1. Use the relation between nuclear size and mass number

For a nucleus, $R = R_0 A^{1/3}$ where $R$ is nuclear radius and $A$ is mass number.

Given the ratio of nuclear sizes is $1 : 2^{1/3}$ So, $A_1^{1/3} : A_2^{1/3} = 1 : 2^{1/3}$ Cubing both sides, $A_1 : A_2 = 1 : 2$ Thus the masses of the two fragments are in the ratio $m_1 : m_2 = 1 : 2$ (since nuclear mass is proportional to mass number).

2. Apply conservation of momentum

Initially, the nucleus is at rest, so total momentum is zero. After disintegration, the two fragments move in opposite directions with equal magnitudes of momentum: $m_1 v_1 = m_2 v_2$

Substitute $m_1 : m_2 = 1 : 2$: $1 \cdot v_1 = 2 \cdot v_2$ So, $v_1 : v_2 = 2 : 1$

3. Compare with the given ratio

Given the speed ratio is $n : 1$ Hence, $n = 2$

Therefore, the required integer is: $\boxed{2}$