy (term ∼ ˆ ~ L ) correctly.To determine the mean energy (3.25), we need to calculate E and E . In whatfollows, we are going to show that both E and E are identically zero. First, the term h φ | δ ( q − ˆ Q )( ˙ˆ Y a − B ˆ J a ) | φ i in the integrand of E can be rewritten as h φ | δ ( q − ˆ Q )( ˙ˆ Y a − B ˆ J a ) | φ i ≃ h φ | δ ( q − ˆ Q ) | φ ih φ | ˙ˆ Y a − B ˆ J a | φ i + X σµ h φ | δ ( q − ˆ Q ) | φ σµ ih φ σµ | ˙ˆ Y a − B ˆ J a | φ i (3.29) where σ, τ, . . . refer to unoccupied (particle) states, while µ, λ, . . . refer to occupied(hole) states. Using the condition (3.28), we see that the term h φ | δ ( q − ˆ Q )( ˙ˆ Y a − In this approximate expression, we have neglected the two-body part of the operator ˙ˆ Y a ≡ (cid:2) ˆ H, i ˆ Y a (cid:3) . ˆ J a ) | φ i becomes equal to h φ | δ ( q − ˆ Q ) | φ ih φ | ˙ˆ Y a − B ˆ J a | φ i . Now, using the fact that | φ k i is an eigenfunction to ˆ J and that the action of both ˙ˆ Y a and ˆ J a ( a = 1 , on | φ k i generate | φ k ± i , we can ascertain that h φ | ˙ˆ Y a − B ˆ J a | φ i is itself identically zero,and hence E is equal to zero. To see this, note that ( ˙ˆ Y a − B ˆ J a ) have non-zeromatrix elements only between | φ k i and h φ K ± | . So, h φ k | ˙ˆ Y a − B ˆ J a | φ k i = 0 ( a = 1 , , (3.30) and also h φ − k | ˙ˆ Y a − B ˆ J a | φ k i = 0 , (3.31) except for K = but, in our case, K is always an integer (because we are dealingwith an even nucleus).Second, E is identically zero, since both B and B can be shown to be equalto zero. To see this, using these expressions, h ˆ J , i ˆ Y i = ˆ Q − ˆ Q q − q , h ˆ J , i ˆ Y i = ˆ Q q − q , (3.32) h ˆ J , i ˆ Y i = ˆ Q − ˆ Q q − q , h ˆ J , i ˆ Y i = − ˆ Q q − q , (3.33) we can easily show the following important relation: h φ | [ ˆ J a , i ˆ Y b ] | φ i = δ ab ( a = 1 , , (3.34) since h ˆ Q ab i = δ ab . Now, applying this relation to the variational principle (3.29), wecan verify that h φ | [ ˙ˆ Y a , ˆ Y b ] | φ i is equal to Bδ ab , i.e. , B ab = B h φ | [ ˆ J a , i ˆ Y b ] | φ i = Bδ ab ( a = 1 , . (3.35) Now, since both of E and E are zero, and using the relation ( ˆ J − ˆ L ) | φ i D (Ω) =0 of (3.24) and (3.28), we can show that the mean energy (3.25) reduces to h ψ IM | ˆ H | ψ IM i = Z dτ d Ω g ∗ (Ω) h φ | δ ( q − ˆ Q ) (cid:20) ˆ H − B ( ˆ J + ˆ J + ˆ J (cid:21) | φ i g D (Ω)+ B Z dτ d Ω h φ | δ ( q − ˆ Q ) | φ i g ∗ D ∗ (Ω) h ˆ L + ˆ L + ˆ L i g D (Ω) . (3.36) Using the approximation h φ | δ ( q − ˆ Q ) (cid:18) ˆ H − B ~J (cid:19) | φ i ≃ h φ | δ ( q − ˆ Q ) | φ ih φ | ˆ H − B ~J | φ i , (3.37)10 e can rewrite (3.36) in the following simpler form E I = h ψ IM | ˆ H | ψ IM ih ψ IM | ψ IM i ≈ BI ( I + 1) + h φ | ˆ H − B ( ˆ J + ˆ J + ˆ J ) | φ i , (3.38) where we have used he fact that ˆ L D IMK = I ( I + 1) D IMK . Note that the energyexpression (3.38) has a term, − B h ˆ ~J i which represents a substraction of a mean-rotational energy. This term is familiar from the standard Peierls-Yoccoz angularmomentum projection method. We expect this approximate treatment of the δ -function in (3.38) to overestimate the mean-energy by a term of the order of half thezero-point vibration energy. Moment of Inertia
Let us now look at the moment of inertia, which is given by B − . The inertialparameter B can be determined from eqs. (3.29) and (3.34); i.e. , it is given by thetwo equations h φ | h ˆ H, i ˆ Z a i | φ ph i = h φ | ˆ J a | φ ph i , B − = h φ | h ˆ Z a , i ˆ J a i | φ i , (3.39) where ˆ Z a = ˆ Y a /B . This expression for B − is of the well-known Thouless-Valantinform[13]. Note that if we neglect the residual two-body interactions from the Hamil-tonian, expressions (3.39) would give rise to Inglis cranking formula[14] B ≈ X σµ |h φ | ˆ J a | φ σµ i| E σ − E µ . (3.40) This approximate formula is well-known to overestimate the moment of inertia quitebadly.In what follows, we are going to examine the connection between the BO methodand the large body of (rotational) data[15], e.g. , the moment of interia increaseswith angular momentum I . First, note that Eq. (3.38), which was derived for time-reversal invariant | φ i , describe a rotational spectrum with constant moment of inertia, B − , in disagreement with data. To see this, consider the case K = 0 for which h φ | ˆ J i | φ i = 0 ( i = 1 , , . Hence, the energy expression (9.2.20) becomes E I = 12 BI ( I + 1) + h ˆ H i − ∆ E fℓ , (3.41) Recall that we have omitted the vibrational energy part, E . ith ∆ E fℓ = B h ∆ ˆ J + ∆ ˆ J i , the fluctuation energy which is generated by angularmomentum fluctuations. In this case, therefore, the energy spectrum is that of arigid rotor, since the moment of inertia B − , as given by (3.39), is constant. Thiscontradicts, of course, the experimental facts.Second, note that the failure of (3.38) to generate a moment of inertia, B − ,which increases with angular momentum is due to a restrictive assumption on | φ i ,the time-reversal invariance of | φ i for K = 0 . In what follows, we are going toshow that the BO approach has a natural mechanism for introducing a moment ofinertia which increases with I , provided the restrictive assumption on | φ i is dropped.In addition, we will show that the energy we obtain for this case is lower than theenergy, (3.41), obtained with a time-reversal invariant | φ i . To this end, let us considera symmetry-violating | φ i for which h ˆ J i is not zero but for which h ˆ J i and h ˆ J i areboth zero. In this analysis, we will restrict ourselves to the simplest case: K = 0 ,and hence | φ i is an eigenfunction to ˆ J with eigenvalue zero, ˆ J | φ i ≡ . In thiscase, the energy expression (3.38) reduces to E I = 12 BI ( I + 1) + h ˆ H i − B h ˆ J i − ∆ E fℓ . (3.42) Note that this energy is lower than the energy, (3.41), obtained with a time-reversalinvariant | φ i (provided the fluctuation energy is unchanged). In what follows, weshall neglect the angular momentum fluctuations, ∆ E fℓ , in the determination of themean field | φ i . The energy expression (3.42) then provides a basis for a variationaldetermination of the symmetry violating | φ i : δ h ˆ H − ω ˆ J i ω = 0 , ω = B h ˆ J i ω , (3.43) where the notation h ˆ0 i ω is used to abbreviate h φ ( ω ) | ˆ0 | φ ( ω ) i . The parameter ω has,obviously, the significance of an angular velocity.Now, we are in a position to show that B ( ω ) decreases when the angular momen-tum increases. To see this, using the relation B ( ω ) = ω h ˆ J i ω of (3.43), we have dB ( ω ) dω = 1 h ˆ J i ω " − h ˆ J i /ω d h ˆ J i ω dω . (3.44) Since h ˆ J i ω is well-known to increase with the angular velocity ω , and as shown inFig. 1, the slope d h ˆ J i ω dω is always larger than h ˆ J i ω ω . Thus, the slope, dBdω of B ( ω ) isnegative and, hence, B ( ω ) would behave as shown in Fig. 2: B ( ω ) decreases as theangular velocity ω increases. ω ✻ h ˆ J i ω ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ α (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) β h ˆ J i ω ω Figure 1: h ˆ J i ω increases with the angular velocity ω . Therefore, we conclude that the moment of inertia B − ( ω ) increases, indeed, withangular velocity ω , and hence with angular momentum also.In what follows, we are going to show that there exists a non-zero value, ω c , of ω at which the energy E I ( ω c ) of (3.42) is equal to its lowest value. To this end, letus write the energy expression (3.42) in the following form (from which we omit thefluctuation term, ∆ E fℓ ): E I ( ω ) ≃ h ˆ H − ω ˆ J i ω + 12 B ( ω ) h h ˆ J i ω + I ( I + 1) i = h ˆ H − ω ˆ J i ω + 12 " ω B ( ω ) + B ( ω ) I ( I + 1) . (3.45) First, note that the derivative, dE I ( ω ) dω = 12 h I ( I + 1) − h ˆ J i ω i dB ( ω ) dω , (3.46) of E I ( ω ) vanishes at a value ω c which is determined by h ˆ J i ω c = I ( I + 1) , i.e. , dE I ( ω ) dω (cid:12)(cid:12)(cid:12)(cid:12) ω c = 0 = ⇒ h ˆ J i ω c = I ( I + 1) . (3.47) Second, we can easily show that the second derivative of E I ( ω ) , d E I ( ω ) dω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω c = ωB ( ω ) " ωB ( ω ) (cid:18) dB ( ω ) dω (cid:19) − dB ( ω ) dω ω = ω c , (3.48)13 ω ✻ B ( ω ) B ( ω )Figure 2: The inertial parameter B ( ω ) decreases as the angular velocity ω in-creases, since dB/dω < is positive, since, as shown above ( c.f. Eq. (3.44)), dBdω is negative. Finally, weconclude that, using a trial function | φ ( ω ) i whose time-reversal symmetry is broken,one obtains, indeed, lower values for the inverse moment of inertia, B ( ω ) , and forthe energy than those calculated with a T · R invariant mean field. Calculation of the energy difference: ∆ E I Let us now calculate the energy difference, ∆ E I , between E I ( ω = 0) and E I ( ω c ) .Using Eq. (3.46), we can show that ∆ E I = E I ( ω c ) − E I (0) = Z ω c dE I ( ω ) dω dω = 12 [ B ( ω c ) − B (0)] I ( I + 1) + 12 Z ω c ω ddω (cid:18) B ( ω ) (cid:19) dω . (3.49) This expression can, after a partial integration, be reduced to ∆ E I = − B I ( I + 1) + Z √ I ( I +1)0 ω (cid:16) h ˆ J i (cid:17) d h ˆ J i = − Z √ I ( I +1)0 ω (cid:16) h ˆ J i (cid:17) (cid:20) BB ( ω ) − (cid:21) d h ˆ J i . (3.50) This expression shows that E I ( ω c ) is, indeed, lower than E I (0) , since B is largerthan B ( ω ) . So, if we know the dependence of the angular velocity ω on h ˆ J i ω , we ω ✻ E I ( ω ) + ∆ E fl E (0) E (0) ❄✻ ∆ E E (0) E ( ω c ) ❄✻ ∆ E E ( ω c ) E (0) ❄✻ ∆ E Figure 3: Behavior of E I ( ω ) as a function of the angular velocity ω , where | ∆ E I | = | E I (0) − E I ( ω c ) | is an increasing function of the angular momentum I can easily calculate the energy difference between E I (0) and E I ( ω c ) . Note that, theenergy difference | ∆ E I | is an increasing function of the angular momentum I . Thequalitative behavior of the energy E I ( ω ) , for various values of I , is plotted in Fig. 3. Let us summarize what we have achieved in this work. First, we have shown that themoment of inertia generated by the NBO method is identical to the Thouless-Valantinform. Second, the two relations (3.43) and (3.47) determine the intrinsic (symmetry-breaking) function | φ ω i and the value, ω c , of ω where E I ( ω c ) is equal to its lowestvalue, respectively. These two relations provide a bridge (connection) between theNBO method, which is a truly quantum mechanical description of collective motion,and the semi-classical approaches based on the idea of self-consistent cranking (SCC).Thus, we have established a connection between the NBO approach and the largebody of experimental data, since the two relations (3.43) and (3.47) are known toprovide reasonable descriptions of vast amounts of empirical data ranging from low- ying rotational states to high angular momentum states. [16]–[21] So, the present(NBO) method appears to be well-equipped to describe low as well as high lyingrotational states. Additionally, we should mention that work has been started toapply the NBO method to the description of the backbending phenomenon whichwas first observed by Johnson and his collaborators.[22]In summary, we have studied here the rotational spectrum of even, axially-symmetricnuclei within the framework of the NBO method. We have made use of trial functionsin which the intrinsic structure is described within a mean-field approximation. Wehave shown that the NBO formalism gives back the Thouless-Valantin moment ofinertia. Then, we have established a connection between the NBO method and theSCC model, which has been successful in reproducing vast amounts of experimentaldata. Finally, we have shown that the introduction of a non time-reversal invariantintrinsic function both lowers the energy for a given I , and provides a moment ofinertia that increases with the angular momentum I . Acknowledgments
Supported in part by the Alabama Commission on Higher Education.
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