Neutron Slowing Down.. 1
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Lesson 7: Neutron Slowing Down
Study of an Elastic Collision
Slowing Down Probabilities
Average Logarithmic Energy Loss
Lethargy
Moderator Characteristics
Slowing Down Source (Slowing Down Density)
Fundamental Equations of Slowing Down
Neutron Slowing Down.. 2
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Slowing Down
Till now, we have discussed the behaviour of monoenergetic neutrons • E.g. thermal neutrons, with appropriately averaged cross-sections…
A thermal reactor, however, has n’s between ~ 2 MeV and ~ 0.01 eV • One needs to study how changes from ~ 2 MeV to 3/2 kT • Slowing down process determines the “thermal -neutron source”
In the case of a fast reactor, there is also slowing down • changes from ~ 2 MeV to ~ 100 keV • Neutron spectrum depends strongly on core composition
→ In any case, one needs to determine the neutron energy spectrum for evaluating the different reaction rates.
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Neutron Slowing Down.. 3
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Study of an Elastic Collision
Most important slowing-down mechanism: elastic scattering by moderator nuclei • Inelastic scattering also plays a role, but only for fast neutrons (E ≥ 1 MeV) • Consider the most common situation • Nucleus at rest, of mass A (rel. to the neutron mass)
Advantageous to consider the C- System • A single parameter, θc , characterises the collision (instead of 2, in the L - System)
L - System C - System
Neutron Slowing Down.. 4
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Study of an Elastic Collision (contd.)
CM velocity For the neutron: For the nucleus:
In the C - System , conservation of momentum: conservation of energy:
Eliminating Vc , and then → The velocities remain the same in the C - System (only the direction changes)
(conservation of momentum)
Neutron Slowing Down.. 5
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Study of an Elastic Collision (contd.2)
For the change in neutron energy in the L - System,
Thus,
with
Neutron Slowing Down.. 6
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Study of an Elastic Collision (contd.3)
For θc = 0 , E = E’ (no loss of energy) For θc = π , E = αE’ (maximal energy loss)
The energy loss depends on θc , but also strongly on A
• E.g. For H1 , A = 1 , α = 0 → A loss of 100% is possible in a single collision
For H2 , A = 2 , α = 1/9 → Max. loss possible in a single collision ~ 89%
Neutron Slowing Down.. 7
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Study of an Elastic Collision (contd.4)
One may also consider the relation between θc , θ
We have:
With and →
Alternatively,
Neutron Slowing Down.. 8
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Probability P1 (E’ → E) dE
In the majority of cases, scattering isotropic in C - System
Using as variable µ = cos θc , no. of n’s scattered between µ, µ+ dµ ∝ width dµ
Max. interval: (-1, +1) ⇒ max. width: Δµ = 2 , i.e. fraction betn. µ, µ+ dµ : dµ/2
Differentiating , one has
Thus, probability for a n to have an energy betn. E, E+dE :
Neutron Slowing Down.. 9
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Probability P2 (E’ → E) dE
Probability that the energy of the neutron is < E
P2 (E → E’) = 1 for E = E’ • That E lies betn. E’ , αE’ is certain
P2 decreases linearly (until 0 for E = αE’) The loss of energy after a given, single collision is stochastic, as is µ , or θc
Neutron Slowing Down.. 10
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Average Energy Loss
Average energy loss:
Average logarithmic energy loss:
Average no. of collisions for E → E’
⇒ Result depends on energy
With
With
⇒ ξ not dependent on energy, only on A (For A > 10 , )
Neutron Slowing Down.. 11
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Moderator Characteristics For thermal reactors,
Thus, avg. no. of collisions necessary:
For a mixture of isotopes:
Macroscopic Slowing-down Power:
Moderating Ratio:
A ξ
H 1 1 18
H2O - 0.92 20
D 2 0.725 25
D2O - 0.509 36
Be 9 0.209 87
C 12 0.158 115
O 16 0.120 152
… … … …
… … … …
U 238 0.00838 2172
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Neutron Slowing Down.. 12
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Lethargy
With reference to the intial energy E0 , the lethargy is
The increment Δu corresponds to a logarithmic decrease in energy ΔE
The energy E0 corresponds to u = 0 (E0 → Eth implies for u : 0 → 18.2)
ξ is the average lethargy increment per collision
Other relationships:
Neutron Slowing Down.. 13
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Spectral Densities
Corresponding to an energy band between E , E+dE
Description of a system during slowing down needs
E.g. fission rate at in the band E , E+dE :
For calculating the heat source at each point, viz. , one needs , i.e. … distribution of the spectral density of the flux
densities w.r.t. energy (units: n.cm-2.s-1.MeV-1)
Thus, at ,
Neutron Slowing Down.. 14
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Fundamental Slowing-down Equations
For the energy band E , E+dE , the neutron balance equation is:
Q dE … total sources between E , E+dE • “True” (fission, isotopic sources,… ), as well as those resulting from slowing down
(neutrons of energy > E are scattered into the band E , E+dE)
Considering the n’s between E’ , E’+dE’ , scattering rate is
No. scattered with an energy < E is
Total no. scattered below E at
Slowing-down source (cm-3.s-1)
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Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Slowing-Down Equations (contd.)
With
Diff. gives slowing-down source in band E , E+dE
Thus, neutron balance eqn.:
After division by dE and taking the limit dE → 0 ,
… (1)
… (2)
⇒ (1), (2) : Fundamental Slowing-down Equations
Neutron Slowing Down.. 16
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Slowing-Down Equations (contd.2)
In practice, one works with Eqns. (1), (2), but one can show that is indeed well defined, e.g. by eliminating q from these equations and then using Fick’s Law…
Considering Eq. (1), i.e.
one has:
Neutron Slowing Down.. 17
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Slowing-Down Equations (contd.3)
Using Eq. (2),
With (Fick’s Law) , one may eliminate
(Diffusion Equation for the band E , E+dE → yields the spectral flux density
Neutron Slowing Down.. 18
Laboratory for Reactor Physics and Systems Behaviour
Neutronics
Summary, Lesson 7
Slowing Down via Elastic Collisions
Average Logarithmic Energy Loss per Collision
Lethargy
Moderator Characteristics
Spectral Flux Density
Slowing Down Source
Fundamental Equations of Slowing Down