# Sommerfeld Parameter Confusion

In almost every reference site I can find, the Sommerfeld parameter $\eta$ is explicitly stated as

$$\eta=\frac{Z_1Z_2e^2}{\hbar \nu}=\frac{Z_1Z_2e^2}{\hbar }\sqrt{\frac{\mu}{2E}}$$

Where $Z_1$ and $Z_2$ are the atomic numbers of the two nuclei involved, $\mu$ is the reduced mass of the two nuclei, and $E$ is the approximate center of mass energy of the the nuclei, and $v$ is the relative velocity of the nuclei.

The problem I have with this is that it is not dimensionless as it should be. The units of what I have just wrote down are

$$\frac{\textrm{(Coulomb)}^2}{\textrm{(Energy)(Length)}}$$

However, when I check numerically with other references, it seems like it is defined to be

$$\eta=\frac{Z_1Z_2e^2}{4\pi\epsilon_{0}\hbar c}\sqrt{\frac{\mu}{2E}}=\alpha Z_1Z_2\sqrt{\frac{\mu}{2E}}$$

Where $\alpha$ is the fine-structure constant, and $\mu$ is the reduced mass in terms of energy/$c^2$. This makes sense, but is not what is explicitly stated as the Sommerfeld parameter.

What exactly is the definition of the Sommerfeld parameter? Must I go through the solution of the Schrodinger equation for a two-nuclei system (woods-saxon + yukawa potential) to derive this for myself?

References:

The first formula is written in the Gaussian unit system, while the second one is in the SI system. In the Gaussian system, the unit of electric charge is $statC =g^{1/2}cm^{3/2}s^{-1}$. So, the Sommefeld parameter in the Gaussian unit system is dimensionless as it would be.
• As a general rule, many of these unit issues can be fixed by a factor of $4\pi\epsilon_0$, which is unity in CGS units but has dimension $\rm (coulomb)^2 (joule)^{-1}(meter)^{-1}$ in SI. – rob Apr 13 '15 at 12:12
Simply multiply by $d/d$, where $d$ is a arbitrary distance, then the unit of $d/(hv)$ will be (Joule)^(-1), and $\frac{Z_1Z_2e^2}{d}$ will be in $J$. Then the product is dimensionless.