Physics 215
Winter 2003
Prof. Ioan Kosztin
Introduction to Modern Physics
Lecture #27
Nuclear Physics
! Properties of nuclei
! Nuclear Magnetic Resonance and
Magnetic Resonance Imaging (MRI)
! Binding energy and nuclear forces
! Nuclear models
! Radioactivity
Atomic Nucleus
d ~ 10−14 − 10−15 m
nucleon = proton or neutron
d = d 0 A1/ 3
Z protons
A
Z
Z=atomic number
X
N=A-Z neutrons
A=mass number
m p ≈ mn ≈ 1u
M nuc ≈ Au
1u = 1.66 ×10−27 kg =
−4
me ≈ 5 ×10 u
m12C
12
1
Nuclear Stability
Nuclei are held together by
very short range (~2 fm),
very strong attractive
nuclear forces acting
between the constituent
nucleons
(nuclear forces are much
stronger than EM forces!)
line of stability
Nuclear Forces
! one of the 4 fundamental forces in nature
(are “charge-blind”)
! very strong and always attractive
! extremely short range (r~10 -15 m)
! saturated (i.e., a nucleon interacts with a
limited number of other nucleons)
! are mediated by exchange of massive
particles called mesons (Yukawa, 1934)
2
Nuclear Spin and Magnetic Moment
LI = I ( I + 1) !
M I = I !, ( I − 1)!,..., − I !
 e! 
µI = 
L
 2m p  I
"#$
= µn
nuclear
magneton
Potential energy in magnetic field
U = − µI ⋅ B = − µ n B M I
Nuclear Magnetic Resonance (NMR)
• Splitting of the nuclear energy levels in magnetic field
• Quantum (resonant) transitions between magnetic
nuclear energy levels
3
Nuclear Magnetic Imaging (MRI)
strong
permanent
magnet
(2-5 T)
MRI signal ~ 10-7 eV
X-ray signal ~ 105 eV
MRI image →
Nuclear Binding Energy
Eb = ( Zm p + Nmn − M A ) c 2
= ( Zm p + Nmn − M A ) × 931.5 MeV / u
4
Binding Energy per Nucleon
Nuclear Models
Liquid-Drop Model
Eb = C1 A − C2 A
2/3
Z ( Z − 1)
( N − Z )2
− C3
− C4
A1/ 3
A
semi empirical formula for the binding energy
Independent-Particle Model
protons and neutrons are
fermions which in nucleus are
filling up single particle states
in the order of increasing
energy (similarly to electrons
in atoms)
5
Radioactivity
Spontaneous emission of nuclear radiation by nuclei
Radiation Detectors
6
Law of Radioactive Decay
dN
= −λ N
dt
decay
constant
N (t ) = N 0 e− λt
R=
dN
= R0e − λt
dt
decay rate
T1/ 2 = ln 2 / λ
half time
1Ci ≡ 3.7 × 10 Bq (decay / s)
10
Radioactive Decay
Alpha decay
A
Z
X→
disintegration energy:
A− 4
Z −2
Y + 24 He
Q = ( M X − M Y − M α )c 2
(Application: smoke detector)
Beta decay
Y + e− +ν
A
Z
X→
A
Z +1
A
Z
X→
A
Z −1
Y + e+ +ν
n → p + e− +ν
Gamma decay
A
Z
X * → ZA X + γ
7
Carbon Dating
C → 147 N + e − +ν
14
6
T1/ 2 ≈ 5, 730 years
12
6
C is a stable carbon isotope
N ( C)
≈ 1.3 × 10
in the atmosphere
N ( C)
14
6
−12
12
6
age of the sample
t = T1/ 2 ln ( Ratmosphere / Rsample )
Problem
A freshly prepared sample of a certain radioactive isotope
has an activity of 10 mCi. After 4 h, its activity is 8 mCi.
(a) Find the decay constant and half-life.
(b) How many atoms of the isotope were contained in the
freshly prepared sample?
(c) What is the sample’s activity 30 h after it is prepared?
8
Problem
9