For a parent at rest, momentum conservation gives equal and opposite daughter and alpha particle momenta. In the nonrelativistic recoil approximation,

Q = T_{α} + T_{D} = \frac{p ^{2}}{2 m _{α}} + \frac{p ^{2}}{2 M _{D}}, T_{α} = Q \frac{M _{D}}{M _{D} + m _{α}} .

Approximating nuclear masses by their mass numbers,

M_{D} ≃ (A - 4) m_{u}

and

m_{α} ≃ 4 m_{u}

, gives

T_{α} ≃ \frac{A - 4}{A} Q .

The printed phrase “atomic number

A

” must here mean mass number. Atomic number alone does not determine the recoil fraction. For example,

^{210} Po

and

^{212} Po

have the same atomic number

84

, but the approximate fractions are

206/210

and

208/212

, respectively.

T_{α} = (\frac{M _{P}^{2} + m _{α}^{2} - M _{D}^{2}}{2 M _{P}} - m_{α}) c^{2} = Q \frac{M _{D} + Q / ( 2 c ^{2} )}{M _{P}},

where

M_{P} = M_{D} + m_{α} + Q / c^{2}

. Expanding for

Q ≪ M_{P} c^{2}

recovers the recoil-sharing expression.

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