IL NUOVO CIMENTO

VOL. 101 A, N. 5

Maggio 1989

The Nuclear Fusion for the Reactions

2H(d, n) 3He, 2H(d, p) 3H, 3H(d, n) 4He. A. SCALIA D i p a r t i m e n t o di F i s i c a dell'Universita - C a t a n i a I N F N - Sezione di C a t a n i a

(ricevuto il 6 0 t t o b r e 1988)

Summary. - - In the present paper the elastic model for the subbarrier fusion is applied to the reactions 2H(d, n)3He, 2H(d, p)3H, 3H(d, n)aHe. A comparison between the values of the reaction rates obtained in the framework of the elastic model and those reported in the literature is shown. PACS 25.70 - Heavy-particle-induced reactions and scattering.

1. - I n t r o d u c t i o n .

The deuterium-induced reactions on deuterium are relevant for the primordial nucleosynthesis during the early stages of the Universe; these reactions and the deuterium-induced reactions on tritium are important, for the concept and design of future fusion reactors. In the present paper we apply the elastic model for the subbarrier fusion to the above-mentioned reactions. A comparison between the values of the reaction rates obtained in the framework of the elastic model and those r e p o r t e d in the literature is shown.

2. - T h e e l a s t i c m o d e l . In the elastic model for the subbarrier fusion(') we assume that i) t h e r e exists an angle of< ~. at which the fusion begins, ii) when the fusion begins it becomes the dominant process,

(') A. SCALIA: N u o v o C i m e n t o A, 98, 571 (198). 795

796

A. SCALIA

so that the particles which fuse are those scattered, in the Rutherford scattering, in the angular region (Of,=) and the fusion cross-section can be written as ~f=

(1)

~R(O)sinOdO=r: -~

ctg2~=~bf,

of

where zR(0) is the Rutherford differential cross-section, v is the Coulomb parameter, K is the wave number, bf is the impact parameter corresponding to the critical angle Of. The angle Ofin the Rutherford scattering can be expressed as (2)

Of= 2 arcsin

so that eq. (1) can be written (3)

~f = r:R~(1 \

Z1 Z2 e2 2ERr - Z1 Z2 e2'

Z1ZEe2/Rf),

where E is the centre-of-mass energy, Rf is the distance of the closest approach of the Rutherford trajectory with b = bf. Following refo (1) R~ can be expressed as

(4) ao = - ~ ,

y-

EB - E

~

,

G(y) = exp [ - exp [exp[y]]],

EB and Es are two parameters expressed in MeV, 2a0 = 2)7/K is the minimum value of the closest approach distance (it is obtained when 1 = b = 0 or 0 = =). The function G(y) is the same for all the systems. G(y) and y are dimensionless. By using eq. (4), eq. (3) can be rewritten (5)

~f= :~

+ G(y))G(y),

~o_

[2V ~2 f - r:l-77I,

EB and Es are determined by fitting the experimental values of fusion crosssection. In ref. (1~)we considered systems with 24 ~
797

THE NUCLEAR FUSION FOR THE REACTIONS ETC. 10

0

10 -1

"~. I0 -2

10-3

10-4

510

I

100 E (keV)

I

I

150

200

Fig. 1. - Comparison between the experimental excitation function (4,5) and the calculated one by using eqs. (4), (5) for the reaction eH(d,n)3He, ~-I(d,p)3H.

I0

10

0

10-1

lO- 2

10

30

50 E (keY)

70

90

Fig. 2. - Same as fig. 1 for reaction ~H(d, n)4He (6).

(4) W. R. ARNOLD, J. A. PHILLIPS, G. A. SAWYER, E. J. STOVALLjr. and J. L. TUCK:

Phys. Rev., 88, 159 (1952). (5) G. PRESTON, P. F. D. SHAW and S. A. YOUNG: Proc. R. Soc. London, Ser. A, 226,206 (1954). (6) E . J . STOVALLjr., W. R. ARNOLD, J. A. PHILLIPS, G. A. SAWYER and J. L. TUCK: Phys. Rev., 88, 159 (1952).

A. SCALIA

798

The values of EB, Es are shown in table I. TABLE I. System

EB (MeV)

Es (MeV)

~H + 2H 2H + 3H

0.196 483 9 0.0417788

0.225 029 7 0.0372916

In ref. (3) we showed that the parameters EB, Es can be connected to the properties of nuclear part of the nucleus-nucleus potential. Following ref. (2.3)the nuclear part of the nucleus-nucleus potential can be expressed for r < Rf as Z(E) r

VN(r, E) = - 7 - [exp [(r - Rf) a(E)] - 1]

(6)

with V

-i

2

(7) Rf = --~ (1 + G(y)),

G(y) = exp [ - exp [exp [y]]],

Y = EB -.___.__EE Es

Now we consider the ratio VN(r, (EB-Es)/2)/VN(r, EB) when r ~ 0 , eqs. (6), (7) it follows (8)

lim Too

from

gN((r, (EB -- Es)/2) EB (1 + 2G(1/2)) 2 VN(r, EB) -- (EB -- Es)/2 1 + 2G(0)

exp [ - 2 1 + G(1/2) ] _ 1 L

~--2G-~-I72-)J

i

1 + G(0) ] _ exp - 2 1 + 2 G ( 0 ) j 1 Equation (8) connects the properties of the potential VN(r, E) to the parameters EB, Es. The values of lira~ [VN(r, (EB -- Es)/2)/VN(r, BB)] vs. n l + n2, where A1 and A2 are the mass numbers of the projectile and target, respectively, are reported in fig. 3. The values of EB and Es for the systems considered in this work are those of table I; for the other systems the values of EB and Es are reported in ref. (3). By inspection of fig. 3 it follows that the expression lim(VN(r, (EB -- Es)/2))/ (VN(r, EB)) can be rewritten (9)

lirm

VN(r, (EB -- Es)/2) = 1 + 2 exp VN(r, EB)

+ A22.789 ----f(A1 -{-Ae), (56) 3/4

799

T H E N U C L E A R F U S I O N FOR T H E REACTIONS ETC.

g

V

---

w

@

5..~c_ ?

i

0

20

i

L

I

I

80

120

I

I

160

I

I

I

200

240

AI +A 2

Fig. 3. - The function lim~ VN(r, (EB - Es)/2)/VN(r, EB) defined in eq. (8) vs. As + A2 for the systems of table I. Solid line: the function f(A, + A2) defined in eq. (9).

the function ? ( n 1 -4-A2) is reported in fig. 3. From eqs. (8), (9) it follows

(10)

2 1

EB

(1 + 2G(1/2)) 2 exp [ - 2 ~ +-2G-G-~)J

d = -~s =

exp - 2 i ~ 12 ~+ ]G(0) -1 ~

(l+2G(0)) 2

3. - N u c l e a r

reaction

1

1 f(As + a2)

rate.

The reaction rate in a thermal equilibrium distribution by weighting the cross-section ~ by the Maxwell-Boltzmann velocity is given

(11)

r12 = nl n2 (o-v}/(1 + ~12)

with (12)

( av } = [8/r~M12(kT)S]1/2; a(E) exp [ - E/kT] E dE, o

where M12 is the reduced mass of particles 1 and 2, nl and n2 are the number

800

A. SCALIA

densities of particles 1 and 2, v is the relative velocity, E is the centre-of-mass energy, k is Boltzmann's constant, T is the temperature. We note that the mean lifetime of particles 1 against destruction by particles 2 is given by (13)

T2(1) = 1



n2

and the rate of nuclear energy production is given by (14)

~12= Q12r12,

Q12 is the Q-value of nuclear reaction. For nonresonant reactions the cross-section it is written usually (~) (15)

z(E) = [S(E)/E] exp [ - 2 ~ ] ,

with (16)

2 ~ = 2r: Z1 Z2 e2/hv = (EG/E) 1/2,

E G is the Gamow energy, v is the Sommerfeld parameter, Z1 and Z2 are integral nuclear charges of the two particles, S(E) is the nuclear or astrophysical S factor. By using eqs. (15), (16), eq. (12) becomes (17)

= [8/=M12(kT)3] le f S(E) exp -

-

dE.

o

The energy dependence of the integrand is determined predominantly by the product of the Maxwell-Boltzmann factor and the penetration factor, leading to a peak of the integrand with a width AE1 and with its maximum at energy El. For S(E) = $1 we have

I E1 -- [r~e2Z1 Z2 kT(M12 c2/2) 1/2hc2]~8 , (18) AE1 = 4(E1 kT/3) '/2 and the cross-section factor is given by (19)

= (2/M12) 1/2AEi(kT) -8/2$1 exp - -3El ~ - j1.

(7) D. D. CLAYTON:Principle of Stellar Evolution and Nucleosynthesis (McGraw-Hill, New York, N.Y., 1968).

THE NUCLEAR FUSION FOR THE REACTIONS ETC.

80t

If we use the elastic model the cross-section z is equal to zf, see eq. (5), and eq. (12) gives for the cross-section factor(8) d

(~v> = M(T) f IT(y) dy

(20) with

( 8 ~ ) 1 / 2 ( Z 1 Z 2 e2)2

M(T) -

(21)

M~e(kT)~/2

'

IT(y)= d l~_yG(y)[l + G(y)]exp [ EB

EB--yEs.]

kT

J'

E~ - E

d=--~s,

Y-

- ~ ~ y ~d.

Es

'

For fixed T the least solution in y of eq. (3) (22)

1 Ts - - d y + -T- = Ml(y)[(1 + 2V(y))/(1 + G(y))]

where (23)

Ts = Es/k ,

MI(y) = [exp [exp [y]]] exp [y]

gives the value Yo at which I~(y) attains the maximum value, so that we have for the cross-section factor

( ~ > = M(T) IT(yo) Ayo.

(24)

In ref. (3) we showed that for kT < E s it is (25)

hyo 81/2~ d2 [

d

;

a-yo

[~y2 L -

73-,/2 JJy=yo

,

so that the total reaction rate is given by

(26)

r12 =

(1 + ~12) -1 n l

89

IT(yo) Ayo,

M(T), IT(yo) are defined in eq. (21), Ayo is defined in eq. (25). For kT < Es eq. (25) is not true. Now we note that the cross-section factor can be rewritten as

(27)

(~v> = M ( T ) f I~(y)dy=M(T)

IT(y)dy +

IT(y)dy .

802

A. SCALIA

TABLE II. 2H + 2H

System T9 (9 0.100 0.273 0.400 0.600 0.800 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000

E1 (9 (keV)

Eo (9 (keY)

NA
NA (*)

NA@V> (0

(em3 mo1-1 s -1)

(em~mo1-1 s -1)

(em3mo1-1 s -1)

26.25 51.39 66.16 86.70 105.03 121.88 193.47 253.52 307.12 356.38 402.44 446.00 487.53 527.35 565.72

30 74.96 111.19 137.07 158.23 224.61 263.99 290.99 311.25 327.22 340.28 351.53 359.18 367.51

4.151. lo 6 8.222.106 1.234.107 1.732.107 2.164.107 3.254.107 3.410.107 3.677- 107 3.015- 107

4.85.105 3.85- 106 Tree) 7.50.106 1.34.107 1.94- 107 2.52.107

5.18.107 7.4 9107 9.27- 107 1.08.10 s 1.22. lOs 1.34. l0 s 1.44.10 s 1.53- 108 1.60.10 s

2.846" 107 2.681- 107 2.522- 107 2.366- 107 2.221 9107

(a) T9 temperature of reactans in unit of 109K. (b) E1 is defined in eq. (18). (c) Eo is defined in the text. (d) NA(~v} is obtained by using eqs. (21)-(25). (e) NA(cv} is obtained by using eq. (21)-(23), (28). (f) Na(~v) is given in ref. (8). (g) Tm is the minimum value of the temperature at which eq. (22) has solution.

W e showed in ref. (~) t h a t for kT < Es I~(y) can be a p p r o x i m a t e d b y a Gaussian function. F o r kT~: Es I~(y) can be a p p r o x i m a t e d b y a Gaussian function for Yo < Y < d; for - ~ < y < Yo this is not possible. W e note t h a t w h e n kT increases Y0 becomes smaller and smaller and G(y) goes to the saturation, so t h a t eq. (27) can be r e w r i t t e n

(I~[EB-E\(

(28)

=M(T)

V

(EB-E~expI_~TJdE =

1 +V\~))exp

-

dE +

(8) G. R. CAUGHLAN, W. A. FOWLER, M. J. HARRIS and B. A. ZIMMERMAN:At. 197 (1985).

Nucl. Data Tables, 32,

Data

THE NUCLEAR FUSION FOR THE REACTIONS ETC.

803

+ ~IG(EI3-E~(I+G(EB-E~exp[--~]dE} E-tI~-s ] \~]] r Eol o ~-M(T)~oG(EB~EsE~ "Es }}expLJ-7-+

G(EB-E~176 }t

+M(T) t

r o ~-M(T) ~ Es ]~ 1 +Gt/EB-Eo Es ))l-~oeXp[--~}+~'(Eo)}"

TABLE III. System

2H + 3H

T9(o)

E1 (b) (keV)

Eo(<) (keV)

NA(cv} (d) ( c m 3 m o l - l s -1)

0.010 0.015 0.030 0.060 0.200 0.400 0.600 0.800 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000

6.01 7.87 12.51 19.85 44.29 70.31 92.14 111.62 129.52 205.60 269.41 326.37 378.72 427.66 473.95 518.07 560.39 601.17

6.18 14.18 22.76 40.47 51.88 58.75 63.37 66.62 75.90 80.19 82.61 84.29 85.48 86.34 87.01 87.57 88.02

2.294- 104 5.391- 105 9.163.106 1.643. l0 s 3.065. l0 s 3.374. l0 s 3.319. l0 s

N A ( ~ } (") (cmamol-ls-1)

NA(CO} (r) ( c m S m o l - l s -1)

3.193. l0 s 2.409. l0 s 1.896. l0 s 1.449. l0 s 1.188- l0 s 1.001 9 l0 s 8.599.107 7.507.107 6.643.107 5.940.107

1.51.103 1.70. l04 Tin(9 5.18.105 7.97- 106 2.14- 108 4.35. l0 s 5.13. l0 s 5.27.108 5.17. l0 s 4.23.10 s 3.53.10 s 3.04. l0 s 2.69- 10s 2.42. l0 s 2.21. l0 s 2.04. l0 s 1.90- l0 s 1.78.10 s

(a) T9 temperature of reactans in unit of 109K. (b) E~ is defined in eq. (18). (c) Eo is defined in the text. (d) NA(~} is obtained by using eqs. (21), (22), (23), (24), (25). (e) NA(~v} is obtained by using eq. (21), (22), (23), (28). (f) NA(av} is given in ref. (8). (g) Tm is the minimum value of the temperature at which eq. (22) has solution.

804

A. SCALIA

F r o m above arguments follows that the reaction rate for k T ~ E s (29)

is given

rlz = (1 + ~?12)-lnln2M(T)G (EB--Eo~.

Es]

9(I+G\(EB : E0~ f/~Eo

F Eol + (~l(Eo)}"

E0 is the value of the energy at which the function I~((EB - E)/Es) attains the maximum value, ~I(E) is the exponential integral. A comparison between the values Of NA(~V> obtained by using eqs. (21), (25), (24), (28)and those obtained by using eq. (17) is shown in tables II, III. Na is Avogadro's number.

9

RIASSUNTO

I1 modello elastico per la fusione sotto la barriera Coulombiana viene applicato alle reazioni 2H(d,n)3He, ZH(d, p)SH, 8H(d, n)4He. Un confronto tra i valori del ,,reaction rate, ottenuti usando il modello elastico e quelli riportati in letteratura viene effettuato per le reazioni in esame.

Pea~nlxx

aaepnoro

cx~reaa

2H(d, n)3He, 2H(d, p)3H, 3H(d, n)4He.

PeamMe (*). - - B aTOfipa6oTe ynpyra~ MO~IeJIbJIJ/a noa6apr, epaoro crIrlTe3a npviMenaeTc~i K peaKanaM ell(d, n)3He, 2H(d, p)3H, 3H(d, n)gHe. Hposoanrcn cpaaaeune pe3abTaTOa aJIn CKOpOCTefipearunfi, KOTOp1,IenosIyqenbi a paMKaX yIIpyrofi uoaenn, c aarInb~Mn, nueromMncn B JirrrepaType /Ina paccMoTpeIm~,LXpearamfi.

(*) Hepe6ec)eno pec)ata~uefi.