DENSJT: Statistical Shell-Model Level Density Computer Code Description and User's Manual

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. . OR NLITM -5486 DENSJT Statistical Shell-Model Level Density Computer Code Description and User's Manual Bill Dalton DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. 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Neither the United States nor the Energy Research and Development Administrat ion/United States Nuclear Regulatory Commission , nor any of their employees , nor any of the ir contractors , subcontractors , or the ir employees , makes any warranty, e:.po'eH u o i on~li~u , uo d••u'""' diiY I"Ydl lidi.Jilily ur '"'IJUII•ii.Jil i ty fur the accuracy, completeness or usefulness of any information , apparatus , product or process d isclosed , or represents that its use would not i nfringe privately owned r ights . , .. ORNL/TM-5486 Contract No. W-7405-eng-26 PHYSICS DIVISION DENSJT Statistical Shell-Model Level Density Computer Code Description and User's Manual Bill Dalton ~-----NOTICE-----· '0 1 U al:j)Urt woo Jlropuocl a.o an O':'"i':>'.'!'! nf ~nr" sponsored by the United States Government. Neither the United States nor the United States Energy Research and Development Administration, nor any of their employees, nor Any nf their contra~ton. subcontractors, or their employees, makes any warranty, express or imp tied, or assumes any legal Uability or responsibility for the accuracy, completeness or usef\Jtness or any infuuuatiuu, .a.ppuatUJ, product or prooeu disclosed, or represents that its use would not infringe privately owned rights. JULY 1976 OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37830 operated by UNION CARBIDE CORPORATION for the ENERGY REGEAROII AND DEVELOPMENT ADNINISTKATION IJISTRIBUTION OF THIS OOCUM£NT IS UNUMlT~ THIS PAGE WAS INTENTIONALLY · LEFT BLANK -~ . I. INTRODUCTION •••••• II. LEVEL DENSITY FORMULA III. FIXED-J AVERAGING IV. FIXED-T DIMENSION iii CONTENTS V. REDUCTION FORMULA AND BAS-IC INPUTS • VI. PROGRAM DESCRIPTION VII. INPUT DATA •. VIII. DESCRIPTION OF OUTPUT IX. TEST ON CODE DENSJT APPENDIX I: BASIC INPUTS APPENDIX II: BASIC SUMS REFERENCES AND FOOTNOTES . . . . . Page 1 2 3 4 5 6 11 12 14 17 23 27 1 I. INTRODUCTION Detailed calculations 1 - 3 have demonstrated that the shell model level density is approximately Gaussian in shape. Consequently, these level densi- ties can be described by calculating the first few low moments of the Hamil- tonian. In many cases such as in Hauser-Feshbach calculations, where de- tails of individual levels are not needed, the statistical method is simpler and more economical than solving the shell-model problem exactly. This is especially true when the number of active nucleons becomes larger than two. The FORTRAN code DENSJT described here is constructed to calculate the appropriate moments and dimensions needed in describing the Gaussian level density for fixed values of the total angular momentum (J) and isobaric spin (T = T ). The input for this code is the same as in the usual shell model 0 codes, namely, the single particle orbital properties (na' ta, ja, E ) and a the antisymmetric two-body matrix elements ~T • To calculate the moments aSJ.I\1 for fixed values of the angular momentum (J) we use the method described by 4 Ginocchio. The averages of operators over the entire configuration space 5-7 are calculated by using the reduction formula of Ayik and Ginocchio. In this formula, all of the terms ("basic inputs" as defined by these authors) can be expressed in terms of the usual shell model input mentioned above·. Ginocchio, with Ayik 7 • 8 and Yen 3 • 9 demonstrate the usefulness of this re- duction formula, but unfortunately their "basic inputs" were derived for re- stricted sets of orbitals, namely those sets for which all of the ja values differ. For use in the present code we have derived all of the "basic inputs" needed in the reduction formula but without any restrictions on the orbitals. 2 These "basic inputs" were first derived using the directed-diagram method of Ayik and Ginocchio 6 to obtain the contractions. To verify that c'ontributing 10 diagrams had not been omitted, we wrote a symbolic code DIGRAM which was used to form the contractions, find the phase, count the number of left con- tractions, count the duplications due to operator symmetries and appropriately label the operator indices. In the following pages the formulae used in DENSJT are described. The basic inputs are listed in convenient tables. A block flow chart showing the general structure and the specific function of each subroutine is included. All necessary information for the input ~~g outPut of this code· is i!P.sr.riheci in Sections VII and VIII. II. LEVEL DENSITY FORMULA J 4 G In recent literature ' ' it has been demonstrated that the Gaussian distribution [ [. ] :>.] - E-t S(J E) = (2J+l)n(J) J ' 1/2 exp 2 (27T) C1J J (1) gives a reasonable description of the fixed-J shell model level density. In these calculations, the dimension n(J) the centroid EJ and thP. winth oJ are calculated from the same truncated single particle basis used in the exact ~hell model calculation. The total number of states at a fixed-J value is the product (2J+l)n(J). The factor (2J+l) represents the magnetic degeneracy. The factor n(J) rep- resents the number of states at a fixed-J value not counting the 2J+l mag- netic degeneracy. That is, n(J) is the fixed-J, fixed-M state density at each M, in the interval [-J,J]. The function n(J) is given by the following 3 formula n(J) = D(M=J) - D(M=J+l). (2) Here, D(M) is the total number of states with magnetic projection M. Com- 3-9 . parisons with exact calculations have demonstrated that the following Gaussian formula with excess correction (fourth moment) gives a good descrip- . . tion of the state distribution over the magnetic projection (assuming that the configurations are D(l:f) symmetric about M = 0). - z2 = _d_ e 2 [1 + * (z 4 -6z 2 +3)]. /2;>. The normalized variable (Z), the width (>.) and excess (y) are given by the following formulae: Here < >means average over all states. III. FIXED-J AVERAGING (3) (4) Using a perturbed distribution method, Ginocchio 4 obtained the following low moment expansion formula for finding the fixed-J average of an operator K. <K>J = <K> [1 +aD~ (x)/(1 + { 4 Y D 2 (x))]. Here, D 1 , D 2 , and a are given by the following formulae . 2 2J+l a = _21 t<~z K> - 1~ X = ----v;-' . .,_ <K> 3 4 X -3, D 2 (x) = X 1oi + 1s. (5) (6) (7) The fixed-J moments 2 EJ and a 3 appearing in (1) are obtained by setting K = H 4 and K = (H- EJ) 2 respectively in (5) and (6). For these calculations, the following operator averages over all states are needed 2 2 2 <J 2 >, <H>, <H >, <Jz H>, The reduction formula used to calculate these averages is described in Section v. IV. FIXED-T DIMENSION The dimension (d) appearing in (3) depends upon 'the isobaric spin speci- fication. With m and m representing the number of active protons and neutrons, p n we define T and m by the formulae 0 with inverses M=M +M,T p n o = M -M P n 2 M (T ) = ! + T , M (T ). = tl - T • p o 2 o n o 2 o The dimension of states with T = T resulting from all states with T > T z 0 - 0 is given by (8) (9) d (T ) = [N l [N l ClO) u Mp(To) Mn(To) Here, N represents the number of single particle states. The d~ension with fixed T T z T is given by 0 d = d(T) - d(T +1). 0 0 (11) With the assumption that the matrix elements of the Hamiltonian are indepen- dent of T one can derive the following relationship z T=T <H> o T =T z 0 (12) .. 5 In order to calculate the level density difference for neighboring T = T z nuclei, one cah run the code DENSJT with T = T = T and then with T = T = T + 1. z 0 z 0 V. REDUCTION FORMULA AND BASIC INPUTS From the work of Ginocchio and Ayik 6 we have the following two-configuration reduction formula for the. average of an operator K M ,M <K> p n p p •D p n t t P n p n l: t =0 n (13) In this formula the indices n and p indicate the neutron and proton configu- rations. The neutron and proton single particle states are assumed to be the p p same so that N p = N n N. The quantities Dtptn are the "basic inputs" as de- 6 p n fined by Ginocchio and Ayik. Here P and t indicate the particle rank of p p the operator and the left contraction count in the proton space. The basic inputs for the isobaric spin basis were derived by Ginocchio, Ayik and Yin 8 ' 9 for the case where the single particle orbitals all differed in their j-values. This restriction amounts to replacing the delta function o( ja,jB) by o(a,B) in the basic inputs where a represents the entire orbital; that is, a : (j ,~ ,N ). To count and categorize the contractions for each a a a operator, these authors used a directed-diagram method. 6 In the present work, we used the directed-diagram method to derive the. basic inputs needed in (13) without the above-mentioned restrictions on the orbitals. As a check on these formulae, we wrote a computer program DIGRA}1 which uses a permutation algorithm to systematically exhaust all possible conLracLlun~ and cound duplications due to·the operator symmetries. 6 2 2 The basic input formulae for operators through H J are listed in z Appendix I. These inputs are expressed in terms of an array func- tion U(N), each element of which is defined in Appendix II. To simplify the expressions in this table, we use the following definitions: - wJ,l e f aBJ.lV L aB J.lV f ae = II + Ha, s) (14) (15) (16) The ~; are the usual antisymmetric two-body matrix elements with symmetry CXt.>J.lV properties j +j -J-T = (-l) a B WJT BaJ.lV (17) VI. PROGRAH DESCRIPTION The entire program is in the FORTRAN IV language. The different parts of the computation are made in separate subroutines. The main routine serves only to direct control in calling the subroutines in sequence. All input data are read in subroutine PUTIN and all output information is written from subroutine ALLOUT. The Block Flow Chart shows the general structure of DENSJT and indicates the purpose of each subroutine. Calculations of those Basic Sums with one index are grouped together in subroutine USUMl, those involving two indices are grouped together in subroutine USUM2, etc. Because of the antisymmetric and Hermitian properties (17) of the two- body matrix elements ~,T , it is only necessary to have a table of the . ala2a3a4 7 unique ones. It is assumed that input data at least contains the unique sub- set of matrix elements WJ,T which satisfy the following conditions (where .ala2a3a4 each a. can range over-the set of single particle orbitals) 1 a ·< a a < a 1 - 2' 3 - 4 (18) (19) The subroutine CHEK checks to see if the called matrix element satisfies con- ditions (18) and (19). If these conditions are not satisfied, then the anti- S)~metric and Hermitian properties in (17) are used to express this matrix element in terms of one that does satisfy (18) and (19). After the above checks, the appropriate matrix element is looked up in the input table. The index (K) which locates the matrix element in this table is computed by the formula K = 225 + 2ZO + a3215 + a4210 + 2J22 + 2T. (20) al a2 This same formula is used in the subroutine PUTlN which makes the table. Our reason for using this packing formula is that it makes it possible to estab- lish the index uniquely using a thirty-two bit computer wo~d. We emphasize here that the input table may contain matr.ix elements in addition to those satisfying (18) and (19). These additional ones will not be used in the computation, however, as they are not needed. Their inclusion will simply make the data table larger and consequently increase the look-up time. To look up the matrix element corresponding to the index K, a binary search method is used in subroutine GETIND. This method consists of determining if the index K is in the upper or lower half of the table. If it is in the upper half, the lower limit on the table is changed to the mid-point. This process 8 is then repeated on the upper half of the table until the search converges to the location of the index K. This look-up method, of course, assumes that the matrix element table has been constructed such that the indices computed according to (20) appear in increasing order. There are two limits on this computer code. The first is the dimensions of the arrays. The present dimensions are adequate to handle a table index K through 5000. The dimensions on the single-particle orbitals are adequate for 20 different orbitals. The second limit on this code is the particular packing formula in (20). The indices a. must be less than 32 = 2 5 if this formula is to be used. Con- l. sequently, this packing formula cannot be used for cases where the number of single-particle orbitals exceeds 31. The Block Flow Chart for DENSJT is given on the following pages. Block Flowchart for DENSJT PUTINI Reads in single particle properties and table of two-body matrix elements. ~ FACTOJ Makes R,n(n') in array FACLOG(N) and makes . 0 ALF(N) . log 10 (n~) in array • USuMJI Makes basic sums U(3), U(4), U(ll), U(13), U (19) and U (38). • USUM2j Makes basic sums U(l), U(2), U(5), U(6), U(l4), U(l5), U(l6), U(22), U(23), U(27), U(28), U(33), U(34) and U(42). • DELONGJ Makes those basic sums involving the delta func- tion o(ja,j~): U(8), U(9), U(lO), U(l7), U(l8), U(25), U(26 , U(29), U(30), U(31), U(32), U(35), · U(39), and U(43) .+ _____ ,CHEK I lcHEK I CAYIK Makes sums W(a,S), V (a , S ) , Q (a , S) , R (a , S) , S (a , S) , T (a , S) + t lcHEK I 9 USUM4 I Makes the basic sums U(7), U(l2), U(20), U(21), U(24). and U_(40) + xxzz I Makes the basic sums U(37J, U(44), and U(45) -- t FORSIX I - Make-s .the ~aeii~ sums U{36)~ U(41). and U(46) : -- - . I JZSQ I Makes average <J 2, I z T=T 0 I HWIDJT I Makes average <H 2 >T=To. Makes I TZHH .. .. • .. - - - - - - - - .. -- I CHEK I I SIXJ I I CHEK I SIXJ I CHEK I SIXJ ··-- ·-" JZPOW I 2 Makes average <J 2 > at given values of M ,M p n AHNP I Makes.average <H> at given values of Mp,Mn ++ L DINO I JZHAMJ ' Makes average <J 2 H> z at given values of M ,M L----~-----P ____ JL t+ I DINO I HDS I 2 Makes average <H > at given values of M ,M p n -~ I DINO I I AHNP I JZHH I Makes average <J 2 2 H 2 > at given values of M ,M p n DINO I I JAHAM I I AHNPI 10 ALLOUT I Prints or punches dimensions, centroids, and widths. I End I CHEK I Applies symmetries and appropriate phases needed to look up two-body matrix elements GETIND I Looks up two-body matrix elements using a binary search method. ·•' 11 VII. INPUT DATA All input read statements are in subroutine PUTIN. The data card organization is given as follows. The format for each card is also given. Card 1 Col. 1-3 Col. 4-6 Col. 7-9 Card 2 Col •. 1-3 Col. 4-6 Col. 7-9 Col. 21-41 Card NSPJ+2 Col. 1-5 . Col. 6-10 Col. 11-15 Col. 16-20 Col. 21-25 Col. 26-30 Col. 36-56 Format {3132 NBAR - isobaric spin T ::::; (M -M )/2 0 p n NUCL number of active nucleons = M +M. p n NSPJ = number of single-particle orbitals. NSPJ+l, Format (3I3,11X,lD20.8) NR(K) radial quantum number for the Kth orbit. NL(K) = orbital angular momentum for the Kth orbit. NJ(K) = twice the angular momentum value (£ ± }) for the Kth orbit. EN(K) = single-particle energy of Kth orbit. LA, (LA,LB,LB,LC, are . ' LB, the left-to-right LC, single-particle orbital LD, indices on the two-body matrix elements, each may range from 1 through NSPJ.) LJ = twice the total angular momentum value of the two- body matrix element. LT = twice the total isobaric spin (T 0 or 1) for the two-body matrtx element. W =value of the two-body matrix element. Using the packing formula in (20), an index KODE(N) is calculated from 12 LA, LB, LC, LD, LJ, and LT. The N-th matrix element is stored in array TMAT(N). The matrix element table is described then by arrays KODE(N) and TMAT(N) where N = 1---Kount. Kount is the total number of two-body matrix elements read in. Kount is found by accumulation on the index N and needs not be read in as separate data. The matrix element table may include all of the two-body matrix elements for the orbitals involved. However, the ~ook-~p P+Q~eQ~~e q~~cttP~Q in VI ., selects out only those matrix elements which satisfy conditions described· in (18) and (19). It is thus only necessary that the table include those matrix elements which satisfy these conditions. The order on the matrix element table is important. The search procedure assumes that the index Kode(N) is larger than the index Kode (N-1). A convenient way to guarantee that one has this increasing order is to construct the input data in the usual odometer order on the indices. This is the order presently used in most matrix ele- ment codes at ORNL. VIII. DESCRIPTION OF OUTPUT All output statements are described in subroutine ALLOUT. The averages of operators needed to calculate the dimension D(M) centroid J and width crJ are calculated before ALLOUT is called and stored in the following names: ZLAM = ') ;>.. ... = <J 2> z SZLAM <J 4 = > z EHM = <H> EHMZ = <J 2 H> z EHMS 2 <;H > DIMTZ = d = d(T ) - d(T +1) 0 0 13 In ·subroutine ALLOUT the following quantities are computed and stored in the indicated names 1/2 WIDM = A = [<J 2 >] = M-space width z EXCS = y = <J 4 .> I A 4 - 3 = M-space excess z ALPE = l [<J 2 H>/A 2 <H> - 1] = a for centroid 2 z ALPWD = l [<J 2 (H - <H>) 2 >/A 2 <(H - <H>) 2 > -1] = a for width 2 z ECTALL = <H> = scalar energy .centroid (all J included) 1/2 EWDALL = [<(H - <H>) 2 ] = scalar energy width (all J included) The above five quantities along with the dimension DIMTZ, the isobaric spin value and the nucleon number are printed out. The dimension D(M) in Eq. (3) is computed in integer steps in the range 0 ~ M ~ 4A in array DIMM(N+l). Notice that D(O) = DIMM(l). The program prints out the following quantities J, n(J), e 3 , a 3 for integer values of J in the range 0 ~ J ~ 4A. If the output is punched as data for another program, it is best to calculate S(J,E) using Eq. (1) in that program. We again emphasize that n(J) here does not include the 2J+l magnetic projection count. 14 IX. TEST ON CODE DENSJT ·. The first test we made on the code DENSJT was to calculate by hand the basic sums for a simple but non-trivial case. We used two orbitals j = 1/2 and j = 3/2. The single-particle energies and two-body matrix elements for the unique subset were set equal to one. The remaining matrix elements were obtained from these using the symmetry relations. The 46 basic sums with the exception of U(36) and U(46) were calculated by hand and useu Lo V4lidate the comp1Jt.er. results for these sums. U(41), U(36), and U(46) are calculated together in subroutine FORSIX. The same loops and look-up values for the matrix elements were used to cal~ulate all Lhtee. Decau8~ of the lonsth of these sums (involving 6 indices) only U(41) was calculated by hand. The basic sums validated using this hand check are listed as follows. Basic Sums for Hand Check 1 l.'l.UU 24 .:;a.oo 2 36.00 2.5 198.00 3 o.uu 26 190.00 4 6.00 27 330.00 5 30.00 28 330.00 6 36.00 29 165.00 7 27.00 30 165.00 8 150.00 31 165.00 q 216.00 32 -66.00 10 180.00 33 0.00 11 5.50 34 27.50 12 66.00 35 -55.00 13 5.50 36 47.71 14 -5.50 37 8.47 15 33.00 38 10.38 16 27.50 39 0.00 17 137.50 40 -9.0 18 137.50 41 40.71 19 5.50 42 o.oo 20 -8.50 43 0.00 21 50.50 44 14.56 22 27.50 45 30.50 23 -5.50 46 42.00 The second test on the program as well as on the basic inputs is a 15 comparison we made with the calculations of Gino'cchio 4 ·for 20 Ne. In both calculations, the standard 63 Kuo matrix elements for the s-d shell were used. The orbitals are Od 5 iz (-4.15)·; ls 112 (-3.28), and Od 312 (+.93). The fixed-J, fixed-T dimension, centroid, and width are given below, where, for comparison, the excess was set equal to zero. When plotted on the same scale, these three curves coincide with the dashed curves in the plots of Ginocchio. 20 Results for Ne Comparisons (y - 0.0) J Dimension Energy Centroid (MeV) Energy Width (MeV) 0 .161D+02 -.170D+02 .76D+Ol 1 .423D+02 -.172D+02 .76D+Ol 2 .542D+02 -.175D+02 .75D+Ol 3 • 511D+02 -.180D+02 .74D+Ol 4 .388D+02 -.186D+02 • 71D+Ol 5 .245D+02 -.194D+02 .67D+Ol 6 .131D+02 -.204D+02 .61D+Ol 7 .603D+Ol -.215D+02 .50D+Ol ·a .238D+Ol -.228D+02 .• 29D+Ol In the present version of DENSJT, the excess is calculated and generally differs from zero. In order to double check copies of the computer code, the follow- ing results with calculated excess are included. 20 Ne Check Results Using Kuo Matrix Elements J Dimension Energy Centroid (MeV) Energy Width (MeV) 0 .136D+02 -.167D+02 • 77D+Ol 1 .370D+02 -.1700+02 .76D+Ol 2 .500D+02 -.174D+02 .75D+Ol 3 • .J03D+02 -.i80D+02 .74D+Ol 4 .406D+02 -.186D+02 • 71D+Ol 5 .268D+02 -.194D+02 .68D+Ol 6 .144D+02 -.203D+02 .62~01 7 .629D+Ol -.214D+02 .51D+Ol 8 .212D+Ol -.233D+02 .76D+OO 16 The following CPU run times on the CDC Cyber-70 computer were obtained: 3 orbitals in the s-d shell: CPU = 1 second. 10 orbitals in the s-d-f-p shells: CPU = 97 seconds. 1) 2) 3) 4) 5) 6) 17 APPENDIX I: BASIC INPUTS 2 2 . . 2 2 2 for <J 2 >, <H>, <J 2 >, <H >, and <J 2 H > 2 <Jz> <H> D 01 (H ) = U(3) 01 n D 02 (V ) = U(1) 02 n Dii (W) = U(2) D 03 (J 2 H ) = U(l3) 01 Zn n D 03 (J 2 H ) = U(3) U(11) - U(13) 02 Zn n 7) ~ 12 (J 2 H ) = U(11) U(3) 11 Zn p ~) D 22 (J 2 V ) = U(11) U(1) . 21 Zn p 9) 10) 11) 12) 13) 14) D 04 (J 2 V ) = U(14) + U(_16) 02 Zn n 04 2 D 03 CJ 2 nVn) = U(1) U(11) - U(16) D 13 (J 2 W) = U(15) 11 Zn D 13 (J 2 W) = U(l1) U(2) - U(15) 12 Zn 18 15) o~;(H~) = U(3) U(3) Oll(H H ) . 16) = U(3) U(3) ll np 17) o 03 (H V ) 02 n n = U(5) 18) o 03 (H V ) 03 n n = U(3) U(1) 19) o 21 (H V ) 21 n p = U(3) U(l) 20) o 04 ciJ 02 n = U(7) 21) 0 o4.(V2) 03 n = U(8) 22) 0 o4(V2J 04 n = U(l) U(l) 23) o;;cvnvp) = U(l) U(1) 24) 0 22(W2) 11 = U(l2) 25) 0 22 (W2) 12 = U(9) = 0 22 (W2) 7.1 26) 0 22 (W2) 22 = U(2) U(2) 27) o 12 (H W) 11 n = U(6) · 28) o 12 (H W) 12 n = U(3) U(2) 29) 13 '\2(VnW) = 11(10) 30) o 13 (V W) 13 n = U(l) U(2) <J2H2> z 31) 022Ci H2) 11 Zn p = U(ll) U(4) 32) 0 22(J2 H2) 21 Zn p = U(ll) U(3) U(3) 19 33) 0 42 (J2 V2) 21 Zn p = U(ll) U(7) 34) 0 42 (J2 V2) 31 Zn p =. U(ll) U(S) 35) 042(i V2) 41 Zn p = u (11) U(l) U(l) 36) 032(J2 H V ) 21 Zn p p = U(ll) U(S) 37) . 032 (J2 H V ) 31 ·Zn p p = U(ll) U(3) U(l) 38) o 13 (i H H ) 11 Zn n p = U(13) U(3) 39) o 1 ~(i H H ) = [U(3) U(ll) - U(13)] U(3) 12 Zn n p 40) 014(J2 V H ) = [U(14) + U(16)] U(3) 12 Zn n p 41) 014(J2 V H ) = [U(l) U(ll) - U(16)] U(3) 13 Zn n p 42) 023(J2 H V ) 21 Zn n p = u (13) U(l) 43) 023 (J2 H V ) = [U(3) U(ll) - U(l3)] U(l) 22 Zn n p 44) 024(J2 V V ) = [U(l4) + U(16)] U(1) 22 Zn n p 45) 024(J2 V V ) = [U (1) U(ll) - U(16)] U(l) 23 Zn n p 46) 0 04 (J2 H2) 01 Zn n = U(l9) 4 7) 0 04 (J2 H2) 02 Zn n = u(11) U(4) + 2U(13) U(3) - 4U(19) 48) 0 04 (J2 H2) 03 Zn n = U(ll) U(3) u (3) + U(l9) - 2U(13) U(3) 49) 0 06(J2 V2) 02 Zn n = U(20) + U(21) SO) 006(i V2) 03 Zn n = u(18) + 2U(l7) + 2U(45) + 2U(35) + U(7) U(ll) - 2U(46) - 2U(21) 20 51) o 06 (J 2 v 2 ) = 2 [U(14) + U(16)] U(l) + U(ll) U(8) 04 Zn n + U(20) + U(21) - 4U(18) - 2U(17) - 2U(35) 52) o 06 (J 2 V 2 ) = U(ll) U(l) U(l) + U(18) - 2U(16) U(l) OS Zn n 53) o 05 (J 2 H V ) = U(22) + U(34) + 2U(23) 02 Zn n n 54) o 05 (J 2 H V ) = U(13) U(l) + U(ll) U(S) + U(3) U(l4) 0~ Zn n n + U(3) U(16) - 4U(22) - 2U(23) - U(34) 05 2 55) 0 04 (JznHn Vn) = U(l) U(3) U(ll) + U(22) - U(13) U(l) - U(3) U(16) 58) o 24 (J 2 W 2 ) = U(ll) U(12) + 2U(26) + 2U(37) - 2U(24) - 2U(36) 12 Zn 59) O~~(J~ W 2 ) = U(ll) U(9) + 2U(15) U(2) - 4U(25) ~...,.. ""'n 60) o 24 (J 2 w 2 ) = U(ll) U(9) + U(24) - 2U(26) 13 Zn '14 , ') 61) o" (Jl. W'") = U(ll) U(2) 0(2) + U(2SJ - <!U(J.SJ U(2) ' 23 Zn 62) o 2 7i(J 2 WH ) = U(27) 11 Zn p 63) o~f(J~nWllp) ;;; U(3) U(J..S) 64) o 23 (J 2 WH ) = U(ll) U(6) - U(27) 12 Zn p 65) o 23 (J 2 WH ) = U(ll) U(3) U(2) - U(3) U(lS) 22 Zn p 66) o 14 (J 2 H W) = U(28) 11 Zn n 67) o 14 (J 2 H W) = U(ll) U(6) + U(13) U(2) + U(3) U(lS) - 4U(28) 12 Zn n 21 68) o 14 (J 2 H W) = U(11) U(3) U(2) + U(28) - U(3) U(15) - U(13) U(2) 13 Zn n 69) 0 33(J2 WV ) 21 Zn p = U(29) 70) D33(J2 WV ) 22 Zn p = U(11) U(IO) - U(29) 71) D33(J2 WV ) 31 Zn p = U(l) U(IS) 72) D33(J2 WV ) 32 Zn p = U(l) U(ll) U(2) - U(1) U(15) 73) o 15 (} WV) 12 Zn n = U(31) + U(30) + U(32) 74) o 15 (J 2 V W) = U(11) U(10) + U(16) U(2) + U(14)U(2) + U(1)U(15) 13 Zn n - 4U(30) - U(31) - U(32) 75) o 15 (J 2 V W) = U(11) U(1) U(2) + U(30) - U(16) U(2) - U(1) U(15) 14 Zn n :S3 2 78) o 22 CJznJZpW ) = 2U(33) U(2) + U(40) - 4U(39) 79) D~~(JZnJZpl-lpW) = U(42) 80) D~i(JZnJZpHpW) = U(3) U(33) - U(42) 42 81) o 21 CJznJZpVpW) = U(43) + U(44) 82) 42 D 31 CJz J 7 V W) = U(1) U(33) - U(43) . n ~P p 83) o 04 (J 4 ) = U(38) 01 Zn 84) o 04 (J 4 ) = 3U(ll) U(ll) - 4U(38) 02 Zn 85) 86) o 04 ci J = uc38) 03 Zn 22 o 22 (J 2 J 2 ) = U(ll) U(ll) 11 Zn Zp 23 APPENDIX II: BASIC SUMS U(1) = } E (2J+1) ~;a~ U(2) = r (2J+1) w~eae U(3) = E (2j +1) E a a 1.!(4) = E (2j +1) E2 a a U(S) = E (2J+1) E VJ a aeae U(6) = E (2J+1) E WJ a aeae U(7) 1 (2J+1) (VJ ]2 =- E 4 cxS~v V(a,S) = E (2J+1) v~~e~ l!,J U(9) = E o(ja,je) [W(a, B)] 2 a e 2j +1 a W(a,S) = E (2J+1) WJ ~ • ..r a~B~ U(ll) = !_ E (2j + 1) j (j +1) 3 a a· a U(l2) = E (2J+1) (WJ ] 2 . ~e~v U(13) = - 3 1 E (2J+1) E,. (2j +1) j (j +1) '"" a a a 24 U(14) = ~ E (2J+1) ~SaS D(J,a,S) U(l6) = - 3 1 E(2J+1) j (j +1) VJ a a aSaS 1 U(17) = ~ E o(j,_,,ju) Q(r.t.,R) V(a,R) ..)aS ..,, P 2j +1 a j (j +1) [V(a,S)] 2 a a U(:ZO) = l~ 1: (2._T.tl) [V~t3Jl..)i' D(J,u,S) U(21) 1 (2J+1) j (j +1) [~SJ.l..) 2 =- E 6 0: 0: U(22) 1 (2J+1) j (j +1) E VJ = - E 3 a a a aSaB U(23) 1 . (2J+1) E VJ D(J,a,S) = - E 6 a aSaS u (24) 1 (2J+1) js(js+1) [~all') 2 = - r 3 U(25) 1 o(ja,jS) j (j +1) [W(~,{3) ]2 = - E 3 a a (2j a +1) U(26) 1 o(jo.,js) R(a,S) W(a,S) = - E 3 2j +1 a 25 R(a,a) = E (2J+l) j (j +1) WJ ).l,J J.l J.l a).laJ.l U(27) 1 (2J +1) jaUa+1) E WJ =- E 3 a aaaa U(28) 1 (2J+1) j (j +1) E WJ = 3 E a a a aaaa U(29) 1 E o(ja,ja) R(a, a) V(a,a) =- 3 a,a 2j +1, a U(30) 1 E o(ja,je) j (j +1) V(rt,B) W(a, a) =- 3 a,a a a 2j +1 a · U(31) 1 E o(ja,ja) Q(a,a) W(a,~) =- 3 a,a (2j +1) a. U(32) 1 E o(ja,ja) W(a,a) S(a,a) = 3 . a, a 2j +1 a S(a,a) = E ; (2J+1) VJ D(J,a,J.l) u .• J a).l8J.l U(33) 1 (2J +1) WJ D(J,a,B) = - E 6 . aaaa U(34) 1 (2J+1) jaUe+l) E VJ =- E 3 a aaaa 1 U(35) ~ IE o(ja,ja) V(a,a) S(a,a) 2j +1 a 1 . .J J I U(36) = 3 I: w~a~.i\> wctlhi\! (2.T+l) (2.T'+J.) g(,l) g(f3) U(37) = .!_ E [Z(a,8) ] 2 3 a a Z(a,B) = E W~J.laJ.l (2J+l} g(J.l) J.i,J 1-' j +jQ+J ( -1) J.l 1-J 26 U(38) 1 U(39) = ~ I 5(jajB) W(a,B) T(a,B) 2j +1 a T(a,B) = I )..I,J W~aJ..IB (2J+1) D(J,)..I,a) 1.1(40) = .!. I (WJ ] 2 f.2J+l.) D( J a 8) 6 al3 )..1\J · ' ' [(2j +2) (2j +1) 2j·]'/2 g(a) a a = 4 U(42) 1 (2J+1) WJ E D(J a 13) =-I 6 aBaB a • • U(43) =~I 6(ja~j~) V(a,B) T(a,p) 2j +1 a U(44) = ~ I Z(a,B) X(a,B) a,S X(a,B)= I ~ f3 (2J+l) g(J..I) ct)..I.]J u,J U(45) = ~ E {X(a,l3)] 2 a,S u (46) = .!. I ~ J' v 3 aBJ..IV a8J..IV j + j + j + j +2J I ( _ 1 ) a B J..l v (2J+ 1) JJ' 1 ·j)\)j).l (2J'+l) JJ'l j Bja j +j +J (-1) P B g(v) g(B) 27 REFERENCES AND FOOTNOTES • 1. F. S. Chang, J. B. 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McVoy, Dept. of Physics, Univ. of Wisconsin, Madison, WI 53706 63. J. G. Belinfante, Georgia Inst. of Technology, Atlanta, GA 30332 64. T. L. Talley, Los Alamos Scientific Lab., P. 0. Box 1663, Los Alamos, NM 87545 65. T. Tamura, Center for Nuclear Studies, Univ. of Texas, Austin, TX 78712 66-92. Technieal Information Center, Oak Ridge, TN 37830