Nuclear Physics

The energy levels, $E_{n}$, of the hydrogen atom are given by

$$ E_{n}=\frac{-2.16 \times 10^{-18}}{n^{2}} \mathrm{~J}, \text { where } n \text { is a positive integer. } $$

(i) What is the ionization energy of the atom?

(ii) What is the wavelength of the $\mathrm{H}_{\alpha}$ line, which is due to transitions from the $n=3$ to $n=2$ level? [4]

Determine the binding energy per nucleon, in $\mathrm{MeV}$, of an alpha particle. [3]

$$ \begin{aligned} \text { Mass of proton } &=1.0080 \mathrm{u} \\ \text { Mass of neutron } &=1.0087 \mathrm{u} \\ \text { Mass of alpha particle } &=4.0026 \mathrm{u} \\ 1 \mathrm{u} &=930 \mathrm{MeV} / \mathrm{c}^{2} \end{aligned} $$

A radioactive substance, with a half-life of $T$, contains a particular nucleus that has NOT decayed over an observational period of $5 T$. What is the probability that it will decay over a further period of (i) $T$ and (ii) $3 T$ ?

Determine the half-life, $T$, of a substance with activity that decreases by $1.00 \%$ in $10^{8} \mathrm{~s}$ (approximately 3 years). [5]

Sketch, on the same diagram, the paths of three alpha particles, of the same energy, which are directed towards a fixed nucleus so they are deflected through (i) $10^{\circ}$, (ii) $90^{\circ}$, and (iii) $180^{\circ}$. If the nucleus in (iii) is not fixed, what is the relative velocity of the two particles at the distance of closest possible approach? [5]

Determine the half life of uranium given that $3.23 \times 10^{-7} \mathrm{~g}$ of radium is found per gram of uranium in ancient minerals. The half life of radium is 1,600 years. The atomic weights of uranium and radium are 238 and 226 respectively. All the radium arises from the uranium. [5]

Wood from the coffin of an Egyptian mummy showed a specific activity of $1.2 \times 10^{2}$ $\mathrm{s}^{-1} \mathrm{~kg}^{-1}$. Comparable living wood has a value of $2.0 \times 10^{2} \mathrm{~s}^{-1} \mathrm{~kg}^{-1}$. The half life of carbon-14 is $5.70 \times 10^{3}$ years. What is the time interval, $T_{B}$, in years, since the burial? [5]

Protons are accelerated from rest through a p.d. of $2.0 \times 10^{6} \mathrm{~V}$ and fired at a gold $\left({ }_{79}^{197} \mathrm{Au}\right)$ foil. What is the distance of closest approach of a proton to the gold nucleus? [4]

What mass of radium, mass number 226, half-life of 1620 years, and an $\alpha$ emitter, is required to produce an average of $10 \alpha$ particles per second? [4]

The activity of polonium, Po, fell to one eighth of its initial value in 420 days. Calculate the half-life, $t_{h}$, of polonium.

Give the numerical values of $a, b, c, d, e$, and $f$ in the nuclear equation. [4]

$$ \underset{\mathrm{b}}{\mathrm{a}} \mathrm{Po} \rightarrow \quad{ }_{\mathrm{d}}^{\mathrm{c}} \alpha+{ }_{82}^{206} \mathrm{~Pb}+\underset{\mathrm{f}}{\mathrm{e}} \gamma $$

An isotope of polonium, ${ }_{84}^{210}$ Po decays by alpha particle emission with a half-life of 138 days. A mass of $5 \mathrm{mg}$ of this isotope is in the form of a thin film in a very thin-walled glass container such that the alpha particles can escape. By how much will the mass of the thin film of polonium be reduced after 100 days? [6]

The ${ }_{92}^{238} \mathrm{U}$ decays according to

$$ { }_{92}^{238} \mathrm{U} \rightarrow{ }_{90}^{234} \mathrm{Th}+{ }_{2}^{4} \mathrm{He} $$

Determine the kinetic energy of the emitted $\alpha$-particle in $\mathrm{MeV}$.

$c=2.998 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$

Mass of the ${ }_{92}^{238} \mathrm{U}$ nucleus is $3.85395 \times 10^{-25} \mathrm{~kg}$

Mass of the ${ }_{90}^{234} \mathrm{Th}$ nucleus is $3.78737 \times 10^{-25} \mathrm{~kg}$

Mass of the $\alpha$-particle is $6.64807 \times 10^{-27} \mathrm{~kg}$ [5]

A short pulse of $10^8$ neutrons is fired through a vacuum at a target. If the bunch of neutrons is travelling at a speed $v=2200 \mathrm{~m} \mathrm{~s}^{-1}$ and the half life of a neutron is $880 \mathrm{~s}$, how many neutrons will decay whilst travelling a distance of $11 \mathrm{~m}$ towards the target? [4]

In an experiment, the count rate of a Geiger counter is the sum of a constant background rate $B$ and that due to the radiation emitted directly from a certain decaying radioactive source. The total count rate was measured over a short time interval at the same time in three consecutive weeks. The rate measurements were $C_1$, $C_2$ and $C_3$ respectively. Obtain an expression for $B$ in terms of $C_1$, $C_2$ and $C_3$. [4]

Radiation is attenuated when it passes through material. The intensity decreases exponentially with distance from the surface of the material. A cuboid is made from a material which halves the radiation at a depth of $1.0\,\text{cm}$ below the surface. When a beam of radiation is directed in turn along the three axes of the cuboid, the intensities are reduced to $60\%$, $70\%$, $80\%$ of the incident. Determine the volume of the cuboid. [4]