7. Let $\alpha, \beta$ be a pair of conjugate complex numbers. If $|\alpha-\beta|=2 \sqrt{3}$ and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha|=$ $\qquad$ .
$$
\frac{\alpha}{\beta^{2}}=\frac{\bar{\alpha}}{\bar{\beta}^{2}} \Rightarrow \alpha^{3}=\bar{\alpha}^{3} \Rightarrow \alpha^{2}+\alpha \bar{\alpha}+\bar{\alpha}^{2}=0 \text {. Let } \alpha=x+y \mathrm{i}, \beta=x-y \mathrm{i}(x, y \in \mathbf{R}) \text {, }
$$

then $x^{2}-y^{2}+2 x y \mathrm{i}+x^{2}+y^{2}+x^{2}-y^{2}-2 x y \mathrm{i}=3 x^{2}-y^{2}=0$.
Also, $|\alpha-\beta|=|2 y i|=2 \sqrt{3} \Rightarrow y^{2}=3 \Rightarrow x^{2}=1$, so $|\alpha|=\sqrt{x^{2}+y^{2}}=2$.