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<h1>Source code for grogupy.magnetism</h1><div class="highlight"><pre>
<span></span><span class="c1"># Copyright (c) [2024] []</span>
<span class="c1">#</span>
<span class="c1"># Permission is hereby granted, free of charge, to any person obtaining a copy</span>
<span class="c1"># of this software and associated documentation files (the &quot;Software&quot;), to deal</span>
<span class="c1"># in the Software without restriction, including without limitation the rights</span>
<span class="c1"># to use, copy, modify, merge, publish, distribute, sublicense, and/or sell</span>
<span class="c1"># copies of the Software, and to permit persons to whom the Software is</span>
<span class="c1"># furnished to do so, subject to the following conditions:</span>
<span class="c1">#</span>
<span class="c1"># The above copyright notice and this permission notice shall be included in all</span>
<span class="c1"># copies or substantial portions of the Software.</span>
<span class="c1">#</span>
<span class="c1"># THE SOFTWARE IS PROVIDED &quot;AS IS&quot;, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR</span>
<span class="c1"># IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,</span>
<span class="c1"># FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE</span>
<span class="c1"># AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER</span>
<span class="c1"># LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,</span>
<span class="c1"># OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE</span>
<span class="c1"># SOFTWARE.</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<div class="viewcode-block" id="blow_up_orbindx">
<a class="viewcode-back" href="../../implementation/grogupy.magnetism.html#grogupy.magnetism.blow_up_orbindx">[docs]</a>
<span class="k">def</span> <span class="nf">blow_up_orbindx</span><span class="p">(</span><span class="n">orb_indices</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;Function to blow up orbital indices to make SPIN BOX indices.</span>
<span class="sd"> Args:</span>
<span class="sd"> orb_indices : np.array_like</span>
<span class="sd"> These are the indices in ORBITAL BOX</span>
<span class="sd"> Returns:</span>
<span class="sd"> orb_indices : np.array_like</span>
<span class="sd"> These are the indices in SPIN BOX</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">orb_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">2</span> <span class="o">*</span> <span class="n">o</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">o</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">orb_indices</span><span class="p">])</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
<span class="k">return</span> <span class="n">orb_indices</span></div>
<div class="viewcode-block" id="spin_tracer">
<a class="viewcode-back" href="../../implementation/grogupy.magnetism.html#grogupy.magnetism.spin_tracer">[docs]</a>
<span class="k">def</span> <span class="nf">spin_tracer</span><span class="p">(</span><span class="n">M</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;Spin tracer utility.</span>
<span class="sd"> This takes an operator with the orbital-spin sequence:</span>
<span class="sd"> orbital 1 up,</span>
<span class="sd"> orbital 1 down,</span>
<span class="sd"> orbital 2 up,</span>
<span class="sd"> orbital 2 down,</span>
<span class="sd"> that is in the SPIN-BOX representation,</span>
<span class="sd"> and extracts orbital dependent Pauli traces.</span>
<span class="sd"> Args:</span>
<span class="sd"> M : np.array_like</span>
<span class="sd"> Traceable matrix</span>
<span class="sd"> Returns:</span>
<span class="sd"> dict</span>
<span class="sd"> It contains the traced matrix with &quot;x&quot;, &quot;y&quot;, &quot;z&quot; and &quot;c&quot;</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">M11</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
<span class="n">M12</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
<span class="n">M21</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
<span class="n">M22</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
<span class="n">M_o</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
<span class="n">M_o</span><span class="p">[</span><span class="s2">&quot;x&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">M12</span> <span class="o">+</span> <span class="n">M21</span>
<span class="n">M_o</span><span class="p">[</span><span class="s2">&quot;y&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="p">(</span><span class="n">M12</span> <span class="o">-</span> <span class="n">M21</span><span class="p">)</span>
<span class="n">M_o</span><span class="p">[</span><span class="s2">&quot;z&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">M11</span> <span class="o">-</span> <span class="n">M22</span>
<span class="n">M_o</span><span class="p">[</span><span class="s2">&quot;c&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">M11</span> <span class="o">+</span> <span class="n">M22</span>
<span class="k">return</span> <span class="n">M_o</span></div>
<div class="viewcode-block" id="parse_magnetic_entity">
<a class="viewcode-back" href="../../implementation/grogupy.magnetism.html#grogupy.magnetism.parse_magnetic_entity">[docs]</a>
<span class="k">def</span> <span class="nf">parse_magnetic_entity</span><span class="p">(</span><span class="n">dh</span><span class="p">,</span> <span class="n">atom</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">l</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;Function to define orbital indexes of a given magnetic entity.</span>
<span class="sd"> Args:</span>
<span class="sd"> dh : sisl.physics.Hamiltonian</span>
<span class="sd"> Hamiltonian from sisl</span>
<span class="sd"> atom : integer or list of integers, optional</span>
<span class="sd"> Defining atom (or atoms) in the unit cell forming the magnetic entity. Defaults to None</span>
<span class="sd"> l : integer, optional</span>
<span class="sd"> Defining the angular momentum channel. Defaults to None</span>
<span class="sd"> Returns:</span>
<span class="sd"> list</span>
<span class="sd"> The orbital indexes of the given magnetic entity</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># case where we deal with more than one atom defining the magnetic entity</span>
<span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">atom</span><span class="p">)</span> <span class="o">==</span> <span class="nb">list</span><span class="p">:</span>
<span class="n">dat</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">atom</span><span class="p">:</span>
<span class="n">a_orb_idx</span> <span class="o">=</span> <span class="n">dh</span><span class="o">.</span><span class="n">geometry</span><span class="o">.</span><span class="n">a2o</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">if</span> <span class="p">(</span>
<span class="nb">type</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span>
<span class="p">):</span> <span class="c1"># if specified we restrict to given l angular momentum channel inside each atom</span>
<span class="n">a_orb_idx</span> <span class="o">=</span> <span class="n">a_orb_idx</span><span class="p">[[</span><span class="n">o</span><span class="o">.</span><span class="n">l</span> <span class="o">==</span> <span class="n">l</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">dh</span><span class="o">.</span><span class="n">geometry</span><span class="o">.</span><span class="n">atoms</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">.</span><span class="n">orbitals</span><span class="p">]]</span>
<span class="n">dat</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a_orb_idx</span><span class="p">)</span>
<span class="n">orbital_indeces</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">(</span><span class="n">dat</span><span class="p">)</span>
<span class="c1"># case where we deal with a singel atom magnetic entity</span>
<span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">atom</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span><span class="p">:</span>
<span class="n">orbital_indeces</span> <span class="o">=</span> <span class="n">dh</span><span class="o">.</span><span class="n">geometry</span><span class="o">.</span><span class="n">a2o</span><span class="p">(</span><span class="n">atom</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">if</span> <span class="p">(</span>
<span class="nb">type</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span>
<span class="p">):</span> <span class="c1"># if specified we restrict to given l angular momentum channel</span>
<span class="n">orbital_indeces</span> <span class="o">=</span> <span class="n">orbital_indeces</span><span class="p">[</span>
<span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">l</span> <span class="o">==</span> <span class="n">l</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">dh</span><span class="o">.</span><span class="n">geometry</span><span class="o">.</span><span class="n">atoms</span><span class="p">[</span><span class="n">atom</span><span class="p">]</span><span class="o">.</span><span class="n">orbitals</span><span class="p">]</span>
<span class="p">]</span>
<span class="k">return</span> <span class="n">orbital_indeces</span> <span class="c1"># numpy array containing integers labeling orbitals associated to a magnetic entity.</span></div>
<div class="viewcode-block" id="calculate_anisotropy_tensor">
<a class="viewcode-back" href="../../implementation/grogupy.magnetism.html#grogupy.magnetism.calculate_anisotropy_tensor">[docs]</a>
<span class="k">def</span> <span class="nf">calculate_anisotropy_tensor</span><span class="p">(</span><span class="n">mag_ent</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;Calculates the renormalized anisotropy tensor from the energies.</span>
<span class="sd"> It uses the grogu convention for output.</span>
<span class="sd"> Args:</span>
<span class="sd"> mag_ent : dict</span>
<span class="sd"> An element from the magnetic entities</span>
<span class="sd"> Returns:</span>
<span class="sd"> K : np.array_like</span>
<span class="sd"> elements of the anisotropy tensor</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># get the energies</span>
<span class="n">energies</span> <span class="o">=</span> <span class="n">mag_ent</span><span class="p">[</span><span class="s2">&quot;energies&quot;</span><span class="p">]</span>
<span class="c1"># calculate the diagonal tensor elements</span>
<span class="n">Kxx</span> <span class="o">=</span> <span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">Kyy</span> <span class="o">=</span> <span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">Kzz</span> <span class="o">=</span> <span class="mi">0</span>
<span class="c1"># perform consistency check</span>
<span class="n">calculated_diff</span> <span class="o">=</span> <span class="n">Kyy</span> <span class="o">-</span> <span class="n">Kxx</span>
<span class="n">expected_diff</span> <span class="o">=</span> <span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="n">consistency_check</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">calculated_diff</span> <span class="o">-</span> <span class="n">expected_diff</span><span class="p">)</span>
<span class="k">return</span> <span class="n">Kxx</span><span class="p">,</span> <span class="n">Kyy</span><span class="p">,</span> <span class="n">Kzz</span><span class="p">,</span> <span class="n">consistency_check</span></div>
<div class="viewcode-block" id="calculate_exchange_tensor">
<a class="viewcode-back" href="../../implementation/grogupy.magnetism.html#grogupy.magnetism.calculate_exchange_tensor">[docs]</a>
<span class="k">def</span> <span class="nf">calculate_exchange_tensor</span><span class="p">(</span><span class="n">pair</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;Calculates the exchange tensor from the energies.</span>
<span class="sd"> It produces the isotropic exchange, the relevant elements</span>
<span class="sd"> from the Dzyaloshinskii-Morilla (Dm) tensor, the symmetric-anisotropy</span>
<span class="sd"> and the complete exchange tensor.</span>
<span class="sd"> Args:</span>
<span class="sd"> pair : dict</span>
<span class="sd"> An element from the pairs</span>
<span class="sd"> Returns:</span>
<span class="sd"> J_iso : float</span>
<span class="sd"> Isotropic exchange (Tr[J] / 3)</span>
<span class="sd"> J_S : np.array_like</span>
<span class="sd"> Symmetric-anisotropy (J_S = J - J_iso * I ––&gt; Jxx, Jyy, Jxy, Jxz, Jyz)</span>
<span class="sd"> D : np.array_like</span>
<span class="sd"> DM elements (Dx, Dy, Dz)</span>
<span class="sd"> J : np.array_like</span>
<span class="sd"> Complete exchange tensor flattened (Jxx, Jxy, Jxz, Jyx, Jyy, Jyz, Jzx, Jzy, Jzz)</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># energies from rotations</span>
<span class="n">energies</span> <span class="o">=</span> <span class="n">pair</span><span class="p">[</span><span class="s2">&quot;energies&quot;</span><span class="p">]</span>
<span class="c1"># Initialize output arrays</span>
<span class="n">J</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="c1"># J matrix calculations</span>
<span class="c1"># J(1,1) = mean([DEij(2,2,2), DEij(2,2,3)])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
<span class="c1"># J(1,2) = -mean([DEij(1,2,3), DEij(2,1,3)])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="c1"># J(1,3) = -mean([DEij(1,2,2), DEij(2,1,2)])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="c1"># J(2,2) = mean([DEij(2,2,1), DEij(1,1,3)])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="c1"># J(2,3) = -mean([DEij(1,2,1), DEij(2,1,1)])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">J</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="c1"># J(3,3) = mean([DEij(1,1,1), DEij(1,1,2)])</span>
<span class="n">J</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="c1"># D vector calculations</span>
<span class="c1"># D(1) = mean([DEij(1,2,1), -DEij(2,1,1)])</span>
<span class="n">D</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="o">-</span><span class="n">energies</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="c1"># D(2) = mean([DEij(2,1,2), -DEij(1,2,2)])</span>
<span class="n">D</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="o">-</span><span class="n">energies</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="c1"># D(3) = mean([DEij(1,2,3), -DEij(2,1,3)])</span>
<span class="n">D</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="o">-</span><span class="n">energies</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="n">J_iso</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">trace</span><span class="p">(</span><span class="n">J</span><span class="p">)</span> <span class="o">/</span> <span class="mi">3</span>
<span class="c1"># based on the grogu output pdf</span>
<span class="c1"># traceless symmetric exchange matrix:</span>
<span class="c1"># Jxx, Jyy, Jxy, Jxz, Jyz</span>
<span class="n">J_S</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">J_iso</span><span class="p">,</span> <span class="n">J</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">J_iso</span><span class="p">,</span> <span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">J</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">J</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="k">return</span> <span class="n">J_iso</span><span class="p">,</span> <span class="n">J_S</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">J</span></div>
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