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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.

Φ( r ,E)

E

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

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

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