Transcript lecture 27
SPH4UI Physics
Nuclear Binding, Radioactivity
Modern understanding: the ``onion’’ picture
Let’s see what’s inside!
3
Nice Try
Introduction: 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) 1911 Rutherford, Geiger and Marsden performed scattering experiments Established the point mass nature of the nucleus
Nuclear force
was a new type of force 1919 Rutherford and coworkers first observed nuclear reactions in which naturally occurring alpha particles bombarded nitrogen nuclei to produce oxygen 1932 Cockcroft and Walton first used artificially accelerated protons to produce nuclear reactions 1932 Chadwick discovered the neutron 1933 the Curies discovered artificial radioactivity 1938 Hahn and Strassman discovered nuclear fission 1942 Fermi achieved the first controlled nuclear fission reactor
Some Properties of Nuclei
All nuclei are composed of protons and neutrons Exception is ordinary hydrogen with just a proton The
atomic number (charge)
, Z , equals the number of protons in the nucleus The
neutron number
, N , is the number of neutrons in the nucleus The
mass number
, A , is the number of nucleons in the nucleus A = Z + N Nucleon is a generic term used to refer to either a proton or a neutron The mass number is not the same as the mass Notation
Z A X
where X is the chemical symbol of the element
Example: 27
Al
13 Mass number is 27 Atomic number is 13 The Z Contains 13 protons Contains 14 (27 – 13) neutrons may be omitted since the element can be used to determine Z
Charge and mass
Charge:
The electron has a single negative charge, -e ( e = 1.60217733 x 10 -19 C ) The proton has a single positive charge, +e Thus, charge of a nucleus is equal to Ze The neutron has no charge Makes it difficult to detect
Mass:
It is convenient to use
atomic mass units ,
u , to express masses 1 u = 1.660559 x 10 -27 kg Based on definition that the mass of one atom of C-12 is exactly 12 u Mass can also be expressed in MeV/c 2 From E R = m c 2 1 u = 931.494 MeV/c 2
Summary of Masses
Masses
Particle
Proton Neutron Electron
kg
1.6726 x 10 -27 1.6750 x 10 -27 9.101 x 10 -31
u
1.007276
1.008665
5.486x10
-4
MeV/c 2
938.28
939.57
0.511
Quick problem: protons in your body
What is the order of magnitude of the number of protons in your body? Of the number of neutrons? Of the number of electrons? Take your mass approximately equal to 70 kg.
An iron nucleus (in hemoglobin) has a few more neutrons than protons, but in a typical water molecule there are eight neutrons and ten protons. So protons and neutrons are nearly equally numerous in your body, each contributing 35 kg out of a total body mass of 70 kg.
N
35
kg
1 nucleon 1.67 10 27
kg
28 10 protons Same amount of neutrons and electrons.
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 The KE of the particle must be completely converted to PE 1 2
mv
2
k e q q
1 2
r
k e
or
d
4
k Ze e
2
mv
2
d
For gold: d = 3.2 x 10 -14 m , for silver: d = 2 x 10 -14 Such small lengths are often expressed in 10 -15 m a fermi) femtometers m where 1 fm = (also called
Size of Nucleus
Since the time of Rutherford, many other experiments have concluded the following Most nuclei are approximately spherical Average radius is
r
r A o
1 3 r o = 1.2 x 10 -15 m Example:
Z A
27
Al
13
has radius
r
15
m
Density of Nuclei
The volume of the nucleus (assumed to be spherical) is directly proportional to the total number of nucleons
This suggests that
all nuclei have nearly the same density
Nucleons combine to form a nucleus as though they were tightly packed spheres
Nuclear Stability
There are very large repulsive electrostatic forces between protons These forces should cause the nucleus to fly apart The nuclei are stable because of the presence of another, short range force, called the
nuclear (or strong) force
This is an attractive force that acts between all nuclear particles The nuclear attractive force is stronger than the force at the short ranges within the nucleus Coulomb repulsive
Nuclear Stability chart
Light nuclei are most stable if N = Z Heavy nuclei are most stable when N > Z As the number of protons increase, the Coulomb force increases and so more nucleons are needed to keep the nucleus stable No nuclei are stable when Z > 83
Isotopes
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
Z A X
atomic number (charge) , Z neutron number , N nucleon number , A ,
Example : 11
C
6 12
C
6 13
C
6 14
C
6
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 separated protons and neutrons Binding Energy per Nucleon
Binding Energy Plot
Iron (Fe) is most binding energy/nucleon. Lighter have too few nucleons, heavier have too many.
Fission Fission = Breaking large atoms into small Fusion = Combining small atoms into large
Binding Energy Notes
Except for light nuclei, the binding energy is about 8 MeV nucleon per 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 curve is slowly varying at A > 40 This suggests that the nuclear force saturates A particular nucleon can interact with only a limited number of other nucleons
Question
Where does the energy released in the nuclear reactions of the sun come from?
(1) covalent bonds between atoms (2) binding energy of electrons to the nucleus (3) binding energy of nucleons
Question
Which element has the highest binding energy/nucleon?
• Neon (Z=10) • Iron (Z=26) • Iodine (Z=53)
Question
Which of the following is most correct for the total binding energy of an Iron atom (Z=26)?
9 MeV 234 MeV 270 MeV 504 Mev For Fe, B.E./nucleon
9MeV
56
Fe
26
has 56 nucleons Total B.E
56x9=504 MeV
Binding Energy
Einstein’s famous equation
E = m c
2 Proton: mc 2 = 938.3MeV
Neutron: mc 2 = 939.5MeV
Adding these, get 1877.8MeV
Deuteron: mc 2 =1875.6MeV
Difference is Binding energy, 2.2MeV
M
Deuteron
= M
Proton
+ M
Neutron
– |Binding Energy|
Big Problem: binding energy
41
Nb
Using atomic mass units
Calculate the average binding energy per nucleon of
93 41
Nb
Given: m p = 1.007276u
m n = 1.008665u
1u=931.5 MeV Find: E b = ?
E b
A
In order to compute binding energy, let’s first find the mass difference between the total mass of all protons and neutrons in Nb and subtract mass of the Nb :
Number of protons:
N p
41 Number of neutrons:
Mass difference:
41
m
p
52
m n
u
m Nb
N n
u
92.9063768
u
0.8425192
u
2
Thus, binding energy is
0.842519
u
931.5
93
MeV u
8.44
MeV
nucleon
Radioactivity
Radioactivity
is the spontaneous emission of radiation Experiments suggested that radioactivity was the result of the decay, or disintegration, of unstable nuclei Three types of radiation can be emitted Alpha particles The particles are 4 He 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
B field into screen
Types of Radioactivity
Radioactive sources detector
a b g
He
particles: electrons rays) Barely penetrate a piece of paper Can penetrate a few mm of aluminum photons (more energetic than x Can penetrate several cm of lead
The Decay Processes – 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 Conservation of mass-energy and conservation of momentum must hold
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
Z A X
A Z
4 2
Y
2 4
He
X is called the
parent nucleus
Y is called the
daughter nucleus
Alpha Decay -- Example
Decay of 226 Ra 226
Ra
88 222
Rn
86 2 4
He
Half life for this decay is 1600 years Excess mass is converted into kinetic energy Momentum of the two particles is equal and opposite
Beta Decay
During beta decay, the daughter nucleus has the same number of nucleons as the parent, but the atomic number is one less In addition, an electron (positron) was observed The emission of the electron is from the nucleus The nucleus contains protons and neutrons The process occurs when a neutron is transformed into a proton and an electron Energy must be conserved
Beta Decay – Electron Energy
The energy released in the decay process should almost all go to kinetic energy of the electron Experiments showed that kinetic energy few electrons had this amount of 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
Beta Decay
Symbolically
Z A X
Z
A
1
Y
e
Z A X
Z
A
1
Y
e
is the symbol for the neutrino is the symbol for the antineutrino To summarize, in beta decay, the following pairs of particles are emitted An electron and an antineutrino A positron and a neutrino
Gamma Decay
Gamma rays are given off when an excited nucleus “falls” to a lower energy state Similar to the process of electron “jumps” to lower energy states and giving off photons The excited nuclear states result from “jumps” made by a proton or neutron The excited nuclear states may be the result of violent collision or more likely of an alpha or beta emission Example of a decay sequence The first decay is a beta emission The second step is a gamma emission 12
B
5 12
C
* 6 12
C
6 *
e
12
C
6 g The C* indicates the Carbon nucleus is in an excited state Gamma emission doesn’t change either A or Z
Decay Rules
1) Nucleon Number is conserved.
2) Atomic Number (charge) is conserved.
3) Energy and momentum are conserved.
a
: example
238
U
92 234 90
Th
a
1) 238 = 234 + 4 2) 92 = 90 + 2 recall
2 4
He
a
Nucleon number conserved Charge conserved
b
: example
0 1 n 1 1 p 0 1
e
0 0 g
: example
Z A P
*
Z A P
0 0 g
Neutrino needed to conserve energy and momentum.
Practice
A nucleus undergoes
a
FALSE?
decay. Which of the following is 1. Nucleon number decreases by 4 2. Neutron number decreases by 2 3. Charge on nucleus increases by 2
a
decay is the emission of
4 2
He
a
A decreases by 4 Ex .
238 U 92 234 90 Th 2 4
He
Z decreases by 2 (charge decreases!)
Practice
Th
b
Which of the following is true?
decay . 1. The number of protons in the daughter nucleus increases by one. 2. The number of neutrons in the daughter nucleus increases by one.
b
decay is accompanied by the emission of an electron: creation of a charge -e.
0 1
e
0 1
e
0 0 0 0
n e
neutrino “escape.”
Decay
Which of the following decays is NOT allowed?
1
238
U
92 234 90
Th
a
2
214 Po 84 210 82 Pb 4 2 He
3
14 6 C 14 7 N g
4
40 19 K 40 20 p 0 1
e
0 0
238 = 234 + 4 92 = 90 + 2 214 = 210 + 4 84 = 82 + 2 14 = 14+0 6 <> 7+0 40 = 40+0+0 19 = 20-1+0
Decay
Which of the following are possible reactions?
(a) and (b). Reactions (a) and (b) both conserve total charge and total mass number as required. Reaction (c) violates conservation of mass number with the sum of the mass numbers being 240 before reaction and being only 223 after reaction.
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
N
λ is called the
decay constant
material will decay and determines the rate at which the The
decay rate
or
activity
, R, of a sample is defined as the number of decays per second
R
N
t
N
Radioactivity
Decays per second, or “activity”
N
t
N
No. of nuclei present decay constant Start with 16 14 C atoms.
After 6000 years, there are only 8 left.
How many will be left after another 6000 years?
1) 0 2) 4 3) 8 Every 6000 years ½ of atoms decay
Decay Curve
The decay curve follows the equation
N
N e
0
t
The
half-life
is also a useful parameter The half-life is defined as the time it takes for half of any given number of radioactive nuclei to decay T 1 2 ln 2 0 .
693
Units
The unit of activity, R , is the
Curie, Ci
1 Ci = 3.7 x 10
10
decays/second
The SI unit of activity is the
Becquerel, Bq
1 Bq = 1 decay / second
Therefore, 1 Ci = 3.7 x 10 10 Bq
The most commonly used units of activity are the mCi and the µCi
Decay Function
N e
0
t
N
0
t T
1/2
time
Practice
The half-life for beta-decay of 14 C is ~6,000 years. You test a fossil and find that only 25% of its 14 C is un decayed. How old is the fossil?
3,000 years At 0 years: 100% remains 6,000 years 12,000 years At 6,000 years: 50% remains At 12,000 years: 25% remains
Radioactivity Quantitatively
Decays per second, or “activity”
N
t
N
No. of nuclei present Survival:
decay constant
N e
0
t
No. of nuclei present at time t No. we started with at t=0 Instead of base e 1/2 : we can use base
t e
t T
1/2
where
T
1/ 2 0.693
Half life Then we can write
N e
0
t
N
0
t T
1/2
Radioactivity Example
The half-life for beta-decay of around in 22920 years? 14 C is 5730 years. If you start with 1000 carbon-14 nuclei, how many will be
t
N
0
t T
1/2
T
1/ 2 0.693
)
N
0
t T
1/2 1000 1 62 .5
22920 573 0
T
1/ 2 0.693
0.693
5730 4
N e
0
t
1000
e
62.5
4 22920)
Uses of Radioactivity
Carbon Dating Beta decay of 14 C is used to date organic samples The ratio of 14 C to 12 C is used Smoke detectors Ionization type smoke detectors use a radioactive source to ionize the air in a chamber A voltage and current are maintained When smoke enters the chamber, the current is decreased and the alarm sounds Radon pollution Radon is an inert, gaseous element associated with the decay of radium It is present in uranium mines and in certain types of rocks, bricks, etc that may be used in home building May also come from the ground itself
Binding Energy
Which system “weighs” more?
1) Two balls attached by a relaxed spring.
2) Two balls attached by a stretched spring.
3) They have the same weight.
M 1 M 2 M 2 = M balls = M balls – M 1 + M + M spring spring = E spring /c 2 + E spring ≈ 10 -16 /c Kg 2
Strong Nuclear Force
Acts on Protons and Neutrons
Strong enough to overcome Coulomb repulsion
Acts over very short distances
Two atoms don’t feel force
Q Values (nice to know)
Energy must also be conserved in nuclear reactions The energy required to balance a nuclear reaction is called the
Q value
of the 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
Problem: nuclear reactions
Determine the product of the reaction What is the Q value of the reaction?
7 3
Li
2 4
He n
Determine the product of the reaction What is the Q value of the reaction?
7 3
Li
2 4
He
Y X
?
0 1
n
Given: reaction Find: Q = ?
It is easier to use atomic mass units rather than kg.
In order to balance the reaction, the total amount of nucleons (sum of must be the same on both sides. Same for the Z -number. A )
7
: A -numbers)
X
1
X
Number of protons ( Z ) :
3 2
Y
0 10 5
Thus, it is B , i.e.
3 7
Li
2 4
He
10
B
5 0 1
n
The Q-value is then
Q
2
m
7
Li
m
4
He
7.016005
u
4.002603
u
m
10
B
n
2 10.012938
u
1.008665
u
931.5
2.79
MeV
Summary
Nuclear Reactions Nucleon number conserved Charge conserved Energy/Momentum conserved a particles = nucleii 2
He
b
-
particles = electrons g particles = high-energy photons
Survival :
N e
0
t T
1/ 2 0.693
Decays Half Life is time for ½ of atoms to decay