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

Link to text

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


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.