Introduction to Reactor Physics

Introduction to Reactor Physics

Vasily Arzhanov

Reactor Physics, KTH

Course Objectives

• Derive and Solve Equations Describing

Multiplying Media in Several Approximations.

• Evaluate Important Reactor Parameters Including Performance and Safety.

• Describe and Compare Various Reactor Designs.

• Characterize Various Fuel Cycles.

• Represent Various Waste Management Strategies.

• Use Industry Adopted Soft- and Hardware for Evaluating Basic Reactor Parameters.

Having finished the course, you will be able to:

Nuclear Engineering

• Nuclear Engineering is an endeavor that makes use of radiation and radioactive material for the benefit of mankind.

• Like their counterparts in chemical engineering, nuclear engineers endeavor to improve the

quality of life by manipulating basic building blocks of matter.

• Unlike chemical engineers, nuclear engineers

works with reactions that produce millions of

times more energy per reaction than any other

known material.

Nuclear Energy

• It is free from the problems of fossil fuels: greenhouse gas emissions.

• A typical 1000 MW coal-burning plant emits yearly:

– 100 000 tons of

SO

– 75 000 tons of

NO

– 5 000 tons of fly ash

• USA generates 20% of the electricity at NPP; it avoided in 1999 the emission of 150 million tonnes of CO

• On contrary, there is still the association of nuclear

power with the tremendous destructive force.

Use of Nuclear Energy

• Energy generation (electricity, heating)

• Propulsion of naval vessels

• Nuclear-powered spacecraft

• Production of radioisotopes

• Activation analysis

The International System of Units

• Le Système International d‘Unités.

• Abbreviated as SI.

• World’s most widely used in everyday life, commerce and science, notable

exceptions: US and UK.

• SI was developed in 1960 from metre- kilogram-second, MKS, rather than

centimetre-gram-second, CGS.

Base Units

Base quantity Name Symbol

Length meter m

Mass kilogram kg

Time second s

Electric current ampere A Thermodynamic temperature kelvin K

Amount of substance mole mol

Luminous intensity candela cd

Some Derived Units

Derived quantity Name Symbol Other SI units Base units

Plane angle radian rad – m·m^-1 = 1

Force newton N – m·kg·s^-2

Energy, work, heat joule J N·m m²·kg·s^-2

Power watt W J/s m²·kg·s^-3

Electric charge coulomb C – s·A

Electric potential volt V W/A m²·kg·s^-3·A^-1

Celsius temperature degree ºC – K

Activity becquerel Bq s^-1

Absorbed dose gray Gy J/kg m²·s^-2

Dose equivalent sievert Sv J/kg m²·s^-2

SI Prefixes

Factor Name Symbol Factor Name Symbol

10²⁴ yotta Y 10^-1 deci d

10²¹ zetta Z 10^-2 centi c

10¹⁸ exa E 10^-3 milli m

10¹⁵ peta P 10^-6 micro μ

10¹² tera T 10^-9 nano n

10⁹ giga G 10^-12 pico p

10⁶ mega M 10^-15 femto f

10³ kilo k 10^-18 atto a

10² hecto h 10^-21 zepto z

10¹ deka da 10^-24 yocto y

Accepted Non-SI Units

Name Symbol Value in SI

minute (time) min 1 min = 60 s

hour h 1 h = 60 min = 3600 s

day d 24 h = 86400 s

degree (angle) º 1º = (π/180) rad

minute (angle) ´ 1´ = (1/60)º = (π/10800) rad second (angle) ˝ 1˝ = (1/60)´ = (π/648 000) rad

liter L 1 L = 1 dm³ = 10^-3 m³

tonne t 1 t = 1000 kg

electronvolt eV 1 eV = 1.60218×10^-19 J unified atomic mass unit amu, u,

m_u

1 u = m_u = 1.66054×10^-27 kg

atomic unit of mass m_e m_e = 9.109382×10^-31 kg Astronomical unit ua 1.49598×10¹¹ m

Currently Accepted Non-SI Units

Name Symbol Value in SI

nautical mile 1852 m

bar bar 1 bar = 0.1 MPa = 100 kPa = 10⁵ Pa ångström (angstrom) Å 1 Å = 0.1 nm = 10^-10 m

barn b 1 b = 100 fm² = 10^-28 m² = 10^-24 cm²

curie Ci 1 Ci = 3.7×10¹⁰ Bq

roentgen R 1R = 2.58×10^-4 C/kg

rad rad 1 rad = 1 cGy = 10^-2 Gy rem rem 1 rem = 1 cSv = 10^-2 Sv 1 kWh = (1000 W)×(3600 s) = 3.6×10⁶ J

1 mmHg = 1 Torr = (1/760) atm = 133.322 Pa

Some Units in Reactor Physics

Very often, centimeter will be used rather than meter.

Mass density in g/cm³: ρ_w ≈ 1 g/cm³

Number density: #/cm³ : n = 10¹² n/cm³; N = 10²⁴ atom/cm³ Velocity in m/s: v_th = 2200 m/s

Energy in eV: ε_f ≈ 200 MeV per 1 fission of ²³⁵U

Fundamental Particles

Particles of Interest to Nuclear Engineering

• Electron, e or e^-

• Positron, e⁺

• Neutrino, ν, ν_e

Classified as leptons

(λεπτος – small, thin) do not

experience the strong interaction.

• Neutron, n

• Proton, p

Classified as hadrons, any

strongly interacting composite subatomic particles (composed of quarks):

Fundamental Constants

http://physics.nist.gov

Name Symbol = Value

Speed of light c = 299 792 458 m/s

Planck constant h = 6.626 068 96 × 10^-34 Js Elementary charge e = 1.602 176 487 × 10^-34 C

Electron mass m_e = 9.109 382 15 × 10^-31 kg Neutron mass m_n = 1.674 927 211 × 10^-27 kg

Proton mass m_p = 1.672 621 637 × 10^-27 kg Atomic mass const. m_u = 1.660 538 782 × 10^-27 kg Avogadro constant N_A = 6.022 141 79 × 10²³ mol^-1 Boltzmann constant k = 1.380 6504 × 10^-23 J K^-1

Molar gas constant R = 8.314 472 J mol^-1 K^-1 Universal

Electromagnetic

Atomic and nuclear

Physico-chemical

Atoms and Nuclei

Atoms (ατομος - indivisible) are building blocks of gross matter.

Simplified helium model

Atomic number, Z, is the total number of protons.

Neutron number, N, is the total number of neutrons.

Atomic mass number, A, is the total number of nucleons.

A = Z+N Q = Ze

4 4

He ≡

He

α-particle

α

≡

4 +2

2

He

Nuclides and Isotopes

Oxygen has three stable isotopes, ¹⁶O, ¹⁷O, ¹⁸O and five known unstable, ¹³O, ¹⁴O, ¹⁵O, ¹⁹O and ²⁰O.

Isotope Abundance

16O 99.8 %

17O 0.037 %

18O 0.204 %

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

≡ ×

+ +

≡ ×

+ +

D

D

17 17

16 17 18

17 17

16 17 18

O O 100%

O O O

O O 100%

O O O

a N

N N N

w m

m m m

In nature:

A particular value of (Z) defines a chemical element.

A particular value of (Z,N) defines a nuclide.

Each nuclide (Z,N) is considered an isotope of the corresponding chemical element (Z).

By extension, two nuclides with the same Z value are called isotopes of each other.

Some Important Nuclides

Z Nuclide Abundance a/o Half-life

0 n 12 m

1

1H

2H

3H

99.985 0.015

12.33 y

5 ¹⁰B

11B

19.6 80.4

6

12C

13C

14C

98.89 1.11

5736 y

92

234U

235U

238U

0.0025 0.72 99.27

2.46 ×10⁵ y 7.04 ×10⁸ y 4.68 ×10⁹ y

Example

Z Nuclide Abundance a/o Half-life

0 n 12 m

1

1H

2H

3H

99.985 0.015

12.33 y

5 ¹⁰B

11B

19.6 80.4 6

12C

13C

14C

98.89 1.11

5736 y 92

234U

235U

238U

0.0025 0.72 99.27

2.46 ×10⁵ y 7.04 ×10⁸ y 4.68 ×10⁹ y Problem.

A glass of water is known to contain 6.6 ×10²⁴ atoms of hydrogen.

How many atoms of deuterium (²H) are present?

Solution.

Isotopic abundance of ²H is 0.015 a/o. The fraction of ²H is therefore 1.5 ×10^-4. The total number of ²H is then

1.5 ×10^-4 × 6.6 ×10²⁴ = 9.9 ×10²⁰

Unified Atomic Mass Unit

"The AME2003 atomic mass evaluation (I). Evaluation of input data, adjustment

procedures". A.H. Wapstra, G. Audi, and C. Thibault. Nuclear Physics A729, 129 (2003).

"The AME2003 atomic mass evaluation (II). Tables, graphs, and references". G. Audi, A.H.

Wapstra, and C. Thibault. Nuclear Physics A729, 337 (2003).

Let m(¹²C) be the mass of neutral ¹²C.

Arbitrarily, we set m(¹²C) = 12 u.

( )

u 6

1u 1 C

m ≡ ≡ 12 m

Atomic Weight

The atomic weight of an atom is the mass of the neutral atom expressed in atomic mass units.

( ) ( ) ( ) ( )

u 6

X X

X 12

C

A A

Z Z

A Z

m m

M ≡ m = × m

(unitless number!!)

( )

^X ( )

^X

m = M × m

The mass of any atom in amu is numerically equal to the atomic weight of atom in question.

( )

^{X ≈}

M A

In practice, it is acceptable:

Atomic Weight of Mixtures

The atomic weight of an element is the average atomic weight of the mixture.

i i i

M ≡ ∑ γ M

M_i is atomic weight of the ith isotope.

Isotope Abundance [a/o] Weight

16O 99.8 % 15.99492

17O 0.037 % 16.99913

18O 0.204 % 17.99916

M(O) = M(O_nat) = 15.99938

Molecular Weight

The total mass of a molecule relative to the mass of neutral

C is called the molecular weight.

To a very good precision, the molecular weight is merely the sum of atomic weights of the constituent atoms.

M(O

) = 2 × 15.99938 = 31.99876

Gram Atomic Weight

Atomic and molecular weights are unitless numbers.

By contrast, gram atomic (molecular) weight is defined as the amount of a substance having a mass, in grams, equal to the atomic (molecular) weight of the substance.

This amount (number of entities) of material is also called a mole.

Thus 1 g.a.w. or 1 mole of

C is exactly 12 grams of this isotope.

1 mole of natural O

≈ 31.99876 g.

Avogadro’s Number

The number of structural elements in one mole

( )

( )

( ) ( )

X X g =

X 10 kg X X

A A

Z Z

A A A

Z Z Z

M

M ⁻ m N M m N

≡

= × = ⋅ = ⋅ ×

The mass of 1 mole of

23 u

0.001 kg #

6.02214179 10

A mol N N

m

⎡ ⎤

= ≡ ≈ × ⎢⎣ ⎥⎦

One mole of any substance contains the

same number of entities, namely N

.

Mole as Unit

A mole is the amount of substance of a system, which contains as many

elementary entities as there are atoms in 0.012 kilogram (or 12 grams) of ¹²C, where the carbon-12 atoms are unbound, at rest and in their ground state.

According to the SI, the mole is not dimensionless, but has its very own dimension, namely "amount of substance", comparable to other dimensions. The SI

additionally defines the Avogadro constant as having the unit reciprocal mole.

The mole (symbol: mol) is the SI base unit that measures an amount of substance. The mole is a counting unit. A mole is much like "a dozen."

Mole

Subatomic

Atomic

(Neutrons, protons, electrons, photons)

(Neutral atoms, ions)

Atomic Radii

Atomic radius is not a precisely defined physical quantity, nor is it constant in all circumstances. The value assigned to the radius of a particular atom will always depend on the definition chosen for "atomic radius", and different definitions are more appropriate for different

situations.

Except for a few of the lightest elements, these average radii are approximately the same for all atoms, about 2 × 10^-10 m.

A reasonable definition is an average distance

Radii and Periodic Table

Nuclear Radii

Various types of scattering experiments suggest that nuclei are roughly spherical and appear to have essentially the same density. The data are summarized in the expression called the Fermi model:

1 3 0

-15

0

1.25 fm = 1.25 10 m r r A

r

=

= ×

The constant density, V ~ A, suggests that nuclei are similar to liquid drops.

Mass and Energy

Mass and energy are equivalent and convertible, one to the other.

2 0

E

= m c

Complete annihilation of m

releases

1 g → E = 9×10

J = 25×10

kWh.

2

0.511 MeV

m c

=

2

u

931.5 MeV

m c =

Particles in Motion

0

2 2

= 1

− m m

v c

2

2 2 2

0 0 2 2

1 1

1

= = +

⎛ ⎞

= − = ⎜ − ⎟

⎜ − ⎟

⎝ ⎠

tot rest k

k

E mc E E

E mc m c m c

v c

2 0

1 when

≈ 2 

E

1 m v

v c 2 when

≈ 

E

m v v c

p = mv

Apparent mass:

Total energy:

Kinetic energy:

Relativistic Effects

Electrons: E

≥ 10 keV (relativistic formula should be used).

Neutrons: E

≤ 20 MeV (classical formula may be used).

2

0

2

k

=

E m v is accurate enough when v ≤ 0.2 c or E

≤ 0.02 E

[ ] [ ]

2

2 eV = 1.383 1 0

m s

=

→ ×

E m v v E

Particle Wavelengths

Einstein:

Planck:

( )

( )

2 2

0

m

E

E mc pc m c p

c

= = + ⎯⎯⎯

→ =

E_k E_rest

hc h E

E h ν c

λ λ

= = ⎯⎯ → =

h

λ = p

Photon:

h h p mv

λ = =

De Broglie:

Neutron Wavelength

2

h λ = m E

tot rest

hc E E λ =

−

Non relativistic: Relativistic:

2.86 10

λ = ^× E

cm eV

Ionization

Pb (Z = 82)

The process of removing an electron from an atom is called ionization.

7.38 eV

88 keV

hc

λ = E 1.24 10

λ = E ^×

m eV

6

1.24 10 11

1.409 10 m 88000 7.38

λ = ^{× −} = × ⁻

−

Atomic Excited States

• In a neutral atom, the electrons can be in a variety of different orbits or states.

• The state of lowest energy is the ground state.

• When the atom possesses more energy, it is said to be in an

excited state or energy level.

• The highest energy state

corresponds to the situation in which the electron has been completely removed from the atom and the atom is ionized.

Energy, eV

10.19 12.07 13.58

0

Nuclear Excited States

• Nucleons in nuclei are also moving in various orbits.

• The orbits are not as well defined and

understood.

• In any case, there is a state of lowest energy, ground state; except for very lightest nuclei, all nuclei have excited states.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Energy, MeV

12C Energy levels

Number (Atom) Density

Mass # Mole Mass

Volume = Volume × # × Mole

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

X X X

X

X X

X X

ρ

ρ ρ

⋅ ⋅

= ≈

⋅ ⋅

= ≈

A A A

Z Z Z

A Z

A A

A A

Z A Z A

A

Z A

N M N A

N N

N N

N M A

Example 1

For water of normal (unit) density, compute:

• the number of H₂O molecules per cm³;

• the atom densities of hydrogen and oxygen;

• the atom density of ²H.

Isotope Abundance

1H 99.985 a/o

2H 0.015 a/o

Solution 1

The atom weights: M_H = 1.00794 M_O = 15.9994

The molecular weight of water is M = 2×M_H + M_O = 18.0153 (natural mix.)

( ) ( )

( )

22 2

22 2

2 4 19

H O 3.343 10 O

(H) 2 H O 6.69 10

( H) 1.5 10 (H) 1.0028 10 ρ

−

= ⋅ = × =

= × = ×

= × × = ×

NA

N N

M

N N

N N

Chemical Composition

Let a substance be given by a chemical formula, X_mY_n, for example Fe₃O₄.

X X

X Y

w mM

mM nM

= +

Then the atom density of X or Y is and

m n m n

X X Y Y X Y

N = ⋅m N N = ⋅n N

The weight fraction of X is easily evaluated as

Weight Percent

Usually, the components of mixtures are given in percent by weight. Let ρ be the physical density of the mixture. Then the density of ith component is

i

w

ρ = ρ

i i

N w N

N M M

ρ ⋅ ρ

= =

to be compared with the case of

isotopic abundance. i ^{i A}

N N

M

= γ ρ

Enrichment in Weight Percent

Isotope Abundance, a/o

234U 0.0025

235U 0.72

238U 99.27

Natural uranium, U_nat must be enriched in ²³⁵U.

Often, we disregard

It is the practice to specify enrichment in weight percent.

The atomic weight of the enriched uranium may be evaluated as follows Total number of U atoms in cm³ _i

i

i A

A

i i

w N

N N

N M M

ρ

ρ _{= = =}

∑ ∑

1 _i

i i

w M =

∑

Atomic weigh of mixture Atomic weigh of ith isotope

Example 2

1) How much ²³⁵U is in the reactor?

2) What are the atom densities of ²³⁵U and ²³⁸U in the rods?

A reactor is fueled with 1500 kg of uranium rods enriched to 20 w/o in ²³⁵U.

The remainder is ²³⁸U. The density of uranium is 19.1 g/cm³.

The atomic weights of ²³⁵U and ²³⁸U are 235.0439 and 238.0508

Solution 2

1) 20 w/o means the that 20% of the total uranium mass is ²³⁵U. The amount of ²³⁵U is therefore 0.20×1500 kg = 300 kg.

2) The atomic weights of ²³⁵U and ²³⁸U are 235.0439 and 238.0508

23 21

235 235

235

23 238 22

238

238

0.20 19.1 6.022 10

9.79 10 235.0439

0.80 19.1 6.022 10

3.86 10 238.0508

ρ

ρ

× × ×

= = = ×

× × ×

= = = ×

A

A

w N

N M

w N

N M

Example 3

The fuel for a reactor consists of pellets of uranium dioxide (UO₂) which has a density of 10.5 g/cm³. The uranium is enriched to 30 w/o in ²³⁵U.

What is the atom density of uranium ²³⁵U in the fuel?

The atomic weights of ²³⁵U and ²³⁸U are 235.0439 and 238.0508

UO₂

Fuel pellet:

1 cm

2 cm U = ²³⁸U+²³⁵U

( )

( )

( ) ( )

2 3

235

235 238

UO 10.5 g cm

U 30%

U U

m

m m

ρ =

+ =

Solution 3

( )

2

235 235

235

235 235 U

U 2

U U

U

UO U O

235 238

U 235 238

UO

2 1

ρ

ρ ρ

ρ ρ

=

=

=

= =

+

= +

A

U

N N

M w w

M M

w M M M

w w

M M M

Maxwellian Distribution

In a gas being at thermal equilibrium, the energies of atoms or molecules are distributed according to the Maxwellian distribution function.

Let N(E) be the density of particles per energy.

( )

N E dE = number of particles per unit volume having energies in dE about E.

( )

( ) 2 N ^{E kT}

N E E kT e

π kT

= −

23 5

1.3806 10 J °K 8.6170 10 eV °K

k = × ⁻ = × ⁻

Boltzmann’s constant:

T is the absolute temperature in ^oK

For solids and liquids, the energy distributions are more complicated.

However, to a first approximation this formula is also valid for solids and liquids. But the parameter T differs somewhat from the actual

0

( ) N N E dE

∞

=

∫

Distribution Function

0 0.5 1 1.5 2 2.5 3 3.5 4

E kT

N(E)

Ep

kT

E kT

0

0

0

1 2

1 3

( ) 2

293.61 °K

0.0253 eV 1 eV 40 Ep kT

E EN E dE kT

N

T kT

∞

=

= =

=

= ≈

∫

Gas Law

M

M A

A

PV n RT

n N R

P T

V N

P NkT

=

⎛ ⎞

⎛ ⎞

= ⎜ ⎝ ⎟⎜ ⎠⎝ ⎟ ⎠

=

Ideal gas:

The END