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					         Neutron Slowing-Down Spectrum


                     B. Rouben
                McMaster University
      Course EP 6D03 – Nuclear Reactor Analysis
                  (Reactor Physics)
                   2009 Jan.-Apr.




2009 January                                      1
                       Contents
   We derive the most commonly referenced
    analytical form of the neutron slowing-down
    spectrum, the 1/E variation, subject to the
    appropriate approximations.
   Reference: Duderstadt & Hamilton, Sections 8.I
    A &B




    2009 January                                2
                   Probability Distribution of Neutron’s
                        Post-Collision Energy in L

   In the previous learning object, we established from
    the kinematics of neutron-nucleus collisions that the
    probability distribution of the neutron’s post-collision
    energy in the laboratory system is uniform in the
    allowed range:
       P EnL  EnL  
                             1
                 '
                                    , EnL  EnL  EnL (1)
                                              '

                        1   EnL




    2009 January                                           3
                                 Approximations
   Let us deal with the simplest case first, hydrogen, i.e.,  = 0, for
    which the probability distribution becomes
             P EnL  EnL  
                                 1
                       '
                                    ,   0  EnL  EnL ( 2)
                                             '

                                EnL
   We will also assume that the slowing down is in an infinite
    medium (or that leakage can be neglected during the slowing
    down, which is a fairly good approximation – leakage represents
    at most 2-3 %). This approximation also means that all space
    dependence disappears.
   We will assume that absorption can be neglected with respect to
    scattering – this is a good approximation for most moderators,
    including hydrogen.
   And we will assume that upscattering is negligible during
    neutron slowing down. This is a very good approximation for
    neutron energies higher than that corresponding to the thermal
    motion of nuclei.
   Also, finally, there is no external neutron source, and fission
    neutrons are born at a single representative high energy, say Eh.
    2009 January                                                     4
               Neutron-Transport Equation in Energy
   The neutron-transport equation as we know it is:
               r, E ,    S r, E ,        S f r, E   t r, E  r, E ,  
                                                  1
                                                 4
                                      s r, E '  E , '    r, E ' , ' dE ' d' (3)
                                      ' E '
   The assumptions we made  r and  disappear; the flux gradient  0; the only
    source term is the fission source term, which we rewrite as S(E-Eh) – we use a 
    function since the source is assumed to be at a single energy. The equation then
    becomes effectively                      E   h

                             s E  E     s E '  E  E ' dE ' S E  Eh  (4)
                                                E

    where s(E) is the total scattering cross section, the only component of t, since
    the absorption cross section has been assumed negligible.

   Note that Eh is the upper limit on the integral, since its is the source energy, and
    we have assumed no upscattering. E is also the lower limit, since neutrons of
    lower energy cannot be upscattered.

       2009 January                                                                              5
            Effective Neutron-Transport Equation
   In the previous learning object, we derived, from the kinematics
    and the assumed isotropic scattering off a light moderator, the
    relationship
                                s E '       s E ' 
               s E  E  
                     '
                                                        for hydrogen (5)
                             1   E    '
                                                 E '


   Then Eq. (4) becomes
                                   s E ' 
                               Eh

                s E  E         '
                                              E 'dE ' S E  Eh    (6)
                                0
                                     E
   If we write R(E) = s(E)(E) (7) we get the new form
                              RE '  '
                          Eh

                   R E       '
                                     dE  S E  Eh  (8)
                            E
                               E

    2009 January                                                               6
          Solving the Neutron-Transport Equation in E
   Since there is a  function on the r.h.s. of Eq. (8), the solution for
    R0(E) will have to have a  function, i.e., we can write
                   RE   R0 E   K E  Eh , where K is a cons tan t (9)

   If now we substitute the form (9) into Eq. (8), we get

    R0 E   K E  Eh  
                               Eh
                                    R E   K E  E  dE  S E  E 
                                          '            '

                                     0                            h      '
                                                                                    h
                                E
                                              E'
                                 R0 E '  ' K
                                     Eh

     R0 E   K E  Eh        '
                                          dE      S E  Eh                   (10)
                               E
                                   E           Eh

   The  functions must be the same on both sides  K = S                                (11)
    And therefore also             R0 E '  ' S
                                 E
                                                  h

                                      R0 E                 '
                                                                       dE         (12)
                                                  E
                                                           E                  E

    2009 January                                                                             7
             Solving the Neutron-Transport Equation in E
   To solve Eq. (12), let’s first differentiate it with respect to E:
    dR0 E    R E                      dR E      dE
              0     , which we can write 0           (13)
      dE         E                         R0 E      E
    and the general solution to this is immediate :
    ln R0 E    ln E   P, where P is a cons tan t    (14)
    We can evaluate P by writing Eq.(12) at Eh , and taking log arithms :

    R0 Eh         ln R0 Eh    ln Eh   ln S 
                 S
                                                             (15)
                 Eh
    and comparing this to Eq.(14)  P  ln S         (16)
    and we can then evaluate R0 E  from Eq.(14) for any E :

    R0 E  
                 S
                            (17)
                 E


    2009 January                                                            8
            The Slowing-Down Spectrum in Hydrogen
   Let’s put all the results together: Eq. (17) for R0(E), Eq. (9)
    for R(E), and Eq. (11) for the value of K, and finally Eq. (7)
    to evaluate . We get:
        R E   R0 E   K E  Eh    S E  Eh  (18 )
                                             S
                                             E
          E                          E  Eh 
                        S         S
                                                     (19 )
                    E s  E   s  E 
   If we make the further approximation that the hydrogen total
    scattering cross section s(E) is independent of energy (or
    very weakly dependent on energy, a good approximation in
    most of the slowing-down range), we see right away from
    Eq. (19) that
         Below Eh , the energy at which the fission neutrons are born,
                                                                  1
         the slowing  down flux in hydrogen is proportional to     ( 20 )
                                                                  E
    2009 January                                                             9
            The Slowing-Down Spectrum in Hydrogen

   Another way of interpreting Eq. (20) is to say that
    the product E(E) is nearly constant with energy
    below the fission energy Eh.
   Statement (20), that the slowing-down flux is
    proportional to 1/E [or its equivalent statement
    above] provides an important, simple, basic formula
    for the slowing-down spectrum – even if it is
    somewhat of an approximation.




    2009 January                                      10
                   Non-Hydrogen Moderator
   The analysis of the slowing-down spectrum in a non-
    hydrogenous moderator is more complicated, but
    under similar approximations the same form of the
    slowing-down spectrum can be found:
                E  
                        S
                                ( 21)
                       E s


   This has the same form for the slowing-down
    spectrum, with an additional factor of  (the average
    gain in lethargy per collision) in the denominator.


    2009 January                                            11
                               Slowing-Down Density
   The “neutron slowing-down density” qsd(E) as a function of the
    energy E is defined as the number of neutrons which slow down
    past the energy E.
   It can be shown that
                       E /
                                            E  E '
          qsd E            s E '             dE '   ( 22)
                        E
                                            E 'E '
          and , without absorption :
                            E /
                                   E  E ' dE '
          qsd E  
                       S
                             
                              E
                                    1   E '2
                                                            ( 23)

          For E below the source, this gives
          qsd E   S ( 24)
          i.e., the slowing  down density is cons tan t below source energies,
          which makes perfect sense when assu min g no absorption.
    2009 January                                                                  12
                    Relaxed Approximations
   The approximations which have been made in the
    above analysis can be relaxed one by one.
   The analysis then becomes progressively more
    complicated; the 1/E factor remains, but other
    factors enter into the final formula.
   For example: Distribution of fission-neutron
    energies. Relaxing the approximation that all
    neutrons are born at a single energy gives back
    the same 1/E form, for energies E well below the
    range of fission energies
    (cont’d)

     2009 January                                13
                    Relaxed Approximations (cont.)
   For another example: Non-negligible absorption cross
    section.
   In this case, it can be shown that the 1/E spectrum is
    modified (below the source energies) to:
                                    Eh a E '  dE ' 
               E  
                         S
                                exp                    (25)
                       Et E      E  t E '  E ' 
                                                      

   The exponential factor in Eq. (25) represents the
    probability that the neutron survives slowing down to
    energy E; i.e., it is the resonance-escape probability to
    energy E, denoted p(E).
     2009 January                                                 14
                        Summary
   Although the real situation in real reactors can
    become quite complicated, as a simple conceptual
    result, we should just remember that the slowing-
    down spectrum has approximately a 1/E form,
    modified by a number of other factors.




    2009 January                                 15
           Flux Spectrum Over Full Energy Range
   Now that we have derived the (approximate) slowing-down
    spectrum, we are able to “piece together” the neutron flux over the
    energy range from fission energies to the thermal range, using:
      the fission spectrum at energies above about 50-100 keV

      the slowing-down spectrum to about 1 eV

      the Maxwellian spectrum at thermal energies, below about 1 eV

       [note that in the thermal energy range neutrons can gain as well
       as lose energy in collisions; to be consistent with the
       approximation of no upscattering in the derivation of the
       slowing-down spectrum, the “boundary” between thermal and
       epithermal energies should be selected sufficiently high to
       ensure negligible upscattering from the thermal region to the
       epithermal: in many applications and computer codes, this
       boundary is taken as 0.625 eV].
     2009 January                                                 16
          Flux Spectrum Over Full Energy Range

   The piecing together of the neutron spectrum
    results in the sketch in the next slide.

   Note that the thermal spectrum is not a perfect
    Maxwellian (which applies to a gas, without
    absorption); the Maxwellian is deformed
    somewhat by neutron absorption.



    2009 January                                   17
      Flux Spectrum Over Full Energy Range




2009 January                            18
              Numerical Spectrum Calculations
   The analytical derivations above are useful to acquire a
    general understanding of the dependence of the neutron
    flux on energy.
   However, since they do depend on some approximations,
    the analytical forms may not be sufficiently accurate for
    precise reactor design and analysis.
   Therefore, most modern reactor analysis relies on
    numerical solutions of the neutron-transport equation in
    the basic lattice, either multigroup calculations using a
    very large number of energy groups (several dozen to
    several hundred), or Monte Carlo calculations in
    continuous energy.
     2009 January                                        19
               END


2009 January         20

				
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