11

Xi Tao（CNDC/CIAE） Chonghai Cai (Naikai University)

Qingbiao Shen( CNDC/CIAE) E-mail:taoxixishi@ciae.ac.cn

2012.08.29

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Outline

APRML code Background Introduction of APRML code R-matrix theory in APRML

n+6Li reaction Reaction channel Input files and parameters Results

Summery

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Background

International 1936, Breit G. and Wigner E. P R, r-matrix theory, describe resonance. 1958, Lane A. M. and Thomas R. G. , a class paper for r-matrix theory. SAMMY（1980-Now）, resonance parameters. EDA： The Energy Dependent Analysis code，developed by D.

Dodder, G. Hale and K. Witte at Los Alamos. n+12C, minus energy level.

In China Chengjiu Zhu(朱诚久)、Guochn Qiu(丘国春)、Weili Sun(孙伟力)、

Zhenpeng Chen(陈振鹏)、Hongwei Wang(王宏伟) RAC code： Zhenpeng Chen(陈振鹏) ，used for light nuclei

CNDC LUNF can calculate light nuclei reaction cross section and double

differential cross section, but can not deal with resonance cross section.

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Introduction of APRML

Adjusting Parameters of R-Matrix for Light nuclei in 1p-shell. Chonghai Cai(Nankai University), Qingbiao

Shen(CNDC/CIAE), Xi Tao(CNDC/CIAE) Base on R-matrix theory Light nuclei in 1p-shell Cross section of 2-bodys reaction Angular distribution (n,tot),(n,el),(n,non),(n,γ),(n,inl),(n,p),(n,t),(n,3He),(n,α),(n,d),(n,5He),(n,X)

Auto adjusting parameters

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Theory 《低能和中能核反应理论》

Low and moderate energy nuclear reaction theory 《高等量子力学讲义》

Advanced quantum mechanics 《中子引发轻核反应的统计理论》

Neutron induced nuclear reaction statistics theory in light nuclei 申老师和蔡老师的报告

Reports of professor Qingbiao Shen and Chonghai Cai

Structure of APRML Codes：APRML.for, plot.for, compiled by fortran Input files：APRMLk.dat ，APRMLi.dat Output files： APRML.dat, APRMLo.dat, APRMLs.dat,APRMsa.dat plot.dat, plot.gnu

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S-matrix

Total wave function after reaction：

Integral cross section formula:

Neutron injected

⎭⎬⎫

+⎩⎨⎧ −

+=Ψ

∑

∑

'''''''''',''''

0)(

)('

1)(1

12

jln

JMjlnll

Jnljjln

JMnljll

iJMIMjm

jmiml

ljJM

t

iFGSv

iFGv

eCClkri

l

Ii

i

i

ααααα

σ

ϕϕ

π

－

2

,'''',''''''

2222,''

ˆˆˆ

1 Jnljjlnnljjln

Jjljlnn SJ

Iik αααααα δπσ −= ∑

( )∑ −=ljJ

Jljt SJ

Ik)Re(1ˆ

ˆ2

22

πσ

∑ ∑ −=ljJ jl

Jljjlljjlel SJ

Ik ''

2

,'',''2

22ˆ

ˆ2δπσ )||1(ˆ

ˆ22

'',''

222

)( ∑∑ −=jl

Jljjl

ljJ

nne SJ

Ikπσ

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R-matrix

Element of S-matrix：

Minus energy level used：

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+−++= ∑+−

λλλλ

λλφφ

Γ∆

ΓΓδ

2

21

21

' ,','

)(0'' iEE

ieSS cccc

icccc

cc

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Γ−−

ΓΓ+= ∑+−

λλλ

λλφφ δ

2

21

21

' ,','

)(' iEE

ieS cccc

icc

cc

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Parameters in element of S-matrix： Phase of hard sphere scattering

Resonance width matrix

⎟⎟⎠

⎞⎜⎜⎝

⎛=

c

cc G

Farctanφ

)( ccc kaFF = )( ccc kaGG =

)( 3/13/1Tpcc AAra +=

2)(2 ccc EP λλ γ=Γ 2 2( ) cc

c c

kaP EG F

=+

Adjusting parametersAdjusting parametersAdjusting parametersAdjusting parameters

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Formula of angular distribution

Ω+

Ω+

Ω=

Ω dd

dd

dd

dd )3(

,'')2(

,'')1(

,'','' nnnnnnnn αααααααα σσσσ

nnnnCnn

vzZef αααα

αα δθµ

δθσ

,''4

2

2

2

,''2

)2(,''

2cosec

2)(

dd

⎟⎟⎠

⎞⎜⎜⎝

⎛==

Ω

( )] )(cos 1Reˆ

2cosec

2ˆˆdd

2sinln

2)(2

2

22

2

22,''

)3(,''

20

θ

θµ

δσ

θη

πσσ

αααα

lJlj

iii

ljJ

nnnn

PSeJ

vzZe

kIi

l

−⎢⎢⎣

⎡

=Ω

⎟⎠⎞

⎜⎝⎛+−−

∑

∑∞

=

+=Ω 0

)1(,'' )(cos)12(

41

dd

LLL

nn PBL θπ

σ αα

( ) ( )[ ]*,'''',''''

)(,'''',''''

)(

2121212121212

''''

122

'

2

2

22222222

'221

11111111

'11

2222211111

Re

)'';''''''();()12()12(ˆˆ)1(

Jjnljlnjnljln

iJjnljlnjnljln

i

JjljlJjljl

IiIi

L

SeSe

LIiJJjjllAiILJJjjllAJJIik

B

llllαααα

σσαααα

σσ δδ

π

−−

++−

=

+−+

−−+

∑

coherent of coulomb and reaction

Coulomb part

Reaction part

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APRML(characteristic) APRML considers all reaction channels. Total width is equal to the sum of all channels.

All parameters correlate. Adjusting one parameter, and all cross section changed.

cc

λ λΓ = Γ∑

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Content

APRML code Background Introduction of APRML code R-matrix theory in APRML

n+6Li reaction Reaction channel Input files and parameters Results

Summery

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Reaction channels

Experimental data

Structure parameters

CalculationComparison

Results

Adjusting parameters

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Particles outgoing process of n+6Li reaction

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n+6Li reaction channels choosed

The 8th channel use a fake 2-bodys channel instead. The 8th channel use a fake 2-bodys channel instead. The 8th channel use a fake 2-bodys channel instead. The 8th channel use a fake 2-bodys channel instead. n+n+n+n+6666LiLiLiLi****The 1st channel include elastic scatteringThe 1st channel include elastic scatteringThe 1st channel include elastic scatteringThe 1st channel include elastic scattering

((((n,tot)(n,el)(nn,tot)(n,el)(nn,tot)(n,el)(nn,tot)(n,el)(n, , , , γ)(n,inl)(n,p)(n,t)(n,a)(n,nda)(n,inl)(n,p)(n,t)(n,a)(n,nda)(n,inl)(n,p)(n,t)(n,a)(n,nda)(n,inl)(n,p)(n,t)(n,a)(n,nda))))

n+6Li→7Li*→

γ+7Li 0

n+6Li 1

p+6He 2

d+5He 6 7

t+α 3 5

n+d+α 8

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Experimental data used

Cross section (n,tot) EXFOR (n,el) EXFOR(in high energy)

ENDF/B-VII.1(low energy) (n, γ) ENDF/B-VII.1 (n,inl) ENDF/B-VII.1 (n,p) EXFOR (n,a) EXFOR, used (n,t) instead (n,nda) d outgoing data in EXFOR

Angular distribution DA of (n,el), 1964 R.O.Lane DA of (n,a), 1974 J.C.Overley

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Elastic scattering distribution in EXFOREntry 年代 作者 文献 能点

10415 1961 R.O.Lane (J,AP,12,135,196102) 0.225,0.235,0.245,0.255,0.265,0.275MeV

10710 1979 H.D.Knox (J,NSE,69,223,197902) 4.08,4.26,4.57,4.83,5.05,5.29,5.54,5.74,6.05,6.37,6.66,6.94,7.32,7.50

10854 1980 P.W.LISOWSKI (R,LA-8342,8010)(P,NEANDC(E)-194,78)

5.96,9.83

10904 1982 A.B.SMITH (J,NP/A,373,305,8201)(R,ANL-NDM-52,8002)

1.50,1.59,1.72,1.82,1.87,2.00,2.10,2.30,2.40……3.70,3.817,3.90,4.00

10914 1979 H.KNOX,R.O.LANE (C,79KNOX,783,7910)(J,BAP,23,942(DC2),7811)

2.30,2.80,3.31,3.83,4.08

11092 1976 H.B.WILLARD (J,PR,101,765,56) 0.210,0.258,0.300

11153 1968 J.C.HOPKINS (J,NP/A,107,139,6801) 4.83,5.74,7.50

11170 1964 R.O.Lane (J,PR,136,B1710,1964) 0.20,0.25,0.30,0.32,0.35,0.40,0.45,0.50,0.60,0.70,0.80,0.90,1.00,1.10,1.20……2.0

20376 1967 H.H.KNITTER (R,EUR-3454E,6704) 1.0,1.1,1.2,1.33,1.4,1.5……2.00,2.09,2.19,2.30

20492 1973 F.DEMANINS (R,INFN/BE-73/2,7306) 1.98,2.24,2.49,2.74,2.98,3.2,4.1,4.64

20749 1977 H.H.KNITTER (R,EUR-5726E,1,7704) 0.233,0.248,0.266,0.296,0.323,0.338,0.353,0.390,0.450,0.500,0.550,0.600,0.650,0.700,0.750,0.800,0.850,0.900,0.950,1.00,1.10,1.20,1.33,1.4,1.5……2.00,2.09,2.19,2.30,2.50,2.60,2.70,2.80,2.90,3.00

21147 1963 R.BATCHELOR (J,NP,47,385,6309) 3.35,4.00,5.15,6.36,7.54

21276 1971 M.CANCE (P,EANDC(E)-140,115,7108) 6.00

21986 1985 S.Chiba (J,NST,22,(10),771,198510) 4.2,5.4,14.2

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Angular distribution of (n,t) in EXFOR年代作者 能点 参考文献 Entry

2007 M.Devlin+ 1.10e6--4.50e6 C,2007NICE,,(563),2007 14159

2006 Guohui Zhang+ 1.05e6 1.54e6 2.25e6 J,NSE,153,41,2006 32651

2003 Guohui Zhang+ 1.85e6 2.67e6 J,NSE,143,(1),86,200301 32646

2000 Guohui Zhang+ 3.67e6 4.42e6 J,NSE,134,312,2000 32544

1988 S.Shirato+ 1.41e7 C,88MITO,,249,198805 22361

1988 A.Trzcinski 1.81e7 T,Trzcinski,1988 30943

1986 C.M.Bartle 2.16e6 1.37e7 J,RE,95,331,1986 30918

1982 H.G.Knox+ 2.00e6 3.50e6 J,BAP,27,703(DE12),8209 12974

1982 S.Higuchi+ 1.41e7 J,NP/A,384,(1),51,198208 21694

1981 J.C.Engdahl+ 2.28e4 J,NSE,78,44,198105 10755

1979 C.M.Bartle 2.16e6 9.66e6 J,NP/A,330,1,197910 10446

1977 E.Rosario-Garcia+ 4.71e6 7.25e6 J,NP/A,275,453,1977 10632

4.37e6 7.27e6 10632

1977 R.E.Brown+ 8.70e4 3.98e5 J,PR/C,16,513,7708 10866

1974 J.C.Overley+ 1.00e5 1.90e6 J,NP/A,221,573,197403 10382

1.00e5 1.90e6 10382

1967 D.Rendic+ 2.70e6 1.44e7 R,ZFK-130,143,196712 30193

1965 G.Robaye+ 2.50e5 3.90e5 6.00e5 C,65ANTWERP,,500(18),6507 20262

1963 Y.Baudinet-Robinet+ 1.90e5 J,JPR,24,803,6311 20196

1960 V.P.Perelygin+ 2.15e6 J,AE,9,(6),488,6012 40455

1959 S.J.Bame Jr+ 1.50e5 5.65e5 J,PR,114,1580,59 11103

1954 G.M.Frye Jr 1.41e7 J,PR,93,1086,195403 11071

1954 J.B.Weddell+ 1.10e6 1.50e6 2.00e6 J,PR,95,117,5407 11076

1953 L.E.Darlington+ 2.70e5 4.00e5 6.00e5 2.00e5 J,PR,90,1049,5306 11065

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Resonance energy level

-2.2175 -1.3967 -0.5758 0.245 1.82 3.18 3.51 3.57 5.17 5.9908 6.8117 7.6325

1minmax

−−

=∆N

EEE

Fake

Real

Fake

Fake energy level formula

EEEEEEEEEEEEEEEEEE

N

N

N

∆+=∆+=∆+=∆−=∆−=∆−=

+

+

+

32

23

min6

min5

min4

min3

min2

min1

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Parameters used in calculation

ridus 4 (n,el) non threshold， 9 (n, γ) non threshold ， 9 (n,inl) 2.186MeV， 4 (n,p) 3.183MeV， 3 (n,a) non threshold ， 9 (n,nda) 1.721MeV， 5

Total parameters: 43

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1st input file

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2nd input file

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Total cross section

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Total cross section

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Elastic scattering cross section

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Elastic scattering cross section

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Cross section of (n,t)

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Cross section of (n,p)

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Cross section of (n,dx)

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200keV 250keV

350keV

450keV

Elastic angular distribution

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600keV 800keV

1.5MeV 2.0MeV

Elastic angular distribution

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(n,a) angular distribution

100keV

150keV

260keV

1.00MeV

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(n,a) angular distribution

1.5MeV 1.7MeV

1.8MeV 1.9MeV

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Summary

APRML is a nuclear reaction code for calculating and fitting light nuclei cross sections, developed at CNDC and NanKai University.

APRML was compiled and adjusted less than a year, many mistakes and bugs have been fixed. The code has been compiled and some functions are under debugging and testing.

APRML is used to calculate and fit n+6Li reaction. Preliminary result is given. The shapes of cross sections roughly agree with experimental data, but the angular distribution result is not well.

In future, We will deal with the problems and fix bugs, and we hope this code will be used for light nuclei reaction evaluation.

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