DPAUserObjectBase

Description

A base class for computing the displacement per atom (DPA) caused by neutron irradiation. The DPA calculation requires uses the following information:

the scalar flux $var element = document.getElementById("moose-equation-56b700fb-e512-4b9f-ae30-0cccc81984ea");katex.render("\\phi(E)", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-933e43d3-7ae6-40ed-90e9-d9ff8c2ce8f5");katex.render("E", element, {displayMode:false,throwOnError:false});$ denotes neutron energy.
the number density of isotope $var element = document.getElementById("moose-equation-237b2ba3-99a5-4bdc-9bd9-e41ac65a4cc6");katex.render("i", element, {displayMode:false,throwOnError:false});$ denoted by $var element = document.getElementById("moose-equation-4d32024b-cc99-4b35-bfa6-67ee0e07656f");katex.render("N_i", element, {displayMode:false,throwOnError:false});$ . The index $var element = document.getElementById("moose-equation-a51e4874-91bf-4460-895e-e375ac29d81c");katex.render("i", element, {displayMode:false,throwOnError:false});$ runs over all isotopes $var element = document.getElementById("moose-equation-70923667-9693-46f7-8d4d-a3cc0939074e");katex.render("i=1,..,I", element, {displayMode:false,throwOnError:false});$ .
the microscopic cross section of isotope $var element = document.getElementById("moose-equation-85bfe85a-ab38-4acd-9056-6cebb3d3d5d2");katex.render("i", element, {displayMode:false,throwOnError:false});$ for reaction type $var element = document.getElementById("moose-equation-c4565e46-4506-4e78-8cff-5415d9d622da");katex.render("x", element, {displayMode:false,throwOnError:false});$ (currently $var element = document.getElementById("moose-equation-53c3abdd-5198-48cd-bec0-077a4201e5c6");katex.render("x", element, {displayMode:false,throwOnError:false});$ can stand for elastic and inelastic scattering).
the atomic number $var element = document.getElementById("moose-equation-ae3684cd-3a08-4c57-8d20-c4d024b2edfd");katex.render("Z_i", element, {displayMode:false,throwOnError:false});$ and mass number $var element = document.getElementById("moose-equation-e0031a26-59a5-4215-993f-4510798cd896");katex.render("A_i", element, {displayMode:false,throwOnError:false});$ of isotope $var element = document.getElementById("moose-equation-dda636a9-9697-408d-9dde-8dbaa5bb1d06");katex.render("i", element, {displayMode:false,throwOnError:false});$ .
the number of displacements of type $var element = document.getElementById("moose-equation-6d4ee377-8824-4793-b0c9-c8db6159f755");katex.render("j", element, {displayMode:false,throwOnError:false});$ that a primary knock-on atom (PKA) of type $var element = document.getElementById("moose-equation-96ac60ef-201d-486f-abaf-797b2f9cc148");katex.render("i", element, {displayMode:false,throwOnError:false});$ and energy $var element = document.getElementById("moose-equation-a84193b6-fc7d-4b03-a4a7-ffc6e98a7b34");katex.render("T", element, {displayMode:false,throwOnError:false});$ causes is denoted by $var element = document.getElementById("moose-equation-b26dba64-20fe-48f2-8b9c-6bfef7ff8ab2");katex.render("\\nu_{i,j}(T)", element, {displayMode:false,throwOnError:false});$ .

The increase in dose $var element = document.getElementById("moose-equation-312a467d-6d04-4f2c-ae5c-1a47d9b768c6");katex.render("\\Delta D", element, {displayMode:false,throwOnError:false});$ measured in dpa after a time interval $var element = document.getElementById("moose-equation-dc7cdef1-5e74-4ed0-b928-aced1e81dc09");katex.render("\\Delta t", element, {displayMode:false,throwOnError:false});$ is given by: $var element = document.getElementById("moose-equation-808eb610-d105-4224-a059-16d8c31bdb25");katex.render(" \\Delta D = \\Delta t \\sum\\limits_x \\sum\\limits_{i=1}^I \\sum\\limits_{j=1}^J N_i \\int_{0}^{\\infty} \\sigma_{i,x}(E) \\phi(E) \\left[ \\int_{T_{\\text{min}, i, x}}^{T_{\\text{max}, i, x}} \\nu_{i,j}(T) H_{i,x}(E,T) dT \\right] dE,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-80654215-6559-4f11-b75b-75ed212b6a21");katex.render("H_x(E,T)", element, {displayMode:false,throwOnError:false});$ is the probability density function that a reaction of type $var element = document.getElementById("moose-equation-c5f5a448-ee2f-4fa0-b96e-eee896dc113b");katex.render("x", element, {displayMode:false,throwOnError:false});$ causes a PKA of type $var element = document.getElementById("moose-equation-d18a2f09-6478-4720-8d9f-493472a8aa18");katex.render("i", element, {displayMode:false,throwOnError:false});$ with energies in the interval $var element = document.getElementById("moose-equation-e9c2ffed-ed95-4c42-9b4d-5cfe0788b653");katex.render("dT", element, {displayMode:false,throwOnError:false});$ about $var element = document.getElementById("moose-equation-24c87234-9a6c-484b-aca2-8ea06d882531");katex.render("T", element, {displayMode:false,throwOnError:false});$ ; $var element = document.getElementById("moose-equation-9165ab3b-ff32-4edf-aca8-47e49e2b34e3");katex.render("T_{\\text{min}, i, x}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-3cdc7bb7-6957-46cd-b473-2fe20f367893");katex.render("T_{\\text{max}, i, x}", element, {displayMode:false,throwOnError:false});$ are the maximum and minimum recoil energy that a neutron in reaction channel $var element = document.getElementById("moose-equation-e93ce161-e91b-4283-9461-e7c5245b28a2");katex.render("x", element, {displayMode:false,throwOnError:false});$ can cause in a collision with isotope $var element = document.getElementById("moose-equation-0bcbb985-068c-4579-a1f5-3edf98e59d40");katex.render("i", element, {displayMode:false,throwOnError:false});$ .

It is convenient to define the parameter $var element = document.getElementById("moose-equation-2c11c25b-08d8-40dd-a1f2-e1c59b235be4");katex.render("\\gamma_i", element, {displayMode:false,throwOnError:false});$ : $var element = document.getElementById("moose-equation-87042952-a11a-4517-94b4-7e8f088ff82c");katex.render(" \\gamma_i = \\frac{4 A_i}{(A_i + 1)^2}", element, {displayMode:true,throwOnError:false});$

For inelastic reactions, the reduced energy is defined: $var element = document.getElementById("moose-equation-ecdb71f4-5533-44af-bfc1-bc3da2658677");katex.render(" \\delta_i = \\frac{Q_i}{E} \\frac{A_i + 1}{A_i}", element, {displayMode:true,throwOnError:false});$

For elastic and inelastic scattering:

Elastic scattering: we assume isotopic scattering in the center of mass frame; then we have $var element = document.getElementById("moose-equation-c0acb314-924a-43da-aece-650e02c29ed8");katex.render("H_{i,x}(E,T) = \\frac{1}{\\gamma_i E}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-418935e2-ea52-4dc1-9c35-2daf73eceaf9");katex.render("T_{\\text{min}, i, x} = 0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-f286c355-039c-475c-80d2-fbdac0c269ab");katex.render("T_{\\text{max}, i, x} = \\gamma_i E", element, {displayMode:false,throwOnError:false});$ .
Inelastic scattering: $var element = document.getElementById("moose-equation-550c1db6-0b9d-45f4-9701-34178098d55a");katex.render("H_{i,x}(E,T) = \\frac{1}{\\gamma_i E\\sqrt(1 - \\delta_i)}", element, {displayMode:false,throwOnError:false});$ ,

$var element = document.getElementById("moose-equation-a13e548c-a4b3-47e2-8583-bedc128d489c");katex.render("\\begin{aligned} T_{\\text{min}, i, x} &=\\frac{\\gamma_i E}{2}\\left(1 - \\sqrt{1 - \\delta_i}\\right)-\\frac{1}{2}\\delta_i \\\\ T_{\\text{max}, i, x} &=\\frac{\\gamma_i E}{2}\\left(1 + \\sqrt{1 - \\delta_i}\\right)-\\frac{1}{2}\\delta_i. \\end{aligned}", element, {displayMode:true,throwOnError:false});$

We observe that in both cases $var element = document.getElementById("moose-equation-7957bc14-2ff8-44a2-9f1e-aec17a8a7636");katex.render("H_{i,x}(E,T) = H_{i,x}(E)", element, {displayMode:false,throwOnError:false});$ .

From deterministic transport theory, usually the group flux $var element = document.getElementById("moose-equation-7a42e67e-f587-4382-92e2-44e286373880");katex.render("\\phi_g", element, {displayMode:false,throwOnError:false});$ is provided. It is defined by $var element = document.getElementById("moose-equation-e26a30c9-4eed-4bb3-a147-23332d01f7db");katex.render(" \\phi_g = \\int_{E_{g+1}}^{E_{g}} \\phi(E) dE,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-5f00f4f3-706e-4b79-b39d-4ac9d6d5c926");katex.render("E_g", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-ee194b0d-c607-4a49-b780-687769cadc8f");katex.render("E_{g+1}", element, {displayMode:false,throwOnError:false});$ are energy group boundaries ordered from high to low, i.e. $var element = document.getElementById("moose-equation-92bee46b-de4e-47de-9de6-1a4b03c5725e");katex.render("E_{g+1} < E_g", element, {displayMode:false,throwOnError:false});$ . We make the approximation: $var element = document.getElementById("moose-equation-2903d99a-fc59-46c1-b43c-232d9c44cc7c");katex.render(" \\phi(E) \\approx \\frac{\\phi_g}{E_{g} - E_{g+1}} \\text{ for } E_{g+1} \\le E < E_{g}.", element, {displayMode:true,throwOnError:false});$ The group-averaged cross section is defined by: $var element = document.getElementById("moose-equation-c8c35eec-f337-4bee-addf-e53f3894e4fc");katex.render(" \\sigma_{g,i,x} = \\frac{1}{\\phi_g} \\int_{E_{g+1}}^{E_{g}} \\sigma_{i,x}(E) \\phi(E) dE.", element, {displayMode:true,throwOnError:false});$

Then the dpa increment $var element = document.getElementById("moose-equation-c16a5479-782c-4956-b786-261672cb8ebe");katex.render("\\Delta D", element, {displayMode:false,throwOnError:false});$ can be approximated by: $var element = document.getElementById("moose-equation-1484e38c-fba0-4bd1-9c5d-9cb9020554b6");katex.render(" D = \\delta t \\Delta t \\sum_{g} \\sum\\limits_x \\sum\\limits_{i=1}^I \\sum\\limits_{j=1}^J \\frac{N_i \\sigma_{g,i,x} \\phi_g}{E_{g} - E_{g+1}} \\Gamma_{i,j,g,x},", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-59d79662-7c33-41f1-b48a-ee8b71ab83db");katex.render(" \\Gamma_{i,j,g,x} = \\int_{E_{g+1}}^{E_{g}} H_{i,x}(E) \\left[\\int_{T_{\\text{min}, i, x}}^{T_{\\text{max}, i, x}} \\nu_{i,j}(T) dT \\right] dE.", element, {displayMode:true,throwOnError:false});$

The approximation of using constant $var element = document.getElementById("moose-equation-b85df9f9-b497-4a4b-9e9a-03588448deb0");katex.render("\\phi(E)", element, {displayMode:false,throwOnError:false});$ within each energy group is expected to lead to large errors if the fast energy range is represented by too few groups. An improvement is needed that accounts for the variation of $var element = document.getElementById("moose-equation-cd64dcf0-d445-4016-9f53-41dfdd515974");katex.render("\\phi(E)", element, {displayMode:false,throwOnError:false});$ within each energy group.

Description