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?