The first induced nuclear reaction was accomplished by none other than Australian physicist Ernest Rutherford, the same guy who bombarded atoms with alpha particles and discovered the nuclear structure of the atom.From his experience with alpha emitters, Rutherford thought that alpha particles might just be energetic enough to breach the nuclear boundary.

Topic 7.3 ExtendedA – Nuclear Reactions

In fact, he did just that in the following nuclear reaction:

14N + 4He 17O + 1H7 2 8 1

nitrogen alpha particle oxygen proton There is an intermediate step in this reaction:

14N + 4He (18F*) 17O + 1H7 2 9 8 1

nitrogen alpha particle fluorine oxygen proton

FYI: In the standard notation, the asterisk (*) represents an excited nucleus. This is comparable to an excited electron in that a nucleon is bumped up to a higher shell in the nucleus (as per Schrödinger).

FYI: The intermediate excited fluorine nucleus rids itself of excess energy by ejecting a proton.

Alchemy, the search for the means of changing metal into gold, has finally come to be one of man's accomplishments.The following reaction turns mercury into gold!

Topic 7.3 ExtendedA – Nuclear Reactions

200Hg + 1H 197Au + 4He80

1

79 2

mercury proton gold -particle Notice that the mercury is bombarded by a proton.

Since protons are not a natural byproduct of radioactive decay, a particle accelerator is used to speed up protons (hydrogen nuclei) and collide them with the mercury nuclei with sufficient energy to penetrate the nucleus.

A Word of Advice: Don't go home asking mom for a particle accelerator so you can make your own bootleg gold. Accelerators cost billions of dollars. Furthermore, it will cost you millions in electric bills just to run the machine to make an ounce of the stuff!

The next slide shows a particle accelerator in the US:

Fermilab National Accelerator Laboratory

A shorthand notation for nuclear reactions of this sort- ELEMENT + particle ELEMENT + particleis

Topic 7.3 ExtendedA – Nuclear Reactions

A + a B + bA + a b + B

A(a , b)B

Element-Particle-Particle-Element Notation

Thus the preceding two reactions look like this:14N(,p)17O

200Hg(p,)197Au

FYI: There are no stable nuclei beyond Z = 92 (uranium). However, elements all the way up to Z = 109 have been made in particle colliders such as Fermilab. These new nuclei are unstable, though.

FYI: Some nuclear physicists believe that if the next magic number (beyond 126) is reached, a new class of superheavy stable element may be created with over 300 nucleons.

CONSERVATION OF MASS-ENERGY AND THE Q VALUE

Topic 7.3 ExtendedA – Nuclear Reactions

In light of Einstein's special theory of relativity (E = m0c2) we can no longer just speak of "conservation of mass" or "conservation of energy."Instead we speak of conservation of mass-energy.The total energy of each of the constituent particles of a nuclear reaction is given by

E = K + m0c2 Relativistic Energy of a Moving Particle

We can then express the conservation of mass-energy as...

Ei = Ef Conservation of Mass-Energy

CONSERVATION OF MASS-ENERGY AND THE Q VALUE

Topic 7.3 ExtendedA – Nuclear Reactions

In the reaction A + a B + b we can invoke the conservation of mass-energy like this:

KA + mAc2 + Ka + mac2 = KB + mBc2 + Kb + mbc2 conservation of mass-energy

(KB + Kb) - (KA + Ka) = (mA + ma)c2 - (mB + mb)c2 why?

Now we define the Q-value of the reaction:

Q = K = (KB + Kb) - (KA + Ka)Or...

Q = (mA + ma)c2 - (mB + mb)c2

This last can be simplified somewhat:

Q = (mA + ma - mB - mb)c2 = mc2 Q-Value of a Nuclear Reaction

FYI: If Q < 0 the reaction ABSORBS ENERGY. It is ENDOERGIC.

FYI: If Q > 0 the reaction RELEASES ENERGY. It is EXOERGIC.

FYI: Endoergic = endothermic. Exoergic = exothermic.

FYI: Radioactive decay reactions are ALWAYS exoergic.

CONSERVATION OF MASS-ENERGY AND THE Q VALUE

Topic 7.3 ExtendedA – Nuclear Reactions

What is the Q-value for the reaction

14N + 4He 17O + 1H7 2 8 1

nitrogen alpha particle oxygen proton 14.003074 u 4.002603 u 16.999133 u 1.007825 u

Q = (mA + ma - mB - mb)c2

Q = (14.00307 + 4.002603 - 16.999133 - 1.007825)uc2

A a B b

Q = -0.001285uc2 Endoergic or exoergic?Since u = 1.660610-27 kg, and c = 3108 m/s,

Q = -1.9210-13 J

FYI: Lifting a 2-kg textbook up from the floor to the desktop takes about U = mgy = 2(10)(1.5) = 30 J (about 1.6 1014 times Q!).

Don't forget, we can convert directly to MeV:

Q = -0.001285uc2 931.5 MeV1 uc2

Q = -1.197 MeV

THRESHOLD ENERGY

Topic 7.3 ExtendedA – Nuclear Reactions

The minimum kinetic energy Kmin that an incident particle needs in order to initiate an endoergic reaction in a stationary target is called the threshold energy and is given by

Kmin = 1 + |Q|ma

MA

Threshold Energy for an Endoergic Reaction

Note that Kmin > |Q|.

FYI: The reason Kmin = |Q| is not sufficient is because of conservation of momentum. Since the initial momentum of the particle is NOT zero and the initial momentum of the target IS zero, the collision must result in SOME velocity for the target. Thus some of the kinetic is energy "needed" to conserve momentum.

What is the threshold energy for the reaction 14N(,p)17O?The notation A(a , b)B tells us what A, a, B, and b are so that Q = (mA + ma - mB - mb)c2 becomesQ = (14.003074 + 4.002603 - 16.999133 - 1.007825)uc2

Q = (-0.001281uc2)(931.5 MeV/uc2) = -1.193 MeV

Kmin = 1 + | -1.193 MeV |4.00260314.003074

Kmin = 1.534 MeV

FYI: Kmin is significantly bigger than |Q|.

REACTION CROSS SECTIONS

Topic 7.3 ExtendedA – Nuclear Reactions

The minimum kinetic energy Kmin can be calculated to produce a particular reaction.If the kinetic energy of the incident particle exceeds Kmin for more than one possible reaction the result can be any of the possible reactions - determined by the probabilities described by Schrödinger's equation.Supercomputers can sometimes be used to calculate the probabilities of all the possible reactions, or they can be determined experimentally.The reaction cross section is a measure of the probability that a particular reaction will occur at a particular incident kinetic energy.

Neutron K (eV)

Rea

ctio

n C

ross

S

ectio

n

10-1 100 101 102

FYI: The peaks where the probabilities of reactions are greatest are called resonances.

resonances

FYI: The resonances occur when the nucleus attains different energy levels (we talked about this - it is analogous to electron energy levels). Then the excited nucleus undergoes one of its possible decays.

FYI: We will return to cross sections when we learn about the search for elementary particles...