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My Chapter 29

Lecture

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Chapter 29: Nuclear Physics

•The Nucleus

•Binding Energy

•Radioactivity

•Half-life

•Biological Effects of Radiation

•Induced Nuclear Reactions

•Fission and Fusion

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§29.1 Nuclear Structure

The atomic nucleus is composed of neutrons and protons. These particles are called nucleons.

The atom’s atomic number (Z) gives the number of protons in its nucleus. It is the atomic number that determines an atom’s identity.

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The nucleon number or mass number is A = Z+N, where N is the number of neutrons.

Masses of atoms are sometimes give in terms of atomic mass units. 1u = 1.66053910-27 kg.

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The mass quoted for an atom in the periodic table is a weighted average over all of the natural isotopes of that element. The weight factors are determined by using the relative abundance on Earth of each isotope.

Atoms of the same element with differing numbers of neutrons are known as isotopes.

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For an atomic nucleus .AV

Am

∝∝

This implies the density of an atomic nucleus is independent of A.

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3

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Ar

ArV

∝

∝= π

As an equality 31

0Arr =

where r0 = 1.210-15 m = 1.2 fm

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Example (text problem 29.2): Calculate the mass density of nuclear matter.

m 102.1 150

31 −×== Arr

Consider a nucleus with one nucleon (A = 1).

The density is

( ).kg/m 103.2

m 102.134

kg 1066.1

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317

315

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3

×=×

×=

=

−

−

π

πρ

r

m

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Example (text problem 29.9): Find the radius and volume

of the nucleus.

( )( ) m. 1070.5107m 102.1 151/3151/30

−− ×=×== Arr

.m 107.73

4 3433 −×== rV π

The radius is

The volume is

Tc10743

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§29.2 Binding Energy

A nucleus is held together by the strong nuclear force. This force only acts over distances of a few fermis.

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The binding energy (EB) of a nucleus is the energy that must be supplied to separate it into individual protons and neutrons.

EB = Total energy of Z protons and N neutrons – total energy of nucleus.

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Total energy of Z protons and N neutrons = (mass of Z protons and N neutrons)c2.

Total energy of nucleus = (mass of nucleus)c2.

These can be used to define the mass defect m = (mass of Z protons and N neutrons) (mass of nucleus) so that

.2mcEB =

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Nucleons also obey the Pauli Exclusion Principle such that only two protons (neutrons) can occupy each proton (neutron) energy level.

Like an atom, a nucleus can be put into an excited state if it absorbs a photon of the correct energy. The nucleus can then emit a photon to go to a lower energy state.

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Example (text problem 29.13): (a) Find the binding energy of the 16O nucleus.

( )

( )u1370046.0

u9949146.15u0086649.1 1.00782508

atom Oneutral of mass

neutrons 8 of mass atoms H 8 of mass16

=

−+=

−

+=Δm

MeV 8.1272 == mcEB

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(b) What is the average binding energy per nucleon?

Example continued:

n.MeV/nucleo 986.7

nucleons ofnumber nucleonper energy Binding

=

= BE

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§29.3 Radioactivity

Some nuclei are unstable and decay. These nuclei are radioactive. A nucleus can emit an alpha ray, beta ray, or a gamma ray during its decay.

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During nuclear reactions:

1. Charge is conserved.

2. The total number of nucleons is constant.

3. Energy is conserved.

Define: disintegration energy = binding energy of radioactive nucleus total binding energy of products. This is the rest mass energy that can be converted into other forms of energy.

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Alpha rays have been identified as helium nuclei.

The reaction for alpha decay is

α4242 +→ −

− DP AZ

AZ

Parent nucleus

Daughter nucleus

Alpha particle

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Example (text problem 29.29): Show that the spontaneous alpha decay of 19O is not possible.

.CO 42

156

198 α+→The reaction is

The mass of the products (including electrons) is 19.01320250u.

The mass of 19O is 19.0035787u.

The mass of the products is larger than the reactant, so this reaction cannot occur spontaneously.

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Beta rays have been identified as either electrons (-) or positrons (+).

The reaction for beta-minus decay is

.e 00

011 ν++→ −+ DP A

ZAZ

The reaction for beta-plus decay is

.e 00

011 ν++→ +−DP A

ZAZ

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The neutrino and antineutrino have no charge and are nearly massless. They do not readily interact with matter.

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During beta-minus decays, a neutron is converted into a proton.

ν0001

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10 epn ++→ −

During beta-plus decays, a proton is converted into a neutron.

ν0001

10

11 enp ++→ +

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During inverse beta decay (electron capture) a proton in a nucleus captures an electron. The reaction is

.npe 00

10

11

01 ν+→+−

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Gamma rays were determined to be high energy photons.

A gamma ray will be emitted when a nucleus is an excited state when making a transition to a lower energy level. For example,

.TlTl 20881

*20881 γ+→

When a nucleus has experienced alpha or beta decay, it is not always left in the ground state.

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§29.4 Radioactive Decay Rates and Half-Lives

The half-life of a sample of unstable nuclei is the time it takes for one-half of the sample to decay. The decay process is quantum mechanical and is based on probability.

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Each radioactive nucleus has a probability per second that it will decay, called the decay constant.

unit time

decay ofy probabilitconstant decay ==λ

The number of nuclei that decay in a short time interval is

.tNN −= λ

There are statistical fluctuations in the number of decays that occur. These fluctuations are of order .N

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The decay rate or activity is the number of radioactive decays that occur in a sample per unit time.

Nt

NR λ=

−== unit timedecays of number

The unit of activity is the bequerel. 1 Bq = 1 decay/sec. Another common unit is the curie. 1 Ci = 3.71010 Bq.

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The number of nuclei remaining in a sample having N0 nuclei at t = 0 is

( ) ./0

τteNtN −=

is the mean lifetime of a nucleus. λ

τ 1=

Note: the above expression for N(t) is a way to determine the number of remaining nuclei only. It does not tell us which nuclei have decayed.

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The activity at time t is ( ) τ/0

teRtR −=

where R0 is the activity at t = 0.

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It is common to write the expressions for N(t) and R(t) in terms of half-life (T1/2).

2ln2/1 τ=T ( ) 2/12/1

2

12 00

Tt

Tt

NNtN ⎟⎠

⎞⎜⎝

⎛=⎟

⎠⎞

⎜⎝⎛=

−

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Example (text problem 29.35): Some bones discovered in a crypt in Guatemala are carbon dated. The 14C activity is measured to be 0.242 Bq per gram of carbon. Approximately how old are the bones?

( ) τ/0

teRtR −=

Solve for t:

( )

( )years270

m0.25bq/gra

bq/gram 242.0ln

2ln

years 5730ln

2ln

ln

0

1/2

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

R

tRT

R

tRt τ

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§29.5 Biological Effects of Radiation

The absorbed dose of ionizing radiation is the amount of radiation energy absorbed per unit mass of tissue. Ionizing radiation is radiation with enough energy to ionize an atom or molecule.

The SI unit of absorbed dose is the Gray. 1 Gy = 1 J/kg. Another common unit is the rad (radiation absorbed dose). 1 rad = 0.01 Gy.

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Different radiation causes different amounts of biological damage. The biologically equivalent dose measures the amount of damage caused by radiation exposure.

Equivalent dose (in sieverts) = absorbed dose (in grays)* QF.

Equivalent dose (in rem) = absorbed dose (in rads)* QF.

QF is a quality factor that is a relative measure of biological damage (200 keV x-rays have QF = 1).

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The sievert is the SI unit of biologically equivalent dose. 1 Sy = 100 rem.

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Alpha, beta, and gamma radiation penetrates to different depths in biological materials.

•Alpha rays are stopped by a few cm of air or about 0.02 mm of aluminum.

•Beta-minus can penetrate a few cm into biological tissue.

•Gamma ray absorption is based on probability so they can penetrate to varying depths.

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Summary

•The Nucleus (atomic & mass numbers)

•Binding Energy

•Radioactive Nuclei

•Alpha, Beta, and Gamma Radiation

•Half-life and Activity

•Absorbed Dose