Nuclear ReactionsNuclear Reactions

AP Physics B

Montwood High School

R. Casao

Nuclear ReactionsNuclear ReactionsNuclear reactions are rearrangements of

nuclear components that result from bombardment by a particle rather than a spontaneous natural process.

Nuclear reactions are subject to several conservation laws:– Charge– Momentum– Energy– Number of nucleons

When two nuclei interact, charge conservation requires that the sum of the initial atomic numbers must equal the sum of the final atomic numbers.

Because of conservation of nucleon number, the sum of the initial mass numbers must also equal the sum of the final mass numbers.

In general, these are not elastic collisions, and therefore, the total initial mass does NOT equal the total final mass.

Reaction EnergyReaction Energy The difference between the masses before and

after the reaction corresponds to the reaction energy, according to the mass-energy relationship E = m·c2.

If initial particles A and B interact to produce final particles C and D, the reaction energy Q is defined as:

Q = (MA + MB – MC – MD) ·c2

To balance the electrons, use the neutral atomic masses.

Kinetic energy can be used if C and D are in the ground state: Q = (KC + KD) – (KA – KB)

Reaction EnergyReaction EnergyWhen Q is positive, the total mass decreases

and the total kinetic energy increases.– Exothermic reaction.

When Q is negative, the total mass increases and the kinetic energy decreases.– Endothermic reaction.– Reaction cannot occur at all unless the initial

kinetic energy is at least equal to Q. – Threshold energy: the minimum kinetic energy

needed to make the reaction take place.

ExampleExampleWhen Li-7 is bombarded by a proton, two

particles (He-4) are produced. Determine the reaction energy.

This is an exothermic reaction because the final total kinetic energy is 17.35 MeV greater than the initial total kinetic energy.

MeV35.17Q

uMeV

5.931)u002603.42(u016003.7u007825.1Q

HeHeLiH 42

42

73

11

ExampleExampleDetermine the reaction energy for the

nuclear reaction:

This is an endothermic reaction.The minimum initial kinetic energy

(threshold energy) for this reaction to occur is 1.191 MeV.

MeV191.1Q

uMeV5.931

u007825.1u999131.16u003074.14u002603.4Q

HONHe 11

178

147

42

The relationship between the threshold kinetic energy KEth and the absolute magnitude of the reaction energy Q can be determined using conservation of energy and momentum:

– where m = mass of incoming particle and M = mass of stationary target particle.

Ordinarily, the stationary N-14 nuclei would be bombarded with particles from an accelerator.

QMm

1KE th

The particles kinetic energy must be greater than 1.191 MeV.

When a particle with mass m and kinetic energy K collides with a stationary particle with mass M, the total kinetic energy Ktotal (the energy available to cause reactions) is:

mMKM

K total

MeV532.1K

u002603.4u003074.14MeV191.1u003074.14

K

total

total

The kinetic energy of the particles must be at least 1.532 MeV.

For a charged particle such as a proton or an particle to penetrate the nucleus of another atom and cause a reaction, it must have enough initial kinetic energy to overcome the potential-energy barrier caused by the repulsive electrostatic forces.

r

qqkU 21

For the reaction of the proton and Li-7, if we treat the proton and the Li-7 nucleus as spherically symmetric charges with radii given by R = Ro·A1/3, their centers will be 3.5 x 10-15 m apart when they touch.

The repulsive potential energy of the proton and the Li-7 nucleus at this separation r is:

Even though the reaction is exothermic, the proton must have a minimum kinetic energy of 1.2 MeV for the reaction to occur.

MeV2.1J10x2U

m10x5.3

C10x602.1C10x602.13C

mN10x9875.8

U

13

15

19192

29

Neutron AbsorptionNeutron AbsorptionAbsorption of neutrons by nuclei forms is

an important class of nuclear reactions.Heavy nuclei bombarded by neutrons in a

nuclear reaction undergo a series of neutron absorptions alternating with beta decays, in which the mass number A increases by as much as 25.

Some of the trans-uranic elements, elements having an atomic number larger than 92, are produced in this way.– These elements have not been found in nature.

Disintegration Energy (Disintegration Energy ( Decay) Decay)

Alpha decay: parent daughter + – Disintegration energy Q = (mp – md – m)·c2

– Disintegration energy appears in the form of kinetic energy in the daughter nucleus and the alpha particle.

– Alpha decay requires Q > 0; if Q < 0, the parent nucleus is stable with respect to alpha decay and alpha decay does not occur.

Disintegration Energy (Disintegration Energy ( Decay) Decay)

Of the energy available to the daughter and alpha particle as kinetic energy:– larger mass has smaller kinetic energy.– smaller mass has greater kinetic energy.

parent

alphadaughter

parent

daughteralpha

m

mQKE

m

mQKE

Disintegration Energy (-Disintegration Energy (- Decay) Decay)Negative beta decay:

parent daughter + -

– Disintegration energy Q = (mp – md)·c2

– Example:

– C-14 emits a negatively charged beta particle (an ordinary electron) and the resulting nucleus is N-14.

– Might expect to determine Q using:

Q = (mp – md – m)·c2

veNC 01

147

146

This does not correctly represent the mass of all the electrons in the parent and daughter atoms.

– mp and md are atomic masses and include the electron masses.

– The mass for C-14 would be accurate because it would include the mass of 6 electrons.

– The mass of N-14 would be incorrect because it would include the mass of 7 electrons.

– In the N-14 formed by the beta decay there are only the 6 electrons from the parent C-14 atom.

– To compensate for the fact that the daughter atom contains the mass of 1 more electron than we need, we do not write the m term in the energy equation.

Correct equation for disintegration energy for negative beta decay: Q = (mp – md)·c2

The nitrogen created in the beta decay of carbon has only 6 orbiting electrons because the parent carbon had only six electrons.

The daughter is nitrogen by definition because it contains 7 protons.

The N-14 is positively ionized and will attract an electron to complete its outer electron shell.

Disintegration Energy (+Disintegration Energy (+ Decay) Decay)

Positive beta decay:parent daughter + +

– Example:

– N-13 emits a positively charged beta particle and the resulting nucleus is C-13.

– The mass for the parent N-13 atom will be correct.– The mass for the daughter C-13 atom includes the

mass for only the 6 electrons for a normal C-13 atom.

veCN 01

136

137

In our daughter C-13 atom, there are 7 electrons from the parent N-13 atom.

The daughter C-13 is positively ionized. We have to include an me = m term in the disintegration equation to account for this positive ion.

The mass of the positron is also m.Disintegration energy:

Q = (mp – md – 2·m)·c2

The decay of a free proton into a neutron, a +, and a neutrino is not possible because the combined mass of the neutron and beta particle is greater than the mass of the proton.

However, the same reaction is possible if the proton is bound within the nucleus because the required energy is provided by the motion of the nuclei in the nucleus.

e01

10 venp