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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

http://../Active_Figures/Active%20Figures%20Media/AF_4409.htmlhttp://../Active_Figures/Active%20Figures%20Media/AF_4409.html

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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


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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


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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


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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

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