Conservation Laws in Nuclear Reactions

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

  1. Conservation of nucleons. The total number of nucleons before and after a reaction are the same.
  2. Conservation of charge. The sum of the charges on all the particles before and after a reaction are the same
  3. Conservation of momentum. The total momentum of the interacting particles before and after a reaction are the same.
  4. Conservation of energy. Energy, including rest mass energy, is conserved in nuclear reactions.
 
Types of Nuclear Reactions
Although the number of possible nuclear reactions is enormous, nuclear reactions can be sorted by types. Most of nuclear reactions are accompanied by gamma emission. Some examples are:
  • Elastic scattering. Occurs, when no energy is transferred between the target nucleus and the incident particle.

 208Pb (n, n) 208Pb

  •  Inelastic scattering. Occurs, when energy is transferred. The difference of kinetic energies is saved in excited nuclide.

 40Ca (α, α’) 40mCa

  • Capture reactions. Both charged and neutral particles can be captured by nuclei. This is accompanied by the emission of ˠ-rays. Neutron capture reaction produces radioactive nuclides (induced radioactivity).

 238U (n, ˠ) 239U

  • Transfer Reactions. The absorption of a particle accompanied by the emission of one or more particles is called the transfer reaction.

4He (α, p) 7Li

  • Fission reactions. Nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts (lighter nuclei). The fission process often produces free neutronsand photons (in the form of gamma rays), and releases a large amount of energy.

235U (n, 3 n) fission products

  • Fusion reactions.  Occur when, two or more atomic nuclei collide at a very high speed and join to form a new type of atomic nucleus.The fusion reaction of deuterium and tritium is particularly interesting because of its potential of providing energy for the future.

3T (d, n) 4He

  • Spallation reactions. Occur, when a nucleus is hit by a particle with sufficient energy and momentum to knock out several small fragments or, smash it into many fragments.
  • Nuclear decay (Radioactive decay). Occurs when an unstable atom loses energy by emitting ionizing radiation. Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. There are many types of radioactive decay:
    • Alpha radioactivity. Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Because of its very large mass (more than 7000 times the mass of the beta particle) and its charge, it heavy ionizes material and has a very short range.
    • Beta radioactivity. Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The beta particles have greater range of penetration than alpha particles, but still much less than gamma rays.The beta particles emitted are a form of ionizing radiation also known as beta rays. The production of beta particles is termed beta decay.
      • Gamma radioactivity. Gamma rays are electromagnetic radiation of an very high frequency and are therefore high energy photons. They are produced by the decay of nuclei as they transition from a high energy state to a lower state known as gamma decay. Most of nuclear reactions are accompanied by gamma emission.
        • Neutron emissionNeutron emission is a type of radioactive decay of nuclei containing excess neutrons (especially fission products), in which a neutron is simply ejected from the nucleus. This type of radiation plays key role in nuclear reactor control, because these neutrons are delayed  neutrons.

        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 = 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, they are of the order of a million times greater than the electron binding energies of atoms.

        Q-value of DT fusion reaction
        Q-value of DT fusion reaction

        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=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 said to be endothermic (or endoergic) and they require a net energy input.

        See also: Q-value Calculator

         
        Example: Exothermic Reaction - DT Fusion
        Q-value of DT fusion reaction
        Q-value of DT fusion reaction

        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

        Example: Endothermic Reaction - Photoneutrons
        In nuclear reactors the gamma radiation plays a significant role also in reactor kinetics and in a subcriticality control. Especially in nuclear reactors with D2O moderator (CANDU reactors) or with Be reflectors (some experimental reactors). Neutrons can be produced also in (γ, n) reactions and therefore they are usually referred to as photoneutrons.

        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 in excess of 6 MeV, which is 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, 9Be6Li, 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.

        Photoneutron sources
        Nuclides with low photodisintegration
        threshold energies.

        In case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:

        Photoneutron - 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:
        • 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.
         
        Equations of Conservation of Momentum and Energy
        Let us assume the one dimensional elastic collision of two objects, the object A and the object B. These two objects are moving with velocities vA and vB along the x axis before the collision. After the collision, their velocities are v’A and v’B. The conservation of the total momentum demands that the total momentum before the collision is the same as the total momentum after the collision. Likewise, the conservation of the total kinetic energy, which demands that the total kinetic energy of both objects before the collision is the same as the total kinetic energy after the collision. Both law may be expressed in equations as:

        equations-of-conservation-of-momentum-energy

        solution-conservation-of-momentum-energy

        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.

        Elastic Nuclear Collision
        See also: Neutron Moderators

        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 moderator of the reactor. The probability of the fission U-235 becomes very large at the thermal energies of slow neutrons. This fact implies increase of multiplication factor of the reactor (i.e. lower fuel enrichment is needed to sustain chain reaction).

        The neutrons released during fission with an average energy of 2 MeV in a reactor on average undergo a number of 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. Using the laws of conservation of momentum and energy and the analogy of collisions of billiard balls for elastic scattering, it is possible to derive the following equation for the mass of target or moderator nucleus (M), energy of incident neutron (Ei) and the energy of scattered neutron (Es).

        equation momentum energy

        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 scattered neutron is taken as the average of energies with scattering angle 0 and 180°.

        Moreover, it is useful to work with logarithmic quantities and 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:logarithmic energy decrement - equationFor heavy target nuclei, ξ may be approximated by following formula:the logarithmic energy decrement per collisionFrom 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 neutron in the carbon.

        ξCARBON = 0.158

        N(2MeV → 1eV) = ln 2⋅106/ξ =14.5/0.158 = 92

        Table of average logarithmic energy decrement for some elements
        Table of average logarithmic energy decrement for some elements.

        For a mixture of isotopes:

        the logarithmic energy decrement for mixtures

        Example: Elastic Nuclear Collision
        A neutron (n) of mass 1.01 u traveling with a speed of 3.60 x 104m/s interacts with a carbon (C) nucleus (mC = 12.00 u) initially at rest in an elastic head-on collision.

        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 of kinetic energy, and apply them to our system of two particles.

        conservation-laws-elastic-collisions

        We can solve this system of equation or we can use the equation derived in 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.

        solution-elastic-collision

        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 the neutron moderation and it significantly depends on the mass of moderator nuclei.

         
        References:
        Nuclear and Reactor Physics:
        1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
        2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
        3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
        4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
        5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
        6. Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN: 978-0471805533
        7. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
        8. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
        9. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

        Advanced Reactor Physics:

        1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
        2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
        3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
        4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

        See above:

        Conservation of Energy