chapter 44 nuclear structure. milestones in the development of nuclear physics 1896: the birth of...
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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)
More Milestones
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
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
More Properties of Nuclei
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
Symbolism
X is the chemical symbol of the element Example:
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
XAZ
Al2713
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 12 13 146 6 6 6C C C C, , ,
Charge
The proton has a single positive charge, e The electron has a single negative charge,
- e The neutron has no charge
Made it difficult to detect in early experiments Easy to detect with modern devices
e = 1.602 177 33 x 10-19 C
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 = 1.602 177 x 10-19 J
Some Masses in Various Units
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
From conservation of energy, the kinetic energy of the particle must be completely converted to potential energy
Active Figure 44.1
Use the active figure to adjust the atomic number of the target nucleus
Also adjust the kinetic energy of the alpha particle
Observe the approach of the alpha particle
PLAYACTIVE FIGURE
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
For gold, he found d = 3.2 x 10-14 m
2
24 e
Zed k
mv
More About Size
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
Size of Nucleus, Final
Since the time of Rutherford, many other experiments have concluded the following: Most nuclei are approximately spherical Average radius is
ro = 1.2 x 10-15 m A is the mass number
1 3or r A
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
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
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
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
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
Binding Energy, cont.
The binding energy can be calculated from conservation of energy and the Einstein mass-energy equivalence principle:
Eb (MeV) = [ZM(H) + Nmn – M (AZX)] x
931.494 MeV/u M(H) is the atomic mass of the neutral hydrogen atom M (A
ZX) represents the atomic mass of an atom of the isotope (A
ZX) Mn is the mass of the neutron
The masses are expressed in atomic mass units
Binding Energy per Nucleon
Notes from the 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 each product nucleus are more
tightly bound to one another than are the nucleons of the original nucleus
More Notes from the Graph
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 has the largest binding energy per
nucleon
6228Ni
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
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
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
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
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
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
Liquid-Drop Model – Binding Energy Effect Summary
Putting these terms together results in the semiempirical binding-energy formula:
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
2
2 31 2 3 41 3
1b
Z Z N ZE C A C A C C
A A
Liquid Drop Model, Final
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
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 A and Z There is a difference between the binding energy
per nucleon given by the semiempirical formula and experiments
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
Features of Binding Energy, cont.
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
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
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
Shell Model
The shell model is also called the independent-particle 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
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 set of allowed states for the protons differs from the set of allowed states for the neutrons
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 spin-orbit effect for nucleons is due to the nuclear force The spin-orbit effect influences the observed
characteristics of the nucleus
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 An extra proton or neutron can be added only at
the expense of increasing the nucleus’s energy This increase in energy leads to greater instability
in the nucleus
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
Marie Curie
1867 – 1934 Polish scientist Shared Nobel Prize in 1903
for studies in radioactive substances Prize in physics Shared with Pierre Curie
and Becquerel Won Nobel Prize in 1911 for
discovery of radium and polonium Prize in chemistry
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
Radioactivity – Types
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
Distinguishing Types of Radiation
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
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
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
λ is called the decay constant and determines the rate at which the material will decay
N is the number of undecayed radioactive nuclei present
No is the number of undecayed nuclei at time t = 0
gives λto
dNλN N N e
dt
Decay Curve
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
1 2
ln 2 0693.T
λ λ
Active Figure 44.9
Use the active figure to adjust the half-life
Observe the decay curve
PLAYACTIVE FIGURE
Decay Rate
The decay rate R of a sample is defined as the number of decays per second
Ro = Noλ is the decay rate at t = 0 The decay rate is often referred to as the activity
of the sample
λto
dNR λN R e
dt
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
Decay Processes The blue circles are the stable
nuclei seen before Above the line the nuclei are
neutron rich and undergo beta decay (red)
Just below the line are proton rich nuclei that undergo beta (positron) emission or electron capture (green)
Farther below the line the nuclei are very proton rich and undergo alpha decay (yellow)
Active Figure 44.10
Click on any colored dot
Study the decay modes and decay energies
PLAYACTIVE FIGURE
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 X is called the parent nucleus Y is called the daughter nucleus
4 42 2X Y HeA A
Z Z
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
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
Alpha Decay, Example
Decay of 226 Ra
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
226 222 488 86 2Ra Rn He
Active Figure 44.11
Use the active figure to observe the decay of radium-226
PLAYACTIVE FIGURE
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
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
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
Beta decay is not completely described by these equations
1
1
X Y e
X Y e
A AZ Z
A AZ Z
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 Energy must be conserved
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?
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
Beta Decay – Completed
Symbolically
is the symbol for the neutrino 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
1
1
X Y e
X Y e
A AZ Z
A AZ Z
ν
ν
ν
Beta Decay – Examples
Active Figure 44.15
Use the active figure to observe the decay of Carbon-14
PLAYACTIVE FIGURE
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
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:
In most cases, a K-shell electron is captured, so this is often referred to as K capture
01 1X e YA A
Z Z ν
Electron Capture, Detection
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
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
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
The X* indicates a nucleus in an excited state Typical half-life is 10-10 s
X X*A AZ Z γ
Gamma Decay – Example
Example of a decay sequence The first decay is a beta emission The second step is a gamma emission
Gamma emission doesn’t change Z, N, or A The emitted photon has an energy of hƒ equal to
E between the two nuclear energy levels
12 125 6
12 126 6
B C e
C C
*
*
ν
γ
Summary of Decays
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
Radioactive Series, Overview
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
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
Nuclear Reactions, cont.
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
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
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
Conservation Rules for Nuclear Reactions
The following must be conserved in any nuclear reaction Energy Momentum Total charge Total number of nucleons
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
( 1)I I
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
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
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