1. Criticality of Nuclear reactors and Geometric Buckling
Derek Smith
Eastern Illinois University

2. How do you make Nuclear energy safe?
1. Understand a basic reactor.
Step1: let’s first look inside a nuclear reactor

3. The inside of a basic nuclear reactor consists of a reactor core that contains the fuel rods usually fissionable Uranium or Plutonium. In the reactor core there also are control rods, control rods will be an important focus to maintaining reactor criticality which will be explained later. Outside the core is typically heavy water that generates steam, which then turns the turbines and then generates power. That steam is condensed again into water to be used again in the reactor process.

4. How to understand Nuclear Energy
1. Understand a basic reactor.
2. Understand how the control rods maintain reactor criticality.
Step1: let’s first look inside a nuclear reactor

5. Farther down= more neutrons absorbed & less heat
The control rods are used as neutron absorbers and the farther they go down the more they absorb and the temperature goes down as well. The farther up the control rods go there are less neutrons absorbed and a higher heat in the reactor core. These control rods are essential to “controlling” the criticality of a nuclear reactor
Farther down= more neutrons absorbed & less heat
Farther up= less neutrons absorbed & higher heat

6. This is a photo of a nuclear reactor core
This is a photo of a nuclear reactor core. The water is glowing blue because of Cherenkov radiation which is light emitted from electrons of the reactor core that move faster that the speed of light in water allowing their wavelengths to emit blue light.

8. How to understand Nuclear Energy
1. Understand a basic reactor.
2. Understand how the control rods maintain reactor criticality.
3. what is criticality?
Okay so now we’ve came to an understanding of what the nuclear reactor is, how the control rods in that reactor work, the third step is to understand exactly what criticality of a nuclear reactor is

9. Criticality
Criticality may be defined as the “attainment of physical conditions such that a fissile material will sustain a chain reaction”
Accidental criticality is the highest hazard a health physicist deals with.
This can be maintained with efforts to prevent accidental criticality with Criticality control or Nuclear safety
Accidental criticality- during the chain reaction the nuclei of the fissile material splits, thereby liberating tremendous amounts of energy in the form of radiation or producing large quantities of radioactive fission products

10. Accidental criticality
U nucleus
Alpha particle
This is an example of a chain reaction from accidental criticality from a small scale you can see that the original parent nucleus is unstable oscillating to a point where alpha particles as well as fast protons are released causing the same reaction in another U235 nucleus.

11. Uncontrolled chain reaction: from accidental criticality
This is a demonstration on a larger scale that shows how fast accidental criticality can get out of hand.

12. Criticality
Subcritical- if more neutrons are lost by escape or nonfission absorption than are produced, and the chain reaction isn’t self sustaining and dies out
Supercritical- a sustained chain reaction with the rate of fission neutron production exceed the rate of loss
Critical- when exactly one neutron per fission is available for initiating another fission.
Make all three larger

13. Sub-critical

14. Criticality
Subcritical- if more neutrons are lost by escape or nonfission absorption than are produced, and the chain reaction isn’t self sustaining and dies out
Supercritical- a sustained chain reaction with the rate of fission neutron production exceed the rate of loss
Critical- when exactly one neutron per fission is available for initiating another fission.
Make all three larger

15. Super-Critical
This is a demonstration on a larger scale that shows how fast accidental criticality can get out of hand.

16. Criticality
Subcritical- if more neutrons are lost by escape or nonfission absorption than are produced, and the chain reaction isn’t self sustaining and dies out
Supercritical- a sustained chain reaction with the rate of fission neutron production exceed the rate of loss
Critical- when exactly one neutron per fission is available for initiating another fission.
Make all three larger

17. Critical
This is a demonstration on a larger scale that shows how fast accidental criticality can get out of hand.

18. Criticality Control Accidental criticality depends on the following:
Quantity of the fissile material
Geometry of the fissile assembly
Presence or absence of a moderator
Presence or absence of a neutron reflector
Presence or absence of a strong neutron absorber (poison)
Concentration of fissile material, if the fissile material is in solution
Interaction between two or more assemblies or arrays of fissile material, each one of which is subcritical by itself. Consideration of this possibility is important in
the transport and storage of fissile materials.

19. Criticality control
Nuclear safety can be assured by limiting at least one of the factors that determines criticality
Mass control- limiting the mass of fissile material to less than the critical mass under any conceivable condition
Geometry control- having a geometric configuration that can never become critical because the surface-to-volume ratio is such that excessive neutron leakage makes it impossible to attain a multiplication factor as great as 1.
Concentration control- if the solution of fissile material is sufficiently dilute, absorption of neutrons by the hydrogen atoms makes a sustained chain reaction impossible. The degree of enrichment of 235U is important to this control.

20. How to understand Nuclear Energy
1. Understand a basic reactor.
2. Understand how the control rods maintain reactor criticality.
3. what is criticality?
4. Understanding Fission
Okay so now we’ve came to an understanding of what the nuclear reactor is, how the control rods in that reactor work, the third step is to understand exactly what criticality of a nuclear reactor is

21. Nuclear Fission: Uranium relation
Nuclei with odd numbers of nucleons are more easily fissioned than those with an even number of nucleons. For example which fissions after capturing a thermal neutron,
Whereas which can also capture a thermal neutron, is transformed into an even-odd nucleus and rids itself of its excitation energy by emitting a gamma ray

22. Nuclear Fission: fission fragments
When an atom fissions, it splits into two fission fragments plus several neutrons (the mean number of neutrons per fission of is 2.5) plus gamma rays according to the conservation equation:
An approximate distribution of this energy is as follows:
Fission fragments, kinetic energy MeV
Neutron kinetic energy
Fission gamma rays
Radioactive decay
Beta particle
Gamma rays
neutrinos
200 MeV
When that atom fissions it releases energy the following:

23. Nuclear Fission: Spontaneous fission
For the possibility of fission the following mass- energy relationship must hold:
This condition can only be met by isotopes whose atomic number and atomic mass are such that:
Although its likelihood is very small spontaneous fission (can cause accidental criticality) is very important in criticality control.
If an isotope : the nucleus is unstable toward fission and would undergo spontaneous fission.
This goes back to accidental criticality

24. Uncontrolled chain reaction: from accidental criticality
This is a demonstration on a larger scale that shows how fast accidental criticality can get out of hand.

25. Nuclear Fission: Rate of Fission
Most of the energy dissipated in the critical assembly is heat energy. Using a mean value of 190 MeV (million- electron volts) heat energy per fission, the rate of fission to generate one watt of power is calculated as follows:
This slide isn’t much detail it just shows a value as to how much energy is

26. How to understand Nuclear Energy
1. Understand a basic reactor.
2. Understand how the control rods maintain reactor criticality.
3. what is criticality?
4. Understanding Fission
5.Putting a value on criticality
Okay so now we’ve came to an understanding of what the nuclear reactor is, how the control rods in that reactor work, the third step is to understand exactly what criticality of a nuclear reactor is

27. Multiplication factor: The Four-Factor Formula
Criticality, also known as the value of Keff depends on the supply of neutrons of proper energy to initiate fission and also on the availability of fissile atoms.

28. Four factor Formula: Infinite Multiplication Factor
ηis the mean number of neutrons emitted per absorption of Uranium, so n thermal neutrons will result in ηn fission neutrons.
Є=fast fission factor with max value= 1.29
p=Resonance capture is called Resonance escape probability or p and is defined as the fraction of the fast, fission produced neutrons that finally become thermalized. The value of p depends on the ratio of moderator to fuel.
f=The fraction of the total number of thermalized neutrons absorbed by the fuel (including all the uranium) is called the thermal utilization factor, f

29. Reactivity and Reactor Control
Increase in the neutron multiplication factor >1 is called excess reactivity, defined by:
For neutrons in one generation, we have additional neutrons in succeeding generation. The time rate of change of neutrons is:
, is the lifetime of the neutron generation

30. Reactivity and Reactor Control
0.001s is the mean lifetime of a neutron from its birth to its absorption in pure 235U
When the excess reactivity is 0.1%, that is ∆k = 0.001, the reactor period is: T=0.001/0.001= 1s and the power level increases by a factor of e, or each second.
If ∆k were increased to 5% then: T= 0.001/0.005= 0.2s ,and the power lever increases in 1s would be by a factor of 150.
What is e

31. How to understand Nuclear Energy
1. Understand a basic reactor.
2. Understand how the control rods maintain reactor criticality.
3. what is criticality?
4. Understanding Fission
5.Putting a value on criticality
6.Multiplying medium
Okay so now we’ve came to an understanding of what the nuclear reactor is, how the control rods in that reactor work, the third step is to understand exactly what criticality of a nuclear reactor is

32. Reactor Physics Multiplying Medium
A multiplying medium is one in which fission, either thermal or fast or both, does occur.
= absorption
= fission
both terms have the same mathematical form cross section times a flux
Thermal- room temp. neutron having energy 0.25 eV fast neutrons have energy of 1MeV

33. How to understand Nuclear Energy
1. Understand a basic reactor.
2. Understand how the control rods maintain reactor criticality.
3. what is criticality?
4. Understanding Fission
5.Putting a value on criticality
6.Multiplying medium
7. Buckling
Okay so now we’ve came to an understanding of what the nuclear reactor is, how the control rods in that reactor work, the third step is to understand exactly what criticality of a nuclear reactor is

34. Bare Slab Reactor y = flux boundaries X Z Extrapolation distance (d)
Center line x y X Z
Draw diagram and small explanation phi= neutron flux. The size=a in the posotive x direction . All three directions can be equal or different in the y direction.
Extrapolation distance (d)
(-a/2-d) a/ a/ (a/2+d)

35. Buckling
The neutron diffusion equation for the bare slab reactor can be written as would be:       
      
in which B1 is called Buckling. Buckling is the measurement of extent to which the flux curves or "buckles".
buckling can be used to infer leakage. The greater the curvature the more leakage expected. For critical reactivity the material buckling should be equal to geometrical buckling.
Hence reactivity can be controlled with proper buckling incorporated in reactor’ s design.
(brief description of eigenvalue) Geometric buckling is a measure of neutron leakage, while material buckling is a measure of neutron production minus absorption. Thus, in the simplest case of a bare, homogeneous, steady state reactor, the geometric and material buckling must be equal.

36. How to understand Nuclear Energy
1. Understand a basic reactor.
2. Understand how the control rods maintain reactor criticality.
3. what is criticality?
4. Understanding Fission
5.Putting a value on criticality
6.Multiplying medium
7. Buckling
8. Determination of reactor’s critical dimension
Okay so now we’ve came to an understanding of what the nuclear reactor is, how the control rods in that reactor work, the third step is to understand exactly what criticality of a nuclear reactor is

37. Reactor’s critical dimension
Rearranging the Buckling equation we get:
We can then solve for Rex

38. Sources
Ferguson, C. D. (2011). Nuclear Energy- what everyone needs to know. New York, New York: Oxford University Press, Inc.
Cotton, S. (n.d.). Uranium Hexafluoride - UF6. Retrieved February 26, 2012, from chm.bris.ac.uk:
Hewitt, P. G. (2006). Conceptual Physics 10th edition. St. Petersburg: Pearson-Addison Wesley.
Moniz, E. (2011). Why We Still Need Nuclear Power. Foreign Affairs ,
  Nuclearfiles.org. (n.d.). from nuclear proliferation to nuclear testing. Retrieved February 25, 2012, from Nuclearfiles: project of the nuclear age peace foundation: