Nuclear physics

23.1 Mass defect and nuclear binding energy

Mass and energy

Mass and energy are equivalent:

$E = mc^2$

So a change in energy means a change in mass: $\Delta E = c^2\Delta m$. A nuclear reaction that releases energy leaves the products with less total mass.

Nuclear equations

Nucleon number and proton number balance on both sides:

$^{14}_{7}\text{N} + ^{4}_{2}\text{He} \rightarrow\, ^{17}_{8}\text{O} + ^{1}_{1}\text{H}$

Mass defect and binding energy

• Mass defect: the difference between the total mass of the separate nucleons and the mass of the nucleus. The nucleus is always lighter.
• Binding energy: the energy needed to separate a nucleus completely into its individual nucleons. It equals $c^2$ × the mass defect.

Binding energy per nucleon

A higher binding energy per nucleon means a more stable nucleus. The peak is near iron-56.

Fusion and fission

Fusion needs very high temperatures, so that the positive nuclei are moving fast enough to overcome their electrostatic repulsion.

23.2 Radioactive decay

Spontaneous and random

• Random: you cannot predict which nucleus will decay next or when, though each one has a constant probability per unit time.
• Spontaneous: decay is not affected by external factors such as temperature or pressure.

The fluctuations in the count rate from a radioactive source are evidence for its randomness.

Activity and the decay constant

Activity $A$ is the number of decays per unit time, measured in becquerels (1 Bq = 1 decay per second).

The decay constant $\lambda$ is the probability per unit time that a nucleus will decay.

$A = \lambda N$

Half-life

Half-life $t_{1/2}$ is the time taken for the number of undecayed nuclei of a nuclide (or the activity) to fall to half.

$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{t_{1/2}}$

Exponential decay

$x = x_0 e^{-\lambda t}$

where $x$ can be the activity, the number of undecayed nuclei, or the received count rate.

Taking logs gives $\ln A = \ln A_0 - \lambda t$. So a plot of $\ln A$ against $t$ is a straight line with gradient $-\lambda$.