B..2 Evaluation of the Effective Potential at the point of maximum expansion

With the above results at hand, we now evaluate the second derivative of the effective potential, i.e.

$$\tilde{V} (x) '' = \frac{ 3 x ^{2} \tilde{\lambda} ^{2} +... ...x ^{3} \left( x ^{2} + \tilde{\lambda ^{2}} \right) ^{3/2}} ;$$ (35)

we are interested in its sign at the value $x _{\mathrm{min}}$, the maximum radius of the shell given in (33): a positive sign will indicate that the potential is ``$\cup$-shaped'' and thus the shell stable. To get the result, we can restrict the study to the numerator $\tilde{N} (x _{\mathrm{min}})$ of (36),

$$\tilde{N} (x _{\mathrm{min}}) = 3 x _{\mathrm{min}} ^{2} ... ...\lambda} ^{2} + 6 \tilde{\lambda} ^{4}) \vert \Theta \vert ,$$

since the denominator does not contribute to the sign. We also remember that the quantity $x _{\mathrm{min}}$ depends on $\Theta$, since (31) comes from (20) with $a = \vert \Theta \vert$.

Even with the above simplification, the detailed study of the sign of the quantity under consideration is complicated, mainly because of the non-trivial $\Theta$ dependence; a graphical analysis is also of little help, since $\tilde{N} (x _{\mathrm{min}})$ depends on the three variables $\Theta$, $\tilde{\epsilon}$ and $\tilde{\lambda}$, namely the adimensional charge $Q/H$, the charge/mass ratio $q / \mu$ of the test particle and the adimensional angular momentum per unit mass $L / (H \mu)$ of the test particle. Thus we will not search for the most general result, i.e. we will not give necessary and sufficient conditions for the stability of the shell; we will show, instead, that in some physically reasonable situations the shell itself is indeed stable against single charged particle decay.

In particular we see that in the units we are using, the adimensional parameter $\tilde{\epsilon}$ is a large number, i.e. the charge/mass ratio for an elementary particle is very large. Thus we consider the behaviour of the numerator for large $\tilde{\epsilon}$:

$$\lim _{\tilde{\epsilon} \to \infty} \tilde{N} (x _{\mathrm{min}}) = + \infty ;$$

the limit has always the ``$+$'' sign, since we consider emission of charges of the same sign of those composing the shell; thus in this limit the second derivative of the effective potential is positive, which shows stability under charged elementary particle decay.

As a second case, we see what happens for radial emission of particles:

$$\lim _{\tilde{\lambda} \to 0} \tilde{N} (x _{\mathrm{min}}... ...a \vert ( \vert \tilde{\epsilon} \vert - x _{\mathrm{min}}) ;$$

we again used the fact that ejected particles have the same charge as the shell, so that $\tilde{\epsilon} \Theta = \vert\tilde{\epsilon}\vert \cdot \vert\Theta\vert$. In this case also we see that, for elementary particle emission, we certainly can realize the situation $\tilde{\epsilon} > x _{\mathrm{min}}$, so the sign is again positive.

Even restricting the study to the two cases above, we can thus conclude that, shell configurations which are stable against single particle decay can be realized.