UniaxialAFMSublattice

Calculates a residual contribution for an uniaxial AFM sublattice

Overview

Here we consider the first example from the work of Rezende et al. (2019) of a uniaxial antiferromagnet $var element = document.getElementById("moose-equation-00e8fa4d-1693-4956-8356-fcee641389cf");katex.render("\\mathrm{MnF}_2", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ . The total free energy density of the material is,

$var element = document.getElementById("moose-equation-fd8a9174-d06e-4490-bf37-3f314ecc845c");katex.render(" \\begin{aligned} f = - H_0 \\left(M_{1z} + M_{2z}\\right) + \\frac{H_\\mathrm{E}}{M_s} \\mathbf{M}_1 \\cdot \\mathbf{M}_2 - \\frac{H_\\mathrm{A}}{2 M_s} \\left(M_{1z}^2 + M_{2z}^2\\right). \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

We look for normalized solutions for $var element = document.getElementById("moose-equation-4949ce0b-0dd9-4856-a6bf-5eec5e4c10f6");katex.render("\\mathbf{m}_\\eta = \\mathbf{M}_\\eta / M_s", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ to the two-sublattice LLG-LLB equation in the presence of zero Gilbert damping ( $var element = document.getElementById("moose-equation-2913d6fb-939a-4085-932d-1b091748dc9f");katex.render("\\alpha = 0", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ ),

$(1)var element = document.getElementById("moose-equation-a860e038-0fe7-4701-86e8-1e69dd84ce77");katex.render(" \\begin{aligned} \\frac{d \\mathbf{m}_\\eta}{dt} = -\\gamma \\left(\\mathbf{m}_\\eta \\times \\mathbf{H}_\\eta\\right) + \\frac{\\gamma\\tilde\\alpha_\\parallel}{(1+\\alpha^2)} m_{\\eta}^2 \\left[ m_{\\eta}^2 - 1 \\right]\\mathbf{m}_\\eta, \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

We ignore the time dependence and the longitudinal damping term (from the LLB approximation). Then, we multiply both sides by a test function $var element = document.getElementById("moose-equation-56421a14-b568-4d88-b421-2e4dfbc495be");katex.render("\\psi_h", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ and integrate over the volume, which yields,

$var element = document.getElementById("moose-equation-2c4b0992-1168-43db-b40d-02779a5da772");katex.render(" \\begin{aligned} -\\left(\\psi_h, \\gamma \\mathbf{m}_\\eta \\times \\mathbf{H}_\\eta \\right) = 0 \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

The effective field is computed as,

$var element = document.getElementById("moose-equation-b0ac5b25-7241-4e1b-ace4-452f01c2cda3");katex.render(" \\begin{aligned} \\mathbf{H}_\\eta = -\\frac{1}{\\mu_0 M_s}\\left(\\frac{\\delta f}{\\delta \\mathbf{m}_\\eta}\\right). \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

This gives the residual contribution (in index notation) for $var element = document.getElementById("moose-equation-9c25e238-641f-4aba-8dc0-a980dd0e7582");katex.render("m_{\\eta k}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ to be

$var element = document.getElementById("moose-equation-9b644e67-080d-4ad0-8b3f-9a36d265863d");katex.render(" \\begin{aligned} \\mathcal{R}_{m_{\\eta k}} &= -\\gamma \\left(\\psi_h, \\epsilon_{ijk} m_{i\\eta} H_{j\\eta} \\right)\\\\ &= \\frac{\\gamma}{\\mu_0 M_s} \\left(\\psi_h, \\epsilon_{ijk} m_{i\\eta} \\frac{\\delta f}{\\delta m_{\\eta j}} \\right)\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

The variational derivatives of $var element = document.getElementById("moose-equation-7c0b4db0-8e64-46bd-b588-a0748d25bb21");katex.render("f", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ can be computed as,

$var element = document.getElementById("moose-equation-e5b2ad03-aea2-4e6d-ac60-811076ca0e5a");katex.render(" \\begin{aligned} \\frac{\\delta f}{\\delta m_{\\eta j}} &= M_s \\frac{\\partial}{\\partial m_{\\eta j}} \\left(- H_0 \\left(m_{1z} + m_{2z}\\right) + H_\\mathrm{E} m_{1 \\beta} m_{2 \\beta} - \\frac{H_\\mathrm{A}}{2} \\left(m_{1z}^2 + m_{2z}^2\\right)\\right)\\\\ &= M_s \\left(- H_0 \\delta_{zj} + H_\\mathrm{E} \\delta_{\\beta j} m_{\\eta \\beta} - H_\\mathrm{A} \\delta_{zj} m_{\\eta z}\\right)\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

Therefore,

$var element = document.getElementById("moose-equation-fecffd23-05f3-47d4-9900-6d8f15f1290b");katex.render(" \\begin{aligned} \\mathcal{R}_{m_{\\eta k}} &= -\\frac{\\gamma}{\\mu_0} \\left(\\psi_h, \\epsilon_{ijk} m_{i\\eta} \\left\\{- H_0 \\delta_{zj} + H_\\mathrm{E} \\delta_{\\beta j} m_{\\eta \\beta} - H_\\mathrm{A} \\delta_{zj} m_{\\eta z}\\right\\} \\right)\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

Example Input File Syntax

Input Parameters

componentAn integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).
mag1_xThe x component of the constrained 1st sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The x component of the constrained 1st sublattice magnetization vector
mag1_yThe y component of the constrained 1st sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The y component of the constrained 1st sublattice magnetization vector
mag1_zThe z component of the constrained 1st sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The z component of the constrained 1st sublattice magnetization vector
mag_subAn integer corresponding to the sublattice this Kernel acts on
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the sublattice this Kernel acts on
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
displacementsThe displacements
C++ Type:std::vector<VariableName>
Controllable:No
Description:The displacements
mag2_xThe x component of the constrained 2nd sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The x component of the constrained 2nd sublattice magnetization vector
mag2_yThe y component of the constrained 2nd sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The y component of the constrained 2nd sublattice magnetization vector
mag2_zThe z component of the constrained 2nd sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The z component of the constrained 2nd sublattice magnetization vector
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

References

S. M. Rezende, A. Azevedo, and R. L. Rodríguez-Suárez. Introduction to antiferromagnetic magnons. Journal of Applied Physics, 2019.

@article{Rezende2019,
    author = "Rezende, S. M. and Azevedo, A. and Rodr\'{i}guez-Su\'{a}rez, R. L.",
    title = "{Introduction to antiferromagnetic magnons}",
    journal = "Journal of Applied Physics",
    volume = "126",
    eid = "151101",
    year = "2019"
}

(tutorial/AFMR_MnF2_ex.i)


##see J. Appl. Phys. 126, 151101 (2019);

Nx = 2
Ny = 2
Nz = 2

xMax = 0.01  # in micrometers = 10 nm
yMax = 0.01
zMax = 0.01


[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 3
    nx = ${Nx}
    ny = ${Ny}
    nz = ${Nz}
    xmin = 0.0
    xmax = ${xMax}
    ymin = 0.0
    ymax = ${yMax}
    zmin = 0.0
    zmax = ${zMax}
    elem_type = HEX8
  []
[]

[GlobalParams]
  mag1_x = mag1_x
  mag1_y = mag1_y
  mag1_z = mag1_z

  mag2_x = mag2_x
  mag2_y = mag2_y
  mag2_z = mag2_z
[]

[Materials]
  [./constants_kOe] # Constants used in other material properties
    type = GenericConstantMaterial
    prop_names = ' H0         Ms        g0          He        Ha      '
    prop_values = '8.0e-5    1.0      2.8e9      0.000526     8.2e-6      '
  [../]
  [./constants] # Constants used in other material properties
    type = GenericConstantMaterial
    prop_names = ' alpha     mu0   nx ny nz   long_susc t'
    prop_values = '0.0       1256   0  0  1          1.0     0'
  [../]

  [./a_long]
    type = GenericFunctionMaterial
    prop_names = 'alpha_long'
    prop_values = 'bc_func_1'
  [../]
[]

[Functions]
  [./bc_func_1]
    type = ParsedFunction
    value = 'st'
    vars = 'st'
    vals = '0.5'
  [../]
[]

[Variables]
  [./mag1_x]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi1
      theta = polar_theta1
      M0s = 1.0 #amplitude of the RandomConstrainedVectorFieldIC
      component  = 0
    [../]
  [../]
  [./mag1_y]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi1
      theta = polar_theta1
      M0s = 1.0
      component  = 1
    [../]
  [../]
  [./mag1_z]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi1
      theta = polar_theta1
      M0s = 1.0
      component  = 2
    [../]
  [../]

  [./mag2_x]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi2
      theta = polar_theta2
      M0s = 1.0
      component  = 0
    [../]
  [../]
  [./mag2_y]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi2
      theta = polar_theta2
      M0s = 1.0
      component  = 1
    [../]
  [../]
  [./mag2_z]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi2
      theta = polar_theta2
      M0s = 1.0
      component  = 2
    [../]
  [../]

[]

[AuxVariables]

  #--------------------------------------------#
  #                                            #
  #  field to seed IC that obeys constraint    #
  #                                            #
  #--------------------------------------------#

  [./azimuth_phi1]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0.3
      max = 0.35
      seed = 2
    [../]
  [../]
  [./polar_theta1]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0.0001
      max = 0.0002
      seed = 37
    [../]
  [../]

  [./azimuth_phi2]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0.3
      max = 0.35
      seed = 2
    [../]
  [../]
  [./polar_theta2]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 3.1415
      max = 3.1416
      seed = 37
    [../]
  [../]

  [./mag1_s]
    order = FIRST
    family = LAGRANGE
  [../]
  [./mag2_s]
    order = FIRST
    family = LAGRANGE
  [../]

  [./Neel_L_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./Neel_L_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./Neel_L_z]
    order = FIRST
    family = LAGRANGE
  [../]

  [./SSMag_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./SSMag_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./SSMag_z]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  #---------------------------------------#
  #                                       #
  #          Time dependence              #
  #                                       #
  #---------------------------------------#

  [./mag1_x_time]
    type = TimeDerivative
    variable = mag1_x
  [../]
  [./mag1_y_time]
    type = TimeDerivative
    variable = mag1_y
  [../]
  [./mag1_z_time]
    type = TimeDerivative
    variable = mag1_z
  [../]

  [./mag2_x_time]
    type = TimeDerivative
    variable = mag2_x
  [../]
  [./mag2_y_time]
    type = TimeDerivative
    variable = mag2_y
  [../]
  [./mag2_z_time]
    type = TimeDerivative
    variable = mag2_z
  [../]

  #---------------------------------------#
  #                                       #
  #     AFM resonance kernel terms        #
  #                                       #
  #---------------------------------------#

  [./afmr1_x]
    type = UniaxialAFMSublattice
    variable = mag1_x
    mag_sub = 0
    component = 0
  [../]
  [./afmr1_y]
    type = UniaxialAFMSublattice
    variable = mag1_y
    mag_sub = 0
    component = 1
  [../]
  [./afmr1_z]
    type = UniaxialAFMSublattice
    variable = mag1_z
    mag_sub = 0
    component = 2
  [../]

  [./afmr2_x]
    type = UniaxialAFMSublattice
    variable = mag2_x
    mag_sub = 1
    component = 0
  [../]
  [./afmr2_y]
    type = UniaxialAFMSublattice
    variable = mag2_y
    mag_sub = 1
    component = 1
  [../]
  [./afmr2_z]
    type = UniaxialAFMSublattice
    variable = mag2_z
    mag_sub = 1
    component = 2
  [../]

  #---------------------------------------#
  #                                       #
  #          LLB constraint terms         #
  #                                       #
  #---------------------------------------#

  [./llb1_x]
    type = LongitudinalLLB
    variable = mag1_x
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 0
  [../]
  [./llb1_y]
    type = LongitudinalLLB
    variable = mag1_y
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 1
  [../]

  [./llb1_z]
    type = LongitudinalLLB
    variable = mag1_z
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 2
  [../]

  [./llb2_x]
    type = LongitudinalLLB
    variable = mag2_x
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 0
  [../]
  [./llb2_y]
    type = LongitudinalLLB
    variable = mag2_y
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 1
  [../]

  [./llb2_z]
    type = LongitudinalLLB
    variable = mag2_z
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 2
  [../]
[]


[AuxKernels]
  [./mag1_mag]
    type = VectorMag
    variable = mag1_s
    vector_x = mag1_x
    vector_y = mag1_y
    vector_z = mag1_z
    execute_on = 'initial timestep_end final'
  [../]

  [./mag2_mag]
    type = VectorMag
    variable = mag2_s
    vector_x = mag2_x
    vector_y = mag2_y
    vector_z = mag2_z
    execute_on = 'initial timestep_end final'
  [../]


  [./Neel_Lx]
    type = VectorDiffOrSum
    variable = Neel_L_x
    var1 = mag1_x
    var2 = mag2_x
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]
  [./Neel_Ly]
    type = VectorDiffOrSum
    variable = Neel_L_y
    var1 = mag1_y
    var2 = mag2_y
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]
  [./Neel_Lz]
    type = VectorDiffOrSum
    variable = Neel_L_z
    var1 = mag1_z
    var2 = mag2_z
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]

  [./smallSignalMag_x]
    type = VectorDiffOrSum
    variable = SSMag_x
    var1 = mag1_x
    var2 = mag2_x
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]
  [./smallSignalMag_y]
    type = VectorDiffOrSum
    variable = SSMag_y
    var1 = mag1_y
    var2 = mag2_y
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]
  [./smallSignalMag_z]
    type = VectorDiffOrSum
    variable = SSMag_z
    var1 = mag1_z
    var2 = mag2_z
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]

[]

[Kernels]
  #---------------------------------------#
  #                                       #
  #          Time dependence              #
  #                                       #
  #---------------------------------------#

  [./mag1_x_time]
    type = TimeDerivative
    variable = mag1_x
  [../]
  [./mag1_y_time]
    type = TimeDerivative
    variable = mag1_y
  [../]
  [./mag1_z_time]
    type = TimeDerivative
    variable = mag1_z
  [../]

  [./mag2_x_time]
    type = TimeDerivative
    variable = mag2_x
  [../]
  [./mag2_y_time]
    type = TimeDerivative
    variable = mag2_y
  [../]
  [./mag2_z_time]
    type = TimeDerivative
    variable = mag2_z
  [../]

  #---------------------------------------#
  #                                       #
  #     AFM resonance kernel terms        #
  #                                       #
  #---------------------------------------#

  [./afmr1_x]
    type = UniaxialAFMSublattice
    variable = mag1_x
    mag_sub = 0
    component = 0
  [../]
  [./afmr1_y]
    type = UniaxialAFMSublattice
    variable = mag1_y
    mag_sub = 0
    component = 1
  [../]
  [./afmr1_z]
    type = UniaxialAFMSublattice
    variable = mag1_z
    mag_sub = 0
    component = 2
  [../]

  [./afmr2_x]
    type = UniaxialAFMSublattice
    variable = mag2_x
    mag_sub = 1
    component = 0
  [../]
  [./afmr2_y]
    type = UniaxialAFMSublattice
    variable = mag2_y
    mag_sub = 1
    component = 1
  [../]
  [./afmr2_z]
    type = UniaxialAFMSublattice
    variable = mag2_z
    mag_sub = 1
    component = 2
  [../]

  #---------------------------------------#
  #                                       #
  #          LLB constraint terms         #
  #                                       #
  #---------------------------------------#

  [./llb1_x]
    type = LongitudinalLLB
    variable = mag1_x
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 0
  [../]
  [./llb1_y]
    type = LongitudinalLLB
    variable = mag1_y
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 1
  [../]

  [./llb1_z]
    type = LongitudinalLLB
    variable = mag1_z
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 2
  [../]

  [./llb2_x]
    type = LongitudinalLLB
    variable = mag2_x
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 0
  [../]
  [./llb2_y]
    type = LongitudinalLLB
    variable = mag2_y
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 1
  [../]

  [./llb2_z]
    type = LongitudinalLLB
    variable = mag2_z
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 2
  [../]
[]

[BCs]
  #---------------------------------------#
  #                                       #
  #  periodic magnetization distribution  #
  #                                       #
  #---------------------------------------#

  [./Periodic]
    [./xyz]
      auto_direction = 'x y z'
      variable = 'mag1_x mag1_y mag1_z mag2_x mag2_y mag2_z'
    [../]
  [../]
[]

[Postprocessors]
   [./dt]
     type = TimestepSize
   [../]

  #---------------------------------------#
  #                                       #
  #       Average |M| and along other     #
  #       directions                      #
  #                                       #
  #---------------------------------------#

  [./<M1>]
    type = ElementAverageValue
    variable = mag1_s
    execute_on = 'initial timestep_end final'
  [../]
  [./<M2>]
    type = ElementAverageValue
    variable = mag2_s
    execute_on = 'initial timestep_end final'
  [../]

  [./<m1x>]
    type = ElementAverageValue
    variable = mag1_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<m1y>]
    type = ElementAverageValue
    variable = mag1_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<m1z>]
    type = ElementAverageValue
    variable = mag1_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<m2x>]
    type = ElementAverageValue
    variable = mag2_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<m2y>]
    type = ElementAverageValue
    variable = mag2_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<m2z>]
    type = ElementAverageValue
    variable = mag2_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<Lx>]
    type = ElementAverageValue
    variable = Neel_L_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<Ly>]
    type = ElementAverageValue
    variable = Neel_L_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<Lz>]
    type = ElementAverageValue
    variable = Neel_L_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<SSmx>]
    type = ElementAverageValue
    variable = SSMag_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<SSmy>]
    type = ElementAverageValue
    variable = SSMag_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<SSmz>]
    type = ElementAverageValue
    variable = SSMag_z
    execute_on = 'initial timestep_end final'
  [../]
[]


[Preconditioning]
  #---------------------------------------#
  #                                       #
  #            Solver options             #
  #                                       #
  #---------------------------------------#

  [./smp]
    type = SMP
    full = true
    petsc_options = '-snes_ksp_ew'
    petsc_options_iname = ' -ksp_gmres_restart -snes_atol -snes_rtol -ksp_rtol -pc_type'
    petsc_options_value = '    121               1e-8      1e-6      1e-6      bjacobi'
  [../]
[]

[Executioner]
  type = Transient            
  solve_type = 'NEWTON'
  [./TimeIntegrator]
    type = NewmarkBeta
  [../]
  dtmin = 1e-11
  dtmax = 1.0e-3
  [./TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 25
    iteration_window = 2000
    growth_factor = 1.3
    cutback_factor = 0.8
    dt = 12.0e-9
  [../]
  verbose = true

  num_steps = 18000
[]

[Outputs]
  print_linear_residuals = false
  [./out]
    type = Exodus
    file_base = out_AFMR_ex
    elemental_as_nodal = true
    interval = 500
  [../]
  [./outCSV]
    type = CSV
    file_base = out_AFMR_ex
    interval = 1
  [../]
[]

(test/tests/magnets/AFMR_MnF2_ex.i)


##see J. Appl. Phys. 126, 151101 (2019);


Nx = 2
Ny = 2
Nz = 2

xMax = 0.01  # = 10 nm
yMax = 0.01
zMax = 0.01


[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 3
    nx = ${Nx}
    ny = ${Ny}
    nz = ${Nz}
    xmin = 0.0
    xmax = ${xMax}
    ymin = 0.0
    ymax = ${yMax}
    zmin = 0.0
    zmax = ${zMax}
    elem_type = HEX8
  []
  [./cnode]
    input = gen

    ############################################
    ##
    ##   additional boundary sideset (one node) 
    ##   to zero one of the elastic displacement vectors 
    ##   vectors and eliminates rigid body translations 
    ##   from the degrees of freedom
    ##
    ##   NOTE: This must conform with the about
    ##         [Mesh] block settings
    ##
    ############################################

    type = ExtraNodesetGenerator
    coord = '0.0 0.0 0.0'
    new_boundary = 100
  [../]
[]

[GlobalParams]
  mag1_x = mag1_x
  mag1_y = mag1_y
  mag1_z = mag1_z

  mag2_x = mag2_x
  mag2_y = mag2_y
  mag2_z = mag2_z
[]

[Materials]
  [./constants_kOe] # Constants used in other material properties
    type = GenericConstantMaterial
    prop_names = ' H0         Ms        g0          He        Ha      '
    prop_values = '8.0e-5    1.0      2.8e9      0.000526     8.2e-6      '
  [../]


  [./constants] # Constants used in other material properties
    type = GenericConstantMaterial
    prop_names = ' alpha     mu0   nx ny nz   long_susc t'
    prop_values = '0.0       1256   0  0  1          1.0     0'
  [../]

  [./a_long]
    type = GenericFunctionMaterial
    prop_names = 'alpha_long'
    prop_values = 'bc_func_1'
  [../]
[]

[Functions]

  ##############################
  ##
  ## Define the ramping function
  ## expression to be used
  ##
  ##############################

  [./bc_func_1]
    type = ParsedFunction
    value = 'st'   #*tanh(sl*t)+1.0'
    vars = 'st' #sl'
    vals = '0.5' # 795.775'
  [../]
[]

[Variables]
  [./mag1_x]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi1
      theta = polar_theta1
      M0s = 1.0 #amplitude of the RandomConstrainedVectorFieldIC
      component  = 0
    [../]
  [../]
  [./mag1_y]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi1
      theta = polar_theta1
      M0s = 1.0 #amplitude of the RandomConstrainedVectorFieldIC
      component  = 1
    [../]
  [../]
  [./mag1_z]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi1
      theta = polar_theta1
      M0s = 1.0 #amplitude of the RandomConstrainedVectorFieldIC
      component  = 2
    [../]
  [../]

  [./mag2_x]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi2
      theta = polar_theta2
      M0s = 1.0 #amplitude of the RandomConstrainedVectorFieldIC
      component  = 0
    [../]
  [../]
  [./mag2_y]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi2
      theta = polar_theta2
      M0s = 1.0 #amplitude of the RandomConstrainedVectorFieldIC
      component  = 1
    [../]
  [../]
  [./mag2_z]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomConstrainedVectorFieldIC
      phi = azimuth_phi2
      theta = polar_theta2
      M0s = 1.0 #amplitude of the RandomConstrainedVectorFieldIC
      component  = 2
    [../]
  [../]

[]

[AuxVariables]
  #--------------------------------------------#
  #                                            #
  #  field to seed IC that obeys constraint    #
  #                                            #
  #--------------------------------------------#
  [./azimuth_phi1]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0.3
      max = 0.35
      seed = 2
    [../]
  [../]
  [./polar_theta1]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0.2
      max = 0.21
      seed = 37
    [../]
  [../]

  [./azimuth_phi2]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 0.3
      max = 0.35
      seed = 2
    [../]
  [../]
  [./polar_theta2]
    order = FIRST
    family = LAGRANGE
    [./InitialCondition]
      type = RandomIC
      min = 2.92
      max = 2.93
      seed = 37
    [../]
  [../]

  [./mag1_s]
    order = FIRST
    family = LAGRANGE
  [../]
  [./mag2_s]
    order = FIRST
    family = LAGRANGE
  [../]

  [./Neel_L_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./Neel_L_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./Neel_L_z]
    order = FIRST
    family = LAGRANGE
  [../]


  [./SSMag_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./SSMag_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./SSMag_z]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxKernels]
  [./mag1_mag]
    type = VectorMag
    variable = mag1_s
    vector_x = mag1_x
    vector_y = mag1_y
    vector_z = mag1_z
    execute_on = 'initial timestep_end final'
  [../]

  [./mag2_mag]
    type = VectorMag
    variable = mag2_s
    vector_x = mag2_x
    vector_y = mag2_y
    vector_z = mag2_z
    execute_on = 'initial timestep_end final'
  [../]


  [./Neel_Lx]
    type = VectorDiffOrSum
    variable = Neel_L_x
    var1 = mag1_x
    var2 = mag2_x
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]
  [./Neel_Ly]
    type = VectorDiffOrSum
    variable = Neel_L_y
    var1 = mag1_y
    var2 = mag2_y
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]
  [./Neel_Lz]
    type = VectorDiffOrSum
    variable = Neel_L_z
    var1 = mag1_z
    var2 = mag2_z
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]

  [./smallSignalMag_x]
    type = VectorDiffOrSum
    variable = SSMag_x
    var1 = mag1_x
    var2 = mag2_x
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]
  [./smallSignalMag_y]
    type = VectorDiffOrSum
    variable = SSMag_y
    var1 = mag1_y
    var2 = mag2_y
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]
  [./smallSignalMag_z]
    type = VectorDiffOrSum
    variable = SSMag_z
    var1 = mag1_z
    var2 = mag2_z
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]

[]

[Kernels]
  #---------------------------------------#
  #                                       #
  #          Time dependence              #
  #                                       #
  #---------------------------------------#

  [./mag1_x_time]
    type = TimeDerivative
    variable = mag1_x
  [../]
  [./mag1_y_time]
    type = TimeDerivative
    variable = mag1_y
  [../]
  [./mag1_z_time]
    type = TimeDerivative
    variable = mag1_z
  [../]

  [./mag2_x_time]
    type = TimeDerivative
    variable = mag2_x
  [../]
  [./mag2_y_time]
    type = TimeDerivative
    variable = mag2_y
  [../]
  [./mag2_z_time]
    type = TimeDerivative
    variable = mag2_z
  [../]


  #---------------------------------------#
  #                                       #
  #     AFM resonance kernel terms        #
  #                                       #
  #---------------------------------------#

  [./afmr1_x]
    type = UniaxialAFMSublattice
    variable = mag1_x
    mag_sub = 0
    component = 0
  [../]
  [./afmr1_y]
    type = UniaxialAFMSublattice
    variable = mag1_y
    mag_sub = 0
    component = 1
  [../]
  [./afmr1_z]
    type = UniaxialAFMSublattice
    variable = mag1_z
    mag_sub = 0
    component = 2
  [../]


  [./afmr2_x]
    type = UniaxialAFMSublattice
    variable = mag2_x
    mag_sub = 1
    component = 0
  [../]
  [./afmr2_y]
    type = UniaxialAFMSublattice
    variable = mag2_y
    mag_sub = 1
    component = 1
  [../]
  [./afmr2_z]
    type = UniaxialAFMSublattice
    variable = mag2_z
    mag_sub = 1
    component = 2
  [../]

  #---------------------------------------#
  #                                       #
  #          LLB constraint terms         #
  #                                       #
  #---------------------------------------#

  [./llb1_x]
    type = LongitudinalLLB
    variable = mag1_x
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 0
  [../]
  [./llb1_y]
    type = LongitudinalLLB
    variable = mag1_y
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 1
  [../]

  [./llb1_z]
    type = LongitudinalLLB
    variable = mag1_z
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 2
  [../]

  [./llb2_x]
    type = LongitudinalLLB
    variable = mag2_x
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 0
  [../]
  [./llb2_y]
    type = LongitudinalLLB
    variable = mag2_y
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 1
  [../]

  [./llb2_z]
    type = LongitudinalLLB
    variable = mag2_z
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 2
  [../]
[]

[BCs]
  #---------------------------------------#
  #                                       #
  #  periodic magnetization distribution  #
  #                                       #
  #---------------------------------------#

  [./Periodic]
    [./xyz]
      auto_direction = 'x y z'
      variable = 'mag1_x mag1_y mag1_z mag2_x mag2_y mag2_z'
    [../]
  [../]

[]

[Postprocessors]
   [./dt]
     type = TimestepSize
   [../]

  #---------------------------------------#
  #                                       #
  #       Average |M| and along other     #
  #       directions                      #
  #                                       #
  #---------------------------------------#

  [./<M1>]
    type = ElementAverageValue
    variable = mag1_s
    execute_on = 'initial timestep_end final'
  [../]
  [./<M2>]
    type = ElementAverageValue
    variable = mag2_s
    execute_on = 'initial timestep_end final'
  [../]


  [./<m1x>]
    type = ElementAverageValue
    variable = mag1_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<m1y>]
    type = ElementAverageValue
    variable = mag1_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<m1z>]
    type = ElementAverageValue
    variable = mag1_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<m2x>]
    type = ElementAverageValue
    variable = mag2_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<m2y>]
    type = ElementAverageValue
    variable = mag2_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<m2z>]
    type = ElementAverageValue
    variable = mag2_z
    execute_on = 'initial timestep_end final'
  [../]


  [./<Lx>]
    type = ElementAverageValue
    variable = Neel_L_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<Ly>]
    type = ElementAverageValue
    variable = Neel_L_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<Lz>]
    type = ElementAverageValue
    variable = Neel_L_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<SSmx>]
    type = ElementAverageValue
    variable = SSMag_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<SSmy>]
    type = ElementAverageValue
    variable = SSMag_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<SSmz>]
    type = ElementAverageValue
    variable = SSMag_z
    execute_on = 'initial timestep_end final'
  [../]
[]


[Preconditioning]
  #---------------------------------------#
  #                                       #
  #            Solver options             #
  #                                       #
  #---------------------------------------#

  [./smp]
    type = SMP
    full = true
    petsc_options = '-snes_ksp_ew'
    petsc_options_iname = ' -ksp_gmres_restart -snes_atol -snes_rtol -ksp_rtol -pc_type'
    petsc_options_value = '    121               1e-8      1e-6      1e-6      bjacobi'
  [../]
[]

[Executioner]
  type = Transient            
  solve_type = 'NEWTON'
  [./TimeIntegrator]
    type = ImplicitEuler
  [../]
  dtmin = 1e-11
  dtmax = 1.0e-3
  [./TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 25
    iteration_window = 2000
    growth_factor = 1.3
    cutback_factor = 0.8
    dt = 4.0e-9
  [../]
  verbose = true

  num_steps = 2
[]

[Outputs]
  print_linear_residuals = false
  [./out]
    type = Exodus
    file_base = out_AFMR_ex
    elemental_as_nodal = true
    interval = 1
  [../]
  [./outCSV]
    type = CSV
    file_base = out_AFMR_ex
    interval = 1
  [../]
[]