Measuring Vibration Excitation Radius & Alpha Parametar

[Petar Mali]

When we have vibration excitation then the radius of nucleus is define like:
R=R_0[1+\sum^{\infty}_{\lambda=0}\sum^{\lambda}_{\mu=-\lambda}\alpha_{\lambda\mu}Y^{\lambda}_{\mu}(\theta,\phi)]

where \alpha_{\lambda,\mu}=\alpha_{\lambda,-\mu} and \alpha_{\lambda,\mu}=\alpha_{\lambda,\mu}(t)

How you measure this \alpha parametar?

Y^{\mu}_{\lambda}=\frac{(-1)^{\mu+\lambda}}{2^{\lambda}\lambda!}\sqrt{\frac{2\lambda+1}{4\pi}\frac{(\lambda-\mu)!}{(\lambda+\mu)!}}e^{i\mu\varphi}(sin\Theta)^{\frac{\mu}{2}}\frac{d^{\mu+\lambda}}{d(cos\Theta)^{\mu+\lambda}}sin^{2\lambda}(\Theta)

And more:
Kinetic energy of system is define like:

T=\frac{1}{2}\sum_{\lambda,\mu}B_{\lambda}|\frac{d \alpha_{\lambda,\mu}}{d t}|^2

Rayleight use \rho=\frac{3M}{4R^3_0\pi}, and get B_{\lambda}=\frac{3MR^2_0}{4\pi\lambda}. How?

Thanks for answers

 

[malawi_glenn]

Which book, what have you tried? ..

 

[Petar Mali]

Well this is from book "Osnovi nuklearne fizike" - Lazar Marinkov. I tried Burcham and some book of Gamov. From the Marinkov's book I think that this is given in reference P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, New York, Heidelberg, Berlin, 1980) but I don't have this book.

 

[hamster143]

The first equation is just a decomposition of a generic function defined on a sphere in terms of spherical harmonics Y. Like a Fourier transform, but on a sphere. \alpha's are decomposition coefficients.

Y's, though scary looking, are normalized so that the integral of |Y|^2 over the entire sphere is something simple (there are a few different definitions, one common definition is that \int |Y|^2 d\Omega = 1. I can't tell right away which one is used by your book.) If you assume that only one of \alpha's is nonzero and make certain assumptions about the nuclear matter, perhaps that non-excited nucleus is a homogeneous sphere of density \rho, deformed according to the formula above, and make assumptions about velocity distribution, and you compute kinetic energy by integrating over the entire volume, you'll get an equation that expresses B in terms of \rho.

 

[Petar Mali]

Thanks for answering.
In that series is \alpha perhaps complex functions in general?