Cover Image

Saved in:
Bibliographic Details
Main Author: Caspary, Reinhard
Format: Recurso digital
Language:English
Published: Zenodo 2026
Online Access:https://doi.org/10.5281/zenodo.20401184
Tags: Add Tag
No Tags, Be the first to tag this record!
Table of Contents:
  • <p> This dataset provides angular matrices of spherical tensor operators for the f<sup>12</sup> configuration (lanthanide ion Tm<sup>3+</sup>) in exact arithmetic. The matrices are derived from the software package <a href="https://github.com/reincas/AMELI">AMELI</a> (Angular Matrix Elements of Lanthanide Ions). They have been calculated in the determinantal product state space and transformed to $LS$-coupling. Each $LS$-state is classified by a complete set of irreducible representations matching those published by C. W. Nielson and G. F. Koster. The Hamilton matrices are comprehensively compared and verified with the values published by B. R. Judd and W. T. Carnall. </p> <h2>Matrix Storage Format</h2> <p> The value $v$ of each non-zero matrix element is represented by three values $s$, $n$, and $d$ as signed square root of a rational number: $$v=(-1)^s\sqrt{\frac{n}{d}}\ .$$ The storage format leverages the sparsity of the matrices. It is closely related to the COO standard format for sparse matrices with three vectors for the row index, column index, and value index of each non-zero matrix element. For symmetric matrices only the lower triangle matrix is stored. The actual values are stored in a separate list of unique values, the length of which is usually much smaller than the total number of non-zero elements. Storage follows the canonical COO order with row as primary and column as secondary index. The list of unique values is sorted in ascending order. If a matrix contains integer values $n$ or $d$ that exceed the 64-bit maximum of HDF5 data types, its values are split bit-wise and stored in multiple parts. </p><p> Each matrix is stored in a <a href="https://scidatacontainer.readthedocs.io/en/latest/index.html">SciDataContainer</a> file with filename extension <code>.zdc</code>. This container file is a standard ZIP folder with a certain inner structure of metadata in JSON files following the <a href="https://en.wikipedia.org/wiki/FAIR_data">FAIR</a> principles of modern research data management. The matrix data is stored in the HDF5 file <code>data/matrix.hdf5</code> inside the container. A detailed description of the data structures is given in the JSON file <code>meta.json</code>. </p> <h2>Content of Dataset Record</h2> <p> All tensor operator matrices are grouped with respect to their state spaces. The following files are included in this dataset record: </p><table> <tbody><tr> <th>File</th><th>Description</th> </tr> <tr> <td><code>product.zip</code></td> <td>Product state tensor operator matrices in the subfolder <code>product</code>. Contains also the matrix <code>transform.zdc</code> used to transform product states or product state matrices into the $LS$-coupling scheme. Intended for energy level fits and transition calculations for lanthanide ions in <strong>crystalline materials</strong>.</td> </tr> <tr> <td><code>sljm.zip</code></td> <td>Tensor operator matrices in $LS$-coupling in the subfolder <code>sljm</code>.</td> </tr> <tr> <td><code>slj.zip</code></td> <td>Tensor operator matrices of the $J$-multiplets in $LS$-coupling. These are matrices from <code>sljm.zip</code> collapsed to the stretched states $M_J=J$ in the subfolder <code>slj</code> and matrices containing reduced matrix elements in the subfolder <code>sljm_reduced</code>. Intended for energy level fits and Judd-Ofelt calculations for lanthanide ions in <strong>amorphous materials</strong>.</td> </tr> <tr> <td><code>support.zip</code></td> <td>This special file is only useful when the AMELI package is extended to generate new custom operator matrices. In this case the included data containers with intermediate results optimize the calculation speed within the AMELI package.</td> </tr> <tr> <td><code>hashes.json</code></td> <td>The hash codes in this file are used for the automized version management algorithm.</td> </tr> </tbody></table> <p></p> <h2>Matrix Files</h2> <p> Each set of tensor operator matrices contains the following files: </p><table> <tbody><tr><th>Filename</th> <th>Description</th> <th>Tensor Operator</th> <th>Radial Integral</th> </tr><tr><td><code>U_{k}_{q}.zdc</code></td><td>Component $q$ of the total unit tensor operator of rank $k$ in the orbital angular momentum space</td><td>$\mathrm{{U}}^{{(k)}}_q$</td><td></td></tr> <tr><td><code>T_{k}_{q}.zdc</code></td><td>Component $q$ of the total unit tensor operator of rank $k$ in the spin space</td><td>$\mathrm{{T}}^{{(k)}}_q$</td><td></td></tr> <tr><td><code>UU_{k}.zdc</code></td><td>Squared total unit tensor operator of rank $k$ in the orbital angular momentum space</td><td>$(\mathrm{{U}}^{{(k)}}\cdot\mathrm{{U}}^{{(k)}})$</td><td></td></tr> <tr><td><code>TT_{k}.zdc</code></td><td>Squared total unit tensor operator of rank $k$ in the spin space</td><td>$(\mathrm{{T}}^{{(k)}}\cdot\mathrm{{T}}^{{(k)}})$</td><td></td></tr> <tr><td><code>UT_{k}.zdc</code></td><td>Scalar product of the total unit tensor operators of rank $k$ in the orbital and spin spaces</td><td>$(\mathrm{{U}}^{{(k)}}\cdot\mathrm{{T}}^{{(k)}})$</td><td></td></tr> <tr><td><code>C_{k}_{q}.zdc</code></td><td>Component $q$ of the Coulomb operator of rank $k$</td><td>$\mathrm{{C}}^{{(k)}}_q$</td><td></td></tr> <tr><td><code>L_{q}.zdc</code></td><td>Component $q$ of the total orbital angular momentum operator</td><td>$\mathrm{{L}}_q$</td><td></td></tr> <tr><td><code>S_{q}.zdc</code></td><td>Component $q$ of the total spin angular momentum operator</td><td>$\mathrm{{S}}_q$</td><td></td></tr> <tr><td><code>J_{q}.zdc</code></td><td>Component $q$ of the total angular momentum operator</td><td>$\mathrm{{J}}_q$</td><td></td></tr> <tr><td><code>LL.zdc</code></td><td>Squared total orbital angular momentum operator</td><td>$(\mathrm{{L}}\cdot\mathrm{{L}})$</td><td>$\alpha$</td></tr> <tr><td><code>SS.zdc</code></td><td>Squared total spin angular momentum operator</td><td>$(\mathrm{{S}}\cdot\mathrm{{S}})$</td><td></td></tr> <tr><td><code>JJ.zdc</code></td><td>Squared total angular momentum operator</td><td>$(\mathrm{{J}}\cdot\mathrm{{J}})$</td><td></td></tr> <tr><td><code>LS.zdc</code></td><td>Scalar product of the total orbital and spin angular momentum operators</td><td>$(\mathrm{{L}}\cdot\mathrm{{S}})$</td><td></td></tr> <tr><td><code>C2.zdc</code></td><td>Casimir operator of the special group $G_2$</td><td>$\mathrm{{C}}_2(G_2)$</td><td>$\beta$</td></tr> <tr><td><code>CR.zdc</code></td><td>Casimir operator of the special orthogonal (rotational) group in $2l+1$ dimensions</td><td>$\mathrm{{C}}_2(SO(2l+1))$</td><td>$\gamma$</td></tr> <tr><td><code>H1_{k}.zdc</code></td><td>Coulomb first order perturbation Hamiltonian of rank $k$</td><td>$\mathrm{{f}}_k$</td><td>$F^k$</td></tr> <tr><td><code>H2.zdc</code></td><td>Spin-orbit first order perturbation Hamiltonian</td><td>$\mathrm{{z}}$</td><td>$\zeta$</td></tr> <tr><td><code>H4_{c}.zdc</code></td><td>Effective Coulomb second order perturbation Hamiltonian</td><td>$\mathrm{{t}}_c$</td><td>$T^c$</td></tr> <tr><td><code>Hss_{k}.zdc</code></td><td>Spin-spin first order perturbation Hamiltonian of rank $k$</td><td>$\mathrm{{m}}_{{k,ss}}$</td><td></td></tr> <tr><td><code>Hsoo_{k}.zdc</code></td><td>Spin-other-orbit first order perturbation Hamiltonian of rank $k$</td><td>$\mathrm{{m}}_{{k,soo}}$</td><td></td></tr> <tr><td><code>H5_{k}.zdc</code></td><td>Spin-spin and spin-other-orbit first order perturbation Hamiltonian of rank $k$</td><td>$\mathrm{{m}}_k$</td><td>$M^k$</td></tr> <tr><td><code>H6_{k}.zdc</code></td><td>Effective electrostatic spin-orbit second order perturbation Hamiltonian of rank $k$</td><td>$\mathrm{{p}}_k$</td><td>$P^k$</td></tr> <tr><td><code>Hcf_{k}_{q}.zdc</code></td><td>Component $q$ of the crystal field perturbation Hamiltonian of rank $k$</td><td>$\mathrm{{H}}_{{\mathrm{{cf}},q}}^{{(k)}}$</td><td>$B^k_q$</td></tr></tbody></table> <p></p> <p> The component part <code>_{q}</code> is missing in the filenames of the set of matrices containing reduced matrix elements. Reduced matrix elements are an entity introduced by the <a href="https://en.wikipedia.org/wiki/Wigner%E2%80%93Eckart_theorem">Wigner-Eckart theorem</a>, which describes the angular behaviour of any spherical tensor operator $\mathrm{A}^{(k)}$: $$ \langle J' M' | \mathrm{A}^{(k)}_{q} | J M \rangle = (-1)^{J'-M'} \begin{pmatrix} J' & k & J \\ -M' & q & M \end{pmatrix} \langle J' || \mathrm{A}^{(k)} || J \rangle $$ </p> <h2>State Representation</h2> <p> Each matrix data container includes the quantum numbers of all states in the JSON file <code>data/matrix.json</code>. For product states these are the quantum numbers $n$, $l$, $m_l$, $s$, and $m_s$ of each electron. States in $LS$-coupling are characterised by irreducible representations which in most cases are connected to eigenvalues of certain tensor operators. </p><table> <tbody><tr><th>Key</th> <th>Irreducible Representation</th> <th>Related Tensor Operator</th> </tr><tr><td><code>S2</code></td><td>Multiplicity $2S+1$ derived from the eigenvalue $S(S+1)$</td><td>Squared operator of the total spin $(\mathrm{S}\cdot\mathrm{S})$</td></tr> <tr><td><code>C7</code></td><td>3-tuple $W=(w_1w_2w_3)$</td><td>Casimir operator $\mathrm{C}_2(SO(7))$ of the special orthogonal group in 7 dimensions</td></tr> <tr><td><code>C2</code></td><td>2-tuple $U=(u_1u_2)$</td><td>Casimir operator $\mathrm{C}_2(G_2)$ of the special group $G_2$</td></tr> <tr><td><code>L2</code></td><td>Spectral character of $L$ derived from the eigenvalue $L(L+1)$</td><td>Squared operator of the total orbital angular momentum $(\mathrm{L}\cdot\mathrm{L})$</td></tr> <tr><td><code>J2</code></td><td>Quantum number $J$ derived from the eigenvalue $J(J+1)$</td><td>Squared operator of the total angular momentum $(\mathrm{J}\cdot\mathrm{J})$</td></tr> <tr><td><code>Jz</code></td><td>Eigenvalue $M_J$</td><td>Operator of the axial component of the total angular momentum $\mathrm{J}_0$</td></tr> <tr><td><code>sen</code></td><td>Seniority number $\nu$</td><td>Operators $(\mathrm{S}\cdot\mathrm{S})$ and $\mathrm{C}_2(SO(7))$</td></tr> <tr><td><code>num</code></td><td>Index of different states with same quantum numbers $L$ and $S$</td><td></td></tr> <tr><td><code>tau</code></td><td>Index $\tau$ of different states with same set of quantum numbers</td><td></td></tr></tbody></table> <p></p> <h2>Version History</h2> <p> The dataset is published with a semantic version number in the scheme 'major.minor.patch', starting at '1.0.0'. The <strong>patch number</strong> is incremented for every update of the datasets metadata without any modification of the included files. The <strong>minor number</strong> identifies a dataset release with modified files. However, these are only non-functional modifications of metadata in the data containers. Modifications may include updated or corrected titles or descriptions in the metadata of data containers, or new metadata attributes. The <strong>major number</strong> is incremented with functional modifications of data containers or additional tensor operators. </p> <h3>Version 1.0.0</h3> <p> </p><ul> <li>Initial version</li> </ul> <p></p> <h3>Version 2.0.0</h3> <p> </p><ul> <li>Replaced proprietary sign fix inside $J$-multiplets by ladder operator</li> <li>Coulomb operators added</li> <li>Crystal field operators added</li> </ul> <p></p> <h3>Version 3.0.0</h3> <p> </p><ul> <li>Fixed <code>Hcf/{k},0</code> matrices (wrong factor 2)</li> </ul> <p></p>

Similar Items