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Additional: Nuclear Reactions and Q-value

Balancing Nuclear Reactions

A nuclear reaction is a process in which atomic nuclei are transformed from one state to another. This can involve collisions between nuclei or between nuclei and subatomic particles (like neutrons, protons, or electrons), or spontaneous transformations like radioactive decay. Unlike chemical reactions which involve rearrangements of electrons, nuclear reactions involve changes in the composition or energy state of the nucleus itself.

Nuclear reactions are represented by nuclear equations, similar to chemical equations. However, instead of balancing atoms and charges of ions, we must balance the number of nucleons and the total charge.

Conservation Laws in Nuclear Reactions

In any nuclear reaction, certain quantities are conserved:

  1. Conservation of Mass Number (A): The total number of nucleons (protons + neutrons) before the reaction must be equal to the total number of nucleons after the reaction. The sum of the mass numbers of the reactants equals the sum of the mass numbers of the products.
  2. Conservation of Atomic Number (Z): The total number of protons (or total charge) before the reaction must be equal to the total number of protons (or total charge) after the reaction. The sum of the atomic numbers (charges in units of $e$) of the reactants equals the sum of the atomic numbers of the products.
  3. Conservation of Energy and Momentum: The total relativistic energy (including rest mass energy) and the total momentum are conserved in nuclear reactions. The energy released or absorbed in a nuclear reaction is often related to a change in the total rest mass ($E=mc^2$).
  4. Conservation of Angular Momentum: The total angular momentum (orbital and spin) of the system is conserved.
  5. Conservation of Lepton Number and Baryon Number: More advanced conservation laws involve lepton number (e.g., for electrons, muons, neutrinos) and baryon number (for protons, neutrons, etc.). These are particularly important in beta decay and other particle physics reactions.

Writing and Balancing Nuclear Equations

A general nuclear reaction can be written as:

Target nucleus + Projectile particle $\longrightarrow$ Product nucleus + Ejected particle(s) + Energy

Using the notation $_Z^A X$ for nuclei and particles:

$_ {Z_1}^{A_1} X_1 + _ {Z_2}^{A_2} X_2 \longrightarrow _{Z_3}^{A_3} X_3 + _{Z_4}^{A_4} X_4 + \text{Energy (Q)} $

To balance the nuclear equation, we apply the conservation laws for mass number (A) and atomic number (Z):

Sum of mass numbers of reactants = Sum of mass numbers of products: $ A_1 + A_2 = A_3 + A_4 $

Sum of atomic numbers of reactants = Sum of atomic numbers of products: $ Z_1 + Z_2 = Z_3 + Z_4 $

We can often determine an unknown particle or nucleus in a reaction by applying these conservation rules.

Common particles involved in nuclear reactions and their notation:

Example 1. Complete the following nuclear reaction: $_{13}^{27}Al + _2^4He \longrightarrow _{15}^{30}P + ? $

Answer:

Given reaction: $_{13}^{27}Al + _2^4He \longrightarrow _{15}^{30}P + _Z^A X$ (Let the unknown particle be $_Z^A X$).

Apply the conservation of mass number (A):

$ A_{Al} + A_{He} = A_P + A_X $

$ 27 + 4 = 30 + A $

$ 31 = 30 + A \implies A = 1 $

Apply the conservation of atomic number (Z):

$ Z_{Al} + Z_{He} = Z_P + Z_X $

$ 13 + 2 = 15 + Z $

$ 15 = 15 + Z \implies Z = 0 $

The unknown particle has mass number $A=1$ and atomic number $Z=0$. This corresponds to a neutron ($_0^1n$).

The complete nuclear reaction is:

$ _{13}^{27}Al + _2^4He \longrightarrow _{15}^{30}P + _0^1n $

(This is a historically significant reaction, the first artificial transmutation of an element, performed by Rutherford in 1919, although the reaction here is slightly different, showing the production of a neutron from bombarding Aluminium with alpha particles, as later studied by Chadwick.)

Q-value of a Nuclear Reaction (Energy Release/Absorption)

Nuclear reactions typically involve changes in the total binding energy of the system. According to mass-energy equivalence, a change in binding energy corresponds to a change in the total rest mass. The energy released or absorbed in a nuclear reaction is called the Q-value of the reaction.

Definition of Q-value

The Q-value of a nuclear reaction is defined as the energy released or absorbed during the reaction. It is calculated from the difference in the total rest mass of the reactants and the total rest mass of the products.

Consider a nuclear reaction: $ X_1 + X_2 \longrightarrow X_3 + X_4 $

Let $M_1, M_2, M_3, M_4$ be the rest masses of the particles $X_1, X_2, X_3, X_4$, respectively.

According to the conservation of energy, the total energy before the reaction equals the total energy after the reaction. Total energy includes the rest mass energy ($mc^2$) and the kinetic energy ($K$) of the particles.

$ (M_1 c^2 + K_1) + (M_2 c^2 + K_2) = (M_3 c^2 + K_3) + (M_4 c^2 + K_4) $

Rearranging the terms:

$ (K_3 + K_4) - (K_1 + K_2) = (M_1 + M_2)c^2 - (M_3 + M_4)c^2 $

$ \Delta K = \Delta M \, c^2 $

Where $\Delta K$ is the change in total kinetic energy and $\Delta M$ is the change in total rest mass, $\Delta M = (M_1 + M_2) - (M_3 + M_4)$.

The Q-value is defined as the energy released or absorbed, which is equal to the change in total kinetic energy:

$ Q = (K_{products}) - (K_{reactants}) = (M_{reactants} - M_{products}) c^2 $

$ Q = [(M_1 + M_2) - (M_3 + M_4)] c^2 = \Delta m_{defect} \, c^2 $

Where $\Delta m_{defect} = (M_1 + M_2) - (M_3 + M_4)$ is the "mass defect" or change in total rest mass in the reaction.

Exoergic and Endoergic Reactions

The sign of the Q-value indicates whether energy is released or absorbed:

The Q-value is typically expressed in MeV. As with binding energy, if the mass difference is in atomic mass units (u), the Q-value in MeV is:

$ Q = \Delta m \, (in \, u) \times 931.5 \, MeV/u $

Threshold Energy for Endoergic Reactions

For an endoergic reaction ($Q < 0$), energy must be supplied. This energy comes from the kinetic energy of the reactant particles. For the reaction to occur, the minimum kinetic energy of the reactants must be sufficient to provide the absorbed energy $|Q|$ and also conserve momentum. If particle $X_2$ is incident on a stationary target $X_1$, the minimum kinetic energy of $X_2$ required for the reaction to proceed is called the threshold energy ($K_{th}$).

$ K_{th} = |Q| \left(1 + \frac{M_2}{M_1}\right) $

The factor $\left(1 + \frac{M_2}{M_1}\right)$ accounts for the fact that some kinetic energy is always required to conserve momentum, even at the threshold, so the products are not at rest. The total initial kinetic energy must be converted into rest mass energy plus the minimum kinetic energy of the center of mass of the system.

Example 2. Calculate the Q-value of the fusion reaction $_1^2H + _1^3H \longrightarrow _2^4He + _0^1n$. Given the atomic masses: $m(_1^2H) = 2.014102 \, u$, $m(_1^3H) = 3.016049 \, u$, $m(_2^4He) = 4.002603 \, u$, $m(_0^1n) = 1.008665 \, u$. ($1 \, u = 931.5 \, MeV/c^2$).

Answer:

Given reaction: $_1^2H + _1^3H \longrightarrow _2^4He + _0^1n$

Given atomic masses:

$M_{reactants} = m(_1^2H) + m(_1^3H) = 2.014102 \, u + 3.016049 \, u = 5.030151 \, u$

$M_{products} = m(_2^4He) + m(_0^1n) = 4.002603 \, u + 1.008665 \, u = 5.011268 \, u$

The mass defect ($\Delta m$) in the reaction is $M_{reactants} - M_{products}$.

$ \Delta m = 5.030151 \, u - 5.011268 \, u = 0.018883 \, u $

Since $\Delta m$ is positive, mass is lost in the reaction, meaning energy is released. This is an exoergic (exothermic) reaction.

The Q-value of the reaction is $Q = \Delta m \cdot c^2$. Using the energy equivalent of 1 u = 931.5 MeV/c$^2$:

$ Q = 0.018883 \, u \times 931.5 \, MeV/u $

$ Q \approx 17.59 \, MeV $

The Q-value of the Deuterium-Tritium fusion reaction is approximately +17.59 MeV. This is the energy released per fusion event.