The Effect of Enhanced δLi(α, n)11 B Reaction Rate on Primordial Abundance of 7Li

U P B

— an international journal

The effect of enhanced *Li ( a , n)^^B reaction rate on primordial abundance of ^Li

Titus K Mathew^ Sowmya Ramani* and N C Rana

inter U niversity Centre for Astronom y and A strophysics, Post B ag-4, Ganeshkhind, P u n e-4 1 1 0 0 7 , India

Received J4 December 1993, accepted 2R December 1993

A b s t r a c t : U sin g B oyd

et al

K e y w o r d s : Astrophysical 5-factor, lith iu m -a reactions, big bang nucleosynthesis.

P A C S N o s . : 9 5 .3 0 .C q , 9 7 .1 0 .C v, 9 8 .8 0 .-k

1. Introduction

The standard big-bang model [1] is thought to be a successful model for predicting the primordial abundance of nuclei up to and gives an arguably valid prediction for ^Li. But recent work [2] on "*He has raised questions about the agreement between theory and observation, even then the standard model fits well the observations to some extent. The ultimate test of the homogeneous model has to come from the prediction of the heavier elements like ** B. Heavier elements are produced mainly through the cycle [3]:

“He(^H,y)’Li (n,y)*Li (a,n)"B(n.y)'^B (p)'^C (n,y)"*C

The most important reaction in this series is ^i(a,n)** B which determines the abundance of

“ B and through which other heavy elements can subsequently form. To know the dynamics of the above series, it is very important to know the cross section and reaction rate of each

^ On leave from Department o f Physics. Cochin U niversity o f Science and T echnology, C ochin-682 0 22, India A sum m er school student from Department o f Physics. I IT, M adras-600 0 3 6 . India

© 19941ACS

component reaction. In the case of the reaction *Li(a,n) B, having a half-life o f ®Li as low as 840.3 ms» it is very difficult to produce it in the existing laboratory conditions. So what used to be done was to measure the reaction rate of the inverse process and apply the principle of detailed balance in order to infer the reaction rate of the forward process. Moreover, the temperature at which the reaction had to take place in the early Universe is quite low compared to the centre of mass energy for the reaction that can be carried out in the laboratory. So the only alternative that remains is an extrapolation of the laboratory data available in the high energy region to the low energy region of astrophysical interest.

This kind of extrapolation is usually done in astrophysics through the so-called astrophysical 5-factor formalism used by Wagoner, Fowler and Hoyle [1]. It is assumed that the astrophysical 5-factor for non-resonant processes is a slowly varying function of energy.

For the first time, Boyd et al [4] have been able to measure the direct reaction cross- section for ®Li(a,n)“ B using ®Li radioactive beams up to a C.M. energy of as low as 1.5 MeV, which shows that the 5-factor derived from the direct reaction is about 5-8 times larger than those obtained by Paradellis et al [5] from a study of its usual reverse reaction

** B(n,a)^Li. The strong dependence on energy and existence of several resonances are noted and therefore the assumption of existence of no resonance structure in the low energy region leading to the concept of 5(0) factor is basically invalid. But since the big bang nucleosynthesis took place in the energy range of 0.1 to 1 MeV, one has to extrapolate the data to get the correct value of the astrophysical 5-factor for the reaction.

In this work, using the latest data due to Boyd et al [4] we are doing an extrapolation by a new method and comparing the results with that obtained using the standard method.

The presentation of the work is organised as follows. In Section 2 we give the astrophysicist's way of calculating reaction cross section and reaction rates. In Section 3 we are presenting our calculation of the astrophysical 5-factor followed by the conclusion.

2. Nuclear reaction rate

If projectile nuclei of mass m \ each are allowed to hit target nuclei of mass m2 each, the reaction rate characterised by the average of the cross section and velocity,

100

Titus K Mathew, Sowmya Ramani and N C Rana

c a v > = J f{v)va{v)dv

0

where / (v) is the normalised Maxwell-Boltzmann distribution function of velocity of the reactant nuclei, since we are assuming that in the big bang model the nuclear reactions take place in a thermally equilibrium environment with the Maxwell-Boltzmann distribution of velocities of reactant nuclei. <J(v) is the cross section for the reaction. Both / ( v) and CT(v) can be expressed as

/ ( v ) =

2 n k ^ ] ^{exp 1 -}

2 k ^ ⁽²⁾

(3)

where // = (l/m | + l/m^X^, the reduced mass of the colliding system expressed in atomic mass unit (1 amu m„ == 1.6605 x 10'^^ Kg), kg is the Boltzmann constant, T is the temperature, 5(v) is the astrophysical 5-factor, a = ^ he is the atomic fine structure constant, c is the speed of light, Zje and Z2e are the electric charge of the reactant nuclei, is the permittivity of vacuum. Thus in the astrophysical contexts, eq. (1) can be written as

(o v ) = 47t

2trkgTj

J ^

0 2 * ^ J

dv. (4)

In terms of energy E, eq. (4) can be expressed as {av) =

niim

M2

n J

k^T b Te

dE

where,

I Itt e ^ Z .Z . b = v 2um„ ---

'' ^ " 4 e h

= 31.290Z,Z2/r 1/2 KeV.

(5)

(

)

The integrand of eq. (5) has a peak called the Gamow peak, whose maximum occurs at an energy Eq,and its width called the effective width or the Sommerfeld width is A. Both Eq

and A can be expressed as the functiqns of temperature : Eq = bk„T^{x 2 /3}

= 1.220 KeV,

A = 4 = 0.749 (z fZ 2 r 6 )‘'*KeV,

(7)

(8)

where is the temperature in units of 10* K. The value of the integral in eq. (5) has an approximated value as

(ov> = ^1/2 J7T “^eff (^ o) exp k ^ (9)

where SggiEo) is the effective 5-factor at the peak value

E

,

5ctr (^o) = -SCO)

5 k j S(0) / 35 , ^ 36 ^Eq 5(0) I ” 36 ®

(

10

where 5(0) is the astrophysical 5-factor at Eg = 0. provided S(E) is expanded in Taylor's series, 5 ( £ ) » 5(0) + £ 5 (0 ) + ( £ 7 2 ) 5 ( 0 ) + ...

102 Titus K Mathew, Sowmya Ramani and N C Rana

3 , Calculation of

In this section we are calculating the value of Scff(Eo) and hence S^ff (T), by extrapolating the data of Boyd et al [4]. The data of Boyd et al is given in Table 1. Since the data of Boyd et al

T a b le 1. Data o f Boyd

et al

£(M eV ) a(m b) 5 (£ )M e V b a m

1.62 2.00 2.19 2.38 2.76 2.95 3.24 3.52 3.81 4.09 4.38 4.66 4.86 5.24 5.43 5.71 5.90 6.19 6.47

381 509 519 545 462 419 472 381 276 424 414 271 333 305 133 195 104 186 162

1258 968 796 695 436 349 335 235 151 209 186 113 132 113 46 64 33 57 47

for ®Li(a, n)’’B reaction is available only up to the lowest energy of 1.5 MeV, so we consider the data from Paradellis et el [5] for the same reaction at energies less than 1.5 MeV, but we modified the data of Paradellis et al in view of the direct data of Boyd et al by multiplying it by an average factor that is derived from the comparison between the two data sets in the overlapping domain of the centre of mass energy. In the low energy range where there is no data available, we took S(E) as a constant.

T a b ic 2 . V alues o f integrand o f eq (5) at different values o f temperature.

£ i n MeV

S(E)

bam

a t r = 4 X 10* a t r = 8 X 10* a t r = 1 0 x l 0 ' '

0.2000 (-01) 0.5433 (05) 0.2168 (-03) 0.1644 (-03) 0.6527 (-04) 0.2500 (-01) 0.5433 (05) 0.2817 (00) 0.2167 (00) 0.8859 (-01) 0.5000 (-01) 0.5433 (05) 0.1251 (08) 0.1035 (08) 0.4889 (07)

0.1000(00) 0.5433 (05) 0.2000(13) 0.1913(13) 0.1208(13)

0.1500 (00) 0.5433 (05) 0.2698 (i5 ) 0.2983(15) 0.2518(15) 0.2000 (00) 0.5433 (05) 0.3746 (16) 0.4788 (16) 0.5401 (16) 0.2500 (00) 0.5433 (05) 0.1792(17) 0.2647 (17) 0.3992(17)

E i n M cV 5 ( E ) in M e V bam

atr = 4 X 10^ a t r = 8 x l 0 » a t r = 1 0 x l 0 ^ 0.5971 (00)

0.6119(00) 0.6545 (00) 0.7158 (00) 0.7652 (00) 0.7771 (00) 0.8747 (00) 0.9352 (00) 0.9563 (00) 0.1000(01) 0.1109(01) 0.1231 (01) 0.1269(01) 0.1366(01) 0.1494 (01) 0.1618 (01) 0.1999 (01) 0.2190 (01) 0.2380 (01) 0.2761 (01) 0.2951 (01) 0.3237 (01) 0.3522 (01) 0.3808 (01) 0.4094 (01) 0.4379 (01) 0.4665 (01) 0.4855 (01) 0.5141 (01) 0.5427 (01)

0.5433 (05) 0.4688 (05) 0.3228 (05) 0.2070 (05) 0.9863 (04) 0.2588 (05) 0.6310(04) 0.4246 (04) 0.3622 (04) 0.3083 (04) 0.2075 (04) 0.1770 (04) 0.2851 (04) 0.1919(04) 0.1396 (04) 0.1258 (04) 0.9678 (03) 0.7960 (03) 0.6949 (03) 0.4361 (03) 0.3493 (03) 0.3347 (03) 0.2350 (03) 0.1512(03) 0.2091 (03) 0.1865 (03) 0.1127 (03) 0.1318 (03) 0.1127 (03) 0.4645 (02)

0.1099(18) 0.8915(17) 0.4989(17) 0.2223 (17) 0.7536(16) 0.1812(17) 0.2018 (16) 0.7945(15) 0.5578(15) 0.3136(15) 0.7055 (14) 0.1644(14) 0.1752(14) 0.3930(13) 0.6421 (12) 0.1298 (12) 0.8569 (09) 0.6043 (08) 0.4358 (07) 0.1713(05) 0.1050 (04) 0.2062 (02) 0.2876 (00) 0.3582 (-02) 0.9389 (-04) 0.1560 (-05) 0.1730 (-07) 0.1397 (-08) 0.2149 (-10) 0.1575 (-12)

0.4444(18) 0.3763 (18) 0.2383 (18) 0.1268(18) 0.4963 (17) 0.1235 (18) 0.1826(17) 0.8567 (16) 0.6395 (16) 0.4082 (16) 0.1261 (16) 04181 (15) 0.4969 (15) 0.1478 (15) 0.3501 (14) 0.1016(14) 0.2023(12) 0.2479(11) 0.3105 (10) 0.3683 (08) 0.3922 (07) 0.1764 (06) 0.5634 (04) 0.1606 903) 0.9643 (01) 0.3669 (00) 0.9315 (-02) 0.1307 (-02) 0.4604 (-04) 0.7729 (-06)

0.5020 (19) 0.4632 (19) 0.3755(19) 0.2852 (19) 0.1486(19) 0.3964(19) 0 1032 (19) 0.6879(18) 0.5804(18) 0.4773 (18) 0.2782 (18) 0.1869(18) 0.2759(18) 0.1443(18) 0.7187 (17) 0.4294 (17) 0.7790 (16) 0.2880(16) 0.1089(16) 0.1176(15) 0.3781 (14) 0.8917(13) 0.1493(13) 0.2232 (12) 0.7024(11) 0.1401 (11) 0.1865(10) 0.7899 (09) 0.1458 (09) 0.1285 (08)

We calculated the integral of eq. (5) for different temperatures ranging from 0.2 x 10®

K to 22 X 10® K. We have presented a sample of our calculated data^ in Table 2. On having this data, we have integrated the equation graphically. The value of the integral at different temperatures are given in Table 3.

Next, we calculated the right hand side of eq. (9) without the factor 5eff(^o)» is,

<ov> / S ^ (Eq) at different temperatures and the results are tabulated in Table 4. We compare

68B-(2)

104

Titus K Mathew, Sowmya Ramani and N C Rana

Table 3 and Table 4 at corresponding temperatures and thu:

different temperatures. The newly found values of S'effC^

Table 3 . V alues o f the imegrai at different temperatures.

r X I C K Integral

0.6 2.01 X itr’*

4 5..34 ^X 10”

8 5 3.S X 10-”

10 I6.36X 10-^'

20 2.S.24X 10“

22 .34.92 ^X 10-“

T a b le 4. V alu es o f — — at different temperatures.

T ^X 10*K (<TV)

s^ff(£;,)

4 739,3 X l(T

8 ^{lO.W x} l(f

10 4064 ^X in'’

14 2.536 ^X 10“’

18 8626 ^X 10'"

20 1.39.3 ^X 10"

22 2115X 10"

T a b le 5 . V alu es o f (7 ) at different temperatures.

T

1 2.01.5 ^X 10*

4 2.027 X 10*

8 2.03.5 ^X 10*

10 2.038 ^X 10*

20 2.049 ^X 10*

22 2.051 ^X 10*

shown in Figure 1. On extrapolating the curves to lower temperatures, we found that the value of Seff (0) is 2.0 (±0.5) x 10^ MeV barn. The values of S^ffiT) are then calculated for different temperatures and are given in Table 5.

T in 10^ K

F ig u r e 1. Plot o f lo g

{ S ^ T))

T)

The new values of 5-factor will affect the old values of reaction rate of *^Li(a, n)‘' B substantially. According to Malaney and Fowler [3], the old reaction rate of ®U(a, n)*‘B in required astrophysical range of energies, is given by

N^{(Xv) = 8.62 X lo' Y ,'* exp 19.461 pl/3

^ 9A

2 -1 - I

cm s mole , (1 1)

where is the A\ ogadro number, Tg is the temperature in units of 10*^ K and

= Ta (

12

15.1

On comparing eqs. (11) and (9) one can find that the 5eff (£q) Malaney and Fowler, is 8.40 X 10*^ MeV barn. One can note the difference between our value and their value of

By incorporating our value of S^ffiEo) into the reaction rate equation instead of Malaney-Fowler's value, the reaction rate becomes.

N ^{av) = 2.05 X lO'Vg^® exp 19.46l"' 9A y

cm^j 'mole ' (13)

Probably due to the above reaction rate, the abundance of " B and ’Li should also change.

106 Titus K Mathew, Sowmya Ramani and N C Rana

We modified Wagoner’s [6] code by incorporating our new reaction rate for

*Li(a,n)*‘ B and also some new important reactions of Li, which were not included in a recent work by Smith

et al

Acknowledgments

TKM and SR are thankful to lUCAA for the support and hospitality during their stay there.

References

[1] R V Wagoner, W A Fowler and F Hoyle Astrophys. J. 148 3 (1967); T P Walker, G Stcigman, K A Olive and H Kang Astrophys 7 281 493 (1984)

[2] G J Mathews, R N Boyd and G N Fuller Astrophys J. 403 65 (1993) [3] R A Malaney and W A Fowkr Astrophys, J Utt. 345 L5 (1989)

[4] R N Boyd, I Tanihata, N Inabc. T Kubo, T Nakkagawa, T Suzuki, M Yonokura, X X Bai, K Kimura.

S Kubono, S Shimoura, H S Xu and D Hirala Phys, Rev. Lett. 68 1283 (1992)

[5] T Paradeliis, S Kossionides, G Donkellis, X Aslanoglou, P Assimakopolous, A Pakou, C Rolf and K Lanankc Z Physik. A337 221 (1990)

[6] R V V^Sigoncr Astrophys. J. Suppl. Ser 18 247 (1969)

[7] M S Smith, L H Kawano and R A Malaney Fermilab preprint (1992)