The angular momentum distribution of the deformed intrinsic nuclear state

Nuclear Physics A212 (1973) 18--26; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE ANGULAR M O M E N T U M DISTRIBUTION OF THE DEFORMED INTRINSIC NUCLEAR STATE + R. K. B H A D U R I

Physics Department, MeMaster University, Hamilton, Ontario, Canada and S. DAS G U P T A

Physics Department, McGill University, Montreal, Quebec, Canada Received 14 May 1973 (Revised 19 June 1973) Abstract: In a ttartree-Fock or Itartree-Fock-Bogoliubov calculation of a deformed intrinsic state, one obtains a distribution o f angular momentum states. Using an analogy from statistical mechanics, we obtain an expression for this distribution. The concept of an average temperature for the intrinsic state is introduced, which is directly related to the rotational energy content of the intrinsic state. The relationship of this temperature to microscopic particle-hole calculations is clarified. Assuming a rotational spectrum for the ground-state band of an axially symmetric doubly even nucleus, it is demonstrated that the deduced distribution o f the angular momentum states gives rise to an overlap function of the intrinsic wave function which falls off, for small angles of rotation, as a Gaussian. Finally, the Yoccoz formula for the moment of inertia is derived using classical statistical mechanics, and semiclassical corrections to it are obtained.

1. Introduction For a given nuclear Hamiltonian, there are well-established quantum mechanical methods of obtaining the nuclear intrinsic state. Most common are the Hartree-Fock (HF) or Hartree-Fock-Bogoliubov (HFB) calculations that yield an intrinsic state I~) which describes, in general, the equilibrium properties of a band. For a deformed nucleus, the intrinsic state [~) is not an eigenfunction of the total angular momentum operator j z, and may be written as an admixture of the physical states ]0s) of a band: I~) = ~ asl~Ps),

(1)

J

where we have suppressed the other quantum numbers. The states [~/s) are eigenfunctions of the nuclear Hamiltonian and the angular momentum operator j2, and the distribution of these angular momentum states in 1~) is determined by lajI2. For the ground state band of an axially symmetric doubly even nucleus, it may be assumed that H[~pj> =

/

Eo+ 2-~J(J+l)

}

]~Os),

* Research supported by the National Research Council of Canada. 18

(2)

DEFORMED INTRINSIC STATE

19

where H is the nuclear Hamiltonian and E 0 the energy of the J = 0 ground state; in what follows we put E 0 = 0, i.e., we measure all energies with respect to Eo without any loss of generality. Numerical values of [as] 2 and the projected energies are generally obtained by the Peierls-Yoccoz projection technique 1). Numerical calculations 2) show that for well-deformed nuclei the overlap function (q~le-i°S, lq~) is well approximated by a Gaussian form in the variable 0, and the projected energies form a rotational spectrum. Peieds and Yoccoz 1) actually showed that a Gaussian overlap function implies a rotational projected spectrum. Yoccoz 3) derived a simple formula for the moment of inertia p a r a m e t e r / f o r a given band assuming the validity of the Gaussian overlap. In the present work, we make a connection between the quantum mechanical m a w - b o d y method and a method based on the statistical mechanics of a rotor. Specifically, we assume the existence of a rotational spectrum for the band and find by the statistical method a closed expression for the coefficients [asl2. This is done in sect. 2, where we also show that the statistical formula for lasl 2 gives close agreement with the numerical values of the projection method. In sect. 3, we show that this distribution of the angular momentum states implies a Gaussian form for the overlap function for small angles of rotation. In the statistical method, the concept of an average temperature for the rotational band is introduced, which is directly related to the total rotational energy content of the intrinsic state. In sect. 4, the nature of this temperature is further clarified in the language of m a w - b o d y particle-hole formalism. FinalIy, in sect. 5, the Yoccoz formula is derived from classical statistical mechanics, and corrections to it are obtained by making a semiclassical approximation.

2. The statistical method

In this section we confine our attention to the ground state band of a doubly even nucleus. In the quantum mechanical self-consistent calculations of the intrinsic state of the system, one essentially obtains, by minimizing the energy, an equilibrium configuration in a situation where different angular momentum states of the band are allowed to mix. As such, the intrinsic states is an admixture of physical states at equilibrium. The H F o r HFB theories to which we are referring here are of course formulated at zero temperature. There is, however, an analogous situation of a rotor at a finite temperature which we now describe. Suppose there is a rotor with available angular momentum states J = 0, 2, 4 , . . . and that it is at equilibrium with a thermostat at some temperature T. The equilibrium properties of this system are describable in statistical mechanics by a partition function Z; and by taking the zero of energy to be the J -- 0 ground state, we have h2 " Z(fl) =s~e (2J + 1) exp E--fl 2II J(J ÷1)1

,

(3)

20

R.K. BHADURI AND S. DAS GUPTA

with fl = 1/kBT, k B being the Boltzmann constant, and I the moment of inertia for the band. Note that different angular momentum states contribute to Z(fl), their equilibrium admixture being of the canonical form for a given ft. Note that both in the zero temperature quantum mechanical self-consistent method and in the finite temperature rotor problem, an equilibrium situation is being described where different angular momentum states can contribute to an expectation value. In particular, for the rotor at temperature T, the ensemble average ( j 2 ) is given by

(j2)

= s ~'even(2J+ 1)J(J+ 1)exp

-fl ~] J(J+

1/ Z(fl).

1)

(4)

If, on the other hand, we calculate the expectation value o f J 2 with respect to the normalized intrinsic state given by eq. (1), we get (~lJ2l 4~) = E

[ajl2j(d+l) •

(5)

d even

We now push the analogy between the two situations further by assigning an equilibrium temperature to the intrinsic state, and assuming that identical results are obtained by calculating ( ~ [Je I~ ) and the rotor ensemble average ( J a ) , provided the intrinsic state I~) is adequate to describe the equilibrium condition. In most cases, the H F state suffices for this purpose, but in situations where pairing is important, the HFB state should do better. Note that the intrinsic state contains a distribution of angular momentum states; we assert that this distribution is the same as that of a rotor in thermal equilibrium at a suitably chosen temperature. We shall presently see that this temperature is not a free parameter, but is determined uniquely by the properties of the intrinsic state and in fact does not appear explicitly in the final equations. Using this ansatz, we can equate eqs. (4) and (5) to obtain a solution,

hz laal 2 = ( 2 J + 1) exp I - fl

J(J + 1)] /Z(fl).

(6)

For a spherical nucleus, we know that laol 2 = 1 and [asl 2 = 0 for J ~ 0. This corresponds to fl = oo or T = 0. For deformed nuclei in the s-d shell, on the other hand, (jz) is typically between 15 and 20; from eq. (4) it follows that flh2/2I<< 1 in such eases. For deformed nuclei in the rare-earth region, (j2) may be of the order of 100, and flh2/2I should be very much smaller than unity. We would now like to evaluate fl for a given intrinsic state and eliminate it from eq. (6). For a deformed nucleus, since flh2/2I << 1, many angular momentum state3 contribute to Z([3) in eq. (3). In such cases, a semiclassical expansion of Z(fl) can be made by making use of the Euler-Maclaurin approximation 4),

f(n) =

f(n)dn+-}[f(O)-f(oo)]

-~[f'(O)-f'(oo)]

n=O 1

tit

+Y2-o-[f (O)--f

tit

(oo)]-~-o--~4o[f(v)(o)-f(vl(oo)]+ . . . .

(7)

DEFORMED

INTRINSIC

STATE

21

which yields the following series for Z(fl): z(~) = ~

1

v

+ + + + o f f _1 + + ~ f l ~

+ ....

(8)

For a well-deformed nucleus, since flh2/I << 1, the first term of the series, which is the classical partition function, dominates over all others. The above equation could be recast in the form

Z(fl) = Zc(fl)[l+~-h2fl ++O~4(~)2+~_+~h6(~)3+ "1 o.

,

with the classical partition function denoted by Zc(fl) = I/flh 2.

(9)

This is an expansion 5) in powers of h, and may be termed a semiclassical series. It should be borne in mind that we have used a constant moment of inertia parameter I for the entire band. The rotational energy content of the band is given by tz2

1 ,~z

(lO)

For a well-deformed nucleus, since the classical term dominates the series (8) for Z(fl), we obtain h2 1 aZc 1 2-7 (J~) ~ , (11)

Zc ~fl fl

which is merely a statement of the classical equipartition theorem, since an axially symmetric rotor has two degrees of freedom. We could easily make semiclassical corrections to eq. (11) by retaining the next term in the series (8) for Z(fl), thereby obtaining 1 hz _ ~,~ _ ( ( j 2 )

fl

j_~).

(12)

21

Note that eq. (11) or (12) relates the temperature directly with the rotational energy content of the intrinsic state. From eqs. (10) and (11), we see t that the intrinsic state may be regarded as a "heated u p " ground state; its energy relative to the J = 0 ground state band head being k T = l/ft. Since (j2) >> 1 for a deformed nucleus, the semiclassical correction is small; however, we shall see that it has important consequences on the normalization of the overlap function. Using the above equations, we can easily eliminate fl and Z(fl) from (6); taking the classical expressions (9) and (11), we obtain

]as[ 2 _ 2(2J+ 1) e_S(s+a)/
(j2)

t T h e a u t h o r s a r e i n d e b t e d t o t h e r e f e r e e f o r this c o m m e n t .

(13)

22

R . K . B H A D U R I A N D S. DAS G U P T A

while taking the next order corrections in the series yields the improved semiclassical formula [aj[ 2 = 2(2J + 1)e- s(s + 1)/( + ~) ((j2) + 9) (14) Note that statistical mechanics does not determine ( j 2 ) , which must be calculated from a quantum mechanical intrinsic state. If ( j 2 ) >> 1, then the distribution of the angular momentum states is completely specified by the statistical formula (14), or its approximation (!3). Formulae (:13) and (14) are tested numerically in table 1. The axially symmetric HF and HFB intrinsic states for some nuclei in the s-d shell have been obtained by Goeke e t a L 6) using the G-matrix elements of the Yale potential. They also calculate ( j 2 ) and obtain [as[2 by Peieds-Yoccoz projection, which are displayed in the first TABLE 1 The laj[ 2 obtained from a microscopic projection calculation 6) are compared with the predictions of the statistical formulae (13) and (14) for some s-d shell nuclei 8

Nucleus


22Ne prolate I-IF

18.26

22Ne prolate I-IFB (T = 1 pairing) 24Ne prolate I4F 2*Ne prolate HFB (T = 1 pairing) 2sSi prolate H F 2sSi prolate HFB (T = 0 pairing) a°Si prolate I-IF a°Si prolate I4FB (T = 1 pairing)

Calc. method

[aoI2

[a~l2

la,] 2

[at?

lasI ~

~ la,? f= o

Projection Eq. (13) Eq. (14)

0.098 0.1095 0.1057

0.373 0.3943 0.3826

0.334 0.3297 0,3244

0A51 0.1427 0.1435

0.0361 0.0374

1.0123 0.9936

15.07

Projection Eq. (13) Eq. (14)

0.124 0.1327 0.1271

0.426 0.4456 0.4304

0.319 0.110 0.3168 0.1063 0.3122 , 0.1081

0.0190 0.0202

1.0204 0.9980

17.09

Projection Eq. (13) Eq. (14)

0.107 0.1170 0.1126

0.384 0.4120 0.3991

0.319 0.3268 0.3216

0.132 0.1303 0.1314

0.0295 0.0307

1.0156 0.9954

12.59

Projection Eq. (13) Eq. (14)

0.150 0.1589 0.1509

0.473 0.4932 0.4742

0.289 0.2920 0.2889

0.076 O.0735 0.0760

0.0089 0.0098

1.0265 0.9998

25.96

Projection Eq. (13) Eq. (14)

0.070 0.0770 0.0751

0.288 0.3057 0.2989

0.319 0.3209 0.3159

0.208 0.1986 0.1977

0.0818 0.0826

0.9840 0.9702

25.95

Projection Eq. (13) Eq: (14)

0.071 0.0771 0.0751

0.289 0.3058 0.2990

0.319 0.3209 0.3160

0.208 0.1986 0:1976

0.0817 0.0825

0.9841 0.9702

21.66

Projection E q . (13) Eq. (14)

0.085 0.0923 0.0896

0.332 0.3500 0.3409

0.331 0.3301 0.3247

0.182 0.1727 0.1725

0.0565 0.0576

1.0016 0.9854

20.59

Projection Eq. (13) Eq. (14)

0.094 0.0971 0.0941

0.354 0.3629 0.3532

0.333 0.3310 0.3260

0.171 0.1642 0.1643

0.0500 0.0512

1.0052 0.9888

DEFORMED INTRINSIC STATE

23

row of the table for each nucleus. The corresponding lajI 2 calculated by eqs. (13) and (14) are shown in the next two rows. Note that the results displayed are for the ground state bands of 22Ne and 24Ne, and for the excited state bands of 28Si and 3°Si. In general the results agree within a few per cent, in particular the agreement is better with the semiclassical formula (14) when applied to the HFB intrinsic state. In the latter case, with the exception of 28Si, agreement is seen to be within three per cent for the lajI 2, this despite the fact that in the projection calculations deviations from the rotational spectrum are observed 6).

3. Microscopic interpretation of the intrinsic state temperature One interpretation of temperature, already noted, is that it gives the rotational energy content of the intrinsic state. In this section, we give a related microscopic interpretation to the same quantity. Let us assume that I~) has been obtained by a H F calculation for a given H. From eqs. (1) and (2) it is clear that the intrinsic state I~) is an eigenstate of the operator (H-(hZ/2I)JZ). This implies that h2

-- (mX, nX', iX, jX'[J2[~) = (reX, nX', iX, jX'IH[~), 21 where X, X ' are isospin indices; mX, nX' are particle orbitals; iX, j X ' are hole orbitals and [mX, nX', iX, j X ' ) is a two-particle-two-hole state t. Writing j 2 = j x~+2j 2 +Jz2 a n d recalling that j z [ ~ ) = 0, we obtain h2

2-1 {2(Jx)m,(Jx),~+ 2(Y,)m,(J,)nj - 26XX'(Jx)mj(J~),i-- 26xx'(J,)mi(J,),~} = V.mntj(a')", (15) where the right side is the antisymmetrised potential matrix element. Let us multiply b o t h sides ofeq. (15) by (J~),j and sum over nj. We note that Z

= Z

=

nj

nj

X

=Y

nj

= <

IS,

= 0.

nj

The exchange terms are of the type Z ~XX'(Jx)mj(Jx)ni(Jx)nj' nj

which is a sum of incoherent contributions and are, therefore, neglected. We thus have

h22(~ljzx[~)(jx)m~ ~ (a) -= gn~nij(Jx)nj, 21 ,j which means

(a)

nj

V~,~j(J~,),j = ~(J~)~,,

The argument of this section closely follows ref. 7).

(16)

24

R . K . B H . A D U R I A N D S. D A S G U P T A

where z = kBT is the temperature of the intrinsic state and we have used eq. (1!). According to Thouless' theorem 8), however,

Z

= Y

nj

nj

(t7)

where Fm~,,~ is the Tamm-Dancoff matrix, i.e.,

Fro,.,~ = (e,, - e,)6,,, 6,j + v~a~j.

(lS)

In the above equation, e,,, e~ are the self-consistent single particle energies. Combining eqs. (16) and (17) we get (see also sect. 4 of ref. 9))

Z Fm~,ng(d~)nJ = ~C(Jx)mi,

(19)

nj

which shows that z is one of the eigenvalues obtained in a particle-hole TammDancoff calculation of the H F solution. From eq. (16), it is clear that the column vector (Jx),,i is an eigenvector of the F-matrix with eigenvalue h2 -- -- 2 ( ~ l J 2 I~). 2I For reasonable interactions, the numerical value o f , for s-d shell nuclei is of the order of 2 MeV [see table 3 of ref. 7)]. Numerically, we have found that z is the lowest of the eigenvalues of the Tamm-Dancoffmatrix, although we cannot prove it analytically. 7. The overlap function The overlap function in the Peierls-Yoccoz theory is defined as

f(O) = (~le-Z°Jyl~),

(20)

which in turn determines the coefficients [asl2:

la,l

= ½(2J+1)

Iod o(O)I(O)

sin 0d0,

(21)

where we are considering the K = 0 band of a doubly even nucleus, and dgo(O) = Pj(cos 0). Inverting eq. (21), and noting that lasl 2 = 0 for odd ar, we get

f(O) = 2 laslZPs( c°s 0),

0 _< 0 -< 7r.

(22)

d even

The above equation immediately implies that f ( ~ - 0 ) = f(O). We can evaluate f(O) directly from this equation by substituting for [asl z from the statistical formulae (13) or (14). Using the classical expression (13), we get

f(O) -- ( j22 ) s ~. . . . ( 2 J + 1)e -s(s+ 1)/Ps(cos0).

DEFORMED. INTRINSIC STATE

25

For small 0, we can make the approximation t Pa(cos 0) g

Jo(2~jJ(J+ 1) sin ½0),

(23)

where Jo is the cylindrical Bessel function of order zero. This expression is good for all J up to order sin2½0. Note that the classical expression for [as]2 was obtained by ignoring the Euler-Maclaurin correction terms in Z(fi); to the same order of approximation we get, for small 0,

I(O) ~ ~

(4j + l)e-2J(2J÷wJo(2~/2f(2j+ l) sin ½0)dj = exp ( - s i n 2 ½0(j2)) ,.~ exp (-¼02(j2)).

(24)

The semiclassical expression (14) for [aj[ 2 was obtained by retaining the leading Euler-Maclaurin correction term in Z(fl); to the same order of approximation we get, using eq. (14),

f(O) ,~ 1 q_~ -4-+

(25)

Note that higher order terms in (24) or (25) could be obtained by using, instead of (23), the more complicated approximation to Pj(cos 0) correct to order sin4 ½0: Ps(cos 0) ~ Jo(2X/J(J-~+1) sin

½0)-½x/J(J+ 1) sin a ½0Jl(2x/J(J+ 1) sin ½0),

(26)

which introduces additional terms; for example (24) gets modified to

f(O) ~

[1--½ sin 4 ½0]e-Si.= +o

(27)

The main point that we want to make in this section is that the statistical expressions for lasl 2 lead naturally to a Gaussian form for the overlap function, which is found to be closely satisfied numerically 2). Peierls and Thouless 1o) have proved, in the case of bosons, that the overlap function is a Gaussian. 5. The moment of inertia We start with the general statistical mechanical expression 11) -

a = - &. @

(28)

For the axially symmetric rotor, using the classical expression (11) for = B-~ = \ ~ / . t The authors are indebted to Dr. C. S. Warke for pointing out this line o f attack.

26

R. K. BHADURI AND S. DAS GUPTA

hZlZ _ (HZ>-(H> z

Hence

~/

(ju)a

(29)

'

which may be rewritten as

(HJ 2) -- ( H ) ( J 2 )

h2

(30)

Eq. (30) is the Yoccoz formula 3). It could also be derived from the identity

9)

h2 = ( H J 2 > - < H > ( j a > 21 ( j 4 > _ (j2>2 ' and noting that for the classical partition function (9), ( j 4 > = 2 ( j 2 ) 2 . Thus the Yoccoz formula follows from classical statistical mechanics, and we can easily get the semiclassical corrections to it by using the series (8) for Z(fl). In this case, a little algebra shows that

o8 = y + Eliminating

1/fl

by eq. (12), we finally get, from (28), (H2>-2

or

[h2~

t/

"~

= (h2t2 \ ~ / {( J 2 > 2 ÷ + ( J z > + l - ~ } (Hj2>_(H>
~

2

2

2

2

2

{ +~-(J >+-vs-}

"

(31)

This is a generalisation of the Yoccoz result for the axially symmetric rotor, assuming, o f course, the validity of eq. (3). The authors are indebted to Dr. C. S. Warke for discussions leading to the derivation of the overlap function, to Prof. D. W. L. Sprung for drawing attention to eq. (26), and to Mr. C. K. Ross f o r a critical reading of the manuscript. The comments of the referee were found to be very useful.

References 1) R. E. Peierls and J. Yoccoz, Phys. Soc. A70 (1957) 381 2) G. Kipka, Advances in auclear physics, ed. M. Baranger and E, Vogt, vol. 1 (Plenum Press, New York, 1968) p. 183 3) J. Yoccoz, Proc. Phys. Soc. AT0 (1957) 388 4) R. K. Pathria, ia Statistical mechanics (Pergamon Press, Oxford, 1972) p. 162 5) L. D. Landau and E. M. Lifshitz, in Statistical physics (Addison-Wesley,Readiag, Mass., 1958) p. 96 6) K. Goeke, A. Faessler and H. H. Wolter, Nucl. Phys. A183 (1972) 352 7) M. K. Banerjee, D. d'Olivera arid G. J. Stephenson, Jr., Phys. Rev. 181 (1969) 1404 8) D. J. Thouless, Nucl. Phys. 21 (1960) 225; D. J. Thouless, in The quantum mechanics of many body systems (Academic Press, New York, 1961) p. 89 9) S. Das Gupta and A. van Ginneken, Phys. Rev. 174 (1968) 1316 10) R. E. Peierls and D. J, Thouless, Nucl. Phys. 38 (1962) 154 11) R. K. Pathria, in Statistical mechanics (Pergamoa Press, Oxford, 1972) p. 70