Pc Chapter 44
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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 nucleus) 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
Milestones, final
1930: Cockcroft and Walton first observed nuclear reactions using artificially accelerated nuclei 1932: Chadwick discovered the neutron 1933: Curies discovered artificial radioactivity 1938: Hahn and Strassmann discovered nuclear fission 1942: Fermi and collaborators achieved the first controlled nuclear fission reactor
Some Properties of Nuclei
All nuclei are composed of protons and neutrons
The atomic number Z equals the number of protons in the nucleus
Exception is ordinary hydrogen with just a proton
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 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
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 12 13 14 Example: 11 C , C , C , 6 6 6 6C
Charge
The proton has a single positive charge, e The electron has a single negative charge, - e The neutron has no charge
Makes it difficult to detect
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
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
(SLIDESHOW MODE ONLY)
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 4ke Ze 2 d mv 2
For gold, he found d = 3.2 x 10-14 m For silver, he found d = 2 x 10-14 m
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
r ro A
13
ro = 1.2 x 10-15 m
A is the mass number
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
Since r3 would be proportional to A
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 It is the strongest force in nature Very short range
Independent of charge
It falls to zero when the separation between particles exceeds about several fermis The nuclear force on p-p, p-n, n-n are all the same Does not affect electrons
Its magnitude depends on the relative spin orientations of the nucleons
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 = (Zmp + Nmn – MA) x 931.494 MeV/u
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
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
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
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
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
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
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
Liquid-Drop Model – Binding Energy Effect Summary
Putting these terms together results in the semiempirical binding-energy formula:
Eb = C1A – C2A2/3 – C3Z(Z - 1)/A1/3 – C4(N - Z)2/A
The four constants are adjusted to fit the theoretical expression to the experimental data
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 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 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
Beta particles
The particles are 4He nuclei 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
Beta particles
Barely penetrate a piece of paper 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 dN λN gives N Noe λt dt
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
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
T1 2
ln 2 0.693 λ λ
Active Figure 44.9
(SLIDESHOW MODE ONLY)
Decay Rate
The decay rate R of a sample is defined as the number of decays per second dN RλN R e dt
λt o
Ro = Noλ is the decay rate at t = o The decay rate is often referred to as the activity of the sample
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
(SLIDESHOW MODE ONLY)
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 A Z
A 4 Z 2
Y 42 He
X is called the parent nucleus Y is called the daughter nucleus
Decay – General Rules
When one element changes into another element, the process is called spontaneous decay or transmutation 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 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 226 88
Ra→
222 86
Rn+ He 4 2
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
Active Figure 44.11
(SLIDESHOW MODE ONLY)
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
A Z
X
A Z 1
Ye
A Z
X
A Z 1
Y e
Beta decay is not completely described by these equations
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
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
X
A Z 1
A Z
X
A Z 1
Y e ν
ν is the symbol for the neutrino is the symbol for the antineutrino ν To summarize, in beta decay, the following pairs of particles are emitted An electron and an antineutrino A positron and a neutrino
Y e ν
A Z
Beta Decay – Examples
Active Figure 44.15
(SLIDESHOW MODE ONLY)
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: A Z
X e 0 1
Y ν
A Z 1
In most cases, a K shell electron is captured, so this is often referred to as K capture
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 highenergy photon A Z
X* X γ A Z
The X* indicates a nucleus in an excited state
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 C* e ν
12 6
C* 126 C γ
12 6
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
Summary of Decays
Natural Radioactivity
Classification of nuclei
Unstable nuclei found in nature
Nuclei produced in the laboratory through nuclear reactions
Exhibit artificial radioactivity
Three series of natural radioactivity exist
Give rise to natural radioactivity
Uranium Actinium Thorium
Some radioactive isotopes are not part of any decay series
Radioactive Series, Overview
Decay Series of 232Th
Series starts with 232 Th Processes through a series of alpha and beta decays The series branches at 212Bi Ends with a stable isotope of lead, 208Pb
Nuclear Reactions
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
I ( I 1)h
NMR, cont.
Nuclear magnetic moments will precess when placed in an external magnetic field 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, protons in different parts of the body precess at different frequencies The resonance signal can provide information about the positions of the protons
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 nucleus) 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
Milestones, final
1930: Cockcroft and Walton first observed nuclear reactions using artificially accelerated nuclei 1932: Chadwick discovered the neutron 1933: Curies discovered artificial radioactivity 1938: Hahn and Strassmann discovered nuclear fission 1942: Fermi and collaborators achieved the first controlled nuclear fission reactor
Some Properties of Nuclei
All nuclei are composed of protons and neutrons
The atomic number Z equals the number of protons in the nucleus
Exception is ordinary hydrogen with just a proton
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 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
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 12 13 14 Example: 11 C , C , C , 6 6 6 6C
Charge
The proton has a single positive charge, e The electron has a single negative charge, - e The neutron has no charge
Makes it difficult to detect
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
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
(SLIDESHOW MODE ONLY)
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 4ke Ze 2 d mv 2
For gold, he found d = 3.2 x 10-14 m For silver, he found d = 2 x 10-14 m
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
r ro A
13
ro = 1.2 x 10-15 m
A is the mass number
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
Since r3 would be proportional to A
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 It is the strongest force in nature Very short range
Independent of charge
It falls to zero when the separation between particles exceeds about several fermis The nuclear force on p-p, p-n, n-n are all the same Does not affect electrons
Its magnitude depends on the relative spin orientations of the nucleons
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 = (Zmp + Nmn – MA) x 931.494 MeV/u
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
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
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
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
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
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
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
Liquid-Drop Model – Binding Energy Effect Summary
Putting these terms together results in the semiempirical binding-energy formula:
Eb = C1A – C2A2/3 – C3Z(Z - 1)/A1/3 – C4(N - Z)2/A
The four constants are adjusted to fit the theoretical expression to the experimental data
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 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 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
Beta particles
The particles are 4He nuclei 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
Beta particles
Barely penetrate a piece of paper 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 dN λN gives N Noe λt dt
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
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
T1 2
ln 2 0.693 λ λ
Active Figure 44.9
(SLIDESHOW MODE ONLY)
Decay Rate
The decay rate R of a sample is defined as the number of decays per second dN RλN R e dt
λt o
Ro = Noλ is the decay rate at t = o The decay rate is often referred to as the activity of the sample
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
(SLIDESHOW MODE ONLY)
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 A Z
A 4 Z 2
Y 42 He
X is called the parent nucleus Y is called the daughter nucleus
Decay – General Rules
When one element changes into another element, the process is called spontaneous decay or transmutation 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 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 226 88
Ra→
222 86
Rn+ He 4 2
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
Active Figure 44.11
(SLIDESHOW MODE ONLY)
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
A Z
X
A Z 1
Ye
A Z
X
A Z 1
Y e
Beta decay is not completely described by these equations
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
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
X
A Z 1
A Z
X
A Z 1
Y e ν
ν is the symbol for the neutrino is the symbol for the antineutrino ν To summarize, in beta decay, the following pairs of particles are emitted An electron and an antineutrino A positron and a neutrino
Y e ν
A Z
Beta Decay – Examples
Active Figure 44.15
(SLIDESHOW MODE ONLY)
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: A Z
X e 0 1
Y ν
A Z 1
In most cases, a K shell electron is captured, so this is often referred to as K capture
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 highenergy photon A Z
X* X γ A Z
The X* indicates a nucleus in an excited state
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 C* e ν
12 6
C* 126 C γ
12 6
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
Summary of Decays
Natural Radioactivity
Classification of nuclei
Unstable nuclei found in nature
Nuclei produced in the laboratory through nuclear reactions
Exhibit artificial radioactivity
Three series of natural radioactivity exist
Give rise to natural radioactivity
Uranium Actinium Thorium
Some radioactive isotopes are not part of any decay series
Radioactive Series, Overview
Decay Series of 232Th
Series starts with 232 Th Processes through a series of alpha and beta decays The series branches at 212Bi Ends with a stable isotope of lead, 208Pb
Nuclear Reactions
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
I ( I 1)h
NMR, cont.
Nuclear magnetic moments will precess when placed in an external magnetic field 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, protons in different parts of the body precess at different frequencies The resonance signal can provide information about the positions of the protons
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