SEMICLASSICAL TREATMENT OF ANGULAR MOMENTUM

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SEMICLASSICAL TREATMENT OF ANGULAR MOMENTUM

R. Hasse

To cite this version:

R. Hasse. SEMICLASSICAL TREATMENT OF ANGULAR MOMENTUM. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-305-C2-313. �10.1051/jphyscol:1987247�. �jpa-00226516�

Colloque C2, supplQment au n o

SEMICLASSICAL TREATMENT OF ANGULAR MOMENTUM

R.W. HASSE

Gesellschaft fur Schwerionenforschung (GSI),

0-6100 Darmstadt 11, F.R.G. and Kernforschungszentrum Karlsruhe, 0-7500 Karlsruhe, F.R.G.

RQsum6 - Nous formulons la semiclassique pour les modules et z-composants d u moment angulaire domes. En utilisant comme fonctions d'onde des ondes planes proj6tk sur bon moment angulaire nous derivons des expressions pour le propagateur B une particule et des quantitCs dkrivkes comme la fonction de partition, la matrice de densit6 nonlocale, la densitk B une particule et la densite de niveaux B une particule - un trou. Dans l'espace de Wigner ils deviennnent proportionels aux contraintes d u moment angulaire 6(IRxPI/A - l ) et d((RxP),/h - m). Comme applications nous calculons les densitks B une particule et les densitks de niveaux B une particule - un trou, de m&me la partie imaginaire du potentiel optique avec une force B deux corps B portke finie et les nombres d'occupations par la partie r6elle.

Abstract - Semiclassics is formulated for fixed modulus and z-component of the angular momentum. By using angular momentum projected plane waves as wave functions, we derive semiclassical expressions for the single-particle propagator and therefrom derived quantities like the partition function, the nonlocal density matrix, the single-particle density and the one particle- one hole level density. In Wigner space they become proportional to the angular momentum constraints G(IRxPl/h - l ) and 6((RxP),/h - ^m).

As applications we calculate the single-particle and one particle- one hole level densities as well as the imaginary part of the optical potential with a zero range two-body interaction and the occupation num- bers via the real part.

1. Introduction

Thomas-Fermi theory or semiclassics is a convenient method to formulate average ground state properties of nuclei, like matter and level densities, ground state energy or kinetic energy density;

for an overview, see Ring and Schuck[l]. The same methods were also applied to properties of excited nuclei, like multiphonon level densities[2,3], response functions[4], or the optical poten- tial and quantities derived therefrom[5]. Only a few applications, however, treat good angular momentum like Jensen and LuttingerIG] who calculate angular momentum distributions in the Thomas-Fermi model. Other quantities like multiphonon excitation level densities and response functions are also experimentally accessible for given l and correlated occupation numbers and the effective mass of l-states could be compared with those of a Brueckner-Hartree-Fock calcu- lation.

In this paper, we therefore formulate semiclassics for given angular momentum and given z- component or summed over the z-components. In section 2, we derive the propagator for given l , m and other quantities. In section 3 the transformation to Wigner space is discussed. Section 4 contains applications to one-particle and one-particle one-hole level densities and section 5 to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987247

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the imaginary part of the optical potential and to the occupation numbers and the last section summarizes the paper. More details of the theory can be found in ref.171.

2. Semiclassics in coordinate space

For spherically symmetric potentials which commute with 2' ^and iz the projection onto good angular momentum !fand its z-component m is achieved by employing the eigenstates of j2 and jz,

< r l k l m >= 4aie jt(kr) Y&,(&)Y~,(F.). _{( I )} Here and in the foIlowing indices L, m or f! denote fixed modulus and z-component of the angular momentum or summed over the z-component, respectively. In semiclassics, the single particle propagator in coordinate representation

cfrn(r, r') =< rf!rnle-BHlr'f!m > ₍₂₎

is the central quantity. Here, P is a Lagrange parameter to constrain the energy, which mostly serves as Laplace-transformed energy, L1;Ac. ^For P = 0 one obtains the unit operator

cfcO

The same result can be obtained from the constraint

< r ( ( 2 l f 1)6(i2 - f!(f!+ 1)) 6(& - m)lrl >= P!,(r, r') 6(r - r') . (5)

As a result, the semiclassical traces become

TI-, = 4 /d3r/d3r' Plm(r,rl) 6(r - r') . (6) The factor of four comes from summation over spin and isospin whereas in the local quantities (not under traces) this is not taken into account. In treating the potential as locally constant the propagator can be evaluated by neglecting ti-corrections, i.e. using the asymptotic form of the Bessel function with the argument rnr2/Pti2,

Here and in the following f!(f!

+

strn(.> r') =< ^{rt.m16(~ - H)Irlf!rn} >= ,:jl Cfm(r, r'), (8) the partition function

zfrn ^{= Tr}cfrn(r,rl) ,

the density of states

g L r n ( ~ ) = Ttlm 6 ( ~ - ^{H ) =} l;:,

zfm,

where X is the Fermi kinetic energy measured from the bottom of the potential. Here, relative and center-of mass coordinates are defined as

r

+

2 ' ^s = r - r', ₍₁₂₎

and the potential V(R) is normalized to V(R = 0) = 0. With the propagator (7) the nonlocal density becomes

1 sin ( ~ r - r l l p $ ( ~ ) / h )

ptrn(r,r1) = -Ptm(r,rl) I

s Ir - rrl

where the local Fermi momentum is defined a s

and the same local assumption has been made as in the potential, i.e. rr' has been replaced by R2. From this one derives the local density

which vanishes beyond the classical turning point Rx where p F ( R ~ ) = 0 and the kinetic energy density

Since the quantities given above do not depend on the magnetic quantum number rn, the corresponding m-summed quantities simply obtain by multiplying by a factor of ( 2 l + I). The single-particle level density, summed over m reads

The limits of integration are determined by the classically allowed region defined by the step function. The level density for given angular momentum can also be found in ref. [6] in inte- grated form.

3. Semiclassics in Wigner space

Semiclassics is also conveniently formulated in Wigner space (R, P), where the relative coordi- nate s is Fourier transformed into PI

and the local limit is reached by integrating over all momenta. Since the Wigner transform of the Hamiltonian is equal to the classical Hamiltonian, the relevant quantities obtain simple forms.

Quantum mechanically and also semiclassically only l a n d m are good quantum numbers whereas classically all three components of the angular momentum are fixed. This means that the

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quantum average over the x, y-components of the angular momentum is represented by the azimuthal average

The classical limit then obtains from (81 e,,L,, < tz = t. The Wigner transform of eq.(5), hence is given by the classical counterpart

8 ( R x P/A - 1) = 6(RPsin 8lii - L) 6((R x P),/ti - m), ₍₂₀₎ where cos8 = R . P I R P . Then the trace becomes

Since in Wigner space and neglecting ti-corrections the expectation value of the product of two operators is equal to the product of the expectation values, all single particle quantities become proportional to 8 ( R x P / h - I). The propagator for instance becomes

which, summed over m, can be rewritten as

P;: t2ti2

c ~ ( R , P) = b(PRsin Blh - 1) exp -/3

(- ⁺

⁺

2m 2mR2

This now contains the familiar radial kinetic energy by virtue of PR = PCOS 0 and the centrifugal potential. The nonlocal spectral density

gim(R, P ) = 6(R x P / h - 1) 6(c - H ( P , R)) ₍₂₄₎ simply reflects energy and momentum conservation and the nonlocal density reads

pt,(R, P ) = 8 ( R x Plh. - 1) O(X - H(P, R)). (25) Experimentally, however, only angle averaged quantities are accessible. Hence, averaging over the polar angles 8, q5 the angular momentum constraint becomes

+I d cos e tie O(RP/~L - e)

S'(R, P )

-- /

-1 2 ~ R P J ~ '

The nonlocal density, for instance, averaged over all angles, becomes

The quantities derived above by Wigner transforming eq.(5) can also be obtained by direct Wigner transform of the semiclassically asymptotic quantities.

4. Densities ^and level densities

In this section, we calculate densities and level densities for given angular momentum by employ- ing an harmonic oscillator potential. The oscillator constant is denoted by hw = 41 M ~ V / N ' / ~ ,

A = ( 3 ~ / 2 ) ' / ~ h w . The partition function can be evaluated to give

and the level density becomes

The level density of the harmonic oscillator simply consists of step functions for given energy.

The local densities of an harmonic oscillator for fixed angular momentum is shown in Fig. 1.

One notices that only the f = 0 states contribute at the center. The maximum angular momen- tum for given energy follow from the step function in eq.(14), L(E) = E / ~ w .

Figure 1: Spin-isospin summed density of 200 particles for given angular momentum ( l = 0...6).

Next, we calculate the one-particle one-hole level density. Integrated over all angular momenta it is defined as

giplh(e) = 9 1 n z @(A - Hz) @ ( H i

-

+

9 f z h ( ~ ) = n i ~ z / d e i S d & @ ( A - H z ) @ ( H I - A ) 6 ( ~ - H l + H z )

x 6 ( R 1 x P l / h - l l ) 6 ( R 2 x P 2 / h - lZ)$(l - (Il

+

g:pmlh(~) = Wl?2JdL~Jd12JdmlJdm2@(A- Hz) @ ( H i - A ) S(E- H l + H z )

x 6 ( R I Pl sinBl/h - l l ) 6 ( & P 2 sinB2/fi - .t2)A(el&, e), ₍₃₂₎

where

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is the angle averaged triangular relation. By using the methods of refs.[2,3] it is transformed into a folding expression for fixed angular momentum

Here g 6 and gf,, are the one-particle and one-hole level densities, i.e. the single particle level densities restricted to energies above and below the Fermi energy, respectively,

Note that our result eq.(34) differs from the expression recently proposed by Krappe(91 who did not account for the angle averaging in the angular momentum conserving delta function but it is conform with the semiclassical limit of the response function of Delafiore[lO]. The result is shown in Fig. 2. The levelling off which starts at the Fermi energy and the later onset of higher angular momenta is due to the maximal angular momenta at the Fermi energy.

4.0

N

52 3.5 -

Figure 2: One-particle one- hole excitation level density o f 200 particles for given angular momentum ( L = 5...20).

I . I .

0 U1 20 30 40 50 60 70 86 90 100

Energy /MeV

5. Optical potential

Finally we calculate the imaginary part of the optical potential as well as the occupation num- bers. The on-shell value of the imaginary part of the optical potential including polarization and correlation graphs is calculated with a zero range two-body interaction

Without angular momentum constraints in coordinate space it is obtained from the imaginary part of the mass operator[5],

t , m in Wigner space it becomes

Since the force (36) conserves the momentum and, furthermore, since it puts all intrinsic coordi- nates at the same position R it conserves also the angular momentum. As a result, Wtrn(e, R, P) remains a single particle quantity and, hence,

The on shell value of the latter has been calculated in ref.151 as

where C = v ; r n / ( 2 ~ h ~ ) ~ and p ~ = p r ( R ) is the local Fermi energy of eq. (14) and the on-shell condition is P,2/2m = E - V(R). With the angle averaged angular momentum constraint the imaginary part of the optical potential for fixed l becomes

W'(E, R) = bL(Po, R) W ( E , R). ₍₄₁₎ As strength for the two-body interaction, va = 745 MeV fm3 is used which reproduces the experimentally known quadratic behaviour around the Fermi energy and a harmonic oscillato~

potential is employed. The result is shown in Fig.3.

Figure 3: Imaginary part o f the optical potential of 200 particles at R=5 frn in a har- monic oscillator for given an- gularmomenturn (!! = 2...16).

The Fermi energy is at 47 MeV.

Energy / MeV

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One observes that due to the classical condition

for energies below the Fermi energy only a few angular momenta contribute. The higher partial bound states have a larger W L and since the single-particle spreading width is proportional to the imaginary part of the optical potential also a larger spreading width. This seems to be in agreement with recent observations of Langevin-Joliot et. al. Ill] . At high energy the imaginary part of the optical potential behaves like the centrifugal potential.

The correlation and polarization graphs of the mass operator, i.e the real part of the optical potential, usually is obtained from the imaginary part via the dispersion relations 151,

where P denotes the principal value. For fixed l , m the occupation numbers are then defined as

where the on-shell condition is given by

In using the same methods as in ref.[5] the real part of the optical potential has been calculated and the local density is obtained with the help of the Perey-Buck potential. The resulting occu- pation numbers are shown in Fig.4. One observes that the occupation gap becomes smaller for increasing angular momentum, i.e. 0.9 ... 0.5 for [ = 0...6 . This can modify the calculations of charge densities by Hartree-Fock methods with the inclusion of semiclassical correlations where usually the occupation numbers are not [-dependent [ 1 2 ] .

1.0

Figure 4: Occupation numbers o f 200 particles in a Perey-Buck potential for given angular momentum (.t = O...G). E

The Fermi energy is at -8 MeV. Each 0.5

curve is shifted by 0.1(6 - l ) .

0.0

-50 0 50

E / MeV

Semiclassics has been generalized to cases where the the modulus and the z-component of the angular momentum is fixed. We derived the propagator and other quantities in (r,rl) - space and in Wigner space (R, P). In coordinate space quantities under traces are easily evaluated because angular momentum projection can be done solely in the traces. In Wigner space all single-particle quantities are proportional to an angle averaged angular momentum conserving delta function. As applications, densities and one-particle one-hole excitation level densities have been calculated for an harmonic oscillator potential and the imaginary part of the opti- cal potential has been obtained for a zero range two-body interaction and with an harmonic oscillator potential and the correlated occupation numbers were obtained from the real part of the optical potential with an underlying Perey-Buck potential. The schemes outlined are very flexible and can be applied to arbitrary potentials and work is under way to employ realistic Woods-Saxon potentials and also deformed potentials which do not commute with j2, iZ.

Acknowledgement

The author likes to thank Dr. P. Schuck for the suggestion of this subject.

References

[I] P. Ring and P. Schuck 'The nuclear many-body problem' (Springer, New York 1980) ch. 13 121 G. Ghosh, R.W. Hasse, P. Schuck and J. Winter, Phys. Rev. Lett. 50 (1983) 1250 [3] A. Blin, B. Hiller, R.W. Hasse and P. Schuck, Nucl. Phys A456 (1986) 109.

[4] P. Schuck, G . Ghosh and R.W. Hasse, Phys. Lett. 118B (1982) 237

[5] R.W. Hasse and P. Schuck, Nucl. Phys A438 (1985) 157; Nucl. Phys A445 (1985) 205 [6] J.H.D. Jensen and J.M. Luttinger, Phys. Rev. 86 (1952) 907

[7] R.W. Hasse, Nucl. Phys. , in print

[8] L.I. Schiff 'Quantum mechanics' (McGraw-Hill, New York 1968); A.R. Edmonds 'Angular momen- tum in quantum mechanics' (Princeton Univ. Press, Princeton 1957)

[9] H.J. Krappe, Phys. Lett. B181 (1985) 191 [lo] A. Delafiore and F. Martera, Firenee preprint 1986

[ll] H. Langevin-Joliot, E. Gerlic, J. Guillot, J . Van de Wiele, Y. Van der Werf and N. Blasi, Nucl. Phys A462 (1987) 221

(121 R.W. Hasse, B. Friman and D. Berdichevsky, Phys. Lett. B181 (1985) 5