For the 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 is the same.
- Conservation of energy. Energy, including rest mass energy, is conserved 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 as 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 the charge, momentum, angular momentum, and energy (including rest energies). Additional conservation laws not anticipated by classical physics are electric charge, lepton number, and baryon number. Certain of these laws are obeyed under all circumstances, and others are not. We have accepted the 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, we shall find that these principles must be extended when we consider relativistic nuclear energies or those involving weak interactions.
Some conservation principles have arisen from theoretical considerations, and 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 the 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 is 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 mass and energy is described by Einstein’s famous formula E = mc2.
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, and they are a million times greater than the electron binding energies of atoms.
The Q-value of that reaction determines the energetics of nuclear reactions. 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 target A give place to two products, B and b. This can also be expressed in the notation we have used so far, a + A → B + b, or even in a more compact notation, A(a,b)B.
See also: E=mc2
The Q-value of this reaction is given by:
Q = [ma + mA – (mb + mB)]c2
which is the same as the excess kinetic energy of the final products:
Q = Tfinal – Tinitial
= Tb + TB – (Ta + TA)
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 endothermic (or endoergic), and they require net energy input.
See also: Q-value Calculator.
The DT fusion reaction of deuterium and tritium is particularly interesting because of its potential of providing energy for the future. Calculate the reaction Q-value.
3T (d, n) 4He
The atom masses of the reactants and products are:
m(3T) = 3.0160 amu
m(2D) = 2.0141 amu
m(1n) = 1.0087 amu
m(4He) = 4.0026 amu
Using the mass-energy equivalence, we get the Q-value of this reaction as:
Q = {(3.0160+2.0141) [amu] – (1.0087+4.0026) [amu]} x 931.481 [MeV/amu]
= 0.0188 x 931.481 = 17.5 MeV
A high-energy photon (gamma-ray) can, under certain conditions, eject a neutron from a nucleus. It occurs when its energy exceeds the binding energy of the neutron in the nucleus. Most nuclei have binding energies above 6 MeV, above the energy of most gamma rays from fission. On the other hand, there are few nuclei with sufficiently low binding energy to be of practical interest. These are 2D, 9Be, 6Li, 7Li, and 13C. As can be seen from the table, the lowest threshold have 9Be with 1.666 MeV and 2D with 2.226 MeV.
threshold energies.
In the case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:
The reaction Q-value is calculated below:
The atom masses of the reactant and products are:
m(2D) = 2.01363 amu
m(1n) = 1.00866 amu
m(1H) = 1.00728 amu
Using the mass-energy equivalence, we get the Q-value of this reaction as:
Q = {2.01363 [amu] – (1.00866+1.00728) [amu]} x 931.481 [MeV/amu]
= -0.00231 x 931.481 = -2.15 MeV
The energy released in a nuclear reaction can appear mainly in one of three ways:
- The 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 also very important 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 the total momentum is conserved in the collision of two objects such as billiard balls. The assumption of conservation of momentum and 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, they are found to be the same. We say that the total kinetic energy is conserved.
- Large-scale interactions between satellites and planets are perfectly elastic, like the slingshot type gravitational interactions (also known as a planetary swing-by or a gravity-assist maneuver).
- 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 deflect by the electromagnetic force.
- Rutherford scattering is the elastic scattering of charged particles also by the electromagnetic force.
- A neutron-nucleus scattering reaction may also be elastic, but in this case, the neutron is deflected by a strong nuclear force.
The relative speed of the two objects after the collision has the same magnitude (but opposite direction) as before the collision, no matter what the masses are.
It is known the fission neutrons are of importance in any chain-reacting system. All neutrons produced by fission are born as fast neutrons with high kinetic energy. Before such neutrons can efficiently cause additional fissions, they must be slowed down by collisions with nuclei in the reactor’s moderator. The probability of the fission U-235 becomes very large at the thermal energies of slow neutrons. This fact implies an increase in the multiplication factor of the reactor (i.e., lower fuel enrichment is needed to sustain a chain reaction).
The neutrons released during fission with an average energy of 2 MeV in a reactor on average undergo many collisions (elastic or inelastic) before they are absorbed. During the scattering reaction, a fraction of the neutron’s kinetic energy is transferred to the nucleus. It is possible to derive the following equation for the mass of target or moderator nucleus (M), the energy of incident neutron (Ei), and the energy of scattered neutron (Es) using the laws of conservation of momentum and energy and the analogy of collisions of billiard balls for elastic scattering.
where A is the atomic mass number, in case of the hydrogen (A = 1) as the target nucleus, the incident neutron can be completely stopped. But this works when the direction of the neutron is completely reversed (i.e., scattered at 180°). In reality, the direction of scattering ranges from 0 to 180 °, and the energy transferred also ranges from 0% to maximum. Therefore, the average energy of the scattered neutron is taken as the average of energies with scattering angles 0 and 180°.
Moreover, it is useful to work with logarithmic quantities. Therefore one defines the logarithmic energy decrement per collision (ξ) as a key material constant describing energy transfers during a neutron slowing down. ξ is not dependent on energy, only on A and is defined as follows:
For heavy target nuclei, ξ may be approximated by the following formula:
From these equations, it is easy to determine the number of collisions required to slow down a neutron from, for example from 2 MeV to 1 eV.
Example: Determine the number of collisions required for thermalization for the 2 MeV neutrons in the carbon.
ξCARBON = 0.158
N(2MeV → 1eV) = ln 2⋅106/ξ =14.5/0.158 = 92
For a mixture of isotopes:
What are the velocities of the neutron and carbon nucleus after the collision?
Solution:
This is an elastic head-on collision of two objects with unequal masses. We have to use the conservation laws of momentum and kinetic energy and apply them to our system of two particles.
We can solve this system of equations or use the equation derived in the previous section. This equation stated that the relative speed of the two objects after the collision has the same magnitude (but opposite direction) as before the collision, no matter what the masses are.
The minus sign for v’ tells us that the neutron scatters back of the carbon nucleus because the carbon nucleus is significantly heavier. On the other hand, its speed is less than its initial speed. This process is known as neutron moderation, and it significantly depends on the mass of moderator nuclei.
