User`s guide

Parametric Fitting
3-47
It is sometimes useful to describe a variable expressed as a function of angle in
terms of Legendre polynomials
where P
n
(x) is a Legendre polynomial of degree n, x is cos(θ
α
), and a
n
are the
coefficients of the fit. Refer to MATLAB’s
legendre function for information
about generating Legendre polynomials.
For the alpha-emission data, you can directly associate the coefficients with the
nuclear dynamics by invoking a theoretical model, which is described in [8].
Additionally, the theoretical model introduces constraints for the infinite sum
shown above. In particular, by considering the angular momentum of the
reaction, a fourth-degree Legendre polynomial using only even terms should
describe the data effectively.
You can generate Legendre polynomials with Rodrigues’ formula:
The Legendre polynomials up to fourth degree are given below.
Table 3-4: Legendre Polynomials up to Fourth Degree
nP
n
(x)
01
1 x
2 (1/2)(3x
2
– 1)
3 (1/2)(5x
3
– 3x)
4 (1/8)(35x
4
– 30x
2
+ 3)
yx() a
n
P
n
x()
n 0=
∞
∑
=
P
n
x()
1
2
n
n!
------------
d
dx
-------


n
x
2
1–()
n
=