Chapter 3
R-matrix approach for Astrophysical S-Factor and Nuclear Cross-section
Chapter 3
R-matrix approach for Astrophysical S-Factor
and Nuclear Cross-section
For interaction of charged particles, the astrophysical S-factor is defined as
S(E)=σ(E)E exp(2πη)
Where S(E) and σ(E) are the astrophysical S-factor and total cross-section of the reaction,
respectively; E is the collision energy in CM frame; η is Sommerfeld parameter defined as

Z1 Z 2 e 2
,
v
Here Z1 and Z2 are the charge numbers of a projectile and target nuclei, respectively; v is their
relative velocity; ћ is the plank’s constant. The total capture cross-section is a sum over the
mutlipolaity λ and the final state total angular momentum Jf which is given by
σ(E) =    , Jf ( E ),
Jf ,
Where the cross-section σ(E) is further calculated by R-matrix approach.
Chapter 3
R-matrix approach for Astrophysical S-Factor and Nuclear Cross-section
R-matrix Treatment of the Resonant Cross-section
For the atomic and nuclear R-matrix approach is one of the popular treatment.
It follows fundamental scheme for the R-matrix is to split the space into two regions by a
boundary. The internal region where the short-range nuclear potential has a dominant role and
the extremal region where the Coulomb potential is dominant. In the R-matrix method the crosssection is totally dependent on the channel radius b and the Coulomb functions. The
multichannel R-matrix is define as
Rif(E) =
N
 i
 E  E
n
f

The number of poles is specifying by the reduce width γλ and energy Eλ. The γλi and γλf are the
entrance and exit reduce widths of the channel. The effect of the poles are related to the
resonance energy and their widths. The nuclear cross- section can easily be determined from the
R-matrix for various partial waves.
R-matrix are applicable in two ways. (I) The calculation depend R-matrix: In which
the parameters such as Eλ, γλi and γλf are calculated. This method is used in the microscopic
calculations. The technique is entirely systematic in scattering problems, microscopic
calculations, and many body interactions. It has various application in the field of atomic and
nuclear physics. (II) The phenomenological R-matrix method: In which the mentioned
parameters ae fitted upon the experimental data. The phenomenological R-matrix and the
calculation depended on R- matrix is quite the same in origin at certain extents. But they have
somewhat different applications. However, in the nuclear astrophysics, certain experimental
familiar quantities like nuclear cross section, astrophysical S-factor, and phase shift are
parameterized by the R-matrix method.
The R-matrix approach is applicable for both transfer and radiative capture (resonance
and direct) Processes. The non-resonant contributions are simulated by a high-energy pole. They
are referred background contribution.
According to the R-matrix method, the resonant Cross-section of the radiation capture
reaction is parameterized by the equation
 r (E) 
p ( E ) ( E )


,
K 2 ( E  E R ) 2  ( E ) 2 / 4
Chapter 3
R-matrix approach for Astrophysical S-Factor and Nuclear Cross-section
k is the wave number of incoming particle in the CM system, P (E ) ,  (E ) and (E ) are the
proton partial width, radiative partial width and total width, respectively: is the statistical term
  1   ij 
2 J  1 ,
2 I1  12 I 2  1
The Kronecker delta  ij accounts for the identity of interacting particles, I1, I2 and J are the spins
of interacting particles and total spin in resonance state .
The energy dependent proton width I’,{E) and radiative capture gamma width TAE are defined
as
P1 ( E )
P ( E R ),
P2 ( E )
p ( E ) 
and
 E F
 ( E )  
 ER   F



2  1
P ( E R ),
Where P ( E R ) and  ( E R ) are the experimental particle and radiative widths, respectively,
l,  and  f represents the orbital angular momentum, electric multi-polarity and final bound
state energy of compound nucleus, respectively. The penetrability Pl(E) is defined as
P1 ( E ) 
kb
Fl (k , b)  Gl2 (k , b)
2
Where b, Fl (k,b} and Gl(k, b} are the channel radius, regular and irregular Coulomb functions.
Nuclear Physics Fact of CNO Cycle
For stellar evolution and burning processes, the nuclear reaction rate is the key factor. The
reaction rate explains the nuclear interaction probability between two interacting nuclei or two
nuclear clusters. The reaction rates depend on the reaction cross-section on the stellar conditions.
The reaction rates is defined below
Chapter 3
R-matrix approach for Astrophysical S-Factor and Nuclear Cross-section
 8
v  
 
1

2 1
 E 

 ( E ) E  exp  dE
3 
 kT 
 kT 2 0
Nuclear Reaction Rates
The understanding of thermonuclear reaction rates is the key to understand the elemental
production in the stars. Thermonuclear rates are calculated in terms of stellar temperature T. It is
directly dependent on the nuclear cross-section. The proton capture processes have an
importance in the nucleosynthesis of low and intermediate-mass nuclei in the stellar
environments. Nuclear reactions rates are important for the description of stellar burning models.
They are strongly dependent on the presence of resonance positions in the low energy regions.