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Vibration excitation

  1. Jul 11, 2009 #1
    When we have vibration excitation then the radius of nucleus is define like:
    [tex]R=R_0[1+\sum^{\infty}_{\lambda=0}\sum^{\lambda}_{\mu=-\lambda}\alpha_{\lambda\mu}Y^{\lambda}_{\mu}(\theta,\phi)][/tex]

    where [tex]\alpha_{\lambda,\mu}=\alpha_{\lambda,-\mu}[/tex] and [tex]\alpha_{\lambda,\mu}=\alpha_{\lambda,\mu}(t)[/tex]

    How you measure this [tex]\alpha[/tex] parametar?

    [tex]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)[/tex]

    And more:
    Kinetic energy of system is define like:

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

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

    Thanks for answers
     
  2. jcsd
  3. Jul 11, 2009 #2

    malawi_glenn

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    Which book, what have you tried? ..
     
  4. Jul 11, 2009 #3
    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.
     
  5. Jul 14, 2009 #4
    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. [tex]\alpha[/tex]'s are decomposition coefficients.

    Y's, though scary looking, are normalized so that the integral of [tex]|Y|^2[/tex] over the entire sphere is something simple (there are a few different definitions, one common definition is that [tex]\int |Y|^2 d\Omega = 1[/tex]. I can't tell right away which one is used by your book.) If you assume that only one of [tex]\alpha[/tex]'s is nonzero and make certain assumptions about the nuclear matter, perhaps that non-excited nucleus is a homogeneous sphere of density [tex]\rho[/tex], 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 [tex]\rho[/tex].
     
  6. Jul 17, 2009 #5
    Thanks for answering.
    In that series is [tex]\alpha[/tex] perhaps complex functions in general?
     
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