- Email: [email protected]
Solution of bound state problems in nuclear shell models with momentum dependent potentials
Computer Physics Communications 10 (1975) 182—193 © North-Holland Publishing Company
SOLUTION OF BOUND STATE PROBLEMS IN NUCLEAR SHELL MODELS WITH MOMENTUM DEPENDENT POTENTIALS M.A.K. LODHI and B.T. WAAK Department of Physics, Texas Tech University, Lubbock, Texas 79409, USA Received 4 June 1974, revised manuscript received 9 September 1975
PROGRAM SUMMARY Title of program: NUCLEAR POTENTIAL Catalogue number: ACWK Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: IBM 360/50;Installation: Texas Tech
the Schr~dingerequation is analytically solvable [3]. The wave function and eigenvalue are then expressable in terms of the parameters of the analytically solvable function. The program fmds the values of the parameters that give a “best” fit to the effective potential and then computes the corresponding eigenvalue.
Operating system: OS 360
Restrictions on the complexity of the problem The parameters of the effective potential are chosen to be suit-
Programming language used: FORTRAN IV (G)
able for nuclei along the beta stable line and with A > 10.
High speed storage required: 21,553 words No. of bits in a word: 32 Overlay structure: None Other peripherals used: Card Reader and Punch; Line printer No. of cards in combined program and test deck: 1349 Card punching code: EBCDIC Keywords: Nuclear, energy levels, wave functions, Morse function, analytic solution, Schr~dingerequation, nonlocal potential, momentum-dependent potential, shell model. Nature of physical problem Investigations of the structure of nuclei show that a nonlocal effect should be included in the nucleon—nucleus or nucleon— nucleon interaction [1]. A nonlocal potential can be understood as reflecting the correlations existing in nuclear matter, whereby the presence of a particle at position r influences the probability of finding another nucleon at a point r’ in the neighborhood of r [2]. This in turn affects the energy of the particle at r’. The nonlocal potential can be expressed as an effective energydependent potential. From the effective potential, the singleparticle wave functions and eigenvalues are calculated. Method of solution The effective patential is approximated by a function for which
Typical running time (IBM 360/50) The running time (IBM 360/50) for one single-particle state can range from 1—IS minutes. The variation in running time strongly depends on the number of iterations required for convergences and the number of points considered in the matching procedure. A good initial energy estimate significantly reduces the number of iterations, so an energy formula has been provided which generates a reasonably accurate estimate. The running time for the example is 1.3 minutes (2.8 minutes including compilation). Unusual features of the program The input data is read under the NAMELIST option of FORTRAN IV (G). The variable DUMMY is used for the sole purpose of reading the card identification/sequence field in columns 77—80. Under normal running conditions, the data cards will not be sequenced and the variable DUMMY can be eliminated. References [1] Intern. Conf. on The Nucleon—Nucleon Interaction, Rev. Mod. Phys. 39 No. 3 (1967). [2] MA. Preston, Physics of the Nucleus (Addison-Wesley, Reading, Mass., 1963) p. 192. [3] M.A.K. Lodhi, Phys. Rev. 182 (1968) 1061; M.A.K. Lodhi and B.T. Waak, Phys. Rev. Lett. 29 (1972) 301.
M.A.K. Lodhi, B. 7’. Waak/Nonlocal effects in nuclear shell models
183
LONG WRITE-UP 1. Introduction
The following definitions are used in (1) and (2): Xn(X) = [1 + 6 f(x)] ‘12G~1(x)where G~1(x)is the usual radial
Nonlocal potentials are appearing more frequently in nucleon—nucleon and nucleon—nucleus studies as methods of handling them are developed. Most of these studies used numerical techniques in solving dinger’shave equation and the numerical methods haveSchrö-
wavefunction;x = na where r is the radial distance and a is the unit of length; V(x) = —V0 1(x) and V~(x)= _V~p(x)where V0 and V~are strength parameters; 1 \—1 R f(x) = (~l+ expX~/ 1a 1’3 + C [2] ; p(x) is the perturbation and wherebeRstate-dependent; = r0A may e~= V 0/E0, c~= V /E0, and = E/E0 where E is the eigenvalue;E0 = h~/(2ma2) is the unit of energy, m = ~(mN + m~)(A 1)/A where mN and mp are the neutron and protonand masses respec2 1(2112 ) = b2e~/(4a2) is a measure tively; 6 = V0mb of the degree of momentum dependence, where b is the “range” of the nonlocality; and 1 is the orbital angular momentum quantum number. The fifth term in eq. (2) is the centrifugal term arising from the kinetic energy operator and the second, third, fourth, and sixth terms arise from the nonlocality. The sixth term is the energy dependent term. The perturbation term e~p(x) in (2) is a sum of terms which include the spin—orbit splitting, the nuclear symmetry energy, the Coulomb energy for protons, and the proton potential anomaly. We express the perturbation then as follows
proved to be adequate. However, analytic methods for solvingthewill Schrodinger equationunderstanding containing a nonlocal potential promote physical and be helpful in application. In principle, a computer program can be designed to approximate an effective nuclear potential to a potential for which the Schrödinger equation isofsolvable in some known parameters the analytically solvablefunctions. potential The are adjusted so that the “best” approximation is achieved. The eigenvalue is then expressable in terms of the parameters in a close form. We present here a method which provides an approximation of an effective nuclear potential by the so-called Morse function whose parameters are adjusted to give the “best” approximation.
2. Energy-dependent effective potential The SchrOdinger equation with a nonlocal potential can be transformed into a radial wave equation with an energy-dependent effective potential by expanding the nonlocal potential in a power series in the momentum [1—31.The radial wave equation can be put into the form [2,3] 2
x~1(x) (v(x;e~.) —
—
where v(x~e2)= E
—
~~ö)Xni(~)
=
0,
2
~
—
1
+
+ 1 6 f”(x) 6 J(x) 4 1 + 6 f(x)
2 ÷1 6f’~x) 1 [6f’~x)] ~ [1 +6flx)]2 2 [1 -l-6.tIx)]x 1(1+ 1) 6e~. f(x)—1 e~p(x) x x
(1)
—
~
2
=
e~p(x)
2
e
15p15(x) + symPsym(X) 2 2 + CcPc(X) + capa(x) These terms are discussed in the order they appear. The spin—orbit interaction of the Thomas type is used [41 e~p15(x)=
211 2x) — —
~ (I s)f’(x)/(a e~Lf’(x)/(a2x),
(3)
where = V 15/E0, 1715 is theand spin—orbit strength, a~0 is the spin—orbit parameter, L
=
41,
when /
1+4,
=4(/+1), whenll—+, 0, whenl0. (4) The nuclear symmetry energy produces a stronger central potential for protons than for neutrons and it
184
MA.K. Lodhi, B. T. Waak/Nonlocal effects in nuclear shell models
is given by [2]
3. Morse function and solution of radial equation
~ymPsym(X) = ±2 sym
N—Z ~ ~1(x),
(5)
where the upper sign refers to the proton case, the lower 2
A class of simple potentials is available for which the Schrodinger equation can be solved in terms of special functions [5]. The Morse function is used here since its general shape is neraly the same as the nucleon—nucleus
one to the neutron, sym sym/E’o~and Vsym is the strength parameter. By considering the nucleus to be a uniformly charged sphere and the proton to be a point charge, the Coulomb
potential it replaces. It can be written in the dimensionless form [6]
energy for protons [4] is
VM(x)
2R e~p~(x) = ~c
x >R/a,
~‘
(6) —c~ ~— ~ 2x2~ x ~
~
tractive potential which cancles approximately onehalf the classical Coulomb potential acting upon an individual pioton. Considering the magnitude of the effect and for computational simplicity, the Coulomb potential plus the proton potential anomaly is taken as onehalf the Coulomb energy [2]. The values of the parameters used for generating the potential for neutron states are taken from ref. [2] They are as follows: RN = l.2A~3fm, d0.6485 fm, .
D
D0 {1 —exp[—~(x—x1)] }2
=~-
(7)
where D,D0, 13 andx1 are the adjustable parameters. Substituting vM(x) for v(x;e~)in (1) and defining the change of variable Z(x) = (2/~)(D/E 0)e_~x_x~), (8) yields ~“(Z)
+~
1
2
x~
k
1(Z)— (—i
—~
E0
~) Xnl(Z)
=
0
(9) where 2 =( D k
D 0 —
e~
/132a2
(10)
—
The solution of (9) 2kkZ) is in terms of the generalized Laguerre polynomial [2] L~ n!
~ls
=
=
70MeV,
2,
a~
X(
0= 4fm Vsym
=
2k + 1)n’~n [Z(x)] and the corresponding energy is
22 MeV.
The proton parameters differ from the neutron ones in only two ways; the Coulomb term and the halfway 3 + C, for radius of theAproton distribution R~r= r0~A~ nuclei with > 60. The parameters and C are deop
terminedso that the well-known proton magic numbers arethe produced. systems A ~ distr~bution. 60R is the same as halfway For radius of thewith neutron For systems A > 60, the halfway radius of the proton distribution is chosen to be, R~+ 1.0 133 A1”3 + 0.67957. Other parameters used are:
(11)
E= —(1 +6){D0—2f~..fb~(n+4) 2a2(n +4)2E + 13 0), (12) where (r)n = r(r + 1)~ (n +n — 1), (r) 0 = 1, n = 0, 1,2, and D—D0 ~ —E/(1 + 6) ~ D0 provided Z0 = 1~-~i is large compared that to unity. Z(0) = (2/L3a)V~k~ e are thus related to the Morse The energy eigenvalues parameters in the analytic form (12). .
...,
4. Iteration procedure alfm,
b—4fm,
F 0
=
20.7351 and 6 =-~-V0/E0.
The effective potential given by (2), computed for
M.A.K. Lodhi, B.T. Waak/Nonlocal effects in nuclear shell models
185
an arbitrary state in a nucleus, is approximated by an analytically solvable potential vM(x), and thus (1) can be solved analytically. Sincef(x) approaches unity in the interior of the nucleus, the factor [f(x) 1] in the energy dependent term is usually a relatively small residual in bound-state problems. This is an advantageous condition which makes v(x;e~)rather insensitive to the value used for e~.[2]. Since the energy dependence of the effective potential is small, the initial value used for is relatively unimportant. This value of estimated from guess, theory, or experiment is used to generate the potential v(x;e~.)given by (2). The effective potential v(x;e~)thus generated is replaced by the analytically solvable potential vM(x) by a matching procedure. The matching procedure yields the relevant parameters which specify vM(x) completely and gives the eigenvalue E from (12) readily. From the eigenvalue F thus obtained the effective potential u(x;e~.)is regenerated which in turn is replaced by vM(x). The procedure is iterated until the assumed e~.in v(x;e~)agrees with the eigenvalue calculated from the parameters of vM(x). In this way the procedure is self-consistent. The assumed and calculated energies are said to agree in this work if they are with in 0.4 MeV or 10% of each other.
leads to the standard least square method. The minimization is done by setting the first partial derivative of ~ with respect to each Morse parameter equal to zero. The resulting formulas are given in Appendix A. For matching vM(x) to v(x;e~.)there are two important factors to be considered viz., the weighting function and the range of fitting. A selection of an appropriate combination of the weighting function and the range of fitting must be made before the nuclear systematics can be generated. The limits ofx in the sum involved in the matching procedure must be carefully chosen because the results may change significantly with small changes in the limits. Since v(x;e~)must be examined before an appropriate range of x for the matching procedure can be determined, v(x; e~)is generally computed for a greater range of x than is considered in the matching procedure. It has been determined that the initial value of x used in the matching is best specified as that value of x such that v(x; CE)2 ~ 0. Thus, the repulsive core of non sstates is excluded and the matching for s-states starts close to x = 0 since the point x = 0 is excluded for cornputational reasons. Likewise, it has been determined that the final value ofx used in the matching is best specified by the following two methods: 1) The matching ends at a point x = x’ + ~(x’—x 1)where 5. Matching procedure v(x;e~.)and v(x’,~)are the first minimum and maximum value of v(x;~)respectively such that The Morse function VM(x), chosen as the analytically x’ ‘‘x1 0. Thus, the matching ends a short distance solvable potential in this work, is matched to the effecbeyond the maximum v(x’;e~). tive potential v(x; e~)given by (2). The matching pro2) The matching ends near the surface region where cedure used here does not assume any particular funcv(x;4) is approximately constant which is at xx’ tional form for v(x; ei), thus the nuclear potential can + -~(x x1). be arbitrarily chosen and does not have to be of the For the following combinations of weighting functype in (2). The well-known lest square method is slight- tion and range of fitting, the resulting Morse function ly modified by introducing a weighting factor F(x). Thus, is a good approximation to the effective potential. For the function to be minimized is / ~ 0 states, the weighting factor (l4a) and method 2 -
—
=
~
[v,~(x)
x
2 2 v(x; CE)] F(x)
—
i.~1
)
where the reasonable choices for F(x) have been narrowed to the following two forms: =
F(x) =
1 1/x
(1 a) l4b~ “
~‘
The minimization of ~ using F(x) as specified by (l4a)
are used.1 For thethese weighting factor (14b) method are s-states used. For combinations, smalland changes in the range of fitting change the results only slightly or not at all. In fact, the two combinations of weighting factor and range of fitting give almost identical values of E except for s-states. For these levels, the combination with yields better A different weighting factor(14b) for s-states may beeigenvalues. expected since the s-states do not have a repulsive core. Since it is found that the two cornbinations give almost the same eigenvalue for non s-states, and s-states are definitely better obtained by the com-
c
Enter
ncre— rr.er?t to next orbit?
Increment yes
Orbit
no Write data and parametric values
Set •m~ult parameters
Fead Data
D
calculate effective potential End of Data?
yes Terminate .iol
Calculate range of matching
no Calculate other
parameters S
Calculate sore e parameters
S
nOISe orbit for Data Set?
A
Has ener~ converged?
no
yes yes F C Punched Output?
es
Write resuits on cards.
no
plot
tive tial function. potenand Morse
5 Plot no
GO to next rb it yes B
Fig. 1. Flow chart of the program NU2POT.
effec—
A
M.A.K. Lodhi, B. 7’. Weak/Nonlocal effects in nuclear shell models
bination with (l4b), the results are nearly the same as using the latter combination throughout.
6. Program structure The structure of the program is illustrated in fig. 1. It has been found to be necessary to perform certain checks while performing the matching. In order to guarantee that a zero of eq. (A—6) (see Appendix A) exists between the specified limits, the numerical procedure requires the values of the left side of (A—6) evaluated at the initial limits to have opposite signs. For the solution to be realistic, the resulting Morse parameters D and D0 must be positive. If no solution is found, all of the Morse parameters are set equal to zero and the energy eigenvalues set to —2 MeV. Since the shape of the curve by the left side of (A—6) is unknown, the possibility of more than one solution must be investigated, Thus, the full range of 13 is searched during each iteration, and if more than one solution is found, the one corresponding to the minimum value of ~ given by (13) is accepted. After the above checks and restrictions are included, there has been very little problem (which is easily e.g.,no byproblem varying the parameters) with nomanageable solutions and withinput multiple solutions. The subroutine SEP and EQ4 are written in double precision. The double precision was necessary in a minority of the cases, and in a computer with a greater word length, single precision may be sufficient. The subroutine EQ4 accounts for the majority of the time used, thus it may be very profitable to write this subroutine in a lower level language.
7. Program I/O variables 7.].
Input data
The data is read in with a namelist read statement. Only the first two variables (A and Z) must be specified in the data set. The other variables are defined before the data set is read, so their default values may be used or changed as desired. A detailed description of the input variables, their default values, the available options, and the use of the program is contained in the program listing. There are 20 variables that may be specified in the data. They are listed as follows:
187
A Z PART
Mass number of the nucleus. Atomic number of the nucleus. Particle in the specified state, either a “P” or an “N” for proton or neutron respectively. LEVEL Principal quantum number of the state — a real variable. L Orbital angular momentum quantum number — a real variable. J Total angular momentum quantum number — a real variable. MOMENT The number of the state in the predetermined sequence. MSTATE The last state to be considered in the calculation (the maximum value of MOMENT).
NSTATE
The number of additional states to be cornputed above the last orbit needed to hold all of the appropriate particles. The states are ordered by the predetermined sequence, not by energy. ENERGY Vector containing initial energy guesses to the states in the chosen nucleus. W Initial energy guess for the eigenvalue of the state specified by LEVEL, L, and J. RO A constant in the formula 3+ C. for the halfway radius R = R0*A~ C A constant in the formula for the halfway radius R = R0*A113 + C. VU Strength parameter of the central potential. Vl Strength parameter of the nuclear symmetry term. VS Strength parameter of the spin—orbit interaction (— V~ 5). ASO A spin—orbit parameter. DL A parameter in the form factor f(x)(d). CARD Determines whether the results are to be punched on cards. CARD may take on the values “YES” or “NO”. PLOTT Determines whether the potentials will be plotted. PLOTT may take on the values “YES” or “NO”. It may be appropriate to make a few remarks about the use of some of the variables defined above. (i) The sequence of states used in the program is the ordering of levels given by Mayer and Jensen [7] and the ordering is given in the program listing. (ii) If LEVEL = 0 or is not read in, the program increments the orbitals in the predetermined sequence. The first and last orbitals of the sequence to be considered
188
MA.K.
Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models
are determined by MOMENT and MSTATE or NSTATE respectively. (iii) The values of MOMENT is incremented for each orbital, so in input, MOMENT ÷1 is the beginning orbital in the sequence. (iv) NSTATE is used only if MSTATE is unspecified. (v) If any element of ENERGY is not specified, the program calculates an energy guess from a simple formula given in the program listing. (vi) The variables MOMENT, MSTATE, NSTATE and ENERGY are used only if the option (LEVEL = 0) of letting the program increment the states is chosen. 7.2. Output The output is in the form of printed output and an optional punched card output. The printed output occurs for each iteration, whereas the punched output occurs for the final iteration only. The printed output is self-explanatory and utilizes 131 spaces across the page. Two of the input variables may be changed in the printout. On output, NSTATE is its input value plus the number of states needed to hold all of the appropriate particles. Thus, it is the last state to be considered in the calculation. On output, MOMENT is the number of the being state in the predetermined sequence is curtently processed. The variables in the that punched output are explained in the program listing.
8. Subprograms and COMMON variables 8.]. Subprognams The program is divided into a MAIN program and five subprograms. The subroutine POTEN allows easy replacement of the effective potential if a matching to another potential is desired. The subprograms perform the following functions: POTEN calculates the effective potential (2). SEP performs the sums for a 1 andf in (A—2) in double precision. RTMI solves the nonlinear equation (A—6) (Obtamed from the System/360 Scientific Subroutine Package, Version II). EQ4 evaluates the left side of (A—6) for a given /3, in double precision. PLOT plots the resulting potentials. .
&2. COMMON vaniab/es There are three labeled commons and no blank cornmon used. The vectors V, X, and F are the effective potential (2), radius values, and weighting function (14) respectively and their vector sizes must be the same. NFIRST and NLAST are the first and last values of the radius to be considered in the matching procedure. DPRIME, DDO, DBETA and DXI are the Morse parameters D, D0, j3, and X1 respectively. PESA and PESF are the variables calculated in the subroutine SEP, they are a1 and fin (A—2). Rand DL are the form factor parameters defined in sect. 2 (DL = d). INDEX is a parameter that distinguishes between protons and neutrons. ALPHA is the Coulomb parameter and includes the proton potential anomaly.2/(E ALPHA = 0 for neutrons and ALPHA = 4e~~4(vc—l)e 0R) for protons (6). DELTA (6), EPSIO2 (c~),and EPSIW2 (e~)are defined in sect. 2 and EPSIO2 includes the nuclear symmetry energy perturbation. L is the orbital angular momentum quantum number and EPSI = e~a~L[see(3) and (4)].
9. Comments 40Ca. testenergy run is guess for theis 1—14.9 d512 state a proton TheThe initial MeV,ofthere is noinpunched output, and a plot is specified. The computations take one iteration and about 1.3 minutes as execution time (2.8 minutes including compilation) on IBM 360/50. The running time strongly depends on the number of iterations required, the range of fit, and the weighting function. The output is self-explanatory.
10. Results A comparison of the effective potential and the Morse function is illustrated in the test output. The Morse function is seen to give a very good average fit to the effective potential. In table 1 the calculated energies and experimental values [8—10] are shown for a few sample nuclei. The calculated and experimental values are seen to compare rather favorably below the Fermi levels. As a check on the accuracy of the matching, the “effective potential” has been taken as a Morse function, and the output Morse parameters of the potential have been recovered to three decimal places (see table 2).
MA. K Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models Table 1 Co~r~parison of the calculated and and experimental separation en2C, 160 40Ca are for proton states ergies. The energies for ‘ and the 208Pb values are neutron energies
Nucleus
State
Separation energies (MeV) Cal.
189
Table 2 Check on of thea accuracy of the matching. parameters are those given Morse function used The as aninput “effective potential”. The output parameters are then obtained by using the matching method described in sect. 5. D and D 0 are in May, and 13 and x1 are in fm~and fm respectively
D
Exp.”
13
x1 2.84
l2~
lP3/2
14.85
14 — 17
Input
40.5
26.3
0.628
160
1P3/2
18.89
19.1
±1.4
Output
40.5000
26.3000
0.6280002.84000
40Ca
lpl/2 1P3/2
11.77 30.76
12.7 33.3
± ± 1.4
512 Id312 1h9/2
16.12 8.47 9.75
14.9 ±2.5 8.4 ±0.5 10.7 d
2f7~2 2fs~2 2g9/2 3d5/2 3d3~2
10.59 7.87 3.75 3.51 2.03
6.5 b
--
-
1d
a As quoted in ref. [8] b ref. [9], d as quoted in ref. [4].
9.5 7.8 d 2.4 1.4
References [1] W.E. Frahn and R.H. Lemmer, Nuovo Cimento 6 (1957)
d d
664. [2] A.E.S. Green. T. Sawada, and D.S. Saxon, The nuclear independent particle model (Academic Press, New York, 1968). [3] M.A.K. Lodhi, Phys. Rev. 182 (1969) 1061;M.A.K. Lodhi
cI d
as quoted in ref. [10];
Essentially this work provides a simple method which describes the whole range of nuclei (with A 10 to all the way up to superheavy nuclei) within a reasonable accuracy with a fixed set of parameters. However, other users have the option of using a wider variety of parameters. In a separate publication we are presenting results for several nuclei using this technique. ~‘
B.T. and Waak, Phys. Rev. Lett. 29 (1972) 301.(W.A. Ben[4] and A. Bohr B.R. Mottelson, Nuclear structure jamin, New York, 1969), pp. 238, 173, 325 (in the order of appearance). [5] A.K. Bose, Nuovo Cimento 32 (1964) 679. [6] P.M. Morse and H. Feshback, Methods of theoretical physics (McGraw-Hill, New York, 1953) p. 1672. [71 M.G. Mayer and J.H.O. Jensen, Elementary theory of nuclear shell structure (Wiley, New York, 1964) p. 58. [8] G. Jacob and Th.A.J. Mans, Rev. Mod. Phys. 38 (1966) 121. [9] A,N. James, PT. Andrews, P. Kirkby and B.G. Lowe, Nucl. Phys. A138 (1969) 145. [10] M. Riou, Rev. Mod. Phys. 37 (1965) 375.
Appendix A. Minimization of~ BXv~x;e~) The formulas that minimize ~ with respect to D, D0 f3, and X1 are given in terms of the following definitions: D’
=
D~= D0/E0,
DIE0,
F(x)
x
e
=
xBXv~x~) d=~
‘
~ xB~v(x;e~) F(x)
~
.1(x)’
~
x
(A-i) A
=
exp(j3ax1),
B
=
exp(—j3a),
and
g
~ F(x)’
~ F(x)’
3x
a
v(x;e~~) 1
B~v(x;e~)
=
~B
_________
~
F(x)
‘
~
F(x)
‘
x
F(x)’
x
190
k
=
M.A.K. Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models
~ xBx F(x)’ ..-
m
= ~
/
=
1
xB~ ~ F(x)’
=
B. Analytical summations for the case F(x) 1 For the weighting function F(x) = 1, the terms =
F(x)’ ~ ~ F(x)
(A-2)
through n of A(-2) can be summed analytically. Let xM/s, NFMNL, ~
The minimization formulas are now written in terms of the above definitions. 2i — A31) D~= [D’~ — 3Ah + 3A —b+Ac]/(g—Ah). (A-3) This value of D’ minimizes ~ with respect to x 1 and 9, /3, and x is in terms of D 1. 2bh—A2cg+Acf—Aa D’(A 1h +a1g—bf) 4hi — A4gj + 2A3gi +A3f1 — 3A3h2 X(A + 3A2gh 3A2fi+ 2Afh — 2Ag2)1. (A-4)
f
x1N1/s,
whereM is the summation index, s is a scale factor, and NF andofNL correspond respectively first and last values x in the summation. Using to thethe relations N1
~i::RM= 1_RN
(B-i)
M=0
and
~I MRM
N-i
=
R [1 (l—R)2
RN _NRN_l (i—R)] ,(B-2)
—
yields This equation allows D’ to be calculated once f3 and x 1 are shown. 2 +bfi+cg2 —a A 2(a1h 1gi—cfh —bgh) 2~1. (A-5) X(cgh+a1hi+bff—a1gf—cfi—bh Once a value for /3 is assumed, x 1 can be calculated from (A-5). 4D’A [(D’ D~)k— A(3D’ —
f NI. — NF + ~,
(B-3)
~ BkX
(B-4)
Ek(NF) l_E~L_NF+1~ 1_Fk
X
Fk~~ 2 {NF s(1_Fk)
~DXBkX =
—
(NF — l)Ek
(B-5)
—
(NL + 2)/C}. 1~(NL-NF+1)k + (NL)E~_N~ Therefore, for F(x) = 1, the sums in (A-2) become —
+
3A2D’m —A3D’n
—
d +Ae]
=
0.
(A-6)
Every quantity in (A-6) can be written in terms of ~ thus, the problem is to solve (A-6) for j3. Since (A-6) contains terms like ~ x~(e13)mxv(x; e~),the possibility of an algebraic solution is excluded. Therefore, a numerical procedure is used to obtain /3 from (A-6). The numerical procedure assumes values for /3 and each time computes the left side of (A-6) until the proper value of /3 is determined. The values of D’, D~,and x 1 are determined in the following manner. For each value assumed for /3, x1 is calculated from (A-5), D’ is calculated from (A-4), and D~is calculated from (A-3). When a value 0 is found such that the left side of(A-5) is zero, the other parameters D’, D~,and x1 are also known.
a
=
1
b
NL ~ M=NF NL
~
=
~
=
~
~
=
s
~)‘
E~ v
M=NF NL
d
2
~
EMV
M=NF NI.
~
(M
M=NF NL
~
~)‘ ~),
(M)EM ~ S
e~SMNF ~ (M)E2Mv(~c~,),
MA.K. Lodhi, B. T. Waak/Nonlocal effects in nuclear shell models
f=NL—NF+1,
E2NF
2)2 {NF—(NF—1)E2
1
s(l—E
gE~
1ENL NFl 1—F
191
2~~ —
h=E2~ l_F2LNF1), l—E2
÷(NL)~2~L_N~2)},
(NL÷l)E E3NF 32 ~NF—(NF—1)E3
i=E3~~ l_E3~L~l) i—F3
—
(NL+
3~~2~’}, 1~3(NL—NF+i)+ (NL)E
/ = E4NF
k=
lE4(NL-NF+l) i—F4
ENF 2 {NF — (NF— l)E
~NF
s(l—F4)2 {NF—(NF—1)E4
—
NL+l)F4
L-NF + i) + (NL)E4~_N~2)}.
(B-6)
s( 1-F) —
(NL+l)FNL_~~ +(NL)E~~~2},
The numerical procedures for finding /3 and the calcula-
tions forD’, D~,and x 1 are much faster for the weighting factor F(x) = 1 when using (B-6) rather than (A-2).
M.A.K. Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models
192 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * *
a 15
a
7 ‘5 0 0 15 .
l 51 4 U _i
a
..i I 4 0— 0-51 —
za
0-5) 0— I 0— Ui > — 0• 0 ii
a
a a a z lu
C
• > UI 7 F— ‘0 ‘0 ‘0 — Si ii
e
UI
24$*
~
0
~
U,
* * N * — * * * * * — * — * IX * * * 27 * * 0> * — I * *0* * U X * * Z —4
—
*
5-OW tO7~- z 10 0 7
C~ * * * * * * * * Z * — * * Z * 0 * 0* 0 * 7 * 0. * * * .5 * * * 0 * *
4 U 00 4 (51
a
UI 04 05) N S.
11
0 —
Ui * 0.
* * 0.
* * * * * * * * * * *
* * * * *
* * * * * *
1-
• 7 II.
Ca
I •
—
a
0> U. W
—
I~1
77 0 0 20 III X
0-
* * * *
* * * * *
* *
Ui
o-
7 — 7
0. —
Z 0
**
Z
4* 4* *4 4* ~* *4 4* 8* *4 ** *4 *8 8* *8 4*
0
*4
a W ~
~J ..1 — 3
— IU Z
0
0
•
00
+ — (5) ‘
0 0 0’ LI) ‘a 4 U) a 0
—
u
* *
.5
0 0’OW ‘00 407
—ow I II 3~ (fl U) LU 0
00 N Z 511
a
UI 0
a a 0 7
0
4
*8 8* *4 4* *4 ** ** *8 *8
.J 4 .a
*4 4* 4* 4*
Z 11 0-
4* *4 04* ** 18* U ** Z *8 ~** *4
Ui I 0-
a 7
I—
a a
Ui > 0U
•
a a
a U. Ui
0*~ 0. *4 Z *4
t~. 0
U. 0
7
4
‘5
UI
0-
—
WLU m U (0 OW
C) _i a.
c
LU
4*
U.
I 0-
z a
4*
*4 *4 **
.J
0*0-_i 5 * — 711 * Z > ..i * —
4*
2
a
112
70 1102 7 4 UI 7
*4 4* *8 *8 ** *4 *4 4* 4* *4
~
4* 44 .J *4 ..J *4 ‘—4* 34* 4* 514* 04* a *4 4*4 U **
0
*
.1 40.
• X
I-I
0 7 0.
a
Ui 0-0 OW 0.0 —
4
W>
u
>0 —m
0
~
— ~0 Ui
* * *11 * *0oW *51 *
m
U • Wa)U005.0— UJ~.~Z 0 UJlS
‘~ 4 7 — 5.
I-.a
a
0-0Z
Ui — >
aoo
0—a —7 ZU4
Ui 7
ozo.
a
07 0-5.0 (0 SUJ
—ISa
I 3
u~
007 • 77.54 a .j 51 • 15 6
Z
* * * * * * * * * * * * * * * * * * * * * *0
Si
-a
*8 **
—
a
* *
a
Z UI
U 2
U
ZI-
m
0.11 4151
a 0a s
o-’
II
.~7N
N
a 75-00
* * * * * * * * * * * * * * * * * * * * *
0
4*
•
7
5. 0
z
**
o
a
— I7 .5 I-
*4
a
*
• > Ui
*
** ** 4* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *4 4* *-* *4 *4 4* *4 *8 *4 4* ** *4 8* *4 *4 ** ~* *4 *4 4* 4* 4* ** 4* ** 8* *4
.
— * II* U * — Z * SC 0 * — U. * > * 104 z * — * 0* I * tO * 0* * 3 * I * II * * 51 * * SC * * . * — * * * — * SC * * SC * ‘— * * * ~ * 0* 11 * 0 * I — * — * a — * XX *
0
z
• 0— —
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
‘I) 6
Ui
*4
6 0
74
0 N
*0*4 *0 * 451 * *1 *1.* *5) *Z
7
00
~
*0
6 0 0 5.
—
.5 7 4
a aIII
• CS
7
I Ui 711) — — 7. 0 0-’) 0 • 4 X ui!l
a I I— 0 0-
0UI 7 0 0. 7 Ui 5.
—
0 I0 ~ U ISe
0 Z — 0-1fl4 U 0 ‘5 Si F- 70-
7
a oI
a
a.
0 02 U
•ou
V)~ Ui (51 7
0..>
0
a a
ii I 55<7 (0UI s04 — — 3404-. Ui Ui C — — LU F0— 0 — 4 .
“) F• —
.J •
II)
._j
a
LU N
‘0
7 >>li. LUUI”7 77.511.
I
2 0
m U. —
*‘
*1*4 *7 4W *08-’ * *0 * *7 *11 *0 *2 *~ *Z * * 4. ~oo *
*
*5)4 4’. * *5, *Z .Jn) *04 *I1 *0-0*4 Z.J *7 114 I-’.
*a *0-00-
II
SC
70.504
Ui
7
* *1.
— Q’514F7 .4)500 4.500 7 N0)5
4 0 .,~ >
0 11 >
*70*0>0 *
—a.
UJUJW >>>
OW W5) am 1.0 1117
712
—
l009F••~~ OF-ON T’)N
(0
a
a
8LU *0 *7 *0 *Z * 45 *0 *7
11
0-
*4 ..J1 *70.0.
7 UI 10
7 0 51
—
—
UI
I.-
5.
11111111
04 001-SC U
a
Ui 1111-
‘occe ,Sr’)F-
1.5. 00
11111$
*0-0*5< 00
LU XZLO 4.Z 714
x.~m
MA.K Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models X
=
J*STEP/SCALE +.•..+..
10. II. 12. 13. 14. 15. 16. 17. (8. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 3*. 37. 38. 39. 40. 41. 42. 43. 44. 45 46. 47. 48. 49. 50, 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 85. -
• -
• -
9’ 01 102 103: 104.
107. 108.
—
..+....+•.
NLOW*(STEPI).’SCALE ..+..
•.+_
(J*
(31/
100.00
..•+••..4_.
_..,...•+.
—
(
1Ø)*(
..0....+....+.
...+..
121/ ..+.
193
100.00
=
...f..
•..+•••.+•
J*( .•+.
0.130) .•_+.
.02 .1 2 .22 .32 .42 .52 .62 .72 .8 2 .92
.oB
A A A B
A AS BA B B A B A B A 8 A BA 2 46 A B A B A B A 6 A 8 A A A
B B
A
(.200
—
...+.
.1 .2 .3 .4 .5 .6 .7 .8 .9
.0 I .2 .4 .6 .8 .9
8
.0
B B
A
.1 .2 .3 .4 .5 .6 .7 .8 .9
B
A
8 A
B A B 2 BA B
A A
8
.0
A
B B
A B
A B
A B
A B
A B B
A B 8 B 8 B
A A
A A A A B A B A B A 6 A BA BA BA BA BA BA BA 114 BA 2 2 2 2 AS AB AS AS AB AS AS AB AB AB AB AS AS
.1 .2 .3 .4 .5 .6 .7 .8 .9
.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 .0 .1 .2 .3 .4 .5
:~ .6
.9 .0 .1 .2 .3 .4 .5 .6 .7
A A A A A
AS 6 B B 8 5
.9 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9
A A A
8 B B
.0 .1 .2
A A A
8 B B
:5 .6 .7
SOLUTION OF BOUND STATE PROBLEMS IN NUCLEAR SHELL MODELS WITH MOMENTUM DEPENDENT POTENTIALS M.A.K. LODHI and B.T. WAAK Department of Physics, Texas Tech University, Lubbock, Texas 79409, USA Received 4 June 1974, revised manuscript received 9 September 1975
PROGRAM SUMMARY Title of program: NUCLEAR POTENTIAL Catalogue number: ACWK Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: IBM 360/50;Installation: Texas Tech
the Schr~dingerequation is analytically solvable [3]. The wave function and eigenvalue are then expressable in terms of the parameters of the analytically solvable function. The program fmds the values of the parameters that give a “best” fit to the effective potential and then computes the corresponding eigenvalue.
Operating system: OS 360
Restrictions on the complexity of the problem The parameters of the effective potential are chosen to be suit-
Programming language used: FORTRAN IV (G)
able for nuclei along the beta stable line and with A > 10.
High speed storage required: 21,553 words No. of bits in a word: 32 Overlay structure: None Other peripherals used: Card Reader and Punch; Line printer No. of cards in combined program and test deck: 1349 Card punching code: EBCDIC Keywords: Nuclear, energy levels, wave functions, Morse function, analytic solution, Schr~dingerequation, nonlocal potential, momentum-dependent potential, shell model. Nature of physical problem Investigations of the structure of nuclei show that a nonlocal effect should be included in the nucleon—nucleus or nucleon— nucleon interaction [1]. A nonlocal potential can be understood as reflecting the correlations existing in nuclear matter, whereby the presence of a particle at position r influences the probability of finding another nucleon at a point r’ in the neighborhood of r [2]. This in turn affects the energy of the particle at r’. The nonlocal potential can be expressed as an effective energydependent potential. From the effective potential, the singleparticle wave functions and eigenvalues are calculated. Method of solution The effective patential is approximated by a function for which
Typical running time (IBM 360/50) The running time (IBM 360/50) for one single-particle state can range from 1—IS minutes. The variation in running time strongly depends on the number of iterations required for convergences and the number of points considered in the matching procedure. A good initial energy estimate significantly reduces the number of iterations, so an energy formula has been provided which generates a reasonably accurate estimate. The running time for the example is 1.3 minutes (2.8 minutes including compilation). Unusual features of the program The input data is read under the NAMELIST option of FORTRAN IV (G). The variable DUMMY is used for the sole purpose of reading the card identification/sequence field in columns 77—80. Under normal running conditions, the data cards will not be sequenced and the variable DUMMY can be eliminated. References [1] Intern. Conf. on The Nucleon—Nucleon Interaction, Rev. Mod. Phys. 39 No. 3 (1967). [2] MA. Preston, Physics of the Nucleus (Addison-Wesley, Reading, Mass., 1963) p. 192. [3] M.A.K. Lodhi, Phys. Rev. 182 (1968) 1061; M.A.K. Lodhi and B.T. Waak, Phys. Rev. Lett. 29 (1972) 301.
M.A.K. Lodhi, B. 7’. Waak/Nonlocal effects in nuclear shell models
183
LONG WRITE-UP 1. Introduction
The following definitions are used in (1) and (2): Xn(X) = [1 + 6 f(x)] ‘12G~1(x)where G~1(x)is the usual radial
Nonlocal potentials are appearing more frequently in nucleon—nucleon and nucleon—nucleus studies as methods of handling them are developed. Most of these studies used numerical techniques in solving dinger’shave equation and the numerical methods haveSchrö-
wavefunction;x = na where r is the radial distance and a is the unit of length; V(x) = —V0 1(x) and V~(x)= _V~p(x)where V0 and V~are strength parameters; 1 \—1 R f(x) = (~l+ expX~/ 1a 1’3 + C [2] ; p(x) is the perturbation and wherebeRstate-dependent; = r0A may e~= V 0/E0, c~= V /E0, and = E/E0 where E is the eigenvalue;E0 = h~/(2ma2) is the unit of energy, m = ~(mN + m~)(A 1)/A where mN and mp are the neutron and protonand masses respec2 1(2112 ) = b2e~/(4a2) is a measure tively; 6 = V0mb of the degree of momentum dependence, where b is the “range” of the nonlocality; and 1 is the orbital angular momentum quantum number. The fifth term in eq. (2) is the centrifugal term arising from the kinetic energy operator and the second, third, fourth, and sixth terms arise from the nonlocality. The sixth term is the energy dependent term. The perturbation term e~p(x) in (2) is a sum of terms which include the spin—orbit splitting, the nuclear symmetry energy, the Coulomb energy for protons, and the proton potential anomaly. We express the perturbation then as follows
proved to be adequate. However, analytic methods for solvingthewill Schrodinger equationunderstanding containing a nonlocal potential promote physical and be helpful in application. In principle, a computer program can be designed to approximate an effective nuclear potential to a potential for which the Schrödinger equation isofsolvable in some known parameters the analytically solvablefunctions. potential The are adjusted so that the “best” approximation is achieved. The eigenvalue is then expressable in terms of the parameters in a close form. We present here a method which provides an approximation of an effective nuclear potential by the so-called Morse function whose parameters are adjusted to give the “best” approximation.
2. Energy-dependent effective potential The SchrOdinger equation with a nonlocal potential can be transformed into a radial wave equation with an energy-dependent effective potential by expanding the nonlocal potential in a power series in the momentum [1—31.The radial wave equation can be put into the form [2,3] 2
x~1(x) (v(x;e~.) —
—
where v(x~e2)= E
—
~~ö)Xni(~)
=
0,
2
~
—
1
+
+ 1 6 f”(x) 6 J(x) 4 1 + 6 f(x)
2 ÷1 6f’~x) 1 [6f’~x)] ~ [1 +6flx)]2 2 [1 -l-6.tIx)]x 1(1+ 1) 6e~. f(x)—1 e~p(x) x x
(1)
—
~
2
=
e~p(x)
2
e
15p15(x) + symPsym(X) 2 2 + CcPc(X) + capa(x) These terms are discussed in the order they appear. The spin—orbit interaction of the Thomas type is used [41 e~p15(x)=
211 2x) — —
~ (I s)f’(x)/(a e~Lf’(x)/(a2x),
(3)
where = V 15/E0, 1715 is theand spin—orbit strength, a~0 is the spin—orbit parameter, L
=
41,
when /
1+4,
=4(/+1), whenll—+, 0, whenl0. (4) The nuclear symmetry energy produces a stronger central potential for protons than for neutrons and it
184
MA.K. Lodhi, B. T. Waak/Nonlocal effects in nuclear shell models
is given by [2]
3. Morse function and solution of radial equation
~ymPsym(X) = ±2 sym
N—Z ~ ~1(x),
(5)
where the upper sign refers to the proton case, the lower 2
A class of simple potentials is available for which the Schrodinger equation can be solved in terms of special functions [5]. The Morse function is used here since its general shape is neraly the same as the nucleon—nucleus
one to the neutron, sym sym/E’o~and Vsym is the strength parameter. By considering the nucleus to be a uniformly charged sphere and the proton to be a point charge, the Coulomb
potential it replaces. It can be written in the dimensionless form [6]
energy for protons [4] is
VM(x)
2R e~p~(x) = ~c
x >R/a,
~‘
(6) —c~ ~— ~ 2x2~ x ~
~
tractive potential which cancles approximately onehalf the classical Coulomb potential acting upon an individual pioton. Considering the magnitude of the effect and for computational simplicity, the Coulomb potential plus the proton potential anomaly is taken as onehalf the Coulomb energy [2]. The values of the parameters used for generating the potential for neutron states are taken from ref. [2] They are as follows: RN = l.2A~3fm, d0.6485 fm, .
D
D0 {1 —exp[—~(x—x1)] }2
=~-
(7)
where D,D0, 13 andx1 are the adjustable parameters. Substituting vM(x) for v(x;e~)in (1) and defining the change of variable Z(x) = (2/~)(D/E 0)e_~x_x~), (8) yields ~“(Z)
+~
1
2
x~
k
1(Z)— (—i
—~
E0
~) Xnl(Z)
=
0
(9) where 2 =( D k
D 0 —
e~
/132a2
(10)
—
The solution of (9) 2kkZ) is in terms of the generalized Laguerre polynomial [2] L~ n!
~ls
=
=
70MeV,
2,
a~
X(
0= 4fm Vsym
=
2k + 1)n’~n [Z(x)] and the corresponding energy is
22 MeV.
The proton parameters differ from the neutron ones in only two ways; the Coulomb term and the halfway 3 + C, for radius of theAproton distribution R~r= r0~A~ nuclei with > 60. The parameters and C are deop
terminedso that the well-known proton magic numbers arethe produced. systems A ~ distr~bution. 60R is the same as halfway For radius of thewith neutron For systems A > 60, the halfway radius of the proton distribution is chosen to be, R~+ 1.0 133 A1”3 + 0.67957. Other parameters used are:
(11)
E= —(1 +6){D0—2f~..fb~(n+4) 2a2(n +4)2E + 13 0), (12) where (r)n = r(r + 1)~ (n +n — 1), (r) 0 = 1, n = 0, 1,2, and D—D0 ~ —E/(1 + 6) ~ D0 provided Z0 = 1~-~i is large compared that to unity. Z(0) = (2/L3a)V~k~ e are thus related to the Morse The energy eigenvalues parameters in the analytic form (12). .
...,
4. Iteration procedure alfm,
b—4fm,
F 0
=
20.7351 and 6 =-~-V0/E0.
The effective potential given by (2), computed for
M.A.K. Lodhi, B.T. Waak/Nonlocal effects in nuclear shell models
185
an arbitrary state in a nucleus, is approximated by an analytically solvable potential vM(x), and thus (1) can be solved analytically. Sincef(x) approaches unity in the interior of the nucleus, the factor [f(x) 1] in the energy dependent term is usually a relatively small residual in bound-state problems. This is an advantageous condition which makes v(x;e~)rather insensitive to the value used for e~.[2]. Since the energy dependence of the effective potential is small, the initial value used for is relatively unimportant. This value of estimated from guess, theory, or experiment is used to generate the potential v(x;e~.)given by (2). The effective potential v(x;e~)thus generated is replaced by the analytically solvable potential vM(x) by a matching procedure. The matching procedure yields the relevant parameters which specify vM(x) completely and gives the eigenvalue E from (12) readily. From the eigenvalue F thus obtained the effective potential u(x;e~.)is regenerated which in turn is replaced by vM(x). The procedure is iterated until the assumed e~.in v(x;e~)agrees with the eigenvalue calculated from the parameters of vM(x). In this way the procedure is self-consistent. The assumed and calculated energies are said to agree in this work if they are with in 0.4 MeV or 10% of each other.
leads to the standard least square method. The minimization is done by setting the first partial derivative of ~ with respect to each Morse parameter equal to zero. The resulting formulas are given in Appendix A. For matching vM(x) to v(x;e~.)there are two important factors to be considered viz., the weighting function and the range of fitting. A selection of an appropriate combination of the weighting function and the range of fitting must be made before the nuclear systematics can be generated. The limits ofx in the sum involved in the matching procedure must be carefully chosen because the results may change significantly with small changes in the limits. Since v(x;e~)must be examined before an appropriate range of x for the matching procedure can be determined, v(x; e~)is generally computed for a greater range of x than is considered in the matching procedure. It has been determined that the initial value of x used in the matching is best specified as that value of x such that v(x; CE)2 ~ 0. Thus, the repulsive core of non sstates is excluded and the matching for s-states starts close to x = 0 since the point x = 0 is excluded for cornputational reasons. Likewise, it has been determined that the final value ofx used in the matching is best specified by the following two methods: 1) The matching ends at a point x = x’ + ~(x’—x 1)where 5. Matching procedure v(x;e~.)and v(x’,~)are the first minimum and maximum value of v(x;~)respectively such that The Morse function VM(x), chosen as the analytically x’ ‘‘x1 0. Thus, the matching ends a short distance solvable potential in this work, is matched to the effecbeyond the maximum v(x’;e~). tive potential v(x; e~)given by (2). The matching pro2) The matching ends near the surface region where cedure used here does not assume any particular funcv(x;4) is approximately constant which is at xx’ tional form for v(x; ei), thus the nuclear potential can + -~(x x1). be arbitrarily chosen and does not have to be of the For the following combinations of weighting functype in (2). The well-known lest square method is slight- tion and range of fitting, the resulting Morse function ly modified by introducing a weighting factor F(x). Thus, is a good approximation to the effective potential. For the function to be minimized is / ~ 0 states, the weighting factor (l4a) and method 2 -
—
=
~
[v,~(x)
x
2 2 v(x; CE)] F(x)
—
i.~1
)
where the reasonable choices for F(x) have been narrowed to the following two forms: =
F(x) =
1 1/x
(1 a) l4b~ “
~‘
The minimization of ~ using F(x) as specified by (l4a)
are used.1 For thethese weighting factor (14b) method are s-states used. For combinations, smalland changes in the range of fitting change the results only slightly or not at all. In fact, the two combinations of weighting factor and range of fitting give almost identical values of E except for s-states. For these levels, the combination with yields better A different weighting factor(14b) for s-states may beeigenvalues. expected since the s-states do not have a repulsive core. Since it is found that the two cornbinations give almost the same eigenvalue for non s-states, and s-states are definitely better obtained by the com-
c
Enter
ncre— rr.er?t to next orbit?
Increment yes
Orbit
no Write data and parametric values
Set •m~ult parameters
Fead Data
D
calculate effective potential End of Data?
yes Terminate .iol
Calculate range of matching
no Calculate other
parameters S
Calculate sore e parameters
S
nOISe orbit for Data Set?
A
Has ener~ converged?
no
yes yes F C Punched Output?
es
Write resuits on cards.
no
plot
tive tial function. potenand Morse
5 Plot no
GO to next rb it yes B
Fig. 1. Flow chart of the program NU2POT.
effec—
A
M.A.K. Lodhi, B. 7’. Weak/Nonlocal effects in nuclear shell models
bination with (l4b), the results are nearly the same as using the latter combination throughout.
6. Program structure The structure of the program is illustrated in fig. 1. It has been found to be necessary to perform certain checks while performing the matching. In order to guarantee that a zero of eq. (A—6) (see Appendix A) exists between the specified limits, the numerical procedure requires the values of the left side of (A—6) evaluated at the initial limits to have opposite signs. For the solution to be realistic, the resulting Morse parameters D and D0 must be positive. If no solution is found, all of the Morse parameters are set equal to zero and the energy eigenvalues set to —2 MeV. Since the shape of the curve by the left side of (A—6) is unknown, the possibility of more than one solution must be investigated, Thus, the full range of 13 is searched during each iteration, and if more than one solution is found, the one corresponding to the minimum value of ~ given by (13) is accepted. After the above checks and restrictions are included, there has been very little problem (which is easily e.g.,no byproblem varying the parameters) with nomanageable solutions and withinput multiple solutions. The subroutine SEP and EQ4 are written in double precision. The double precision was necessary in a minority of the cases, and in a computer with a greater word length, single precision may be sufficient. The subroutine EQ4 accounts for the majority of the time used, thus it may be very profitable to write this subroutine in a lower level language.
7. Program I/O variables 7.].
Input data
The data is read in with a namelist read statement. Only the first two variables (A and Z) must be specified in the data set. The other variables are defined before the data set is read, so their default values may be used or changed as desired. A detailed description of the input variables, their default values, the available options, and the use of the program is contained in the program listing. There are 20 variables that may be specified in the data. They are listed as follows:
187
A Z PART
Mass number of the nucleus. Atomic number of the nucleus. Particle in the specified state, either a “P” or an “N” for proton or neutron respectively. LEVEL Principal quantum number of the state — a real variable. L Orbital angular momentum quantum number — a real variable. J Total angular momentum quantum number — a real variable. MOMENT The number of the state in the predetermined sequence. MSTATE The last state to be considered in the calculation (the maximum value of MOMENT).
NSTATE
The number of additional states to be cornputed above the last orbit needed to hold all of the appropriate particles. The states are ordered by the predetermined sequence, not by energy. ENERGY Vector containing initial energy guesses to the states in the chosen nucleus. W Initial energy guess for the eigenvalue of the state specified by LEVEL, L, and J. RO A constant in the formula 3+ C. for the halfway radius R = R0*A~ C A constant in the formula for the halfway radius R = R0*A113 + C. VU Strength parameter of the central potential. Vl Strength parameter of the nuclear symmetry term. VS Strength parameter of the spin—orbit interaction (— V~ 5). ASO A spin—orbit parameter. DL A parameter in the form factor f(x)(d). CARD Determines whether the results are to be punched on cards. CARD may take on the values “YES” or “NO”. PLOTT Determines whether the potentials will be plotted. PLOTT may take on the values “YES” or “NO”. It may be appropriate to make a few remarks about the use of some of the variables defined above. (i) The sequence of states used in the program is the ordering of levels given by Mayer and Jensen [7] and the ordering is given in the program listing. (ii) If LEVEL = 0 or is not read in, the program increments the orbitals in the predetermined sequence. The first and last orbitals of the sequence to be considered
188
MA.K.
Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models
are determined by MOMENT and MSTATE or NSTATE respectively. (iii) The values of MOMENT is incremented for each orbital, so in input, MOMENT ÷1 is the beginning orbital in the sequence. (iv) NSTATE is used only if MSTATE is unspecified. (v) If any element of ENERGY is not specified, the program calculates an energy guess from a simple formula given in the program listing. (vi) The variables MOMENT, MSTATE, NSTATE and ENERGY are used only if the option (LEVEL = 0) of letting the program increment the states is chosen. 7.2. Output The output is in the form of printed output and an optional punched card output. The printed output occurs for each iteration, whereas the punched output occurs for the final iteration only. The printed output is self-explanatory and utilizes 131 spaces across the page. Two of the input variables may be changed in the printout. On output, NSTATE is its input value plus the number of states needed to hold all of the appropriate particles. Thus, it is the last state to be considered in the calculation. On output, MOMENT is the number of the being state in the predetermined sequence is curtently processed. The variables in the that punched output are explained in the program listing.
8. Subprograms and COMMON variables 8.]. Subprognams The program is divided into a MAIN program and five subprograms. The subroutine POTEN allows easy replacement of the effective potential if a matching to another potential is desired. The subprograms perform the following functions: POTEN calculates the effective potential (2). SEP performs the sums for a 1 andf in (A—2) in double precision. RTMI solves the nonlinear equation (A—6) (Obtamed from the System/360 Scientific Subroutine Package, Version II). EQ4 evaluates the left side of (A—6) for a given /3, in double precision. PLOT plots the resulting potentials. .
&2. COMMON vaniab/es There are three labeled commons and no blank cornmon used. The vectors V, X, and F are the effective potential (2), radius values, and weighting function (14) respectively and their vector sizes must be the same. NFIRST and NLAST are the first and last values of the radius to be considered in the matching procedure. DPRIME, DDO, DBETA and DXI are the Morse parameters D, D0, j3, and X1 respectively. PESA and PESF are the variables calculated in the subroutine SEP, they are a1 and fin (A—2). Rand DL are the form factor parameters defined in sect. 2 (DL = d). INDEX is a parameter that distinguishes between protons and neutrons. ALPHA is the Coulomb parameter and includes the proton potential anomaly.2/(E ALPHA = 0 for neutrons and ALPHA = 4e~~4(vc—l)e 0R) for protons (6). DELTA (6), EPSIO2 (c~),and EPSIW2 (e~)are defined in sect. 2 and EPSIO2 includes the nuclear symmetry energy perturbation. L is the orbital angular momentum quantum number and EPSI = e~a~L[see(3) and (4)].
9. Comments 40Ca. testenergy run is guess for theis 1—14.9 d512 state a proton TheThe initial MeV,ofthere is noinpunched output, and a plot is specified. The computations take one iteration and about 1.3 minutes as execution time (2.8 minutes including compilation) on IBM 360/50. The running time strongly depends on the number of iterations required, the range of fit, and the weighting function. The output is self-explanatory.
10. Results A comparison of the effective potential and the Morse function is illustrated in the test output. The Morse function is seen to give a very good average fit to the effective potential. In table 1 the calculated energies and experimental values [8—10] are shown for a few sample nuclei. The calculated and experimental values are seen to compare rather favorably below the Fermi levels. As a check on the accuracy of the matching, the “effective potential” has been taken as a Morse function, and the output Morse parameters of the potential have been recovered to three decimal places (see table 2).
MA. K Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models Table 1 Co~r~parison of the calculated and and experimental separation en2C, 160 40Ca are for proton states ergies. The energies for ‘ and the 208Pb values are neutron energies
Nucleus
State
Separation energies (MeV) Cal.
189
Table 2 Check on of thea accuracy of the matching. parameters are those given Morse function used The as aninput “effective potential”. The output parameters are then obtained by using the matching method described in sect. 5. D and D 0 are in May, and 13 and x1 are in fm~and fm respectively
D
Exp.”
13
x1 2.84
l2~
lP3/2
14.85
14 — 17
Input
40.5
26.3
0.628
160
1P3/2
18.89
19.1
±1.4
Output
40.5000
26.3000
0.6280002.84000
40Ca
lpl/2 1P3/2
11.77 30.76
12.7 33.3
± ± 1.4
512 Id312 1h9/2
16.12 8.47 9.75
14.9 ±2.5 8.4 ±0.5 10.7 d
2f7~2 2fs~2 2g9/2 3d5/2 3d3~2
10.59 7.87 3.75 3.51 2.03
6.5 b
--
-
1d
a As quoted in ref. [8] b ref. [9], d as quoted in ref. [4].
9.5 7.8 d 2.4 1.4
References [1] W.E. Frahn and R.H. Lemmer, Nuovo Cimento 6 (1957)
d d
664. [2] A.E.S. Green. T. Sawada, and D.S. Saxon, The nuclear independent particle model (Academic Press, New York, 1968). [3] M.A.K. Lodhi, Phys. Rev. 182 (1969) 1061;M.A.K. Lodhi
cI d
as quoted in ref. [10];
Essentially this work provides a simple method which describes the whole range of nuclei (with A 10 to all the way up to superheavy nuclei) within a reasonable accuracy with a fixed set of parameters. However, other users have the option of using a wider variety of parameters. In a separate publication we are presenting results for several nuclei using this technique. ~‘
B.T. and Waak, Phys. Rev. Lett. 29 (1972) 301.(W.A. Ben[4] and A. Bohr B.R. Mottelson, Nuclear structure jamin, New York, 1969), pp. 238, 173, 325 (in the order of appearance). [5] A.K. Bose, Nuovo Cimento 32 (1964) 679. [6] P.M. Morse and H. Feshback, Methods of theoretical physics (McGraw-Hill, New York, 1953) p. 1672. [71 M.G. Mayer and J.H.O. Jensen, Elementary theory of nuclear shell structure (Wiley, New York, 1964) p. 58. [8] G. Jacob and Th.A.J. Mans, Rev. Mod. Phys. 38 (1966) 121. [9] A,N. James, PT. Andrews, P. Kirkby and B.G. Lowe, Nucl. Phys. A138 (1969) 145. [10] M. Riou, Rev. Mod. Phys. 37 (1965) 375.
Appendix A. Minimization of~ BXv~x;e~) The formulas that minimize ~ with respect to D, D0 f3, and X1 are given in terms of the following definitions: D’
=
D~= D0/E0,
DIE0,
F(x)
x
e
=
xBXv~x~) d=~
‘
~ xB~v(x;e~) F(x)
~
.1(x)’
~
x
(A-i) A
=
exp(j3ax1),
B
=
exp(—j3a),
and
g
~ F(x)’
~ F(x)’
3x
a
v(x;e~~) 1
B~v(x;e~)
=
~B
_________
~
F(x)
‘
~
F(x)
‘
x
F(x)’
x
190
k
=
M.A.K. Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models
~ xBx F(x)’ ..-
m
= ~
/
=
1
xB~ ~ F(x)’
=
B. Analytical summations for the case F(x) 1 For the weighting function F(x) = 1, the terms =
F(x)’ ~ ~ F(x)
(A-2)
through n of A(-2) can be summed analytically. Let xM/s, NFMNL, ~
The minimization formulas are now written in terms of the above definitions. 2i — A31) D~= [D’~ — 3Ah + 3A —b+Ac]/(g—Ah). (A-3) This value of D’ minimizes ~ with respect to x 1 and 9, /3, and x is in terms of D 1. 2bh—A2cg+Acf—Aa D’(A 1h +a1g—bf) 4hi — A4gj + 2A3gi +A3f1 — 3A3h2 X(A + 3A2gh 3A2fi+ 2Afh — 2Ag2)1. (A-4)
f
x1N1/s,
whereM is the summation index, s is a scale factor, and NF andofNL correspond respectively first and last values x in the summation. Using to thethe relations N1
~i::RM= 1_RN
(B-i)
M=0
and
~I MRM
N-i
=
R [1 (l—R)2
RN _NRN_l (i—R)] ,(B-2)
—
yields This equation allows D’ to be calculated once f3 and x 1 are shown. 2 +bfi+cg2 —a A 2(a1h 1gi—cfh —bgh) 2~1. (A-5) X(cgh+a1hi+bff—a1gf—cfi—bh Once a value for /3 is assumed, x 1 can be calculated from (A-5). 4D’A [(D’ D~)k— A(3D’ —
f NI. — NF + ~,
(B-3)
~ BkX
(B-4)
Ek(NF) l_E~L_NF+1~ 1_Fk
X
Fk~~ 2 {NF s(1_Fk)
~DXBkX =
—
(NF — l)Ek
(B-5)
—
(NL + 2)/C}. 1~(NL-NF+1)k + (NL)E~_N~ Therefore, for F(x) = 1, the sums in (A-2) become —
+
3A2D’m —A3D’n
—
d +Ae]
=
0.
(A-6)
Every quantity in (A-6) can be written in terms of ~ thus, the problem is to solve (A-6) for j3. Since (A-6) contains terms like ~ x~(e13)mxv(x; e~),the possibility of an algebraic solution is excluded. Therefore, a numerical procedure is used to obtain /3 from (A-6). The numerical procedure assumes values for /3 and each time computes the left side of (A-6) until the proper value of /3 is determined. The values of D’, D~,and x 1 are determined in the following manner. For each value assumed for /3, x1 is calculated from (A-5), D’ is calculated from (A-4), and D~is calculated from (A-3). When a value 0 is found such that the left side of(A-5) is zero, the other parameters D’, D~,and x1 are also known.
a
=
1
b
NL ~ M=NF NL
~
=
~
=
~
~
=
s
~)‘
E~ v
M=NF NL
d
2
~
EMV
M=NF NI.
~
(M
M=NF NL
~
~)‘ ~),
(M)EM ~ S
e~SMNF ~ (M)E2Mv(~c~,),
MA.K. Lodhi, B. T. Waak/Nonlocal effects in nuclear shell models
f=NL—NF+1,
E2NF
2)2 {NF—(NF—1)E2
1
s(l—E
gE~
1ENL NFl 1—F
191
2~~ —
h=E2~ l_F2LNF1), l—E2
÷(NL)~2~L_N~2)},
(NL÷l)E E3NF 32 ~NF—(NF—1)E3
i=E3~~ l_E3~L~l) i—F3
—
(NL+
3~~2~’}, 1~3(NL—NF+i)+ (NL)E
/ = E4NF
k=
lE4(NL-NF+l) i—F4
ENF 2 {NF — (NF— l)E
~NF
s(l—F4)2 {NF—(NF—1)E4
—
NL+l)F4
L-NF + i) + (NL)E4~_N~2)}.
(B-6)
s( 1-F) —
(NL+l)FNL_~~ +(NL)E~~~2},
The numerical procedures for finding /3 and the calcula-
tions forD’, D~,and x 1 are much faster for the weighting factor F(x) = 1 when using (B-6) rather than (A-2).
M.A.K. Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models
192 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * *
a 15
a
7 ‘5 0 0 15 .
l 51 4 U _i
a
..i I 4 0— 0-51 —
za
0-5) 0— I 0— Ui > — 0• 0 ii
a
a a a z lu
C
• > UI 7 F— ‘0 ‘0 ‘0 — Si ii
e
UI
24$*
~
0
~
U,
* * N * — * * * * * — * — * IX * * * 27 * * 0> * — I * *0* * U X * * Z —4
—
*
5-OW tO7~- z 10 0 7
C~ * * * * * * * * Z * — * * Z * 0 * 0* 0 * 7 * 0. * * * .5 * * * 0 * *
4 U 00 4 (51
a
UI 04 05) N S.
11
0 —
Ui * 0.
* * 0.
* * * * * * * * * * *
* * * * *
* * * * * *
1-
• 7 II.
Ca
I •
—
a
0> U. W
—
I~1
77 0 0 20 III X
0-
* * * *
* * * * *
* *
Ui
o-
7 — 7
0. —
Z 0
**
Z
4* 4* *4 4* ~* *4 4* 8* *4 ** *4 *8 8* *8 4*
0
*4
a W ~
~J ..1 — 3
— IU Z
0
0
•
00
+ — (5) ‘
0 0 0’ LI) ‘a 4 U) a 0
—
u
* *
.5
0 0’OW ‘00 407
—ow I II 3~ (fl U) LU 0
00 N Z 511
a
UI 0
a a 0 7
0
4
*8 8* *4 4* *4 ** ** *8 *8
.J 4 .a
*4 4* 4* 4*
Z 11 0-
4* *4 04* ** 18* U ** Z *8 ~** *4
Ui I 0-
a 7
I—
a a
Ui > 0U
•
a a
a U. Ui
0*~ 0. *4 Z *4
t~. 0
U. 0
7
4
‘5
UI
0-
—
WLU m U (0 OW
C) _i a.
c
LU
4*
U.
I 0-
z a
4*
*4 *4 **
.J
0*0-_i 5 * — 711 * Z > ..i * —
4*
2
a
112
70 1102 7 4 UI 7
*4 4* *8 *8 ** *4 *4 4* 4* *4
~
4* 44 .J *4 ..J *4 ‘—4* 34* 4* 514* 04* a *4 4*4 U **
0
*
.1 40.
• X
I-I
0 7 0.
a
Ui 0-0 OW 0.0 —
4
W>
u
>0 —m
0
~
— ~0 Ui
* * *11 * *0oW *51 *
m
U • Wa)U005.0— UJ~.~Z 0 UJlS
‘~ 4 7 — 5.
I-.a
a
0-0Z
Ui — >
aoo
0—a —7 ZU4
Ui 7
ozo.
a
07 0-5.0 (0 SUJ
—ISa
I 3
u~
007 • 77.54 a .j 51 • 15 6
Z
* * * * * * * * * * * * * * * * * * * * * *0
Si
-a
*8 **
—
a
* *
a
Z UI
U 2
U
ZI-
m
0.11 4151
a 0a s
o-’
II
.~7N
N
a 75-00
* * * * * * * * * * * * * * * * * * * * *
0
4*
•
7
5. 0
z
**
o
a
— I7 .5 I-
*4
a
*
• > Ui
*
** ** 4* ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** *4 4* *-* *4 *4 4* *4 *8 *4 4* ** *4 8* *4 *4 ** ~* *4 *4 4* 4* 4* ** 4* ** 8* *4
.
— * II* U * — Z * SC 0 * — U. * > * 104 z * — * 0* I * tO * 0* * 3 * I * II * * 51 * * SC * * . * — * * * — * SC * * SC * ‘— * * * ~ * 0* 11 * 0 * I — * — * a — * XX *
0
z
• 0— —
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
‘I) 6
Ui
*4
6 0
74
0 N
*0*4 *0 * 451 * *1 *1.* *5) *Z
7
00
~
*0
6 0 0 5.
—
.5 7 4
a aIII
• CS
7
I Ui 711) — — 7. 0 0-’) 0 • 4 X ui!l
a I I— 0 0-
0UI 7 0 0. 7 Ui 5.
—
0 I0 ~ U ISe
0 Z — 0-1fl4 U 0 ‘5 Si F- 70-
7
a oI
a
a.
0 02 U
•ou
V)~ Ui (51 7
0..>
0
a a
ii I 55<7 (0UI s04 — — 3404-. Ui Ui C — — LU F0— 0 — 4 .
“) F• —
.J •
II)
._j
a
LU N
‘0
7 >>li. LUUI”7 77.511.
I
2 0
m U. —
*‘
*1*4 *7 4W *08-’ * *0 * *7 *11 *0 *2 *~ *Z * * 4. ~oo *
*
*5)4 4’. * *5, *Z .Jn) *04 *I1 *0-0*4 Z.J *7 114 I-’.
*a *0-00-
II
SC
70.504
Ui
7
* *1.
— Q’514F7 .4)500 4.500 7 N0)5
4 0 .,~ >
0 11 >
*70*0>0 *
—a.
UJUJW >>>
OW W5) am 1.0 1117
712
—
l009F••~~ OF-ON T’)N
(0
a
a
8LU *0 *7 *0 *Z * 45 *0 *7
11
0-
*4 ..J1 *70.0.
7 UI 10
7 0 51
—
—
UI
I.-
5.
11111111
04 001-SC U
a
Ui 1111-
‘occe ,Sr’)F-
1.5. 00
11111$
*0-0*5< 00
LU XZLO 4.Z 714
x.~m
MA.K Lodhi, B. T. Weak/Nonlocal effects in nuclear shell models X
=
J*STEP/SCALE +.•..+..
10. II. 12. 13. 14. 15. 16. 17. (8. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 3*. 37. 38. 39. 40. 41. 42. 43. 44. 45 46. 47. 48. 49. 50, 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 85. -
• -
• -
9’ 01 102 103: 104.
107. 108.
—
..+....+•.
NLOW*(STEPI).’SCALE ..+..
•.+_
(J*
(31/
100.00
..•+••..4_.
_..,...•+.
—
(
1Ø)*(
..0....+....+.
...+..
121/ ..+.
193
100.00
=
...f..
•..+•••.+•
J*( .•+.
0.130) .•_+.
.02 .1 2 .22 .32 .42 .52 .62 .72 .8 2 .92
.oB
A A A B
A AS BA B B A B A B A 8 A BA 2 46 A B A B A B A 6 A 8 A A A
B B
A
(.200
—
...+.
.1 .2 .3 .4 .5 .6 .7 .8 .9
.0 I .2 .4 .6 .8 .9
8
.0
B B
A
.1 .2 .3 .4 .5 .6 .7 .8 .9
B
A
8 A
B A B 2 BA B
A A
8
.0
A
B B
A B
A B
A B
A B
A B B
A B 8 B 8 B
A A
A A A A B A B A B A 6 A BA BA BA BA BA BA BA 114 BA 2 2 2 2 AS AB AS AS AB AS AS AB AB AB AB AS AS
.1 .2 .3 .4 .5 .6 .7 .8 .9
.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 .0 .1 .2 .3 .4 .5
:~ .6
.9 .0 .1 .2 .3 .4 .5 .6 .7
A A A A A
AS 6 B B 8 5
.9 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9
A A A
8 B B
.0 .1 .2
A A A
8 B B
:5 .6 .7