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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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Φ( r ,E)

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E

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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N

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)

Neutron Slowing Down.. 15

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