Nuclear reactions and the Q value

Changing one nucleus into another by a collision

A radioactive nucleus changes by itself. In a nuclear reaction we change a nucleus by hitting it. A projectile $a$ strikes a target nucleus $X$, and the products are $Y$ and $b$:

A short way of writing this is $X(a, b)Y$. Rutherford made the first one in 1919, when alpha particles from a radioactive source hit nitrogen and gave oxygen-17 and a proton:

Look at the numbers. The mass numbers add up to $4 + 14 = 18 = 17 + 1$, and the atomic numbers add up to $2 + 7 = 9 = 8 + 1$. These two sums must always agree. The reaction is also ruled by the conservation of energy (with the rest energy of the nuclei included), the conservation of momentum and the conservation of angular momentum.

Two more reactions made history in 1932. In one, alpha particles on beryllium gave carbon and a particle with no charge, the neutron, which Chadwick discovered: ${}^{4}_{2}\text{He} + {}^{9}_{4}\text{Be} \to {}^{12}_{6}\text{C} + {}^{1}_{0}n$. In the other, Cockcroft and Walton used protons that they had speeded up, and split lithium: ${}^{1}_{1}\text{H} + {}^{7}_{3}\text{Li} \to 2\,{}^{4}_{2}\text{He}$.

The Q value

The energy released in a reaction is the Q value. It is the loss of mass times $c^2$:

The second form says that $Q$ is also the gain in kinetic energy. Use atomic masses and the electrons look after themselves, provided that the total of the atomic numbers is the same on both sides, as it is here.

• If $Q > 0$, the products are lighter than the reactants. The reaction is exothermic, and it releases energy.
• If $Q < 0$, the products are heavier. The reaction is endothermic. Energy has to be supplied, and the reaction does not start if the projectile has too little energy.

Worked example: the neutron reaction. Beryllium-9 ($9.012183$ u) and helium-4 ($4.002603$ u) give carbon-12 ($12.000000$ u) and a neutron ($1.008665$ u).

1. Mass before. $9.012183\ \text{u} + 4.002603\ \text{u} = 13.014786\ \text{u}$.
2. Mass after. $12.000000\ \text{u} + 1.008665\ \text{u} = 13.008665\ \text{u}$.
3. The loss. $0.006121\ \text{u}$, so $Q = 0.006121\ \text{u} \times 931.494\ \text{MeV/u} = 5.70\ \text{MeV}$.

$Q$ is positive, so the reaction releases energy and goes even with slow alpha particles. This is why a mixture of beryllium and an alpha emitter is a handy source of neutrons.

Worked example: splitting lithium. Hydrogen-1 ($1.007825$ u) and lithium-7 ($7.016004$ u) give two helium-4 nuclei ($2 \times 4.002603$ u). The loss is $8.023829 - 8.005206 = 0.018623$ u, so $Q = 17.35$ MeV. If the proton comes with little energy, the two alpha particles go off in opposite directions with the same energy, about $\dfrac{17.35}{2} = 8.7$ MeV each. A proton of $0.5$ MeV would add its energy to the pot, and each alpha particle would then have $\dfrac{17.35 + 0.5}{2} = 8.9$ MeV.

How the energy is shared

Suppose a parent nucleus at rest breaks into two products. The total momentum is zero, so the products go in opposite directions with equal momentum $p$. Kinetic energy is $K = \dfrac{p^2}{2m}$, so the lighter product has the larger energy:

For an alpha decay, $K_\alpha = Q\,\dfrac{A - 4}{A}$, as in the last course.

Set $Q = 5$ MeV and $A = 226$. Which product takes almost all the energy? Now lower $A$ to $100$. Does the daughter get more or less?

The threshold energy

For an endothermic reaction we must give the projectile enough energy. You might think that $|Q|$ is enough, but it is not. The target is at rest and the projectile has momentum, so after the reaction the products must still move forward. This part of the energy cannot be used. A calculation in the frame of the centre of mass gives the least kinetic energy of the projectile, the threshold energy:

This is for a target at rest and speeds much lower than the speed of light.

Worked example. A proton hits lithium-7 and makes beryllium-7 ($7.016929$ u) and a neutron. The masses before are $1.007825\ \text{u} + 7.016004\ \text{u} = 8.023829$ u, and after, $7.016929\ \text{u} + 1.008665\ \text{u} = 8.025594$ u. The products are heavier by $0.001765$ u, so $Q = -1.644$ MeV. Then $K_\text{th} = 1.644\ \text{MeV} \times \dfrac{1.007825 + 7.016004}{7.016004} = 1.88\ \text{MeV}$. A proton with $1.7$ MeV, which is more than $|Q|$, still cannot start the reaction.

Practise with new numbers

Try it in Python

The program finds $Q$ for four reactions from the atomic masses, and says which release energy.

Common mistakes

• Using the Q value as the threshold. The threshold is higher, by the factor $\left(1 + \dfrac{m_a}{m_X}\right)$.
• Forgetting to check $A$ and $Z$. A reaction with a wrong total of mass number or charge is not allowed, whatever the masses say.
• Mixing up the sign of $Q$. $Q =$ mass before minus mass after. A positive $Q$ means energy released.
• Mixing atomic masses and nuclear masses. Use one kind, and with atomic masses count the electrons on both sides.
• Giving the heavier product the larger share of the energy. With equal momenta, the lighter product has the larger kinetic energy.
• Leaving out the kinetic energy of the projectile. The products share $Q + K_\text{in}$.

Remember these

• A nuclear reaction conserves mass number, charge, energy, momentum and angular momentum.
• $Q = (\sum m_\text{before} - \sum m_\text{after})c^2$. $Q > 0$: exothermic. $Q < 0$: endothermic.
• From rest, two products share the energy in the inverse ratio of their masses.
• $K_\text{th} = |Q|\left(1 + \dfrac{m_a}{m_X}\right)$ for a target at rest.
• Examples: $Q = 5.70$ MeV for beryllium and an alpha particle, $17.35$ MeV for lithium and a proton, and $-1.19$ MeV for nitrogen and an alpha particle.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Complete the reaction ${}^{27}_{13}\text{Al} + {}^{4}_{2}\text{He} \to {}^{30}_{15}\text{P} + X$. What is X?

2. A reaction has $Q = -3.0$ MeV. A projectile with a kinetic energy of exactly $3.0$ MeV is used on a target at rest. Does the reaction happen?

3. A nucleus of mass number $200$ at rest splits into two parts of mass numbers $50$ and $150$. Which has the larger kinetic energy, and what is the ratio?

4. The masses before a reaction add to $20.0000$ u and the masses after add to $19.9950$ u. What is $Q$?

References

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei, on nuclear energy and the Q value.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics, on nuclear reactions.
• AME2020 atomic mass evaluation (Wang and others, Chinese Physics C, 2021), for the atomic masses.