PHYS-E0460 INTRODUCTION TO REACTOR PHYSICS - SEPPO SIPILÄ - MYCOURSES

PHYS-E0460
Introduction to Reactor Physics

Seppo Sipilä

Basic concepts of
atomic and nuclear physics

Seppo Sipilä

Learning goal for today
 The learning goal of this
 lecture is to
 1) get acquainted with
 basic concepts of the
 atomic nucleus and its
 interactions with
 neutrons
 2) understand some key
 concepts, e.g. law of
 radioactive decay,
 binding energy, cross

 Ron Leishman
 sections, mean free
 path.

 PHYS-E0460 Introduction to Reactor Physics (2020) 3

Fundamental particles of matter
 Charge Mass [kg]

electron, e- -q 9.1096 × 10-31
proton, p +q 1.6726 × 10-27 (1836 me)
neutron, n 0 1.6749 × 10-27 (1839 me)
photon, g 0 0
neutrino, n 0 ~0. Six different types, of which the
 electron neutrino and electron antineutrino
 are of interest to nuclear engineering.
Furthermore: antiparticles
• same mass, opposite charge
• especially the positron e+ is important (emitted in b+ activity, many applications)

(q = elementary charge, 1.602 × 10-19 C)

 PHYS-E0460 Introduction to Reactor Physics (2020) 4

Fundamental interactions
• Strong interaction (“strong nuclear force”)
 – holds the quarks of hadrons together, also acts between hadrons
 (e.g. protons & neutrons, which are baryons) holding the atomic nucleus together
 – very short range (~10-15 m, the diameter of a light nucleus), gluon as force carrier.

• Electromagnetic interaction
 – acts between electrically charged particles (e.g. e, p)
 – infinite range, photon as intermediary particle (force carrier).

• Weak interaction
 – between “all” particles (quarks, leptons, neutrinos)
 – heavy W and Z bosons as force carriers (~80-90 proton mass)
 – extremely short range (~10-18 m, ca. 0.1% of nucleus diameter).
 – can change quarks or their flavor, e.g. cause of beta activity:
 n → p + e- + n (by quark: udd → uud + W- ; W- → e- + n)
 p → n + e+ + n (by quark: uud → udd + W+ ; W+ → e+ + n)

 PHYS-E0460 Introduction to Reactor Physics (2020) 5

Structure of the atom
• The atomic nucleus consists of nucleons, i.e. protons
 (Z of them) and neutrons (N of them). Z + N = A.
 – radius of the nucleus R  1,25 fm (10-15 m) * A1/3
• The nucleus is surrounded by a cloud of electrons
 – number of electrons = Z of the nucleus
 → neutral atom
 – radius of the order of 200 pm (10-12 m)
• Almost all of the atom’s mass (>99.9 %) is in the nucleus
 Note! Wrong proportions.
• The strong nuclear force binds the nucleons together, If this nucleus was the size of a
 but table tennis ball, the electron
 – electric repulsion of protons weakens the structure cloud would be ~3-4 kilometers
 away. Electrons are also very
 – neutrons act as “glue” through the strong interaction. light compared to the nucleons:
 me  0.00054mp.

 PHYS-E0460 Introduction to Reactor Physics (2020) 6

Elements and isotopes
• The numbers describing the nucleus are
 charge number Z = number of protons
 neutron number N = number of neutrons
 mass number A =Z+N

• The chemical properties of the elements depend on the electron cloud’s structure.
 – in other words, Z defines what element is in question.

• A nucleus of given Z can occur with several mass numbers A: isotopes
 – in other words, isotopes have a different number of neutrons, N
 – there are lots of unstable isotopes; typically 1-10 stable ones per element
 – e.g. oxygen: stable isotopes A = 16,17,18
 unstable isotopes A = 13,14,15 (b+);19,20 (b-)
 – e.g. hydrogen: stable isotopes A = 1,2 (hydrogen and deuterium)
 unstable isotope A = 3 (tritium, b-)

 PHYS-E0460 Introduction to Reactor Physics (2020) 7

Isotope notation

• E.g. oxygen: full notation

 mass number A state of ionization
 18 2-
 charge number Z
 8 O 2 number per molecule

 18
• Short notation (most typically used):
 O

 PHYS-E0460 Introduction to Reactor Physics (2020) 8

Atomic mass M
• Defined using the isotope 12C: M(12C) = 12 (exactly)
 – atomic mass M = 12 * (mass of the atom) / (mass of 12C atom)
 – for a naturally occurring isotope mix of an element, the atomic mass M = σ gi Mi
 • gi = share of isotope i atoms of all atoms (0…1)
 • Mi = atomic mass of isotope i
 – Similarly for molecules (e.g. compounds), the molecular mass M = σ Mi

• 1 mol = NA particles (atoms, molecules...)
 – NA = Avogadro’s number, 6.022 × 1023 Note! This picture of one
 oxygen atom and 16
 hydrogen atoms contains
 • mass of 1 mol = M grams a grave error; if we think
 of nuclei, there are three
 – 1 mol 12C ↔ 12 g (exactly) errors.

 – 1 mol 16O2 ↔ 31.99876 g 1) O-16 is not composed of 16 protons
 2) Can’t keep 16 individual protons on the scales due to Coulomb repulsion
 Amedeo Avogadro (1776–1856) 3) Even if we had 8 protons and 8 neutrons, the scales would not
 be in balance due to the mass defect (cf. slide 16)!

 PHYS-E0460 Introduction to Reactor Physics (2020) 9

Charged states of atoms and nuclei; radiation
• Atoms: the electrons around the nucleus
 are on electron shells (K, L, M etc.)
 – ground state of an atom: electrons are
 at lowest possible energy state (shell)
 – excited state: electron(s) at shell(s)
 higher than the ground state.
• Excited states are not stable: they return to
 ground state as an electron ”drops” to a Discharge of a nuclear excited state:
 L
 (1) g emission, or:
 lower shell. The excitation energy is (2) internal conversion
 (4) K
 (2) e
 released by photon emission (typically in (the energy is quantum-
 mechanically transferred
 e
 the visible or X-ray part of the spectrum). directly to an inner-shell
 electron)
 vacancy
• Nuclei also have internal energy states Internal conversion (2) leads
 to the emission of an X-ray e
 – the energy differences are large (MeV) quantum (3), or an equal
 (1)
 amount of energy is transferred
 – the excited state discharges by g to an outer-shell electron (4) g (3)
 X-ray
 which is ejected (Auger effect).
 emission or by internal conversion. quantum

 PHYS-E0460 Introduction to Reactor Physics (2020) 10

Radioactivity
• Elements heavier than calcium (Z > 20) have clearly more neutrons than protons in their nuclei
• these are needed as ”glue” to
 compensate for the electric
 repulsion between protons
 a emitters
• if there are too many or too
 few neutrons, the nucleus is
 unstable, i.e. radioactive.
• too few neutrons
 → b+ activity (e+)
• too many neutrons isotopes of tin (Sn)
 → b- activity (e-)
• nuclei heavier than lead
 (Z > 82) also have a activity
 (4He), which is a quantum-
 mechanical tunneling process
 governed by the strong
 nuclear force.

 PHYS-E0460 Introduction to Reactor Physics (2020) 11

Types of radioactivity
Alpha radiation:
• the emitted a particle (4He) has a discrete, isotope-specific energy
• mass number A diminishes by 4
 92 U → Th + 42 He
• charge number Z diminishes by 2. 238 234
 90

Beta radiation:
• the emitted electrons or positrons have a continuous energy spectrum
• a neutrino n (b+) or antineutrino n (b-) is also emitted
 55 Cs → 56 Ba + e + ν e
 137 137 _
• mass number A does not change
 +
• charge number Z increases (b-) or decreases (b+) by 1. 22
 11Na → 22
 10 Ne + e + νe
Gamma radiation (not really a decay reaction):
• in a and b emission, the daughter nucleus is often left in excited state, which
 typically discharges almost immediately by the emission of one or several g quanta.
• in some cases, the excited state lasts longer (> 1 ns) and is called metastable: these
 are isomeric states. The discharge of an isomeric state is called isomeric transition.

 PHYS-E0460 Introduction to Reactor Physics (2020) 12

Law of radioactive decay
For all radioactive decay processes, the decay
probability of a radioactive nucleus per unit l = 2.91610-13 1/s
time is a time-independent constant.
 – This constant is called the decay constant and l = 1.37210-11 1/s
 is denoted with the letter l [1/s].
Consider a radioactive isotope sample with n(t)
atoms left at time t.
 – over a differential time step dt, the number of
 atoms decaying is dn = λn(t )dt
 – The decay rate ln(t) [1/s] is called the sample’s
 activity: α (t ) = dn dt = λn(t )
The modern SI unit of activity is the Becquerel: 1 Bq = 1 decay/second.
The traditional unit is the Curie: 1Ci = 3.7 × 1010 decays/second.

 PHYS-E0460 Introduction to Reactor Physics (2020) 13

The activity of a sample vs. time
At time t, the rate of change in the number of nuclei in the sample is dn = − λn(t )dt .
So, dn(t ) dt = − λn(t ) . Integrating this we get

 a (kBq)
 80
n(t) = n0 e – λt
 70
 127g 99Tc (b-)

where n0 is the original number of atoms. 60
Multiplying each side by l we get the activity 50
 40
α(t) = α 0 e – λt
 . 30
 20
The time over which the sample’s activity
diminishes to half is called the half-life T1/2. 10
 0
A formula for T1/2 can be derived by writing 0 3 6 9 12 15 18 21 24 27 30
 time (d)
α0 / 2 = α0 e – λT1/2

and taking the logarithm of both sides, we get T1/2 = ln(2) / λ.

 PHYS-E0460 Introduction to Reactor Physics (2020) 14

Nuclear reactions
Nuclear reaction (strict definition): two nuclear particles (two nuclei, or a nucleus and
a nucleon, or a nucleus and a photon, i.e. a target and a projectile) interact, and two
or more nuclear particles and/or g quanta are formed.
Note: this excludes e.g. decay processes such as a, b and spontaneous fission.

Broader definition: any change in the state of the nucleus.

Four basic laws:

1. The number of nucleons is conserved
2. Charge is conserved
3. Momentum is conserved
4. Mass-energy is conserved

 PHYS-E0460 Introduction to Reactor Physics (2020) 15

Binding energy of a nucleus
When a nucleus is formed from nucleons, a few MeV e.g. 1H + n → 2H + g
of energy per nucleon is released: the nucleus is m(1H) + m(n)  m(2H) + 2.23 MeV
lighter than the summed-up mass of its free nucleons. /nucleon starts to diminish with heavier nuclei. Why?
 – mass defect  = Zmp + Nmn – mnucleus

 Binding energy c2 / nucleon (MeV, short notation /nucleon)
 – breaking up the nucleus to nucleons 235U
 requires E = c2 = binding energy 92Kr
 141Ba

 – even Z & N nuclei are the most stable.
 Approximative example: fission of U-235
In heavy nuclei, the electric repulsion
between protons is strong. 235U + n → 141Ba + 92Kr + 3n + g
 – c2/nucleon diminishes with growing A
 235 = 7.6 MeV/nucleon  235 nucleons = 1786 MeV
 – maximum c2/nucleon is found at the
 iron group (Z~26, A~50…60). 141 = 8.3 MeV/nucleon  141 nucleons = 1170 MeV
 92 = 8.7 MeV/nucleon  92 nucleons = 800 MeV
Production of nuclear energy is based sum: 1970 MeV
on releasing nuclear binding energy.
 The released energy is 1970−1786 MeV =184 MeV.
 – fission: by splitting heavy nuclei
 – fusion: by combining light nuclei

 PHYS-E0460 Introduction to Reactor Physics (2020) 16

Von Weizsäcker’s semiempirical
mass formula (1935)
Mass M of nucleus = NMn + ZMp – [aA – bA2/3 – gZ2 / A1/3 – z(A-2Z)2 / A – d ]. […] = Eb

 mass of neutrons Spin coupling term:
 surface effect – N Z d
 Coulomb
 correction to the even even 0
 force) the surface asymmetry term (Pauli’s
 tension. exclusion principle): as Constants
 (in MeV):
Based on the liquid drop model (Nucleons at the N/Z grows, Eb diminishes
 Mn = 939.573
of the nucleus (Gamow 1928), surface of the because fermions can’t Mp = 938.280
which treats the nucleus as an nucleus are less be in identical quantum a = 15.56
incompressible drop of liquid. state in the nucleus; b = 17.23
 tightly bound to it). g = 0.697
 some neutrons have to z = 23.285
Radius of the nucleus R ~ A1/3.
 be at higher energy shell d = 12.0

 PHYS-E0460 Introduction to Reactor Physics (2020) 17

Maxwell-Boltzmann distribution
 The most probable energy
If the system is not disturbed from the outside, the Ep can be calculated from the
distribution of atoms or molecules in a gaseous substance derivative’s zero dn/dE = 0:
will stabilize to thermal (Maxwellian) distribution. Ep = kT / 2.
 The average energy Ē is
If n(E) is the number density of particles at energy E, obtained by integration
then n(E)dE is the density of particles at E…E+dE. 
 1
 n0 0
 E= En(E)dE
In a Maxwellian distribution n(E) has the form

 n(E)
 3/2 = 3kT / 2.
 E  1 
n(E ) = n0  2   e − E / kT ,
 π  kT 
where n0 is the total density of particles, the Boltzmann
constant is k = 1.3806×10-23 [J/K], and T is the
temperature of the gas [K].

(This formula applies reasonably well also for liquids and
solids when T > 300 K.)
 E/kT

 PHYS-E0460 Introduction to Reactor Physics (2020) 18

Interaction of radiation with matter
Charged particles (a, b, ions): interactions with electrons (ionization); a positively charged
particle scattering from a nucleus can also cause a Coulomb excitation of the nucleus
(nuclear reaction, discharges by g emission).

 incident g photon dislodged electron
 (from outer shell)

 incident g photon incident g photon
 scattered g photon positron
 (E > 1.022 MeV)
 Photoelectron
 (in g energy range electron
 from inner shell)

 Pair production (interaction with the electric
 Photoelectric effect Compton scattering field of the nucleus)

 g photon
Gamma: absorption/scattering from electrons (ionization); slow

energetic g: pair production and photonuclear reactions, neutron

e.g. 25Mg(g,p)24Na; there’s also photofission.
 e.g. radiative
Neutrons: interaction only with nuclei. capture:

 PHYS-E0460 Introduction to Reactor Physics (2020) 19

Neutron interaction mechanisms
Scattering: Notation
 – elastic (excitation state of target nucleus does not change) AZ(n,n)AZ

 – inelastic (excitation of target nucleus) AZ(n,n’ )AZ*

Neutron capture: AZ(n,g)A+1Z

Charged particle reactions: AZ(n,p)AY,

 AZ(n,a)A-3X …

Neutron-producing reactions: AZ(n,2n)A-1Z

 AZ(n,3n)A-2Z ...

Fission: AZ(n,f)BX + CY + n n
 (A+1 = B + C + n)

 PHYS-E0460 Introduction to Reactor Physics (2020) 20

Atom density
In nuclear engineering it is often necessary to calculate the atom density N
(= number of atoms in unit volume) of an isotope:

N = r NA g / M [1/cm3] e.g. natural UO2:
 e.g. natural U:

 r density of substance [g/cm3] 18.9 g/cm3 10.5 g/cm3
 NA Avogadro’s number [1/mol] 6.022 × 1023
 g atom fraction of isotope [0…1] g = 0.0072 (235U)
 M atomic mass of substance [g/mol] 238.0289 270.0277

 N235 = 3.44 × 1020 N235 = 1.69 × 1020

Atom density is important e.g. in defining the probabilities of nuclear reactions caused
by neutrons, i.e. cross sections.

 PHYS-E0460 Introduction to Reactor Physics (2020) 21

Cross section v v
 v 1m2
• Neutron beam: density n [1/m3], velocity v [m/s]: v v
 v v
• Beam intensity: I = n v [1/m2s]
• Single atom as a target in a neutron beam: reaction rate r = s I [1/s]
 – s = microscopic cross section [m2]
 • “probability” of a reaction per atom (typical to each element)
 A
 • most common unit: barn (b). 1 b = 10-28 m2
• Target (many atoms) in a neutron beam:
 x
 – Target’s area A, thickness x, atom density N (i.e. NAx atoms altogether)
  reaction rate R = s I N A x [1/s] (assuming no multiparticle correlations)

• Reaction density F in the target (per unit volume): F = R V = R ( Ax ) = σ N I [1/m 3 s]

• Let’s define the macroscopic cross section Σ = σN [1/m]  F = Σ I
 Σ is the reaction probability per travelled unit distance in the target (or medium).

 PHYS-E0460 Introduction to Reactor Physics (2020) 22

Cross section of a mixture
 Y
• Medium with two atom species, X and Y: Y
 v v
 – cross sections σ X and σ Y [m2] v v
 X
 v Y
 – atom densities NX and NY [1/m3] v v Y
 – mixture’s Σ = NX σ X + NY σ Y [1/m] X
 Y

• Molecule XnYm: Y
 – molecular density N [1/m3] Y
 Y X
 – effective microscopic cross section Y X2Y4
 X
 σ = Σ / N = (nN σ X + mN σ Y ) / N = n σ X + m σ Y Y
 Y
 X
(assuming neutron interactions with X and Y to be independent
 – does not hold for elastic scattering of low-energy neutrons
 from molecules and solids)

 PHYS-E0460 Introduction to Reactor Physics (2020) 23

Cross sections for different reactions
• Scattering: elastic se (Se)
 inelastic si (Si)
• Radiative capture: sg (Sg)
• Fission: sf (Sf)
• Particle-producing reactions: sp, sa,... (Sp, Sa,...)

• Absorption cross section: sa = sg+ sf +sp+sa,... (Sa)
• Total cross section: st = sa +se +si (St)

For small neutron energies (that are of interest in most fission reactors)
 si = 0 (E < 1 keV) as neutron’s energy must exceed 1st excitation energy of target
 se = 4pR2 (E = 0.1 eV…10 keV) for potential scattering (no intermediate nucleus);
 there’s also resonance elastic scattering off heavy nuclei (with intermediate nucleus).
 σ γ  1 E  1 v (E < 1 eV) for most nuclei, called 1/v absorbers because of this.
 For many fissile nuclei also sf behaves like 1/v.

 PHYS-E0460 Introduction to Reactor Physics (2020) 24

Example: neutron absorption by 235U
Two possible absorption reactions (neutron’s kinetic energy E = 0.0253 eV):
 – radiative capture sg = 99 b
 – fission sf = 582 b

Absorption cross section
 sa = sg+ sf = 99 b + 582 b = 681 b

Probability of fission
 pf = sf / sa = 582 b / 681 b = 85,5 %

Probability of radiative capture
 pg = sg / sa = 99 b / 681 b = 14,5 %

 PHYS-E0460 Introduction to Reactor Physics (2020) 25

Energy dependence of cross sections
The cross sections vary Fission cross sections of uranium and plutonium
with neutron energy: resonance region

s = s(E) 1000
 100

 sf (b)
Many reactions where 10
the energies of the 1
reacting particles are
 10-8 10-6 10-4 10-2 1 10
small, are compound incident neutron energy (MeV)
nucleus reactions:
 A + B → C* → D + E e.g. 235U(n, g)236U: 235U + n → 236U* → 236U + g
• the reaction probability is increased if the projectile A’s kinetic energy + its binding
 energy in the compound nucleus C equals the energy of an excited state of C.
 – resonant structure (‘peaks’ in the cross section)
 – excited states with energies close to A’s binding energy are especially significant.

 PHYS-E0460 Introduction to Reactor Physics (2020) 26

Attenuation of neutron beam intensity
 x
in a target
 x=0
 I0
Let the original intensity of a monoenergetic I0e − Σt L
neutron beam be I0, and the thickness of the
target is L.
Denote by I(x) the intensity of those neutrons L
that have made it to depth x in the target
without interacting. (note: in reality also scattered
 neutrons would come through).
From there on, over the differential distance dx,
 The probability that a neutron at x
the intensity of the beam of such neutrons
 that hasn’t interacted yet will
decreases as dictated by the reaction density: interact during the next distance dx:
dI( x) = −Nσt I( x)dx = −Σt I( x)dx. dI( x )
 = Σt dx . So, Σ t is the probability
 I( x) of interaction per
By integrating this we get
 travelled unit distance.
 − Σt x
I( x) = I0e .
 PHYS-E0460 Introduction to Reactor Physics (2020) 27

Mean free path
 I( x )
The probability that a neutron makes it to x without interacting is = e −Σt x
 I0
Let’s calculate how far on average the neutron makes it in the target before
interacting. Let p(x)dx be the probability of the first interaction happening along the
differential distance dx after x.

So, p(x)dx is the conditional probability that
a neutron that has made it to x interacts p( x)dx = e − Σt x Σt dx .
along the next infinitesimal distance dx:
From this we get the average for x, i.e. the mean free path λ :
  

  xp( x)dx = Σt 
 −Σt x
λ= x = xe dx = 1/ Σt .
 x =0 x =0
 λ describes the average
So, we can also write I ( x ) = I0e − x/λ . distance travelled by the neutron
 in the medium before interacting.

 PHYS-E0460 Introduction to Reactor Physics (2020) 28