# lecture29

```Nuclear physics
1. Structure
•
•
At the center of an atom there is a tiny but massive nucleus.
Nuclei are made up of protons and neutrons
Proton has a positive charge with the same magnitude as electron (+e=1.6x10-19C)
and a mass mp = 1.67262x10-27 kg
Neutron is electrically neutral (q=0)
and has a mass mn = 1.67493x10-27 kg that is very slightly lager the mass of the proton
Nucleons: protons and neutrons
Different nuclei are referred as nuclides
2. Terminology
Z - the atomic number or proton number (the number of protons in the nucleus)
N - the neutron number (the number of neutrons in the nucleus)
A - the atomic mass number (the number of nucleons in the nucleus)
A=Z+N
X - chemical symbol for the element; it contains the same information as Z but in a
more easily recognizable form
A
Standard notation: Z X
Isotopes – nuclei that contain the same number of protons but different number
of neutrons
Because of wave-particle duality, the size of the nucleus is somewhat fuzzy.
V  43 r 3  A

r  1.2 10
15

mA
1
3
Example: Estimate the diameter of the following nuclei:
d H  2rH  21.2 10

m 235   15 10
15
dU  2rU  21.2 10 15
m 1  2.4 10 15 m
1
1
H,
1
3
1
3
15
m
dU
 6.17
dH
Example: Compare the density of nuclear matter to the density
of normal solids
 nucl mnucl / Vnucl r


 atom matom / Vatom r
3
atom
3
nucl
3
 10 
  15   1015
 10 
10
235
92
U.
4. Atomic mass units
Masses of atoms are measured with reference to the carbon-12 atom
12
A neutral 6 C atom is given the precise value 12.000000 u (unified atomic mass units)
1 u = 1.6605x10-27 kg = 931.5 MeV/c2
Proton:
mp = 1.67262x10-27 kg = 1.007276 u = 938.27 MeV/c2
Neutron: mn = 1.67493x10-27 kg = 1.008665 u = 939.57 MeV/c2
1H atom:
mH = 1.67353x10-27 kg = 1.007825 u = 938.78 MeV/c2 (proton +electron)
Electron: me = 9.1094x10-31 kg = 0.00054858 u =0.51100 MeV/c2
What it is MeV/c2 ?
E  mc 2
8.98740 1016 J
1kg  c  1kg(2.9979 10 m / s )  8.98740 10 J 

19
1.6022 10 J / eV
1kg  8.98740 1016 J / c 2  5.6094 1035 eV / c 2  560.94 10 27 MeV / c 2
2
8
2
16
Binding Energy
The total mass of a stable nucleus is always less than the sum of
the masses of its separate protons and neutrons.
Where has the mass gone?
It has become energy, such as radiation or kinetic energy,
released during the formation of the nucleus.
This difference between
the total mass of the
constituents and the
mass of the nucleus is
called the total binding
energy of the nucleus.
Binding Energy and Nuclear Forces
•The higher the binding energy per
nucleon, the more stable the nucleus.
•More massive nuclei require extra
neutrons to overcome the Coulomb
repulsion of the protons in order to be
stable.
•The force that binds the nucleons
together is called the strong nuclear force.
•It is a very strong, but short-range, force.
It is essentially zero if the nucleons are
more than about 10-15 m apart.
•The Coulomb force is long-range; this is
why extra neutrons are needed for stability
in high-Z nuclei.
•Nuclei that are unstable decay; many such decays are governed by another
force called the weak nuclear force.
(Discovered in 1896 by Henry Becquerel)
Most of the known nuclides are unstable (radioactive).
transforming itself into a different nuclide.
Radioactive rays were observed to be of three types:
1. Alpha rays, which could barely penetrate a piece of paper
2. Beta rays, which could penetrate 3 mm of aluminum
3. Gamma rays, which could penetrate several centimeters of lead
Alpha and beta rays are bent
in opposite directions in a
magnetic field, while gamma
rays are not bent at all.
1. Alpha rays are helium nuclei
2. Beta rays are electrons
3. Gamma rays are electromagnetic radiation
Individual particles such as electrons, neutrons, and protons cannot be seen directly,
so their existence must be inferred through measurements.
The Geiger counter
When a charged particle
passes through, it ionizes the
the wire, producing a pulse.
A scintillation counter uses a scintillator – a material
that emits light when a charged particle goes through it.
The scintillator is made light-tight, and the light flashes
are viewed with a photomultiplier tube, which has a
photocathode that emits an electron when struck by a
photon and then a series of amplifiers.
A cloud chamber contains a
supercooled gas; when a
charged particle goes
through, droplets form along
its track. Similarly, a bubble
chamber contains a
superheated liquid, and it is
bubbles that form. In either
case, the tracks can be
photographed and measured.
Alpha Decay
Alpha decay can be written as
(transmutation
of one element
into another)
M P c 2  M D c 2  m c 2  Q
parent
nucleus
daughter
nucleus
disintegration
energy
parent
nucleus
daughter
nucleus
Alpha decay occurs when the strong nuclear force cannot hold a large
nucleus together. The mass of the parent nucleus is greater than the sum of
the masses of the daughter nucleus and the alpha particle; this difference is
called the disintegration energy.
Alpha decay is so much more likely than other forms of nuclear
disintegration because the alpha particle itself is quite stable.
Beta Decay
Beta decay occurs when a nucleus emits an electron or positron.
Example 1: the decay of carbon-14:
•The Greek letter v is the symbol for neutrino - another particle born in this
process. The symbol  means antineutrino.
•The nucleus still has 14 nucleons, but it has one more proton and one fewer
neutron.
•This decay is an example of an interaction that proceeds via the weak nuclear
force.
•The electron in beta decay is not an orbital electron; it is created in the decay.
•The fundamental process is a neutron decaying to a proton, electron, and
neutrino: n  p  e  
•The need for a particle such as the neutrino was discovered through analysis
of energy and momentum conservation in beta decay – it could not be a twoparticle decay.
•Neutrinos are notoriously difficult to detect, as they interact only weakly, and
direct evidence for their existence was not available until more than 20 years
Example 2: Beta decay can also occur where the nucleus emits a
positron rather than an electron:
Example 3: A nucleus can capture one of its inner electrons:
Gamma Decay
Gamma rays are very high-energy
photons. They are emitted when a
nucleus decays from an excited state
to a lower state, just as photons are
emitted by electrons returning to a
lower state.
Conservation of Nucleon Number and Other
Conservation Laws
A new law that is evident by
that the total number of
nucleons cannot change.
Decay Series
A decay series occurs
isotope decays to
isotope, which decays
to another, and so on.
This allows the
creation of nuclei that
otherwise would not
exist in nature.
Half-Life and Rate of Decay
Nuclear decay is a random process; the decay of any nucleus is not
influenced by the decay of any other.
Therefore, the number of decays in a short time interval is proportional
to the number of nuclei present and to the time:
using calculus
N
 N   N 
 e  t

 
 
 t   t  0 N 0
λ - decay constant (characteristic of that particular nuclide)
N / t  N 

ln 2
N
T1
2

ln 2 1.0  10 22


7
5730 yr  3.156  10 s / yr


 3.8  1010
The half-life is the time it takes for half the nuclei in a given sample to decay.
It is related to the decay constant:
1


Radioactive dating can be done by analyzing the fraction of carbon in
organic material that is carbon-14.
The ratio of carbon-14 to carbon-12 in the atmosphere has been roughly
constant over thousands of years. This ratio is about 1.3x10-12.
N0
 C  N  C   N  C  N  C   1.3 10
14
6
12
0 6
14
0 6
12
6
12
A living plant or tree will be constantly exchanging carbon with the
atmosphere, and will have the same carbon ratio in its tissues.
When the plant dies, this exchange stops. Carbon-14 has a half-life of
about 5730 years; it gradually decays away and becomes a smaller and
smaller fraction of the total carbon in the plant tissue.
This fraction can be measured, and the age of the tissue deduced.
Objects older than about 60,000 years cannot be dated this way – there
is too little carbon-14 left.
Other isotopes are useful for geologic time scale dating.
Uranium-238 has a half-life of 4.5 x 109 years, and has been used to date
the oldest rocks on Earth as about 4 billion years old.
Ex: The mass of carbon in an animal bone fragment found in an archeological
site is 200 g. If the bone registers an activity 16 decays/s, what is its age?
m  200 g
N t  16 s
N0
1
m
 C N C
14
6
 1.3  10
12
6
N
12
T 1  5730 y
2
   12
6 C   12.0 g
N A  6.02  10
 C   m C   m
14
6
12
6
 C   1.3  10
14
0 6
N
12
N
t 0  N 0  
N
 C  N
12
6
 C   1.3 10
12
6
12
m
A
NA

m

 1.00  10 25
 1.3  1013
ln 2
m
1.3  10 12 N A  50 s 1
T1

2
23
t ?
N t 
N t 0
 e  t

 N t 0  T12  N t 0 
t  ln 

ln 

  N t   ln 2  N t  
1
5730 y   50s 1 
3
t
 ln 

9
.
4

10
yr
1 
ln 2
  16s 
Stability and Tunneling
When a nucleus decays through alpha emission, energy is released.
Question: Why is it that these nuclei do not decay immediately?
Answer: Although energy is released in the decay, there is still an energy
barrier
The alpha particle can escape
through the barrier - quantum
mechanical phenomenon
called tunneling.
As stated in the Heisenberg
uncertainty principle, energy
conservation can be violated
as long as the violation does
not last too long:
The higher the energy barrier, the less time the alpha particle has to get
through it, and the less likely that is to happen. This accounts for the
extremely wide variation in half-lives for alpha decay.
```