8.5.1.2. Conservation Laws

Table of Contents

Why conservation laws matter in nuclear reactions

In any nuclear reaction, some quantities must remain unchanged before and after the reaction. These are called conservation laws. They are the basic checks that tell us whether a proposed reaction is possible. If even one required conserved quantity does not balance, the reaction cannot occur.

A nuclear reaction is often written in the form

$$
a + A \rightarrow b + B
$$

where $a$ is an incoming particle, $A$ is the target nucleus, $b$ is an outgoing particle, and $B$ is the product nucleus. Conservation laws compare the total quantities on the left side with the total quantities on the right side.

Main conserved quantities

For the nuclear reactions studied at an introductory level, the most important conserved quantities are electric charge, nucleon number, energy, and momentum. In many cases angular momentum is also conserved.

Quantity	What must stay the same
Electric charge	Total charge before = total charge after
Nucleon number	Total number of protons and neutrons before = total after
Energy	Total energy before = total after
Momentum	Total momentum before = total after
Angular momentum	Total angular momentum before = total after

These laws do not mean that each individual object stays the same. Instead, the total for the whole system stays the same.

For every allowed nuclear reaction,
$$
\text{total conserved quantity before} = \text{total conserved quantity after}
$$
If this fails for charge, nucleon number, energy, or momentum, the reaction is forbidden.

Conservation of electric charge

Electric charge must always be the same before and after the reaction. The total number of positive charges on one side must match the total on the other side.

If a nucleus with atomic number $Z$ participates in a reaction, its charge is $+Ze$. A proton has charge $+e$, a neutron has charge $0$, and an alpha particle has charge $+2e$.

Consider

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

Check the charge:

Left side, $7 + 2 = 9$

Right side, $8 + 1 = 9$

So charge is conserved.

Conservation of nucleon number

In nuclear physics, the total number of nucleons, meaning protons plus neutrons, is conserved in ordinary nuclear reactions. This is often checked using the mass number $A$.

For the same reaction,

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

the total mass number is

Left side, $14 + 4 = 18$

Right side, $17 + 1 = 18$

So nucleon number is conserved.

This is one of the easiest and most useful checks when reading reaction equations.

In a nuclear reaction equation,
$$
\sum A_{\text{before}} = \sum A_{\text{after}}
$$
and
$$
\sum Z_{\text{before}} = \sum Z_{\text{after}}
$$
These two balances are the first checks of any reaction.

Conservation of energy

Energy is always conserved, but in nuclear reactions it is important to remember that mass is part of energy. A reaction may produce kinetic energy because a small amount of mass is converted into other forms of energy.

The total energy includes rest energy, kinetic energy, and sometimes excitation energy of nuclei. So the energy balance is more than just comparing masses.

In symbolic form,

$$
E_{\text{total,before}} = E_{\text{total,after}}
$$

If the products have less total rest mass than the reactants, the difference can appear as kinetic energy. If the products have greater total rest mass, extra kinetic energy must be supplied by the incoming particles.

This idea leads directly to reaction energetics and the $Q$ value, which is treated separately. Here, the key point is only that total energy never disappears and never appears from nothing.

Conservation of momentum

Momentum must also be conserved in every nuclear reaction. Since nuclei and particles move in space, the total vector momentum before the reaction must equal the total vector momentum after the reaction.

In vector form,

$$
\vec{p}_{\text{before}} = \vec{p}_{\text{after}}
$$

For a reaction

$$
a + A \rightarrow b + B
$$

the momentum law is

$$
\vec{p}_a + \vec{p}_A = \vec{p}_b + \vec{p}_B
$$

If the target nucleus is initially at rest, then $\vec{p}_A = 0$, so

$$
\vec{p}_a = \vec{p}_b + \vec{p}_B
$$

This explains why outgoing particles often emerge in different directions. Their momenta must add up correctly.

Rendering scientific sketch…

Momentum balance in a nuclear reaction

The diagram shows the idea that the final momentum vectors can combine to equal the initial momentum vector.

Momentum is a vector quantity, so direction matters:
$$
\sum \vec p_{\text{before}} = \sum \vec p_{\text{after}}
$$
You must balance momentum in each spatial direction, not just the magnitudes.

Conservation of angular momentum

Angular momentum is also conserved in nuclear reactions. For beginner-level reaction writing, this law is often less visible than charge or nucleon number, but it becomes important when determining whether a transition or reaction channel is allowed.

The total angular momentum includes orbital angular momentum and the intrinsic angular momentum, or spin, of the particles and nuclei involved.

In general,

$$
\vec{L}_{\text{before}} + \vec{S}_{\text{before}} = \vec{L}_{\text{after}} + \vec{S}_{\text{after}}
$$

At an introductory stage, it is enough to know that some reactions that appear to satisfy charge and nucleon number can still be restricted by angular momentum conservation.

Using conservation laws to complete a reaction

Conservation laws help identify an unknown particle or nucleus in a reaction. Suppose we have

$$
{}^{27}_{13}\mathrm{Al} + {}^{4}_{2}\mathrm{He} \rightarrow {}^{30}_{15}\mathrm{P} + X
$$

Use nucleon number:

$$
27 + 4 = 30 + A_X
$$

$$
A_X = 1
$$

Use charge:

$$
13 + 2 = 15 + Z_X
$$

$$
Z_X = 0
$$

A particle with $A=1$ and $Z=0$ is a neutron, so

$$
X = {}^{1}_{0}\mathrm{n}
$$

This is a standard use of conservation laws in nuclear notation.

Example of an impossible reaction

Now consider

$$
{}^{14}_{7}\mathrm{N} \rightarrow {}^{14}_{6}\mathrm{C} + {}^{1}_{1}\mathrm{p}
$$

Check nucleon number:

Left side, $14$

Right side, $14 + 1 = 15$

Nucleon number is not conserved, so this reaction is impossible as written.

Even before worrying about energy or momentum, the equation already fails a basic conservation test.

Summary of reaction balancing

When checking or constructing a nuclear reaction, the simplest order is to begin with particle counting, then remember motion and energy.

Step	Question
1	Is total charge conserved?
2	Is total nucleon number conserved?
3	Can momentum be conserved?
4	Can total energy be conserved?
5	Are there angular momentum restrictions?

A correctly written nuclear reaction must satisfy at least these balances:
$$
\sum Z_{\text{before}} = \sum Z_{\text{after}}
$$
$$
\sum A_{\text{before}} = \sum A_{\text{after}}
$$
$$
\sum \vec p_{\text{before}} = \sum \vec p_{\text{after}}
$$
$$
E_{\text{total,before}} = E_{\text{total,after}}
$$

Final perspective

Conservation laws are the bookkeeping rules of nuclear physics. They do not by themselves tell us how likely a reaction is, or how much energy it releases, but they tell us whether the reaction is even possible in principle. In practice, physicists use them first, because they are powerful, simple, and universal.

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