アトミックGS 12

FOLD / DWHI
を使う
巨大共鳴の計算
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
2009 年 4 月 16 日
於 本郷理学部 1 号館 314 号室
この資料には
1. fold/dwhi 計算のための準備
(a) Z 係数 (OBTD) の計算
(1) ckz
(2) oxbash
(3) normod
(b) oxbash-dens: 1 粒子軌道エネルギーの計算
2. fold/dwhi 計算
(a) wasw: 1 粒子殻模型波動函数の動径方向成分の計算
(b) fold: 遷移密度と形状因子の計算
(c) dwhi: 微分散乱断面積の計算
をこの順に書いた
Z 係数の計算 (1)
p-shell 模型空間, Cohen-Kurath 相互作用
(Nucl. Phys. 73 (1965) 1)
例) •
•
•
•
実行例
N(1+ , gs) → 12 C(0+ , gs)
12
B(1+ , gs) → 12 C(0+ , gs)
10
C(0+ , gs) → 10 B(0+ , IAS)
10
Be(0+ , gs) → 10 B(0+ , IAS)
12
$ g77 ckz.for -o ckz
$ ./ckz
12
7 5
T =
1
J =
1
1
EX = 0.000
⇒ CKZ (K. Muto (TITech)) を用いるのが簡便
• 始状態の (A, Z , T , J, #) を指定
• 終状態の Z を指定
• Cohen-Kurath 相互作用を指定
((6-16)BME, (8-16)POT, (8-16)BME)
指定した始状態から全ての可能な終状態への
Z 係数が計算される
PROGRAM MAIN
IMPLICIT REAL*8 (A-H,O-Z)
CALL ZCOUT(12,7,3,3,1,6,2) !
STOP
END
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
:
:
:
:
:
12
MASS NUMBER OF THE INITIAL AND FINAL STATES
ATOMIC NUMBER OF THE INITIAL NUCLEUS
2T+1 OF THE INITIAL STATE
2J+1 OF THE INITIAL STATE
ASCENDING SERIAL NUMBER OF THE INITIAL
STATE AMONG STATES WITH THE SAME A, T AND J
NZ2 : ATOMIC NUMBER OF THE FINAL NUCLEUS
IPOT : PARAMETER WHICH SPECIFIES ONE OF THE
THREE COHEN-KURATH INTERACTIONS
IPOT = 1 : (6-16)2BME
IPOT = 2 : (8-16)POT
IPOT = 3 : (8-16)2BME
以下省略
C
6 6
0
0
2)
13.458
COHEN-KURATH
(8-16)POT INTERACTION
3)
22.763
1
1
1
1
3
3
1
1
3
1
3
1
4)
28.907
5)
29.583
-0.024826 -0.076292
-0.060119 0.030587
-0.064947 -0.034098
-0.001696 -0.052878
N(gs) → 12 C(gs)
N
7 5
T =
1
1
EX = 0.000
1
1
1
1
12
J 2J->2J'
1 3
3 -0.044084 -0.003437 -0.042137
1 3
1
0.195925 0.172568 0.145348
1 1
3
0.398462 -0.301410 0.040239
1 1
1 -0.033553 -0.008460 -0.065921
N(gs) → 12 C を (8-16)POT を用いて計算
SUBROUTINE ZCOUT(NA,NZ1,ITZ1,JJZ1,NOZ1,NZ2,IPOT)
THIS SUBPROGRAM CALCUALATES AND PRINTS
Z-COEFFICIENTS OF THE SPECIFIED TRANSITIONS.
INPUT;
NA
NZ1
ITZ1
JJZ1
NOZ1
T
1
1
1
1
-->
1)
0.000
12
12
N
-->
12
C
6 6
1
1)
2)
19.591
35.911
-0.093603 0.049688
0.011794 -0.036449
-0.057377 -0.010339
0.024495 0.028870
COHEN-KURATH
(8-16)POT INTERACTION
以下省略
• サブルーチン zcout の呼び出し部分を書き換え,
その度ごとにコンパイルして実行
• GNU Fortran Compiler (g77),
Intel Fortran Compiler (v11) で動いた
• 計算結果は標準出力に
• 出力された Z 係数はそのまま DWBA コードに入力できる
Z 係数の計算 (2)
OXBASH を使う
実行例
12
Linux 版:
ftp://ftp.nscl.msu.edu/以下略
Windows 版:
http://www.nscl.msu.edu/以下略
Oxbash for Windows PC を Linux で使う
Wine (Linux における Windows の互換レイヤー) を使う
準備
0. Wine では ~/.wine/drive_c/ が Windows の C:\ に相当する
1. ~/.wine/drive_c/aaa/ というディレクトリを作り、
この下に oxbash.zip を unzip する
2. ~/.wine/drive_c/aaa/oxbash/help/help.pdf を読む
3. ~/.wine/drive_c/aaa/oxbash/login/ 以下のファイルを修正
• login/login.bat
set disk=c:
set oxarea=\aaa\oxbash\
set nudir=\aaa\oxbash\
set path=%path%;%oxarea%exe\
copy oxbash-dir.dat %disk%\oxbash-dir.dat
• login/oxbash-dir.dat
1
!c:\aaa\oxbash\exe\ ! directory with the exe files
!c:\aaa\oxbash\sps\ ! directory with old sps and int files
!c:\aaa\oxbash\spsn\ ! directory with new sps and int files
default では drive letter が h になっていた
C の基底状態波動函数の計算
(p-shell 内で実現される A = 12, J = 0, T = 0 である全状態の波動函数の計算)
~$ cd ~/.wine/drive_c/aaa/oxbash/login
~/.wine/drive_c/aaa/oxbash/login$ wine cmd
CMD Version 1.0.1
C:\aaa\oxbash\login\へ移動
command prompt を起動
C:\aaa\oxbash\login>login
login.bat を実行
C:\aaa\oxbash\login>set disk=c:
C:\aaa\oxbash\login>set oxarea=\aaa\oxbash\
C:\aaa\oxbash\login>set nudir=\aaa\oxbash\
C:\aaa\oxbash\login>set path=C:\windows\system32;C:\windows;\aaa\oxbash\exe \
C:\aaa\oxbash\login>copy oxbash-dir.dat c:\oxbash-dir.dat
C:\aaa\oxbash\login>sd rsh
cd ../rsh/
C:\aaa\oxbash\rsh>md c12
mkdir c12
C:\aaa\oxbash\rsh>sd c12
C:\aaa\oxbash\rsh\c12>copy c:\oxbash-dir.dat .
対話的に batch file を作成
C:\aaa\oxbash\rsh\c12>shell
found oxbash-dir.dat in current directory
this is oxbash for windows
( using label.dat
from area c:\aaa\oxbash\sps\
)
directory for exe files > c:\aaa\oxbash\exe\
directory for old sps and int files > c:\aaa\oxbash\sps\
directory for new sps and int files > c:\aaa\oxbash\spsn\
|------------ oxbash (hpc version 2005-12) -----------|
this version allows up to 248 m-states and 42 j-states
name for batch file : c12
option (lpe, den, st or h) : lpe
波動函数の計算
single-particle state file : p
詳細は sps/label.dat に
( using p.sps
from area c:\aaa\oxbash\sps\
)
core は 4 He ゆえ価核子は 8 個
total number of valence particles : 8
any restrictions ? (y/n) :
interaction filename : ckpot
min. j, max. j : (2f) : 0.,0.
スピン
min. t, max. t : (2f) : 0.,0.
アイソスピン
bad j, t values (2f) (or <cr> for no more) :
parity (0=+ve, 1=-ve, 2=both) : 0
パリティは正
( using label.dat
from area c:\aaa\oxbash\sps\
)
a0008a
:basis +proj +oper +matrix +lanczos
option (lpe, den, st or h) : st
file c12.bat has been created
C:\aaa\oxbash\rsh\c12>c12
stop
12c.bat を実行
Z 係数の計算 (2)
実行例
12
N(1+ , gs) → 12 C(0+ , gs) への 1 粒子遷移密度 (OBTD) の計算
1. 12 N(1+ , gs) の波動函数を計算
2. 12 C(0+ , gs) の波動函数を計算
3. 2 個の波動函数を用いて OBTD を計算
~/.wine/drive_c/aaa/oxbash/login$ wine cmd
name for batch file : n12c12
option (lpe, den, st or h) : lpe
single-particle state file : p
( using p.sps
from area c:\aaa\oxbash\sps\
)
total number of valence particles : 8
any restrictions ? (y/n) :
interaction filename : ckpot
min. j, max. j : (2f) : 1.,1.
min. t, max. t : (2f) : 1.,1.
bad j, t values (2f) (or <cr> for no more) :
parity (0=+ve, 1=-ve, 2=both) : 0
( using label.dat
from area c:\aaa\oxbash\sps\
)
a2208a
:basis +proj +oper +matrix +lanczos
option (lpe, den, st or h) : lpe
single-particle state file : p
total number of valence particles : 8
any restrictions ? (y/n) :
interaction filename : ckpot
min. j, max. j : (2f) : 0.,0.
min. t, max. t : (2f) : 0.,0.
bad j, t values (2f) (or <cr> for no more) :
parity (0=+ve, 1=-ve, 2=both) : 0
a0008a
:basis +proj +matrix +lanczos
option (lpe, den, st or h) : den
d,c,1,2,t,4,is,iv,it,at,bt or h for help ? : t
initial state m-scheme eigenfunctions filename : a2208a
min.,max. vectors reqd. (-1 for all) :1,1
final state m-scheme eigenfunctions filename : a0008a
min.,max. vectors reqd. (-1 for all) :1,1
a2208a
min. j, max. j : (f) : 1.,1.
a2208a
min. t, max. t : (f) : 1.,1.
bad j, t values (2f) (or <cr> for no more) :
a0008a
min. j, max. j : (f) : 0.,0.
a0008a
min. t, max. t : (f) : 0.,0.
bad j, t values (2f) (or <cr> for no more) :
restrict coupling for operator :
mvec a2208a
mvec a0008a
tramp a2208a a0008a
a2208a000.trd
option (lpe, den, st or h) : st
file n12c12.bat has been created
C:\aaa\oxbash\rsh\n12c12>n12c12
実行結果
~/.wine/drive_c/aaa/oxbash/rsh/n12c12$ cat a2208a000.obd
dt format, made from oxbash binary output file: a2208a000.trd
! model-space name = p
! interaction name = ckpot
!
!
---------------------------------!
| ai <-> af =
12 <->
12 |
!
| ji <-> jf =
1 <->
0 |
!
| ti <-> tf =
1 <->
0 |
!
| pi <-> pf =
+ <->
+ |
!
| #i <-> #f =
8 <->
5 |
!
---------------------------------!
! one-body transition densities a(dj,dt) =
! = <f||| [a+(k1)a(k2)]^(dj,dt) |||i>/sqrt[(2dj+1)(2dt+1)]
! for input to some dwba codes
! z = a(dj,dt) * <ti,tiz,dt,dtz|tf,tfz>sqrt(2dt+1)/sqrt[(2ji+1)(2tf+1)]
!
where <|> is the clebsch-gordan and dtz=tf-ti
! with edmonds (de-shalit talmi) reduced matrix element convention
!
! for n,l,j =
1.0 1.0 1.5 label k =
2
! for n,l,j =
1.0 1.0 0.5 label k =
3
!
! ji,
jf,
ti,
tf,
! dj,
ni,
nf,
ef,
ei,
exi,
exf,
!
1.0, 0.0, 1.0, 0.0, 0.0, 0.0,
1.0,
1.,
1., -71.045, -55.960,
0.000,
0.000,
3, 3,
0.00000, -0.05824, ! k1,k2,a(dt=0),a(dt=1)
3, 2,
0.00000,
0.33933,
2, 3,
0.00000,
0.69022,
2, 2,
0.00000, -0.07627,
0,
Z 係数 (CKZ) ↔Wildenthal 遷移振幅 (OXBASH) の変換
Zj∆J,∆T
p ,jh
=

Jf Tf ||| [a†jp × ãjh ]∆J,∆T ||| Ji Ti 
2∆T + 1
√
√
Ti T3i ∆T ∆T3 |Tf T3f 
(2Ji + 1)(2Tf + 1)
2∆T + 1 2∆J + 1

∆T
1
1
1
1
∆J
1
1
1
1
1
2·1+1
1 1 1 −1|0 0 = √
(2 · 1 + 1)(2 · 0 + 1)
3
initial
p3/2
p3/2
p1/2
p1/2
final
p3/2
p1/2
p3/2
p1/2
CKZ
−0.044084
0.195925
0.398462
−0.033553
OXBASH
−0.07627
1
0.33933
×√
3 0.69022
−0.05824
2
+2 2
+1 +1 1 +1
1×1
1
+1
+ 1 0 1/2 1/2
0 + 1 1/2 − 1/2
2
0
1
0
0
0
+ 1 − 1 1/6 1/2 1/3
0 0 2/3
0 − 1/3 2
− 1 + 1 1/6 − 1/2 1/3 − 1
1
−1
0 − 1 1/2 1/2 2
− 1 0 1/2 − 1/2 − 2
−1 −1 1
Z 係数の計算 (3)
Normal mode 波動函数による集団励起の記述 (Bohr&Mottelson)
µ
励起状態 |LJM を IS/IV 型 tensor 演算子 OLJM
の作用への応答とし,
†
これを 1p1h 状態 [ap ah ]LJM |0 で展開:
|LJM =

ph
LJM †
Zph
[ap ah ]LJM |0
展開係数が Z 係数 (OBTD) であり
µ
ph; LJM|OLJM
|0
µ
2
ph |ph; LJM|OLJM |0|
LJM
Zph
=
µ
Tensor 演算子 OLJM
• non-spin-flip 遷移

µ
O(L=J)JM
=
riJ YJM (r̂i )tµ (i)
i
• spin-flip 遷移
µ
=
OLJM

i
riL [Y (r̂i ) ⊗ σi ]JM tµ (i)
Remark:
• The normal modes exhaust the full multipole strength
(Non-energy-weighted sum rule is exhausted).
• Only part of residual interactions between valence nucleons
are taken into account
→ over prediction of strength
computer code
NORMOD
by S. Y. van der Werf (KVI)
PROGRAM TO CALCULATE SCHEMATIC COLLECTIVE WAVE FUNCTION OF
R**(L+M)*YL OR R**(L+M)*[SIGMA*YL].
CARD 1: IRCONV = 0 FOR DW81, =1 FOR DWBA98.
SINGLE PARTICLE RADIAL WAVE FUNCTIONS ARE DEFINED TO BE POSITIVE
TOWARDS INFINITY IN DW81. IN DWBA91 THEY ARE DEFINED TO BE POSITIVE
NEAR THE ORIGIN.
CARD 2: NUMP= NUMBER OR PARTICLE SHELLS
CARDS
o
o
o
o
o
o
o
3: ( x NUMP)
FNP = NUMBER OF NODES IN PATICLE RADIAL WAVEF.
FLP = ANGULAR MOMENTUM OF PARTICLE
F2JP = TWICE TOTAL ANG. MOM. OF PARTICLE
U2= EMPTINESS OF PARTICLE-SHELL.
NPTYPE=0 FOR NEUTRON, 1 FOR PROTON
NPFLAG = MAJOR SHELL NUMBER OF PARTICLE
ISHELP = NUMBER OF THE SHELL, USED TO WRITE THE WAVE FUNCTION
TO DEVICES # 7-10, IN THE FORMAT : ISHELP, ISHELH, COEFF
CARD 4: NUMH= NUMBER OR HOLE SHELLS
CARDS
o
o
o
o
o
o
o
5: ( x NUMH)
FNH= NUMBER OF NODES IN HOLE RADIAL WAVEF.
FLH= ANGULAR MOMENTUM OF HOLE
F2JP(H)= TWICE TOTAL ANG. MOM. OF HOLE
V2= FULLNESS OF THE HOLE -SHELL.
NHTYPE=0 FOR NEUTRON, 1 FOR PROTON
NHFLAG = MAJOR SHELL NUMBER OF PARTICLE
ISHELH = NUMBER OF THE SHELL, USED TO WRITE THE WAVE FUNCTION
TO DEVICES # 7-10, IN THE FORMAT : ISHELP, ISHELH, COEFF
CARDS 6: CHARACTERISTICS AND RESTRICTIONS OF THE TRANSFER OPERATOR
(READS CASES UNTIL EOF)
o M= EXPONENT OF R IS (L+M), WHERE L=JTRAN FOR NATURAL PARITY AND
L=JTRAN-1 OR L=JTRAN+1 FOR UNNATURAL PARITY TRANSITIONS
o JTRAN= TRANSFERRED TOTAL ANG. MOM.
o PARITY= PRODUCT OF PARITIES OF INITIAL AND FINAL STATE
o AMASS = MASS OF TARGET,
o ISPIN = ISOSPIN TRANSFER, ISPIN3 =3-COMPONENT OF ISPIN
o REACTION: TYPE
ISPIN
ISPIN3
(P,P) T=0
0
0
(P,P) T=1
1
0
(P,N)
1
1
(N,P)
1
-1
o IDIFL= MINIMUM DIFFERENCE BETWEEN MAJOR SHELL NUMBERS
o IDIFH= MAXIMUM DIFFERENCE BETWEEN MAJOR SHELL NUMBERS
Z 係数の計算 (3)
計算例
計算結果
IVSM via the 90 Zr(12 B, 12 C)90 Y(1+ ) reaction
2ω, ∆J π = 1+ , ∆T = ∆S = 1, ∆L = 0
0i
2p
5ω
odd
1f
0h
4ω
even
3ω
odd
2ω
even
1ω
odd
0
even
2s
1d
2p1/2
2p3/2
0h9/2
0h11/2
1d3/2
1d5/2
0g
1p
0f
1s
0d
0p
0s
1p1/2
1p3/2
1s1/2
0i 13/2
14 126
2 112
1f 5/2
1f 7/2
4 110
6 106
2s1/2
0g7/2
0g9/2
0f 5/2
0f 7/2
0d3/2
0d5/2
0p1/2
0p3/2
0s1/2
Possible 1p1h configurations
ν
ν
π
π
−1
1d5/2 0d5/2
1f 7/2 0f 5/2−1
1d3/2 0d5/2−1
1f 5/2 0f 5/2−1
1d5/2 0d3/2−1
2p3/2 1p3/2−1
1d3/2 0d3/2−1
2p1/2 1p3/2−1
2s1/2 1s1/2−1
2p3/2 1p1/2−1
1f 7/2 0f 7/2−1
2p1/2 1p1/2−1
1f 5/2 0f 7/2−1
8 100
10 92
12 82
2 70
4 68
6
64
8
58
10 50
2 40
6
4
8
2
2
0
3
3
1
1
28
2
16
2
8
4
6
5
3
1
5
7
1
3
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0
0
0
0
0
0
0
4
4
4
5
5
5
5
1
2
3
4
5
6
7
ν(1d5/2)
ν(1d3/2)
ν(2s1/2)
ν(1f 5/2)
ν(1f 7/2)
ν(2p1/2)
ν(2p3/2)
2 5 1.0 1 2 8
π(0d5/2−1
)
−1
2 3 1.0 1 2 9
π(0d3/2−1
)
0 1 1.0 1 2 10
π(1s1/2−1 )
3 7 1.0 1 3 11
π(0f 7/2 )
3 5 1.0 1 3 12
π(0f 5/2−1
)
1 3 1.0 1 3 13
π(1p3/2−1
)
−1
1 1 1.0 1 3 14
π(1p1/2 )
1 1 90 1 -1 2 2
./normod < zrnpivsm.inp > zrnpivsm.out
cat zrnpivsm.out
M= 2 JTRAN= 1 PARITY= 1 ISPIN= 1 ISPIN3=-1
AMASS= 90 IDIFL= 2 and IDIFH= 2 hbar w
32
20
2
cat zrnpivsm.inp
38
4
6
$
1
7
1
1
2
1
1
2
2
7
0
0
1
0
0
1
1
2
$
$
14
2
NP LP 2JP TYPE
NH LH 2JH TYPE
S=0 L=1
1
1
1
1
1
2
2
1
1
1
1
1
2
2
2
2
2
0
0
0
0
1
0
1
0
0
1
0
0
1
1
0
1
1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
STREN
STREN
STREN
STREN
2
2
2
2
2
0
0
3
3
3
3
3
1
1
1
1
1
5
5
3
3
3
1
1
5
5
5
7
7
1
1
3
3
3
(1)
(2)
(3)
(4)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
=
=
=
=
0.0000E+00
0.7192E+03
0.0000E+00
0.1127E+07
OVERLAP <1!1>
OVERLAP <2!2>
OVERLAP <3!3>
OVERLAP <4!4>
OVERLAP <1!3>
OVERLAP <2!4>
<T=0!T=1> <1!
<T=0!T=1> <2!
<T=0!T=1> <3!
<T=0!T=1> <4!
1
2
2
2
2
0
2
0
3
3
1
3
3
1
1
3
1
1
=
=
=
=
=
=
=
=
=
=
5
3
5
3
1
3
1
7
5
3
7
5
3
1
5
3
1
EL
EL
EL
EL
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.0000E+00
0.1000E+01
0.0000E+00
0.1000E+01
0.0000E+00
-.1267E-06
0.0000E+00
-.1000E+01
0.0000E+00
-.1000E+01
=
=
=
=
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
S=1 L=0
-0.2556
0.2733
-0.2733
0.1366
0.0000
0.0000
-0.2582
-0.3703
0.2070
0.0000
-0.3207
0.3703
-0.2880
0.1018
0.0000
-0.3220
0.2880
FM**
FM**
FM**
FM**
S=1 L=1
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
S=1 L=2
-0.1052
-0.0984
0.0984
0.1968
0.2899
-0.1941
0.0000
0.1630
0.2916
0.5915
-0.1882
-0.1630
0.1268
0.3586
-0.3348
-0.1134
-0.1268
6
4
6
8
1 粒子殻模型波動函数の, 動径成分の境界条件は
DW 81: 漸近的に正
FOLD: 原点付近で正
(WASW の出力する波動函数がそうなっている)
であり, 双方とも
• Y と σ s の結合順は  + s = j
• 角度成分には Ym の他に i  を含まない
ことから, FOLD と DW 81 とでは,
入力する Z 係数の位相が逆らしい (要確認！)
1 粒子軌道エネルギーの計算
実行例
実行結果
Zr(12 B, 12 C)90 Y 反応 ((n, p) 型) を念頭に置き,
90
Zr の陽子, 90 Y の中性子, の 1 粒子軌道エネルギーを計算する
(OXBASH の dens で基底状態密度を計算する)
~/.wine/drive_c/aaa/oxbash/rsh/zr90$ cat dens.dao
(type of potential),x = ( sk20 ), 0.0000
90
~/.wine/drive_c/aaa/oxbash/login$ wine cmd
C:\aaa\oxbash\rsh\zr90>dens
initialized with ws2 potential and ia,iz =
initialized from hbwtab, hbw =
(previous input option =
16
8
13.260
)
new tty option (h for help) : cp
cp
: change type of potential
[ type of potential: ws,ho,ht,hr,hl,skn,bh,wq,wen (h for help) ], x
sk20
skn,x : (x=0) skyrme(n) n=20=(skx) bab prc58 220, 1998
initialized from hbwtab, hbw =
(previous input option = cp
ia,iz
90,40
13.260
)
new tty option (h for help) : az
az : read ia,iz
(default previous values =
16
8)
ia,iz =
90
40
----- proton bound state results ----k n l j
e
ie occ
1 1 s 1/2 -34.404 1 2.00 52.26 53.54 6.00
2 1 p 3/2 -28.984 1 4.00 52.26 53.34 6.00
3 1 p 1/2 -27.576 1 2.00 52.26 53.29 6.00
4 1 d 5/2 -22.295 1 6.00 52.26 53.09 6.00
5 1 d 3/2 -19.262 1 4.00 52.26 52.98 6.00
6 2 s 1/2 -17.428 1 2.00 52.26 52.91 6.00
7 1 f 7/2 -14.610 1 8.00 52.26 52.80 6.00
8 1 f 5/2 -9.644 1 6.00 52.26 52.62 6.00
9 2 p 3/2 -8.464 1 4.00 52.26 52.58 6.00
10 2 p 1/2 -7.035 1 2.00 52.26 52.52 6.00
11 1 g 9/2 -6.107 1 0.00 52.26 52.49 6.00
12 1 g 7/2 -0.200 3 0.00 53.47 53.47 6.00
13 2 d 5/2 -0.200 3 0.00 53.11 53.12 6.00
14 2 d 3/2 -0.200 3 0.00 56.06 56.06 6.00
15 3 s 1/2 -0.200 3 0.00 56.04 56.05 6.00
16 1 h11/2 -0.200 3 0.00 56.62 56.62 6.00
17 1 h 9/2 -0.200 3 0.00 67.73 67.73 6.00
18 2 f 7/2 -0.200 3 0.00 66.44 66.45 6.00
19 2 f 5/2 -0.200 3 0.00 70.59 70.60 6.00
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.268
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
1.100
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
0.650
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
1.227
ia,iz =
90
39
----- neutron bound state results ----k n l j
e
ie occ
1 1 s 1/2 -40.288 1 2.00 45.28 44.16 6.00 1.251 1.100 0.731 0.650 1.222
(previous input option = az
) new tty option (h for help) : gd
2 1 p 3/2 -33.932 1 4.00 45.28 44.34 6.00 1.251 1.100 0.731 0.650 1.222
gp
: calculate wf and ground state (gs) point densities
3 1 p 1/2 -32.696 1 2.00 45.28 44.37 6.00 1.251 1.100 0.731 0.650 1.222
gd
: "gp" plus folding of finite nucleon size into gs densities
4 1 d 5/2 -26.880 1 6.00 45.28 44.53 6.00 1.251 1.100 0.731 0.650 1.222
途中の標準出力を省略
5 1 d 3/2 -24.062 1 4.00 45.28 44.61 6.00 1.251 1.100 0.731 0.650 1.222
----- (type of potential,x) = (sk20, 0.0000) ------- a,z =
90 40
6 2 s 1/2 -23.760 1 2.00 45.28 44.62 6.00 1.251 1.100 0.731 0.650 1.222
7 1 f 7/2 -19.219 1 8.00 45.28 44.75 6.00 1.251 1.100 0.731 0.650 1.222
(previous input option = gd
) new tty option (h for help) : az
8 1 f 5/2 -14.444 1 6.00 45.28 44.88 6.00 1.251 1.100 0.731 0.650 1.222
9 2 p 3/2 -15.063 1 4.00 45.28 44.86 6.00 1.251 1.100 0.731 0.650 1.222
ia,iz
(default previous values =
90 40)
10 2 p 1/2 -13.485 1 2.00 45.28 44.91 6.00 1.251 1.100 0.731 0.650 1.222
90,39
11 1 g 9/2 -11.002 1 10.00 45.28 44.98 6.00 1.251 1.100 0.731 0.650 1.222
12 1 g 7/2 -4.289 1 0.00 45.28 45.16 6.00 1.251 1.100 0.731 0.650 1.222
initialized from hbwtab, hbw =
8.797
13 2 d 5/2 -6.480 1 1.00 45.28 45.10 6.00 1.251 1.100 0.731 0.650 1.222
14 2 d 3/2 -4.302 1 0.00 45.28 45.16 6.00 1.251 1.100 0.731 0.650 1.222
(previous input option = az
) new tty option (h for help) : gd
15 3 s 1/2 -5.086 1 0.00 45.28 45.14 6.00 1.251 1.100 0.731 0.650 1.222
途中の標準出力を省略
16 1 h11/2 -2.222 1 0.00 45.28 45.22 6.00 1.251 1.100 0.731 0.650 1.222
17 1 h 9/2 -0.200 3 0.00 54.43 54.43 6.00 1.251 1.100 0.731 0.650 1.222
----- (type of potential,x) = (sk20, 0.0000) ------- a,z =
90 39
18 2 f 7/2 -0.200 3 0.00 49.21 49.20 6.00 1.251 1.100 0.731 0.650 1.222
19 2 f 5/2 -0.200 3 0.00 53.38 53.38 6.00 1.251 1.100 0.731 0.650 1.222
(previous input option = gd
) new tty option (h for help) : st
20 3 p 3/2 -0.200 3 0.00 51.88 51.88 6.00 1.251 1.100 0.731 0.650 1.222
the output file is dens.dao
21 3 p 1/2 -0.200 3 0.00 53.11 53.10 6.00 1.251 1.100 0.731 0.650 1.222
22 1 i13/2 -0.200 3 0.00 56.56 56.56 6.00 1.251 1.100 0.731 0.650 1.222
initialized from hbwtab, hbw =
8.797
1 粒子軌道エネルギーの計算
N USHELL の speplot を使うと,
dens.dao から 1 粒子軌道エネルギーを読んで plot することができる
C:\aaa\nushell\rsh\zr90>ren dens.dao zr90.spe
C:\aaa\nushell\rsh\zr90>speplot zr90
90
40
sk20-skx
found proton data
s 1
-34.404
p 3
-28.984
p 1
-27.576
d 5
-22.295
40
d 3
-19.262
s 1
-17.428
f 7
-14.610
f 5
-9.644
p 3
-8.464
p 1
-7.035
g 9
-6.107
g 7
-0.200
d 5
-0.200
d 3
-0.200
s 1
-0.200
h11
-0.200
h 9
-0.200
f 7
-0.200
f 5
-0.200
0
0.000
found neutron data
s 1
-41.081
p 3
-34.642
p 1
-33.452
d 5
-27.465
d 3
-24.678
s 1
-24.161
f 7
-19.722
f 5
-14.958
p 3
-15.648
p 1
-14.112
g 9
-11.461
g 7
-4.741
d 5
-7.076
d 3
-4.878
s 1
-5.403
h11
-2.652
h 9
-0.200
f 7
-0.200
f 5
-0.200
protons
p 3
-0.200
出力を適当に抜粋
0
0.000
10
90 Zr
E (MeV)
-20
-30
90 Y
sk20−skx
sk20−skx
50
0
-10
http://www.nscl.msu.edu/ (中略) NuShell (後略)
http://www.nscl.msu.edu/以下略
http://www.nscl.msu.edu/以下略
h11
d3
g7
s1
d5
g9
p1
p3
f5
f7
s1
d3
d5
g9
p1
f5
p3
f7
f7
s1
d3
f7
d5
s1
d3
d5
d5
p1
p3
p1
p3
s1
h11
d3
g7
s1
d5
p1
f5
p3
s1
d3
p1
p3
d5
51
g9
p1
p3
f5
g9
-40
-50
39
p1
p3
s1
s1
s1
− parity
+ parity
neutrons
protons
− parity
+ parity
neutrons
1 粒子殻模型波動函数の動径方向成分の計算
• Woods-Saxon ポテンシャルを用いて計算.
• 1 粒子束縛エネルギーを入力. ポテンシャルの深さが fit される.
• “参加” する軌道の波動函数を全て求めておく.
B → 12 C
p 1p3/2, 1p1/2
n 1p3/2, 1p1/2
–
12
–
208
Pb → Tl
n 3s1/2, 2d5/2, 2d3/2, 1g9/2, 1g7/2
p 2s1/2, 1d5/2, 1d3/2, 0g9/2, 0g7/2
208
$ cat wsaw12b12c.inp
0.1
20.
1
B12C12
11.
5.
60.
15.9570
1.
1.
11.
5.
60.
15.9570
1.
1.
11.
5.
60.
3.3704
1.
1.
11.
5.
60.
3.3704
1.
1.
-1.
$ cat wsaw208pb208tl.inp
0.1
20.
1
PB208TL
207.
81.
60.
0.654
1.
0.
207.
81.
60.
1.192
1.
2.
207.
81.
60.
0.405
1.
2.
207.
81.
60.
3.462
1.
4.
207.
81.
60.
1.000
1.
4.
207.
81.
60.
7.692
1.
0.
207.
81.
60.
10.070
1.
2.
207.
81.
60.
8.513
1.
2.
207.
81.
60.
12.190
1.
4.
-1.
150
0
.65
0.
.65
0.
.65
0.
.65
0.
150
1.25
1.
1.25
1.
1.25
0.
1.25
0.
1.25
0.5
1.25
1.5
1.25
0.5
1.25
1.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
0.
1.25
1.
1.25
1.
1.25
1.
1.25
1.
1.25
0.5
1.25
2.5
1.25
1.5
1.25
4.5
1.25
3.5
1.25
0.5
1.25
2.5
1.25
1.5
1.25
3.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
7.0
.5
0
.65
3.
.65
2.
.65
2.
.65
1.
.65
1.
.65
2.
.65
1.
.65
1.
.65
0.
• 1 粒子軌道のエネルギーは
– Sp (12 C), Sn (12 B) の実験値を用いる
–
OXBASH ,
などする
N USHELL の dens を用いて計算する
1 粒子殻模型波動函数の動径方向成分の計算
計算結果
0.6
12
Rn(r )
0.4
B,
12
C
208
Pb,
208
Tl
0.2
0.0
となって,
-0.2
ϕnm (r) = Rn (r)Ym(θ, φ)
と変数分離し, size parameter
0.5
rRn(r )
b≡
を導入すれば, 動径方向成分は
0.0

-0.5
∞
0
1
Rn (r) =
(2 + 1)!!
dr r 2 (Rn (r ))2 = 1
0.0 2.5 5.0 7.5 10.0 12.515.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0
r (fm)
r (fm)
参考: 3 H, 3 He の 1s 波動函数
1.0
Green’s function Monte Carlo
Woods-Saxon
(J. Carlson, PRC 36, 2026 (1987), etc.)
Rn(r )
参考: 調和振動子ポテンシャルに対する厳密解
調和振動子ポテンシャル
1
U(r) = mω 2 r 2
2
に対しては, Schrödinger 方程式は



1
mω 2
2
∇ +
r ϕ(r) = Eϕ(r)
Hϕ(r) = ω
2
mω


mω
1/2  
r 
b


3 r2
−r 2 /2b2
×e
1 F1 1 − n,  + ; 2
2 b
2−n+3 (2n + 2 − 1)!!
b3 π 1/2 (n − 1)!
となる. 1 F1 は Kummer の合流型超幾何函数で, 級数展開は


3 r2
F
1
−
n,

+
;
1 1
2 b2
n−1

(2 + 1)!!  r 2k
(n − 1)!2k
(−1)k
=
(n − k − 1)!k! (2 + 2k + 1)!! b
k=0
である. エネルギー固有値は




3
1
ω = N +
ω
En = 2n +  −
2
2
0.5
0.0
0.0


5.0
r (fm)
10.0
15.0 0.0
5.0
r (fm)
10.0
15.0
である.
遷移密度, 形状因子の計算
軌道の名前 (番号と軌道の対応)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
0s1/2
0p1/2
0p3/2
1s1/2
0d3/2
0d5/2
1p1/2
1p3/2
0f5/2
0f7/2
2s1/2
1d3/2
1d5/2
0g7/2
0g9/2
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
2p1/2
2p3/2
1f5/2
1f7/2
0h9/2
0h11/2
3s1/2
2d3/2
2d5/2
1g7/2
1g9/2
0i11/2
0i13/2
3p1/2
3p3/2
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
2f5/2
2f7/2
1h9/2
1h11/2
0j13/2
0j15/2
4s1/2
3d3/2
3d5/2
2g7/2
2g9/2
1i11/2
1i13/2
0k15/2
0k17/2
有効相互作用
$ cat franey_270
4
3
4
.25
5890.37
.40
-1361.95
.55
-229.750
1.4
-10.5000
.25
-6999.55
.40
349.040
.55
-68.9521
.15
80527.5
.25
-7432.99
.40
-29.5714
.70
-60.2151
3
3
4
.25
2212.64
.40
-900.238
.55
85.9426
.25
-807.467
.40
559.786
.55
-4.87654
.15
127002.
.25
-10469.4
.40
385.365
.70
-4.85957
1.00
1.00
3470.90
-1347.93
104.810
-10.5000
-3321.20
-294.795
22.3046
15605.2
-1091.07
177.657
16.1522
-51517.2
16778.0
-2599.18
31.5000
5809.93
-1035.73
-50.7069
3.50000
3629.68
-676.925
-279.248
424.916
-61.2376
9.69778
-108413.
7476.58
-284.082
3.72577
-2206.25
-351.375
-6.27354
-2866.75
228.178
-50.1090
実部
虚部
103
1.00
1.00
|t(q)| (MeV · fm3 )
$ cat fold12b208pbjp1jt1.inp
1
1PBBIVSM
600 0.03
3000.
12.
1
1
1
0.0+
1.0+
0.0
+0.0
1.0
+1.0
3
3 0.000
2
2
1 0.0
-0.033553
Cohen-Kurath
2
3
1 0.0
0.195925
3
2
1 0 0
0.398462
(8-16)POT
3
3
1 0.0
-0.044084
-1
-1
B12C12
1.0+
0.0+
23.0
+23.0
22.0
22.0
3
3 0.000
22
11
1 0.0
-0.3681
24
13
1 0.0
-0.4032
24
12
1 0.0
0.4311
23
13
1 0.0
-0.4311
Normal Modes
23
12
1 0.0
0.2155
26
14
1 0.0
0.4586
25
14
1 0.0
0.2713
-1
-1
PB208TL
0.897
3.67
1.000
franey_270
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
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
10
2
C
tστ
tτC
tτT
101
10
0
10−1
0
Fourier 変換されて
遷移密度計算に用いられる
(inter.for)
tτLS
2
4
6
q (fm)
8
Love and Franey, PRC 24 1073 (1981)
遷移密度, 形状因子の計算
形状因子とそのノルム
遷移密度とその Fourier 変換
0.01
0.00
-0.01
-0.02
12
ρ(r )
-0.04
-0.05
0.02
B→
12
C
ρ(q)
-0.03
0.01
0.00
-0.01
-0.02
-0.03
208
-0.04
-0.05
4πr 2 ρ (fm−1 )
4
2
Pb → 208 Tl
0 1 2 3 4 5 6 7 8 9 10
r (fm)
208
Pb → 208 Tl
-4
1.0
12
B→
12
C
Im
0.0
dr r ρ(r ) = 0
0 1 2 3 4 5 6 7 8 9 10
r (fm)
Im
Re
101
208
10
Pb(12 B, 12 C)208 Tl
10−1
10−2
208
0
1
10−3
Pb → 208 Tl
2
3
−1
q (fm )
4
v (q) = 4π
4π
Re
0
0
2
Pb(12 B, 12 C)208 Tl
0.5
K

208
1.5
5
10−4
0
2
角運動量代数を除いて
 ∞
F (r ) ∼
dq q 2 jJr v (q)ρab (q)ρAB (q)
0
-2
3
2
1
0
-1
-2
-3
-4
-5
3
2
1
0
-1
-2
-3
-4
-5
|F (r )|
0.02
ρL (q) = 4π

∞
0 ∞
0
4
6 8 10 12 14
r (fm)
形状因子
ds s 2 jK (qs)v K (s)
有効相互作用
dr r 2 jL (qs)ρL (r )
遷移密度
正しい定式化は
J. Cook, K. W. Kember, et al., PRC 30, 1538 (1984) を参照のこと
微分散乱断面積の計算
微分散乱断面積の計算結果
0.
0.991
0.
1.186
0.
0.
0.
0.991
0.
1.186
0.
0.
• Reaction Q-value (Qgg ) for 208 Pb + 12 B: 8369.92 keV (208 Tl + 12 C)
• IVSM の励起エネルギーを 8369.92 keV とした
(IVSM の Q-value を 0 MeV とした)
光学模型ポテンシャル
101
Pb(12 B, 12 C)208 Tl
Ex = 8369.92 keV
208
100
dσ/dΩ (mb/sr)
$ cat dwhi12b208pbjp1jt1.inp
1210000041000000
208PB(12B,12C)208TL
PBBIVSM
161.
0.
0.05
300 2 2 0 0 2
0.03
600
3000. 12.
5.
208.
82.
1.00
1.
-10. 0.803 0.815 0.
-35.82
0.
0.0
12.
6.
208.
81.
1.00
1.
-10. 0.803 0.815 0.
-35.82
0.
0 2 2
0.
0.
0.
1.
2 2 2
0.
0.
0.
1.
12b208pbjp1jt1.plot
011
10−1
211
10−2
10−3
(Jr Jp Jt )
0
J. Y. Hostachy, et al.,
“Elastic and inelastic scattering of 12 C ions at intermediate energies”
Nucl. Phys. A490 (1988) 441-470.
(Experiment at the Lboratoire National Saturne at Sacley)
1
2
3
4
Θcm (deg)
5
6