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Quoting from 'Nuclear Physics - Theory and Experiment' by RR Roy, BP Nigam 2005 edition

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How did the author arrive at equations (23a, 23b,23c)?


Chapter 8 Nuclear model II, 8.7 Quadrupole Deformation, page 286

In the body-fixed reference frame in which the coordinate axes coincide with the principal axes, we denote the deformation parameters $\alpha_{2\mu}$ by $a_{2\mu}$. The relationship between deformation in the two coordinate systems is $$\begin{align} \sum_\mu a^*_{2\mu}Y_{2\mu}(\theta,\phi)&=\sum_\nu a^*_{2\nu}Y_{2\nu}(\theta,\phi)\\ &=\sum_\nu a^*_{2\nu}\sum_\mu D^2_{\mu\nu}(\theta,\phi,\psi)Y_{2\mu}(\theta',\phi') &(22c) \end{align}$$ so that $$\begin{align} \alpha_{2\mu}&=\sum_\nu a_{2\nu}D^{2*}_{\mu\nu}(\theta,\phi,\psi) \qquad\qquad\qquad\qquad\qquad\qquad&(22d) \end{align}$$ Since, in terms of the principal axes, the product of inertia is zero, we define the following $$\begin{align} a_{20}&=\beta\cos\gamma \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&(23a)\\ a_{21}&=a_{2-1}=0&(23b)\\ a_{22}&=a_{2,-2}=\tfrac1{\sqrt 2}\beta\sin\gamma&(23c) \end{align}$$ where $\beta$ and $\gamma$ are new parameters whereby the $a$'s are defined. The deformations $\delta R_j$ along the principal axes $j=1,2,3$ (nuclear body-fixed axes) are obtained from $$\delta R(\theta,\phi)=R_0 \sum^2_{\mu=-2}a^*_{2\mu}Y_{2\mu}(\theta,\phi)$$


and how can we write the equations $\delta R_1$?


page 287

and are given by $$\begin{align} \delta R_1\left(\frac\pi2,0\right)&=\sqrt{\frac5{4\pi}}\beta R_0\cos\left(\gamma-\frac{2\pi}3\right)\\% i am so depressed \delta R_2\left(\frac\pi2,\frac\pi2\right)&=\sqrt{\frac5{4\pi}}\beta R_0\cos\left(\gamma-\frac{4\pi}3\right)\\ \delta R_3\left(0,\phi\right)&=\sqrt{\frac5{4\pi}}\beta R_0\cos\left(\gamma\right)\\ \end{align}$$


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It looks for me like a definition of convenience, rather than a deduction. This looks for me like a spherical problem of determining the effects of deformation of a nucleus from perfect spheric symmetry, as there are also spherical harmonics in the computation. I think these parameter definitions in the eqn. 23a)-c) are intended to take into account the deformation from the sphere.

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