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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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