Nuclear Reactions. This is an example of nuclear reaction. Now consider a chemical reaction

Nuclear Reactions. Introduction REDs are important while yellow is less. Nuclear reactions are the transformations that occur when two nuclei collide. The first such reaction was observed by Rutherford in 99 when the bombarded nitrogen nuclei with alpha particles from a natural radioactive source and made possible the reaction 4 4 7 He + 7 N H + 8 O.. (i) This is an example of nuclear reaction. Now consider a chemical reaction NO + O 3 N O 3 + O.. (ii) Here molecules (or atoms and molecules) collide and rearrange their constituent atoms. However, there is an important practical difference between nuclear reactions and chemical reactions: Under normal conditions on earth, chemical reactions occur naturally and abundantly whereas nuclear reactions occur hardly. For two atoms or molecules to react chemically, they only need to approach one another closely enough for their outer electrons to overlap, and this condition is easily met at normal temperatures and densities. But even in the most violent chemical reactions the nuclei, buried deep inside their protective shells of electrons, remain very far apart compared to the nuclear force range. Even if the two nuclei cross their electrons cloud to come closer, the Coulomb repulsion between their positive charges do not let them so close to have the nuclear reactions. For example, the energy needed to bring a proton to the surface of an alpha particle is about MeV. For this reason, nuclear reactions occur naturally only very rarely on earth. However, they do occur in stars, where temperature is so high that the kinetic energy of thermal motion is sufficient to overcome the Coulomb repulsion. To produce a nuclear reaction in the laboratory, it is needed first produce nuclei with energies of the order MeV or more and then direct them at other nuclei. It is the particle accelerators which made it possible to accelerate many different nuclei to an energy sufficient to induce nuclear reactions. On the other hand, since neutrons are electrically neutral, even of low -energy, they can approach and penetrate a nucleus. In this sections we will discuss some nuclear reactions with their types and energy involved during such reactions. Later on, we introduce nuclear reactors and the reactions involved on such reactors. Representation of a nuclear reaction Typically an equation representing a nuclear reaction may be written as x + X Y + y In words this would read like: when an incident projectile x hits the target nucleus X, a nuclear reaction takes place and as a result there is a new nucleus Y and outgoing particle y. The above reaction can also be written in short form as X(x, y)y. For example 4 7 N (α, p) 7 8 O stands for a reaction between an 4 4 incident α particle ( He) and a 7 N nucleus with the emission of a proton(p = H) and formation of a new nucleus 7. O 8

. The balance of Mass and energy in nuclear Reactions Consider a nuclear reaction represented by the equation x + X Y + y. (.0) where X is the target nucleus, x is the incident projectile, Y the product nucleus, and y be the outgoing particle. Also consider X is initially at rest so that it has no kinetic energy. Since the total mass and energy is conserved, we have (E x + m x c ) + M X c = (E Y + c ) + (E y + m y c ).. (.) where E x is the kinetic energy of the projectile, m x c be its rest mass energy and similarly M X c, E Y, c, E y and m y c. Now we introduce the quantity Q, which represents the difference between the kinetic energy of the products of the reaction and that of the incident particle, Q= E Y + E y E x. (.) The quantity Q is called the energy balance of the reaction or Q value. This difference in total kinetic energy in a nuclear reaction is also called disintegration energy. From equation (.) and (.) we get: Q = E Y + E y E x = [(m x + M x ) ( + m y )]c.(.3) Thus we see that Q is also the change in the total rest mass. From equation (.3), we see that if Q is positive, the kinetic energy of the product is greater than that of the reactants, the reactions is then said to be exoergic or exothermic. In this case total mass of the reactants is greater than that of the products. If the value of Q is negative, the reaction is endothermic or endoergic..3the Q equation The analytical relationship between the kinetic energy of the projectile and outgoing particle and the nuclear disintegration energy Q is called the Q equation. Consider an incident particle of mass m x moving with velocity v x collide with a target nucleus of mass M X at rest. After collision, outgoing particle of mass m y is emitted with velocity v y at angle of θ. Let be the rest mass of product nucleus which is emitted with velocity v Y at angle of φ, figure (.). According to conservation of linear momentum in the plane of paper, we have equations: m x v x = v Y cos Φ + m y v y cos θ. (.30), along the direction of incident particle.

Figure. 0 = v Y sin Φ m y v y sin θ,. (.3) along the direction perpendicular to the incident particle. Rearranging both equations, we get: m x v x m y v y cos θ = v Y cos Φ, and m y v y sin θ = v Y sin Φ Squaring and adding, we get: v Y = m x v x + m y v y m x v x m y v y cos θ.. (.3) Using the kinetic energy relations: E Y = v Y, E x = m xv x and E y = m yv y Substituting these values in equation (.3) we get: E Y = m x E x + m y E y p x p y cos θ,where p x and p y be the momentum of particles x and y respectively. E Y = m x E x + m y E y 4 m x m y E x E y cos θ [ E x = p x and E m y = p y x Or, E Y = m xe x + m ye y m x m y E x E y cos θ. (.33) m y ]

We have Q value equation (equation.): Q= E Y + E y E x Putting the value of E Y from equation (.33) we get: Q= m xe x + m ye y m x m y E x E y cos θ + E y E x Or, Q = E y ( + m y ) E x ( m x ) m x m y E x E y cos θ (.34) This is the standard form of Q equation. If θ = 90 0 then equation (.34) becomes Q = E y ( + m y ) E x ( m x ). (.35).4 Threshold Energy For an endoergic reaction, the energy needed to excite the compound nucleus is Q so that nuclear reaction takes place. The energy is supplied in the form of kinetic energy of the incident particle. However, all the kinetic energy is not available for excitation because some energy is used to provide momentum to the compound nucleus. This momentum is distributed among the products of the reaction. Consequently, for Q to be available for excitation of the compound nucleus, we must supply some energy in addition to Q. Thus the minimum kinetic energy which the incoming particle should possess so that nuclear reactions may take place is called the threshold energy. Let s get calculated the threshold energy: Let M c and V c denote the mass and velocity of the compound nucleus, then according to conservation of momentum: m x v x = M C V C V c = m x M c v x.. (.40) The part of the kinetic energy of the incident particle needed for excitation of the compound nucleus is Q = m xv x M cv c From equation (.40) Q = m xv x M (m x ) c v (M c ) x Or, ( Q) = m xv x ( m x M c ) But, M c = M X + m x and hence Q = m xv x ( M X ) M X +m x

Threshold energy is then given by m xv x = ( Q) ( M X+m x M X ) Or, E th = m xv x = ( Q) ( + m x M X )... (.4).5 Types of nuclear reactions To know more about nuclear reactions, we have to know more about the mechanism by which such reactions commonly occur. We consider the following categories of the simple types of nuclear reactions:. Elastic Scattering. The incident particle strikes the target nucleus and leaves without energy loss. In general, the direction of the incident particle get altered. For example, scattering of α- particles in gold is a good example of this process. 97 97 79Au 4 4 He + 79Au + He. Inelastic Scattering. The scattered particle may loss kinetic energy in excess of that required for an elastic collision with the nucleus. This lead the excitation of target nucleus to a higher allowed energy level. The excited nucleus later decays to the ground state, radiating the excess energy in the form of a γ ray photon. We have well known example. 7 7 3Li + H ( 3Li) + H and then 7 3Li 7 3Li + γ 3. Disintegration. On striking the target nucleus, the incident particle is absorbed and a different particle is ejected. The product nucleus differs from target nucleus. An example is the disintegration of nitrogen by α particle. 4 4 7 N + He O + H 7 8 4. Photo disintegration. The target nucleus absorbs radiations which results the compound nucleus in excited state. This nuclei generally get rid of the excess excitation energy through neutron emission. For example H + γ H + 0 n 5. Radiative Capture. In this case, a particle combine with a nucleus to produce a new nucleus or a compound nucleus. The new nucleus is in then excited state. The excess energy is emitted in the form of γ ray photons. For example.6 Conservation law 4 6C + H 7N + γ Any nuclear reaction occurs with conservation of certain quantities. Some of the nuclear reactions which are valid in ordinary nuclear reactions discussed as follows:. Conservation of Energy. The total energy of the products, including both mass energy and kinetic energy of the particles plus energy involved must be equal to the mass energy of the initial ingredients plus the kinetic energy of the bombarding particles.

. Conservation of Momentum. The total linear momentum of the products must be equal to the linear momentum of the bombarding particle, the target nucleus is usually taken to be at rest. 3. Conservation of charge. The total electric charge of the products must be equal to the total electric charge of initial particles. 4. Conservation of Nucleons. The law of conservation of nucleons states that total number of nucleons entering and leaving the reaction is constant. Hence for an isolated system, total number of nucleons remains constant. 5. Conservation of Spin. The spin characteristic of the closed system can not be changed i.e. total spin before interaction is equal to the total spin after interaction. 6. Conservation of Parity. The parity of the system is the product of intrinsic parities of the target nucleus and bombarding nucleus. No violation of the parity has been observed in a nuclear reaction. Angular momentum and isotopic spin are the other physical quantities which are also conserved in a nuclear reaction. 7. Quantities not conserved. The quantities which are not conserved in nuclear reactions are the magnetic dipole moments and the electric quadrupole moments of the reacting nuclei. These moments depends up on the internal distribution of mass, charge and current with in the nuclei involved and do not follow conservation laws..7 Nuclear fission When uranium( 35 9U ) is bombarded with a neutron, two large fragments barium( 40 56 Ba ) and krypton( 36 9 Kr ) along with the production of three neutrons and large amount of energy. The reaction is as follow 35 36 + n U 40 Ba + 9 Kr + 3 n + Q.. (.70),where Q is the energy released 9U 0 9 56 during the fission reaction 36 0 Meitner and Frisch described this reaction as a fission reaction. It is defined as the process of breaking of up of the nucleus of a heavy atom into two, more or less equal fragments with the release of a large amount of energy. The release of energy is due to the mass deficiency in the product nuclei than that of reactants. This difference in mass is converted into energy according to Einstein s mass energy relation. We can now estimate the energy released during the fission reaction such as (.70). From the binding energy curve [figure (.4)] in.4, we can see that B/A for the two fragments(with A 00) is nearly MeV higher than that of the parent nucleus(a 00). Hence the binding energy per nucleon increases by about MeV, so the kinetic energy released from a single fission is roughly about 00 MeV. Now we calculate the energy released for kg of Uranium( Number of atoms in 35 gram of uranium= 6.03 0 3 For gram of uranium= 6.03 03 35 Hence kg of uranium = 6.03 06 35 Then energy produced by kg of uranium during fission= 6.03 06 35 35 9U) as follows: 00 MeV = 5.3 0 6 MeV.

This energy is enormous energy. This is the reason why nuclear energy is used to generate electricity. Bohr and Wheeler s theory of nuclear fission Bohr and Wheeler had explained the phenomena of nuclear fission as like the breaking of liquid drop. They treated a charged nucleus analogous to a liquid drop. The spherical shape of liquid-drop nucleus depends on a balance involving the surface tension forces and the Coulomb repulsive forces. Initially the nucleus is spherical in shape (stage A). If energy is added to the drop, Figure. as in the excitation energy resulting from the capture of a slow neutron in the nucleus, oscillations are set up within the drop; these tend to distort the spherical shape so that the drop may become ellipsoidal in shape(stage B).The surface tension force tends to make the drop return to its original shape, while the excitation energy tends to distort the shape still further. If the excitation energy is sufficiently large, the drop may attain the shape of a dumbbell (Stage C). The coulomb repulsive forces then push the two bells apart until the dumbbell splits into two similar drops, each of which then becomes spherical in shape (Stage D). The possible steps in the process of nuclear fission according to the liquid-drop model are shown in figure...8 Chain Reaction The chain reaction is a self-sustaining process that once started need no additional agents to keep it going. Generally in the chain reaction, the number of neutrons goes on multiplying rapidly almost in geometrical progression during fission till whole of fissile material is disintegrated. Example: Suppose a single slow neutron which incident on uranium nucleus ( 35 9U), prompt a fission reaction resulting three neutrons. These three neutrons may cause other three fission reactions producing nine neutrons. These nine neutrons in turn may cause fission in nine uranium nuclei producing 7 neutrons and so on. The number of neutrons produced in n such steps is 3 n. The ratio of secondary neutrons produced to the original neutrons is called the multiplication factor (K). Two types of chain reaction are possible. In one, the chain reaction is first accelerated so that the neutrons are build up to a certain level and there after the number of fission producing neutrons is kept constant. This is called controlled chain reaction. In nuclear reactors, such chain reactions occur. The other type of chain reaction is called uncontrolled chain reaction. In such reaction, number of neutrons goes on multiplying in geometric progression and huge amount of energy is released in very short interval of time. This type of chain reaction takes place in atom bombs. When multiplication factor K= then chain reaction will be steady or critical while for k > it is building up,called supercritical state. and when k <, the chain reaction will be dying down called subcritical state. Critical size for maintenance of chain reaction Each fission of uranium nucleus ( 35 9U) produces.5 neutrons on the average, all of them are not available for further fission. There may be leakage of neutrons through the surface of the system. The

loss of the neutrons is also caused by radiative capture of neutrons by the uranium and other nonfission capture by the different substances in the system and by impurities. If the loss of neutrons is less than the neutrons produced, a chain reaction takes places. The escape of neutrons takes place from the surface of the reacting body and fission occurs throughout its volume. Hence the escape rate varies with the surface area 4π r and production rate varies with the volume 4 3 πr3. Escape rate Production rate r [ 4π r 4 3 πr3 = r ] For the body of larger size, radius r is large and hence escape rate is small. Thus increasing the volume of the system, the loss of neutrons due to escape from the surface is reduced. In such situation, the production of neutrons will be more than the loss and a chain reaction can be maintained. Thus, there should be certain size, called critical size of the system for sustainable chain reaction. Critical size of a system containing fissile material can be defined as the minimum size at which the number of neutrons produced in the fission process is just balance those lost by leakage and non-fission capture. Chain reaction in natural uranium 38 Natural uranium consists of 99.8% 9U and 0.7% of 35 9U. Most of the mass of the natural Uranium consists of U 38 and neutrons produced will bombard mostly to the nuclei U 38, very few will bombard 35 9U. The uranium U 38 undergoes fission only by fast electrons of energy MeV or more while the fission of U 35 occurs even by neutrons of small energy i.e. slow neutrons. It has found that very few neutrons can cause fission of U 38. Hence the chain reaction can be developed only by slowing neutrons before they are lost through non-fission capture in the uranium and then securing striking to U 35 so that fission reaction builds up. The neutron can be slowed down by sending neutrons among some material called moderator. The function of moderator is to slow down the neutrons produced from fission by elastic collision. Commonly used moderator are heavy water (D O), graphite, beryllium, beryllium oxide, hydrides of metals, organic liquids etc. These materials absorbs neutrons only by very few amount. Figure.3 shows a self-sustaining chain reaction.

Figure: Sustainable chain reaction in U 35.9 Nuclear reactors(imp) In uncontrolled chain reaction like that in atom bomb, very large amount of energy is emitted within an extremely small interval of time. Hence it is not possible to direct this energy for any useful purpose. Nuclear reactor are constructed to bring uncontrolled reaction under human controlled. In such reaction, a steady state is established in which the rate of energy produced is a constant at a desire level and produced energy is directed for some useful purpose. Nuclear reactors have been constructed in all parts of world. They have different designs for a variety of purposes but some of the important elements that all reactors possess are as follows:. Fuel. Moderator 3. Neutron reflector 4. Cooling system 5. The safety and control systems.. Fuel.

The material containing the fissile isotope is called the reactor fuel. The commonly used fissionable material are the uranium isotopes U 33, U 38, the thorium isotope Th 3 and the plutonium isotopes Pu 39, Pu 40 and Pu 4. The form in which a fuel is used in a reactor depends upon various circumstances. Solid fuel elements are used in most of the reactors. Some reactors have been designed which make use of fluid fuels.. Moderator Material used to reduce the neutron energy, known an moderators, are graphite, light water, heavy water (D O),beryllium and its oxide and possibly certain organic compounds. Ideally, moderators have low atomic weight and low absorption cross-section for neutrons. 3. Neutron reflector. By the use of reflectors on the surface of reactors, leakage of neutrons can be very much reduced and the neutron flux in the interior can be increased. Materials of high scattering cross-section and low absorption cross-section are good reflectors. Generally material of high mass number are used as a reflector. 4. Cooling system. The function of cooling system is to remove the heat developed in the reactor core. The materials used of the cooling system are known as coolants. These materials are circulated through the core where they abstract heat from the core and transfer it to the outside of the core. An ideal coolant should have as little effect on the neutrons, should not react chemically with the other materials which it contacts in the system, should have a low vapour pressure at the operating temperature of the reactor, should be reasonably easy to handle, should be able to remove large amounts of heat for small expenditure of pumping power, should not be costly should acquire long lived radioactivity during its passages through the reactor and should not react chemically with the other materials which it contacts in the system. At ordinary temperatures both heavy and light water are good coolants. For reactors operating at high temperatures, high pressure are required to prevent boiling. Liquid metals have been proposed for use at higher temperatures. For example sodium metals. But it is very reactive with water and oxygen and becomes radioactive due to neutron capture when passed through the reactor. As a compromise between water and liquid sodium, certain organic compounds e.g. polyphenyls, were suggested. However, heat extracting property of such hydrocarbons are low than those of water or liquid sodium. In several of earliest reactors, air was used as a coolant. But it is not satisfactory at high temperatures because it reacts chemically with so many materials. Helium is a good coolant but it is very costly. Thus different coolants are suitable for different conditions, but to get an ideal coolant it is very difficult. 5. Control and safety system In control system, control is achieved by a neutron absorbing materials. The system enable the chain reaction to be controlled and prevent it from spontaneously running away. The control elements are commonly located in the core in the form of either rods or plates, but in some reactors it is more convenient to have the control elements in the reflector close to the core. These rods are of a material having a large neutron-absorption cross-section. In thermal reactor the control rods are moved in to decrease the fission rate or neutron flux and out to increase it. The safety systems protect the space surrounding the reactor against intensive neutron flux and gamma rays existing in the reactor core. This is achieved by surrounding the reactor with massive walls of concrete and lead which would absorb neutrons and gamma rays.

.0 Types of Nuclear Reactors (just read only, not in details) Reactors can be classified in a wide variety of ways. For example, they may classified by the manner in which the fuel and the moderator are mixed. In a homogenous reactor the fuel and moderator are mixed in the form of a solution. Most of the nuclear reactors are of the heterogenous type, in which fuel is concentrated in plates, rods, or hollow cylinders, which are distributed in a regular pattern i.e. lattice with in the moderator. Generally reactors are classified according to their purpose. Some of the types according to their purpose are: (a) research reactors (b) production reactors, and (c) power reactors. (a) Research reactors. Research reactors are used primarily to supply neutrons for physical and radioisotope manufacture. In these reactors, the total energy liberated is comparatively small; since they operate at a low level of reactivity. In these cases, cooling is required only to prevent over heating. Some types of research reactors: () Graphite Moderated Research Reactor. The graphite moderated, heterogenous, natural uranium reactor has the distinction of being the first reactor built and operated. This first reactor, now called CP- (Chicago Plile-), was put into operation in 94. This was operated without coolant flow at a power level of 00 watts. Similar assemblies wit air as coolant, like the GLEEP (graphite low energy experimental pile) and BEPO (British experimental pile) at Harwell (England) and the X-0 Clinton pile at Oak Ridge and BNL (Brookhaven National Laboratory) are well examples as they have been the work horses of reactor technology and research. All these reactors use fuel in the shape of cylindrical rods, or variations on this shape, clad usually in aluminum. Several hundred tons of graphite are used as moderator and reflector, with several hundred cylindrical channels passing right through the graphite to allow the introduction and positioning of the fuel elements. Cooling gas is made to pass over the elements through the channels at low pressure. () Water Boiler Type reactor. In this type of reactor, the water boiler is usually a homogenous mixture of a highly enriched uranium salt dissolved in ordinary water. The name arises from the fact that a sudden increase in power will cause the formation of steam bubbles in the solution, which in turn will shut down the reactor quickly. Under normal conditions, the solution does not boil as its temperature is kept below 80 0 C by circulating water through the coils inside of the core vessel. The water boiler reactor has proved to be versatile research tool providing intense sources of neutrons and γ rays. It is a great merit of simplicity of design and construction. Its use is somewhat limited because of its low power rating. (b) Production Reactors. The purpose of a production reactor is to convert fertile into fissile material. The fissible materials are used as a fuel in other reactors. In Hanford works production reactors, the natural uranium is used as a fuel, graphite as a moderator and water as a coolant. The efficiency of conversion process can be increased by decreasing resonance escape probability. It is mainly by capture of neutrons in the resonance region. (c) Power Reactors Power reactor (Important remember it)

The primary purpose of a power reactor is the utilization of the fission energy produced in the reactor core and to convert it into useful power. In such reactor, a quantity having enriched uranium in the form of pure metal or solution of a soluble salt in water constitutes the centre of the heat energy source, figure.4. During the fission process in the core, heat is produced. The cadmium rods are the controlled rods here, which regulate the temperature to a pre-determined value. If it is to lower the Figure.4: A schematic diagram of a power reactor temperature, the cadmium rods are pushed down further as to absorbs more neutrons. If it is desired to increase the temperature, the rods are pulled up little. A fluid is circulated through the shielded reactor and heat exchanger. The hot fluid, while flowing through the heat exchanger, converts water into steam. The steam produced runs turbines to produce electricity. Now we discuss some of the reactor types which appear to show the greatest promise for power production.. Nuclear Fusion and Thermonuclear Reactions An alternative to the fission reaction as a source of energy is its reverse process is known as fusion process, in which the lighter nuclei fuse together and produce heavier nucleus. In this process, there is also some mass is less in the product than that of reactant. This results release of energy. It has been calculated that the sun (our nearest star) emits electromagnetic energy at a rate of about 0 6 joules per second. Astronomical and geological evidences show that the sun has been radiating energy at about its present rate for several billion years. Chemical reactions cannot possibly be the source of this energy, because even it the sun is supposed to be consisted of pure carbo, its complete combustion would supply energy to maintain these radiations only for few thousand years. The question arises how can the sun have maintained this energy output for so long and what is the source of all stellar energy? Many ideas were generated about the stellar energy in the past. Eddington in 90, suggested that the stellar energy was liberated in the formation of helium from hydrogen. But he could explain the detail mechanism of it. In 99, Atkinson and Houtermans, in Germany, considered that energy might be liberated in the very high stellar temperatures.

At very high temperature (of the order 0 7 to 0 9 K) two nuclei have enough kinetic energy to overcome their mutual Coulomb repulsion and hence they can come closer to enter the zone of nuclear attractive forces. Such reactions are called thermonuclear reactions. H.A. Bethe in the United States suggested in 939 that the production of stellar energy is by thermonuclear reactions in which helium-4 nuclei are synthesized from four protons (the nucleus of hydrogen). For many years, it was held that the major portion of the sun s energy was derived form the carbon-nitrogen cycle. According to the suggestion of Bethe, the carbon nitrogen cycle was the major nuclear reactions for release of energy in the sun. The cycle is as follows: 3 6C + H 7N + γ 3 7N 3 6 C + 0 e + ν 3 4 6C + H 7 N + γ 4 5 7N + H 8O + γ 5 8O 5 7 N + 0 e + ν 5 4 7N + H 6C + He 4 4 H He + 0 e + ν + Energy The loss in mass is calculated as follows: 4 4 H = 4.00785 = 4.03300, He = 4.00603 and 9. 0 3 0 e = = 5.48.66 0 7 0 4 amu = 0.000964 amu [ m e = 9. 0 3 kg and amu =.66 0 7 kg] Loss in mass = 0.076006 amu Energy released = 0.076006 93 = 5.7 MeV. It is found that in one million years the sun loses about 0 7 of its mass by the above process. Taking the mass of the sun as 0 30 kg and its present age as 0 0 years, it is estimated that the C-N cycle keep going for another 30 billion years. Proton-Proton Cycle Recent modification of the estimates of the central temperature of the sun now favor the proton-proton cycle chain. In the p-p chain, two protons first fuse to produce a deuterium nucleus which combines with another proton to yield He 3. Two He 3 nuclei interact and form He 4 and two protons. These reactions can be represented by the equations: H + H ( He) 0 H + e + ν + 0.4 MeV H + H ( 3 He) 3 He + γ + 5.5 MeV 3 3 He + He ( 6 4Be) 4 He + H + H +.8 MeV 4 0 4 H He + e + ν + γ + 5.7 MeV A star is able to control thermonuclear fusion in its core because of its strong self-gravity. The thermonuclear reactions in the core of the sun causes high temperatures which generate strong

outward pressures, these act against the sun s own gravity, preventing it from contracting, and holding it in equilibrium. Hence the sun is emitting large amount of energy for billions of years in controlled manner without any change of its shape and size with appreciable amount. Workout Examples. Find (i) mass defect (ii) binding energy (iii) binding energy per nucleon for a helium nucleus. Given the mass of helium nucleus= 4.00509 a.m.u., mass of proton=.00777 a.m.u. and mass of neutron =.0086666 a.m.u. Solution: (i) Mass defect, m = [Zm p + (A Z)m n ] M = [.00777 +.0086666] 4.00509 = 0.030378 a.m.u. (ii) Binding energy, E= mc (iii) = 0.030378 (3 0 8 ).67 0 7 =4.56 0 J = 4.56 0 = 8.53.6 0 9 06 ev = 8.53 MeV Binding energy Binding energy per nucleon= = 7.3 MeV = 8.53 No.of nucleons 4 38. Binding energy per nucleon for 9U is about 7.5 MeV, whereas it is about 8.5 MeV for nuclei of 38 half that mass. If a 9U nucleus were to split into two equal size nuclei, about how much energy would be released in the process? Solution: There are 38 nucleons involved. Each nucleon will release about 8.5-7.5= MeV of energy when the nucleus undergoes fission. The total energy liberated is therefor about 38 MeV. 9 4 6 3. The precise mass in the reaction H + 9 F He + 8 O have been determined by mass spectrometer and are m(h)=.00785 u, m(he) = 4.00603 u, m( 9 9 F ) = 8.998405u; m( 6 8 O ) = 5.99495 u. Determine the Q and the nature of the reaction. 9 4 6 H + 9 F He + 8O + Q.00785 + 8.998405 4.00603 + 5.99495 Or, Q = (.00785 + 8.998405) (4.00603 + 5.99495 ) = 8.7 0 3 u = 8.7 0 3 93MeV = 8. MeV 3 4. The Q value of the Na (n, α) F 0 reaction is 5.4 MeV. Determine threshold energy of the neutrons for this reaction. Given mass of Na 3 is.9898 u, Solution: Q= 5.4 MeV, mass of incident particle i.e. of proton(h ), m i =.008665 u, mass of target sodium, m t =.9898 u. Now threshold energy E th = Q ( m i+m t ) = +5.4 (.008665+.9898 ) = 5.636 MeV. m t.9898