Reactor physics

Reactor training course Institut für Kernchemie

www.kernchemie.uni-mainz.de

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

Neutron induced fission

Abstand

Po

ten

tiel

le E

ner

gie

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

Nucleus

Compound nucleus

Necessaryneutron energy for fission in MeV

slow neutrons

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

U235

Number of fission products nf for 235U

Fission product nf [%]131I (8.05 d) 3.1132Te (77 h) 4.7133Sb (4.1 min) 4.0133Te (63 min) 4.9133I (21 h) 6.9133Xe (5.27 d) 6.6134Te (44 min) 6.9135I (6.7 h) 6.1137Cs (29 a) 6.15140Ba (12.8 d) 6.44143Ce (33 h) 5.7144Ce (285 d) 6.0

a) thermal fission of 233U and 239Pu

b) thermal und 14 MeV-fission of 235U

c) fission by prompt neutrons of 232Th and 238U

Activation

• Structural components of the reactorAluminumStainless steelConcrete

• 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 EwithE = excitation energy for the neutrons

Fission spectrum

Energy spectrum of the prompt neutrons by thermal fission of 235U

Maxwellverteilung mit

EW = 0,7 MeV

Ē = 2 MeV

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

For small neutron energies in thermal reactors is , p and 2n 0

scattering

absorption

fission

capture

-radiation

elastic

inelastic

2n-emission

p-emission

Cross section

Cross sections for 235U

fission

fission

total

total

capture

scattering

Cross section for 238U

totalcapture

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 ) 2.47 0.837 1.33

U (5% U-235) 2.47 0.272 1.94

U (20% U-235) 2.47 0.202 2.05

U-235 2.47 0.183 2.09

U-233 2.55 0.132 2.29

Pu-239 2.91 0.416 2.02

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

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 = 1.33Fast 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

withi = 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 TT >> , 1 << T, ρ <<

ρ = / T ρ 1 / T

Inhour equation

Beispiel: TRIGA ρ = / T

= 0.0073

= 12.3 s

T = 1 h

= 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 - Pulsρ = 2 (2 $) = 0.0146, T = 13.5 ms

Inhour equation for the TRIGA

Rod calibration

Fission of 235U with Thermal Neutrons

1. Neutron absorber rods (k=1, steady state)

2. Neutron moderator (1 MeV 0.025 eV)

1 MeV

0.025 eV

TRIGA Fuel Moderator Elements

TRIGA Fuel:

91 % Zr

1 % H

8 % U (20% U-235)

TRIGA Fuel:

91 % Zr

1 % H

8 % U (20% U-235)

Atomic Ratio:

Zr/H 1/1

Protons in U-Zr-Matrix act as Moderator

Atomic Ratio:

Zr/H 1/1

Protons in U-Zr-Matrix act as Moderator

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

TRIGA Fuel Moderator Elements

Prompt negative temperature coefficient

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 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)

Xen

on

co

nce

ntr

atio

n

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 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