Transcript Chapter 44

Chapter 44
Nuclear Structure
Milestones in the Development of Nuclear Physics
1896: the birth of nuclear physics
 Becquerel discovered radioactivity in uranium compounds
Rutherford showed the radiation had three types:
 alpha (He nuclei)
 beta (electrons)
 gamma (high-energy photons)
1911 Rutherford, Geiger and Marsden performed scattering experiments
 Established that the nucleus could be treated as a point mass and a point
charge
 Most of the atomic mass was contained in the nucleus
 Nuclear force was a new type of force
Introduction
Some Properties of Nuclei
All nuclei are composed of protons and neutrons.
 Exception is ordinary hydrogen with a single proton
The atomic number Z equals the number of protons in the nucleus.
 Sometimes called the charge number
The neutron number N is the number of neutrons in the nucleus.
The mass number A is the number of nucleons in the nucleus.
 A=Z+N
 Nucleon is a generic term used to refer to either a proton or a neutron
 The mass number is not the same as the mass.
Section 44.1
Symbolism
A nuclide is a specific combination of atomic number and mass number that
represents a nucleus.
A
Z
X
 X is the chemical symbol of the element.
Example:
27
13
Al




Mass number is 27
Atomic number is 13
Contains 13 protons
Contains 14 (27 – 13) neutrons
 The Z may be omitted since the element can be used to determine Z.
Section 44.1
More Properties
The nuclei of all atoms of a particular element must contain the same number of
protons.
They may contain varying numbers of neutrons.
 Isotopes of an element have the same Z but differing N and A values.
 The natural abundance of isotopes can vary.
 Isotope example:
11
6
C, 126 C, 136 C, 146 C
Section 44.1
Charge
The proton has a single positive charge, e.
The electron has a single negative charge, - e.
 e = 1.6 x 10-19 C
The neutron has no charge.
 Made it difficult to detect in early experiments
 Easy to detect with modern devices
Section 44.1
Mass
It is convenient to use atomic mass units, u, to express masses.
 1 u = 1.660 539 x 10-27 kg
 Based on definition that the mass of one atom of 12C is exactly 12 u
Mass can also be expressed in MeV/c2.
 From ER = mc2
 1 u = 931.494 MeV/c2
 Includes conversion 1 eV = 1.602 176 x 10-19 J
Section 44.1
Some Masses in Various Units
Section 44.1
The Size of the Nucleus
First investigated by Rutherford in scattering experiments
He found an expression for how close an alpha particle moving toward the
nucleus can come before being turned around by the Coulomb force.
 He used the isolated system (energy) analysis model to find d.
From conservation of energy, the kinetic energy of the particle must be
completely converted to potential energy.
Section 44.1
Size of the Nucleus, cont.
d is called the distance of closest approach.
 d gives an upper limit for the size of the nucleus.
Rutherford determined that
Ze2
d  4ke
mv 2
 For gold, he found d = 3.2 x 10-14 m
Rutherford concluded that the positive charge of the atom was concentrated in a
sphere whose radius was no larger than about 10-14 m.
 He called this sphere the nucleus.
These small lengths are often expressed in femtometers (fm) where 1 fm = 10-15
m.
 Also called a fermi
Section 44.1
Size of Nucleus, Final
Since the time of Rutherford, many other experiments have concluded the
following:
 Most nuclei are approximately spherical.
 Average radius is
r  a A1 3
 a = 1.2 x 10-15 m
 A is the mass number
Section 44.1
Density of Nuclei
The volume of the nucleus (assumed to
be spherical) is directly proportional to
the total number of nucleons.
This suggests that all nuclei have
nearly the same density.
Nucleons combine to form a nucleus as
though they were tightly packed
spheres.
Section 44.1
Nuclear Stability
There are very large repulsive electrostatic forces between protons.
 These forces should cause the nucleus to fly apart.
The nuclei are stable because of the presence of another, short-range force,
called the nuclear force.
 This is an attractive force that acts between all nuclear particles.
 The nuclear attractive force is stronger than the Coulomb repulsive force at
the short ranges within the nucleus.
Section 44.1
Features of the Nuclear Force
Attractive force that acts between all nuclear particles
Very short range
 It falls to zero when the separation between particles exceeds about several
fermis.
Independent of charge
 The nuclear force on p-p, p-n, n-n are all the same
 Does not affect electrons
Section 44.1
Nuclear Stability, cont.
Light nuclei are most stable if N = Z.
Heavy nuclei are most stable when N >
Z.
 Above about Z = 20
 As the number of protons
increases, the Coulomb force
increases and so more neutrons
are needed to keep the nucleus
stable.
No nuclei are stable when Z > 83.
Section 44.1
Binding Energy
The total energy of the bound system (the nucleus) is less than the combined
energy of the separated nucleons.
 This difference in energy is called the binding energy of the nucleus.
 It can be thought of as the amount of energy you need to add to the nucleus to
break it apart into its components.
The binding energy can be calculated from conservation of energy and the
Einstein mass-energy equivalence principle:
Eb = [ZM(H) + Nmn – M (AZX)] x 931.494 MeV/u
 M(H) is the atomic mass of the neutral hydrogen atom
 M (AZX) represents the atomic mass of an atom of the isotope (AZX)
 Mn is the mass of the neutron
 The masses are expressed in atomic mass units and Eb will be in MeV.
Section 44.2
Binding Energy per Nucleon
Section 44.2
Notes from the Binding Energy Graph
The curve peaks in the vicinity of A = 60.
 Nuclei with mass numbers greater than or less than 60 are not as strongly
bound as those near the middle of the periodic table.
There is a decrease in binding energy per nucleon for A > 60.
 Energy is released when a heavy nucleus splits or fissions.
 Energy is released since the nucleons in each product nucleus are more tightly
bound to one another than are the nucleons of the original nucleus.
The binding energy is about 8 MeV per nucleon for nuclei with A > 50.
 This suggests that the nuclear force saturates.
 A particular nucleon can interact with only a limited number of other
nucleons.

62
28
Ni has the largest binding energy per nucleon.
Section 44.2
Final Notes About Binding Energy
When separate nucleons are combined to form a nucleus, the energy of the
system is reduced.
The change in energy is negative.
The absolute value of the change in energy is the binding energy.
Be careful of the signs.
 For example, an increase in binding energy corresponds to a decrease in the
energy of the system.
Section 44.2
Nuclear Models
Two models of the nucleus will be discussed.
Liquid-drop model
 Provides good agreement with observed nuclear binding energies
Shell model
 Predicts the existence of stable nuclei
Section 44.3
Liquid-Drop Model
Nucleons are treated like molecules in a drop of liquid.
The nucleons interact strongly with one another.
They undergo frequent collisions as they jiggle around in the nucleus.
The jiggling motion is analogous to the thermally agitated motion of molecules in
a drop of liquid.
Section 44.3
Liquid-Drop Model – Effects Influencing Binding Energy, 1
The volume effect
 The nuclear force on a given nucleon is due only to a few nearest neighbors
and not to all the other nucleons in the nucleus.
 The total binding energy is proportional to A and therefore proportional to the
nuclear volume.
 This contribution to the binding energy of the entire nucleus is C1A.
 C1 is an adjustable constant.
 It can be determined by fitting the prediction of the model to experimental results.
Section 44.3
Liquid-Drop Model – Binding Energy Effect 2
The surface effect
 Nucleons on the surface have fewer neighbors than those in the interior.
 Surface nucleons reduce the binding energy by an amount proportional to
their number.
 The number of nucleons is proportional to the surface area.
 The surface term can be expressed as –C2A2/3.
 C2 is a second adjustable constant.
Section 44.3
Liquid-Drop Model – Binding Energy Effect 3
The Coulomb repulsion effect
 Each proton repels every other proton in the nucleus.
 The potential energy associated with the Coulomb force is proportional to the
number of protons, Z.
 The reduction in the binding energy due to the Coulomb effect is
–C3Z(Z - 1)/A1/3
 C3 is another adjustable constant.
Section 44.3
Liquid-Drop Model – Binding Energy Effect 4
The symmetry effect
 Any large symmetry between N and Z for light nuclei reduces the binding
energy.
 For larger A, the value of N for stable nuclei is larger.
 The effect can be described by a binding energy term in the form
–C4(N - Z)2 / A
 For small A, any large asymmetry between N and Z makes the term large.
 For large A, the A in the denominator reduces the value of the term so
that it has little effect on the overall binding energy.
 C4 is another adjustable constant.
Section 44.3
Liquid-Drop Model – Binding Energy Effect Summary
Putting these terms together results in the semiempirical binding-energy formula:
Eb  C1A  C2 A
2 3
Z  Z  1
N  Z 
 C3

C
4
A1 3
A
2
The four constants are adjusted to fit the theoretical expression to the
experimental data.
 For A 15, C1 = 15.7 MeV; C2 = 17.8 MeV; C3 = 0.71 MeV; and C4 = 23.6
MeV
Section 44.3
Liquid Drop Model, Final
The graph shows the fit between the
theoretical curve and some sample
experimental values.
The equation fits the known nuclear
mass values very well.
Does not account for some of the finer
details of nuclear structure
 Stability
 Angular momentum
Section 44.3
Features of Binding Energy
When binding energies are studied closely it is found that:
 Most stable nuclei have an even value of A.
 Only 8 stable nuclei have odd values for both Z and N.
 There is a difference between the binding energy per nucleon given by the
semiempirical formula and experiments.
 Peaks occur at magic numbers
Studies of nuclear radii show deviations from the expected values
 Graphs of the data show peaks at values of N equal to the magic numbers
A group of isotones is a collection of nuclei having the same value of N and
different values of Z.
 When the number of stable isotones is graphed as a function of N, there are
peaks at the magic numbers.
Section 44.3
Features of Binding Energy – Magic Numbers
The disagreement between the semiempirical formula and experiments is plotted.
Peaks appear in the graph.
These peaks are at the magic numbers of
Z or N = 2, 8, 20, 28, 52, 82
Section 44.3
Features of Binding Energy, final
Several other nuclear measurements show anomalous behavior at the magic
numbers.
The peaks are reminiscent of the peaks in graphs of ionization energy of atoms
and lead to the shell model of the nucleus.
 Also called the independent-particle model
Section 44.3
Maria Goeppert-Mayer
1906 – 1972
German scientist
Best known for her development of the
shell model of the nucleus
Shared the Nobel Prize in 1963
 Shared with Hans Jensen who
simultaneously developed a similar
model
Section 44.3
Shell Model
In this model, each nucleon is assumed to exist in a shell.
 Similar to atomic shells for electrons
The nucleons exist in quantized energy states.
There are few collisions between nucleons.
Section 44.3
Shell Model, cont.
Each state can contain only two protons
or two neutrons.
 They must have opposite spins.
 They have spins of ½, so the
exclusion principle applies.
The proton states differ from the
neutron states because the two species
move in different potential wells.
Section 44.3
Shell Model, final
Proton energy levels are farther apart than those for neutrons due to the
superposition of the Coulomb force and the nuclear force for the protons.
The nuclear spin-orbit effect for nucleons is due to the nuclear force.
 The spin-orbit effect influences the observed characteristics of the nucleus.
The nuclear spin-orbit effect is much stronger than in the atomic case and has an
opposite sign.
When these effects are taken into account, the shell model is able to account for
the observed magic numbers.
Section 44.3
Shell Model Explanation of Experimental Results
Nuclei with even numbers of protons and neutrons are more stable.
 Any particular state is filled when it contains two protons or two neutrons
having opposite spins.
 An extra proton or neutron can be added only at the expense of increasing
the nucleus’ energy.
 This increase in energy leads to greater instability in the nucleus.
Section 44.3
Shell Model Explanation of Experimental Results, cont.
Nuclei tend to have more neutrons than protons.
 Proton energy levels are higher.
 As Z increases and higher states are filled, a proton level for a given
quantum number will be much higher in energy than the neutron level for the
same quantum number.
 It is more energetically favorable for the nucleus to form with neutrons in the
lower energy levels than protons in the higher levels.
 So, the number of neutrons is greater than the number of protons.
Section 44.3
Marie Curie
1867 – 1934
Polish scientist
Shared Nobel Prize in Physics in 1903
for studies in radioactive substances
 Shared with Pierre Curie and
Becquerel
Won Nobel Prize in Chemistry in 1911
for discovery of radium and polonium
Section 44.4
Radioactivity
Radioactivity is the spontaneous emission of radiation.
 Discovered by Becquerel in 1896
 Many experiments were conducted by Becquerel and the Curies.
Experiments suggested that radioactivity was the result of the decay, or
disintegration, of unstable nuclei.
Section 44.4
Radioactivity – Types of Decay
Three types of radiation can be emitted.
 Alpha particles
 The particles are 4He nuclei.
 Beta particles
 The particles are either electrons or positrons.
 A positron is the antiparticle of the electron.
 It is similar to the electron except its charge is +e.
 Gamma rays
 The “rays” are high energy photons.
Section 44.4
Distinguishing Types of Radiation
All three types of radiation enter a
region where there is a magnetic field.
The gamma particles carry no charge.
The alpha particles are deflected
upward.
The beta particles are deflected
downward.
 A positron would be deflected
upward, but would follow a different
trajectory than the α due to its
mass.
Section 44.4
Penetrating Ability of Particles
Alpha particles
 Barely penetrate a piece of paper
Beta particles
 Can penetrate a few mm of aluminum
Gamma rays
 Can penetrate several cm of lead
Section 44.4
Terminology Notes
Radiation is the term used historically for all emanations from a radioactive
nucleus.
Alpha and beta radiation are actually emissions of particles with nonzero rest
energy.
 Although these are not forms of electromagnetic radiation, the term radiation
is still used.
The symbol N has many uses, so be sure to consider the context in which the
symbol is used.
Section 44.4
The Decay Constant
The number of particles that decay in a given time is proportional to the total
number of particles in a radioactive sample.
dN
  λN gives N  Noe  λt
dt
 λ is called the decay constant and determines the probability of decay per nucleus
per second.
 N is the number of undecayed radioactive nuclei present.
 No is the number of undecayed nuclei at time t = 0.
Section 44.4
Decay Rate
The decay rate R of a sample is defined as the number of decays per second.
R
dN
 λN  Roe  λt
dt
 Ro = Noλ is the decay rate at t = 0.
 The decay rate is often referred to as the activity of the sample.
Section 44.4
Decay Curve and Half-Life
The decay curve follows the equation N
= Noe-λt .
The half-life is also a useful parameter.
 The half-life is defined as the time
interval during which half of a given
number of radioactive nuclei
decay.
ln 2 0.693
T1 2 

λ
λ
Section 44.4
Half-Live, cont.
During the first half-life, ½ of the original material will decay.
During the second half-life, ½ of the remaining material will decay, leaving ¼ of
the original material remaining.
Summarizing, the number of undecayed radioactive nuclei remaining after n halflives is N = No (½)n
 n can be an integer or a noninteger.
Section 44.4
Units
The unit of activity, R, is the curie (Ci)
 1 Ci ≡ 3.7 x 1010 decays/s
The SI unit of activity is the becquerel (Bq)
 1 Bq ≡ 1 decay/s
 Therefore, 1 Ci = 3.7 x 1010 Bq
The most commonly used units of activity are the millicurie and the microcurie.
Section 44.4
Decay Processes
The black circles are the stable nuclei
seen before.
Above the line the nuclei are neutron
rich and undergo beta decay (blue).
Just below the line are proton rich nuclei
that undergo beta (positron) emission or
electron capture (red).
Farther below the line the nuclei are
very proton rich and undergo alpha
decay (tan).
Section 44.5
Alpha Decay
When a nucleus emits an alpha particle it loses two protons and two neutrons.
 N decreases by 2
 Z decreases by 2
 A decreases by 4
Symbolically
A
Z
X  AZ42Y  42 He
 X is called the parent nucleus.
 Y is called the daughter nucleus.
Section 44.5
Decay – General Rules
The sum of the mass numbers A must be the same on both sides of the
equation.
The sum of the atomic numbers Z must be the same on both sides of the
equation.
When one element changes into another element, the process is called
spontaneous decay or transmutation.
Relativistic energy and momentum of the isolated parent nucleus must be
conserved.
Section 44.5
Disintegration Energy
The disintegration energy Q of a system is defined as
Q = (Mx – My – Mα)c2
The disintegration energy appears in the form of kinetic energy in the daughter
nucleus and the alpha particle .
It is sometimes referred to as the Q value of the nuclear decay.
Section 44.5
Alpha Decay, Example
Decay of 226 Ra
226
88
4
Ra  222
Rn

86
2 He
If the parent is at rest before the decay,
the total kinetic energy of the products
is 4.87 MeV.
In general, less massive particles carry
off more of the kinetic energy.
Section 44.5
Alpha Decay, Notes
Experimental observations of alpha-particle energies show a number of discrete
energies instead of a single value.
 The daughter nucleus may be left in an excited quantum state.
 So, not all of the energy is available as kinetic energy.
A negative Q value indicates that such a proposed decay does not occur
spontaneously.
Section 44.5
Alpha Decay, Mechanism
In alpha decay, the alpha particle
tunnels though a barrier.
For higher energy particles, the barrier
is narrower and the probability is higher
for tunneling across.
 This higher probability translates
into a shorter half-life of the parent.
Section 44.5
Beta Decay
During beta decay, the daughter nucleus has the same number of nucleons as
the parent, but the atomic number is changed by one.
Symbolically
A
Z
X  Z A1Y  e
A
Z
X  Z A1Y  e
 Beta decay is not completely described by these equations.
Section 44.5
Beta Decay, cont.
The emission of the electron or positron is from the nucleus.
 The nucleus contains protons and neutrons.
 The process occurs when a neutron is transformed into a proton or a proton
changes into a neutron.
 The electron or positron is created in the process of the decay.
The nucleon number and the total charge are both conserved.
Energy of the isolated system must be conserved.
Section 44.5
Beta Decay – Particle Energy
The energy released in the decay
process should almost all go to kinetic
energy of the β particle.
 Since the decaying nuclei all have
the same rest mass, the Q value
should be the same for all decays.
Experiments showed a range in the
amount of kinetic energy of the emitted
particles.
Were conservation laws violated?
Section 44.5
Neutrino
To account for this “missing” energy, in 1930 Pauli proposed the existence of
another particle.
Enrico Fermi later named this particle the neutrino.
Properties of the neutrino:
 Zero electrical charge
 Mass much smaller than the electron, probably not zero
 Spin of ½
 Very weak interaction with matter and so is difficult to detect
The neutrino was detected experimentally in 1956.
Section 44.5
Beta Decay – Completed
Symbolically
Y  e  ν
A
Z
X
A
Z 1
A
Z
X
A
Z 1
Y  e  ν
  is the symbol for the neutrino.
 ν is the symbol for the antineutrino.
To summarize, in beta decay, the following pairs of particles are emitted.
 An electron and an antineutrino
 A positron and a neutrino
Section 44.5
Beta Decay – Examples
Section 44.5
Beta Decay, Final Notes
The fundamental process of e- decay is a neutron changing into a proton, an
electron and an antineutrino.
In e+, the proton changes into a neutron, positron and neutrino.
 This can only occur within a nucleus.
 It cannot occur for an isolated proton since its mass is less than the mass of
the neutron.
Section 44.5
Electron Capture
Electron capture is a process that competes with e+ decay.
In this case, a parent nucleus captures one of its own orbital electrons and emits
a neutrino:
A
Z
X  01e  ZA1Y  ν
 In most cases, a K-shell electron is captured, so this is often referred to as K
capture.
 Because the neutrino is very hard to detect, electron capture is usually observed
by the x-rays given off as higher-shell electrons cascade downward to fill the
vacancy created in the K shell.
Section 44.5
Q Values for Beta Decay
For e- decay and electron capture, the Q value is Q = (Mx – MY)c2.
For e+ decay, the Q value is Q = (Mx – MY - 2me)c2.
 The extra term, -2mec2, is due to the fact that the atomic number of the
parent decreases by one when the daughter is formed.
 To form a neutral atom, the daughter sheds one electron.
If Q is negative, the decay will not occur.
Section 44.5
Gamma Decay
Gamma rays are given off when an excited nucleus decays to a lower energy
state.
The decay occurs by emitting a high-energy photon called gamma-ray photons.
A
Z
X*  ZA X  γ
 The X* indicates a nucleus in an excited state.
Typical half-life is 10-10 s
The only change in the nucleus is that it ends up in a lower energy state.
 No changes in Z, N or A occur
Section 44.5
Gamma Decay – Example
Example of a decay sequence:
 The first decay is a beta emission.
 The second step is a gamma emission.
12
5
B  126 C*  e  ν
12
6
C*  126 C  γ
 Gamma emission doesn’t change Z, N, or A
 The emitted photon has an energy of hƒ equal to DE between the two
nuclear energy levels.
Section 44.5
Summary of Decays
Section 44.5
Natural Radioactivity
Classification of nuclei
 Unstable nuclei found in nature
 Give rise to natural radioactivity
 Nuclei produced in the laboratory through nuclear reactions
 Exhibit artificial radioactivity
Three series of natural radioactivity exist.
 Uranium
 Actinium
 Thorium
Some radioactive isotopes are not part of any decay series.
Section 44.6
Radioactive Series, Overview
Section 44.6
Decay Series of 232Th
Series starts with 232Th
Processes through a series of alpha
and beta decays
The series branches at 212Bi
Ends with a stable isotope of lead,
208Pb
Section 44.6
Nuclear Reactions
The structure of nuclei can be changed by bombarding them with energetic
particles.
 The changes are called nuclear reactions.
As with nuclear decays, the atomic numbers and mass numbers must balance on
both sides of the equation.
A target nucleus, X, is bombarded by a particle a, resulting in a daughter nucleus
Y and an outgoing particle b.
 a+XY+b
The reaction energy Q is defined as the total change in mass-energy resulting
from the reaction.
 Q = (Ma + MX – MY – Mb)c2
Section 44.7
Q Values for Reactions
The Q value determines the type of reaction.
 An exothermic reaction
 There is a mass “loss” in the reaction.
 There is a release of energy.
 Q is positive.
 An endothermic reaction
 There is a “gain” of mass in the reaction.
 Energy is needed, in the form of kinetic energy of the incoming particles.
 Q is negative.
 The minimum energy necessary for the reaction to occur is called the
threshold energy.
Section 44.7
Nuclear Reactions, final
If a and b are identical, so that X and Y are also necessarily identical, the
reaction is called a scattering event.
 If the kinetic energy before the event is the same as after, it is classified as
elastic scattering.
 If the kinetic energies before and after are not the same, it is an inelastic
scattering.
Section 44.7
Conservation Rules for Nuclear Reactions
The following must be conserved in any nuclear reaction:
 Energy
 Momentum
 Total charge
 Total number of nucleons
Section 44.7
Nuclear Magnetic Resonance (NMR)
A nucleus has spin angular momentum.
Shown is a vector model giving
possible orientations of the spin and its
projection on the z axis.
The magnitude of the spin angular
momentum is
I ( I 1)
I is the nuclear spin quantum number.
Section 44.8
NMR, cont.
For a nucleus with spin ½, there are
only two allowed states
 Emax and Emin
It is possible to observe transitions
between two spin states using NMR.
Section 44.8
MRI
An MRI (Magnetic Resonance Imaging)
is based on NMR.
Because of variations in an external
field, hydrogen atoms in different parts
of the body have different energy
splittings between spin states.
The resonance signal can provide
information about the positions of the
protons.
Section 44.8