Rutherford, Geiger, Marsden
226 222 218
88 Ra → 86 Rn +α → α + 84 Po
rate of alpha scattering at angle θ from nucleus of charge
R(θ) ∝ 2 4
mα vα sin (θ/2)
Rutherford model (1911): Electrons orbit the nucleus like
planets orbit the Sun.
Bohr model of the atom (1913): Electrons stay in the
atom on special orbits (orbitals).
Experimentally veriﬁed by James Franck and Gustav
Ludwig Hertz in 1914. Atoms only absorb certain
“chunks” of energy.
principal quantum number: n = 1, 2, 3, . . .
e− most strongly bound at n = 1.
example: sodium (Na) has 11 electrons. In ground state,
2 electrons are in n = 1 level, 8 in n = 2, and 1 in n = 3.
V (r ) = −
4π 0 r
En = − eV
(Bohr formula, 1913)
hydrogenic (1 electron, Ze nuclear charge):
En = −
Besides n, we have orbital angular momentum
quantum number l.
l = 0, 1, 2, . . . , n − 1
letters: s, p, d, f, g, h, . . .
Then, there is spin quantum number s.
Quantum angular momentum
total angular momentum quantum number j:
values jump in integer steps:
|l − s| ≤ j ≤ l + s
Quantum angular momentum
for the electron, s = 1/2. if l = 1, what are possible values
s = 1/2 and l = 3?
What are all possible j values for electron in n = 4 level?
Proton (1919) was discovered by Rutherford.
Protos = ﬁrst
Chadwick’s Neutron Discovery
• Existence suggested since 1920 by Rutherford.
• Finally found via experiments in 1932.
4 Be5 +4 He2+
2 2 −→ 12
6 C +1 n1
or (α, n) reaction
mass: neutron 939.6 MeV/c2 ≈ proton 938.3 MeV/c2
Fast neutrons = high-energy neutrons. E > 1 eV.
Thermal neutrons = those with average thermal energy
corresponding to room temperature (T = 300 K).
Eth = kB T ≈ eV
where kB = 1.38 × 10−23 J/K.
Energy and Velocity
For a nucleon of kinetic energy 15 MeV, the velocity can
be calculated via
T = mv 2
2T 2 · 15
v= ≈c ≈ 0.18c
de Broglie wavelength of this nucleon is
h 4.1 × 10−21 MeV s
λ= = ≈ 7.3 fm
mv 938MeV c−2 · 0.18c
Electric ﬁeld far away does not know of particle’s
The electric ﬁeld form a wavefront consisting radial
(Coulomb) and transverse components.
q 2 a2
radiated power = P = Larmor’s equation
6π 0 c 3
binding energy of most nuclei ∼ 8 MeV/nucleon
electrons are bound at ∼ 10 eV to atoms.
removing a proton:
Z XN −→ Z −1 YN
removing a neutron:
Z XN −→ Z YN−1
Separation energy (S) is the difference between binding
energies (B) of initial nucleus and ﬁnal nucleus.
S > 0 when we change a stable nucleus (high B) into a
less stable nucleus (low B).
B = ( mconstituents − matom )c 2
S ≡ Bi − Bf
Sp = B(A XN ) − B(A−1 YN )
Z Z −1
Sn = B(A XN ) − B(A−1 YN−1 )
Subatomic particles can be described by quantum
States are represented by wave function ψ(x, t).
Particles = Wave packets = superpositions of waves.
Wave = non-localized state.
∆x · ∆p >
(Heisenberg uncertainty relation)
To get the wave function and its evolution, solve
i = − +V ψ
|ψ(x, t)|2 dx = 1
At any given time, the particle has to be somewhere.
x = ψ ∗ (x)ψ dx
p = ψ ∗ (p)ψ dx
de Broglie wavelength of a (non-zero mass) particle of
Experimental veriﬁcation: Davisson and Germer (1954).
Davisson and Germer used 54-eV electron beam to
scatter of a nickel crystal. An interference peak was
observed, similar to Bragg peak in x-ray diffraction.
∼ 1900: Blackbody radiation study led Planck to think
about nature of electromagnetic energy.
1905: Einstein proposed that light consists of photons,
each possessing a certain lump of energy.
Total energy = multiples of this number.
Planck-Einstein relation gives energy of a photon:
E = hν = ω =
ν and ω are frequency and angular frequency,
h = 6.63 × 10−34 J s = 4.14 eV s
for λ given in angstrom:
Characteristic radiation of atoms which has only certain
values are due to the fact that the atoms only exist in
certain stable states of discrete energies.
excitation (and de-excitation)
hν + Am ↔ An
ionization (and recombination)
hν + A ↔ A+ + e−
Fermions and Bosons
Protons, neutrons, and electrons belong to the fermion
Quarks and leptons are also fermions.
They have odd half-integer spins: s = 1/2, 3/2, 5/2, . . ..
Bosons have integer spin: s = 0, 1, 2, . . ..
examples: photons (s = ±1) and 4 He atoms (s = 0)
Electrons are identical fermions. At a given orbital
(n, l, m), only two electrons can occupy the same state
(one spin-up, one spin-down)
For each l, there are 2l + 1 values of ml . For each (l, ml ,
there is two spin states (ms = ± 1 ).
Exercise: What are maximum number of electrons for
l = 0, 1, 2, 3?
Periodic table shows an integer increase of protons and
electrons. Shells are ﬁlled, from low to high energies.
• H: (1s 1 )
• He: (1s 2 )
• Li: (He)(2s 1 )
• Be: (He)(2s 2 )
• B: (He)(2s 2 )(2p 1 )
t is time. N(t) is number of nuclei. λ is decay constant.
N(t) = N0e−λt
N0 = number of nuclei at the starting time.
decay constant is inversely proportional to the half-life:
A parent nuclide decays and yields a daughter nuclide.
increase in number of daughter (D) = decrease in number
of parents (P)
Df − Di = Pi − Pf
Decays aren’t always 1-to-1:
A → B (55% of the time)
→ C (40%)
→ D (5%)
For branched decays, the total decay constant is just the
sum of each mode constant:
λtot = λ1 + λ2 + λ3 + . . .
For a given decay constant λ, the lifetime of the state is
It is the time taken the state to drop from N0 to
N0 /e ≈ 0.37N0 .
λ1 + λ2 + . . .
A≡− = λN = −λN0 e−λt = A0 e−λt
A is also called “decay rate” or “disintegration rate.”
units: becquerel (1 s−1 ) or curie (3.7 × 1010 s−1 )
Henri becquerel discovered radioactivity from uranium ore
At Cambridge, Rutherford studied these unknown rays
and published results in 1899.
Those that got absorbed by a sheet of paper or a few cm
of air was named alpha rays.
The more penetrating ones were called beta rays.
Alpha (α) = 2p&2n bound state
Z XN −→ Z −2 YN−2 + 4 He2
Alpha emitters with large Q tend to have short half-lives.
ln λ(E) = a − b √
Geiger-Nuttall law. λ is the decay constant; a and b are
constants; Z is the atomic number; E is the decay energy.
W. Pauli: There must be a neutrino. (1930)
Cowan and Reines observed it. (1956)
n → p + e− + νe
¯ β − decay
p → n + e+ + νe β + decay (rare)
p + e− → n + νe e capture (ε)
234 234 −
90 Th144 → 91 Pa143 + e + νe
53m 53 +
27 Co → 26 Fe + e + νe
15 − 15
O+e → N + νe
Charged particles that decelerate create electromagnetic
radiation. This process is known as bremsstrahlung.
Photons can excite or ionize atoms.
Subsequent atomic transitions can produce additional
X-ray photons. This process is called X-ray
If an atomic electron absorbs such X-ray photon, it can be
ejected. These electrons are called Auger (oh-zhay)
A year after Rutherford discovered α and β rays, Paul
Villard discovered a more penetrating radiation from
radium. This is the gamma (γ) ray.
Excited nuclear states can decay via γ emission. Typical
energies ∼ 0.1 − 10 MeV.
43 Tc → 43 Tc + γ isomeric transition
27 Co → 60
28 Ni + e + νe + γ
¯ with β −
An excited nucleus can interact with an orbital electron,
transferring energy Eex .
The electron gets ejected with energy
Ee = Eex − Eb
where Eb is the binding energy of the electron.
The gamma decay and internal conversion decay
contribute to total decay probability:
λ = λγ + λe
quantity description units
activity (A) decay rate curie (Ci), becquerel (Bq)
exposure (X ) air ionization roentgen (R), coulomb/kg
absorbed dose (D) absorbed energy rad, gray (Gy)
dose equivalent (DE) bio. effects rem, sievert (Sv)
1. What kind of radiation does not come from a
nucleus? [choices: α, β, x-ray, γ]
2. Be-7 decays by capturing an electron. What is the
3. 15.1% of natural samarium is 147 Sm, which decays by
emitting α. 10 grams of natural samarium gives 120 α
per second. Calculate activity per gram of 147 Sm.
Reaction Cross Section
a + X −→ Y + b
ﬂuxincident · densitytarget
rate of detecting b
(ﬂux of a) · (X areal density)
Suppose you want to use a short-lived nuclide produced
from a reactor. But you are far away from the reactor.
What can you do?
Prepare the parent nuclide which has longer half-life, in a
device that can separate the daughter from the parent.
22 Ti (t1/2 = 6 y) ⇒ 21 Sc (t1/2 = 3.9 h)
37 Rb (86 d) ⇒ 36 Kr (1.8 h)
42 Mo (66 h) ⇒ 43 Tc (6 h)