1
Thermonuclear Reactions
Non-resonant Reaction Rates
Schrödinger equation:
h̄2 2
− ∇ Ψ + V (r) Ψ = EΨ.
2µ
µ = m1m2/(m1 + m2), E is incident kinetic energy.
V (r) =
Z1Z2e2/r
−V0
r>R
r<R
χ (r) m
Ψ= `
Y` (θ, φ)
r
Assume radial symmetry, ignore angular parts. Define χ:
#
"
2
2
h̄2 00
h̄ 00 ` (` + 1) h̄
+ V (r) − E χ` (r) = − χ` +f (r) χ` (r) = 0.
− χ` +
2µ
2µ
2µr2
<0
r > R0
,
>0
r < R0
The turning point R0 is defined by:
f (r)
Z1Z2e2 ` (` + 1) h̄2
+
E=
.
R0
2µR02
Define φ:
χ` (r) = Aeiφ(r)/h̄ ;
ih̄φ00 − φ0
2
+ 2µ [E − V (r)] = 0.
Lowest order approximation: Neglect φ00:
p Z rp
E − V (r 0)dr0,
φ (r) ' ± 2µ
R
which is valid as long as r is not near R0. The integrand is real
for r > R0 and imaginary for r < R0. The constant A is set
2
by normalization, so χ∗(r)χ(r) gives the probability per unit
radial distance that incoming nucleus is at r. The penetration
factor P`(E, r) is then
!
Z rr
∗
χ (R) χ (R)
2µ
0 ) − E]dr 0 .
P` (E, r) = ∗
∝ exp −2
[V
(r
χ (r) χ (r)
h̄2
R
Setting R ≈ 0, r ≈ R0 (the solution oscillates for r > R0), and
for the case ` = 0, the integral is
√
Z R0 r
2µ
2µE Z1Z2e2 π
Z 1 Z 2 e2
0
0
=π
,
[V (r ) − E]dr =
2
h̄
E
2
h̄v
h̄
0
using v 2 = 2E/µ. This forms the Gamow factor.
Cross Section: (Reactions/s)/(Incident Flux), units of area.
Number of reactions per unit volume per unit time is n1n2vσ
at a given energy (relative velocity v). Integrating over all energies and correcting for double counting of identical particles:
Z Z
3
3
r=
n1 (v1) n2 (v1) vσ (v) d v1d v2/ 1 + δ1,2 .
Maxwellian distributions:
Note that
m 3/2 m1v12
1
n1 (v1) = n1
e− 2kT .
2πkT
Z ∞
n1 (v1) d3v1 ≡ n1.
0
The relative velocity v and the center-of-mass velocity V are:
m1
m2
v,
v2 = V +
v.
v1 = V −
m1 + m 2
m1 + m 2
n1 (v1) d3v1 n2 (v2) d3v2 =
#
"
3/2 (m +m )V 2
µ 3/2 µv2
1
2
m1 + m 2
e− 2kT
e− 2kT d3v .
d3 V
n1 n2
2πkT
2πkT
3
Integration over d3V of the first bracket is unity, and so
Z ∞
µ 3/2 µv2
n1 n2
e− 2kT d3v
r=
vσ (v)
1 + δ1,2 0
2πkT
r
Z ∞
Eσ (E) − E
n n
8
= 1 2
e kT dE.
1 + δ1,2 µπkT 0
kT
The Gamow factor is most significant part of nuclear cross
sections. Also, since maximum quantum mechanical geometrical cross section ∝ λ2 ∝ 1/E, it is convenient to write
S (E) −b/√E
,
e
σ (E) =
E
where S(E) is slowly varying. For ` = 0, A = µ/mb ,
√
√
π 2µZ1Z2
b=
= 31.3Z1Z2 A keV1/2.
h̄
r
Z ∞
√
8
1
n1 n2
E dE
−E/kT
−b/
S (E) e
r=
1 + δ1,2 µπkT kT 0
r
2 !
Z ∞
n n
E − E0
8
1 −3E0/kT
= 1 2
exp −
e
S (E0)
dE
1 + δ1,2 µπkT kT
∆/2
0
r
2 ∆ −3E0/kT
n n
e
S (E0) .
= 1 2
1 + δ1,2 µπkT kT
We approximated the integrand as a Gaussian with centroid
1/3
bkT 2/3
2
2
2
E0 =
= 1.22 Z1 Z2 AT6
keV
2
and width (note T6 = T /106 K)
r
E0kT
2 2
5 1/6
= 0.75 Z1 Z2 AT6
keV.
∆=4
3
4
Define
τ=
3E0
= 42.5
kT
√
one has ∆ = 4 τ kT /3 and
Z12Z22 A
T6
!1/3
n1 n2 2 τ 2
S (E0) e−τ
r=
1 + δ1,2 3µ 9b
n n S (E0) 2 −τ
τ e .
=4.5 · 1014 1 2
1 + δ1,2 AZ1Z2
Electron Screening:
Modified Coulomb potential because of electron screening:
Z1e −r/λD Z1e Z1e
e
≈
−
r
r
λD
with λD the Debye length
s
s
kT
T6
−9
cm.
'
9.2
·
10
λD =
ρYe (Z1 + 1)
4π (Z1 + 1) e2ne
Then reaction rate increased by
!
s
2
ρYe (Z1 + 1)
Z1 Z2 e
' exp 0.17Z1Z2
exp
.
3
kT λD
T6
Effective Thermonuclear Rate:
Near a temperature T0, one can write r = r0(T /T0)n, where
d ln τ d ln r τ − 2 d ln r n=
=
=
.
d ln T T0 d ln T d ln τ T0
3 T0