Reactor physics
Reactor training course
Institut für Kernchemie
www.kernchemie.uni-mainz.de
Binding energy
• Binding energy
EB = E - E = - E
Fission
Fusion
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
Fusion:
1 to3.5 MeV / nucleus
 ca. 20 MeV / fusion
Fission: about 1 MeV / nucleon
 ca. 200 MeV / fission
m = Z mp + N mn – EB / c²
Neutron induced fission
Potentielle Energie
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
Fission of heavy nucleons with neutrons
slow neutrons
Nucleus
Compound nucleus
Necessary
neutron energy for
fission in MeV
fast neutrons
Neutron energies
• Slow (thermal) neutrons
Etherm ≤ 0.4 e V
typical thermal energy Etherm = 0.025 eV
• Epithermal neutrons
0.4 eV < Eepi < 10 keV
• Fast neutrons
Efast ≥ 10 keV
Application of
research reactors
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
Fission products
a)
thermal fission of
233U
b)
c)
and 239Pu
thermal und 14 MeVfission of 235U
fission by prompt
neutrons of 232Th
and 238U
Number of fission
products nf for 235U
Fission product nf [%]
131I
132Te
133Sb
133Te
133I
133Xe
134Te
U235
135I
137Cs
140Ba
143Ce
144Ce
(8.05 d)
(77 h)
(4.1 min)
(63 min)
(21 h)
(5.27 d)
(44 min)
(6.7 h)
(29 a)
(12.8 d)
(33 h)
(285 d)
3.1
4.7
4.0
4.9
6.9
6.6
6.9
6.1
6.15
6.44
5.7
6.0
Activation
• Structural components of the reactor
Aluminum
Stainless steel
Concrete
• Air, water
Fission of 235U – operation of reactors
Prompt fission neutrons
 = average number of
prompt neutrons
produced by the
fission
For U-235:
(E)= 2.43 + 0.106 x E
with
E = excitation energy
for the neutrons
Fission spectrum
Maxwellverteilung mit
EW = 0,7 MeV
Ē = 2 MeV
Energy spectrum of the prompt neutrons by
thermal fission of 235U
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.
Properties of delayed neutrons
Group
Half decay
time
s
Mean
energy
keV
Fractional yield for thermal fission of
233U
235U
239Pu
%
%
%
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
0.009
0.258
0.641
0.209
Cross section 
Probabilities for the neutron
Total cross section  =
Probabilities for the interaction
Neutron - nucleus
Unit: Barn (b),
1 b = 10-24 cm²
Different microscopic cross sections for the
different processes
Microscopic cross sections
elastic
scattering
Cross
section
inelastic
fission
capture
absorption
-radiation
p-emission
2n-emission
For small neutron energies in thermal
reactors is , p and 2n  0
Cross sections for 235U
total
fission
total
capture
scattering
fission
Cross section for 238U
total
capture
fission
Cross section for Cadmium
Measurements with and without Cd:
Separation of thermal and epithermal neutrons
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 + )
Neutron regeneration for thermal neutrons
Fuel



U( natural )
U (5% U-235)
U (20% U-235)
U-235
U-233
Pu-239
2.47
2.47
2.47
2.47
2.55
2.91
0.837
0.272
0.202
0.183
0.132
0.416
1.33
1.94
2.05
2.09
2.29
2.02
Multiplication factor k
k=
number of fissions in one generation
number of fissions in previous generation
k < 1  under critical power
level
k = 1  constant power level
k > 1  over critical power level
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 $ = 13699 ¢
Reaktivität ρ
 < 0  under critical power level
 = 0  constant power level
 > 0  over critical power level
Four – factor formula
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
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 = 1.33 . 1 . 0.6 = 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 = 1.33 . 1.03 . 0.8 = 1.09
 Possible to construct such a reactor
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 $
Reactor period T
• 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
Reactor period T
Example :
Fission neutrons in natural Uranium: l = 0.001 s
Increase of power level of keff = ½ % = 0.005
Reactor period:
T = l / keff = 0.001 s / 0.005 = 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
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 % = 0. 005
Reactor period:
T = l / keff = 0.1 s / 0.005 = 20 s
With delayed neutrons
Increase of the power per second of exp(0.05)
 Control of the reactor is possible
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
0
1
2
5
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.
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.
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
Inhour equation
Beispiel: TRIGA
ρ=/T
 = 0.0073
 = 12.3 s
T=1h
 = 0.0073 . 12.3 / 3600  2.5 . 10-5 (inverse hour)
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.
Inhour equation
Für TRIGA
1 dollar [$] =  = 0.0073 ,
1 cent = 0.01 dollar
2 Dollar - Puls
ρ = 2  (2 $) = 0.0146, T = 13.5 ms
Inhour equation for the TRIGA
Rod calibration
Fission of 235U with Thermal Neutrons
0.025 eV
1 MeV
1. Neutron absorber rods (k=1, steady state)
2. Neutron moderator (1 MeV  0.025 eV)
TRIGA Fuel Moderator Elements
Abstandshalter
Abstandshalter
Aluminium Halterung
Aluminium Halterung
Graphit
Graphit
TRIGA Fuel:
Abbrennbarer
Abbrennbarer
Neutronenabsorber
Neutronenabsorber
91 % Zr
1%H
8 % U (20% U-235)
355 mm
37,3 mm
722 mm
37,3 mm
722 mm
355 mm
Zirkoniumhydrid
Zirkoniumhydrid
8 Gew.% Uran
8 Gew.% Uran
(20% U-235)
(20% U-235)
Atomic Ratio:
Zr/H  1/1
Graphit
Aluminium Halterung
Aluminium Halterung
Graphit
Protons in U-Zr-Matrix
act as Moderator
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 = - 1.64 $
For comparision: 1 ¢ per kW increase of the power level
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)
Xenon concentration
Xenon poisoning
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)
Core excess reactivity
At a power level of 100 kW following contributions a necessary
for the core excess reactivity:
Up to
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
1.00 $
for compensation of the neutron absorption
in samples for irradiation positions
__________
 2.70 $
Summary
1. Nuclear fission
2. Reactivity
3. Ideal (infinite) and real (finite) reactor
4. Inhour equation
5. TRIGA fuel (moderation, prompt negative
temperature coefficient, inherent safe
reactor)
6. Xenon poisoning
More information to the operation of the
TRIGA Mainz
1. Structure
2. Instrumentation
3. Cooling- and purification circuits
4. Radiation protection
5. Safety
6. Checks (internal – external)
7. Special incidents
8. Documentation
9. Organization