Chapter 2 Nuclear Reactor Physics

Ryan Schow

OBJECTIVES 1. Describe and provide basic characteristics of

fundamental atomic particles including charge and

approximate mass.

2. Describe the interactions of particles and atoms with

matter.

3. Describe radioactive decay.

4. Define microscopic cross section, macroscopic cross

section, and mean free path.

5. Describe the neutron moderation process and the

characteristics of reactor neutron flux.

OBJECTIVES

6. Define reaction rate.

7. Describe neutron activation and the characteristics of

radioactivity buildup in a reactor core.

8. Solve problems involving mass to energy conversion,

weight density, number density, weight fraction, volume

fraction, atom fraction, radioactive decay, and reaction

rate calculations to determine power level.

HELIUM ATOM

ELECTRONS

IN VARIOUS

ORBITS

PARTICLE MASS RELATIVE

CHARGE

PROTON 1.00727 AMU +1

NEUTRON 1.00866 AMU NO CHARGE

ELECTRON 0.00055 AMU -1

e

e N P

P N

STANDARD NOTATION FOR AN ATOM

A = Atomic Mass Number

(Number of protons

and neutrons) X = Element Symbol

Z = Atomic Number (Number of

Protons)

Where:

XAZ

Helium

Boron

Carbon

Oxygen

Uranium

Plutonium

He42

B115

C126

O168

U23892

Pu23994

ISOTOPES OF OXYGEN

NATURAL ABUNDANCE - ATOM PERCENT

16 17 18 8 8 8

O O O

8 PROTONS 8 NEUTRONS

8 PROTONS 9 NEUTRONS

8 PROTONS 10 NEUTRONS

99.757% 0.038% 0.205%

COULOMB AND NUCLEAR FORCES Like charged particles slightly separated will experience a

coulomb force of repulsion:

Nuclear forces of attraction are produced when adjacent

nucleons are involved:

ELECTRONS

- -

PROTONS

+ +

PROTONS

+ +

PROTON AND NEUTRON

+ n

NEUTRONS

n n

NEUTRON-TO-PROTON RATIO

0 20 40 60 80 100 120 140 0

20

40

60

80

100 N

UM

BE

R O

F P

RO

TO

NS

(Z

)

NUMBER OF NEUTRONS

(N=A-Z)

LINE OF

STABILITY

Z

N1

RADIATION TYPES ALPHA

NEUTRON

BETA MINUS

GAMMA

a

b

g

HELIUM NUCLEUS

ELECTRON

ELECTROMAGNETIC RADIATION

CHARGE: + 2 MASS: 4 AMU

CHARGE: 0 MASS: 1 AMU

CHARGE: -1 MASS: 1 AMU

1,800

CHARGE: 0 MASS: 0

He42

e01

n10

-

n

+ n +

n

EQUATION

A = activity or decay rate (decay over time or dps decay per second)

λ = decay constant (lower case Greek letter lambda) (s-1)

N = total number of atoms of the nuclide present in the sample

Where: NA

EQUATION

λ = decay constant (s-1)

N = total number of atoms of the nuclide present in the sample

N0 = total number of atoms of the nuclide that were present at time 0

t = total time elapsed since time 0 (s)

Where:

t

0eNN

EQUATION

21

21

21

21

21

21

t

693.0

t693.0

t693.5.ln

eln2

1ln

e2

1

eN2

NN

t

t

t

00

RADIOACTIVE NUCLIDE CONCENTRATION VS TIME (NO NEW PRODUCTION)

TIME (Half-lives)

100.00%

50.00%

25.00%

12.50%

6.25%

0.10% 0.20% 0.39% 0.78% 1.56% 3.13%

0.00%

25.00%

50.00%

75.00%

100.00%

0 1 2 3 4 5 6 7 8 9 10

Pe

rce

nt

of

Ori

gin

al

Nu

cli

de

Co

nce

ntr

ati

on

MASS DEFECT

Dm = mass defect (AMU)

Z = atomic number (number of protons)

mH = mass of H-1 atom (1.007825032 AMU)

A = atomic mass number (number of nucleons)

mn = mass of a neutron (1.008664923 AMU)

M = mass of the atom

Where:

Mm)ZA(Zmm nH D

EXAMPLE 1-1 Calculate the mass defect of the U-238 atom. Uranium

has an atomic number of 92.

M = 238.050785 AMU

Mm)ZA(Zmm nH D

050785.238

)008664923.1)(92238()007825032.1(92m

D

050785.2382651.1477199.92m D

AMU9342.1m D

MASS-ENERGY EQUIVALENCE

E = energy released (J or MeV)

m = mass (Kg or AMU)

c = speed of light (m/sec)

Where:

2mcE

MASS-ENERGY EQUIVALENCE

TWO FORMS OF THE SAME THING

Eq 1-7

Fig. 1-7 2mcE

D

AMU

MeV5.931AMUmMeVE

MASS ENERGY

MASS DEFECT

Mass of all of the individual particles is

greater than the mass of the combined

nucleus.

The difference is called the mass defect.

INDIVIDUAL

PARTICLES

COMBINED

NUCLEUS

p

p p

p

p n n

n

n

e

e e

e

n

e

EXAMPLE 1-2 Determine the energy equivalence of the mass defect

of a U-238 atom. Recall that the mass defect for a

U-238 atom is 1.9342 AMU.

D

AMU

MeV 5.931AMUmE

MeV7.1801E

5.9319342.1E

EXAMPLE 1-3 How much energy is released from the following fission event?

g n3XeSrnU 10139

54

94

38

1

0

235

92

8593.2350526.236m D

)]0087.1(39178.1389154.93[

)0087.10439.235(m

D

AMU1933.0m D

EXAMPLE (cont’d) Using mass to energy equivalence:

AMU mAMU

MeV5.931E D

1933)(931.5)(0. E

MeV1.180E

BINDING ENERGY OF A NEUTRON

Binding Energy of Neutron = 6.5 MeV

COMPOUND

NUCLEUS

TARGET

NUCLEUS

NEUTRON

Dm = (1.0087 + 235.0439) - (236.0456) = 0.0070 AMU

n

U23592 *U236

92

AMU mAMU

MeV5.931E D

MeV6.5 0070)(931.5)(0. E

EXAMPLE 1-4 Calculate the binding energy per nucleon for U-238.

MeV6.7238

7.1801

A

BE

binding energy per nucleon A

BE

BINDING ENERGY PER NUCLEON vs. MASS NUMBER

0

MASS NUMBER

25 50 75 100 125 150 175 200 225 250 2

3

4

5

6

7

8

9

BIN

DIN

G E

NE

RG

Y

PE

R N

UC

LE

ON

(M

eV

)

ELASTIC SCATTERING

n = NEUTRON

X = TARGET NUCLEUS

n

n

X

X

INELASTIC SCATTERING

X* = EXCITED TARGET NUCLEUS

n

n

X

g-ray

X * X

RADIATIVE CAPTURE

n

X

g-ray

Y * Y

Y = ISOTOPE OF TARGET NUCLEUS

Y* = EXCITED ISOTOPE OF TARGET

FISSION OF U-235

gray

FISSION

FRAGMENT 1

FISSION

FRAGMENT 2

n

n

n

gray U-236* U-235

MICROSCOPIC CROSS SECTION MODEL

“TARGET” ATOMS

NEUTRONS

CHARACTERISTIC ABSORPTION CROSS SECTION

SLOW

(THERMAL)

FAST INTERMEDIATE

(EPITHERMAL)

RESONANCE

PEAK sa

eV

MeV

DIFFERENTIAL ENERGY

10-2 1 10-1 10 102 103 104 105 106 107

10-8 10-6 10-7 10-5 10-4 10-3 10-2 10-1 1 10

CROSS SECTION RELATIONSHIP

sT

ABSORPTION

sa

TOTAL CROSS SECTION

sc sf FISSION CAPTURE

ss

SCATTERING

INELASTIC ELASTIC

ssi sse

MICROSCOPIC CROSS SECTIONS

saT sss

fca sss

sises sss

BORON-10 MICROSCOPIC CROSS SECTION VERSUS NEUTRON ENERGY

1

10

100

1,000

10,000

0.01 0.1 1 10 100 1,000 10,000 100,000

Neutron Energy (eV)

Micr

osco

pic C

ross

secti

on (b

arns

) M

icro

sco

pic

Cro

ss S

ectio

n (

ba

rns)

Neutron Energy (eV) 10-2 10-1 100 101 102 103 104 105

104

103

102

101

100

MACROSCOPIC CROSS SECTION

Where:

= macroscopic cross section (cm-1)

N = atomic density (atoms/cm3)

s = microscopic cross section (barns)

s N

MEAN FREE PATH

Where:

= mean free path (cm)

= macroscopic cross section (cm-1)

1

EXAMPLE 1-6 Given that the macroscopic cross section is = 0.5 cm-1.

Calculate the mean free path.

1cm5.0

1

cm2

1

FISSION LIQUID DROP MODEL 1) Neutron incident on

fissionable nucleus

NEUTRON

2) Neutron is absorbed and

energy of system is raised

by binding energy and

kinetic energy of neutron

4) Separation into one possible

fragment pair combination

3) Distortion of compound

nucleus

MODERATION

2 MeV

NEUTRON

Ef

Ei

COLLISION

0.025 eV

NEUTRON

COMPARISON OF MODERATORS

H2O 0.948 0.66 103

MATERIAL COLLISIONS

TO

THERMALIZE

s a

MODERATING

RATIO x

s s

MICROSCOPIC

CROSS

SECTION

(BARNS)

D2O 0.570 0.001 13.6

Be 0.209 0.0092 7.0

C 0.158

19

35

86

114 0.003 4.8

148

7,752

159

253

NEUTRON FLUX DESCRIPTION

seccm

neutrons thermal2

n

n

n

n

n

1 SQUARE

CENTIMETER

n

n

TRACK LENGTH DESCRIPTION OF NEUTRON FLUX

1 CUBIC

CENTIMETER

NEUTRON

DENSITY

NEUTRON

VELOCITY

NEUTRON

FLUX

seccm

NEUTRONS

sec

cm

cm

NEUTRONS23

FLUX DISTRIBUTION IN MODERATOR

NEUTRON ENERGY (eV)

FL

UX

INTERMEDIATE FAST

THERMAL 1 105

SLOW

REACTION RATE

Where:

R = reaction rate (reactions/cm3 sec)

N = atomic density (atoms/cm3)

s = microscopic cross section (cm2)

= neutron flux (neutrons/cm2 sec)

= macroscopic cross section (cm-1)

s NR

REACTOR POWER

Where:

P = thermal power output (MWt)

G = thermal energy produced per fission

(3.2 10-17 MWt sec/fission)

N = atomic density (atoms/cm3)

sf = microscopic fission cross section (cm2)

V = volume of the core (cm3)

= neutron flux (neutrons/cm2 sec)

s VGNP f