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Operators

Operators. B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec. Contents. Introduction of operator notation. Various Energy-Group Methodologies. In reactor physics, we are often dealing with one-group or multigroup models or methodologies:

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Operators

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  1. Operators B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec

  2. Contents • Introduction of operator notation.

  3. Various Energy-Group Methodologies In reactor physics, we are often dealing with one-group or multigroup models or methodologies: • One-group model to learn and study basic concepts • Multigroup (many-group, tens-or-hundreds-of-groups) models in transport theory for lattice-physics calculations • Often, 2-group (or few-group) models in diffusion theory for full-reactor calculations

  4. 1-Group Model • In the simplest, 1-energy-group, model, all energies are lumped into a single group. • We can think of the flux (r) as a scalar – a number (function of position). • Similarly, we can also think of the various cross sections as scalar operators (i.e, simple numbers, but functions of position) which will multiply the flux to produce reaction rates: • Thus, in the 1-group model, the flux and cross sections are all scalar – i.e., simple numbers, most likely functions of position. • There can also be other types of operators, e.g., the differential operator d/dx acts on a function (x) and provides its derivative, d/dx.

  5. 2-Group Model • In the 2-energy-group model, there are 2 fluxes, 1 and 2, one for each energy group (each flux a function of position). • (Note: Group 1  “Fast Group”, Group 2  Thermal Group) • We can combine these fluxes into a column vector of 2 quantities: • Note: We may often drop the position label r for simplicity. (cont.)

  6. 2-Group Model (cont’d) • Some operators are diagonal (i.e., they don’t mix the groups, whereas others are non-diagonal, i.e., they work on the flux of one group and produce a result in another group: • e.g., the fission cross section may operate on the thermal flux and produce neutrons in the fast group • or, the moderation cross section will “down-scatter” neutrons from the fast group to the thermal group • there may also be up-scatter. • Thus, in the 2-group model, we can think of operators as 2x2 matrices which multiply the flux vector to produce another vector, e.g., as an example (remember how matrix multiplication works?), • Thus, in 2 groups, we work with 2-vectors and 2x2 operators.

  7. Convention • The conventional naming of operators is as follows: • The yield – or production - operator (essentially f, in whatever number of groups) is written F. • The “absorption, scattering, and leakage” operator is written M.

  8. n-Group Model • Similarly, in the n-energy-group model we work with n-vectors and n x n operators (matrices):

  9. Interactive Discussion/Exercise • In the 2-group model, (in matrix form, and in the same manner as the example 3 slides back) write down the equation for the combined yield operator (involving f1 and f2) operating on the flux vector, and the resulting vector. • In the 3-group model, write the equation in matrix form for the combined down-scattering operator from group 1 (the highest-energy group) to group 2 and from group 2 to group 3 (the lowest-energy, i.e., thermal, group), and the result of this operator acting on the 3-group flux. • In the 2-group model, write the neutron-absorption operator. What is different, in this operator, from those in the previous 2 exercises?

  10. Solution of Execise

  11. Operators in Continuous Energy • The operators we have defined in the previous slides work on distinct energy groups, i.e., groups 1, 2, 3… • In some methodologies the energy range is not subdivided into distinct groups – energy is treated as a continuous variable. • In this case, the analog of a matrix operator becomes a “kernel”, which operates within an integral (e.g., on energy). • The kernel, which we shall call K, operates on a function of energy and yields another function of energy, e.g.: • Note that, as was the case in the matrix operators, the kernel can in general change energy (or scatter) – e.g., from E’ to E.

  12. Adjoints • Column vectors and operators can have adjoints: • The adjoint of a column vector is a row vector: • The adjoint of a matrix operator is its transpose, obtained by transposing columns into rows:

  13. Scalar (or Dot) Product • We may have occasion to use the scalar or dot product of a column vector and a row vector, e.g., defined (in 2 groups) as: • When dealing with continuous functions, the scalar product is an integral over the domain of the functions:

  14. END

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