野 地 俊 平東京大学大学院理学系研究科物理学専攻物理学教室原子核実験グループ

2008年 5月 20日於 CNSプレハブ 2階会議室

FOLD/DWHIを使う

13C(12N, 12C)13N反応と 13C(12C, 12B)13N反応とを例に

2008年 5月 21日一部改訂

References

J. Cook and J. A. Carr, computer code FOLD,Florida State University, (1988) unpublished.

Based on:F. Petrovich and D. Stanley,Microscopic interpretation of 7Li + 24Mg inelastic scattering at 34 MeV,NPA, 275, (1977), 487–508.

Modified described in:J. Cook, K. W. Kemper, et al.,16O(7Li, 7Be)16N reaction at 50 MeV,PRC 30, 1538–1544 (1984).

R. G. T. Zegers, et al.,The (t , 3He) and (3He, t) reactions as probes of Gamow-Teller strength,PRC 74, 024309 (2006).

References (cont.)

PHYSICAL REVIEW C 74, 024309 (2006)

The (t, 3He) and (3He, t) reactions as probes of Gamow-Teller strength

R. G. T. Zegers,1,2,3 H. Akimune,4 Sam M. Austin,1,3 D. Bazin,1 A. M. van den Berg,5 G. P. A. Berg,6,7

B. A. Brown,1,2,3 J. Brown,8 A. L. Cole,1,3 I. Daito,9 Y. Fujita,10 M. Fujiwara,11,12 S. Gales,13 M. N. Harakeh,5H. Hashimoto,12 R. Hayami,14 G. W. Hitt,1,2 M. E. Howard,3,15 M. Itoh,16 J. Janecke,17

T. Kawabata,18 K. Kawase,12 M. Kinoshita,4 T. Nakamura,19 K. Nakanishi,12 S. Nakayama,14 S. Okumura,12

W. A. Richter,20 D. A. Roberts,17 B. M. Sherrill,1,2,3 Y. Shimbara,1,3 M. Steiner,1 M. Uchida,21 H. Ueno,22

T. Yamagata,19 and M. Yosoi12

計算の概要

遷移行列要素

Tβα = χ(−)β —F

(τ)βα —χ

(+)α

有効相互作用V (τ) = (Vτ + Vστσa · σA + VTτS12)τa · τA遷移密度

ρLSJ =

np

b — —[a†nap] — —a[φ∗pφn]

断面積dσ

dΩ=

µ

2π2

2 kf

ki

—Tαβ —

2

F(τ)βα (R) =

dξadξA ρab(ξa)V

(τ)(R, ξa, ξA)ρAB(ξA)

形状因子 (2重畳み込みモデル)

1粒子波動関数φp, φn b — —[a

†nap] — —a

1体遷移密度 (OBTD)

光学ポテンシャル

OXBASH, Nushell, ckz,...

Feaney-Love,Love-Franey,...

wsaw

fold

dwhi

各々のコードについて

(1) wsaw.for: 1粒子波動関数の計算

• Woods-Saxonポテンシャル中での動径方向の波動関数を計算• 1粒子の束縛エネルギーを入力する

(2) fold.for: 遷移密度, double-foldingポテンシャルの計算

• 1粒子波動関数 (wsaw.forの出力)を入力する• Franey-Loveなどの残留相互作用を遷移密度を用いて double-foldする• Projectileの運動エネルギーを入力する• Projectile系, target系のスピン,アイソスピンと, Z 係数を入力する• 可能な全角運動量移行の組合せ (後述)を全て入力する

(3) dwhi.for: DWBA計算 (DWUCKに基づく)

• Projectileの運動エネルギー,反応のQ値を入力する• 入口と出口の光学ポテンシャルを入力する• 計算に用いる部分波の個数を指定する• 可能な全角運動量移行の組合せを全て入力する

角運動量移行について

Jr L

.

,は軌道角運動量移行

相対座標空間における全角運動量移行 Jrは,projectile, targetについての角運動量移行 Jp, Jtを用いて

Jr = Jp + Jt

と書ける

中心力とテンソル力のみを考慮するときにはに等しくなる:

Jr = L.

cf.)

L = Jr, S = Jp, J = Jt

Anantaraman, et al., PRC 44, 398–414 (1991)p. 405, Sec. IV, 2段落目

例題

Consider the reactions studied here with

Jπ(12Ngs) = 1+, Jπ(12Bgs) = 1+, Jπ(12Cgs) = 0+, Jπ(12Cexc) = 2+,Jπ(13Cgs) = 1/2−, Jπ(13N) = 1/2− or 3/2−,

where 12Cexc stands for the first 2+ state at 4.4 MeV.

15.06

11.7410.839.488.92

3.50

0

4.44

0

Ex (MeV)3/2−

1/2−3/2−

3/2−1/2−

3/2−

1/2−

J π

1/2−0+

1+

1+

2+

125B

126C

127N

137N

136C

Qβ−13.37

QEC17.34

例題の続き

The reactions studied here allow the following combinations (Jr, Jp, Jt):

(1) in the projectile-ejectile system,

(i) for 12N → 12Cgs (1+ → 0+) and 12C → 12Bgs (0+ → 1+) transitions,Jp can take on the value Jp = 1,

(ii) for 12N → 12Cexc (1+ → 2+) transition,Jp can take on the value Jp = 1, 2, 3,

(2) in the target-residual system,

(i) for 13C → 13N (1/2− → 1/2−) transition,Jt can take on the value Jp = 0, 1,

(ii) for 13C → 13N (1/2− → 3/2−) transition,Jt can take on the value Jp = 1, 2.

可能な角運動量移行の組み合わせ

(Jπt )i → (Jπ

t )f (Jπp )i → (Jπ

p )f Jr Jp Jt

1/2− → 1/2−

1+ → 0+

0+ → 1+

0 1 1

2 1 1

1+ → 2+

0 1 1

2 1 1

2 2 0

2 2 1

2 3 1

4 3 1

1/2− → 3/2−

1+ → 0+

0+ → 1+

0 1 1

2 1 1

2 1 2

1+ → 2+

0 1 1

2 1 1

2 1 2

2 2 1

0 2 2

2 2 2

4 2 2

2 3 1

4 3 1

2 3 2

4 3 2

表の見方： 例えばある反応について

(Jπt )i → (Jπ

t )f (Jπp )i → (Jπ

p )f Jr Jp Jt

1/2− → 3/2− 0+ → 1+

0 1 1

2 1 1

2 1 2

13C ( 12C , 12B ) 13N(3.50 MeV)1/2− 0+ 1+ 3/2−

13C(12C, 12B)13N反応において、13Nの終状態がEx = 3.50 MeVの状態の場合

反応に関与する粒子の Jπは

ゆえ,表の以下の部分を見る

doc_wsaw.lis Documentation for August 1989 version of WSAW at MSU NSCL ---------------------------------------------------------

This program produces a file containing Woods-Saxon radial functions where each is prefaced by a header that can be interpreted by FOLD. Based on a program written by Petrovich in the good old days.

----------------------------------------------------------------------

Input structure:

Most input formats are I5 and F10

1) RMESH=0.1, RMAX=20., NPUNCH=1, NBPUNCH=150, IDBUG=0

2) FILENAM

3) TMC, TIZC, V0, A, R0, RC, VSO

TMC = core mass

TIZC = core charge V0 = starting value for volume potential depth (will be fit) A = diffuseness for this potential

R0 = radius parameter for this potential RC = coulomb radius parameter

VSO = spin-orbit potential strength to us

4) EBIND, TMP, TL, TNODE, TIZP, XJ, XS

EBIND = binding energy of particle

TMP = mass of particle

TL = orbital angular momentum (L) of particle orbit

TNODE = number of interior nodes (starts at zero)

TIZP = charge of particle

XJ = total angular momentum (J) of particle orbit XS = spin (S) of particle

arbitrary number of card 3+4 sets terminated by TMC = -1

----------------------------------------------------------------------

The following is from a standard reference run for C-12(C-12,N-12)B-12 at 840 MeV using Woods-Saxon radial wavefunctions. SET VERIFY echoes the VMS DCL commands and comments but not the input to the code. The SET DEF is required to put the result file in the place FOLD expects to find it for a later run. Note that a second run is needed to get C12N12 file that is also required; it uses 18.722 for neutron energy and 0.601 for proton energy instead of the 15.957 proton and 3.370 neutron energy used below.

$ SET VERIFY=(PROC,NOIMAGE)$!$! obtain the Woods-Saxon w-fcns for (C-12,B-12) and (C-12,N-12)$!$ SET DEF [JAC.WORK]$!$ RUN [JAC.BIN]WSAW0.1 20. 1 150 0C12B1211. 5. 60. .65 1.25 1.25 7.0 15.957 1. 1. 0. 1. 0.5 .5 11. 5. 60. .65 1.25 1.25 7.0 15.957 1. 1. 0. 1. 1.5 .5 11. 5. 60. .65 1.25 1.25 7.0 3.370 1. 1. 0. 0. 0.5 .5 11. 5. 60. .65 1.25 1.25 7.0 3.370 1. 1. 0. 0. 1.5 .5 -1. $!$! \eof1$!

----------------------------------------------------------------------

doc_fold.lis Documentation for August 1989 version of FOLD at MSU NSCL --------------------------------------------------------- This program produces direct and exchange double-folded potentials based on the Petrovich, Philpott, Carpenter and Carr formalism with central and tensor forces only. [see NP A425 (1984) 609] The original model for the program is the Petrovich and Stanley folding code which was also a model for ALLWRLD, but they have diverged in general structure and detail. This version handles complex t-matrices and the simple Golin Fermi-motion correction to SNKE in the AEA. The option to use EXCHNG to actually execute this correction exactly via Moshinsky transforms and the like is untested and has not been converted to use the complex t-matrices; it should not be used. May 2005 - updates by R.G.T. Zegers NSCL The correction factors for the strengths were taken out (still listed in output but not used in calculation. The Z coefficients should be input like in DW81, i.e. the OBTD's from OXBASH multiplied by the common scale factors The SKNE approximation was updated: the kA parameter (see Love and Franey 1985) is input as FNRM2. The difference is large for light target nuclei. FNRM2 is a common scale factor for the transformation of tNN to tNA. At 420 MeV the correction is about 10% in cross section. ----------------------------------------------------------------------

Input structure:

Most everything is read in I5 or F10 formats.

1) KEXCHG, KPUNCH=1, FILNAM(char*8)

KEXCHG =0(direct), =1(ZREA for C), =2(EXCHNG -- do not use!)

2) NR, H, ELAB, APROJ, IPRTR, IPRTQ, IPRTF

3a) FJF, PARF, FJI, PARI --- for the PROJECTILE ----

3b) TF, TFM, TI, TIM

3c) NTYPF, KOPTN, ALPHA

NTYPF =1(static), =2(inelastic), =3(charge exchange)

KOPTN =1(S[T]), =2(S[pn]), =3(Z[T]), =4(Z[pn]) =5(Wildenthal trans amp)

3d) IDF, IDI, JX, Z1, Z2 (terminated by -1,-1)

the Z's are the Raynal/ALLWRLD definition for MT=0 conversion to actual MT can be wrong for special cases IDI corresponds to OXBASH's "created", IDF to "destroyed"

3e) FILEN (read and used if ALPHA=0)

4a) FJF, PARF, FJI, PARI --- for the TARGET ----

4b) TF, TFM, TI, TIM

4c) NTYPF, KOPTN, ALPHA

4d) IDF, IDI, JX, Z1, Z2 (terminated by -1,-1)

4e) FILEN (read and used if ALPHA=0)

5) FNRM1, FNRM2, FNRM3, FRCEFILE(char*48)

6) NFORM (number of SETS of cards 7 and 8)

7) JR, JP, JT, KFORCE KFORCE = -1 for Central + Tensor

0 for Central only 1 for LS (spin-orbit) 2 for Tensor only

8) DNORM(1), ..., DNORM(7) --- card for each T, proj and targ

these scale rho-m; rho-s for L=J-1,J,J+1; rho-l for L=J-1,J,J+1

---------------------------------------------------------------------- The following is from a standard reference run for MG-26(C-12,N-12)AL-26 at 840 MeV in the laboratory using a corrected M3Y interaction. The Woods-Saxon radial wavefunction files must be in the directory pointed to by the SET DEF command. The radial functions are produced by WSAW. The force files are in the directory NSCL_LIBRARY:[DWBA.SCRI.FORCEFILES] which should be logically assigned to FOLD_LIBRARY: The SET VERIFY forces echo of the VMS DCL commands and comments but not of the input data read by the program.

$ SET VERIFY=(PROC,NOIMAGE)$!$! run the following MG26(C12,B12) test case$!$! this is the standard reference case with corrected M3Y$!$ SET DEF [WINFIELD.CHEX.WORK]$ ASSIGN NSCL_LIBRARY:[DWBA.SCRI.FORCEFILES] FOLD_LIBRARY:$!$ RUN NSCL_LIBRARY:[DWBA.SCRI.BIN]FOLD 1 1FOLDMG26 600 0.03 840. 12. 1 1 0 1.0+ 0.0+ 1.0 +1.0 0.0 0.0 3 3 0.000 2 2 1 0.0 0.05797 3 2 1 0.0 0.33968 2 3 1 0.0 0.69032 3 3 1 0.0 0.07651 -1 -1C12B12 1.0+ 0.0+ 0.0 -0.0 1.0 1.0 3 3 0.000 6 6 1 0.0 0.44326 6 5 1 0.0 0.07954 4 4 1 0.0 0.10097 4 5 1 0.0 0.04468 5 6 1 0.0 0.01643 5 4 1 0.0 -0.03803 5 5 1 0.0 -0.02952 -1 -1MG26AL26 0.922 0.829 0.340 M3Y_REEO_COR. 2 0 1 1 -1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2 1 1 -1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00$!$! \eof1$! ----------------------------------------------------------------------

doc_fold_addtl.lisSome additional notes on the MSU NSCL version of FOLD-----------------------------------------------------

Explanation of those input parameters which are not explained in DOC_FOLD.LIS and are obvious:

Card 2) NR : number of integration stepsH : step size (fm)ELAB : bombarding energy in MeV These two parameters only usedAPROJ : projectile mass for SNKE wavenumber determinationIPRTR : set to 1 for printout of r-space densitiesIPRTQ : set to 1 for printout of q-space densitiesIPRTF : set to 1 for printout of formfactors

Card 3a) For the projectile...FJF : spin of final particlePARF : parity of final particle (+, -)FJI : spin of initial particlePARI : parity of initial particle (+, -)

Card 3b) Projectile (contd.)TF : Isospin of the final particleTFM : Isospin projection of the final particle (+ve = neutron excess)TI : Isospin of the initial particleTIM : Isospin projection of the initial particle

Card 3c) Projectile (contd.)NTYPF : see DOC_FOLDKOPTN : see DOC_FOLD. The Z-coeff definition used by FOLD is

dJ,DT CG(Ti Tim dT dTm | Tf Tfm) HAT(dT)Z = ---------------------------------- * BHW j,j' HAT(Ji) * HAT(Tf)

where,HAT(j) = SQRT (2j + 1)

and BHW is the Wildenthal transition amplitude (as outputby OXBASH, for example):

+ ~ dJ,dT < Wf ||| (aj x aj') ||| Wi>BHW = -------------------------------- HAT(dJ) * HAT (dT)

Card 3d) Projectile (contd.)IDF : sub-orbital ID number for the final (destroyed) particle or holeIDI : sub-orbital ID number for the initial (created) particle or hole

1 = 0s1/2 16 = 2p1/2 31 = 2f5/2 2 = 0p1/2 17 = 2p3/2 32 = 2f7/2 3 = 0p3/2 18 = 1f5/2 33 = 1h9/2 4 = 1s1/2 19 = 1f7/2 34 = 1h11/2 5 = 0d3/2 20 = 0h9/2 35 = 0j13/2 6 = 0d5/2 21 = 0h11/2 36 = 0j15/2 7 = 1p1/2 22 = 3s1/2 37 = 4s1/2 8 = 1p3/2 23 = 2d3/2 38 = 3d3/2 9 = 0f5/2 24 = 2d5/2 39 = 3d5/2 10 = 0f7/2 25 = 1g7/2 40 = 2g7/2 11 = 2s1/2 26 = 1g9/2 41 = 2g9/2 12 = 1d3/2 27 = 0i11/2 42 = 1i11/2 13 = 1d5/2 28 = 0i13/2 43 = 1i13/2

14 = 0g7/2 29 = 3p1/2 44 = 0k15/2 15 = 0g9/2 30 = 3p3/2 45 = 0k17/2

Cards 4a-d) As per cards 3a-d, but for target system.

OLD CARD 5----------Card 5) FNRM1 : Normalization of SNKE Yukawa of 1st range given in force file

FNRM2 : " " " " " " 2 " " " "FNRM2 : " " " " " " 3 " " " "FRCEFILE : name of file containing n-n force parametersFor a list of available force files, see [DWBA.SCRI.FORCEFILES]

NEW CARD 5----------R.G.T. Zegers Card 5) has been replaced: FNRM1 : normalization of SNKE Yukawas (all ranges) for transformation from tNN to tNA FNRM2 : kA (momentum of projectile in NA frame) instead of lab momentum Especially important for very light (A<10) target nuclei. FNRM3: not used

Card 7) JR : "relative" spin transfer (known in DWHI as LTRT)JP : spin transfer in the projectile system (2*JP = ISTRT in DWHI)

JT : spin transfer in the target system (2*JT = JTRT in DWHI)

doc_dwhi.lis Documentation for August 1989 version of DWHI at MSU NSCL --------------------------------------------------------- This program performs the DWBA calculation using the double-folded scattering potentials produced by FOLD. It is patterned after DWUCK with the addition of better coulomb routines for the heavy ion case by Julian Cook. It produces plot files suitable for PLOTIT.

----------------------------------------------------------------------

Input structure:

Most everything is read in I3 and F7 formats, see example below. Definitions generally follow the DWUCK code this was originally based upon, with a few variations. Check code when in doubt.

1) ICON(1), ..., ICON(20), TITLE

ICON(1)=1 to use the kind of ANGLE input shown for card 2

ICON(2)=2 to use a microscopic form factor

ICON(3)=1 to use incoherent sum of the different LTR cases ICON(4)=0 to print form factor used, non-zero to suppress this

ICON(5)=2 to print complex T-L, complex S-L, and S-L magnitude =1 to print no elastic scattering info =0 to print complex T-L only

ICON(9)=4 to always produce a 4-cycle semi-log graph

ICON(10)=0 to use non-relativistic kinematics

1a) FILNAM(char*8) -- read if ICON(2) is not 1, contains form factor

2) ANGLE(1), ANGLE(2), ANGLE(3)

ANGLE(1) = number of angles ANGLE(2) = initial angle ANGLE(3) = angle step size

3) L, NFF, ISA, ISB, JA, JB

L = number of partial waves for elastic

NFF = number of form factors to expect -- must match number produced by FOLD; must also be used in correct order.

ISA,ISB = 2 * projectile spin in initial (A) and final (B) channel

JA,JB = 2 * target spin in initial (A) and final (B) channel

4) DR, NNR -- must match corresponding card in FOLD

5a) E, FM, Z, FMA, ZA, RY, FS, QCD --- incoming channel

E = lab energy

FM,Z = projectile mass and Z

FMA,ZA = target mass and Z

RY = coulomb radius (multiplies TARGET mass to 1/3)

FS = 0

QCD = 0

5b) FZ, VR, RY, AR, VSOR, VI, RZ, AI, VSOI, PWR (until FZ=0)

FZ = potential option (1=WS, 2=surface WS, 3=second derivative)

VR,RY,AR,VSOR = real volume, radius, diffuseness, and LS values

VI,RZ,AI,VSOI = imaginary volume, radius, diffuseness, and LS

PWR = 0 for cases we use

6a) E, FM, Z, FMA, ZA, RY, FS, QCD --- outgoing channel

E = Q-value here

6b) FZ, VR, RY, AR, VSOR, VI, RZ, AI, VSOI, PWR (until FZ=0)

7) LTRT, ISTRT, JTRT (there are NFF pairs of 7 and 8)

LTRT = JR, ISTRT = 2*JP, JTRT = 2*JT

8) BETAR=0, BETAI=0, BETAC=0, FNORM=1

9) PLOTFILE(char*16)

----------------------------------------------------------------------

The following is from a standard reference run for MG-26(C-12,N-12)AL-26 at 840 MeV in the laboratory using the Roussel-Chomaz optical potential and the potentials produced by the standard case in the documentation for FOLD. The SET VERIFY and SET DEF are redundant if this input follows a FOLD run in a single batch file, as is normally done.

$ SET VERIFY=(PROC,NOIMAGE)$!$! this is the standard reference case with corrected M3Y$!$ SET DEF [WINFIELD.CHEX.WORK]$!$ RUN NSCL_LIBRARY:[DWBA.SCRI.BIN]DWHI1210000040000000 MG26(C,B) STATE 1 ROUSSEL-CHOMAZ (O+SI) POTLFOLDMG26 61. 0. 0.2160 2 0 2 0 2 0.03 600 840. 12. 6. 26. 12. 2.300 0. 0. 1. -100. 1.581 0.905 0. -50.5 1.759 0.78 0. 0. 0. -18.43 12. 5. 26. 13. 2.300 2. 0. 1. -100. 1.581 0.905 0. -50.5 1.759 0.78 0. 0. 0. 0 2 2 0. 0. 0. 1. 2 2 2 0. 0. 0. 1.MG26S1_STD.PLOT $!$! \eof2$! ----------------------------------------------------------------------

実践

• WSAW

– 入射核–出射核系: wsaw12c12b.inp– 標的核–残留核系: wsaw13c13n.inp

• FOLD

– 異なる (Jp,Jt)の組み合わせ毎に別のインプットファイルを用意∗ fold12c13cjp1jt1.inp

∗ fold12c13cjp1jt2.inp

– OBTD (Z係数)を別に計算 (上のインプットファイルに書き入れる)∗ OXBASH, NUSHELL, CKZ,...

– 有効相互作用のファイル (FRCEFILE)• DWHI

– 異なる (Jp,Jt)の組み合わせ毎に別のインプットファイルを用意∗ dwhi12c13cjp1jt1.inp

∗ dwhi12c13cjp1jt2.inp

– 光学ポテンシャルパラメター

wsaw12c12b.inp

0.1 20. 1 150 0C12B1211. 5. 60. .65 1.25 1.25 7.0 15.9570 1. 1. 0. 1. 0.5 .5 11. 5. 60. .65 1.25 1.25 7.0 15.9570 1. 1. 0. 1. 1.5 .5 11. 5. 60. .65 1.25 1.25 7.0 3.3704 1. 1. 0. 0. 0.5 .5 11. 5. 60. .65 1.25 1.25 7.0 3.3704 1. 1. 0. 0. 1.5 .5 -1.

wsaw13c13n.inp

0.1 20. 1 150 0C13N1312. 6. 60. .65 1.25 1.25 7.0 4.946310 1. 1. 0. 0. 0.5 .5 12. 6. 60. .65 1.25 1.25 7.0 4.964310 1. 1. 0. 0. 1.5 .5 12. 6. 60. .65 1.25 1.25 7.0 1.9435 1. 1. 0. 1. 0.5 .5 12. 6. 60. .65 1.25 1.25 7.0 1.9435 1. 1. 0. 1. 1.5 .5 -1.

Separation Energies

21 5 B

Q −β 9.86331

1+ sm 02.02

−β

4.0733Sn

49041Sp

21 6 C

.00+

0.75951Sp

5.12781Sn

21 7 N

Q CE 1.83371

1+ sm 000.11

CE

0.106Sp

09651Sn

31 6 C

2/1 –

013.6494Sn

9.23571Sp

31 7 N

Q CE 4.0222

2/1 – m 569.9

CE

5.3491Sp

0.46002Sn

system Sp,n (MeV)V0 (MeV)

0p1/2 0p3/2

12N → 12Cp 0.601 39.91 37.87n 18.722 68.42 65.40

12C → 12Bp 15.957 68.64 65.59n 3.370 39.68 37.68

13C → 13Np 4.946 41.09 39.20n 1.944 40.96 39.01

fold12c13cjp1jt1.inp

1 1CC11 600 0.03 1200. 12. 1 1 1 1.0+ 0.0+ 1.0 +1.0 0.0 +0.0 3 3 0.000 2 2 1 0.0 -0.058116 2 3 1 0.0 -0.690156 3 2 1 0 0 -0.339352 3 3 1 0.0 -0.076357 -1 -1C12B12 1.5- 0.5- 0.5 -0.5 0.5 0.5 3 3 0.000 2 2 1 0.0 -0.029368 2 3 1 0.0 -0.467747 3 2 1 0.0 0.096996 3 3 1 0.0 -0.100972 -1 -1C13N13 0.965 2.08 1.000 franey_100 2 0 1 1 -1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2 1 1 -1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

fold12c13cjp1jt2.inp

1 1CC12 600 0.03 1200. 12. 1 1 1 1.0+ 0.0+ 1.0 +1.0 0.0 +0.0 3 3 0.000 2 2 1 0.0 -0.058116 2 3 1 0.0 -0.690156 3 2 1 0 0 -0.339352 3 3 1 0.0 -0.076357 -1 -1C12B12 1.5- 0.5- 0.5 -0.5 0.5 0.5 3 3 0.000 2 3 2 0.0 0.097698 3 2 2 0.0 0.233858 3 3 2 0.0 0.076253 -1 -1C13N13 0.965 2.080 1.000 franey_100 1 2 1 2 -1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

One Body Transition Densities

FOLDに入力する Z 係数の定義:

Z ∆J,∆Tjp,jh =

2∆T + 1

(2Ji + 1)(2Tf + 1)TiT3i∆T∆T3|Tf T3f

×Jf Tf |||[a†jp × ajh ]∆J,∆T |||JiTi

√2∆T + 1

√2∆J + 1

.

あるいは同じくZJ(jp, jh) = (Ji J)−1

f(P†jp ⊗ Njh)J

i

,

と書ける.

Taddeucci, et al.,(p, n) reactions on 14C and 14N and the effective nucleon-nucleon interaction,PRC 29, 764 (1984)

武藤氏 (東工大)の CKZを用いて計算• Model space: p shell• Interaction: Cohen-Kurath (8-16)POT, Ref.: Nucl. Phys. 73 (1965) 1.

One Body Transition Densities (cont.)

(i) 12N → 12Cgs

∆T ∆J initial final g.s.1 1 0p3/2 0p3/2 −0.0440841 1 0p3/2 0p1/2 0.1959251 1 0p1/2 0p3/2 0.3984621 1 0p1/2 0p1/2 −0.033553

(ii) 12C → 12Bgs

∆T ∆J initial final g.s.1 1 0p3/2 0p3/2 −0.0763571 1 0p3/2 0p1/2 −0.6901561 1 0p1/2 0p3/2 −0.3393521 1 0p1/2 0p1/2 −0.058116

(iii) 12N → 12Cexc

∆T ∆J initial final 4653 MeV1 1 0p3/2 0p3/2 0.0869261 1 0p3/2 0p1/2 −0.1627201 1 0p1/2 0p3/2 −0.0271721 1 0p1/2 0p1/2 −0.4179851 2 0p3/2 0p3/2 −0.1773961 2 0p3/2 0p1/2 −0.1718051 2 0p1/2 0p3/2 0.0745911 3 0p3/2 0p3/2 0306295

Projectile-ejectile system Target-residual system

(i) 13C → 13N[Jπ = 1/2−, T = 1/2]

∆T ∆J initial final g.s. 8.780 MeV 13.806 MeV1 0 0p3/2 0p3/2 0.173925 0.044096 0.2414521 0 0p1/2 0p1/2 0.461140 −0.062361 −0.3414651 1 0p3/2 0p3/2 −0.071319 −0.059372 −0.1189131 1 0p3/2 0p1/2 0.016346 −0.447696 0.1776301 1 0p1/2 0p3/2 −0.016346 −0.161756 0.0282951 1 0p1/2 0p1/2 0.562934 −0.055857 −0.212706

(ii) 13C → 13N[Jπ = 3/2−, T = 1/2]

∆T ∆J initial final 3.587 MeV 10.432 MeV 14.007 MeV1 1 0p3/2 0p3/2 −0.100972 −0.023650 −0.1334991 1 0p3/2 0p1/2 −0.467747 0.388477 0.0535881 1 0p1/2 0p3/2 0.096996 0.125681 −0.0690991 1 0p1/2 0p1/2 −0.029368 −0.000253 −0.0353321 2 0p3/2 0p3/2 0.076253 0.054859 0.2276971 2 0p3/2 0p1/2 0.097698 0.384981 −0.1186501 2 0p1/2 0p3/2 0.233858 −0.043158 −0.071197

(iii) 13C → 13N[Jπ = 3/2−, T = 3/2]

∆T ∆J initial final 14.793 MeV1 1 0p3/2 0p3/2 −0.0023101 1 0p3/2 0p1/2 0.3562431 1 0p1/2 0p3/2 0.1090181 1 0p1/2 0p1/2 0.0037821 2 0p3/2 0p3/2 0.0009421 2 0p3/2 0p1/2 −0.3551201 2 0p1/2 0p3/2 0.029328

Franey-Love t-matrix interaction strength at 100 MeV(Franey and Love, PRC 31, 488 (1985)) のファイル

franey_100

3 2 4 .25 10598.4 8894.17 -18226.9 7698.88 .40 -3212.64 -2796.83 2536.74 -1609.06 1.4 -10.50 -10.50 31.50 3.50 .25 -11401.5 -1685.70 .40 -597.390 -614.658 .25 68965.0 903.713 .40 -10217.4 80.9495 .55 1643.27 -3.15586 .70 -232.416 21.4986 2 2 4 .25 645.678 11318.7 7345.67 2418.39 .40 -343.385 -3624.79 -1408.26 -683.690 .25 -19562.0 -1543.04 .40 2187.28 241.376 .25 17293.4 -8431.41 .40 -2559.65 1124.87 .55 349.963 -155.646 .70 -30.3573 14.1551

変換係数

Franey-Loveの相互作用を用いる際には, N-A系での運動量 kA

k2A = m2Aβ

1 + α

1 + β

,

α ≡ Ep

2m , β ≡ 4αA(A + 1)2

と tNN から tNAへの変換のSNKE Yukawasの規格化定数

tNA = 20

pttNN ,

2

0 = m2(1 + α), 2p = m2 + k2

A, 2t = m2 + (kA/A)2

とを入力する. Epは入射粒子の核子当りの運動エネルギー.

この測定の場合にはA = 13, Ep = 100 MeV

で,kA = 2.08, 2

0pt

= 0.965 .

Love and Franey, PRC 24 1073 (1981) の (19a–c) 式を見よ.

dwhi12c13cjp1jt1.inp

1210000041000000 13C(12C,12B)13NCC11 161. 0. 0.05160 2 0 2 1 3 0.03 600 1200. 12. 7. 13. 6. 1.30 2. 0. 1. -120. 0.71 0.840 0. -34.0 0.960 0.620 0. 0. 0. -19.09 12. 6. 13. 7. 1.30 0. 0. 1. -120. 0.71 0.840 0. -34.0 0.960 0.620 0. 0. 0. 0 2 2 0. 0. 0. 1. 2 2 2 0. 0. 0. 1.12c13cjp1jt1.plot

dwhi12c13cjp1jt2.inp

1210000041000000 13C(12C,12B)13NCC12 161. 0. 0.05160 1 0 2 1 3 0.03 600 1200. 12. 7. 13. 6. 1.30 2. 0. 1. -120. 0.71 0.840 0. -34.0 0.960 0.620 0. 0. 0. -19.09 12. 6. 13. 7. 1.30 0. 0. 1. -120. 0.71 0.840 0. -34.0 0.960 0.620 0. 0. 0. 2 2 4 0. 0. 0. 1.12c13cjp1jt2.plot

光学ポテンシャル

Anantaraman, et al. PRC 44, 398 (1991) と同じものを用いた.

(12C, 12B) and (12C, 12N) reactions at E/A = 70 MeV

Ref. 20: Buenerd, et al., NPA 424, 313 (1984)

実行例

$ cat run.sh#!/bin/bash./wsaw < wsaw12c12b.inp > wsaw12c12b.out./wsaw < wsaw13c13n.inp > wsaw13c13n.out./folding < fold12c13cjp1jt1.inp > fold12c13cjp1jt1.out./folding < fold12c13cjp1jt2.inp > fold12c13cjp1jt2.out./dwhih < dwhi12c13cjp1jt1.inp > dwhi12c13cjp1jt1.out./dwhih < dwhi12c13cjp1jt2.inp > dwhi12c13cjp1jt2.out$ sh run.sh

結果： 断面積の角度分布

13C(12C, 12B)13NEx = 3.50 MeV

Jπ = 3/2−dσ

/dΩ

(mb/

sr)

θcm (deg)

101

100

10−1

10−2

10−30 1 2 3 4 5

011

211

212

• 破線には(JrJpJt)の 3つ組でラベルしてある

• 実線の微分断面積は3本の破線 (011, 211, 212)の微分断面積の incoherent sum(絶対値の和).

ほかの遷移も同様にして計算した