ANCs for 10B+n/p configurations from the (d,t) and (3He,d) reactions and their using for calculation of the astrophysical S-factors of nucleon radiative capture by 10B

(1)

ANCs FOR 10B+n/p CONFIGURATIONS FROM THE (d,t) AND (3He,d) REACTIONS AND THEIR USING FOR CALCULATION OF THE ASTROPHYSICAL S-FACTORS OF NUCLEON RADIATIVE

CAPTURE BY 10B

S.V. Artemov1_{, N. Burtebaev}2_{, G.K.Nie}1_{, R. Yarmukhamedov}1_,

S.B. Igamov1_{, A.A. Karakhodzhaev}1_{, A.G. Bajajin}1_{, D.T. Burtebaeva}2_,_E.A.

Zaparov1, M.A. Kayumov1, G.A. Radyuk1, G.S. Isabekova2,

N. Amangeldy2_{, D.M. Zazulin}2_{, O. Juraev}1

1 Institute of Nuclear Physics, Academy of Sciences, Tashkent, Uzbekistan

2 Institute of Nuclear Physics, National Nuclear Center, Almaty, Kazakhstan

(2)

Motivation

• Possibility to obtain the reliable ANC (NVC) with use

of the modified DWBA analysis of the peripheral

nucleon transfer reactions and some additional

information;

• Finding-out how correlate the ANCs for mirror states

of the neighbour nuclei;

• Large interest to the astrophysical nuclear reactions

and to the possibility of ANC using for astrophysical

(3)

Contents of the report

• Experimental method and the measured differential cross sections of the 11B(d,t)10B and 10B (3He,d)11C reactions

(E_d= 25 MeV; E_3He=34 MeV – Cyclotron U-150M INP NNC RKaz).

• Results of analysis of the experimental differential cross sections within the framework of the modified DWBA approach taking into account the method of equivalent nuclear potentials (EPN-method).

• Results of calculation of the S-factors S(E) for the direct nucleon radiative capture reactions 10_B(n,)11_{B and}10_B(p,)11_{C at extremely low}

(4)

Two-dimensional ΔE-Е - spectrum and spectra of the projections

(single-charged particles)

10

_B+

3

_{Не (34 MeV, 16°).}

(5)

Modified DWBA method of the analysis

For the peripheral nucleon transfer reaction A(x,y)B, x=y+N

and B=A+N:

(1)

(2)

Here the C’s are the ANCs at two relevant vertexes, the b’s

are the single-particle ANCs for (A+N) and (y+N) proton

bound state wave functions.

),

,

(

)

,

(



2

2 



E

C

R

E

x x B B x B

j

l

j

l

yN

j

AN





2 2

)

,

(

)

,

(

X j X YNl B j B ANl DW x j x l B j B l x x B B

b

E

j

l

j

l

E

R









(6)

Criteria of peripheral character and one-step

mechanism prevailing in the particle transfer reactions

i).

for {r

0

, a

diff

} within the “physically reliable” intervals.

ii).

at the E

proj

values where the stripping mechanism is dominated.

So one should check the behavior of the R(b) functions and compare

the obtained ANCs at several projectile energies (or reaction

channels).

.

)

,

(

E

const

R

_b _peak N A





.

)

,

(

)

,

(

)

(

exp 2

_const

E

R

E

C

peak b peak er N A N A



 





(7)

We consider the spectroscopic factors of the mirror states for

11

B –

11

C nuclei are equal.

Then from the relation C

2N

=S

N

b

2N

one obtains

(G.K. Nie. Bull. RAS.

Ser. Phys. (2005)

):

(b

_n

/b

_p

)

2

= (C

_n

/C

_p

)

2

.

This is a consequence of the condition of the equivalence of

the proton and neutron nuclear potentials in mirror states

(EPN) and it turned out that the ratio is stable to the wide

variety of the parameters of the binding potential.

(8)

Analysis of the differential cross sections of the

11

B(d,t)

10

B and

10

B (

3

He,d)

11

C reactions

and

obtaining spectroscopic information

Analyzed experimental data:

E

_d

=18 MeV

[ I.R.Gulamov, A.M.Mukhamedzhanov and G.K.Nie. Yad. Fiz. 1995 ]

and 25 MeV

(this work)

E

_3He

=21 MeV

[J.R. Comfort et al. Phys.Rev. 1971]

(9)

Experimental differential cross sections of the reaction 11_В(d,t)10_{B (E*=0.0)}

at the energies Е_d=18 and 25 MeV. The calculated cross sections (curves) are given with different sets of the optical potentials

(10)

Testing functions for the reaction 11_B(d,t)10_{B at Ed =25 MeV}_.

The function R(b) is weakly sensitive to a variation of the values of b, so the obtained empirical values VC and ANC do not depend on the geometry

(11)

Differential cross sections of the

reaction

10

В(

3

He,d)

11

C. The curves are the calculated

cross sections for different runs

of the optical parameters.

(12)

Testing functions for reactions 10_B(3_He,d)11_{C at E}

3He = 34 MeV

For the chosen energies the function R(b) is weakly sensitive

to variation of the values of b, so the obtained empirical values VC and ANC do not depend on the parameters of the bound state potential.

(13)

Bound state potentials

Configuration, (Е*, J₎ l, j

r

C

,

fm V₀,MeV r₀, fm; a=0.65 fm B_1j, fm -1/2 11_B→10_{B+n (0.00, 3/2-)} 11C→10B+p (0.0, 3/2-)

1, 3/2 1.3 -55.457

_1.217

4.27 _4.77

11_B→10_{B+n (4.44, 5/2-)} 11_C→10_{B+p (4.32, 5/2-)}

1, 1/2 1.3 -53.909

_1.267

2.443 _2.634

11_B→10_{B+n (6.48,7/2-)} 11_C→10_{B+p (6.74, 7/2-)}

1, 1/2 1.3 45.720

_1.312

1.709 _1.876

(14)

Е_3He, MeV States, (Е*, J₎ l, j b 1j, fm-1/2 G2_, fm C2 , fm1 S 21 11C→10B+p (0.0, 3/2-) 11_B→10_{B+n (0.0, 3/2-)} 1, 3/2 1, 3/2 4.9104.270 6.85.1 40.330.5 1.71.7 34 11_C→10_{B+p (0.0, 3/2-)} _{1, 3/2 4.910} _1.50 _8.9 _0.39 21 11_C→10_{B+p (4.32, 5/2-)} 11_B→10_{B+n (4.45, 5/2-)} 1, 1/2 1, 1/2 2.6342.443 0.350.30 2.071.78 0.270.30 34 11C→10B+p (4.32, 5/2-) 1, 1/2 2.742 0.19 1.14 0.15 21 11_C→10_{B+p (6.48, 7/2-)} 11_B→10_{B+n (6.74, 7/2-)} 1, 1/2 1, 1/2 1.8761.709 0.620.51 3.693.06 1.181.05 34 C→ B+p (6.48, 7/2-) 1, 1/2 1.769 0.21 1.28 0.38

Empirical values VC

G|

2

_{, ANC C}

2

_{and SF S of the mirror}

configurations

11

_B



10

_{B+n and}

11

_С



10

_{B+p for the bound}

(15)

Direct astrophysical S–factor and the

10

B(n,

)

11

B reaction rate

Here we present the results of calculation of the astrophysical

S–factor of the direct radiative capture 10_B(n,)11_{B reaction} populating the ground (0, J=3/2-) and two excited (E*=4.44 MeV; J=5/2-) and E*=6.74 MeV; J=7/2- ) states of the nucleus 11_{B with taking into account the М1-, Е1- and Е2 – transitions.}

(16)

Direct astrophysical S–factor and the

10

B(n,

)

11

B reaction rate

Here we present the results of calculation of

the astrophysical S–factor of the direct

radiative capture reaction

10

B(n,

)

11

B

populating the ground (0, J



=3/2-) and two

**excited (E*=4.44 MeV; J**



**=5/2-) and E*=6.74**

MeV; J



=7/2- ) states of the nucleus

11

B. The М1-, Е1- and Е2 – transitions were taken

into account

(17)

The partial astrophysical S-factor for the radiative

capture of a neutron

A(n,)B

is related to the cross

section as

),

(

)

(

E

k

2 E

S

_l





l



_l

where

l

is the orbital angular momentum in the scattering

channel,

k

and

ν

are relative momentum and velocity

of the neutron

n

and the nucleus

A

in the initial state

and

σ

is a partial cross section.

(18)

.

)

(

)

(





2 l

l

_S

_E

k

E

S

The total astrophysical S – factor S(E) is determined as

In the framework of the modified two-particle potential

approach, the partial

astrophysical S – factor S(E) is

presented as

),

,

(

)

(

2

B B B

ll

l

E

C

R

E

b

S



(19)

The results of calculations of the partial (total) astrophysical S

– factor for the direct radiative capture 10B(n,)11B reaction

going via the E1 – transition from the initial state with l=0

into the bound states with l

_B

=1 (the M1-, E1- and E2-

transitions) are presented in the following figure

(20)

(21)

10-3 10-2 10-1 100 101 106 107 108

N

A

<



>

(

cm

3

m

ol

-1

s

-1

)

T₉

Dependence of the calculated

10

B(n,)

11

B reaction rate on the

(22)

Deduction

1. Dependence of the partial astrophysical S – factors on

the variable energy E is weak.

2. The total astrophysical S – factor:

a slight enhance in the energy region with E>1 MeV,

can be associated with increase of contributions of the

M1-, E2 – transitions and E1 – transition from the initial

(23)

Calculation of astrophysical S – factors and

rates of the proton radiative capture in the

10

B(p,γ)

11

C reaction within the R-matrix

(24)

Astrophysical S - factor for the radiative capture reaction 10_B(р,)11_С. Contribution of the direct radiative capture is shown by dashed line. ANC 11_С _→10_{B+p C}2_{=30.5 fm}-1_{was used.}

(25)

Parameters of the resonances

*

)

Wiescher M., et al. Nucl. Phys. – 1983

Resonance states of

C

11

J



_{, E}

res

(MeV)

Total widths

Г

_i

(keV)

Resonance

strengths

(eV)

Adopted by us

From

*

)

5/2

+,

_0.010

_15.

_14.55

_3.84×10

-17

5/2

+,

_0.56

_500.

_0.5658

3/2

-,

_1.0

_250.

_209.

_2.880

5/2

-,

_1.2

_150.

_236.

_0.440

7/2

-,

1.41

100.

118.

0.012 3/2

-,

_{2.5 (background)}

_50.

_-

₂₀₀

(26)

CONCLUSIONS

• The experimental differential cross sections have been measured:

11В(d,t_G.S.)10В reaction at the energies E_d= 25 MeV

10В(3He,d)11C reaction at the energy E_3He =34 MeV

(E*11C = 0.0 MeV, 3/2-; 4.32 MeV, 5/2- and 6.48 MeV, 7/2-).

• It is shown that these reactions are pure peripheral.

• It is found that in the 11В(d,t)10В (g.s.) reaction the pole diagram of the proton

transfer dominates.

• The values of ANC and S have been obtained although the dominance of the pole

diagram of the proton transfer in the 10В(3He,d)11C (E*=0.0, 4.44, 6.478 MeV) at the

studied energy in 34 MeV is under question.

• It is shown that the relation of the quantities (bn/bp)2=(Cn/Cp)2, in which the value

(bn/bp)2 is calculated theoretically at the EPN-condition, is true when the pole

diagram of the nucleon transfer dominates.

• The astrophysical S - factors at extremely low energies and rats of the

reactions of radiative capture 10B(р,)11С and 10B(р,)11B have been

calculated. A new data about resonant widths and the radiative strengths, which are

(27)