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Skyrme’s interaction in the asymptotic basis
P. Quentin
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P. Quentin. Skyrme’s interaction in the asymptotic basis. Journal de Physique, 1972, 33 (5-6),pp.457-463. <10.1051/jphys:01972003305-6045700>. <jpa-00207271>
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457
SKYRME’S INTERACTION IN THE ASYMPTOTIC BASIS
P. QUENTIN (*)Service de la Métrologie et de la Physique Neutroniques Fondamentales
BP n° 2, 91-Gif-sur-Yvette, France
(Reçu le 12 octobre 1971, révisé le 23 novembre 1971)
Résumé. 2014 Les éléments de matrice des parties à deux et trois corps, et de la partie spin-orbitede l’interaction de Skyrme sont calculés analytiquement dans la base des nombres quantiquesasymptotiques. Ces éléments de matrice ne font intervenir que des combinaisons de fonctionssimples, ce qui facilite grandement leur évaluation numérique. Nous donnons également, danscette même base, les éléments de matrice d’une interaction gaussienne. Et enfin nous proposonsune méthode de calcul des éléments de matrice d’une interaction de Yukawa (en particulier deCoulomb) à partir de ceux d’une interaction gaussienne.
Abstract. 2014 Matrix elements of the two-body, three-body and spin-orbit parts of Skyrme’sinteraction are calculated analytically in the asymptotic basis. These matrix elements only involvecombinations of some simple functions which makes their numerical evaluation rather easy.In the asymptotic basis, we have also calculated matrix elements of a gaussian interaction. Finallya method is proposed for the derivation of matrix elements of the Yukawa (and in particularCoulomb) interaction type from gaussian matrix elements.
LE JOURNAL DE PHYSIQUE ’
TOME 33, MAI-JUIN 1972,
Classification
Physics Abstracts12.00
1 Introduction. - Skyrme’s interaction [1] has beenrecently used in spherical Hartree Fock calcula-tions [2]. Promising results have been obtained inthe study of some super heavy nuclei [3]. It is probablymore efficient, when breaking spherical symmetry(but conserving axial symmetry), to perform suchcalculations in a deformed basis [4]. The purpose ofthis article is to calculate analytically the matrix ele-ments of Skyrme’s interaction in the basis of theso-called asymptotic quantum numbers [5] (in orderto simplify we shall refer to it as the asymptotic basis).With a somewhat analogous method we derive alsothe matrix elements of a gaussian interaction. Thistype of interaction has been taken often as an effec-tive interaction in Hartree Fock calculations [6], [7].But one of its interests is, in particular, that one canderive from the knowledge of its matrix elements,those of a Yukawa or Coulomb interaction.
1.1 THE ASYMPTOTIC BASIS. - The kets of thisbasis are formed by the eigenvectors of the axiallysymmetrical harmonic oscillator hamiltonian and thethird components of the orbital and spin angularmomenta. A ket of this basis will be labelled withobvious notations 1 nz > nl A > 117 >. DefiningI = ni/2 and M = .4/2, one knows [8] that the ket1 IM > is a ket of the standard basis (with just aphase factor) of the (2 1 + 1) dimensional represen-tation of the rotation group. Besides [4], [8], thereexist boson operators b:, b; in such a way that :
(*) Present address : Institut de Physique Nucléaire, Divisionde physique théorique, 91-Orsay.
where
Let us recall also the existence of boson operators a/with
In coordinate representation the corresponding (1)wave functions are :
where
(1) It can be noticed that the phase factor sgn(a - p) is omittedin Ref. [8]. Indeed, in that case, there was a degeneracy withrespect to the change of Q = A + E into - Q, and further-more for a given Q there was no coupling between states ofboth signs of A.In the present paper, as it will be seen later, the Moshinsky
transformation allows, a priori, every possible couplingsbetween states of both signs of A. Indeed, excepted for the L+and the L- operators, all the operators considered in this work,conserve the value of A. Therefore, in general, this phase factoris cancelled.An alternative way to introduce this phase factor is to keep, the
wave function of the Ref. [8] and to replace in the Relation (1)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01972003305-6045700
458
The functions un(x) and Lm(x) are defined by theirgenerating functions :
I.2 SKYRME’S INTERACTION. - This interaction canbe written as
i) The interaction V(2) = ¿ vJ) is a two bodyKj
interaction, composed of the first terms of the shortrange expansion of a central force [9]. Its expressionin coordinate representation will be :
where r = (ri - r2)/ 2 ; V is the gradient operatorassociated with r.
The coefficients ao, ai, a2, can be spin operatorsof the following type :
where Pa is the spin exchange operator.ii) The interaction V(3) = ¿ vù3J is a three body
ijk
short range interaction, the role of which is to simu-late many body effects :
iii) The interaction
is a two-body spin orbit interaction. It is also ashort range interaction. We shall consider, for ins-tance [9], [10] :
II. Moshinsky transformation brackets. - Weshall recall briefly some results concerning transfor-mation brackets between particle states of coordinatesri, r2 and relative and « center of mass » (2) states ofcoordinates : r = (rl - r2)/’2, R = (ri + r2)/-1/2-One dimensional brackets have been already calcu-lated in reference [11] and reference [12] :
This transformation bracket is just a rotation matrixelement di, m,, as it can be shown without any calcu-lations [13] owing to transformation properties ofthe kets |IM > (defined previously) under rotations.Using Messiah’s phase conventions [14] one finds
Transposition of n, and n2 involves no change ofthe transformation bracket n, n2 Il nN > but a
phase factor (-)". This property will be used inthe antisymmetrisation of the two-body matrixelements.An expression of three dimensional transformation
brackets is given in reference [4] and reference [11].Nonetheless one can apply the previous result to
the case of three independant oscillators correspon-ding to boson operators a+, b:, b; and then onesimply writes :
III. Two-body part of Skyrme’s interaction. -
III.1 a0 TERM. - Starting with the generating func-tion of un(x) defined in the Relation (4), one can show
where :
In the same way, one obtains for the remainder ofthe wave-function (x being the projection of r on x0yplane) :
(2) Of course in this case, R is only proportional to the real’center of mass coordinate. This coordinate is just taken foranalytical convenience.
459
One then deduces (i standing for :
with
where C0 = a/mào/h is the spherical harmonic oscil-0 0 3 2
lator constant ; and where coo is defined by Wo = W.L WZ.From a given n (resp. a and b) one deduces n’, N(resp. a’, A and b’, B) by means of selection rules(defined previously in Formula (11)) for the Moshin-sky coefficients.
It is worthwhile to notice that this matrix elementis independent of the deformation of the basis. Besides,because of selection rules valid for the Moshinskycoefficient, one can easily verify the conservation ofangular momentum projection.
II.2 al TERM. - To evaluate this matrix element
one can use the following relations between ôx ope-rators and bosons operators.
First of all, we apply the gradient operator on brasand kets and then we use the quadratures (13) and (14) :
III.3 a2 TERM. - From relations (16) one easily demonstrates
where
Using the techniques defined in the previous subsection one finds
IV. Three-body part of Skyrme’s interaction. - IV .1 ONE DIMENSIONAL PROBLEM. - We have to calculate
By means of the generating functions of un(x) defined in relation (4), the following expansion canbe derived
(3) In Sections III, IV, VI and VII we will ignore trivial spin dependence, of the form : >X1X3 >X2I4°
460
where
Moreover the C coefficients are equal to zero except for three numbers n, nl, n2, which verify the triangleconditions.We can now rewrite eq. (20) as
Let us notice that parity conditions on C coefficientsinvolve in particular that the previous matrix element
6
will be equal to zero for an odd ni quantity.i=1
IV. 2 Two DIMENSIONAL PROBLEM. - We shall usea method, already proposed in an other context [13],to reduce the two dimensional problem to the singledimensional one.
One can rewrite the defining Relation (1) as :
The operators b:, b§ are defined as linear combi-nations of the operators ai, ai [8] :
Replacing the b+ by the a+ operators, we trans-form kets |IM > into a sum of kets
The coefficients of such a transformation are, bydefinition [15], the Wigner rotation matrix elementsD£M.
Using Messiah’s phase conditions [14], one gets :
We can then deduce the two dimensional matrixelement. Making use of selection rules for the onedimensional matrix elements and of the angularmomentum conservation condition, one obtainsthese matrix elements as functions of the a and j8numbers :
where
V. Spin orbit part of Skyrme’s interaction. - Letus define S = s 1 + S2 ; L = iV + x c5(r) V .Thus the considered interaction can be writtenas :
where
By means of Relation (16) one can deduce
461
Thus one gets :
(It can be noticed that the minus sign in front ofthe first term in the sum, comes from the phasefactor sgn introduced in eq. 3).From this expression the corresponding matrix ele-
ments of L - are just obtained by interchanging(n, a, b) with (n’, a’, b’).When coupling bras and kets with the Moshinsky
coefficients
we obtain the following selection rule (with
In a Hartree Fock calculation, one has only tocalculate matrix elements with Qi = S23 and Q2 = Q4(Sl being the third projection of the total angularmomentum).The relation (30) can be written in that particular
case
Let ut define an operator 0 by L. = 0 + 0 1 ;thus lu = 0; b(r) (iôy). By means of relation (16),0 can be written as :
Making use of the selection rule (30), we get forthe matrix element of tu in the plane xOy :
Owing to the reality and symmetry of these matrixelements, one can deduce easily that the matrix ele-ments of L- will be given by :
When including the spin dependence, the matrixelements of Vs.o. are found equal to :
VI. Matrix éléments of a gaussian interaction. -We shall give now, the analytical expression for thematrix elements of a gaussian interaction. This inter-action, as previously noticed, ’is interesting for itselfand also for the calculation of matrix elements ofthe Yukawa (and in particular Coulomb) interac-tion type. We shall treat this question in the nextsection.
Let us consider such an interaction :
In the asymptotic basis the matrix elements of v12are factorized into elements on Oz axis and elementson x0y plane. In the one dimensional case from rela-tions (21) and (4) one can easily show that
In the same way, one finds for the two-dimensional matrix éléments :
462
By means of the generating function defined in relation (4) the W coefficients can be calculated, as follows (4) :
where C’ (n > p) is a coefficient of the binomial expansion. Finally one deduces for the total matrix element
VII. Yukawa interaction and Coulomb interac-
tion. - The Yukawa interaction can be reduced to
an integral of gaussian interactions [13] in the follo-wing way
with r relative coordinate as defined in Relation (6).Thus the matrix elements of the Yukawa interac-
tion can be written as
where p is always evenThe integral I,M,n is defined by :
Let us put some restrictions on the integer num-bers q, l, m, n when using the integrals ,1, " n found inFormula (40) :
From these conditions, one can deduce that inte-
grals of type 1 are always convergent (even in the
case p = 0). Introducing the deformation parameteroc = C2/C2 = wz/ W 1. one gets for the integrals of
type 1 :
The remaining integrals can be calculated by recur-rence relations. Two cases have to be distinguished :the prolate deformation case (oc 1) and the oblatedeformation case (oc > 1). We write a - 1 - - 8b2(8 = 1 for C( 1, 8 = - 1 for C( 1).
In the particular case of the coulomb force (p = 0),we have to evaluate the following integrals
(4) This result is found for a-b > 0. If a-b 0, one can usethe invariance of the W coefficient in the interchange of a(resp. a’) and b (resp. b’) (as it can be seen in relation (36)).
with
The restrictions previously recalled involve : that
m n - 1 and that n > 0.
The following recurrence relations for the J integralsare easily found :
463
and in the limiting case m = 0 ; n = 1 :
VIII. conclusion. - The definition of the asymp-totic basis in terms of boson occupation numbershas allowed us to calculate very easily an analyticexpression of matrix elements in the case of the
Skyrme’s interaction. This analytic expression israther interesting from a numerical computation pointof view. Indeed, such matrix elements only involvesimple combinations of functions independent offorce range and deformation parameters ; (precisely :functions A(p) and C(n, m, p) of the text and matrixelements d,,m,(n/2». The main part of such a simpli-fication lies in the breaking up of the Moshinskycoefficients into three similar parts. This simplicityis also due to the force we have considered. For
example this interaction gives a rather simple defor-mation dependence of its matrix elements (5).
Gaussian interaction and, all the more, the Yukawa(or Coulomb) interaction involve in addition to theformer functions, functions depending on the rangeof the force or on the deformation of the basis (func-tions W(;,’:,a’, b’) and J§§, n(b)). Nevertheless, for a giveninteraction, and for a given basis, these latter func-tions can be tabulated rather easily in considerationof the small number of arguments they possess.The author would very much like to thank
Dr. M. Gaudin for his initiation to some mathema-tical techniques used in this work. He is also greatlyindebted to Professor M. Veneroni who has suggest-ed this study. He acknowledges many stimulatingdiscussions with R. Babinet, H. Flocard and D. Gogny.He would like, at last, to express his thanks to C. Titin-Schnaider for a critical reading of the manuscript.
(5) For a given set of states 1, 2, 3, 4, we have some selectionrules. First of all, those derived from relation (30) which is justthe conservation of angular momentum projection. For thematrix elements we have considered here, we also get on theOz axis : ni + n2 + n3 + n4 = even (excepted for L+ andL- operators, where this sum = odd).
References
[1] SKYRME (T. H. R.), Phil. Mag., 1956, 1, 1043 ; Nucl.Phys., 1959, 9, 615.
[2] VAUTHERIN (D.) and BRINK (D. M.), Phys. Lett.,1970, 32B, 149.
[3] VAUTHERIN (D.), VENERONI (M.) and BRINK (D. M.),Phys. Lett., 1970, 33B, 381.
[4] TUERPE (D. R.), BASSICHIS (W. H.) and KERMAN(A. K.), Nucl. Phys., 1970, A 142, 49.
[5] MOTTELSON (B. R.) and NILSSON (S. G.), Mat. Fys.Skr. Dan. Vid. Selsk., 1959, 1, n° 8.
[6] VOLKOV (A. B.), Nucl. Phys., 1965, 74, 33.
[7] BRINK (D. M.) and BOEKER (E.), Nucl. Phys., 1967,A 91, 1.
[8] QUENTIN (P.) and BABINET (R.), Nucl. Phys., 1970,A 156, 365.
[9] VAUTHERIN (D.), Thesis, faculté des sciences, Orsay,1969.
[10] BELL (J. S.) and SKYRME (T. H. R.), Phil. Mag.,1956, 1, 1055. SKYRME (T. H. R.), Nucl. Phys.,1959, 9, 635.
[11] CHASMAN (R. R.) and WAHLBORN (S.), Nucl. Phys.,1967, A 90, 401.
[12] MUTUKRISHNAN (R.), Nucl. Phys., 1967, A 93, 417.[13] GAUDIN (M.), Private communication.[14] MESSIAH (A.), Mécanique quantique, Dunod, Paris,
1959, Appendix C.[15] WIGNER (E. P.), Group theory, Academic press, New
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