The Atomic Nucleus
Bloomfield
Sections 14.1, 14.2,
and 14.3 (download)
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What is matter made of?
Physics is a reductionist science.
Beneath the surface, nature is simple!
All matter is composed of elements.
There are 91 stable elements, and about 20
more unstable elements which decay by
radioactive processes.
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1
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The Atom
How big is an atom?
How can we measure
the size of an atom?
What fundamental factors
determine the size of an atom?
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2
Jean Baptiste Perrin and Avogadro’s number N
Perrin used Brownian motion to measure
Avogadro’s number, in 1909. The theory
had been worked out in 1905 by Einstein.
Brownian motion is the random jittery
motion of a small particle suspended in a
fluid, due to collisions with the molecules of
the fluid. The properties of the motion
depend on temperature, molecular weight,
and number density. From the observations
Perrin could deduce the value of N. He got
the Nobel prize in 1926.
Today N is determined by X-ray scattering.
N = 6.02 x 1023 atoms
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How big is an atom?
The mass density of solid
carbon (diamond) is 3.52
g/cm3.
Calculate the
number density
The volume occupied by a
single carbon atom in
diamond is V = 1/n ; the
atomic radius R is related
to V by V = 4/3 π R3.
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n=
ρ
atomic mass
=
3.52 g/cm 3
12 g/mol
mol 6.02 × 10 23 atoms
= 0.293
×
cm 3
1 mol
23
= 1.77 × 10 atoms/cm 3
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1/ 3
3V ⎞
atomic radius = ⎛⎜
⎟
⎝ 4π ⎠
= 1.10 × 10 −8 cm
6
3
Result: The typical order of magnitude of an atomic
radius is 10−10 m = 0.1 nm.
What determines this size?
It comes from the quantum dynamics of electrons
interacting with the Coulomb force of attraction to
the nucleus.
Simplest example is a hydrogen atom.
potential energy V (r ) =
kinetic energy K =
− e2
4πε 0 r
p2
1 ⎛ h ⎞
=
⎜
⎟
2m 2m ⎝ 2πr ⎠
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Heisenberg
Uncertainty Principle
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kinetic energy
K(r) ∝ 1/r2
The state of lowest energy
has
radius = Bohr radius
h2 4πε 0
=
me 2 (2π )2
E(r)
V(r) ∝ −1/r
= 0.53 × 10 −10 m
potential energy
Full quantum theory – the Schroedinger equation.
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The Atomic Nucleus
The discovery of the nuclear
structure of the atom.
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Rutherford scattering
Scatter alpha particles (produced by a decay of
polonium) from a thin gold foil. The observation of
large-angle scattering implies that the atom has a very
small massive core — the nucleus.
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corresponding scattering angle
α – Au scattering
Rutherford calculated the
scattering rate based on
classical mechanics for the
scattering process,
assuming all the positive
charge of the Au atom is
concentrated in a nucleus.
varying impact
parameter
The result is perfect agreement
with the measured scattering rate.
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How big is the nucleus?
What determines its size?
From electron-nucleus scattering,
we know that the radius R of a
nucleus is approximately given by
the formula
R = (1.2 fm )A1/ 3
A = the number of nucleons
That is, A ∝ R3 ; because the nucleons are packed
pretty densely in the nucleus.
nuclear radius ~ 10−5 x atomic radius
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Dictionary
Nucleus: The massive and positively charged
particle at the center of an atom
Electron: The negatively charged constituent of atoms.
Proton: The positively charged constituent of nuclei.
Neutron: The electrically neutral constituent of nuclei.
particle
Charge
Mass
Electron
−e
9.109 381 88(72) × 10−31 kilogram
Proton
+e
1.672 621 58(13) × 10−27 kilogram
Neutron
0
1.674 92
x 10−27 kilogram
The electron is much lighter than the proton.
The neutron is slightly heavier than the proton.
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Isotopes
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Each nucleus is characterized by 2 numbers:
Z = number of protons
A = number of nucleons (protons and neutrons)
Chart of the Nuclides
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proton
neutron
Isotopes
hydrogen
1
1
H
3
1
He
4
2
H
3
2
helium
carbon
2
1
H
12
6
C
Z=1
He
13
6
C
Z=2
14
6
C
Z=6
Z=92
uranium
235
92
U
238
92
U
and others
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9
Radioactive decays
238
92
U →
Th + α
234
90
α decay
9
half - life = 4.5 × 10 yr
14
6
C →
14
7
N + e +ν
β decay
half - life = 5730 yr
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Half life
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We cannot predict with certainty when
an unstable nucleus will decay.
Radioactive decay is a random process like dice.
The half life τ is the mean
time for one-half of the
atoms in a radioactive
sample to decay.
Dating
t
N (t ) ⎛ 1 ⎞ τ
=⎜ ⎟
N (0 ) ⎝ 2 ⎠
" age" = t = τ ×
log[N (0) / N (t )]
log[2]
Measure N(t)/N(0), i.e., fraction of the unstable
isotope that remains at time t. The “age” is the
time that has passed since the death of the plant.
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THE STRONG NUCLEAR FORCE
α particle
= a nucleus of helium
P N
N P
What holds it together?
The electric force of two protons is a repulsion
(like charges repel). But the nuclear force
(between pp or pn or nn) is an attraction, and is
much stronger than the electric force.
What are the fundamental forces?
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