DCP2 - Manual

Antisymmetric Distorted Wave Impulse Approximation
Calculations for Composite Particle ScatteringbyT. Udagawa, A. Schulte and F. Osterfeld
Nuclear Physics, A474 (1987) 131-154

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Contents
1 What is DCP2?

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2 What can DCP2 calculate?
2.1 Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Reaction mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Outline of calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Structure of the Program
3.1 Flow of the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Compile and Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 How to make a Input Data File
4.1 Outline of input form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Detailed description of input data . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Calculation options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Output options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Nuclei in each channel, and incident beam lab energy . . . . . . . . . . .
4.2.4 Angular momenta in the reaction . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Optical model parameters (OMP) . . . . . . . . . . . . . . . . . . . . . .
4.2.6 Partial wave expansions in each channel . . . . . . . . . . . . . . . . . . .
4.2.7 Radial integration information . . . . . . . . . . . . . . . . . . . . . . . .
4.2.8 Angle information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.9 Form factor calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.10 Love-Franey NN interaction information . . . . . . . . . . . . . . . . . . .
4.2.11 Single particle and hole state information . . . . . . . . . . . . . . . . . .
4.3 Definition of input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Optical model potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Love-Franey NN effective interaction . . . . . . . . . . . . . . . . . . . .
4.3.3 Single particle potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Input Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 12 C(p, n)12 N (1+ , 18.1 MeV) at Elab =200 MeV . . . . . . . . . . . . . . .
4.4.2 90 Zr(3 H, t)90 Nb, (0− , 27.9 MeV) at Elab =600 MeV . . . . . . . . . . . . .
4.5 Output Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 12 C(p, n)12 N (1+ , 18.1 MeV) at Elab =200 MeV . . . . . . . . . . . . . . .
4.5.2 90 Zr(3 H, t)90 Nb, (0− , 27.9 MeV) at Elab =600 MeV . . . . . . . . . . . . .

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5 Formulas
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5.1 Differential cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2 Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.1 Direct form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.2 Direct form factor for the nucleon-nucleus scattering . . . . . . . . . . . . 27
5.2.3 Exchange form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2.4 Exchange form factor for the nucleon-nucleus scattering . . . . . . . . . . 28
5.2.5 Exchange form factor in the no-recoil approximation . . . . . . . . . . . . 29
5.2.6 Exchange form factor in the plane wave approximation . . . . . . . . . . . 29
5.3 Relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Projectile Density distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4.1 Deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4.2 Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.5 Folding optical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A Subroutines and their functions
A.1 Structure of the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Functions of the subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.1 DCP2 -Main Routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.2 BSAXON and UNCPST - Single particle states . . . . . . . . . . . . . . .
A.2.3 DCHECK1 - Checking field lengths of variables . . . . . . . . . . . . . . .
A.2.4 OPT, FLGLCH, POTEN, and DISWAVE - Distorted waves . . . . . . . .
A.2.5 AFACAL - α-coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.6 EFFINT - Effective interaction components . . . . . . . . . . . . . . . . .
A.2.7 PDENST - Projectile density function . . . . . . . . . . . . . . . . . . . .
A.2.8 FFCALD - Direct form factor . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.9 PACALD - Direct overlap integral and transition amplitudes . . . . . . .
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A.2.10 XLMCAL - Distortion factor . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.11 FFCALDM - Modified direct form factor . . . . . . . . . . . . . . . . . .
A.2.12 DCHECK2 - Checking field lengths of variables for the exchange part . .
A.2.13 TRECAL, DENST, TRECALM, DENSTM - Non-local projectile and target densities for exchange part . . . . . . . . . . . . . . . . . . . . . . . .
A.2.14 GFAC, GFACM - Vector coupling factors for exchange part . . . . . . . .
A.2.15 FFCALE - Exchange form factors . . . . . . . . . . . . . . . . . . . . . .
A.2.16 TENSOR - Tensor interaction for exchange part . . . . . . . . . . . . . .
A.2.17 PACALE - Exchange overlap integral and transition amplitudes . . . . .
A.2.18 FFCALEM - Exchange overlap integral and transition amplitudes . . . .
A.2.19 CROSS - The differential cross sections . . . . . . . . . . . . . . . . . . .
A.2.20 YLCAL, GAUSF, BESSEL, CLEB, CLEBZ, RAC7, NINEJ, DSPLS3 Supporting subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B Variables and their definitions
B.1 PARAMETER variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 COMMON variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.1 CNST - Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.2 CNTRL - Control of the calculation and output . . . . . . . . . . . . . .
B.2.3 SPSTAT - Particle-hole states in the target . . . . . . . . . . . . . . . . .
B.2.4 BSX, UNCPSA, CBST - Bound state wave functions . . . . . . . . . . . .
B.2.5 DWCC, POTCC, COUCC, RELKIN, OPTO, DISW - Distorted waves . .
B.2.6 ANGCC - Angle information . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.7 SPECFC - α-coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.8 FFCC, CDENS CCKF - Preparation of form factor calculations . . . . .
B.2.9 CGFACA, CGFACB, TREDEN - Exchange form factor calculations . . .
B.2.10 CGGR, AMPD, OVDE - Form factors and Transition amplitudes . . . . .
B.2.11 CPWFAC, CGFACC, CXLM - Plane wave approximation for exchange
form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 List of Key variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.4 Variables on the TAPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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What is DCP2?

DCP is the abbreviation for Distorted wave impulse approximation (DWIA) calculations for
Composite Particle scattering. DCP2 is a computer program, which performs a DWIA calculation for the light-ion induced inelastic scattering and charge exchange reactions to the discrete
final states at intermediate energies. It permits the exact calculation of the so-called knock-on
exchange transition amplitudes. It can calculate the exchange part in the no-recoil approximation, and can restore the recoil effect but in the plane wave approximation.
This program is originated from the program DCP1, coded by T. Udagawa, A. Schulte and F.
Osterfeld in the middle of nineteen eighties. The original version of DCP1 was not published,
and was not kept anywhere unfortunately.
B. T. Kim collected some of original subroutines from the Udagawa’s Texas group, rebuilt the
program with helps of Profs. H. Sakai and M. Ichimura during his stay at RIBF, Nishina Center,
RIKEN, Japan in the winters of 2012 and 2013, and renamed it DCP2. Thus, B. T. Kim has
the full responsibility for the present form of DCP2.
The code is written in FORTRAN. The program is written with capital letters. In this manual,
the name consisting of capital letters denotes generally that of a subroutine, a variable, or acommon block.

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What can DCP2 calculate?

2.1

Reactions

DCP2 calculates the differential cross sections for the inelastic scattering
A(a, a0 )B
where A(B) is the target (residual) nucleus, while a is the projectile nucleus and a0 in its excitedstate, and the charge exchange reactions
A(a, b)B
where b is the ejectile nucleus, by exchanging a charge with a, say, (p, n), (3 He, t), etc.

2.2

Reaction mechanism

The antisymmetrized DWIA with the NN effective interactions with direct and exchange partsis the basic reaction mechanism. The effective interaction are assumed to be local and to containcentral and tensor terms with spin-spin and isospin-isospin interactions. The excited and charge
exchanged states are assumed to be in the single particle model state in a Woods-Saxon potential.

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2.3

Outline of calculation

Single particle potential
V (r1 )

NN effective interaction
Vtsk (r)Optical potential
V (ra ) + iW (ra )?

?

Particle-hole state
φp (r1 ), φh (r1 )Distorted wavesχ`a (ra ), χ`b (rb )?

Target densityρT (r1 , r10 )Projectile densityρP (r2 , r20 )

?

?

?

Form factors
f D (ra ), f E (ra , rb )?

Direct Form factor
ρD
(~
r
)
⇒
ρD
r20 , ~r) ⇒ ρDr, ~r1 , ~ra )
P 2
P (~
P (~
D
f (ra ) =R
R
D
drVtsk (r) dr1 ρD
P (ra , r1 , r)ρT (r1 )?

Exchange Form factor
ρE
r1 , ~r10 ) → ρE
r1 , ~r)
T (~
T (~
E
0
0 ,~
ρP (~r2 , ~r2 ) → ρE
(~
r
r
)
→
ρE
rb , ~r1 , ~r)
P 2
P (~G(rb , r) =R
E
Vtsk (r) dr1 ρE
P (rb , r1 , r)ρT (r1 , r)Ef (ra , rb ) ← G(rb , r)
No-recoil approximation(~ra ≈ ~rb )
R
N
R
2+k
f (rb ) = drr G(rb , r)PlaneR wave approximation
drr2+k G(rb , r)j`r (αka r/a)f P W (rb ) =?

?

Transition
amplitudes
RD
T R = Rdra χb (~ra )f D (ra )χa (ra )E
T = dra drb χb (~rb )f E (rb , ra )χa (~ra )?

Differential cross sections
dσ/dΩ = |T D + T E |2

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Structure of the Program

3.1

Flow of the program

Input, Initial Output?

Single particle bound states, ψ(r1 )?

Distorted waves, χ`a (ra ), χ`b (rb )?

Geometrical α-factors?

Direct form factors, f D (ra )Projectile density, ρP (r2 )Target density, ρT (r1 )Effective interaction, Vef f (r)?

?

Exchange form factors, f E (ra , rb )(Exact or No-recoil)
Projectile density, ρP (r2 , r20 )Target density, ρT (r1 , r10 )Effective interaction, Vef f (r)?

Exchange form factors, f E (ra )in the PW approximation
Projectile density, ρP (r2 , r20 )Target density, ρT (r1 , r10 )Effective interaction, Vef f (r)Transition Amplitudes?

Differential Cross Sections?

Output6?

3.2

Compile and Run

DCP2 consists of 8 fortran source files which contain subroutines as follows;
Source file
dcp2.f
cmm.f
crs.f
ffc.f
ffd.f
ffdm.fffe.fffem.fSubroutines
MAIN, DCHECK1, DCHECK2BSAXON, UNCPST, OPT, POTEN, FLGLCH, DISWAVE, RAC7,
CLEB, CLEBZ, NINEJ, GAUSF, BESSEL, YLCAL, DSPLS3CROSSAFACAL, EFFINT, PDENSTFFCALD, PACALDFFCALDM, XLMCALFFCALE, TRECAL, DENST, GFAC, TENSOR, PACALEFFCALEM, TRECALM, DENSTM, GFACM

DCP2 can be compiled by calling a fortran compiler f77 or g77 on the shell of LINUX.
As an example,”g77 dcp2.f cmm.f crs.f ffc.f ffd.f ffdm.f ffe.f ffem.f -o dcp2”
compiles DCP2 and creates a executable file ”dcp2”.
Before the run, one should prepare input data file (unit 7), ”dcp.dat” on the shell. To execute,simply type”./dcp2”

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How to make a Input Data File

4.1

Outline of input form

4.1.1

StructureLine #
1
2
3-4
5-9
10-11
12
13-14
15
16-19
20-32
33-

4.2
4.2.1

Function
1. Calculation options
2. Output options
3. Nuclei in each channel, and incident beam lab energy
4. Angular momenta in the reaction
5. Optical model parameters of each channel
6. Partial wave information in each channel
7. Radial integration information
8. Angle information
9. Form factor calculations
10. NN Love-Franey interaction information
11. Single particle and hole state information

Detailed description of input data
Calculation options

Line 1 :READ(7,10)(KTRLD(N),N=1,9)
FORMAT(24I3)
Stored in COMMON /CNTRL/

Variable
KTRLD(N)

N
1

2
3
4

5
9

Value
0
1
2
0
1
0
1
0
1
0
1
0
1

Explanation
Normal composite particle case with exact finite range form factor.
Recoil effects in the plane wave approximation.
No-recoil calculation is made.
DWIA calculation is made.
PWIA calculation is made.
Normal optical model potential.
Single folding optical model potential.
Tensor force is included.
Tensor force is neglected.
(Tensor force is set equal to zero in EFFINT)
Exchange effect is calculated.
Exchange effect is neglected.
Nonrelativistic kinematics is used.
Relativistic kinematics is used.

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Output options

Line 2 :READ(7,10)(KTLOUT(N),N=1,24)
FORMAT(24I3)
Stored in COMMON /CNTRL/

VariabeKTLOUT(N)

4.2.3

N
1
2
3

Value
1
1
1

4
5
7
8
13

1
1
1
1
10

Explanation
Output of bound state wave functions.
Output of distorted waves.
Output of form factors and transition amplitudes
in PACALD and PACALE.
Output in GFAC.
Output of elastic scattering information in OPT.
Output of Coulomb wave functions in FLGLCH.
Output of ph form factors.
Output of detailed form factors,
at every (KTLOUT(13)×mesh size) point.

Nuclei in each channel, and incident beam lab energy

Line 3 :Variable
TMI
Z1I
PMI
Z2I
ELI
Line 4 :Variable
TMF
Z1F
PMF
Z2F

READ(7,12)
TMI,Z1I,PMI,Z2I,ELI
FORMAT(10F7.3)
Stored in COMMON /DWCC/ by changing the names as
TMASA,TZA,PMASA,PZA, respectively.
Explanation
Mass number of initial target, mA .
Charge number of target, ZA .
Mass number of projectile, ma .
(Presently up to 3. Otherwise, increase parameter ”NXA”)
Charge number of projectile, Za .
Incident beam LAB energy in MeV.
READ(7,12)
TMI,Z1I,PMI,Z2I
FORMAT(10F7.3)
Stored in COMMON /DWCC/ by changing the names as
TMASB,TZB,PMASB,PZB, respectively.
Explanation
Mass number of residual nucleus, mB .
Charge number of residual nucleus, ZB .
Mass number of ejectile, mb .
(Presently up to 3. Otherwise, increase parameter ”NXA”)
Charge number of ejectile, Zb .

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Angular momenta in the reaction

Line 5 :Variable
JATW
ISATW

READ(7,10)
JATW,ISATWFORMAT(24I3)
Stored in COMMON /FFCC/
Explanation
Twice of the total angular momentum of the target A, IA .
Twice of the spin of the projectile a, sa .

Line 6 :READ(7,17)
JT,KPARIT,IST,EET
FORMAT(3I3,3F7.3)
Stored in COMMON /SPSTAT/

VariableJT
KPARIT

Explanation
Transferred total angular momentum, jt .
= 0; No parity change in the reaction,= 1; Parity change.
Transferred spin, st (= s1 = s2 ).
Transferred energy in MeV.

IST
EET

Line 7 :READ(7,10)
NOLTR, (LTR(N),N=1,NOLTR)
FORMAT(24I3)
Stored in COMMON /SPSTAT/

VariableNOLTR
LTR(NOLTR)

Line 8 :Explanation
Maximum number of transferred orbital angular momenta.
Transferred orbital angular momenta, `t .

READ(7,10)
NLSMAX, (ITR(I),ISR(I),L1R(I),I=1,NLSMAX)FORMAT(24I3)
Stored in COMMON /SPSTAT/

Variable
NLSMAXITR(NLSMAX)ISR(NLSMAX)
L1R(NLSMAX)

Explanation
Maximum number of {`1 s1 t1 } sets in the target system.
Isospin, t1 = t2 .
Spin, s1 (= s2 = st ).
Orbital angular momentum, `1 .

Line 9 :READ(7,10)
MXMAX,MJMAXFORMAT(24I3)
Stored in COMMON /FFCC/ for MXMAXStored in COMMON /CTFAC/ for MJMAX

Variable
MXMAX
MJMAX

Explanation
Maximum value of m`t .
Maximum value of mjt in [Eq.(12)].
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4.2.5

Optical model parameters (OMP)Line 10 :Variable
VA
WA
WAS
ARA
AIA
AISA
RZRA
RZIA
RZIAS
RZCA
Line 11 :Variable
VB
WB
WBS
ARB
AIB
AISB
RZRB
RZIB
RZIBS
RZCB
4.2.6

READ(7,12)
VA,WA,WAS,ARA,AIA,AISA,RZRA,RZIA,RZISA,RZCA
FORMAT(10F7.3)
Stored in COMMON /POTCC/
Explanation(Woods-Saxon OMP in the incident channel)
Depth parameter of real potential in MeV.
Depth parameter of imaginary potential.
Depth parameter of imaginary surface potential.
Diffuseness parameter of real potential in fm.
Diffuseness parameter of imaginary potential.
Diffuseness parameter of imaginary surface potential.
Reduced radius parameter of real potential in fm.
Reduced radius parameter of imaginary potential.
Reduced radius parameter of imaginary surface potential.
Reduced radius parameter of Coulomb potential.
READ(7,12)
VB,WB,WBS,ARB,AIB,AISB,RZRB,RZIB,RZISB,RZCB
FORMAT(10F7.3)
Stored in COMMON /POTCC/
Explanation(Woods-Saxon OMP in the exit channel)
Depth parameter of real potential in MeV.
Depth parameter of imaginary potential.
Depth parameter of imaginary surface potential.
Diffuseness parameter of real potential in fm.
Diffuseness parameter of imaginary potential.
Diffuseness parameter of imaginary surface potential.
Reduced radius parameter of real potential in fm.
Reduced radius parameter of imaginary potential.
Reduced radius parameter of imaginary surface potential.
Reduced radius parameter of Coulomb potential.

Partial wave expansions in each channel

Line 12 :READ(7,10)(LDWMIR(I),LDWMXR(I),LDWSTR(I),I=1,2)
FORMAT(24I3)
Stored in COMMON /DWCC/

VariableLDWMIR(1)LDWMXR(1)LDWSTR(1)LDWMIR(2)LDWMXR(2)LDWSTR(2)Explanation(Number of partial waves, `a , `b )Starting partial wave in the incident channel, `a .
Ending partial wave.
Step of partial wave.
Starting partial wave in the exit channel, `b .
Ending partial wave.
Step of partial wave.
114.2.7

Radial integration information

Line 13 :READ(7,11)(NXMIR(I),NXMXR(I),I=1,4),NHDMX,NHEMXFORMAT(14I5)
Stored in COMMON /DWCC/ for NXMIR(I),NXMXR(I)Stored in COMMON /FFCC/ for NHDMX,NHEMX

VariableNXMIR(1)NXMXR(1)NXMIR(2)NXMXR(2)NXMIR(3)NXMXR(3)NXMIR(4)NXMXR(4)
NHDMX
NHEMX

Explanation
Starting mesh point in the incident channel, ra .
Ending mesh point.
Starting mesh point in the exit channel, rb .
Ending mesh point.
Starting mesh point in the projectile system, r2 .
Ending mesh point.
Starting mesh point in the target system, r1 .
Ending mesh point.
Interaction range in the direct form factor calculation, rD .
Interaction range in the exchange form factor calculation, rE .

Line 14 :READ(7,12)(XMESR(I),I=1,4),XMESHD,XMESHEFORMAT(10F7.3)
Stored in COMMON /DWCC/ for XMESRStored in COMMON /FFCC/ for XMESHD,XMESHE

VariableXMESR(1)XMESR(2)XMESR(3)XMESR(4)XMESHDXMESHE

Explanation
Mesh size in the incident channel, ∆ra .
Mesh size in the exit channel, ∆rb .
Mesh size in the projectile system, ∆r2 .
Mesh size in the target system, ∆r1 .
Mesh size in the direct form factor calculation, ∆rD .
Mesh size in the exchange form factor calculation, ∆rE .

4.2.8

Angle information

Line 15 :READ(7,12)
THEB,THEBMX,DTHEB,THMIN,THMAX,THINDFORMAT(10F7.3)
Stored in COMMON /ANGCC/

Variable
THEB
THEBMX
DTHEB
THMIN
THMAX
THIND

Explanation
Starting angle in the differential cross section calculations.
Ending angle.
Step in angle.
Starting angle in elastic cross section calculations.
Ending angle.
step in angle.

124.2.9Form factor calculations

Line 16 :READ(7,10)(KCETN(N),N=1,2)
FORMAT(24I3)
Stored in COMMON /FFCC/

Variable
KCETN(1)

Value
0
1
0
1

KCETN(2)Line 17 :Explanation
Central exchange form factor is calculated.
Not considered.
Tensor exchange form factor is calculated.
Not considered.

READ(7,11)

NBCMI,NBCMX,NBSTPD,NBSTPE
NONAR,NASTEP,N1STEP,NGAUSR,FORMAT(14I5)
Stored in COMMON /FFCC/
Variable
NBCMI
NBCMX
NBSTPD
NBSTPENONAR
NASTEP
N1STEP
NGAUSR

Line 18 :Explanation
Minimum mesh point in the form factor.
Maximum mesh point.
Mesh step in the direct form factor.
Mesh step in the exchange form factor.
Number of mesh point in ra integration for exchange ff.
(Integration is made only around ra = rb with a range NONAR.)
Mesh step in the incident channel radius, ra .
Mesh step in the target system radius, r1 .
(For (p,n) or (p,p’), better put all steps unity.)
Number of gaussian integration points.

READ(7,11)
LAMMXD(1),LAMMXD(2),LAMMXD(3)FORMAT(14I5)
Stored in COMMON /CDENS/ for LAMMXD(I)

Variable
LAMMXD(1)LAMMXD(2)LAMMXD(3)Explanation
Maximum value of λ1 in exchange form factor.
Maximum value of λ2 in exchange form factor.
Maximum value of λ in exchange form factor.

Line 19 :READ(7,14)
LRP1MX,FACNR
FORMAT(I7,6F7.2)
Stored in COMMON /CPWFAC/

Variable
LRP1MX

Explanation
Maximum angular momentum of `r + 1 in the PW approximation.
(Inputting LRP1MX=1, and FACNR=0.0, gives NR approximation.)
Recoil factor α. 1.0 is suggested. (See Formulation 3.3.)

FACNR

13

4.2.10Love-Franey NN interaction information

Line 20 :Variable
KTRL1
VRANG(N)READ(7,14)
KTRL1,(VRANG(N),N=1,6)
FORMAT(I7,6F7.2)
Stored in COMMON /FFCC/ for VRANG(N)Explanation=1; Yukawa type interaction, (Set equal to 1. Actually dummy.)=2; Gaussian type interaction. (Not used in the present program.)
6 different range parameters in the effective NN interaction.

Line 21-32 : READ(7,13)((VSTR(K,N),N=1,6), K=1,12)
FORMAT(7F10.4)
Stored in COMMON /FFCC/
VariableK(tsk)

VSTR(K,N)

K
1
2
3
4
5
6
7
8
9
10
11
12

(000)
(010)
(012)
(100)
(110)
(112)
(000)
(010)
(012)
(100)
(110)
(112)

4.2.11

Strength parameters of Love-Franey interaction[ W. G. Love and M. A. Franey, Phys. Rev. C24 (1981) 1073;
C27 (1983) 438(E); C31 (1985) 488.]
N (=1 to 6) corresponds to 6 different ranges in VRANG(N).
Real SO t-matrix interaction strength.
Real TE t-matrix interaction strength.
Real TNE t-matrix interaction strength.
Real SE t-matrix interaction strength.
Real TO t-matrix interaction strength.
Real TNO t-matrix interaction strength.
Imaginary SO t-matrix interaction strength.
Imaginary TE t-matrix interaction strength.
Imaginary TNE t-matrix interaction strength.
Imaginary SE t-matrix interaction strength.
Imaginary TO t-matrix interaction strength.
Imaginary TNO t-matrix interaction strength.

Single particle and hole state information

Line 33 :READ(7,12)
TMAS,PMAS,ZZT,ZZP
FORMAT(10F7.3)
Stored in COMMON /UNCPSA/

Variable
TMAS
PMAS
ZZT
ZZP

Explanation
Mass of target system. (Target mass -1)
Mass of valence nucleon.
Charge of target system.
Charge of valence nucleon.

14Line 34 :Variable
VSX
VSOR
DFNR
DFNSO
RZR
RXSO
RZC
Line 35 :VariableNOSP
NOSH

READ(7,12)
VSX,VSOR,DFNR,DFNSO,RZR,RZSO,RZC
FORMAT(10F7.3)
Stored in COMMON /BSX/ except VSXStored in COMMON /UNCPSA / for VSX which will be searched in BSAXON.
Bound state potential parameters
Depth parameter.
Spin-orbit depth parameter.
Diffuseness parameter.
Spin-orbit diffuseness parameter.
Reduced radius parameter.
Spin-orbit reduced radius parameter.
Coulomb reduced radius parameter.
READ(7,10)
NOSP,NOSH
FORMAT(24I3)
Stored in COMMON /SPSTAT/
Explanation
Number of particle states.
Number of hole states.

Line 36 :+α
α=NOSP

READ(7,15)(ESP(N),NSP(N),LSP(N),JTWP(N),ITP(N),ITZP(N),
N=1,NOSP)FORMAT(F10.5,5I5)Stored in COMMON /SPSTAT/

Variable
ESP(N)
NSP(N)
LSP(N)
JTWP(N)
ITP(N)ITZP(N)

Single particle state informationenergy.
Number of nodes.
Orbital angular momentum.
Twice of total angular momentum.
Isospin T . (Not used.)= 1; neutron, = −1; proton.

Line 37+α+β
β=NOSH:

Variable
ESH(N)
NSH(N)
LSH(N)
JTWH(N)
ITH(N)ITZH(N)

Single hole state informationenergy.
Number of nodes.
Orbital angular momentum.
Twice of total angular momentum.
Isospin T . (Not used.)= 1; neutron, = −1; proton.

READ(7,15)(ESH(N),NSH(N),LSH(N),JTWH(N),ITH(N),ITZH(N),
N=1,NOSH)
FORMAT(F10.5,5I5)Stored in COMMON /SPSTAT/

15

4.3
4.3.1

Definition of input parameters
Optical model potential

The optical model potential is of Woods-Saxon type, expressed as
U (r) = −V0 f0 (r) − i[WI fI (r) + 4WS fS (r)] + VCoul1
f0 (r) =1 + exp[(r − R0 )/a0 ]1
fI (r) =1 + exp[(r − RI )/aI ]1 + exp[(r − RS )/aS ]fS (r) ={1 + exp[(r − RS )/aS ]}2
Note that no spin-orbit interaction is applied.
4.3.2

Love-Franey NN effective interaction

The Love-Franey nucleon-nucleon effective interaction is a superposition of Yukawa types as
V12 (r) = V C (r) + V T (r)S12
V C (r) =6X

ViC Y (r/Ri ),Y (x) = e−x /xi=1

V T (r) =6X

ViT r2 Y (r/Ri )i=1

3(σ~1 · ~r)(σ~2 · ~r) − (σ~1 · σ~2 )r2
24π 1/2 X∗≡ ()(σ~1 σ~2 )2m Y2m
(r̂)
5
m

S12 =

4.3.3

Single particle potential

The single particle potential of Woods-Saxon type, is expressed as
V (r) = −V0 f0 (r) − λ̄2π (~σ · ~`)VSO fSO (r) + VCoul1
f0 (r) =1 + exp[(r − R0 )/a0 ]1
1 + exp[(r − RSO )/aSO ]fSO (r) =aSO r {1 + exp[(r − RSO )/aSO ]}2√
where λ̄π = h̄/mπ c = 2.

16

4.4
4.4.1

Input Examples
12 C(p, n)12 N (1+ , 18.1 MeV) at E =200 MeVlab

0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0
0 0 0 0
12.0
6.0
1.0
1.0
200.0
12.0
7.0
1.0
0.0
0 1
1 0 1 18.1
2 0 2
2 1 1 0 1 1 2
4 1
11.00 14.00 0.000 0.69
0.59
0.867 1.20
1.17
1.260
11.00 14.00 0.000 0.69
0.59
0.867 1.20
1.17
1.260
0 22 1 0 22 1
1 120
1 120
1
80
1 120 300 300
0.10
0.10
0.10
0.10
0.03
0.150
1.0
50.0
1.0
10.0
70.0
1.0
0 0
1 120
1
1
59
1
1
32
2
0
2
4 1.000
1 1.40
0.70
0.55
0.40
0.25
0.15
31.50
0.000
0.0
-193.752
4714.120.0-10.500.000
0.0-1216.75
3567.350.0
0.00-69.7061
0.0
654.904 -24728.8 325631.0-10.500.00
0.0-2615.66
8396.19
0.0
3.50
0.00
0.0
-1439.49
7313.420.0
0.00
14.85730.0
334.910 -4099.14 50904.3
0.00
0.00
0.0
-563.008-919.824
0.0
0.00
0.00
0.0
-2431.21
7419.73
0.0
0.00-4.704740.0
478.422 -15805.0 212453.0
0.00
0.00
0.0
-469.589
1458.53
0.0
0.00
0.00
0.0
-217.827 -1076.89
0.0
0.00
3.336990.0-303.689
9392.15 -147371.
11.0
1.00
6.0
1.00
46.55 1.91
0.765 1.000 1.260 1.260 1.250
5 2
26.00
0
0
1
1
-1
16.00
0
1
3
1
-1
2.000
0
1
1
1
-1
2.000
0
2
3
1
-1
2.000
0
3
5
1
-1
31.20
0
0
1
1
1
18.72
0
1
3
1
1
*****************the end of the input data

17

1.20
1.20

4.4.290 Zr(3 H, t)90 Nb, (0− , 27.9 MeV) at E =600 MeVlab

0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0
0 0 0 0
90.0
40.0
3.0
2.0
600.0
90.0
41.0
3.0
1.0
0 1
0 1 1 27.9
1 1
1 1 1 14 1
15.31 13.26 0.000 0.724 0.662 0.662 1.276 1.260 1.260
15.31 13.26 0.000 0.724 0.662 0.662 1.276 1.260 1.260
0115 1 0115 1
1 120
1 120
1
80
1 120 300 300
0
0.10
0.10
0.10
0.10
0.03
0.015
0.01
8.00
0.2
10.0
70.0
1.0
0 0
10 120
10
10
59
1
1
32
1
0
1
4 1.0000
1 1.40
0.70
0.55
0.40
0.25
0.15
31.50
0.000
0.0
-193.752
4714.120.0-10.500.000
0.0-1216.75
3567.350.0
0.00-69.7061
0.0
654.904 -24728.8 325631.0-10.500.00
0.0-2615.66
8396.19
0.0
3.50
0.00
0.0
-1439.49
7313.420.0
0.00
14.85730.0
334.910 -4099.14 50904.3
0.00
0.00
0.0
-563.008-919.824
0.0
0.00
0.00
0.0
-2431.21
7419.73
0.0
0.00-4.704740.0
478.422 -15805.0 212453.0
0.00
0.00
0.0
-469.589
1458.53
0.0
0.00
0.00
0.0
-217.827 -1076.89
0.0
0.00
3.336990.0-303.689
9392.15 -147371.
89.0
1.00
40.
1.00
51.00 11.20 0.670 0.670 1.270 1.270 1.270
7 9
5.160
1
1
1
1
-1
2.000
0
4
9
1
-1
1.000
0
4
7
1
-1
1.000
1
2
5
1
-1
1.000
1
2
3
1
-1
1.000
2
0
1
1
-1
1.000
0
5
11
1
-1
14.30
0
3
7
1
-1
8.86
1
1
3
1
-1
7.63
0
3
5
1
-1
5.16
1
1
1
1
-1
19.22
0
3
7
1
1
14.96
1
1
3
1
1
13.42
0
3
5
1
1
12.56
1
1
1
1
1
11.97
0
4
9
1
1
*************The end of the input data*******

18

1.260
1.260

4.5
4.5.1

Output Examples
12 C(p, n)12 N (1+ , 18.1 MeV) at E =200 MeV
lab

DWIA DIRECT-EXCHANGE CROSS SECTION CALCULATION FOR THE REACTION
T
TZ
P
PZ
A-CHANNEL 12.000
6.000
1.000
1.000
B-CHANNEL 12.000
7.000
1.000
0.000
E=200.0 MEVINITIAL STATE JATW,ISATW= 0 1
FINAL STATE J,K,E= 1 0
18.1 MEV
SPIN TRANSFER IST= 1
TOTAL L TRANSFER = 0 2
KTRLD(1-9) = 0 0 0 0 0 0 0 0 1
KTLOUT(1-20)= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0** RELATIVISTIC KINEMATICS IS USEDNN INTERACTION POTENTIAL
KTRL1,VRAN=
1
1.40
0.70
0.55
0.40
0.25
0.15
K= 1 VSTR=
31.50
0.00
0.00
-193.754714.120.00
K= 2 VSTR=-10.500.00
0.00 -1216.75
3567.35
0.00
K= 3 VSTR=
0.00-69.710.00
654.90 -24728.80 325631.00
K= 4 VSTR=-10.500.00
0.00 -2615.66
8396.190.00
K= 5 VSTR=
3.50
0.00
0.00 -1439.49
7313.420.00
K= 6 VSTR=
0.00
14.86
0.00
334.91 -4099.14 50904.30
K= 7 VSTR=
0.00
0.00
0.00
-563.01-919.82
0.00
K= 8 VSTR=
0.00
0.00
0.00 -2431.21
7419.73
0.00
K= 9 VSTR=
0.00
-4.70
0.00
478.42 -15805.00 212453.00
K= 10 VSTR=
0.00
0.00
0.00
-469.591458.53
0.00
K= 11 VSTR=
0.00
0.00
0.00
-217.83 -1076.89
0.00
K= 12 VSTR=
0.00
3.34
0.00
-303.699392.15-147371.00
SINGLE P AND H STATES IN THE TARGET SYSTEMNOSP,NOSH= 5 2
N
L
2J
2S
T
BE
PARTICLE
1
0
0
1
1
-1
26.000PARTICLE
2
0
1
3
1
-1
16.000PARTICLE
3
0
1
1
1
-1
2.000
PARTICLE
4
0
2
3
1
-1
2.000
PARTICLE
5
0
3
5
1
-1
2.000
HOLE
1
0
0
1
1
1
31.200
HOLE
2
0
1
3
1
1
18.720** CALCULATED PH WAVEFUNCTION: KTLOUT(1)NPAIR= 5
NP,NH= 1 1
4 1
2 2
3 2
5 2
NPAIR= 5
NP,NH= 1 1
4 1
2 2
3 2
5 2
PAIR STATES (IT IS L1)=( 1 1 0)
(NP,NH,SF)= 1 1 1.0
4 1 1.0
2 2 1.0
3 2 1.0
5 2 1.0
PAIR STATES (IT IS L1)=( 1 1 2)
(NP,NH,SF)= 1 1 1.0
4 1 1.0
2 2 1.0
3 2 1.0
5 2 1.0
DISTORTED WAVES FOR INITIAL AND FINAL CHANNELSOMP PARAMETERS
V
A
R
W
AI
RI
RC
A-CHANNEL 11.000
0.690
1.200 14.000
0.590
1.170
1.200
0.000
0.867
1.260
B-CHANNEL 11.000
0.690
1.200 14.000
0.590
1.170
1.200
0.000
0.867
1.260
LDMI LDMX LDST NXMI NXMX XMESA-CHANNEL
022
1
1 120 0.100
B-CHANNEL
022
1
1 120 0.100
E,WN,ETA FOR INCIDENT A-CHANNEL= 183.23
2.96
0.08
** OUTPUT OF DW : KTLOUT(2)=NXSTEP, ELASTIC SCAT : KTLOUT(5)** OUTPUT OF COULOMB WF : KTLOUT(7)OM POTENTIAL IN STEPS OF
1.000-0.605E+01 -0.138E+02 -0.561E+01 -0.131E+02 -0.411E+01 -0.102E+02-0.134E+01 -0.460E+01
0.750E+00 -0.115E+01
0.134E+01 -0.227E+00
0.133E+01 -0.423E-01
0.120E+01 -0.779E-02
0.106E+01 -0.143E-02
0.948E+00 -0.263E-03
0.855E+00 -0.482E-04
0.778E+00 -0.886E-05
0.714E+00 -0.163E-05
E,WN,ETA FOR EXIT B-CHANNEL= 165.39
2.80
0.00
OM POTENTIAL IN STEPS OF
1.000-0.108E+02 -0.138E+02 -0.101E+02 -0.131E+02 -0.791E+01 -0.102E+02-0.412E+01 -0.460E+01 -0.136E+01 -0.115E+01 -0.352E+00 -0.227E+00-0.847E-01 -0.423E-01 -0.200E-01 -0.779E-02 -0.470E-02 -0.143E-02-0.110E-02 -0.263E-03 -0.259E-03 -0.482E-04 -0.608E-04 -0.886E-05
19

-0.143E-04 -0.163E-05
ALPHA COEF. PROJ. SYSTEM IN EQ.(13) (AFACAL)ALPHA(NLSK,NLT)=
NLT= 1 ALPHA= -0.33E+00 0.00E+00 0.00E+00 -0.15E+00
NLT= 2 ALPHA= 0.00E+00 -0.15E+00 -0.15E+00 -0.88E-01
FFCALD IS CALLEDINPUT INFORMATION FOR DIRECT FORM FACTOR
N1MIN,N1MAX,N1STEP,XEMSH=
1 120
1NBMIN,NBMAX,NBSTEP,XEMSH=
1 120
1INTRAN,XMESHD=300
MXMAX=
4
FFCALE IS CALLEDINPUT INFORMATION FOR EXCHANGE FORM FACTOR
NAMIN,NAMAX,NASTEP,XEMSH=
1 120
1NBCMI,NBCMX,NBSTPE,XMESH=
1 120
1
N2MIN,N2MAX,N2STEP,XEMSH=
180
1
N1MIN,N1MAX,N1STEP,XEMSH=
1 120
1INTRAN,XMESHE=300
INTEG RANGE FOR RA=RB+59
LAM1MX,LAM2MX,LAMDMX,MXMAX= 20
2
KCETN(1),KCETN(2)=
0
0

0.100
0.100
0.030

0.100
0.100
0.100
0.100
0.150
4

FINAL DIFFERENTIAL CROSS SECTIONS
LT= 0 CONTRIBUTION TO THE CROSS SECTIONSCM ANGLE MOM. TRANS
DIRECT EXCHANGE
TOTAL
0.100E+01 0.166E+00 0.756E+03 0.749E+02 0.358E+03
0.200E+01 0.187E+00 0.730E+03 0.731E+02 0.344E+03
0.300E+01 0.219E+00 0.689E+03 0.703E+02 0.321E+03
0.400E+01 0.256E+00 0.635E+03 0.665E+02 0.293E+03
0.500E+01 0.297E+00 0.572E+03 0.619E+02 0.259E+03
0.600E+01 0.340E+00 0.503E+03 0.569E+02 0.223E+03
0.700E+01 0.386E+00 0.432E+03 0.515E+02 0.187E+03
0.800E+01 0.432E+00 0.362E+03 0.459E+02 0.152E+03
0.900E+01 0.479E+00 0.297E+03 0.404E+02 0.120E+03
0.100E+02 0.526E+00 0.237E+03 0.351E+02 0.910E+02
0.110E+02 0.574E+00 0.185E+03 0.301E+02 0.669E+02
0.120E+02 0.622E+00 0.141E+03 0.255E+02 0.472E+02
0.130E+02 0.671E+00 0.104E+03 0.213E+02 0.319E+02
0.140E+02 0.720E+00 0.749E+02 0.176E+02 0.204E+02
0.150E+02 0.768E+00 0.521E+02 0.143E+02 0.122E+02
0.160E+02 0.817E+00 0.350E+02 0.114E+02 0.672E+01
0.170E+02 0.866E+00 0.225E+02 0.900E+01 0.326E+01
0.180E+02 0.915E+00 0.138E+02 0.696E+01 0.129E+01
0.190E+02 0.964E+00 0.783E+01 0.527E+01 0.341E+00
0.200E+02 0.101E+01 0.403E+01 0.391E+01 0.534E-01
0.210E+02 0.106E+01 0.177E+01 0.282E+01 0.150E+00
0.220E+02 0.111E+01 0.574E+00 0.199E+01 0.437E+00
0.230E+02 0.116E+01 0.784E-01 0.135E+01 0.785E+00
0.240E+02 0.121E+01 0.124E-01 0.893E+00 0.112E+01
0.250E+02 0.126E+01 0.180E+00 0.567E+00 0.139E+01
0.260E+02 0.131E+01 0.444E+00 0.346E+00 0.157E+01
0.270E+02 0.135E+01 0.715E+00 0.202E+00 0.168E+01
0.280E+02 0.140E+01 0.939E+00 0.113E+00 0.170E+01
0.290E+02 0.145E+01 0.109E+01 0.605E-01 0.165E+01
0.300E+02 0.150E+01 0.116E+01 0.314E-01 0.155E+01
0.310E+02 0.155E+01 0.116E+01 0.164E-01 0.141E+01
0.320E+02 0.160E+01 0.109E+01 0.933E-02 0.124E+01
0.330E+02 0.164E+01 0.990E+00 0.641E-02 0.106E+01
0.340E+02 0.169E+01 0.863E+00 0.556E-02 0.884E+00
0.350E+02 0.174E+01 0.728E+00 0.568E-02 0.713E+00
0.360E+02 0.179E+01 0.596E+00 0.626E-02 0.558E+00
0.370E+02 0.183E+01 0.476E+00 0.709E-02 0.424E+00
0.380E+02 0.188E+01 0.371E+00 0.803E-02 0.312E+00
0.390E+02 0.193E+01 0.283E+00 0.897E-02 0.223E+00
0.400E+02 0.198E+01 0.212E+00 0.974E-02 0.154E+00
0.410E+02 0.202E+01 0.155E+00 0.102E-01 0.103E+00
0.420E+02 0.207E+01 0.111E+00 0.101E-01 0.660E-01
0.430E+02 0.212E+01 0.769E-01 0.940E-02 0.408E-01
0.440E+02 0.216E+01 0.516E-01 0.812E-02 0.242E-01
0.450E+02 0.221E+01 0.330E-01 0.643E-02 0.136E-01
0.460E+02 0.226E+01 0.198E-01 0.457E-02 0.722E-02
0.470E+02 0.230E+01 0.110E-01 0.285E-02 0.353E-02
0.480E+02 0.235E+01 0.539E-02 0.150E-02 0.156E-02
0.490E+02 0.239E+01 0.233E-02 0.625E-03 0.639E-03
0.500E+02 0.244E+01 0.103E-02 0.209E-03 0.331E-03
LT= 2 CONTRIBUTION TO THE CROSS SECTIONSCM ANGLE MOM. TRANS
DIRECT EXCHANGE
TOTAL
0.100E+01 0.166E+00 0.126E+01 0.179E+00 0.619E+00
0.200E+01 0.187E+00 0.136E+01 0.191E+00 0.824E+00
0.300E+01 0.219E+00 0.164E+01 0.213E+00 0.130E+01
20

0.400E+01 0.256E+00 0.220E+01 0.249E+00 0.222E+01
0.500E+01 0.297E+00 0.316E+01 0.303E+00 0.370E+01
0.600E+01 0.340E+00 0.456E+01 0.377E+00 0.584E+01
0.700E+01 0.386E+00 0.636E+01 0.475E+00 0.859E+01
0.800E+01 0.432E+00 0.843E+01 0.598E+00 0.118E+02
0.900E+01 0.479E+00 0.106E+02 0.743E+00 0.152E+02
0.100E+02 0.526E+00 0.125E+02 0.909E+00 0.185E+02
0.110E+02 0.574E+00 0.140E+02 0.109E+01 0.213E+02
0.120E+02 0.622E+00 0.149E+02 0.128E+01 0.234E+02
0.130E+02 0.671E+00 0.151E+02 0.147E+01 0.245E+02
0.140E+02 0.720E+00 0.146E+02 0.165E+01 0.247E+02
0.150E+02 0.768E+00 0.134E+02 0.182E+01 0.239E+02
0.160E+02 0.817E+00 0.119E+02 0.196E+01 0.223E+02
0.170E+02 0.866E+00 0.100E+02 0.208E+01 0.202E+02
0.180E+02 0.915E+00 0.809E+01 0.216E+01 0.176E+02
0.190E+02 0.964E+00 0.627E+01 0.220E+01 0.150E+02
0.200E+02 0.101E+01 0.468E+01 0.221E+01 0.124E+02
0.210E+02 0.106E+01 0.337E+01 0.219E+01 0.100E+02
0.220E+02 0.111E+01 0.237E+01 0.213E+01 0.792E+01
0.230E+02 0.116E+01 0.166E+01 0.205E+01 0.617E+01
0.240E+02 0.121E+01 0.119E+01 0.194E+01 0.474E+01
0.250E+02 0.126E+01 0.912E+00 0.182E+01 0.361E+01
0.260E+02 0.131E+01 0.762E+00 0.168E+01 0.273E+01
0.270E+02 0.135E+01 0.696E+00 0.153E+01 0.206E+01
0.280E+02 0.140E+01 0.676E+00 0.138E+01 0.155E+01
0.290E+02 0.145E+01 0.677E+00 0.123E+01 0.117E+01
0.300E+02 0.150E+01 0.681E+00 0.108E+01 0.887E+00
0.310E+02 0.155E+01 0.680E+00 0.945E+00 0.671E+00
0.320E+02 0.160E+01 0.669E+00 0.814E+00 0.510E+00
0.330E+02 0.164E+01 0.648E+00 0.694E+00 0.389E+00
0.340E+02 0.169E+01 0.615E+00 0.584E+00 0.300E+00
0.350E+02 0.174E+01 0.572E+00 0.486E+00 0.234E+00
0.360E+02 0.179E+01 0.522E+00 0.400E+00 0.185E+00
0.370E+02 0.183E+01 0.465E+00 0.326E+00 0.149E+00
0.380E+02 0.188E+01 0.404E+00 0.262E+00 0.121E+00
0.390E+02 0.193E+01 0.342E+00 0.208E+00 0.992E-01
0.400E+02 0.198E+01 0.281E+00 0.163E+00 0.805E-01
0.410E+02 0.202E+01 0.223E+00 0.126E+00 0.644E-01
0.420E+02 0.207E+01 0.172E+00 0.965E-01 0.503E-01
0.430E+02 0.212E+01 0.127E+00 0.728E-01 0.384E-01
0.440E+02 0.216E+01 0.912E-01 0.542E-01 0.285E-01
0.450E+02 0.221E+01 0.631E-01 0.399E-01 0.208E-01
0.460E+02 0.226E+01 0.422E-01 0.290E-01 0.150E-01
0.470E+02 0.230E+01 0.274E-01 0.209E-01 0.110E-01
0.480E+02 0.235E+01 0.175E-01 0.151E-01 0.838E-02
0.490E+02 0.239E+01 0.110E-01 0.109E-01 0.674E-02
0.500E+02 0.244E+01 0.685E-02 0.793E-02 0.576E-02
TOTAL CROSS SECTIONSCM ANGLE MOM. TRANS
DIRECT EXCHANGE
TOTAL
0.100E+01 0.166E+00 0.758E+03 0.751E+02 0.358E+03
0.200E+01 0.187E+00 0.732E+03 0.733E+02 0.344E+03
0.300E+01 0.219E+00 0.691E+03 0.705E+02 0.323E+03
0.400E+01 0.256E+00 0.637E+03 0.667E+02 0.295E+03
0.500E+01 0.297E+00 0.575E+03 0.622E+02 0.263E+03
0.600E+01 0.340E+00 0.507E+03 0.572E+02 0.229E+03
0.700E+01 0.386E+00 0.438E+03 0.519E+02 0.196E+03
0.800E+01 0.432E+00 0.371E+03 0.465E+02 0.164E+03
0.900E+01 0.479E+00 0.307E+03 0.412E+02 0.135E+03
0.100E+02 0.526E+00 0.250E+03 0.360E+02 0.109E+03
0.110E+02 0.574E+00 0.199E+03 0.312E+02 0.882E+02
0.120E+02 0.622E+00 0.156E+03 0.268E+02 0.706E+02
0.130E+02 0.671E+00 0.119E+03 0.228E+02 0.564E+02
0.140E+02 0.720E+00 0.894E+02 0.192E+02 0.451E+02
0.150E+02 0.768E+00 0.656E+02 0.161E+02 0.362E+02
0.160E+02 0.817E+00 0.469E+02 0.134E+02 0.291E+02
0.170E+02 0.866E+00 0.325E+02 0.111E+02 0.234E+02
0.180E+02 0.915E+00 0.219E+02 0.912E+01 0.189E+02
0.190E+02 0.964E+00 0.141E+02 0.748E+01 0.153E+02
0.200E+02 0.101E+01 0.871E+01 0.612E+01 0.124E+02
0.210E+02 0.106E+01 0.514E+01 0.501E+01 0.102E+02
0.220E+02 0.111E+01 0.295E+01 0.412E+01 0.836E+01
0.230E+02 0.116E+01 0.174E+01 0.340E+01 0.695E+01
0.240E+02 0.121E+01 0.121E+01 0.283E+01 0.586E+01
0.250E+02 0.126E+01 0.109E+01 0.238E+01 0.499E+01
0.260E+02 0.131E+01 0.121E+01 0.202E+01 0.431E+01
0.270E+02 0.135E+01 0.141E+01 0.173E+01 0.374E+01
0.280E+02 0.140E+01 0.161E+01 0.149E+01 0.326E+01
0.290E+02 0.145E+01 0.177E+01 0.129E+01 0.283E+01
0.300E+02 0.150E+01 0.184E+01 0.112E+01 0.244E+01
0.310E+02 0.155E+01 0.184E+01 0.961E+00 0.208E+01
0.320E+02 0.160E+01 0.176E+01 0.824E+00 0.175E+01
0.330E+02 0.164E+01 0.164E+01 0.700E+00 0.145E+01
0.340E+02 0.169E+01 0.148E+01 0.590E+00 0.118E+01
0.350E+02 0.174E+01 0.130E+01 0.492E+00 0.947E+00
0.360E+02 0.179E+01 0.112E+01 0.407E+00 0.743E+00
0.370E+02 0.183E+01 0.941E+00 0.333E+00 0.573E+00
0.380E+02 0.188E+01 0.775E+00 0.270E+00 0.433E+00
21

0.390E+02 0.193E+01 0.625E+00 0.217E+00 0.322E+00
0.400E+02 0.198E+01 0.493E+00 0.173E+00 0.234E+00
0.410E+02 0.202E+01 0.378E+00 0.136E+00 0.167E+00
0.420E+02 0.207E+01 0.282E+00 0.107E+00 0.116E+00
0.430E+02 0.212E+01 0.204E+00 0.822E-01 0.792E-01
0.440E+02 0.216E+01 0.143E+00 0.623E-01 0.527E-01
0.450E+02 0.221E+01 0.961E-01 0.463E-01 0.344E-01
0.460E+02 0.226E+01 0.620E-01 0.336E-01 0.223E-01
0.470E+02 0.230E+01 0.384E-01 0.238E-01 0.146E-01
0.480E+02 0.235E+01 0.229E-01 0.166E-01 0.994E-02
0.490E+02 0.239E+01 0.133E-01 0.115E-01 0.738E-02
0.500E+02 0.244E+01 0.788E-02 0.814E-02 0.609E-02

22

4.5.290 Zr(3 H, t)90 Nb, (0− , 27.9 MeV) at E =600 MeV
lab

DWIA DIRECT-EXCHANGE CROSS SECTION CALCULATION FOR THE REACTION
T
TZ
P
PZ
A-CHANNEL 90.000 40.000
3.000
2.000
B-CHANNEL 90.000 41.000
3.000
1.000
E=600.0 MEVINITIAL STATE JATW,ISATW= 0 1
FINAL STATE J,K,E= 0 1
27.9 MEV
SPIN TRANSFER IST= 1
TOTAL L TRANSFER = 1
KTRLD(1-9) = 0 0 0 0 0 0 0 0 1
KTLOUT(1-20)= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0** RELATIVISTIC KINEMATICS IS USEDNN INTERACTION POTENTIAL
KTRL1,VRAN=
1
1.40
0.70
0.55
0.40
0.25
0.15
K= 1 VSTR=
31.50
0.00
0.00
-193.754714.120.00
K= 2 VSTR=-10.500.00
0.00 -1216.75
3567.35
0.00
K= 3 VSTR=
0.00-69.710.00
654.90 -24728.80 325631.00
K= 4 VSTR=-10.500.00
0.00 -2615.66
8396.190.00
K= 5 VSTR=
3.50
0.00
0.00 -1439.49
7313.420.00
K= 6 VSTR=
0.00
14.86
0.00
334.91 -4099.14 50904.30
K= 7 VSTR=
0.00
0.00
0.00
-563.01-919.82
0.00
K= 8 VSTR=
0.00
0.00
0.00 -2431.21
7419.73
0.00
K= 9 VSTR=
0.00
-4.70
0.00
478.42 -15805.00 212453.00
K= 10 VSTR=
0.00
0.00
0.00
-469.591458.53
0.00
K= 11 VSTR=
0.00
0.00
0.00
-217.83 -1076.89
0.00
K= 12 VSTR=
0.00
3.34
0.00
-303.699392.15-147371.00
SINGLE P AND H STATES IN THE TARGET SYSTEMNOSP,NOSH= 7 9
N
L
2J
2S
T
BE
PARTICLE
1
1
1
1
1
-1
5.160
PARTICLE
2
0
4
9
1
-1
2.000
PARTICLE
3
0
4
7
1
-1
1.000
PARTICLE
4
1
2
5
1
-1
1.000
PARTICLE
5
1
2
3
1
-1
1.000
PARTICLE
6
2
0
1
1
-1
1.000
PARTICLE
7
0
5
11
1
-1
1.000
HOLE
1
0
3
7
1
-1
14.300
HOLE
2
1
1
3
1
-1
8.860
HOLE
3
0
3
5
1
-1
7.630
HOLE
4
1
1
1
1
-1
5.160
HOLE
5
0
3
7
1
1
19.220
HOLE
6
1
1
3
1
1
14.960
HOLE
7
0
3
5
1
1
13.420
HOLE
8
1
1
1
1
1
12.560
HOLE
9
0
4
9
1
1
11.970** CALCULATED PH WAVEFUNCTION: KTLOUT(1)NPAIR= 4
NP,NH= 3 5
5 6
4 7
6 8
PAIR STATES (IT IS L1)=( 1 1 1)
(NP,NH,SF)= 3 5 1.0
5 6 1.0
4 7 1.0
6 8 1.0
PROJECTILE DENSITY FOR FOLDING POT. CALCULATIONKMAS, NXMI, NXMX ,XMES3
=
3
1 80
0.100
NORM PROJECTILE,RRMS,G1,FWID= 0.100E+01
0.160E+01
0.675E+00
0.420E+00
NORMAL DENSITY
N2= 10 0.56E+00
N2= 20 0.97E-01
N2= 30 0.52E-02
N2= 40 0.88E-04
N2= 50 0.45E-06
N2= 60 0.73E-09
N2= 70 0.37E-12
N2= 80 0.57E-16
DISTORTED WAVES FOR INITIAL AND FINAL CHANNELSOMP PARAMETERS
V
A
R
W
AI
RI
RC
A-CHANNEL 15.310
0.724
1.276 13.260
0.662
1.260
1.260
0.000
0.662
1.260
B-CHANNEL 15.310
0.724
1.276 13.260
0.662
1.260
1.260
0.000
0.662
1.260
LDMI LDMX LDST NXMI NXMX XMESA-CHANNEL
0 115
1
1 120 0.100
B-CHANNEL
0 115
1
1 120 0.100
E,WN,ETA FOR INCIDENT A-CHANNEL= 578.71
9.39
1.02
** OUTPUT OF DW : KTLOUT(2)=NXSTEP, ELASTIC SCAT : KTLOUT(5)23** OUTPUT OF COULOMB WF : KTLOUT(7)OM POTENTIAL IN STEPS OF
0.333
0.153E+02 -0.133E+02
0.153E+02 -0.133E+02
0.152E+02
0.150E+02 -0.132E+02
0.147E+02 -0.132E+02
0.144E+02
0.141E+02 -0.132E+02
0.136E+02 -0.132E+02
0.132E+02
0.127E+02 -0.130E+02
0.122E+02 -0.128E+02
0.118E+02
0.114E+02 -0.122E+02
0.112E+02 -0.116E+02
0.112E+02
0.115E+02 -0.950E+01
0.119E+02 -0.801E+01
0.125E+02
0.131E+02 -0.475E+01
0.137E+02 -0.334E+01
0.141E+02
0.142E+02 -0.145E+01
0.142E+02 -0.918E+00
0.140E+02
0.137E+02 -0.351E+00
0.134E+02 -0.214E+00
0.130E+02
0.126E+02 -0.791E-01
0.122E+02 -0.479E-01
0.118E+02
0.114E+02 -0.175E-01
0.111E+02 -0.106E-01
0.108E+02
0.104E+02 -0.388E-02
0.101E+02 -0.234E-02
0.984E+01
0.957E+01 -0.856E-03
E,WN,ETA FOR EXIT B-CHANNEL= 550.99
9.14
0.53
OM POTENTIAL IN STEPS OF
1.000
0.378E+00 -0.133E+02
0.201E+00 -0.132E+02 -0.247E+00-0.801E+00 -0.130E+02 -0.903E+00 -0.121E+02
0.680E+00
0.400E+01 -0.444E+01
0.634E+01 -0.133E+01
0.674E+01
0.635E+01 -0.716E-01
0.581E+01 -0.159E-01
0.531E+01
0.488E+01 -0.774E-03
ALPHA COEF. PROJ. SYSTEM IN EQ.(13) (AFACAL)ALPHA(NLSK,NLT)=
NLT= 1 ALPHA= 0.19E+00 0.19E+00
FFCALD IS CALLEDINPUT INFORMATION FOR DIRECT FORM FACTOR
N1MIN,N1MAX,N1STEP,XEMSH=
1 120
1NBMIN,NBMAX,NBSTEP,XEMSH=
10 12010**INTERPOLATION IN R_B IS MADEINTRAN,XMESHD=300
MXMAX=
4
FFCALE IS CALLEDINPUT INFORMATION FOR EXCHANGE FORM FACTOR
NAMIN,NAMAX,NASTEP,XEMSH=
1 120
1NBCMI,NBCMX,NBSTPE,XMESH=
10 12010**INTERPOLATION IN R_b IS MADEN2MIN,N2MAX,N2STEP,XEMSH=
180
1
N1MIN,N1MAX,N1STEP,XEMSH=
1 120
1INTRAN,XMESHE=300
INTEG RANGE FOR RA=RB+59
LAM1MX,LAM2MX,LAMDMX,MXMAX= 1
0
1
KCETN(1),KCETN(2)=
0
0

0.100
0.100
0.030

0.100
0.100
0.100
0.100
0.015
4

FINAL DIFFERENTIAL CROSS SECTIONS
LT= 1 CONTRIBUTION TO THE CROSS SECTIONSCM ANGLE MOM. TRANS
DIRECT EXCHANGE
TOTAL
0.100E-01 0.247E+00 0.159E+03 0.488E+02 0.333E+02
0.210E+00 0.249E+00 0.168E+03 0.490E+02 0.390E+02
0.410E+00 0.256E+00 0.192E+03 0.495E+02 0.540E+02
0.610E+00 0.266E+00 0.225E+03 0.503E+02 0.754E+02
0.810E+00 0.279E+00 0.263E+03 0.511E+02 0.993E+02
0.101E+01 0.296E+00 0.299E+03 0.518E+02 0.121E+03
0.121E+01 0.315E+00 0.326E+03 0.521E+02 0.138E+03
0.141E+01 0.336E+00 0.340E+03 0.518E+02 0.146E+03
0.161E+01 0.359E+00 0.338E+03 0.508E+02 0.144E+03
0.181E+01 0.383E+00 0.321E+03 0.489E+02 0.134E+03
0.201E+01 0.408E+00 0.290E+03 0.461E+02 0.116E+03
0.221E+01 0.434E+00 0.250E+03 0.425E+02 0.938E+02
0.241E+01 0.461E+00 0.204E+03 0.382E+02 0.706E+02
0.261E+01 0.489E+00 0.157E+03 0.334E+02 0.487E+02
0.281E+01 0.517E+00 0.114E+03 0.283E+02 0.304E+02
0.301E+01 0.546E+00 0.775E+02 0.231E+02 0.166E+02
0.321E+01 0.575E+00 0.484E+02 0.181E+02 0.750E+01
0.341E+01 0.604E+00 0.274E+02 0.136E+02 0.248E+01
0.361E+01 0.634E+00 0.136E+02 0.970E+01 0.510E+00
0.381E+01 0.663E+00 0.570E+01 0.651E+01 0.418E+00
0.401E+01 0.694E+00 0.211E+01 0.410E+01 0.116E+01
0.421E+01 0.724E+00 0.126E+01 0.245E+01 0.198E+01
0.441E+01 0.754E+00 0.181E+01 0.150E+01 0.245E+01
0.461E+01 0.785E+00 0.284E+01 0.113E+01 0.247E+01
0.481E+01 0.816E+00 0.373E+01 0.119E+01 0.212E+01
0.501E+01 0.847E+00 0.423E+01 0.153E+01 0.160E+01
0.521E+01 0.877E+00 0.426E+01 0.200E+01 0.111E+01
0.541E+01 0.909E+00 0.392E+01 0.248E+01 0.786E+00
0.561E+01 0.940E+00 0.332E+01 0.285E+01 0.687E+00
0.581E+01 0.971E+00 0.263E+01 0.308E+01 0.788E+00
0.601E+01 0.100E+01 0.194E+01 0.312E+01 0.101E+01
0.621E+01 0.103E+01 0.134E+01 0.298E+01 0.128E+01
24-0.133E+02-0.132E+02-0.131E+02-0.126E+02-0.107E+02-0.636E+01-0.224E+01-0.571E+00-0.130E+00-0.290E-01-0.641E-02-0.142E-02-0.132E+02-0.922E+01-0.318E+00-0.351E-02

0.641E+01 0.106E+01 0.861E+00 0.269E+01 0.149E+01
0.661E+01 0.110E+01 0.521E+00 0.231E+01 0.160E+01
0.681E+01 0.113E+01 0.305E+00 0.188E+01 0.159E+01
0.701E+01 0.116E+01 0.191E+00 0.145E+01 0.147E+01
0.721E+01 0.119E+01 0.149E+00 0.107E+01 0.127E+01
0.741E+01 0.122E+01 0.153E+00 0.766E+00 0.103E+01
0.761E+01 0.125E+01 0.178E+00 0.546E+00 0.780E+00
0.781E+01 0.129E+01 0.206E+00 0.414E+00 0.562E+00
TOTAL CROSS SECTIONSCM ANGLE MOM. TRANS
DIRECT EXCHANGE
TOTAL
0.100E-01 0.247E+00 0.159E+03 0.488E+02 0.333E+02
0.210E+00 0.249E+00 0.168E+03 0.490E+02 0.390E+02
0.410E+00 0.256E+00 0.192E+03 0.495E+02 0.540E+02
0.610E+00 0.266E+00 0.225E+03 0.503E+02 0.754E+02
0.810E+00 0.279E+00 0.263E+03 0.511E+02 0.993E+02
0.101E+01 0.296E+00 0.299E+03 0.518E+02 0.121E+03
0.121E+01 0.315E+00 0.326E+03 0.521E+02 0.138E+03
0.141E+01 0.336E+00 0.340E+03 0.518E+02 0.146E+03
0.161E+01 0.359E+00 0.338E+03 0.508E+02 0.144E+03
0.181E+01 0.383E+00 0.321E+03 0.489E+02 0.134E+03
0.201E+01 0.408E+00 0.290E+03 0.461E+02 0.116E+03
0.221E+01 0.434E+00 0.250E+03 0.425E+02 0.938E+02
0.241E+01 0.461E+00 0.204E+03 0.382E+02 0.706E+02
0.261E+01 0.489E+00 0.157E+03 0.334E+02 0.487E+02
0.281E+01 0.517E+00 0.114E+03 0.283E+02 0.304E+02
0.301E+01 0.546E+00 0.775E+02 0.231E+02 0.166E+02
0.321E+01 0.575E+00 0.484E+02 0.181E+02 0.750E+01
0.341E+01 0.604E+00 0.274E+02 0.136E+02 0.248E+01
0.361E+01 0.634E+00 0.136E+02 0.970E+01 0.510E+00
0.381E+01 0.663E+00 0.570E+01 0.651E+01 0.418E+00
0.401E+01 0.694E+00 0.211E+01 0.410E+01 0.116E+01
0.421E+01 0.724E+00 0.126E+01 0.245E+01 0.198E+01
0.441E+01 0.754E+00 0.181E+01 0.150E+01 0.245E+01
0.461E+01 0.785E+00 0.284E+01 0.113E+01 0.247E+01
0.481E+01 0.816E+00 0.373E+01 0.119E+01 0.212E+01
0.501E+01 0.847E+00 0.423E+01 0.153E+01 0.160E+01
0.521E+01 0.877E+00 0.426E+01 0.200E+01 0.111E+01
0.541E+01 0.909E+00 0.392E+01 0.248E+01 0.786E+00
0.561E+01 0.940E+00 0.332E+01 0.285E+01 0.687E+00
0.581E+01 0.971E+00 0.263E+01 0.308E+01 0.788E+00
0.601E+01 0.100E+01 0.194E+01 0.312E+01 0.101E+01
0.621E+01 0.103E+01 0.134E+01 0.298E+01 0.128E+01
0.641E+01 0.106E+01 0.861E+00 0.269E+01 0.149E+01
0.661E+01 0.110E+01 0.521E+00 0.231E+01 0.160E+01
0.681E+01 0.113E+01 0.305E+00 0.188E+01 0.159E+01
0.701E+01 0.116E+01 0.191E+00 0.145E+01 0.147E+01
0.721E+01 0.119E+01 0.149E+00 0.107E+01 0.127E+01
0.741E+01 0.122E+01 0.153E+00 0.766E+00 0.103E+01
0.761E+01 0.125E+01 0.178E+00 0.546E+00 0.780E+00
0.781E+01 0.129E+01 0.206E+00 0.414E+00 0.562E+00

25

5FormulasThe details of derivation of formulas can be found in a separate note ”DCP2 - Formulation”.

5.1

Differential cross section

The differential cross section, Eq.(21) in the original paper (Hereafter, use as [Eq.(21)].), becomes
X X jsνdσµa µb kb1=|
α t t 1 Tti s ` k` m |2 [Eq.(21)]22dΩ(2πh̄ ) ka (2IA + 1)(2sa + 1) i k` t t1 s1 `1 k`t 1 1 1 t `t
1 1

where µa (µb ) is the reduced mass in the incident (exit) channel and i is i = D for directtransitions and i = E for exchange transitions.
The αtj1t sst1ν`12 k`t coefficients and the transition amplitudes are
αtj1t sst1ν`12 k`t
Tti1 s1 `1 k`t m`t−1†= W (st `t st `1 ; jt k)ŝ−1
t t̂1 < b||[c c]s1 t1˜ν1 ||a >, with ŝ ==(4π)3/2 X
ka kb

`a `b

√
2s + 1[Eq.(13)]i`a −`b +π `ˆa (`a 0`t m`t |`b m`t )Oti1 s1 `1 k`t ,`a `b Y`b m`t (k̂b ) [Eq.(18)]where the direct and exchange overlap integrals areZ

OtD1 s1 `1 k`t ,`a `b= d`a `t `b

OtE1 s1 `1 k`t ,`a `b= Jd`a `t `bdra χ`b (ra )ftD1 s1 `1 k`t (ra )χ`a (ra ) [Eq.(19a)]

Z

≡

Z

drb

dra rb ra χ`b (rb )ftE1 s1 `1 k`t ,`a `b (rb , ra )χ`a (ra ) [Eq.(19b)]1√ `ˆa `ˆt `ˆ−1b (`a 0`t 0|`b 0) [Eq.(19c)]4π

where ftD1 s1 `1 k`t (ra ) and ftE1 s1 `1 k`t ,`a `b (rb , ra ) are direct and exchange form factors, respectively.
χ`a (ra )(χ`b (rb )) are the partial distorted waves in the incident (exit) channel. J is the Jacobian
associated with the transformation of integral coordinate from ~r2 to ~ra . < b||[c† c]s1 t1˜ν1 ||a > isthe usual spectroscopic amplitude for the spin-isospin wave function of projectile system.
The angular momentum couplings are as follows;
Form factor
DirectInteraction
CentralTensorExchange

CentralTensor

Couplings
k = 0, {st `t jt }, δ(s1 , st ), δ(`1 , `t ), s1 = st = 0 or 1,{`a `b `t }, `a + `b + `t = even.
k = 2, {st `t jt }, {s1 `1 jt }, {`1 `t 2}, δ(s1 , st ), s1 = st = 1,{`a `b `t }, `a + `b + `t = even.
k = 0, {st `t jt }, δ(s1 , st ), δ(`1 , `t ), s1 = st = 0 or 1,{`a `b `t }.
k=2, {st `t jt }, {s1 `1 jt }, {`1 `t 2}, δ(s1 , st ), s1 = st = 1,{`a `b `t }.

26

5.2

Form factors

5.2.1 Direct form factor
We write the radial direct form factors as
ftD1 s1 `1 k`t (ra )

= i

−π

(−)`1 `ˆ−1
t

Z

Z
2

r drVtD(r)
1 s1 kZD
r12 dr1 ρDP,k`t `1 (ra , r1 , r)ρT,`1 (r1 ) [Eq.(27)]

2π X ˆ
0∗`t (`t 0`1 m|km) ρD=P,k`t `1 (ra , r1 , µ, r)Ykm (θ2 , 0)Y`1 m (θ, 0)dµ [Eq.(26)]
2
k̂ m
√
1
1
†
ρD
4πd`p `h `1 R`p (r1 )R`h (r1 )T,`1 (r1 ) = X(`p jp , `h jh ; `1 s1 jt ) < IB ||[âjp νp âjh˜νh ]jt ||IA >22

ρDP,k`t `1 (ra , r1 , r)

D(E)

D(E)

where the force components Vt1 s1 k (r) are related to Love-Franey NN interaction Vtsk
Vti1 s1 k (r) =X√ŝ3 t̂3
4πfk ŝ21 t̂21
st


1

 2

1
2
1
2
s1

1

2

 s

1



(r) by



1
1
s 


 2 2 t i11st
Pi Vtsk(r) [Eq.(16)]
2
2






k
t1 t1 0

Where PD = 1 and PE = (−)s+t+1 , while f0 = 1 and f2 =√8. The target density ρDT,`1 (r1 ) is ex-pressed in terms of the particle (hole) wave functions R`p (r1 )(R`h (r1 )), and < IB ||[â†jp νp âjh˜νh ]jt ||IA >is the usual spectroscopic amplitude for the target system. The projectile density ρD
P (r2 ) is written as ρD
(r
,
r
,
r)
in
the
multipole
expansion
method.
P,k`t `1 a 1
The angular momentum couplings from target density which appears both in the direct andexchange factors are
1
1{`p jp }, {`h jp }, {`h `p `1 }, and {jh jp jt }.
2
2
Note that the angular momenta, say, `, `a , λ2 , appeared in the multipole expansions of projectile
density, are reduced to λ2 = k, ` = `t , `q = `1 after angular integrations. (See Formulation 2.1.)
5.2.2

Direct form factor for the nucleon-nucleus scattering

For the nucleon-nucleus scattering, we adopt the following limits for the direct form factorXρP (~r2 ) = δ(~r2 ) =0
λ2
ρDP,λ2 (r2 , r)(−) [Yλ2 Yλ2 ]00

λ2

ρP,λ2 (r20 , r)λ2

= λ̂2 (−) δ(r20 − r)/r2

The radial direct form factor becomes
ftD1 s1 `1 k`t (ra ) = i−π (−)`1 `ˆ−1
t

Z

Z

QD
t1 s1 `1 k`t (ra , r) =r12 dr1 ρDT,`1 (r1 )×µ ≡ cos θ =r2 drVtD(r)QD
t1 s1 `1 k`t (ra , r)
1 s1 k

X

2π
k̂ 2

−1
`ˆt (`t 0`1 m|km)k̂(−)k ()Ykm (θ20 , 0)Y`∗1 m (θ, 0)
r
r
r
1
a
m

ra2 + r12 − r2,
2r1 raµ0 ≡ cos θ20 =27

ra2 + r2 − r122rra

5.2.3

Exchange form factor

The exchange form factor can be written as,
ftE1 s1 `1 k`t ,`b `a (rb , ra ) = J 4π mkaX[λa λb ` α ` β(2k + 1)!
]1/2 δλa +λb ,k (−ra )λa (rb )λb(2λa + 1)!(2λb + 1)!

× X(`α λa `a , `β λb `b ; `1 k`t )d`α λa `a d`β λb `b ct1 s1 `1 k,`α `β (rb , ra ) [Eq.(36)]
ct1 s1 `1 k,`α `β (rb , ra ) =X
2π X ˆ
ˆ
`
(`
m
`
0|`
m
)
`(`0λmα1β`β``1 |`1 m`1 )
1
1
`ˆ2 m
1

Z

dµGkt1 s1 `1 `λ (rb , r)Yλm`1 (θ0 , π)Y`∗α m` (θ, 0)×
Gkt1 s1 `1 `λ (rb , r) =

`λ

`1

1

X
1
√ r−k VtE(r)(−)` λ̂1 λ̂2 (λ1 0λ2 0|λ0)W (λ1 λ2 `1 `; λ`c )1 s1 k4π
λ1 λ2 ` c
ZE
r12 dr1 ρE
P,λ2 ``c (rb , r1 , r)ρT,`1 λ1 `c (r1 , r)×

[Eq.(33b)]

Z

2π X ˆ
0∗`(`0`q m|λ2 m) ρEP,λ2 ``q (rb , r1 , µ, r)Yλ2 m (θ2 , 0)Y`q m (θ, 0)dµ [Eq.(31a)]
2
λ̂2 m
X
1
1
ρE
i`p +`h −π X(`p jp , `h jh ; `1 s1 jt ) < IB ||[â†jp νp âjh˜νh ]jt ||IA > R`p (r1 )T,`1 λ1 `c (r1 , r) =
2
2
ph,η

ρE
P,λ2 ``q (rb , r1 , r) =1

[Eq.(31b)]× (−)η1 `ˆh `ˆc η̂1 (`c 0η1 0|`p 0) W (`c η1 `1 `h ; `p λ1 )
Z
2π X
× 2
η̂1 (η1 0λ1 m1 |`h m1 ) R`h (r10 )Y`h m1 (θ, 0)Yλ∗1 m1 (θ0 , 0)dµ0
`ˆ m
h

1

The angular momentum couplings appeared in the exchange form factor are as follows;
Interaction
CentralTensor

Couplings
δ(`α , `a ), δ(`β , `b ), δ(λa , 0), δ(λb , 0).
{`α λa `a }, {`β λb `b }, {`1 k`t }, {`α `β `1 }, {λa λb k},(`α + λa + `a = even), (`β + λb + `b = even)With this coupling scheme, the central exchange form factor just becomes the c-factor in theabove equation,
ftE1 s1 `1 k=0,`t ,`b `a (rb , ra ) = ct1 s1 `1 k=0,`a `b (rb , ra ).
Note that the angular momenta, say, (`, `q , λ2 ), appeared in the multipole expansions of projectile
density, and (η1 , `c , λ1 ) in target density expansions, are disappeared after integrating over dr̂1
which gives `q = `c and summing over the rest of angular momenta. (See Formulation 2.2.)
5.2.4

Exchange form factor for the nucleon-nucleus scattering

We adopt the following limits for the exchange form factorρP (~r2 , ~r20 ) = δ(~r2 )ρP,λ2 (r20 , r) = λ̂2 (−)λ2 δ(r20 − r)/r2
The radial exchange form factor yields
ftE1 s1 `1 k`t ,`b `a (rb , ra ) = J 4π mkaXλa λb `α `β[(2k + 1)!
]1/2 δλa +λb ,k (−ra )λa (rb )λb(2λa + 1)!(2λb + 1)!

× X(`α λa `a , `β λb `b ; `1 k`t )d`α λa `a d`β λb `b ct1 s1 `1 k,`α `β (rb , ra )28

The projectile density is just replaced byZ

2π X ˆ
0∗`(`0`c m|λ2 m) ρEP,λ2 ``c (rb , r1 , µ, r)Yλ2 m (θ2 , 0)Y`c m (θ, 0)dµ2
λ̂2 m
r2 + r12 − r2
rb − r1 µµ = b, µ0 =
2r1 rbr

ρEP,λ2 ``c (rb , r1 , r) =5.2.5

Exchange form factor in the no-recoil approximation

The no-recoil approximation was originally invented to simplify the cross section calculations forheavy-ion induced one and two nucleon transfer reactions. The essence of the approximation is
to ignore the recoil momentum the target receives in the transfer process. In the same spirit weneglect here the recoil momenta which projectile and target pick up in the knock-on exchange(+)
process. Formally this approximation is obtained by replacing χa (~ka , ~ra ) in [Eq.(15)] through(+) ~χa (ka , ~rb ), i.e., ignoring the difference between the vectors ~ra and ~rb .
We write the radial exchange form factors in the no-recoil approximation as
ftN1 sR1 `1 k`t (rb ) =Z√
dr rk+2 Gkt1 s1 `1 `t λ (rb , r)δ(k, λ) [Eq.(41)]4π(−)k `ˆ1 `ˆ−1tNote that, in the no-recoil approximation, the angular momenta, say, (`, λ), appeared in themultipole expansions are reduced to ` = `t , λ = k. (See Formulation 3.2.)
5.2.6

Exchange form factor in the plane wave approximation

In the plane wave approximation, the recoil effect1 is described by a recoil factor exp(−iα~ka ·r/a).
A simple reason is that in the plane wave approximation the incoming and outgoing waves aredescribed byexp(i~ka · ~ra − i~kb · ~rb ) = exp[i(~ka − ~kb ) · ~rb ] exp[−i~ka · ~r/a]
where we use ~ra = ~rb − ~r/a. Obviously −~ka /a is the change of linear momentum between theexchanged particles.
A possible improvement of the no-recoil approximation is then to replace f N R by the following
f P W that takes into account the recoil factor within the plane wave approximation,
Z

FtP1 sW1 `1 k`t m` (~rb )t=J −1 ftP1 sW1 `1 k`t (rb )Y`t m`t (r̂b )iπ =d~r ftE1 s1 `1 k`t (rb ) exp(−iα~ka · ~r/a)

where a parameter α in the recoil factor is treated as an adjustable parameter. We fit it suchthat the resultant approximate cross section reproduces the exact cross section σ(E) as closely
as possible. It has turned out that a close fit is obtained with α = 1.2
We obtain the radial exchange form factors with recoil effect in the plane wave approximation(See Formulation 3.3 for details.) as,√ X π−` −`ftP1 sW1 `1 k`t (rb ) =4π
i r k̂ λ̂(k0λ0|`r 0)`ˆ1 `ˆr W (`λ`t k : `1 `r )`λ`rZ
k+`1 −`t× (−)(`m`t `r 0|`t m`t )dr rk+2 Gkt1 s1 `1 `λ (rb , r)j`r (αka r/a)

Inputting LRP1MX=1 (`r = 0) and FACNR=0.0 (α = 0.0), yields NR approximation.
1
2

T.Tamura, Phys. Rep. 14C, 59 (1974), Section 4.4.
B. T. Kim, D. P. Knobles, S. A. Stotts, and T. Udagawa, Phys. Rev. C61, 044611 (2000).

29

5.3

Relativistic kinematics

What we want to do is the Lorentz transformation of Lab system with
Elab
k0
mT
mP

:
:
:
:

Kinetic energy of lab system
Wave number
Target mass
Projectile

to c.m. system such that
(E0 + mT , k0 ) → (ω, 0)
where E0 − mP = Elab , and E02 = m2P + k02 . Thus we have
Ã

ω
0

!

Ã

=

βcm =
γcm =

!Ã

γ
−γβ
−γβ
γ

E0 + mT
k0!

~k0(Velocity of c.m. wrt lab frame)
E0 + mT
1
E0 + mT
E0 + mT
p
√
=q
=
2
s
2
1−β
(E0 + mT )2 − k
0

s = ω

2

2

− k02

= (E0 + mT )= (Elab + mP + mT )2 − (Elab + mP )2 + m2P= (mP + mT )2 + 2Elab mT

We write the kinetic energy, wave number and masses in the center of mass system.
√
Ecm = ω − (mT + mP ) = s − (mT + mP )m2
m2
m2T 2 m2T 222
k0 =(E0 − m2P ) = T [(Elab + m2P )2 − m2P ] = T (Elab+ 2Elab mP )
kcm
=
s
s
s
s
E0 + mT
mT√
mcm,T = γcm mT == √ (Elab + mP + mT )ss
12
mcm,P = √ [2Elab+ 4Elab mP + mP (mP + mT ) + Elab mT ]s
1
≈ √ [mP (mP + mT ) + Elab mT ]swith mT >> mP and mT >> Elab .

5.4
5.4.1

Projectile Density distribution
Deuteron

We use the Hulthen wavefunction of the deuteronρd (r) = φ2Hulthen
φHulthen (r) =

1 1
√
4π r

s

2αβ(α + β) −αr(e− e−βr )(β − α)2where α−1 = 4.3 fm, and β = 7α.

30

5.4.2

Helium

We use the 3 He density from a paper by C.W. de Jager, H. de Vries and C. de Vries, Atomic
data and nuclear data tables, 36 (1987) 495.
ρ0 (r) =1z−r2c2 (6b2 − r2 )
−r2
[
exp(
)
−
exp(
)]
4a2
4b7
4b2
8π 3/2 a3

with a = 0.675, b = 0.836, c = 0.366 fm.

5.5

Folding optical potential

The single folding potential is defined asZ

dr1 r12 ρ(r1 )v0 (r, r1 ),U (r) = 4πv0 (r, r1 ) =

1
2

Z 1
−1

v(|~r − ~r1 |)dµ

The last integration is done by the gaussian integration method. Here, the density function isnormalized to the mass of the projectile; for example, Nd = 2.0 for the deuteron and NHe = 3.0
for the helium.

31

A

Subroutines and their functions

A.1

Structure of the program
DCP1

Input

Ini. Output

BSAXONUNCPST

DCHECK1
DISWAVE
POTEN
FLGLCHOPTAFACALEFFINTPDENSTPACALD

DCHECK2
EFFINTTRECALPDENSTDENSTGFACTENSORPACALEFFCALD

FFCALDMFFCALEFFCALEM

CROSS
Supporting Routines
CLEB
CLEBZ
RAC7
NINEJ

YLCAL
GAUSF
DSPLS3
BESSEL

35 SUBROUTINES, 8300 LINES
32

EFFINTPDENSTXLMCALDCHECK2
EFFINTTRECALMPDENST
DENSTM
GFACM

A.2

Functions of the subroutine

A.2.1 DCP2 -Main Routine
The main routine DCP2 controls and performs whole calculation processes.
A.2.1.1

DCP2 defines first constants.

Variable
FACLOG(N)
PI
HBAR
AMAS
WNUNIT
FINE

Definition
Log value of factorial of integer N . Used for vector coupling coefficients.
π = 4 arctan(1).
h̄c = 197.327053 in MeV · fm.
atomic mass unit, mamu = 931.49432
in MeV.
√
wave number unit in 1/fm, 2mamu /h̄c.
fine structure constant, e2 /h̄c = 1/137.0359896.

A.2.1.2

DCP2 reads input data.

A.2.1.3

DCP2 writes initial output.

A.2.1.4

DCP2 calls BSAXON to calculate the single particle and hole states.

A.2.1.5

DCP2 finds Particle-hole pairs, which satisfies

ParityTriangle relations of
Triangle relations of
A.2.1.6`p + `h + π = even(jp , jh , jt )(`p , `h , `1 ) with ~`1 = ~s1 + ~jt

DCP2 calls DCHECK1 to check field lengths of variables.

A.2.1.7 DCP2 calculates the projectile wave functions and density. See Section 5.4 for nuclei
less than 3, otherwise call BSAXON.
A.2.1.8

DCP2 calls OPT to generate distorted waves in the incident and exit channels.

This section first calculates the kinematical variables, such as Ecm , k, η and so on, in the incident
and exit channels.
A.2.1.9

DCP2 calls AFACAL to calculate α coefficients of [Eq.(13)].

A.2.1.10

DCP2 calls XLMCAL to calculate the distortion factor.

A.2.1.11 DCP2 calculates direct form factors. It calls FFCALD or FFCALDM depending on
KTRLD(1). If KTRLD(1)=1, it calls FFCALDM, otherwise FFCALD.
A.2.1.12 DCP2 calculates exchange form factors. It calls FFCALE or FFCALEM depending
on KTRLD(1). If KTRLD(1)=1, it calls FFCALEM, otherwise FFCALE.
A.2.1.13

DCP2 calls CROSS to calculate differential cross sections.

33

A.2.2

BSAXON and UNCPST - Single particle states

A.2.2.1 BSAXON solves the Schrödinger equation to obtain the single particle (hole) radial
wave functions[−h̄2 d2h̄2 `(` + 1)+
+ V (r)]unj`s (r) = −|E0 |unj`s (r)2µ dr22µr2where unj`s (r) is defined asϕ(r, θ, φ) = Rnj`s (r)Y`m (θ, φ) =unj`s (r)Y`m (θ, φ)r

For the details of V (r), see Section 4.3. It searches the central potential depth parameter(Woods-Saxon type) for a given biding energy. (KTRL2=1 was set.)
A.2.2.2 UNCPST generates outward solutions from the origin by using the 4-point Stormer
method and inward solutions from the far outside the well by a single step Stormer method andmatches two solutions at a boundary point.
A.2.3
A.2.3.1

DCHECK1 - Checking field lengths of variables
DCHECK1 checks field lengths of variables with a given input data set.

The meaning of PARAMETER given in the program is following,
Parameter
Name
NHS
NPS
LXA
LXB
LPH
LTL
LM1
LMM
NXA
NXB
NIN
N1X
KFM
KCC
LMJ
KFMAX
KAMAX
KBMAX
KCMAX

Field length of Variable

Max. combinations

NOSH
NOSP
LDWMXR(1)LDWMXR(2)NPHMX
L1R(NLSMAX)+LAMMXD(1)+1
MP1MAX = MIN(LTR(NOLTR),MXMAX)+1
MAX(LAMMXD(1,2,3))
NXMXR(1)*PMASANXMXR(2)
MAX(NHDMX,NHEMX)NXMXR(4)/N1STEP+1
Greater than KFMAX
Greater than KCMAX(MJMAX+1)*3
NLSMAX*7(LAMMX+1)*(LAMMX+2)/2
NLSMAX*LAM1MX*(LAM1MX+1)/2
KBMAX*LAM2MX*(LAM2MX+1)/2

34{t1 s1 `1 k`t }{KF, λ`}{t1 s1 `1 λ1 `0p }{KB, λ2 `}A.2.4

OPT, FLGLCH, POTEN, and DISWAVE - Distorted waves

A.2.4.1 OPT(IDCHNL,NEXPT) calculates distorted waves for the incident(IDCHNL=1) andexit(IDCHNL=2) channels. NEXPT is set to be 1. It solves the Schrödinger equation for χ`with outgoing wave boundary conditionsh̄2 d2 u` (r)h̄2 `(` + 1)
+
[E
−
U
(r)
−
]χ` (r) = 0
2µ dr22µr2where χ` is related to the channel wave function asχ(+) (~k, ~r) =4π X `∗i χ` (r)Y`m` (r̂)Y`m
(k̂)
`
kr `m
`

For the optical model potential U (r), see Section 4.1.
These subroutines were written by Taro Tamura’s group, and how to obtain distorted waves
was written in a paper by T. Tamura and W. R. Coker, ”Computers and the optical model for
nuclear scattering”, in the book edited by S. Fernbach and A. H. Taub, ”Computers and theirrole in the physical sciences” (Gordon and Breach, 1967).
A.2.4.2 FLGLCH calculates the partial Coulomb waves. This subroutine is adopted by a paper
by A. R. Barnett, D. H. Feng, J. W. Steed and L. J. B. Goldfarb, ”Coulomb wave functions for
all real η and ρ”, Comp. Phys. Comm. 8, (1974) 377.
A.2.4.3 POTEN(IDCHNL,IDC) calculates the optical model Woods-Saxon potential U (r) of
Section 4.1. IDC is set to be 1 presently.
One may generate the optical potential by singly folding the potential for target-nucleon system(KTRLD(3)=1). The single folding potential (See Section 5.5.) is obtained by the gaussian
integration method for which a supporting routine
GAUSF(NGAUS,RTS,WGT) (See Section A.2.20.2.), where the abscissas (RTS) and weighting
factors (WGT) for gaussian integration with the number of gaussian points, n=NGAUS are
generated, is called.
A.2.4.4 DISWAVE calculates the distorted partial wave functions and the partial c-matrix bymatching the internal partial wave functions generated from the origin and the Coulomb wave
functions.
A.2.5
A.2.5.1

AFACAL - α-coefficients
AFACAL calculates the expansion coefficient αtj1t sst1ν`11 k`t , [Eq.(13)], defined as
αtj1t sst1ν`11 k`t−1†= W (st `t st `1 ; jt k)ŝ−1
t t̂1 < b||[c c]s1 t1˜ν1 ||a > [Eq.(13)]= W (st `t st `1 ; jt k) × M × L
1 1 1
1 1 1
L = (1 − (−)sa +ta )ŝa ŝb W (sa s1 ; sb )t̂a t̂b W (ta t1 ; tb )
2 2 2
2 2 2
<
t
ν
t
,
−ν
|t
ν
>
M = (−)t1 +ν1 +ta −νa t̂−1
a
a
1
1
b
b
1
The overall phase factor (−)`1 +k+`t is also multiplied.

35

The L-values for different reactions are
Reaction(p, p0 )(p, n)(d, d0 )(d, 2p)(h, t)(α, α0 )

sa
1/2
1/2
1
1
1/2
0

sb
1/2
1/2
1
sb
1/2
0

s1
0
0
s1
s1
s1
0

ta
1/2
1/2
0
0
1/2
0

tb
1/2
1/2
0
1
1/2
0

t1
0
1
0
1
t1
0

L
1
1
p
(3
+
s
)(2
−
s1 )/2δ(t1 , 0)1√
6ŝb W (1 12 s1 21 ; 21 sb )δ(t1 , tb )4δ(t1 , 0)δ(s1 , 0) − (−)t1 +s1
2

q
√
1 1 1
6ŝb W (1 s1 ; sb ) = (3 + s1 )(2 − s1 )/2,2 2 2

sb = 1AFACAL calls supporting subroutines CLEB (See A.2.20.4.) and RAC7 (See A.2.20.6.) whenever Clebsch-Gordan and Racah coefficients, respectively, are needed.
A.2.6

EFFINT - Effective interaction components

A.2.6.1 EFFINT(KEXCH) calculates the nucleon-nucleon effective interaction, Vti1 s1 k (r), of[Eq.(16)], i.e.,


1
2
1
2

 s
1



X
√
i
2 2
3 3
Vt1 s1 k (r) =4πfk ŝ1 t̂1
ŝ t̂
st

1
2
1
2
s1





1
1
s 

 2 2 t 

i
1
1
s
t
Pi Vtsk(r) [Eq.(16)]
2
2



k   t1 t1 0 

where PD√= 1 (KEXCH=0, Direct) and PE = (−)s+t+1 (KEXCH=1, Exchange), while f0 = 1
i (r) is the Love-Franey effective interactions which are input.
and f2 = 8. Vtsk
EFFINT calls a supporting routine NINEJ (See A.2.20.7.) twice for 9-j symbols of the spin andisospin parts.
A.2.7

PDENST - Projectile density function

0
A.2.7.1 PDENST(KEXCH=0, direct form factor) calculates the expansion coefficients ρDP,λ2 (r2 , r)of [Eq.(25)] for the multipole expansion of the local density function ρD
P (r2 ), i.e.,ρD
P (r2 ) =

X

0
λ2
ρD
P,λ2 (r2 , r)(−) [Yλ2 Yλ2 ]00 [Eq.(23)]λ2√ Z 1 DD
0π
ρP,λ2 (r20 , r, µ)Yλ2 0 (θ, 0)dµ, [Eq.(25)]ρP,λ2 (r2 , r) =−1

r22 = (r20 )2 + r2 + 2r20 rµ,µ ≡ cos θ = r̂ · r̂20

For the nucleon scattering (KTRLD(1)=3), it just returns.
It calls a supporting subroutine YLCAL (See Section A.2.20.1.) to obtain the spherical harmonics, and also calls GAUSF(NGAUS,RTS,WGT) (See Section A.2.20.2.) with NGAUS=16 for thegaussian integration. CLEB (See A.2.20.4.) is also called for the Clebsch-Gordan coefficients.

36

A.2.7.2
PDENST(KEXCH=1, exchange form factor) calculates the expansion coefficientsE0
ρP,λ2 (r2 , r) of [Eq.(31a)] for the multipole expansion of the nonlocal density function ρE
r2 , ~r20 ),
P (~
i.e.,
ρE
r2 , ~r20 ) =
P (~

X

0
λ2
0
ρE
P,λ2 (r2 , r)(−) [Yλ2 (r̂2 )Yλ2 (r̂)]00 [Eq.(30a)]

λ2

0
ρE
P,λ2 (r2 , r)=X`ˆ2 η̂2 ω`2 (r20 )(η2 0λ2 0|`2 0)` 2 η22π X
×η̂2 (η2 0λ2 m|`2 m)
2`2 + 1 m(µ = cos θ = ~r2 · ~r20 ,andZω`2 (r2 )Y`2 m (θ, 0)Yλ∗2 m (θ0 , 0)dµcos θ0 = ~r20 · ~r) [Eq.(31a)]A.2.8 FFCALD - Direct form factor
A.2.8.1 FFCALD calls EFFINT(0) (See A.2.6.1.) to calculate the effective interaction for the
direct form factors.
A.2.8.2 FFCALD calls PDENST(0)(See A.2.7.1.) to calculate the projectile density for the
direct form factors.
A.2.8.3 FFCALD calculates the local target density ρDT,`1 (r1 ), (See Formulation 1.5 for details.)
√
1
1
†
ρD
4πd`p `h `1 R`p (r1 )R`h (r1 )T,`1 (r1 ) = X(`p jp , `h jh ; `1 s1 jt ) < IB ||[âjp νp âjh νh ]jt ||IA >22
It calls a supporting routine NINEJ (See A.2.20.7.) to evaluate X-coefficients.
0A.2.8.4
FFCALD further expands the resulting projectile density ρDP,λ2 (r2 , r)Yλ2 µ2 (r̂) of
Eq.(23), obtained in PDENST, once more into
√ X D
0
0
4π
ρP,λ2 `t `1 (ra , r1 , r)[Y`t (r̂a )Y`1 (r̂1 )]km [Eq.(24)]ρDP,k (r2 , r)Ykm (r̂2 ) =
`t `1

ρD
P,k`t `1 (ra , r1 , r)=(r20 )2

=

2π X
k̂ 2

`ˆt (`t 0`1 m|km)m2
r1 + ra2 − 2r1 ra µ

Z
0
∗
ρD
P,k`t `1 (ra , r1 , µ, r)Ykm (θ2 , 0)Y`1 m (θ, 0)dµµ ≡ cos θ = r̂a · r̂1 ,µ0 ≡ cos θ0 = r̂a · r̂20 =r1 µ − ra[Eq.(26)]
r20

Here it calls GAUSF and YLCAL for the integration and CLEB for vector coupling coefficients.
For the NN scattering,ρDP,k`t `1 (ra , r1 , r) =−1
2π X ˆ
)Ykm (θ20 , 0)Y`∗1 m (θ, 0)`t (`t 0`1 m|km)k̂(−)k (
2
r
r
r
1 a
k̂ m

Note that the α-coefficients (See A.2.5) are multiplied in ρDP,k`t `1 (ra , r1 , r) here.
A.2.8.5

FFCALD then calculates the direct form factors, i.e.,

ftD1 s1 `1 k`t ,m` (ra ) = (−)`1 `ˆ−1tt

A.2.8.6ZZ

r2 drVtD(r)
1 s1 kD
dr1 ρDP,k`t `1 (ra , r1 , r)ρT,`1 (r1 ) [Eq.(27)]

FFCALD calls PACALD(1) (See A.2.9.) to obtain the direct transition amplitudes.
37

A.2.9PACALD - Direct overlap integral and transition amplitudes

A.2.9.1 PACALD(IDEXCH) calculates the overlap integrals of the direct form factor (IDEXCH=1)
and the exchange form factor for no-recoil approximation (IDEXCH=2).
Z

OtD1 s1 `1 k`t ,`a `b= d`a `t `b

R
OtE,N
1 s1 `1 k`t ,`a `b= d`a `t `bd`a `t `bZ

dra χ`b (ra )ftD1 s1 `1 k`t (ra )χ`a (ra ) [Eq.(19a)]R
dra χ`b (ra )ftE,N(ra )χ`a (ra )1 s1 `1 k`t

1√ `ˆa `ˆt `ˆ−1b (`a 0`t 0|`b 0)
4π≡

It calls CLEBZ (See Section A.2.20.5.) for vector coupling coefficients.
A.2.9.2

PACALD then calculates direct transition amplitudes.

TtD1 s1 `1 k`t m`t=(4π)3/2 X `a −`b +π ˆi`a (`a 0`t m`t |`b m`t )OtD1 s1 `1 k`t ,`a `b Y`b m`t (k̂b ) [Eq.(18)]ka kb ` `a b

and sums over (t1 s1 `1 , k = 0). Here it calls YLCAL for the spherical harmonics and CLEB forvector coupling coefficients.
A.2.10 XLMCAL - Distortion factor
A.2.10.1 XLMCAL calculates the so-called distortion factor, defined asI`t m`t (ra ) ≡(4π)3/2 X `a −`b +π ˆi`a (`a 0`t m`t |`b m`t )d`a `t `b χ`b (ra )χ`a (ra )ka kb ` `a bThis is a part of integrand of TtD1 s1 `1 k`t m` of Eq. (18) in the original paper, excepting thetform factor in the overlap integral of Eq.(19a). It calls CLEB and CLEBZ for Clebsch-Gordan
coefficients.
A.2.11 FFCALDM - Modified direct form factor
A.2.11.1 FFCALDM obtains exactly the same direct transition amplitudes as those of FFCALD. The order of overlap integrations is interchanged between ra and r1 . According
to the paper by B. T. Kim, D. P. Knobles, S. A. Stotts, and T. Udagawa, Phys. Rev. C61,
044611 (2000), the modified direct form factor is defined as
ftD,M(r1 ) = (−)`1 `ˆ−1t
1 s1 `1 k`t ,m`ZZr2 drVtD(r)
1 s1 kt

dra ρDP,k`t `1 (ra , r1 , r)It1 s1 `1 k`t m`t (ra )where It1 s1 `1 k`t m`t (ra ) is the distortion factor calculated in XLMCAL.
A.2.11.2

FFCALDM then calculates overlap integrals.
OtD,M
1 s1 `1 k`t m`

t

A.2.11.3Z=

r12 dr1 ftD,M(r1 )ρD
T,`1 (r1 )1 s1 `1 k`t ,m`t

FFCALDM finally calculates direct transition amplitudes.
TtD,M
1 s1 `1 k`t m`t=(4π)3/2 X `a −`b +π ˆi`a (`a 0`t m`t |`b m`t )OtD,M
1 s1 `1 k`t ,`a `b
ka kb ` `a b

and sums over (t1 s1 `1 , k = 0).
38

A.2.12DCHECK2 - Checking field lengths of variables for the exchange partA.2.12.1 DCHECK2 checks field lengths of variables for the exchange form factor calculation
with given input data.
The meaning of PARAMETER given in the program is following,
Parameter
NWTMAX
KBMAX
KB
KCMAX
KC
KAMAX
KA
KGMAX
KG
NLS

A.2.13

Field length of Variable(LAM1MX+1)*(LAM1MX+2)/2
NLSMAX*LAM1MX*(LAM1MX+2)/2(NLS,LAM1,LPP)
KBMAX*NOLTR*LAM2MX*(LAM2MX+1)/2(LT,KB,LAM2,LL)
LAMMX*(LAMMX+1)/2(LL,LAM)KCMAX*LAMP1X(KC,LAM)(LT1,L1,S)

Max. Combinations{t1 s1 `1 λ1 `0p }{KB, λ2 `}{KF=(t1 s1 `1 k`t ), λ`}{KC, λ}TRECAL, DENST, TRECALM, DENSTM - Non-local projectile and target densities for exchange partA.2.13.1 TRECAL first calls PDENST(1) (See A.2.7.) and calculates the non-local projectile0
density, ρE
P,λ2 (r2 , r) of [Eq.(31a)],0
ρEP,λ2 (r2 , r) =X

`ˆ2 η̂2 ω`2 (r20 )(η2 0λ2 0|`2 0)` 2 η2×

2π X
η̂2 (η2 0λ2 m|`2 m)
2`2 + 1 m(µ = cos θ = ~r2 · ~r20 ,A.2.13.2andZ

ω`2 (r2 )Y`2 m (θ, 0)Yλ∗2 m (θ0 , 0)dµcos θ0 = ~r20 · ~r)[Eq.(31a)]

TRECAL then calculates the non-local target density, [Eq. (31b)],

ρET,`1 λ1 `c (r1 , r) =X11i`p +`h −π X(`p jp , `h jh ; `1 s1 jt ) < IB ||[â†jp νp âjh νh ]jt ||IA > R`p (r1 )
2
2
ph,η
1

× (−)η1 `ˆh `ˆc η̂1 (`c 0η1 0|`p 0) W (`c η1 `1 `h ; `p λ1 )Z
2π X
×η̂1 (η1 0λ1 m1 |`h m1 ) R`h (r10 )Y`h m1 (θ, 0)Yλ∗1 m1 (θ0 , 0)dµ02`h + 1 m1(µ = cos θ = ~r1 · ~r10 ,andcos θ0 = ~r1 · ~r)[Eq.(31b)]

A.2.13.3 DENST(N,L) is called to calculate the last line of the above equation for a given
nh = N, `h = L.
TRECAL and DENST call supporting subroutines NINEJ, RAC7, CLEBZ for the vector coupling coefficients.
A.2.13.4 TRECALM and DENSTM are basically the same as TRECAL and DENSTM exceptthat r1 is interpolated and that in the target densities the particle states are not added.

39

A.2.14

GFAC, GFACM - Vector coupling factors for exchange partA.2.14.1 GFAC(K) defines various vector coupling factors for the exchange form factors forthe given K=1 for the central interaction and K=2 for the tensor interaction. It first calculates
a coupling factor in the projectile density of [Eq.(32)]Z

2π X ˆ
0∗=`b (`b 0`m|λ2 m) ρEP,λ2 (rb , r1 , µ, r)Yλ2 m (θ2 , 0)Y`m (θ, 0)dµ2λ2 + 1 m
2
2π ˆ`b (`b 0`m2 |λ2 m2 )CLEBQ(KC,M2) =2λ2 + 1
1 + δm2 0
KC = {KB = (t1 s1 `1 λ1 `b ), λ2 `}ρEP,λ2 `b ` (rb , r1 , r)

A.2.14.2

GFAC(K) then calculates a coupling factor in the Gk of [Eq.(33b)]

Gkt1 s1 `1 `b λ (rb , r) =X1
√ r−k VtE(r)(−)`b λ̂1 λ̂2 (λ1 0λ2 0|λ0)W (λ1 λ2 `1 `b ; λ`c )
s
k
1 1
4π
λ1 λ2 ` c
ZE
r12 dr1 ρE
P,λ2 `b `c (rb , r1 , r)ρT,`1 λ1 `c (r1 , r)

×1
√ (−)`b λ̂1 λ̂2 (λ1 0λ2 0|λ0)W (λ1 λ2 `1 `b ; λ`c )4π
KG = {KC, λ}COEF(KG) =A.2.14.3

GFAC(K) finally calculates two coupling factors in c of [(Eq.(35)]
ct1 s1 `1 k,`α `β (rb , ra ) =X
2π X ˆ
`β (`α m`1 `β 0|`1 m`1 )`ˆc (`c 0λm`1 |`1 m`1 )
2
`ˆ m
1

`c λ

`1

Z

×

dµGkt1 s1 `1 `b λ (rb , r)Yλm`1 (θ0 , π)Y`∗α m` (θ, 0)1(µ = cos θ = r̂b · r̂a0 , and cos θ0 = r̂b · r̂)22π ˆ`c (`c 0λm|`1 m)(−)m
2`1 + 1
1 + δm0
KA = {KF = (t1 s1 `1 k`t ), λ`}, `b λ)
NLS = {t1 s1 `1 }CCDD(JLS,M1) = `ˆβ (`α m`β 0|`1 m)
JLS = {`α , `β , NLS}

CLEBF(KA,M1,NLS) =A.2.14.4 GFACM(K) calculates a coupling factor in the form factors for the recoil effects inthe plane wave approximation, instead of the previous section A.2.13.3.
X√
ftP1 sW1 `1 k`t (rb ) =4π
iπ−`r −`b k̂ λ̂(k0λ0|`r 0)`ˆ1 `ˆr W (`b λ`t k : `1 `r )`b λ`rZ

× (−)k+`1 −`t (`b m`t `r 0|`t m`t )dr rk+2 Gkt1 s1 `1 `b λ (rb , r)j`r (αka r/a)

CCDD(JLS) = iπ−`r −`b k̂ λ̂(k0λ0|`r 0)`ˆ1 `ˆr W (`b λ`t k : `1 `r )40

A.2.15FFCALE - Exchange form factors

A.2.15.1 FFCALE calls DCHECK2 (See A.2.12.) to check the field lengths of variables forthe exchange form factors.
A.2.15.2 FFCALE calls EFFINT(1) (See A.2.6.) to calculate the effective interaction for theexchange form factors.
A.2.15.3 FFCALE calls TRECAL (See A.2.13.) to calculate the projectile and target density.
A.2.15.4

FFCALE starts to calculate the central and tensor exchange form factors.

This section has a big DO loop up to the statement number 1000, for the central exchange
form factors (KCT=1) and the tensor exchange form factors (KCT=2). By inputting 1 for
KCETN(KCT) (Input data Line #16), you may exclude the corresponding form factors.
It then calls GFAC(KCT) (See A.2.14.) to calculate the various coupling factors.
It starts next big DO loop up to the state number 900. In the DO loop, the following Section
A.2.15.5 - A.2.15.8 for a given rb are performed.
A.2.15.5

FFCALE performs integration in G-factors.

It first calculates the projectile density, ρE
P,λ2 `a ` (rb , r1 , r), in [Eq. (32)],
2π X ˆ`b (`b 0`m|λ2 m)
λ̂22 mρE
P,λ2 `a ` (rb , r1 , r) =

Z
0
∗
ρE
P,λ2 (rb , r1 , µ, r)Yλ2 m (θ2 , 0)Y`m (θ, 0)dµThe non-local target density from TRECAL (See A.2.13.) is multiplied to the projectile density
and integration over r1 is performed.
ZE
r12 dr1 ρE
P,λ2 `b `c (rb , r1 , r)ρT,`1 λ1 `c (r1 , r)

For a nucleon scattering, it becomesλ2
ρET,`1 `b λ (rb , r) × λ̂2 (−)(See Formulation 3.3.2.)
A.2.15.6

FFCALE calculates the G-factor of [Eq. (33b)],

Gkt1 s1 `1 `b λ (rb , r) =X1
√ r−k VtE(r)(−)`b λ̂1 λ̂2 (λ1 0λ2 0|λ0)W (λ1 λ2 `1 `b ; λ`c )
s
k
1 1
4π
λ1 λ2 ` cZ×E
r12 dr1 ρE
P,λ2 `b `c (r1 , r)ρT,`1 λ1 `c (r1 , r) [Eq. (33b)]

Note that the α-coefficients (See A.2.5) are multiplied in Gkt1 s1 `1 `b λ (rb , r) for the central interaction of the exchange form factor.
A.2.15.7

FFCALE calculates the c-factor of [Eq. (35)].

It first decomposes r into ra and rb in the G-factors,
Gkt1 s1 `1 `b λ (rb , ra ) =2π X ˆ
`c (`c 0λm`t |`t m`t )`ˆ2t `c λ41

ZdµGkt1 s1 `1 `b λ (rb , r)Yλm`t (θ0 , π)Y`∗α m` (θ, 0)t

It finally calculates the c-factor,
ct1 s1 `1 k,`α `β (rb , ra ) =X
2π X ˆ
`
(`
m
`
0|`
m
)
`ˆc (`c 0λm`1 |`1 m`1 )
α
1
β
`
β
`
1
1
`ˆ2 m
1

`c λ

`1

Z

×

dµGkt1 s1 `1 `b λ (rb , r)Yλm`1 (θ0 , π)Y`∗α m` (θ, 0) [Eq. (35)]1

Remember that this is the exchange form factors for central interactions. (See Formulation 2.2.)
It stores the results as ”FFRIT(JLS), JLS=(`α , `β ,NLS)”, and writes on tape. (See B.4 fordetails.)
A.2.15.8 FFCALE also calculates the radial exchange form factor of [Eq. (41)] in the no-recoil
approximation. (See Formulation 3.2.)
ftN1 sR1 `1 k`t (rb ) =Z√
4π(−)k `ˆ1 `ˆ−1
dr rk+2 Gkt1 s1 `1 `t λ (rb , r) [Eq. (41)]t

A.2.15.9 FFCALE calls TENSOR (See A.2.16.) to calculate final form factors for the tensor
interaction.
A.2.15.10 FFCALE calls PACALE (See A.2.17.) to obtain the exchange transition amplitudes.
A.2.15.11 FFCALE calls PACALD(2) (See A.2.9.) to obtain the exchange transition amplitudes in the no-recoil approximation.
A.2.16TENSOR - Tensor interaction for exchange partA.2.16.1
TENSOR calculates exchange form factor ftE1 s1 `1 k`t ,`b `a (rb , ra ) in Eq.(36) in theoriginal paper for k=2, i.e. for tensor interaction,
ftE1 s1 `1 k`t ,`b `a (rb , ra ) = J 4π mkaXλa λb `α `β[(2k + 1)!
]1/2 δλa +λb ,k (−ra )λa (rb )λb(2λa + 1)!(2λb + 1)!

× X(`α λa `a , `β λb `b ; `1 λ`t )d`α λa `a d`β λb `b ct1 s1 `1 k,`α `β (rb , ra ) [Eq. (36)]Note that the multiplication of all geometrical factors becomes unity for central interaction(k = 0) and the form factor becomes ct1 s1 `1 k,`α `β (rb , ra ). (See Formulation 2.2.)
A.2.16.2

TENSOR first calculates the geometrical factor,
4π [(2k + 1)!
]1/2 δλa +λb ,k X(`α λa `a , `β λb `b ; `1 λ`t )d`α λa `a d`β λb `b(2λa + 1)!(2λb + 1)!

and saves as ”C(KJLS)” in a linearized form. Note that the α-coefficients (See A.2.5) are
multiplied in this geometrical factor for the tensor interaction.
A.2.16.3 TENSOR reads c-factor (FFRIT) from Tape 14, multiplies all factors and sums over
λa λb `α `β , and then saves as ”GGRIT(JLS)”, where JLS=(`α , `β ,NLS). If the central interaction
is considered (KCNTN(1).EQ.0), it reads c-factor for the central interaction from Tape 13, andadds to the tensor part. It then finally stores form factors as ”GGRIT(JLS)” for every (rb , ra ),and writes on Tape 15.

42A.2.17

PACALE - Exchange overlap integral and transition amplitudes

A.2.17.1

PACALE calculates the overlap integrals of [Eq.(19b)]
Z

OtE1 s1 `1 k`t ,`a `b

= J

Z

drb

dra rb ra χ`b (rb )ftE1 s1 `1 k`t ,`a `b (rb , ra )χ`a (ra ) [Eq. (19b)]

During the integration processes, it reads exchange form factors ftE1 s1 `1 k`t ,`a `b (rb , ra ) from Tape
15.
A.2.17.2

PACALE finally obtains transition amplitudes,

TtE1 s1 `1 k`t m`t=(4π)3/2 X `a −`b +π ˆi`a (`a 0`t m`t |`b m`t )OtE1 s1 `1 k`t ,`a `b Y`b m`t (k̂b ) [Eq. (18)]ka kb ` `a b

and sums over (t1 s1 `1 , k = 2). Here it calls YLCAL for the spherical harmonics and CLEB forvector coupling coefficients.
A.2.18FFCALEM - Exchange overlap integral and transition amplitudes

A.2.18.1 FFCALEM obtains the exchange form factor with re-coil effects in the plane wave
approximation. As in the case of direct form factor, the order of overlap integrations is interchanged between ra and r1 with the same spirit of the paper by B. T. Kim, et. al.. The first
three steps are the same as FFCALE; namely, it calls DCHECK2 (See A.2.12.) to check thefield lengths of variables for the exchange form factors, EFFINT(1) (See A.2.6.) to calculate theeffective interaction for the exchange form factors, and TRECALM (See A.2.13.) to calculate
the projectile and target density.
A.2.18.2 FFCALEM calls a supporting subroutine BESSEL (See A.2.20.3.) to obtain Bessel
functions which is needed for the tensor exchange form factors in the plane wave approximation.
(See Formulation 2.3.)
A.2.18.3

FFCALEM then calculates the central and tensor exchange form factors.

This section has a big DO loop up to the statement number 1000, for the central exchange
form factors (KCT=1) and the tensor exchange form factors (KCT=2). By inputting 1 for
KCETN(KCT) (Input data Line #16), you may exclude the corresponding form factors.
It then calls GFACM(KCT) (See A.2.14.) to calculate the various coupling factors.
It starts next big DO loop up to the state number 900. In the DO loop, the following Section
A.2.18.4 and A.2.18.5 for a given r1 are performed.
A.2.18.4

FFCALEM calculates the projectile density and integration over rb .

This section calculates the projectile density, ρE
P,λ2 `a ` (rb , r1 , r),ρEP,λ2 `a ` (rb , r1 , r) =2π X ˆ`b (`b 0`m|λ2 m)
λ̂22 mZ
0∗
ρEP,λ2 (rb , r1 , µ, r)Yλ2 m (θ2 , 0)Y`m (θ, 0)dµFor a nucleon scattering, it calculates the projectile density as, (See Formulation 3.1.)
ρEP,λ2 `a ` (rb , r1 , r) =µp =12π X ˆ∗`b (`b 0`m|λ2 m)λ̂2 (−)λ2 (−)Yλ2 m (−µp )Y`m
(µ)
2
r
r
r
λ̂2 m
1 b
r12 − rb2 − r2
r2 + r12 − r2
, µ = b2rrb
2r1 rb43

The distortion factor, It1 s1 `1 k`t m`t (rb ) (See A.2.10.) is multiplied to the projectile density andintegration is performed over rb .
Z
EZP,λ(r1 , r) =2 `b `

A.2.18.4

drb ρEP,λ2 `b ` (rb , r1 , r)It1 s1 `1 k`t m`t (rb )FFCALEM calculates the G factor,

Gkt1 s1 `1 `b λ (r1 , r) =X1
√ r−k VtE(r)(−)`b λ̂1 λ̂2 (λ1 0λ2 0|λ0)W (λ1 λ2 `1 `b ; λ`)1 s1 k4π
λ1 λ2 `Z×E
r12 dr1 ZP,λ(r1 , r)ρE
T,`1 λ1 ` (r1 , r)
2 `b `

A.2.18.5 FFCALEM then defines the modified exchange form factor in the plane wave approximation as,
X
√
ftE,M
(r1 ) =
4πiπ−`r −`b k̂ λ̂(k0λ0|`r 0)`ˆ1 `ˆr W (`b λ`t k : `1 `r )1 s1 `1 k`t`b λ`rZ

× (−)k+`1 −`t (`b m`t `r 0|`t m`t )dr rk+2 Gkt1 s1 `1 `b λ (r1 , r)j`r (αka r/a)

where `r and the recoil factor α values are input as ”LRP1MX” and ”FACNR”, respectively.
(See Section 4.2.9.)
A.2.18.6

FFCALEM then calculates overlap integrals.
OtE,M
1 s1 `1 k`t m`

t

A.2.18.7Z=

r12 dr1 ftE,M(r1 )1 s1 `1 k`t ,m`t

FFCALEM finally calculates direct transition amplitudes.
TtE,M
1 s1 `1 k`t m`t=(4π)3/2 X `a −`b +π ˆi`a (`a 0`t m`t |`b m`t )OtE,M
1 s1 `1 k`t ,`a `b
ka kb ` `a b

and sums over (t1 s1 `1 , k = 2).
A.2.19 CROSS - The differential cross sections
A.2.19.1 CROSS calculates the final differential cross sections of [Eq. (21)],
X X jsνµa µb kb1dσ=|
α t t 1 Tti s ` k` m |22dΩ(2πh̄ )2 ka (2IA + 1)(2sa + 1) i k` t t1 s1 `1 k`t 1 1 1 t `t
1 1

44

[Eq. (21)]A.2.20
A.2.20.1

YLCAL, GAUSF, BESSEL, CLEB, CLEBZ, RAC7, NINEJ, DSPLS3 Supporting subroutines
YLCAL(A,L,M,R) calculates a part of spherical harmonics, Y`m (µ),[

2` + 1 (` − m)! 1/2] P`m (µ),
4π (` + m)!

µ = cos θby utilizing the recurrence relations of the associated Legendre functions, P`m (µ),(2` + 1)µP`m = (` − m + 1)P`+1,m + (` + m)P`−1,m(2` + 1)(1 − µ2 )1/2 P`m = P`−1,m+1 − P`+1,m+1 .
The arguments are A=µ, L=`, M=m, and R = answer.
A.2.20.2 GAUSF(N,R,W) generates abscissas and weighting factors for the Gaussian integration
Z 1−1

f (x)dx =nXwi f (xi ) + ∆

i=1

where
xi = ith zero of Legendre function Pn (x)2[Pn0 (xi )]2wi =1 − x2i
The arguments are N=n, R=xi , and W=wi .
A.2.20.3 BESSEL(A,L,R1,R2) generates the spherical Bessel and Neumann functions, jn (x)and nn (x), by using the recurrence formula,n+1
djn (z)jn (z) +zdzn
djn (z)jn (z) −zdz= jn−1 (z)= jn+1 (z)The arguments are A=x, L=n, R1 = value of spherical Bessel function and R2 = that of
Neumann.
A.2.20.4

CLEB(A,B,C,D,E,F,FACLOG,R) calculates the Clebsch-Gordan coefficients, i.e.,s(j1 m1 j2 m2 |jmj ) = δm,m1 +m2
q××(2j + 1)(j + j1 − j2 )!(j − j1 + j2 )!(j1 + j2 − j)!
(j1 + j2 + j + 1)!

(j + m)!(j − m)!(j1 + m1 )!(j1 − m1 )!(j2 + m2 )!(j2 − m2 )!
X

[

k

(−)k
k!(j1 + j2 − j − k)!(j1 − m1 − k)!(j2 + m2 − k)!
×1](j − j2 + m1 + k)!(j − j1 − m2 + k)!

The summation is extended over all integral k for which the argument of every factorial isnonnegative.
The arguments are A=j1 , B=j2 , C=j, D=m1 , E=m2 , F=m, R = answer, and FACLOG is thelog of integral value of n. n = 500 is set.
45

A.2.20.5 CLEBZ(A,B,C,FACLOG,R) calculates the Clebsch-Gordan coefficients with all m’s
are zero, i.e.,s(j1 + j2 − j)!(j + j1 − j2 )!(j + j2 − j1 )!
g!
(2g + 1)!
(g − j)!(g − j1 )!(g − j2 )!
for j + j1 + j2 = 2g = evenfor j + j1 + j2 = 2g = odd

g+j(j1 0j2 0|j0) = (−)= 0ĵThe arguments are A=j1 , B=j2 , C=j, R = answer, and FACLOG is the same as in CLEB.
A.2.20.6

RAC7(A,B,C,D,E,F,FACLOG,R)calculates the Racah coefficients, i.e.,−1 ˆ−1W (j1 j2 Jj3 ; j12 J23 ) = Jˆ12
J23

X

(j1 m1 j2 m2 |J12 M12 )(J12 M12 j3 m3 |JM )m1 m2 m3

(j2 m2 j3 m3 |J23 M23 )(j1 m1 J23 M23 |JM )The arguments are A=j1 , B=j2 , C=J, D=j3 , E=j12 , F=j23 , R = answer, and FACLOG is thesame as in CLEB. In coding, it does not call CLEB, but use the analytic form of Clebsch-Gordan
coefficients in A.2.20.4 directly.
A.2.20.7

NINEJ(A(9),FACLOG,R) calculates the 9-j symbols as




a b e
X


U  c d e0  = (−)σλ̂2 W (bcef ; λa)W (bcf 0 e0 ; λd)W (ef e0 f 0 ; λg)
λf f0 gσ = a + b + c + d + e + e0 + f + f 0 + g = integer
by calling the Racah coefficient routine RAC7.
The arguments are A(1)=a, A(2)=b, A(3)=c, A(4)=d, A(5)=e, A(6)=e0 , A(7)=f , A(8)=f 0 ,A(9)=g, R = answer, and FACLOG is the same as in CLEB.
A.2.20.8 DSPLS3(X,Y,N,XI,YI,M,Q,AU,IGO) interpolates a given set of {(xi , yi ), i = 1, .., n}
to a set of {(xj , yj ), j = 1, .., m} by the cubic spline interpolation method.
The arguments are X=xi , Y=yi , N=n, XI=xj , YI=yj , and M=m. Q and AU are internallyused, and IGO = 1 is set.

46

B

Variables and their definitions

B.1

PARAMETER variables

Variable
Name

Present
#

KAA
KBB
KCC
KFM
KGG
KJL
KTE

40
30
80
10
125
1200
10000

LAB
LMJ
LMM
LM1
LM2
LPH
LRP
LTL

2000
10
4
5
30
10
11
9

LXA
LXB
L1X
L2X
NHS
NIN
NOU
NPS
NXA
NXB
N1X

150
150
10
150
9
300
300
9
420
140
140

The way counting number.
(DCHEK1 and DCHECK2 check the present number
whether that is good enough for the calculation.)Max. possible combinations of KA={KF=(t1 s1 `1 k`t ), λ`}Max. possible combinations of KB={t1 s1 `1 λ1 `0p }Max. possible combinations of KC={KB, λ2 `}Max. possible combinations of KF={t1 s1 `1 k`t }Max. possible combinations of {KC, λ}Max. possible combinations of {t1 s1 `1 k`t , `a `b }Max. possible combinations of {`α λa `a , `β λb `b ; `1 k`t }which appears in the X-factor of the exchange form factor.
Max. possible combinations of {`t `a `b }(MJMAX+1)*3
Max of (LAMMXD(1,2,3))
Max magnetic quantum in the transferred `t , MP1MAX.
Max. possible combinations of {`h λ1 } in the nonlocal target density.
Max of ph pairs in the target, NPHMX.
Max # of partial waves in the PW approximation.
Max total angular momentum in the target,
L1R(NLSMAX)+LAMMXD(1)+1.
Max of partial waves in the incident channel, LDWMXR(1).
Max of partial waves in the exit channel, LDWMXR(2).
NOLTR*MP1MAXNOLTR*MP1MAX*NPAIRD
Max # of hole states, NOSH.
Max of (NHDMX,NHEMX)
NHEMX
Max # of particle states, NOSP.
NXMXR(1)*PMASANXMXR(2)NXMXR(4)/N1STEP+1

47

B.2

COMMON variables[Variables with * are input data and those in bold face are the key quantities.]
B.2.1

CNST - Constants

Variable name
FACLOG(N)
PI
HBAR
AMAS
WNUNIT
FINE

Definition
Log value of factorial of integer N . Used for vector coupling coefficients.
Presently, N=500.
π = 4.0 arctan(1.0).
h̄c = 197.327053 in MeV · fm.
atomic mass unit, mamu = 931.49432
in MeV.
√
wave number unit in 1/fm, 2mamu /h̄c.
fine structure constant, e2 /h̄c = 1/137.0359896.

Used subroutines : All subroutines except FLGLCH, DCHECK1,DCHECK2, and 8 supporting
subroutines.
B.2.2

CNTRL - Control of the calculation and output

Variable name
KTRLD(9)*KTLOUT(24)*KRTYPE

Definition
Calculation options. See 4.2.1 for details.
Output options. See 4.2.2 for details.
= 21; for (d,2p) reaction,= 31: for (He,t) reaction

Used subroutines : All subroutines except DISWAVE, XLMCAL, and 8 supporting subroutines.
B.2.3

SPSTAT - Particle-hole states in the target

Variable name
EET*
SPECTA(LPH)
ESP(LPH)*
ESH(LPH)*
NSP(LPH)*
LSP(LPH)*
JTWP(LPH)*
NSH(LPH)*
LSH(LPH)*
JTWH(LPH)*
IPAIR
NOSP*
NOSH*
NPST(LPH)
NHST(LPH)
NPAIRD
L1R(8)*
ISR(8)*ITR(8)*NLSMAX*

Definition
Transferred energy
Spectroscopic factor for ph pair. Presently set 1.0. (DCP2)
Binding energy of particle state.
Binding energy of hole state.
Number of nodes of particle state.
Orbital angular momentum of particle state.
Twice of total angular momentum of particle state.
Number of nodes of particle state.
Orbital angular momentum of particle state.
Twice of total angular momentum of particle state.
Dummy
Number of particle states.
Number of hole states.
Particle state number of each ph pair. (DCP2)
Hole state number of each ph pair. (DCP2)
Number of ph pair. (DCP2)
Orbital angular momentum, `1 .
Spin, s1 (= s2 = st ).
Isospin, t1 = t2 .
Number of {`1 s1 t1 } sets in the target system.

48

Continued
Variable nameJT*
MJMAX*KPARIT*IST*NOLTR*N1
LTR(8)*ITP(LPH)*
ITH(LPH)*
ITZP(LPH)*
ITZH(LPH)*

Definition
Transferred total angular momentum, jt .
Maximum magnetic quantum number of jt , mjt .
= 0; No parity change in the reaction,= 1; Parity change.
Transferred spin, st (= s1 = s2 ).
Maximum number of transferred orbital angular momenta.
Dummy
Transferred orbital angular momenta, `t .
Isospin T of particle state. (Not used.)
Isospin T of hole state.
= 1; neutron, = −1; proton of particle state.
= 1; neutron, = −1; proton of hole state.

Used subroutines : DCP2, DCHECK1, DCHECK2, AFACAL, FFCALD, PACALD, FFCALDM,XLMCAL, FFCALE, TRECAL, DENST, GFAC, TENSOR, PACALE,FFCALEM, TRECALM, DENSTM, GFACM, CROSS.
B.2.4
B.2.4.1

BSX, UNCPSA, CBST - Bound state wave functions
COMMON block BSX

Variable nameURSAVE(500)

ENEPT
VNEPT
VSOR*
DFNR*
DFNSO*
RZR*
RXSO*
RZC*
XMES2
NODER
LBTR
JBTRTWNXRAWF
KTRL4

KOPOT

Definition
Calculated single particle wave function Rp (r). Max. of mesh points=500.
After calculation, URSAVE transfers to USAVP (particle state) and
USAVH (hole), saves on another COMMON block CBST
Calculated binding energy (KTRL2=0).
Calculated potential depth (KTRL2=1). KTRL2=1 is set in the program.
(Initial value is input as VSX (See 4.2.11.)
Spin-orbit depth parameter.
Diffuseness parameter.
Spin-orbit diffuseness parameter.
Reduced radius parameter.
Spin-orbit reduced radius parameter.
Coulomb reduced radius parameter.
Mesh size. XMESH2=0.1 is set.
Number of nodal points except the origin.
Orbital angular momentum.
Twice of total angular momentum.
Max. mesh point where wave function calculation is terminated.
Rmax = RC + 10.0 ∗ aC is set.
= 0; wave function itself stored in URSAVE. KTRL4=0 is set.
= 1; wave function * V (r).
= 2; wave function * r−` .
= 3; wave function * v(r) * r−` .
Output control for calculations. KOPOT=0 is set. (No output)

Used subroutines : DCP2, BSAXON, UNCPST.

49

B.2.4.2

COMMON block UNCPSA

UNCPSA is only used for communicating between BSAXON and UNSPST.
Variable nameKTRL3
ITBEMX
ACURCYAMUPMU
KTRL2
KTRL8
KEX2
KEX4
KEX40
KEX41
KEX42
KEX43
TMAS*
PMAS*
ZZT*
ZZP*
RMAS
ZZ
XMES1
PERCNT
VSX*
ISTW
NXRA
NXRM
NXRMP1NXRMP2
NXRMP3
NXRMP4
NXRMP5NODEKGES
EGESRD
EGES
EGEST
DELGESFKAPPAFKAPINURRMINURRMEXRGDLIN(3)RGDLEX(3)Definition= 1; R=RZ*(A1**(1/3)+A2**(1/3))
Number of iteration. ITBEMX=15 is set. (DCP2)
Accuracy need for matching. ACURCY=0.001 is set. (DCP2)= 0; Atomic mass unit, =1, proton mass unit. AMUPMU=0.0 is set.
= 0; B.E. is searched,, = 1; VSX is searched. KTRL2=1 is set. (DCP2)
Dummy
With nonzero, NXCPL2=KEX2.
With nonzero, NXRA=KEX4+NXCPL2.
=0; Integration of the internal solution outwards to be matched at Rm .
=1; Integration of the external solution inwards.
=1; Search has been achieved.
Calculated number of the nodal point minus that of previous calculation.
Repeat searching until KEX42=0.
Dummy
Mass of target system. (Target mass -1)
Mass of valence nucleon.
Charge of target system.
Charge of valence nucleon.
Reduced mass =(TMAS*PMAS)/(TMAS+PMAS).
=ZZT*ZZP.
Mesh size for first 8 points in the Stöormer method. =XMESH2/8 is set.
Ratio of searched step and searched value = 0.2 is set.
Depth parameter of the central bound state potential.
Twice of spin of valence nucleon. =1 is set.
Total mesh points for the calculation.
=NXRAWF+KEX2, or =NXRAWF+30 if KEX2=0.
Matching mesh point. =(XBAR+.5*DFNR*FLOAT(NODE))/XMES2.
=NXRM+1.
=NXRM+2.
=NXRM+3.
=NXRM+4.
=NXRM+5.
Number of present nodal point during the search.
Solve three times near a guessed value, (a − δa, a, a + δa)
Do loop number.
Dummy
Given binding energy =ESP(LPH) or ESH(LPH) which are input.
Initial binding energy. Starts with EGES.
δa in KGES explanation.
√
Wave number outside the well. k =p 2µE/h̄c.
Wave number inside the well. k = 2µ(V0 − E)/h̄c.
Value of internal solution at matching point.
Value of external solution at matching point.
Slope of internal solution at matching point.
Slope of external solution at matching point.

50

Continued
Variable nameXMEM(514)
VCENTR(514)
VSPIN(514)
VCOULM(514)
PFORM1(2)PFORM2(2)PFORM3(2)Definition
Radial distance. 14 more points are need for the Stömer method.
Central potential.
Spin-orbit potential.
Coulomb potential.
PFORM1(1)= exp(x − R0 )/a0 , PFORM1(2)=exp(x − RSO )/aSO .
PFORM2(N)=1.0 /(1.0 +PFORM1(N)).
PFORM3(N)=PFORM1(N)*PFORM2(N)*PFORM2(N).

Used subroutines : DCP2, BSAXON, UNCPST.
B.2.4.3

COMMON block CBST

Variable name
USAVP(NPS*NXA)
USAVH(NHS*NXA)Definition
Final particle state functions φp (r1 ),in the linearized form for NPS states.
Final hole state functions φh (r1 ),in the linearized form for NHS states.

Used subroutines :DCP2, FFCALD, FFCALDM, TRECAL, DENST, FFCALEM, TRECALM,
DENSTM.

B.2.5

DWCC, POTCC, COUCC, RELKIN, OPTO, DISW - Distorted waves

B.2.5.1

COMMON block DWCC

The variables with dimension 4, say A(4) has following meaning;
Variable
A(1)
A(2)
A(3)
A(4)

Meaning
Variable in the incident channel.
Variable in the exit channel.
Variable in the target system. Not used.
Variable in the projectile system. Not used.

Variable nameLDWMIR(4)*LDWMXR(4)*WN(4)
LDWSTR(4)*CE(4)
CFUNIR(4)ECM(4)XMESR(4)*NXMXR(4)*NXMIR(4)*NOEXP
MXSTEP

Definition
Starting partial wave.
Ending partial wave.
Wave number.
Step of partial wave.
η = µZ1 Z2 e2 /h̄2 k
Dummy
Energy in the center of mass system.
Mesh size
Maximum (ending) mesh point.
Minimum (starting) mesh point.
Dummy
Dummy

51

Continued
Variable name
TMASA
TMASB
TMASX
TMASY
PMASA
PMASB
PMASX
PMASY
TZA
TZB
TZX
TZY
PZA
PZB
PZX
PZY
XBARD(4)XBARID(4)Definition
Mass of the target, =TMI*.
Mass of the residual nucleus, =TMF*.
Dummy
Dummy
Mass of the projectile, =PMI*.
Mass of the ejectile, =PMF*.
Dummy
Dummy
Charge of the target, =Z1I*.
Charge of the residual nucleus, =Z1F*
Dummy
Dummy
Charge of the projectile, =Z2I*
Charge of the ejectile, =Z2F*
Dummy
Dummy
Radius parameter in the real optical potential.
Radius parameter in the imaginary optical potential.

Used subroutines : DCP2, DCHECK1, DCHECK2, OPT, POTEN, AFACAL,FFCALD, PACALD, PDENST, FFCALDM, XLMCAL, FFCALE,TRECAL, DENST, GFAC, TENSOR, PACALE,FFCALEM, TRECALM, DENSTM, GFACM, CROSS.
B.2.5.2

COMMON block POTCC

The variables with dimension 4, say A(4) has the same meaning in DWCC.
Variable name
VD(4)*
WD(4)*
WSD(4)*
ARD(4)*
AID(4)*AISD(4)*RZRD(4)*RZID(4)*RZISD(4)*RZCD(4)*
IDCH(4)
VRIT(900)
NXMN
NXMX

Definition
Depth parameter of real potential in MeV.
Depth parameter of imaginary potential.
Depth parameter of imaginary surface potential.
Diffuseness parameter of real potential in fm.
Diffuseness parameter of imaginary potential.
Diffuseness parameter of imaginary surface potential.
Reduced radius parameter of real potential in fm.
Reduced radius parameter of imaginary potential.
Reduced radius parameter of imaginary surface potential.
Reduced radius parameter of Coulomb potential.
1/3
1/3
1/3
=0; R0 = r0 AT , =1; R0 = r0 (AT + AP ), Presently zero is set.
Optical model potential in r, U0 (r)Minimum range of VRIT. =1 is set.
Maximum range of VRIT. =NXMXR(ID)+2.

Used subroutines : DCP2, OPT, POTEN, DISWAVE.

52

B.2.5.3

COMMON block COUCC

Variable nameFC(130)
GC(130)
FCP(130)
GCP(130)
L5L6ETA
SIGMAZRDZ
KTOUT7K9STP
ACCR

Definition
Regular Coulomb wave function, F` (r).
Irregular Coulomb wave function, G` (r).
Derivative of regular Coulomb wave function, F`0 (r).
Derivative of irregular Coulomb wave function, G0` (r).
Maximum `.
Dummy
η(= ZT ZP e2 /h̄2 k)-value.
S-wave Coulomb phase shift.
kr.
kRm where Rm is the matching radius.
Nonzero value prints Coulomb functions.
Dummy# of steps in Runge-Kutta integration. =100 is set.
Accuracy parameter of continued fractions. =10−6 is set.

Used subroutines : OPT, FLGLCH, DISWAVE.
B.2.5.4

COMMON block RELKIN

Variable name
RMASA
RMASB
TREL1
TREL2
OMEGR(20)Definition
Reduced mass in the incident channel in the relativistic kinematics.
Reduced mass in the exit channel.
Kinetic energy in the incident channel.
Kinetic energy in the exit channel.
energyUsed subroutines : DCP2, CROSS.
B.2.5.5

COMMON block OPTO

Variable name
FOLD(NXA)

Definition
Folding potential for the optical model potential.
Calculated in DCP2 if KTRLD(3)=1.

Used subroutines : DCP2, POTEN.
B.2.5.6

COMMON block DISW

Variable name
DISWA(LXA*NXA)DISWB(LXB*NXB)Definition
Distorted waves in the incident channel, χ`a (ra )in the linearized form.
Distorted waves in the exit channel, χ`b (rb ).

Used subroutines : DCP2, OPT, PACALD, XLMCAL, PACALE.

53

B.2.6ANGCC - Angle information

Variable name
THEB*
THEBMX*DTHEB*THMIN*THMAX*THIND*

Definition
Starting angle in the differential cross section calculations.
Ending angle.
Step in angle.
Starting angle in elastic cross section calculations.
Ending angle.
step in angle.

Used subroutines : DCP2, OPT, XLMCAL, PACALD, CROSS.

B.2.7

SPECFC - α-coefficients

Variable nameDefinition

Used subroutines : AFACAL, FFCALD, FFCALDM, FFCALE, TENSOR, FFCALEM.
B.2.8
B.2.8.1

FFCC, CDENS CCKF - Preparation of form factor calculations
COMMON block FFCC

Variable name
VRANG(6)*VSTR(12,6)*
VV(NIN,12)XMESHXMESHD*XMESHE*RHED(NIN)
NBCMI*NBCMX*NONB
NBSTEPLASTEP
LBSTEP
MXMAX
MP1MAXNOLANOLBINTRANKCETN(2)*Definition
6 different range parameters in the effective NN interaction.
Strength parameters of Love-Franey interaction. See A.4.2.10 for details.
NN effective interaction of [Eq.(16)], Vti1 s1 k (r).
NIN=r, # of (t1 s1 k) × 2 = 12.
Mesh size of the ff calculation. =XMESHD* or XMESHE*.
Mesh size in the direct ff calculation, ∆rD .
Mesh size in the exchange ff calculation, ∆rE .
Radial points where NN effective interaction is calculated. (DCP2)
Minimum mesh point in the form factor.
Maximum mesh point.
# of real mesh points of rb in the ff calculation.
=(NBCMX-NBCMI)/NBSTEP+1.
Interpolation step in rb .
Interpolation step in `a . =LDWSTR(1)*Interpolation step in `b . =LDWSTR(2)*Max. value of m`t .
= Min(LTR(NOLTR),MXMAX)+1.
# of real `a in the ff calculation. =(LAP1MX-LAP1MI)/LASTEP+1.
# of real `b in the ff calculation. =(LBP1MX-LBP1MI)/LBSTEP+1.
Interaction range of Vti1 s1 k (r). =NHDMX*, or NHEMX*.
KCETN(1)= 0; Central exchange ff is calculated. =1; Not considered.
KCETN(2)= 0; Tensor exchange ff is calculated. =1; Not considered.

54

Continued
Variable name
NONA
NONAH
NONAR*NASTEP*NHDMX*NHEMX*
NGAUSR*
NBSTPD*
NBSTPE*
N1STEP*
JATW*
ISATW*

Definition# of real mesh points of ra in the ff calculation.
Half of NONA.
NONA and NONAH are defined in Sec. 4 of DCP2 (main) in the code.
# of mesh point in ra integration for exchange ff.
(Integration is made only around ra = rb with a range NONAR.)
Mesh step in the incident channel radius, ra .
Interaction range in the direct form factor calculation, rD .
Interaction range in the exchange form factor calculation, rE .
# of gaussian integration points.
Mesh step in the direct form factor.
Mesh step in the exchange form factor.
Mesh step in the target system radius, r1 .
(For (p,n) or (p,p’), better put all steps unity.)
Twice of the total angular momentum of the target A, IA .
Twice of the spin of the projectile a, sa .

Used subroutines : DCP2, DCHEK1, DCHECK2, EFFINT, FFCALD, PACALD, FFCALDM,XLMCAL, FFCALE, TRECAL, DENST, GFAC, TENSOR, PACALE,FFCALEM, TRECALM, DENSTM, GFACM, CROSS.
B.2.8.2

COMMON block CDENS - Projectile density.

Variable name
RHOD(NXA,NIN,LMM)
or
RHOE(NXA,NIN,LMM)
USAVEX(4×NXA)
SPEC(4)
DENSTY(NXA)LAMMXD(3)*LBXD(4)
NOSTX

Definition0
Direct projectile multipole density, ρD
P,λ2 (r2 , r)
0
NXA=r2 , NIN=r, LMM=λ2 . Calculated in PDENST.
0
Exchange projectile multipole density, ρE
P,λ2 (r2 , r)Projectile wave functions, ψP (r2 ).
Spectroscopic factor for the projectile system. =1.0 is set.
Projectile density function, ρP (r2 ).
LAMMXD(1)=Maximum value of λ1 in ff.
LAMMXD(2)=Maximum value of λ2 in ff.
LAMMXD(3)=Maximum value of λ in ff.
Orbital `2 for the composite projectile system .
(In case of mP > 4, BSAXON is called for ψP (r2 ). )# of bounds states in case of mP > 4.

Used subroutines : DCP2, DCHEK1, DCHECK2, PDENST, FFCALD, FFCALDM,XLMCAL, FFCALE, TRECAL, DENST, GFAC, PACALE,FFCALEM, TRECALM, DENSTM, GFACM.

55

B.2.8.3

COMMON block CCKF

Variable name
L1KFD(KFM)
ISKFD(KFM)
KKFD(KFM)
LTKFD(KFM)
KFMAX
ITKFD(KFM)
NLSKFD(KFM)
NLTKFD(KFM)
LTMIM
LTMAX

Definition
Stores t1 , s1 , `1 , k, and `t values in possible combinations of {t1 s1 `1 k`t }.
`1 , =L1R(NLS)*.
s1 , =ISR(NLS)*.
k, =0 or 2.
`t , =LTR(NOLTR)*.
# of all possible combinations of {t1 s1 `1 k`t }.
t1 , =ITR(NLS)*.
Corresponding # (NLS) of {`1 s1 t1 } set.
Corresponding # (NTS) of `t .
Smallest `t value.
Largest `t value.

Used subroutines : DCP2, DCHEK1, DCHECK2, FFCALD, PACALD, FFCALDM, XLMCAL,FFCALE, TRECAL, GFAC, FFCALEM, TRECALM, GFACM, CROSS.
B.2.9
B.2.9.1

CGFACA, CGFACB, TREDEN - Exchange form factor calculations
COMMON block CGFACA

Variable name
KAMAX
KCMAX
KGMAX
JLSMX
KBBSD(10)LAMD(KAA)
KBC(KCC)
LAM2PC(KCC)
LPP1C(KCC)
LLP1D(KCC)
NINTC(KCC)
KAD(KGG,5)
KCD(KGG)
NLSD(KGG)
CLEBF(KAA,5,5)

CLEBQ(KCC,5)

COEF(KGG)

Definition
Max. possible combinations of KA={KF=(t1 s1 `1 k`t ), λ`} in [Eq.(35)].
Max. possible combinations of KC={KB=(t1 s1 `1 λ1 `0p ), λ2 `} in [Eq.(33b)].
Max. possible combinations of {KC, λ} in [Eq.(33b)].
Max. possible number of {t1 s1 `1 k`t , `a `b } in [Eq.(36)].
Stores KB value for a given {t1 s1 `1 } set.
λ for a given KA set.
Stores KB value for a given KC set.
λ2 for a given KC set.
`0p for a given KC set.
` + 1 for a given KC set.
3t1 + s1 + 1 for a given KC set.
Used for choosing correct array of the effective potential.
Stores KA value for a given (KG,NLS) set. NLSMAX=5 is set.
Stores KC value for a given KG set.
Stores NLS value for a given KG set.
Calculated following geometrical factor in [Eq.(35)].
2m
ˆ`(`0λm|`
CLEBF(KA,M1,NLS) = 2`2π
1 m) 1+δm0 (−) .
1 +1Calculated in subroutine GFAC.
Calculated geometrical factor of partial projectile density in [Eq.(33b)].
2
0 m |λ m )ˆ
CLEBQ(KC,M2) = 2λ2π`(`0`p 2 2 2 1+δm2 0 .
2 +1
Calculated in subroutine GFAC.
Calculated following geometrical factor in [Eq.(33b)].
COEF(KG) = √14π (−)` λ̂1 λ̂2 (λ1 0λ2 0|λ0)W (λ1 λ2 `1 `; λ`0p )Calculated in subroutine GFAC.

Used subroutines : DCHECK2, FFCALE, TRECAL, GFAC, TENSOR, PACALE,FFCALEM, TRECALM, GFACM.
56

B.2.9.2

COMMON block CGFACB

Variable name
JLSPMXJLSCMXNLTD(NLS)KKD(KJL)LAD(KJL)LBD(KJL)LYD(KJL)
LTD(KJL)
NND(KJL)
CLEBD(KJL,10)

CCDD(KJL,10)

Definition
JLSMX for tensor part. Max. possible number of {t1 s1 `1 , k = 2, `t , `a `b }.
JLSMX for central part. Max. possible number of {t1 s1 `1 = `t , `a `b }.
Store NLT value for a given NLS. NLSMAX=5 is set.
Store KK=λ*NLSMAX+NLS for a given {t1 s1 `1 k`t , `a `b } set.
`a value for a given {t1 s1 `1 k`t , `a `b } set.
`a value for a given {t1 s1 `1 k`t , `a `b } set.
=0 if `1 = `t , otherwise 1, for a given {t1 s1 `1 k`t , `a `b } set.
`t value for a given {t1 s1 `1 k`t , `a `b } set.
Store LB*NOLTR+NLT for a given {t1 s1 `1 k`t , `a `b } set.
Calculated the following factor in [Eq.(18)] for a given {t1 s1 `1 k`t , `a `b } set.
CLEBD(KJL,M1) = i`a −`b +π `ˆa (`a 0`t m`t |`b m`t ).
Calculated in subroutine PACALE.
Calculated following geometrical factor in [Eq.(33b)].
CCDD(JLS,M1) = `ˆβ (`α m`β 0|`1 m)
Calculated in subroutine GFAC.

Used subroutines : FFCALE, GFAC, TENSOR, PACALE.
B.2.9.3

COMMON block TREDEN

Variable name
GWT(N1X,NIN,LM2)

TRHO(N1X,NIN,KBB)

Definition
The following part of nonlocal target density of [Eq.(31b)].
R
2π P
0
∗
0
0
ˆ0 0
m1 `h (`h 0λ1 m1 |`h m1 ) R`h (r1 )Y`h m1 (θ, 0)Yλ1 m1 (θ , 0)dµ .
2`h +1
Argument (N1X,NIN,LM2) = (r1 , r, {`0h , λ1 }).
Calculated in subroutine DENST.
The nonlocal target density, ρET,`1 λ1 `c (r1 , r), of [Eq.(31b)].
Argument (N1X,NIN,KBB) = (r1 , r, KB = {t1 s1 `1 , λ1 `c }).
Note that `c = `0p in the original paper.
Calculated in subroutine TRECAL.

Used subroutines : FFCALE, TRECAL, DENST.
B.2.10B.2.10.1

CGGR, AMPD, OVDE - Form factors and Transition amplitudes
COMMON block CGGR

Variable name
GGRI(KFM,N1X)
or
GGRIE(KFM,N1X)

Definition
Direct form factor, ftD1 s1 `1 k`t (ra ), of [Eq.(27)].
Argument (KFM,N1X) =({t1 s1 `1 k`t }, ra ).
No-recoil exchange form factor, ftN1 sR1 `1 k`t (rb ).
Argument (KFM,N1X) =({t1 s1 `1 k`t }, rb ).

Used subroutines : FFCALD, PACALD, FFCALE, PACALE.

57

B.2.10.2

COMMON block AMPL

Variable name
AMPD(700,8)

AMPE(700,8)

Definition
The direct overlap integral, OtD1 s1 `1 k`t ,`a `b of [Eq.(19a)] multiplied bya geometrical factor CLEBD(KJL,M1) calculated in PACALD.
Sums over {t1 s1 `1 k, `a } are done.
Argument (LBLT,M1) =({`b `t }, m`t ). LBLT=700, and M1=8 are set.
The same as AMPD, but for the no-recoil exchange form factors.

Used subroutines : PACALD, PACALE, TENSOR.
B.2.10.3

COMMON block OVDE

Variable nameOVDD(50,10,10)
OVED(50,10,10)
TH(50)DELWN(50)Definition
The direct transition amplitude, TtD1 s1 `1 k`t m` , of [Eq.(18)].
t
Argument (NT,LT,M1) =(θb , `t , m`t ). LT=10, and M1=10 are set.
The same as OVDD, but for the no-recoil exchange form factors.
Angle θb .
Momentum transfer corresponding the angle θb .

Used subroutines : PACALD, PACALE, CROSS.
B.2.11B.2.11.1

CPWFAC, CGFACC, CXLM - Plane wave approximation for exchange
form factor
COMMON block CPWFAC

Variable name
TFAC(3,10,10)

Definition
CG coefficient in the transition amplitude q
in [Eq.(12)].
TFAC(MT,MJT,LT)=(st mt jt mjt |`t m`t )/( 1+δ2m ,0 ) for a given (st = 1, jt )t

LRP1MX*FACNR*

Argument (MT,MJT,LT) =(mt , mjt , `t ).
Maximum angular momentum of `r + 1 in the PW approximation.
(By inputting LRP1MX=1, NR approximation can be obtained.)
Recoil factor α. 1.0 is suggested. (See Formulation 3.3.)

Used subroutines : DCP2, DCHECK2, FFCALDM, FFCALEM, GFACM.
B.2.11.2

COMMON block CGFACC

Variable name
KADD(KGG)
CCDDD(KJL)

Definition
Stores KA value for a given KG set for the modified version.
Calculated following geometrical factor for the modified version.
CCDDD(JLS) = iπ−`r −`b k̂ λ̂(k0λ0|`r 0)`ˆ1 `ˆr W (`b λ`t k : `1 `r )Calculated in subroutine GFACM.

Used subroutines : FFCALEM, GFACM.

58

B.2.11.3

COMMON block CXLM

Variable nameXLMRI(LTL,LM1,NXA)

Definition
The distortion factor, I`t m`t (ra ), defined as(4π)3/2 Pi`a −`b +π `ˆa (`a 0`t m` |`b m` )d` ` ` χ` (ra )χ` (ra ).
ka kb

LLP1MX

`a `b

t

t

a t b

b

a

Argument (LTL,LM1,NXA) =(`t , m`t , ra ).
LLP1MX=L1R(NLSMAX)+LAMMXD(1)+1.

Used subroutines : DCHECK2, FFCALDM, XLMCAL, FFCALEM, GFACM.

B.3

List of Key variables
Variable

Function

Current
dimension
NPS*N1XNHS*N1X
LXA*NXALXB*NXB(30,6)

Calculated
subroutine
BSAXON
BSAXONOPTOPTAFACALStoredcommon
CBST
CBST
DISW
DISW
SPECFC

Vti1 s1 k (r)Argument
notation(r1 × # of `p )(r1 × # of `h )(ra × # of `a )(rb × # of `b )(κ, `t )κ = {t1 s1 `1 k}(r, {t1 s1 k})

USAVP(NP*N1)
USAVH(NH*N1)
DISWA(LA*NA)DISWB(LB*NB)
ALPHA(NLSK,NT)R`p (r1 )R`h (r1 )χ`a (ra )χ`b (rb )αtj1t sst1ν`11 k`t

VV(NI,12)(NIN,12)DENTY(NA)RHOD(NX,NI,LM)ρP (r2 )0
ρDP,λ2 (r2 , r)r2
(r20 , r, λ2 )NXA
(NXA,NIN,LMM)

FFCC
I=1, Ex.
CDENS
CDENS

RHOT(NH,LM)

ρD
P,λ2 `t `1 (ra , r1 , r)(fixed ra r1 ; `t `1 )0
ρEP,λ2 (r2 , r)(r, λ2 )(NIN,LMM)

EFFINT(I)I=0, Dir.
DCP2
PDENST(0)0; Dir.
FFCALD(r20 , r, λ2 )(NXA,NIN,LMM)

CDENS(r, λ2 )(NIN,LMM)PDENST(1)1; Ex.
FFCALE(r1 , `1 )(r1 , r, β))
β = {`1 λ1 `c }(γ, ra )(N1X,5)(N1X,NIN,KBB)FFCALDTRECALDim.
TREDEN(KFM,NXA)FFCALD

CGGR(KFM,NXA)(KAA,NOU,L1X)

FFCALEFFCALE

CGGR
Dim.

(KJL)

FFCALE

TAPE(700,8)(700,8)(700,8)(50,10,10)

PACALD(1)PACALEPACALD(2)PACALD

AMPL
AMPL
AMPL
OVED

(50,10,10)

PACALEOVED

RHOE(NX,NI,LM)RHOT(NH,LM)
TRHO(N1,NL)
TRHO(N1,NI,KB)
GGRI(KF,NA)
GGRIE(KF,NA)
GGRI(KA,NH,NL)

ρE
P,λ2 `t `1 (rb , r1 , r)(fixed rb r1 ; `t `1 )ρDT,`1 (r1 )ρET,`1 λ1 `c (r1 , r)fγD (ra )γ = {t1 s1 `1 k`t }fγN R (ra )Gkt1 s1 `1 `b λ (rb , r)(fixed rb )FFRIT(JLS)E
fγ,`(rb , ra )b `a

AMPD(KF,M1)
AMPE(KF,M1)
AMPE(KF,M1)
OVDD(NT,LT,M1)

D
Oγ,`
a `b
E
Oγ,`
a `b
NR
Oγ,`
a `b
T`Dt m` (k̂b )(γ, ra )(α, r, η)
α = {`b λ}η = {t1 s1 `1 }{γ, `b `a }(fixed rb , ra )(γ, m`t )(γ, m`t )(γ, m`t )(θb , `t , m`t )OVED(NT,LT,M1)

T`Et m` (k̂b )(θb , `t , m`t )

t

t

59

Dim.

Dim.

B.4

Variables on the TAPE

Variable

Arg.

Function

WR unit

Tape #

FFRIT

(JLS)

FFRIT

(JLS2)

GGRIT

(JLS)

fγE0 ,`b `a (rb , ra )(fixed rb , ra )γ1 = {t1 s1 `1 ,fγE2 ,`b `a (rb , ra )(fixed rb , ra )γ2 = {t1 s1 `1 ,E + fE
fk=0
k=2(fixed rb , ra ){γ0 , `b `a }JLSCMCk = 0, `t }{γ2 , `b `a }JLSPMXk = 2, `t }{γ0 , `b `a }

60

13

Write
subroutine
FFCALE

Read
subroutine
TENSOR

remarks
Central Ex. ff

14

FFCALETENSOR

Tensor Ex. ff

15

TENSORPACALE

Total Ex. ff