1. Reactor physics Reactor training course Institut für Kernchemie

2. Binding energy Binding energy EB = E - E = - E
E Energy of the free nucleus; E = 0
E Energy of the compound nucleus
E > E
Mass defect
m = EB / c²
Mass of the nuclei
m = Z mp + N mn – EB / c²
Fission
Fusion
Fusion: 1 to3.5 MeV / nucleus  ca. 20 MeV / fusion
Fission: about 1 MeV / nucleon
 ca. 200 MeV / fission

3. Neutron induced fission
Abstand
Ef = limit for fission
Ef  5 MeV for Z > 90
Neutron induced fission
Capture of a free neutron
 excited compound nucleus (ZK) with excitation energy
EA = EB + Ekin,n  Ef
EB(gg-ZK) > EB(ug-,gu-ZK)
ug-nuclei: uneven number of n and even number of p or opposite
gg–nuclei: even number of n and p
Potentielle Energie

4. Fission of heavy nucleons with neutrons
Nucleus
Compound nucleus
Necessary
neutron energy for fission in MeV
slow neutrons
fast neutrons

5. Neutron energies Slow (thermal) neutrons Etherm ≤ 0.4 e V
typical thermal energy Etherm = eV
Epithermal neutrons eV < Eepi < 10 keV
Fast neutrons Efast ≥ 10 keV
Application of research reactors

6. Uranium 1 g 238U 20 spontaneous fissions per hour (tunnel effect)
106 times more -decays
Natural Uranium
0.72 % 235U
99.28 % 238U
Enrichment of 235U
Power reactors: 3-4%
Research reactors:  20 % (LEU)
> 20 % (HEU)
fuel development

7. Fission products Number of fission products nf for 235U
Fission product nf [%]
131I (8.05 d) 3.1
132Te (77 h) 4.7
133Sb (4.1 min) 4.0
133Te (63 min) 4.9
133I (21 h) 6.9
133Xe (5.27 d) 6.6
134Te (44 min) 6.9
135I (6.7 h) 6.1
137Cs (29 a)
140Ba (12.8 d)
143Ce (33 h) 5.7
144Ce (285 d) 6.0
thermal fission of 233U and 239Pu
thermal und 14 MeV-fission of 235U
fission by prompt neutrons of 232Th and 238U

8. Activation Structural components of the reactor Aluminum
Stainless steel
Concrete
Air, water

9. Fission of 235U – operation of reactors

10. Prompt fission neutrons
= average number of prompt neutrons produced by the fission
For U-235:
(E)= x E
with
E = excitation energy for the neutrons

11. Fission spectrum
Maxwellverteilung mit
EW = 0,7 MeV
Ē = 2 MeV
Energy spectrum of the prompt neutrons by thermal fission of 235U

12. Delayed neutrons
After the emission of the 2 n the excitation energy of the nucleus is too small to emit an other neutron.
The stability for the decay products is reached by - decay.

13. Properties of delayed neutrons
Group
Half decay time s
Mean
energy
keV
Fractional yield for thermal fission of
233U U Pu
% % % 1 55
250
0.022
0.021
0.007 2 23
560
0.077
0.140
0.063 3
6,2
430
0.065
0.126
0.044 4
2,3
620
0.072
0.253
0.068 5
0,61
420
0.013
0.074
0.018 6
0,23
-----
0.009
0.027
0.258
0.641
0.209

14. Cross section  Probabilities for the neutron Total cross section  =
Probabilities for the interaction
Neutron - nucleus
Unit: Barn (b),
1 b = cm²
Different microscopic cross sections for the different processes

15. Microscopic cross sections
elastic
scattering
inelastic
Cross section
fission
capture
absorption
-radiation
p-emission
2n-emission
For small neutron energies in thermal reactors is , p and 2n  0

16. Cross sections for 235U
total
fission
total
capture
fission
scattering

17. Cross section for 238U
capture
total
fission

18. Cross section for Cadmium
Measurements with and without Cd:
Separation of thermal and epithermal neutrons

19. Characteristics for fissionable materials
For fissionable materials in reactors three important characteristics are
= average number of neutrons produced per fission
= ratio of the number of neutrons captured by the fuel
 = average number of neutrons produced per neutron capture by the fuel
 =  / (1 + )

20. Neutron regeneration for thermal neutrons
Fuel   
U( natural )
U (5% U-235)
U (20% U-235) U U Pu

21. Multiplication factor k
number of fissions in one generation
number of fissions in previous generation
k =
k < 1  under critical power level
k = 1  constant power level
k > 1  over critical power level

22. Reactivity ρ Definition of the reactivity ρ: ρ = (k - 1) / k = k/k
Unit Percentage [% k/k] or number [k/k] or Dollar [$] and Cent [¢]
Calculation: 1 $ = 100 ¢ = 0,0073 k/k
1 k/k = 137 $ = ¢

23. Reaktivität ρ  < 0  under critical power level
 = 0  constant power level
 > 0  over critical power level

24. Four – factor formula k =  .  . p . f
For large reactors an infinite multiplication factor is defined
k =  .  . p . f
 = number of neutrons produced per neutron absorbed in the fuel
 = fast fission factor, a correction factor to take into account the fact that some fissions will be produced by fast neutrons
p = resonant escape probability, the probability that a neutron will escape from capture while it is being slowed through the resonance energy range (approximately 1 to 100 eV).
f = the thermal utilization, the fraction of thermal neutrons which are absorbed in in the fuel

25. Four – factor formula - examples
Homogenous mixture of natural uranium and graphite, both being powder:
Number of neutrons produced per absorbed neutron  = 1.33
Fast fission factor   1.0,
Product of resonant escape probability p and thermal utilization f: p.f  0.6
k =  .  . p . f = = 0.8
 Impossible to use such a combination in a reactor
If the uranium is used in rods with diameters of 1 to 2 inch in a matrix of solid graphite (heterogeneous system) p is increased and
the product p.f  0.8
Fast fission factor   1.03
k = = 1.09
 Possible to construct such a reactor

26. Effektive multiplication factor
Real reactor:
Escape of neutrons through the surface (neutron leakage),
Absorption of neutrons in 238U or (n,) reactions
keff = k . Ps . Pth
Ps and Pth number of neutrons, which do not escape (probabilities for slow and fast neutrons)
With the reactivity: ρ = (keff - 1) / keff   in the unit $

27. Reactor period T dn/dt = keff . n / l n = n0 . exp( keff . t / l)
Reactor operation at constant power level:  keff = 1,  = 0
Suddenly multiplication factor changed by keff
Increase dn of the number of neutrons n per unit volume dn in the time dt :
dn/dt = keff . n / l
with l = mean lifetime of the neutrons (time between generations)
Solution of the differential equation:
n = n0 . exp( keff . t / l)
with l / keff = T = Reactor period or e-folding time
Neutron flux: n = n0 . exp(t/T)
without delayed neutrons

28. Reactor period T Example :
Fission neutrons in natural Uranium: l = s
Increase of power level of keff = ½ % = 0.005
Reactor period:
T = l / keff = s / = 0.2 s
Neutron flux without delayed neutrons n = n0 . exp(t/T)
Increase of the power level per second of exp(5)
 Control of the reactor is not possible

29. Reactor period T
Delayed neutrons caused an increase of the mean life time l of 0.1 s
Increase of the power level of keff = ½ % =
Reactor period:
T = l / keff = 0.1 s / = 20 s
With delayed neutrons
Increase of the power per second of exp(0.05)
 Control of the reactor is possible

30. Reactor period T n = n0 . exp(t/T) Reactor period T or e-folding time:
Time, in which the neutron flux changed of the factor e = 2.72
Relative changes of the flux = (1 / T) . 100
Reactor period [s]
-50
-100 
100 50 20
Rel. flux changes [% s-1] -2 -1 1 2 5

31. Inhour equation
The „inhour equation“ gives the relationship between the reactivity and the reactor period in terms of the delayed neutrons (regarding 6 groups) and the prompt neutron lifetime.

32. Inhour equation
Using a time dependent diffusion equation and taking into account the delayed neutrons, it is possible to derive the following equation
ρ = (keff - 1) / keff = l / (T keff) +  i / (1 + T/i ) i
with
i = number of delayed neutrons of the group i
i = life time of the delayed neutrons of the group i
System of 7 linear in-homogenous differential equations of first order, the differential equations of the reactor kinetics.

33. Inhour equation Reactivity: ρ = l / (T keff) +  / (1 + T/  )
 = mean lifetime of the delayed neutrons
 =  i i
Case I: Large positive or negative T
T >> , 1 << T, ρ << 
ρ =   / T ρ  1 / T

34. Inhour equation Beispiel: TRIGA ρ =   / T  = 0.0073  = 12.3 s
T = 1 h
 = / 3600  (inverse hour)

35. Inhour equation Reactivity: ρ = l / (T keff) +  / (1 + T/  )
Case II: Small positive T, large reactivities ρ
0 < T <<  , ρ > 
ρ = l /( keff . T) +   T = l / ( keff ( -  ))
If k is not too far from unit, then T = l / ( -  )
If k exceed 1+ß, then the reactor will be critical on prompt neutrons alone. The reactor is prompt critical.

36. Inhour equation Für TRIGA
1 dollar [$] =  = , cent = 0.01 dollar
2 Dollar - Puls
ρ = 2  (2 $) = , T = 13.5 ms

37. Inhour equation for the TRIGA
Rod calibration

38. Fission of 235U with Thermal Neutrons
1 MeV
0.025 eV
Neutron absorber rods (k=1, steady state)
Neutron moderator (1 MeV  eV)

39. TRIGA Fuel Moderator Elements
91 % Zr
1 % H
8 % U (20% U-235)
Atomic Ratio:
Zr/H  1/1
Protons in U-Zr-Matrix act as Moderator

40. TRIGA Fuel Moderator Elements Prompt negative temperature coefficient
Decrease of reactivity:
per °C
 = -1,2 x 10-4 keff/keff .
at T = 100°C
 = -1.2 x 10-2 keff/keff = $
For comparision: 1 ¢ per kW increase of the power level

41. Xenon poisoning Reason for the Xenon poisoning:
Decay of I-135 into Xe-135 (large capture cross section for thermal neutrons)
Operation at constant power level: Production of I-135 and Xe
Increase of the power level: increase of I-135 and Xe
Decrease of the power level: Xe concentration increases due to the decay of the I-135 and neutrons will be captured by Xe. With a delay time of the half lifetime of I-135 the capture of neutrons decreases and also the power decreases slowly.
Operation of the reactor after the shut down is not possible when the absorption by Xe is too large.
Operation possible, when the Xe-135 production is negligible (about 20 h)

42. Xenon poisoning Time (h) Xenon concentration
a start of reactor operation with not poisoned core
b after fast shut down of the reactor
c after reduction of the power level
d after increase of the power level to the previous level
Time (h)
Xenon concentration

43. Core excess reactivity
At a power level of 100 kW following contributions a necessary for the core excess reactivity:
0.40 $ for operation
0.50 $ for compensation of the temperature effect
0.60 $ for compensation of the poison (Xe-135)
0.20 $ for compensation of the burn-up
Up to $ for compensation of the neutron absorption in samples for irradiation positions
__________
 2.70 $

44. Summary Nuclear fission Reactivity
Ideal (infinite) and real (finite) reactor
Inhour equation
TRIGA fuel (moderation, prompt negative temperature coefficient, inherent safe reactor)
Xenon poisoning

45. More information to the operation of the TRIGA Mainz
Structure
Instrumentation
Cooling- and purification circuits
Radiation protection
Safety
Checks (internal – external)
Special incidents
Documentation
Organization