Quasi-bound state wavefunctions

Computer Physics Communications 11(1976)249—256 © North-Holland Publishing Company

QUASI-HOUND STATE WAVEFUNCTIONS R.M. DeVRIES Nuclear Structure Research Laboratory ~‘, University of Rochester, Rochester, New York 14627, USA Received 15 March 1976

PROGRAM SUMMARY Title ofprogram: QUASI-BOUND STATE WAVEFUNC— TIONS Catalogue number: ABMQ Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue)

Computer: IBM 360/65; Installation: University of Rochester

Nature of the physical problems A program has been written which calculates the solution to the time independent Schrddinger equation for particles trapped by a Coulomb barrier but whose energy is greater than zero. Method For fixedofsolution Woods—Saxon radius and diffuseness, the depth of

Program language used: FORTRAN

the nuclear potential is iteratively found starting from a radius where the total potential energy exceeds the particle energy. Using this nuclear potential the quasi-bound wavefunction is generated from the boundary conditions at r = of

High speed store required: 175K (WATFIV compiler)

G(r)/r where G(r) is the irregular Coulomb wavefunction

No. of bits in a word: 64 (double precision) Overlay structure: None

Restriction on the complexity of the problem The program is dimensioned for a maximum of 400 mesh points and a maximum angular momentum of 10. Both these

No. of magnetic tapes required: None

limits are easily increased if desired.

Other peripherals used: Card reader, line printer Number of cards in cvmbined program and test deck: 628

Typical running time 212Po 208Pb + a), 7 sec using the WATFIV For the test case ( compiler — this includes compilation time.

Operating system: OS MVT Release 21.8

Card punching mode: IBM(029) EBCDIC Keywords: Nuclear, wave function, quasi-bound, unbound, Schrodingers equation, resonant state, reaction, Woods— Saxon. *

Supported by the National Science Foundation,

Unusual features of the program Because the solution is exactly on resonance no searching on the phase shift is required. Therefore this technique works even if the resonance widths are extremely narrow (1010 eV for example).

R.M. De Vries/Quasi-bound state wavefunctions

250

LONG WRITE-UP 1. Introduction

The well-known solution for large r (VN ~ 0) is

In nuclear reactions, states are often excited, which may be considered to consist of a core nucleus plus a single nucleon or cluster (an alpha particle, for example). Fig. 1 exemplifies the radial potentials involved; the orbital angular momentum is ignored in fig. 1, but is easily included. Discrete states at various energies En may exist as shown. The states are classifled as ‘bound’ if En <0 (in fig. 1), ‘quasi-bound’ if 0Eb. The solution of the radial Schrodinger equation for each of the binding conditions requires somewhat different techniques. The method of solution for bound states is well known [11. Unbound states and related reaction mechanism calculations have also been studied [2]. The quasi-bound state solutions are usually found by searching on the phase shift (discussed in section 2) and requiring that it go through iij23. A different technique is employed here which is particularly useful if the state involved is very narrow.

~I•~Sifl (kr + 6), (2) where 6 is the ‘phase shift’ which for a sharp state with given L, and Ea~is 6 = fir. The term ‘resonance’ comes from scattering studies. If Coulomb forces are included, the solution to the Schrodinger equation (again for larger) isa linear combination of the regular and irregular Coulomb functions which look like, respectively, / zZe2 F sin ~kr in (kr) +

2. Theory

— —--

and G

cos IIkr

+

VN(r)) i~i(r)= En ~(r).

.

-~

0

(1)

Eb

tions 0

/

/

I

I

R~

R0

r

-~

/

/ i’Z__—~i VNucIear + VCouIOmb

Lii

(3)

v

corresponds to a resonance situation is required. For relative angular momentum zero and in the absence of Coulomb forces, the Schrödinger equation becomes m V2

zZe —h---in (kr) + ~2

Clearly our quasi-bound resonant state wavefunction must behave like G(r) at large radii [boundary condition (a)] The other boundary conditions are: (b) ri~fi-÷ 0 as r 0, and (c) i~tishall have a specified number of internal nodes. It is not possible to satisfy all these conditions with any arbitrary VN(r). If VN(r) is taken to have a Woods—Saxon form

The solution to the Schrodinger equation which

/

—

______

VN(r)

R’/ 1 (4) exp 1~r , then for fixed radius R and diffuseness a, it will be =

~

“

‘

—

tions are satisfied. Of course, could vary any necessary to vary V0 theone three boundary condithe other constants inuntil the problem rather than V of 0. The problem then becomes how to start from some initial (guess) value of V0 and iterate to the final (unique) value for which the three boundary condiare satisfied. One common technique is to start the numerical solution to the Schrodinger equation at the origin, vary the depth V0 until the proper number of (internal) nodes are obtained inside of a selected surface radius, and then integrate on out to large r. V0 is then varied in smaller steps until 6 = 1/2 or equivalently the ratio of the wavefunction in the intenor to exterior (large r) is maximized [31 Vincent [2] starts at large r withG(r) and integrates inward. Likewise an outward mtegration starting with condition (b) above is compared with the inward integra.

Fig. 1. Energetics of a (1 = 0) quasi-bound resonant state systern as a function of radius between the core and particle (cluster).

.

.

.

R.M. Dc Vries/Quasi-bound state wavefunctions

tion

at some matching radius. The difference between the two solutions can be related to a change in V which starts the next iteration. This method may also be used for unbound states. A related but different method is used here. Notice in fig. 1 that between R1 and R0, En < Vtotal, ~ that by starting an inward integration in this region, the usual methods of solving the Schrödinger equa-

Ic’

0 2

6

~

C

~

~

82

251

~ ~

tionV0 foruntil bound states may employed, varying conditions (b)be and (c) abovethereby are satisfied.

~

The problem with this technique is what to use for

I0~~ 116

I

starting values since the usual bound state starting (r~a,~) values are [1] i,D(rmax) exp(—krm~), (5)

10-s Normal (~‘-e~) Starting Ca,ditions

with k = \/~)~z2 and p the reduced mass. Our positive En energy rules out these starting values; however, it is demonstrated in fig. 2 that almost any starting values may in fact be used and only

I0~

Arbitrary starti~gI Conditions IO~

2

4

6

8

0

2

4

6

8

20

the region of only ~tiwilltobe in error. are outermost using this technique obtain V Since we 0 at the ongin, we can use purely arbitrary starting conditions. Obviously one should start close to R0 (rather than

R(fm) 51Cr = ~ v + ~i (E~= — 5.0 Fig. 2. Bound state system of MeV) comparing integration results using normal (asymptot ically correct) and arbitrary (~i(rniax)= 0.97 X 10—30, ~(rmax — 0.1 fm) = 1.0 X 10—30) starting conditions. A few fermis inside of rmax the two methods yield identical results. Each was normalized to unity,

Ri). Having established V0 (and, therefore, VN) we now perform a new integration inwards from Rmax (well beyond R0 in fig. I) using initial values of G(r). The result is our quasi-bound state wavefunction. It is found that the first wavefunction (satisfying condi-

0~

I0~ —

~

I0~ 212

o~

Po (g. s.) = 208 Pb + a

0

(‘.1 -~-

jot

_____

—

100

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-—

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“ 4

8

12

6

20

24

28

—

32

‘

%i”T\I~”YfliT\r~~Y. \[ 36

40

44

48

52

56

R208pb+~(fm) Fig. 3. Quasi-bound resonant

state wavefunction normalized to G(r)/r for large r.

60

252

R.M. De Vries/Quasi-bound state wavefunctions

tions (b) and (c) but with arbitrary starting values at rmax) is in perfect registry with the final result except, of course, for the arbitrary normalization of the for-

Table 1 Input Card No. Parameter Description

men wavefunction. A typical final result is shown in fig. 3. The overall normalization of i,li has two different forms in this program: (a) iP(r) = G(r)/r for large r, and

1

2

R~

(b)

f

i,1i2(r) r2 dr = 1.

3

0

The R~cutoff radius is taken as the first exterior node (see fig. 3, for an example), but any radius in this region yields the same normalization if En is sufficiently below Eb. The program outputs ui normalized in both of these ways.

ICON(12) control integers — not used — see section 3

1211

ALPHA DRF RMAX ALVAL ECM

run label step size maximum radius I (angular momentum) En (center-of-mass energy)

8X,13A4 9F8.4

FM

particle (or cluster) mass (in AMU) particle (or cluster) atomic number

Z FMA ZA RY

3. Program 4

Besides a main program there are four subroutines, The main program handles input of control integers (unused, but available for individual disk writing erations), and other (but not all) input data. It also

FZ

~

VR

processes the normalization (b) above. 3.1. Subroutine FORMF

RY ARI VSOR

This subroutine is a modified version of a subroutine in the program DWUCK [4] Two cards of input parameters are read in first. The Coulomb potential and VN(r) are generated and then the I (angular momentum),j (total angular momentum) and s (spin of the particle or cluster) values are read. These potentials and the calculated rm~value are used to find the required V 0 value by calling subroutine BIND. The step size in this search is fixed at 0.1 fm. FORMF is called (by MAIN) again to generate VN(r) with the correct V0 and using Rmax and the chosen step size. .

3.2. Subroutine BIND

This routine is taken from the program DWUCK [4] As discussed in section 2, it determines the correct V0 by integration until boundary conditions (b) and (c) are met. The method is that of ref. [1].

Format

__________________________________________________

5

FN FL

FJ2

9F8.4

core nucleus mass core nucleus atomic number Coulomb radius Rc=RYFMA’~’3if RY<0 Rc=RY.(FMA~’3+FM”3) if RY >0 type of potential (only volume Woods—Saxon wells included, but FZ eases user modification) V 0 (guess value) 3,ifRY = VR*VTRIAL <0 R = RY*(FMAI /~ + FML’3) R= RY*FMAi~’ if RY > 0 a (diffuseness) Thomas—Fermi spin orbit strength number of radial nodes

9F8.4

(not includingr = 0, °“) angular momentum, 1, of the state — same as card ALVAL, card 2 twice the total angular momentum of the state

FSS

twice the spin of the particle

VTRIAL

first guess for V0 — if 0., VTRIAL = 60. is used

Note: The input of this program follows that used in the Distorted Wave Born Approximation programs DWUCK (P.D. Kunz, unpublished) and LOLA (R.M. DeVries, unpublished).

R.M. De Vries/Quasi-bound state wavefunctions

253

Table 2 Error messages Error message

Subroutine

Cause

RMAX CHANGED TO

MAIN

Rmax < R

0 linputas> 10

Rmax STOP

FORMF BIND

E0 ~ after 16 tries search for V0 (correct) has not converged

STOP STOP

MAXIMUM L VALUE 15 10 YOUR STATE IS UNBOUND OR NEARLY SO-SORRY BUT WE CANNOT HELP YOU BOUND STATE SEARCH FAILS TO CONVERGE IN 16 INTERATIONS

Note: if Rmax/DRF > 400, Rinax

=

Result =

> 400*DRF

> 400DRF.

3.3. Subroutine COU

to one table 2.

This routine was written by Wills [5] and calculates the regular and irregular Coulomb wavefunctions at Rm~and R~~j~DRF when DRF is the step size.

‘.

The program error messages are listed in

3.4. Subroutine OUTER The final resulting quasi-bound wavefunction is generated from starting values of G(r)/r which are obtained from subroutine COU. The method of integration is Fox—Goodwin.

6. Description of the test run = 2o8P~~ + a is calculated. The final result is shown 212Poin The wavefunction appropriate to the state fig. 3 and was used to determine absolute alpha reduced widths from alpha decay and alpha transfer reactions [6] The program has also been tested for proton states.

4. Program notes

Acknowledgements

The program must be run with at least forty bit words for sufficient accuracy. If a resonance profile is required (i.e. the resonance width is non-negligible) the R max starting conditions are easily changed to F(r) sin 4, + G(r) cos 4, since subroutine COU calculates both G(r) and F(r). The additional phase 4, can also be used to make certain that the ratio of ui(intenor) to 4,(exterior) is maximum, as required by a resonance.

C.M. Vincent, I. Towner and H.T. Fortune all provided valuable help to this study.

5. Input/output The input cards are listed in table 1. Normal output is illustrated at the end of this paper. The list of ~ values are the interations in subroutine BIND attempting to find V 0. The inverse of the integration of 4, per normalization (b) above is labeled ‘normalization .~

.

References Eli

S.M. Perez, Report No. 38, Nuclear Physics Theoretical Group, Oxford University; see also ref. [41. [21 C.M. Vincent,.private comrnunication; see also H.T. Fortune and C.M. Vincent, Phys. Rev. 185 (1969) 1401. [3) D.H. Youngblood, R.L. Kozub, R.A. Kenefick and J.C. Hiebert, Phys. Rev. C2 (1970) 477. [41 P.D. Kunz, program DWUCK, unpublished. [5] J .G. Wills, Thesis, University of Indiana, unpublished; see also J.G. Wills, J. Comput. Phys. 8 (1971) 162. [61 R.M. DeVries, Shapi~a,W.G. avies,Lett. G.C.35 Ball, J.S. 835; Forster and W. D. McLatcl~ie, Phys.DRev. (1975) W.G. Davies, R.M. DeVties, G.C. Ball, J.S. Forster, R.E. Warner, D. Shapira and J. Take (to be published).

R.M. DeVries/Quasi-bound state Wavefunct ions

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