pc chapter 44
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Chapter 44
Nuclear Structure
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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)
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More Milestones 1911 Rutherford, Geiger and
Marsden performed scattering
experiments Established that the nucleus could be
treated as a point mass and a pointcharge
Most of the atomic mass was containedin the nucleus
Nuclear force was a new type of force
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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 discoverednuclear fission
1942: Fermi and collaborators achieved the
first controlled nuclear fission reactor
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Some Properties of Nuclei All nuclei are composed of protons and
neutrons Exception is ordinary hydrogen with just a
proton
The atomic numberZequals the numberof protons in the nucleus Sometimes called the charge number
The neutron numberNis the number ofneutrons in the nucleus
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More Properties of Nuclei The mass numberA 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
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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 Zmay be omitted since the elementcan be used to determine Z
XAZ
Al27
13
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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 Zbutdiffering NandA values
The natural abundance of isotopes can vary
Example: 11 12 13 146 6 6 6C C C C , , ,
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Charge The proton has a single positive charge, e
The electron has a single negativecharge, - e
The neutron has no charge
Makes it difficult to detect e = 1.602 177 33 x 10-19 C
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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 of12C is exactly 12 u
Mass can also be expressed in MeV/c2
From ER = mc2
1 u = 931.494 MeV/c2
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Some Masses in Various
Units
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The Size of the Nucleus
First investigated by Rutherford in scatteringexperiments
He found an expression for how close an alpha particlemoving toward the nucleus can come before beingturned around by the Coulomb force
From conservation of energy, the kinetic energy of theparticle must be completely converted to potentialenergy
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Active Figure 44.1
(SLIDESHOW MODE ONLY)
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Size of the Nucleus, cont. dis called the distance of closest
approach
dgives an upper limit for the size of thenucleus
Rutherford determined that
For gold, he found d= 3.2 x 10-14 m For silver, he found d= 2 x 10-14 m
2
24 ek Zed mv
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More About Size Rutherford concluded that the positive
charge of the atom was concentrated in a
sphere whose radius was no larger thanabout 10-14 m He called this sphere the nucleus
These small lengths are often expressedin femtometers(fm) where 1 fm = 10-15 m Also called a fermi
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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
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Density of Nuclei The volume of the nucleus
(assumed to be spherical) isdirectly proportional to the
total number of nucleons This suggests that all nuclei
havenearly the samedensity Since r3 would be proportional
toA
Nucleons combine to form anucleus as though they weretightly packed spheres
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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 thepresence of another, short-range force, calledthe nuclear force
This is an attractive force that acts between allnuclear particles The nuclear attractive force is stronger than the
Coulomb repulsive force at the short ranges withinthe nucleus
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Features of the Nuclear Force Attractive force that acts between all nuclear
particles
It is the strongest force in nature Very short range
It falls to zero when the separation betweenparticles 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
Its magnitude depends on the relative spin
orientations of the nucleons
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Nuclear Stability, cont. Light nuclei are most
stable ifN= Z
Heavy nuclei are most
stable when N> Z Above about Z= 20
As the number of protons
increases, the Coulomb
force increases and somore neutrons are needed
to keep the nucleus stable
No nuclei are stable
when Z> 83
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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 energyyou need to add to the nucleus to break it apart
into its components
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Binding Energy, cont. The binding energy can be calculated
from conservation of energy and the
Einstein mass-energy equivalenceprinciple:
Eb = (Zmp + Nmn MA) x 931.494 MeV/u
The masses are expressed in atomic mass
units
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Binding Energy per Nucleon
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Notes from the Graph The curve peaks in the vicinity ofA = 60
Nuclei with mass numbers greater than or less
than 60 are not as strongly bound as those nearthe middle of the periodic table
The binding energy is about 8 MeV per
nucleon for nuclei withA > 50
This suggests that the nuclear force saturates A particular nucleon can interact with only a limited
number of other nucleons
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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
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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
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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 andnot to all the other nucleons in the nucleus The total binding energy is proportional toA and therefore proportional to the nuclear
volume This contribution to the binding energy of
the entire nucleus is C1A
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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 theirnumber
The number of nucleons is proportional tothe surface area
The surface term can be expressed as
C2A2/3
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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
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Liquid-Drop Model
Binding Energy Effect 4 The symmetry effect
Any large symmetry between Nand Zfor lightnuclei reduces the binding energy
For largerA, the value ofNfor stable nuclei islarger
The effect can be described by a binding energyterm in the form C4(N- Z)
2 / A
For small A, any large asymmetry between Nand Zmakes the term large
For largeA, theA in the denominator reduces the valueof the term so that it has little effect on the overall bindingenergy
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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)/A
1/3 C4(N- Z)2/A
The four constants are adjusted to fit thetheoretical expression to the experimentaldata
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Features of Binding Energy When binding energies are studied
closely it is found that: Most stable nuclei have an even value ofA
Only 8 stable nuclei have odd values for bothA
and Z
There is a difference between the bindingenergy per nucleon given by the
semiempirical formula and experiments
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Features of Binding Energy
Magic Numbers
The disagreement between the semiempiricalformula and experiments is plotted Peaks appear in the graph These peaks are at the magic numbers of
ZorN= 2, 8, 20, 28, 52, 82
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Features of Binding Energy,
cont. Studies of nuclear radii show deviations from
the expected values Graphs of the data show peaks at values ofN
equal to the magic numbers
A group ofisotones is a collection of nucleihaving the same value ofNand differentvalues ofZ When the number of stable isotones is graphed as
a function ofN, there are peaks at the magicnumbers
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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 thenucleus
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Maria Goeppert-Mayer 1906 1972
Best known for her
development of theshell model of the
nucleus
Shared the Nobel Prize
in 1963 Shared with Hans Jensen
who simultaneously
developed a similar model
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Shell Model The shell model is also called the
independent-particle model
In this model, each nucleon is assumed toexist in a shell Similar to atomic shells for electrons
The nucleons exist in quantized energy states
There are few collisions between nucleons
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Shell Model, cont. Each state can contain
only two protons or two
neutrons They must have opposite
spins
They have spins of , so the
exclusion principle applies
The set of allowed statesfor the protons differs
from the set of allowed
states for the neutrons
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Shell Model, final Proton energy levels are farther apart
than those for neutrons due to the
superposition of the Coulomb force andthe 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
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Shell Model Explanation of
Experimental Results Nuclei with even numbers of protons
and neutrons are more stable
Any particular state is filled when itcontains two protons or two neutrons
An extra proton or neutron can be addedonly at the expense of increasing the
nucleuss energy This increase in energy leads to greater
instability in the nucleus
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Shell Model Explanation of
Experimental Results, cont. Nuclei tend to have more neutrons than
protons Proton energy levels are higher As Zincreases and higher states are filled, a
proton level for a given quantum number will bemuch higher in energy than the neutron level forthe same quantum number
It is more energetically favorable for the nucleus toform with neutrons in the lower energy levels thanprotons in the higher levels
So, the number of neutrons is greater than thenumber of protons
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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 in1911 for discovery of
radium and polonium Prize in chemistry
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Radioactivity Radioactivityis the spontaneous
emission of radiation Discovered by Becquerel in 1896 Many experiments were conducted by
Becquerel and the Curies
Experiments suggested thatradioactivity was the result of the decay,
or disintegration, of unstable nuclei
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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
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Distinguishing Types of
Radiation The gamma particles
carry no charge
The alpha particlesare deflected upward
The beta particles are
deflected downward
A positron would bedeflected upward, but
would follow a different
trajectory than the due
to its mass
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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
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The Decay Constant The number of particles that decay in a given
time is proportional to the total number ofparticles in a radioactive sample
is called the decay constant and determines
the rate at which the material will decay Nis the number of undecayed radioactive nuclei
present No is the number of undecayed nuclei at time t= 0
gives to
dN N N N e
dt
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Decay Curve The decay curve follows
the equation N= Noe-t
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 radioactivenuclei decay
1 2
ln 2 0693.T
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Active Figure 44.9
(SLIDESHOW MODE ONLY)
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Decay Rate The decay rateRof a sample is
defined as the number of decays per
second
Ro = No is the decay rate at t= o The decay rate is often referred to as the
activity of the sample
t
o
dNRN R e
dt
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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
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Decay Processes The blue circles are the stable
nuclei seen before Above the line the nuclei are
neutron rich and undergo betadecay (red)
Just below the line are proton richnuclei that undergo beta(positron) emission or electroncapture (green)
Farther below the line the nucleiare very proton rich and undergoalpha decay (yellow)
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Active Figure 44.10
(SLIDESHOW MODE ONLY)
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Alpha Decay When a nucleus emits an alpha particle
it loses two protons and two neutrons
Ndecreases by 2 Zdecreases by 2 A decreases by 4
Symbolically X is called the parent nucleus Y is called the daughter nucleus
4 4
2 2X Y HeA A
Z Z
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Decay General Rules When one element changes into another
element, the process is called spontaneous
decay ortransmutation
The sum of the mass numbersA must be the
same on both sides of the equation
The sum of the atomic numbers Zmust be
the same on both sides of the equation Relativistic energy and momentum of the
isolated parent nucleus must be conserved
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Disintegration Energy The disintegration energy Q of a system
is defined as
Q = (Mx My M)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
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Alpha Decay, Example Decay of226 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
HeRnRa 42222
86
226
88 +
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Active Figure 44.11
(SLIDESHOW MODE ONLY)
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Alpha Decay, Notes
Experimental observations of alpha-particleenergies show a number of discrete energiesinstead 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 aproposed decay does not occurspontaneously
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Alpha Decay, Mechanism
In alpha decay, the alpha
particle tunnels though a
barrier
For higher energyparticles, the barrier is
narrower and the
probability is higher for
tunneling across This higher probability
translates into a shorter
half-life of the parent
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Beta Decay
During beta decay, the daughter nucleus hasthe same number of nucleons as the parent,but the atomic number is changed by one
Symbolically
Beta decay is not completely described by theseequations
1
1
X Y e
X Y e
A A
Z Z
A A
Z Z
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Beta Decay, cont.
The emission of the electron or positronis from the nucleus
The nucleus contains protons and neutrons The process occurs when a neutron is
transformed into a proton or a protonchanges into a neutron
The electron or positron is created in theprocess of the decay
Energy must be conserved
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Beta Decay Particle Energy
The energy released inthe decay processshould almost all go to
kinetic energy of theparticle Since the decaying nuclei
all have the same restmass, the Q value shouldbe the same for all
decays
Experiments showed arange in the amount ofkinetic energy of theemitted particles
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Neutrino
To account for this missing energy, in 1930Pauli proposed the existence of anotherparticle
Enrico Fermi later named this particle theneutrino
Properties of the neutrino Zero electrical charge Mass much smaller than the electron, probably not zero Spin of Very weak interaction with matter and so is difficult to
detect
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Beta Decay Completed
Symbolically
is the symbol for the neutrino is the symbol for the antineutrino
To summarize, in beta decay, the following pairs ofparticles are emitted An electron and an antineutrino A positron and a neutrino
1
1
X Y e
X Y e
A A
Z Z
A AZ Z
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Beta Decay Examples
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Active Figure 44.15
(SLIDESHOW MODE ONLY)
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Beta Decay, Final Notes
The fundamental process of e- decay isa 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 theneutron
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Electron Capture
Electron capture is a process that competes
with e+ decay
In this case, a parent nucleus captures one ofits 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
0
1 1X e YA A
Z Z
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Electron Capture, Detection
Because the neutrino is very hard to
detect, electron capture is usually
observed by the x-rays given off ashigher-shell electrons cascade
downward to fill the vacancy created in
the K shell
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Q Values for Beta Decay
For e- decay and electron capture, the Qvalue is Q = (Mx MY)c
2
For e
+
decay, the Q value isQ = (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
IfQ is negative, the decay will not occur
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Gamma Decay
Gamma rays are given off when anexcited nucleus decays to a lower
energy state The decay occurs by emitting a high-
energy photon
The X* indicates a nucleus in an excitedstate
X X*A A
Z Z
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Gamma Decay Example
Example of a decay sequence The first decay is a beta emission
The second step is a gamma emission
Gamma emission doesnt change Z, N, orA The emitted photon has an energy ofh equal to
Ebetween the two nuclear energy levels
12 12
5 6
12 12
6 6
B C e
C C
*
*
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Summary of Decays
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Natural Radioactivity
Classification of nuclei Unstable nuclei found in nature
Give rise to natural radioactivity
Nuclei produced in the laboratory through nuclearreactions Exhibit artificial radioactivity
Three series of natural radioactivity exist Uranium Actinium Thorium
Some radioactive isotopes are not part of any
decay series
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Radioactive Series, Overview
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Decay Series of232Th
Series starts with232Th
Processes through a
series of alpha and
beta decays
The series branches
at212
Bi Ends with a stable
isotope of lead, 208Pb
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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 mustbalance on both sides of the equation
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Nuclear Reactions, cont.
A target nucleus, X, is bombarded by aparticle a, resulting in a daughter
nucleus Y and an outgoing particle b a + X Y + b The reaction energyQ is defined as
the total change in mass-energy
resulting from the reaction Q = (Ma + MX MY Mb)c
2
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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 endothermicreaction There is a gain of mass in the reaction
Energy is needed, in the form of kinetic energy of theincoming particles
Q is negative The minimum energy necessary for the reaction to occur is
called the threshold energy
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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
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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
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Nuclear Magnetic Resonance
(NMR)
A nucleus has spin angularmomentum
Shown is a vector modelgiving possible orientationsof the spin and itsprojection on the zaxis
The magnitude of the spinangular momentum is
( 1)I I h
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NMR, cont.
Nuclear magnetic
moments will precess
when placed in an
external magnetic field
It is possible to
observe transitions
between two spinstates using NMR
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MRI
An MRI (MagneticResonance Imaging) isbased on NMR
Because of variations inan external field, protonsin different parts of thebody precess at differentfrequencies
The resonance signal canprovide information aboutthe positions of theprotons
top related