Physics and Engineering of Radiation Detection

10 Chapter 1. Properties and Sources of Radiation

isotope is expected to go through. If the energy comes out to be negative then the
decay is not possible unless energy is supplied through an external agent, such as
by bombarding the material with high energy particles. A positiveQ-value signifies
that the isotope is unstable with respect to that particular mode of decay. Note
that if a nucleus has a positiveQ-value for one decay mode, it does not guarantee
that it can decay through other modes as well (see example below).
Since atomic data tables list isotope masses ina.m.u.therefore one must multiply
masses in the above equation bya.m.u.tokgconversion factor. We can also sub-
stitute this conversion factor, the value of c, and the conversion factor for Joules to
MeVin the above relation to transform it into a more computationally convenient
form.

Qd =(mX−mY−md)8. 94 × 1016 J (minkg) (1.3.6)

=(mX−mY−md)931. 502 MeV (mina.m.u.) (1.3.7)

Care should be exercised when substituting masses in the above relation. The
above relation as it stands is valid for nuclear masses. If one wishes to use atomic
masses, the mass of electrons should be properly accounted for as explained later in
this section.
TheQ-value can be used to determine the kinetic energies of the daughter nucleus
and the emitted particle. To demonstrate this, let us substituteT =^12 mv^2 in
equation 1.3.5. This gives

Td=Qd−

1

2

mYv^2 Y, (1.3.8)

wheremY andvY represent the mass and velocity of the daughter nucleus respec-
tively. The velocity of the daughter nucleus can be determined by applying the law
of conservation of linear momentum, which for this case gives

mYvY=mdvd. (1.3.9)

Note that here we have assumed that the parent was at rest before the decay. The
velocityvYfrom this equation can then be substituted into equation 1.3.A to get

Td=

[

mY

mY+md

]

Qd. (1.3.10)

Similary the expression for the kinetic energy of the daughter nucleus is given by

TY=

[

md

md+mY

]

Qd. (1.3.11)

Let us now write theQ-value relations forαandβdecays.

α-decay: Qα=(mX−mY−mα)c^2

β-decay: Qβ=(mX−mY−mβ)c^2

Note that the above relation is valid for nuclear masses only. For atomic masses,
the following equations should be used