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PHY583 – Note 3a – Nuclear Reactions

NUCLEAR REACTIONS

Nuclear reactions – collisions between particles and target nuclei that change the structure of the target nuclei and producing another particle.

Today’s technology in particle accelerators can achieve particle energies higher than

1000 GeV = 1 TeV.

a+X⟶Y +b .............14.1

Eqn. 14.1 can be written as X(a, b)Y

E.g. Li❑7 ( p ,α ) He❑

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This reaction was first observed by Cockroft & Walton in 1932, using proton accelerated to 600 keV.

Conservation laws for nuclear reaction:

1. Conservation of mass number A. i.e. the total number of nucleons before & after each nuclear reaction must be the same.

2. Conservation of charge q. If the charged nuclear particles involved in nuclear reactions are protons, the number of protons before reaction = the number of protons after reactionTotal qbefore = Total qafter

3. Conservation of energy, linear momentum & angular momentum.These quantities are conserved because:a) a nuclear reaction involved only internal forces between a target nucleus & a

bombarding nucleusb) no external forces to disrupt the conservation laws

Reaction energy Q, is the total kinetic energy released (or absorbed) in a nuclear reaction.

Applying conservation of energy to a reaction given by eqn. 14.1, enable reaction energy Q to be calculated:

First, assume:

1. The target nucleus X is initially at rest2. The bombarding particle has kinetic energy Ka

3. The reaction product Y has kinetic energy KY

4. The reaction product b has kinetic energy Kb

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Second, apply conservation of energy to eqn. 14.1:

M x c2+Ka+M a c

2=MY c2+K Y+M b c2+Kb

Since,

Total kinetic energy released = Kinetic energy of Kinetic energy of in the reaction (Q) final particles initial particle

⟹ Q=( KY+K b )−K a=M x c2+M a c

2−M Y c2−M b c2

Q=(M ¿¿ x+M a−M Y−M b)c2¿............14.2

Exothermic reaction, when Q positive, i.e. nuclear mass is converted to kinetic energy of products Y & b.

Endothermic reaction, when Q negative.

For this reaction to occur minimum input kinetic energy from the bombarding particle is required.

The incident (bombarding) particle must have a minimum kinetic energy called threshold energy, Kth.

Kth > |Q|, because it must supply o (a) the excess mass-energy of the products, |Q|, and o (b) kinetic energy to the products

so that momentum is conserved.

For low energy reactions, (i.e. kinetic energies of all the interacting particles are small

compared to their rest energies) non-relativistic K=12mv2 and p = mv can be used to find

the Kth.

When momentum & energy are conserved in low-energy negative Q reaction,

K th=−Q(1+ M a

M x) ........................14.3

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Table 14.1 List of Q-values for reactions involving light nuclei

Example 14.1 p505

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REACTION CROSS SECTION

Cross section is a measure of the probability that a particular nuclear reaction will occur.

When a beam of particle incident on a target in the form of a thin foil, not every particle interacts with a target nucleus.

The probability that a reaction will occur depends on the ratio of the “effective” area of the target nucleus to the area of the foil.

Analogy: darts thrown at random to a large wall upon which many inflated balloons are hanging.

Suppose a beam of particles is incident upon a thin foil target as in Fig. 14.1a.

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Each target nucleus X has an effective area σ called cross section.

σ is at right angle to the direction of motion of the bombarding particles as in Fig. 14.1b.

Note: - The reaction cross section can be greater than, equal to, or less than the actual geometrical cross section of the target nucleus.- In general, the size of σ for a specific reaction may also

depend on the energy of the incident particle.

Notations:

x = thickness of target foilA = area of target foil R0 = rate at which incident particles strike the foil (particle/s)R = rate at which reaction events occur (reactions/s)n = number of target nuclei per unit volume (particles/m3)

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PHY583 – Note 3a – Nuclear Reactions RR0

=σnAxA

=σnx ...........14.4

nAx = total number of target nuclei in the foilσnAx = total area exposed to the incident beam

Eqn. 14.4 shows that the probability that a nuclear reaction will occur is proportional to

1. the cross section σ ,2. the density of the target nuclei n,3. the thickness of the target x.

σ for a particular reaction can be obtained by measuring R, R0, n & x & using eqn. 14.4.

The number of particles that penetrate a foil without undergoing reaction, N:

N=N 0 e−nσx ........14.5

N0 = the number of incident particlesN = the number that emerges from the slab

Eqn. 14.5 gives the number of particles transmitted through a target of thickness x (Fig. 14.2) which decreases exponentially with the target thickness.

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Fig. 14.5N0 = number of particles incident on a targetdx = thickness of targetN0 dN = the number of particles emerging from target

Nuclear cross sections, with dimensions of area, are typically in the order of the square of the nuclear radius, 1028 m2.

1 barn = 1028 m2

In reality, the concept of cross section in nuclear and atomic physics has little to do with the actual geometric area of the target nuclei.

The model we have used is simply a convenient way of describing the probability of the occurrence of any nuclear reaction.

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Cross section varies with the specific reaction considered & with the particle’s kinetic energy, over more than several times the target nucleus’s geometrical area.

E.g.

1. Interaction between antineutrino & proton via nuclear weak interaction.

ν+ p⟶ e+¿+n ¿ (σ=10−19b)

2. Neutron capture reaction

n+ Cd❑113 ⟶ Cd❑

114 +γ (σ=104b)

INTERACTION INVOLVING NEUTRONS

Because the charge of neutron is neutral:

- Neutrons are not subject to Coulomb force- Neutrons interact very weakly with electrons- Matter appears fairly transparent to neutrons

Typical cross sections for neutron-induced reactions increase as the neutron energy decreases.

Free neutrons undergo beta decay with a mean life 10 min.

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Neutrons in matter are usually absorbed by nuclei before they decay.

NUCLEAR FISSION

Occurs when a heavy nucleus such as U❑235 splits (fissions) into

two particles of comparable mass. The total mass of the product particles is less than the original

mass. Initiated by the capture of a thermal neutron by a heavy

nucleus and involves the release of 200 MeV per fission. This energy release occurs because the smaller fission-product

nuclei are more tightly bound by about 1 MeV per nucleon than the original heavy nucleus.

The fission of U❑235 by a thermal neutron:

n01 + U92

235 ⟶ U ¿92236 ⟶ X+Y +neutron ......14.8

U ¿92236 is an intermediate excited state that last 1012 s before splitting into X & Y.

X & Y are called fission fragments. There are many combinations of X & Y that satisfies the

requirement of conservation of mass-energy, charge & nucleon number.

The distribution of fission products (fragments) versus mass number for the fission of U❑

❑235 bombarded with slow neutrons is

given in Fig. 14.1.

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Fig. 14.1 Most probable fission events correspond to fission fragments with mass number A 95 & A 140

The process produces 2 or 3 neutrons, i.e. average 2.5 neutrons released per event.

A typical reaction of this type:n01 + U92

235 ⟶ Ba56141 + Kr36

92 +3 n01 ....14.9

Most of the 200 MeV released in this reaction, most goes into the kinetic energy of the heavy fragments barium & krypton.

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The sequence of events in the fission of U❑235 :

1. U❑235 nucleus captures a thermal (slow-moving) neutron.

2. This capture results in the formation of U ¿❑236 , & the access

energy of this nucleus causes it to undergo violent oscillation.

3. U ¿❑236 becomes highly distorted, & repulsive force between protons in the two halves of the dumbbell shape tends to increase the distortion.

4. The nucleus splits into 2 fragments, emitting several neutrons in the process.

The remaining fragments which still have a lot of neutrons proceed to decay to more stable nuclei through a succession of decay. Gamma rays are also emitted by nuclei in excited state.

Estimation of disintegration energy Q released in a typical fission reaction:

Binding energy per nucleon for heavy nuclei (A 240) 7.6 MeV

Binding energy per nucleon for intermediate mass nuclei 8.5 MeV

Taking mass number of mother nucleus A = 240, energy released per nucleon is:

Q= (240nucleons )(8.5 MeVnucleon

−7.6 MeVnucleon )=200MeV

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85% of this energy appears in the form of kinetic energy in the heavy fragments. This energy is very large compared to energy released in chemical processes.

E.g. 14.3 & 14.4 (p.512 & 513)

NUCLEAR REACTORS

When U❑235 fissions, an average 2.5 neutrons emitted per

event. These neutrons trigger other nuclei to fission, hence the

possibility of a chain reaction as in Fig. 14.6.

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Fig. 14.6


If chain reaction is not controlled, it can result in violent reaction & released very large amount of energy.

If the energy from 1 kg of U❑235 were released, it is

equivalent to detonating 20,000 tons of TNT. Nuclear reactor – a system designed to maintain self-

sustained chain reaction. First achieved by Fermi at the University of Chicago,

with uranium as the fuel. (Fig. 14.7).

Most reactors in operation today use natural uranium as the fuel.

Natural uranium – 0.7% U❑235 isotope & 99.3% U❑

238 isotope. But U❑

238 almost never fissions, it tends to absorb neutrons, producing neptunium & plutonium.

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Reactor fuel must be artificially enriched to contain at a least few % of U❑

235 .

To achieve a self-sustained chain reaction, on average one of the neutrons emitted in U❑

235 fission must be captured by another U❑

235 nucleus and cause it to undergo fission.

A useful parameter for describing the level of reactor operation is the reproduction constant K, defined as the average number of neutrons from each fission event that actually cause another fission event.

Maximum value for K is 2.5 in fission of uranium. In practice K < 2.5

A self-sustained chain reaction is achieved when K=1. Under this condition the reactor is said to be critical.

When K < 1, the reactor is subcritical & the reaction dies out.

When K >> 1, the reactor is supercritical & a runaway reaction occurs.

In a reactor runs by a utility company to provide power, it is necessary to maintain K slightly > than 1.

Fig. 14.8 shows the basic ingredients of a nuclear reactor core. The fuel is enriched uranium.

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Fig. 14.8 A cross section of a reactor core showing the control rods, fuel elements, and moderating material surrounded by a radiation shield.

Neutron Leakage

In any reactor, a fraction of the neutrons produced in fission leak out of the core before inducing other fission events.

The reactor will not operate if the neutron leakage is too large.

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% lost is large if the reactor is very small because leakage is a function of surface area

volume .

critical feature of the design: the correct surface areavolume

ratio in order for a sustained reaction to be achieved.

Regulating Neutron Energies

Neutrons released in fission reaction is very energetic, kinetic energy 2 MeV.

It is necessary to slow these neutrons to thermal energies so that they can be captured & produce fission of other U❑

235 nuclei, because the probability of neutron-induced fission increases with decreasing energy, as shown in Fig. 14.9.

The energetic neutrons are slowed down by a moderator substance surrounding the fuel as in Fig. 14.8.

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Fig. 14.9 The cross-section for neutron-induced fission of U❑235 .

The average cross section for room-temperature neutrons 500 b.

Neutron Capture

In the process of being slowed down, neutrons may be captured by nuclei that do not fission.

The most common is neutron capture by U❑238 which

constitutes > 90% of the uranium in the fuel elements. The probability of neutron capture by U❑

238 is very high when the neutrons have high kinetic energies & very low when the neutrons have low kinetic energies.

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Slowing down the neutrons by the moderators serves two purposes:1. Making them available for reaction with U❑

235 .2. Decreasing their chances of being captured by U❑

238 .

Control of Power Level

Some control is needed to maintain K 1. If K is more than 1, the heat produce in the runaway

reaction would melt the reactor. Control rods (Fig. 14.8) made of material such as

cadmium are inserted into the core reactor to control the power level.

Cadmium absorbs neutron very efficiently. K value can be varied & power level within the design

range can be achieved by adjusting the number & position of the control rods in the reactor core.

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Energy transformation

Kinetic energy of fission fragments ⟶ heats up liquid water under high pressure in primary closed loop ⟶ heats up water in secondary loop which turns into steam ⟶ drives turbine-generator system i.e. heat from steam turns into kinetic energy of turbine ⟶ turns into electrical energy

Safety & Waste Disposal (p516-517)

1979 near-disaster Three Mile Island in Pennsylvania.

1986 Chernobyl disaster.

Nuclear Fusion

Fusion occurs when two light nuclei combine to form a heavier nucleus.

Mass of final nucleus < combined rest mass of original nuclei

the mass loss = release of energy

E.g. Fusion reaction in the sun

H+ H⟶ H12

11

11 + e+¿+ν

10 ¿ ...........14.10

H+ H⟶ H13

12

11 +γ

The 2nd reaction is followed by one of the following reactions:

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PHY583 – Note 3a – Nuclear Reactions H+ He⟶ He2

423

11 + e+¿+ ν

10 ¿

He+ He⟶ He24

23

23 + H+ H1

111

Basic reactions of proton-proton cycle, believed to be one of the basic cycles by which energy is generated in the sun & other stars with abundance of hydrogen.

Most of the energy production takes place in the interior of the Sun, T 1.5 × 107 K

Fusion requires high temperature, the name thermonuclear fusion reactions.

First hydrogen (fusion) bomb was exploded in 1952. All the reactions in proton-proton cycles are exothermic. Overall view ⟹ proton-proton cycle is that 4 protons

combine to form alpha particle & two positrons, releasing 25 MeV of energy.

Fusion Reactions

A lot of effort is directed toward developing a sustained & controllable fusion power reactor.

Advantages of fusion reactor:

1. The fuel is water.2. Comparatively few radioactive by-products are formed.

Unfortunately a thermonuclear (fusion) reactor that can deliver a net power output is currently not yet a reality.

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Proton-proton interaction is not suitable to be used in fusion reactor, because it requires very high pressure & densities.

The process works in the sun because of the extremely high density of protons in the sun’s interior.

The most promising fusion reaction for terrestrial power reactor involve deuterium (H ¿ ¿1

2 & tritium ( H13 ):

H+ H⟶ He23

12

12 + n0

1 , Q = 3.27 MeV

H+ H⟶ He13

12

12 + H1

1 , Q = 4.03 MeV ..............14.11

H+ H⟶ He24

13

12 + n0

1 , Q = 17.59 MeV

One of the major problems in obtaining energy from any fusion reaction is the need to overcome the Coulomb repulsion force between 2 charged nuclei before they could be fused.

The potential energy U(r) as a function of separation distance between two deuterons (each with charge +e) is given in Fig. 14.11.

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Fundamental problem: to give the 2 deuterons enough kinetic energy to overcome the repulsive potential barrier.

Can be accomplished by heating the fuel to extremely high temperature (108 K).

At these high temperatures, the atoms are ionized and the system consists of a collection of electrons & nuclei – referred to as plasma.

Three critical parameters that determine whether or not a thermonuclear reactor will be successful:

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1. High temperature, T2. Ion density, n3. Confinement time, τ

High temperature requirement

Critical ignition temperature – The temperature at which the power generation rate exceeds the loss rate (due to mechanisms such as radiation losses).

Critical ignition temperature for

a. Deuterium-deuterium (D-D) reaction 4 x 108 K (E kBT 35 keV) b. Deuterium-tritium (D-T) reaction 4.5 x 107 K (E 4 keV)

Fig. 14.12 Power generated (or loss) versus temperature for D-D & D-T fusion reactions. When the generation rate Pgen > loss rate Plost, ignition takes place.

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The intersection between the Plost & Pgen curves gives the critical ignition temperature.

Confinement time - the period for which the interacting ions are maintained at a temperature equal to or greater than the ignition temperature.

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Lawson’s criterion states that a net energy output is possible under the following conditions:

nτ ≥1014 scm3 .......(D-T)

nτ ≥1016 scm3 .......(D-D)

Fig. 14.3 The Lawson number n at which net energy output is possible versus temperature for the D-T & D-D fusion reactions. The region above the curves represents favourable conditions for fusion.

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Three basic requirements for successful thermonuclear power reactor:

a. The plasma temperature must be very high – 4.5 x 107 K for D-T reaction

4 x 108 K for D-D reaction b. The ion density must be high. A high density of

interacting nuclei is necessary to increase the collision rate between particles.

c. The confinement time of plasma must be long. To meet Lawson’s criterion n must be large. For a given value of n, the probability of fusion between two particles increases as increases.

Problem: How can a plasma be confined at 108 K for 1 s?

Magnetic field confinement

A combination of 2 magnetic fields (in a devise called tokamak) is used to confine & stabilize the plasma:

1. A strong toroidal field produced by the current in the windings,

2. A weaker “poloidal” field, produced by the toroidal current.

The toroidal field heats the plasma in addition to confining it.

The resultant helical field lines spiral around the plasma & keep it from touching the walls of the vacuum chamber.

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Fig. 14.14 (a) A schematic diagram of a tokamak used in magnetic confinement. The total magnetic field B is the superposition of the toroidal field Bt and the poloidal field Bp. The plasma is trapped within the spiralling field lines.

International Thermonuclear Experimental Reactor (ITER) – international collaboration effort involving Canada, Europe, Japan & Russia (China & US participate in 2003) – to build a fusion reactor ITER.

ITER address the remaining technological & scientific issues concerning the feasibility of fusion power.

The design is completed & site & construction negotiations are underway.

If the planned devise works as expected, the Lawson number of ITER will be six times greater than the current record holder, JT-60U tokamak of Japan.

ITER will generate 1.5 GW of power & the energy content of the alpha particles inside the reactor will be so intense that they will sustain the nuclear reaction.

Inertial Confinement

Makes use of a D-T target that has a very high particle density of 5 × 1025 particles/cm3, or a mass density of 200 g/cm3.

The confinement time is very short ( 1011 to 109 s).

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Because of their own inertia, the particle do not have the chance to move appreciably from their initial positions.

Hence, Lawson’s criterion can be satisfied by combining a high particle density with a short confinement time.

Laser fusion is the most common type of inertial confinement.

A small D-T pellet 1mm in diameter is truck simultaneously by several focused, symmetrically incident, high-intensity laser beams resulting in a large pulse of input energy that causes the surface of the fuel pallet to evaporate (fig. 14-16).

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The escaping particles produce a reaction force on the core of the pellet, resulting in a strong, inwardly moving, compressive shock wave.

This shock wave increases the pressure & density of the core & produces a corresponding increase in temperature.

When the temperature of the core reaches the ignition temperature, fusion reactions cause the pellet to explode.

This process can be viewed as a miniature hydrogen bomb explosion.

Fusion Reactor Design

In D-T fusion reaction,

H+ H⟶ He24

13

12 + n0

1 , Q = 17.59 MeV

The alpha particles carry 20% of the energy, the neutrons carry 80% (14 MeV).

The charged alpha particles are primarily absorbed in the plasma & produced the desired additional plasma heating.

The neutrons pass through the plasma & must be absorbed in a surrounding blanket material to extract their large kinetic energy and generate electric power.

Molten lithium metal is frequently proposed as the neutrons-absorbing material.

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The lithium is to circulate in a closed heat-exchanging loop to produce steam & drive turbine, as in conventional power plant.

Fig. 14.18 is a diagram of such a fusion reactor system.

A blanket of 1-m thick lithium would capture almost 100% of the neutrons from the fusion of a small D-T pellet.

This lithium blanket not only absorb neutrons, but also limiting the dangerous neutron flux to nearby workers.

The capture of neutrons by lithium is described by the following reaction:

n+ Li⟶ H13

36

01 + He2

4

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The kinetic energy of the charged tritium & the alpha particle products are converted to heat in the lithium.

An extra advantage of using lithium as the energy transfer medium is the production of tritium, H1

3 , which may be separated from the lithium & returned as fuel to the reactor.

The process is indicated in the generic fusion reactor as shown in Fig. 14.18.

Advantages & Problems of Fusion

If fusion power can be harnessed, it will give several advantages over fission-generated power:

1. The low cost & abundance of fuel (deuterium),2. The impossibility of runaway accidents3. A lesser radiation hazard than with fission

Some anticipated problems:

1. The unestablished feasibility of fusion reactors,2. The very high proposed plant costs,3. The scarcity of lithium,4. The limited supply of helium needed to cool the

superconducting magnets used to produce strong confining fields (this problem may be reduced by the development of high-temperature superconductors)

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5. Structural damage & induced radioactivity caused neutron bombardment,

6. The anticipated high degree of thermal pollution.

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