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## Conservation Laws in Nuclear Reactions

A **nuclear reaction** is considered to be the process in which two nuclear particles (two nuclei or a nucleus and a nucleon) interact to produce two or more nuclear particles or ˠ-rays (gamma rays). Thus, a **nuclear reaction** must cause a transformation of at least one nuclide to another. Sometimes if a nucleus interacts with another nucleus or particle without changing the nature of any nuclide, the process is referred to a **nuclear scattering**, rather than a nuclear reaction.

In analyzing nuclear reactions, we apply the **many conservation laws**. **Nuclear reactions** are subject to classical **conservation laws for charge, momentum, angular momentum, and energy** (including rest energies). Additional conservation laws, not anticipated by classical physics, are are **electric charge**, **lepton number and baryon number**. Certain of these laws are obeyed under all circumstances, others are not. We have accepted conservation of energy and momentum. In all the examples given we assume that the number of protons and the number of neutrons is separately conserved. We shall find circumstances and conditions in which this rule is not true. Where we are considering non-relativistic nuclear reactions, it is essentially true. However, where we are considering relativistic nuclear energies or those involving the weak interactions, we shall find that these principles must be extended.

Some conservation principles have arisen from theoretical considerations, others are just empirical relationships. Notwithstanding, any reaction not expressly forbidden by the conservation laws will generally occur, if perhaps at a slow rate. This expectation is based on quantum mechanics. Unless the barrier between the initial and final states is infinitely high, there is always a **non-zero probability** that a system will make the transition between them.

For purposes of this article it is sufficient to note four of the fundamental laws governing these reactions.

**Conservation Laws in Nuclear Reactions**

**Conservation of nucleons**. The total number of nucleons before and after a reaction are the same.**Conservation of charge**. The sum of the charges on all the particles before and after a reaction are the same**Conservation of momentum**. The total momentum of the interacting particles before and after a reaction are the same.**Conservation of energy**. Energy, including rest mass energy, is conserved in nuclear reactions.

## Conservation of Energy in Nuclear Reactions

In analyzing nuclear reactions, we have to apply the general law of conservation of mass-energy. According to this law **mass and energy are equivalent and convertible** one into the other. It is one of the striking results of **Einstein’s theory of relativity**. This **equivalence** of the mass and energy is described by Einstein’s famous formula **E = mc**** ^{2}**.

Generally, in both **chemical** and **nuclear reactions**, some conversion between rest mass and energy occurs, so that the products generally have smaller or greater mass than the reactants. In general, the total (relativistic) energy must be conserved. The “missing” rest mass must therefore reappear as kinetic energy released in the reaction. The difference is a measure of the nuclear binding energy which holds the nucleus together.

**The nuclear binding energies** are enormous, they are of the order of a million times greater than the electron binding energies of atoms.

The **energetics of nuclear reactions** is determined by the **Q-value** of that reaction. The **Q-value** of the reaction is defined as the **difference** between the sum of the **masses** of the **initial reactants** and the sum of the **masses** of the **final products**, in energy units (usually in MeV).

Consider a typical reaction, in which the projectile a and the target A gives place to two products, B and b. This can also be expressed in the notation that we used so far, **a + A → B + b**, or even in a more compact notation, **A(a,b)B**.

See also: E=mc^{2}

The **Q-value** of this reaction is given by:

**Q = [m**_{a}** + m**_{A}** – (m**_{b}** + m**_{B}**)]c**^{2}

which is the same as the **excess kinetic energy** of the final products:

**Q = T**_{final}** – T**_{initial}

** = T**_{b}** + T**_{B}** – (T**_{a}** + T**_{A}**)**

For reactions in which there is an increase in the kinetic energy of the products **Q is positive**. The positive Q reactions are said to be **exothermic** (or **exergic**). There is a net release of energy, since the kinetic energy of the final state is greater than the kinetic energy of the initial state.

For reactions in which there is a decrease in the kinetic energy of the products **Q is negative**. The negative Q reactions are said to be **endothermic** (or **endoergic**) and they require a net energy input.

See also: Q-value Calculator

**energy released in a nuclear reaction**can appear mainly in one of three ways:

**Kinetic energy**of the products**Emission of gamma rays**.**Gamma rays**are emitted by unstable nuclei in their transition from a high energy state to a lower state known as gamma decay.**Metastable state**. Some energy may remain in the nucleus, as a metastable energy level.

A small amount of energy may also emerge in the form of X-rays. Generally, products of nuclear reactions may have different atomic numbers, and thus the configuration of their electron shells is different in comparison with reactants. As the **electrons rearrange** themselves and drop to lower energy levels, internal transition X-rays (X-rays with precisely defined emission lines) may be emitted.

## Conservation of Momentum and Energy in Collisions

The use of the **conservation laws for momentum and energy** is very important also in **particle collisions**. This is a very powerful rule because it can allow us to determine the results of a collision without knowing the details of the collision. The law of **conservation of momentum** states that in the collision of two objects such as billiard balls, the **total momentum is conserved**. The assumption of conservation of momentum as well as the conservation of kinetic energy makes possible the calculation of the **final velocities** in two-body collisions. At this point we have to distinguish between two types of collisions:

**Elastic collisions****Inelastic collisions**

## Elastic Collisions

A **perfectly elastic collision** is defined as one in which there is **no net conversion of kinetic energy** into other forms (such as heat or noise). For the brief moment during which the two objects are in contact, some (or all) of the energy is stored momentarily in the form of **elastic potential energy**. But if we compare the total kinetic energy just before the collision with the total kinetic energy just after the collision, and they are found to be the same, then we say that the **total kinetic energy is conserved**.

- Some large-scale interactions like the
**slingshot type gravitational interactions**(also known as a planetary swing-by or a gravity-assist manoeuvre) between satellites and planets are**perfectly elastic**. - Collisions between
**very hard spheres**may be**nearly elastic**, so it is useful to calculate the limiting case of an elastic collision. - Collisions in
**ideal gases**approach perfectly elastic collisions, as do scattering interactions of sub-atomic particles which are deflected by the electromagnetic force. **Rutherford scattering**is the elastic scattering of charged particles also by the electromagnetic force.- A
**neutron-nucleus scattering reaction**may be also elastic, but in this case the neutron is deflected by the strong nuclear force.