Nuclear Physics Part 1: Nuclear Structure & Reactions

Nuclear Physics Part 1: Nuclear Structure & Reactions Last modified: 25/01/2018

Links The Atomic Nucleus Nucleons Strong Nuclear Force Nuclei Are Quantum Systems Atomic Number & Atomic Mass Number Nuclides Isotopes Radioisotopes Line of Stability Binding Energy Mass Defect Binding Energy Non-SI units: u and MeV Nuclear vs Atomic Masses Example Binding Energy Curve Nuclear Reactions Q-value Q-value & Binding Energy Exothermic & Endothermic Reactions Example Example Summary

The Atomic Nucleus Atoms consist of negatively charged electrons orbiting a small, positively charged nucleus. Atomic nuclei vary in size, but always make up a tiny fraction of an atom s volume. If we imagine that a nucleus were 1 mm in diameter, then the atom would be in the range of 30-50 metres across. An atom is mostly empty space. proton (p) neutron (n).1.5 nm 2 15 fm Despite its small size, the nucleus contains nearly all of the atom s mass.

Nucleons The nucleus is made up of nucleons, of which there are two types: the proton, which is positively charged (equal and opposite to the electron charge) and the neutron, which has zero charge. mass (kg) electric charge (C) proton (p) 1.672623 10 27 1.602 10 19 neutron (n) 1.674928 10 27 0 Apart from electric charge, the properties of the two types of nucleon are very similar - in particular they have almost the same mass, with the neutron s being only slightly larger. Remember the mass of the electron is 9.109390 10 31 kg, about 2000 times smaller than the proton/neutron mass.

Strong Nuclear Force The positively charged protons inside a nucleus will experience repulsive Coulomb forces. How does the nucleus stay together? There is another force acting between nucleons, (imaginatively) called the strong nuclear force (or often just strong force). Strong Force Properties: It only acts between nucleons. Electrons are not affected by the strong force. It is always attractive. The type of nucleon doesn t matter. The force between a neutron and a proton is the same as the force between two neutrons or between two protons. It has a very short range - the force between two nucleons drops to zero at a separation of only a few femtometres (1 fm = 10 15 m). This is very different to the familiar Coulomb and gravitational forces, which although they become weaker with distance, never completely disappear.

To form a stable nucleus, the strong forces holding the nucleus together must be greater than the Coulomb forces pushing the protons apart. This will not always happen. For instance, if we bring 2 protons close together, then they will be attracted by the strong force (red), but the repulsive Coulomb force (blue) will win and so they do not form a nucleus. But, after adding a neutron to the two protons, additional strong forces will act, and in this case they can overcome the repulsive Coulomb force and hold the nucleus together. p p Coulomb forces: repulsive Strong forces: attractive n p p This pattern continues as larger numbers of nucleons are combined. In only some combinations will the strong force be able to overcome the Coulomb repulsion between the protons, and so form a nucleus.

Nuclei Are Quantum Systems We have already seen that atoms are quantum systems. Electrons in an atom must exist with one of a limited range of energies and can move between energy levels by absorbing or emitting photons of the appropriate energy. These photons will typically have energies of a few ev or less. Nuclei are also quantum systems. Like electrons in an atom, nucleons can jump between energy levels by emitting or absorbing photons. These photons are much more energetic than those seen with atoms and will typically have energies measured in MeV (gamma rays). Exactly the same terms are used as for atoms - the lowest energy level is the ground state of the nucleus, and higher energy levels are the excited states. Different types of nuclei will each have a unique emission and absorption spectrum, which can be measured experimentally and used to identify them.

Atomic Number & Atomic Mass Number The number of protons in a nucleus is called the atomic number, Z of that nucleus. Z identifies elements in the periodic table, so determines what type of atom the nucleus is part of. For example: Z = 6 carbon, Z = 20 calcium, Z = 92 uranium etc. The number of neutrons is the neutron number, N. The total number of nucleons is A = Z + N, the atomic mass number. Since the mass of the proton and the neutron are very similar, the mass of the nucleus will be approximately A times the nucleon mass.

Nuclides The word nuclide is used to refer to the type of a nucleus (in the same way as the word element is used to describe the type of an atom). A nuclide is represented using the following notation: atomic mass number atomic number A Z X chemical symbol For example, the nuclide with 6 protons and 6 neutrons is: 12 6 C. You may also see this be written as: 12 C, carbon-12 or C12. In speech this nuclide is usually referred to as carbon 12.

Isotopes The atomic number Z determines the number of electrons in the atom, and hence the atom s chemical properties. Adding or removing neutrons from the nucleus, will of course change the mass of the nucleus, but not affect this chemistry (or at least only very slightly). Nuclides with the same atomic number, but different atomic mass number, are known as the isotopes of the element. Only a limited number of isotopes of each element are possible. Some isotopes can exist, but are unstable and will, over time transform into other stable isotopes - usually of a different element. We will look at these processes in more detail in the next lecture. 26 elements (including aluminium, sodium and gold) have only one stable isotope. Other elements have varying numbers up to the ten stable isotopes of tin.

Radioisotopes Naturally occuring elements will usually contain a mixture of stable and unstable isotopes, some of them being much less common than others. For example, there are three possible isotopes of hydrogen: 1 1H, 2 1H (known as deuterium ) and 3 1H ( tritium ). Of these, tritium is unstable and the others stable. Naturally occurring hydrogen atoms are 99.9885% 1 1H, the remainder 2 1H with only very small amounts of tritium. Some unstable isotopes (such as tritium) occur naturally as the result of nuclear processes, while others are only seen when they are produced artificially in a lab. As they transform into other nuclides, unstable nuclei will usually emit some form of radiation, and are said to be radioactive. Radioactive isotopes are known as radioisotopes.

Line of Stability The graph at right shows all observed isotopes plotted with neutron number N vs atomic number Z and shaded to indicate each isotope s lifetime. N 160 Isotope Lifetimes Stable isotopes (shown in black) form a curve known as the line of stability. 140 120 Some observations from this plot: Lifetimes decrease (stability decreases) further away from this line The heaviest stable isotope has Z = 82 (lead) For lighter nuclei (Z 20) the stable isotopes have Z N Heavier stable isotopes have increasing numbers of neutrons, with N/Z 1.5 for the heaviest stable nuclei 100 80 60 40 20 20 40 60 80 100 stable longer life Z

Mass Defect The simplest multi-nucleon nucleus is 2 1H, also known as a deuteron, consisting of one proton and one neutron: p Imagine reaching into a deuteron at rest, grabbing hold of the two nucleons and pulling them apart, to a separation distance beyond the range of the strong force, where they are again at rest. F p n F p n n rest rest rest The applied forces have done work - adding energy to the deuteron. Where has this energy gone?

We began with the deuteron at rest, so it only had rest mass energy. The separated nucleons are also at rest, so also only have rest mass energy. From this we conclude that the rest mass of the deuteron must be less than the combined rest masses of the two nucleons. The difference between the total mass of the constituent nucleons and the mass of a nucleus is called the mass defect of that nucleus. For example, the mass defect of a deuteron is: δm = m proton + m neutron m deuteron = 1.672622 10 27 + 1.674927 10 27 3.343583 10 27 = 0.003966 10 27 kg

Binding Energy The energy equivalent of the mass defect (i.e. δm c 2 ) is called the binding energy of the nucleus. The binding energy of a deuteron is: 0.003966 10 27 (2.99792 10 8 ) 2 = 3.564 10 13 J = 2.225 MeV The general formula for the binding energy BE of the nucleus A ZX with mass m X is: BE = [Z m proton + (A Z) m neutron m X ] c 2 A larger binding energy indicates a smaller nuclear mass, and thus that the nucleus is more tightly stuck together.

Non-SI units: u and MeV The masses of nucleons and nuclei expressed in kilograms are very small and awkward to calculate with. For this reason, Nuclear Physics calculations usually use a more convenient unit of mass - the unified atomic mass unit (u) (also called the dalton (Da)) Unified Atomic Mass Unit 1 u = 1 12 mass of a 12 6C atom = 1.660538921 10 27 kg Some example masses in u: proton (p) 1.00727647 neutron (n) 1.00866492 electron (e) 5.485799 10 4 1 4 238 1H atom 1.00782504 2He atom 4.002602 92U atom 238.050788 The mass in u of the nuclide A ZX is always very close to A. We will soon see that these small differences are very important.

Of course you will not be expected to remember atomic masses (except possibly 12 6C). In exams all necessary masses will always be given. If you ever need to look up a mass, Wikipedia is a good source. If for example you needed the atomic mass of a gold isotope, search for Isotopes of gold. The energy equivalent of 1 u is: (1 u in kg) c 2 = 1.660538921 10 27 (2.99792 10 8 ) 2 = 1.49241 10 10 J = (1.49241 10 10 ) (1.602 10 19 ) ev = 931.5 MeV Binding (and other) energies in Nuclear Physics are usually expressed in MeV.

Nuclear vs Atomic Masses The previous definition for the binding energy of the nucleus A ZX used the nuclear mass, which was represented as m X. Especially for heavier nuclei, it is usually much easier to measure the mass of an atom of A ZX rather than this nuclear mass. What is the difference? Z electrons! If we represent the atomic mass of A ZX by M X, then: M X }{{} atom = m X }{{} nucleus where m e is the mass of an electron. + Zm e }{{} electrons For example, the atomic mass M H of the hydrogen atom 1 1H is M H = m proton + m e

Returning to the definition of binding energy, and expressing the nuclear mass m X in terms of the atomic mass M X : BE (in MeV) of A ZX = [Z m proton + (A Z) m neutron m X ] 931.5 = [Z m proton + (A Z) m neutron (M X Zm e )] 931.5 = [Z (m proton + m e ) + (A Z) m neutron M X ] 931.5 = [Z M H + (A Z) m neutron M X ] 931.5 This has the same form as the original definition, but with the substitutions: m proton M ( 1 1 H) and m X M X We will always use atomic masses so this is the required form of the equation.

Example Calculate the binding energy of 4 2He. 4 2He consists of 2 protons and 2 neutrons. Using atomic masses in u (shown earlier), the binding energy in MeV will be: ( ) BE = 2 M ( 1 1 H) + 2 m n M ( 4 2 He) 931.5 = (2 1.00782504 + 2 1.00866492 4.002602) 931.5 = 0.03037792 931.5 = 28.297 MeV We might feel tempted to round these mass values off to one or two digits.in this case, that would give a very incorrect result: BE = (2 1.0 + 2 1.0 4.0) 931.5 = 0 MeV ALL the given decimal places in mass values must be used!

Calculate the binding energy per nucleon of 4 2He. The previous calculation gave the binding energy for 4 2He to be 28.297 MeV. There are FOUR nucleons in total, so the binding energy per nucleon is calculated simply: 28.297 MeV/4 = 7.074 MeV The average binding energy, or binding energy per nucleon, allows us to compare the binding energies, and hence the stability, of different nuclides. Plotting the results of similar calculations to the above for all stable nuclides gives the following graph:

Binding Energy Curve BE/A (MeV) 10 12 6 C 4 2He 8 6 4 2 16 8 O 56 Fe Notes: For small nuclei, as A increases, the average binding energy also increases - indicating more tightly bound (i.e. more stable) nuclides. The curve peaks in the A = 50 60 range, so nuclei in this region are the most stable. 56 28Fe is generally considered the most stable nuclide. The curve drops gradually for larger A, indicating less stable heavier nuclei. There are several unusually stable light nuclei: 4 2He, 12 6C and 16 8O. 0 0 50 100 150 200 A

Nuclear Reactions A nuclear reaction occurs when a number of nuclei combine together to produce other nuclei (very much like molecules combining in a chemical reaction): A 1 Z 1 X 1 + A 2 Z 2 X 2 +... A 1 Z 1 Y 1 + A 2 Z 2 Y 2 +... For such a reaction, there are two conservation laws which can be used to identify an unknown nuclide: Z 1 + Z 2 +... = Z 1 + Z 2 +... A 1 + A 2 +... = A 1 + A 2 +... For example, to identify the mystery particle X in: 12 6 C + 4 2He 14 7N + A ZX we know 6 + 2 = 7 + Z and 12 + 4 = 14 + A so we must have A ZX 2 1H

Q-value The Q-value for a reaction is the amount of energy released in that reaction. It is the energy equivalent of δm - the difference between the total rest mass of the initial particles and the total rest mass of the final particles. δm = (total mass at start) (total mass at end) = (m X1 + m X2 +...) (m X 1 + m X 2 +...) Q = δm c 2 Using atomic masses expressed in u, then the Q-value in MeV will be: Q = [(M X1 + M X2 +...) (M Y1 + M Y2 +...)] 931.5

Q-value & Binding Energy This formula is very similar to that for binding energy, seen earlier. In fact binding energy can be thought of as a special case of the Q-value. The Binding Energy of the nuclide A ZX is equal to the Q-value of the (hypothetical!) reaction forming the atom: 1 1H + 1 1H +... }{{} Z protons + Z electrons + 1 0n + 1 0n +... }{{} A Z neutrons A ZX }{{} atom

Exothermic & Endothermic Reactions There are two possibilities for the Q-value of a reaction: If Q > 0 then energy is released. The reaction is exothermic. The released energy will appear as kinetic energy of the products. If Q < 0 then energy must be added (in the form of kinetic energy of the initial particles) before the reaction can proceed. The reaction is endothermic. These words are used in exactly the same way as in Chemistry for a chemical reaction.

Example Fluorine-18 ( 18 9F) is a radionuclide often used in medical procedures. It does not occur naturally, but can be produced using a cylotron to collide energetic protons with water containing 18 8O (Oxygen-18). (a) What is the full equation for this reaction? (b) Calculate the Q-value for the reaction. (a) Determining the full equation for the reaction requires a little simple arithmetic to balance the numbers of protons and neutrons before and after: 1 1H + 18 8O 18 9F + 1 0n To highlight the details of how the reaction is achieved, this equation can also be written in a slightly different form: ( 18 8 O 11 H, 1 }{{} 0n ) 18 9 F target }{{} beam }{{} product

(b) The Q-value calculation is straightforward: [ Q = masses (in u) ] masses (in u) 931.5 initial final = [(M H + M O ) (M F + m n )] 931.5 = (1.007825032 + 17.999161001 18.000937956 1.00866492) 931.5 = 2.44 MeV What is the minimum kinetic energy of protons required for this reaction to proceed? A negative Q-value tells us that the reaction can only proceed if we add energy. This can be achieved in the form of kinetic energy of the proton. How much is required? We need the total final energy to be positive, so any amount greater than 2.44 MeV will achieve this. The minimum energy required is of course 2.44 MeV.

Not Just Nuclei in Nuclear Reactions A nuclear reaction can include other particles in addition to nuclei. For example, the capture of neutrons by Cadmium nuclei is important for the operation of most nuclear reactors. This reaction will involve the production of a gamma ray (i.e. a high energy photon). One such reaction is: 111 48 Cd + 1 0n 112 48Cd + γ This can be thought of as a two-stage process. The neutron is absorbed, resulting in a nucleus in an excited state, indicated by a *, which then drops to its ground state by emission of a photon: 111 48 Cd + 1 0n 112 48Cd 112 48Cd + γ Remember, like an atom, a nucleus is a quantum system with quantized energy levels.

Example Find the Q-value for the reaction: 111 48Cd ( 1 0 n, γ ) 112 48 Cd This is the reaction we have just discussed, written to emphasize that the moving neutron collides with the stationary Cadmium nucleus. 111 48 Cd + 1 0n 112 48Cd + γ The presence of non-nuclei makes no difference to the definition or calculation of the Q-value. We need to remember that the rest mass of the photon is zero: Q = [(M Cd-111 + m n ) (M Cd-112 + m γ )] 931.5 = (110.9041781 + 1.00866492 111.9027578 0) 931.5 = 9.39 MeV Most of this energy is carried by the photon.

Summary Nuclides are identified using the standard notation: A Z X Z= atomic number (no. of protons) X = chemical symbol A = atomic mass number (no. of nucleons) The binding energy of a nuclide is the difference in rest mass energy of the nucleus and its constituent nucleons: BE = ( (mass of nucleons) (mass of nucleus) ) c 2 Usually, non-si units are more convenient: BE (in MeV) = ( ) (mass of nucleons) (mass of nucleus) 931.5 }{{} in u

The Q-value for a nuclear reaction is the energy released in the reaction, and is calculated as the difference between the total rest mass energy of the initial reactants and the total rest mass energy of the final products: ( Q = mass ) mass c 2 initial final Again, non-si units are usually used: ( Q (in MeV) = mass ) mass 931.5 initial final }{{} in u If Q < 0, the reaction is endothermic, and energy must be added (as kinetic energy of the initial reactants), in order for the reaction to proceed.