^{1,a)}, A. P. Shpak

^{1}and A. N. Yaresko

^{2}

### Abstract

The present state of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of and compounds is reviewed. Energy band theory based upon the local spin-density approximation (LSDA) describes the XMCD spectra of transition metal compounds with high accuracy. However, the LSDA does not suffice for lanthanide compounds which have a correlated shell. A satisfactory description of the XMCD spectra could be obtained by using a generalization of the LSDA, in which explicitly electron Coulomb correlations are taken into account ( approach). As examples of this group we consider the compound. We also consider uraniumcompounds. In those compounds where the electrons are rather delocalized, the LSDA describes the XMCD spectra reasonably well. As an example of this group we consider . Particular differences occur for uraniumcompounds in which the electrons are neither delocalized nor localized, but more or less semilocalized. Typical examples are (, Rh, and Pt), and UX (, Se, Te). However, the semilocalized ’s are not inert, but their interaction with conduction electrons plays an important role. We also consider the electronic structure and XMCD spectra of the heavy-fermion compounds, , , , and , where the degree of the localization is increased in comparison with other uraniumcompounds. The electronic structure and XMCD spectra of which possesses simultaneously ferromagnetism and superconductivity also presented. Recently achieved improvements for describing compounds are discussed.

I. RARE-EARTH COMPOUNDS

A.

II. URANIUMCOMPOUNDS

A. Intermetallic compounds

1.

2. (, Rh, and Pt

B. Uranium monochalcogenides

1. Band structure

C. Heavy-fermion compounds

1.

2.

3. and

4.

D.

III. SUMMARY

### Key Topics

- Uranium
- 74.0
- Magnetic moments
- 68.0
- Band structure
- 48.0
- Germanium
- 40.0
- Superconducting compounds
- 37.0

## Figures

Self-consistent fully relativistic spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) calculated for with the states treated as: (1) fully localized ( in core); (2) itinerant (LSDA); and (3) partly localized .^{36}

Self-consistent fully relativistic spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) calculated for with the states treated as: (1) fully localized ( in core); (2) itinerant (LSDA); and (3) partly localized .^{36}

Partial density of states of .^{36} The Fermi energy is at zero.

Partial density of states of .^{36} The Fermi energy is at zero.

(a) Theoretically calculated^{36} (dotted lines) and experimental^{35} (circles) isotropic absorption spectra of at the edges. The experimental spectra were measured in an external magnetic field at . (b) Experimental^{35} (circles) XMCD spectra of at the edges in comparison with theoretically calculated ones in the approximation without (dotted lines) and with (full lines) core-hole effect taken into account.

(a) Theoretically calculated^{36} (dotted lines) and experimental^{35} (circles) isotropic absorption spectra of at the edges. The experimental spectra were measured in an external magnetic field at . (b) Experimental^{35} (circles) XMCD spectra of at the edges in comparison with theoretically calculated ones in the approximation without (dotted lines) and with (full lines) core-hole effect taken into account.

(a) Theoretically calculated^{36} (dashed line) and experimental^{42} (circles) isotropic absorption spectra of at the edges (the spectra are shifted towards smaller energy by ). The experimental spectra were measured on a bulklike layer deposited on a Si substrate at in the total fluorescence yield mode. The dotted lines show the theoretically calculated background spectra; the full thick lines are the sum of the theoretical XAS and background spectra. (b) Experimental^{42} (circles) XMCD spectra of at the edges in comparison with theoretically calculated ones in the (full lines) approximation.

(a) Theoretically calculated^{36} (dashed line) and experimental^{42} (circles) isotropic absorption spectra of at the edges (the spectra are shifted towards smaller energy by ). The experimental spectra were measured on a bulklike layer deposited on a Si substrate at in the total fluorescence yield mode. The dotted lines show the theoretically calculated background spectra; the full thick lines are the sum of the theoretical XAS and background spectra. (b) Experimental^{42} (circles) XMCD spectra of at the edges in comparison with theoretically calculated ones in the (full lines) approximation.

(a) The experimental^{35} (circles) isotropic absorption spectrum of at the N edge in comparison with the calculated ones^{36} in the LSDA (full line) and approximations without (dashed line) and with (dotted line) the core-hole effect taken into account. The experimental spectra were measured with external magnetic field at . The dashed-dotted line shows the theoretically calculated background spectrum. (b) Experimental^{35} (circles) XMCD spectrum of at the Gd edge in comparison with theoretically calculated ones in the LSDA (full line) and putting the states in the core (dashed line); (c) experimental (circles) XMCD Gd spectrum in comparison with theoretically calculated in the approximation with (full line) and without (dashed line) the core-hole effect taken into account.

(a) The experimental^{35} (circles) isotropic absorption spectrum of at the N edge in comparison with the calculated ones^{36} in the LSDA (full line) and approximations without (dashed line) and with (dotted line) the core-hole effect taken into account. The experimental spectra were measured with external magnetic field at . The dashed-dotted line shows the theoretically calculated background spectrum. (b) Experimental^{35} (circles) XMCD spectrum of at the Gd edge in comparison with theoretically calculated ones in the LSDA (full line) and putting the states in the core (dashed line); (c) experimental (circles) XMCD Gd spectrum in comparison with theoretically calculated in the approximation with (full line) and without (dashed line) the core-hole effect taken into account.

empty partial DOSs (in arbitrary units) for the surface layer in the 9/3 MLS (full line) and the bulk structure (dashed line)^{36} in comparison with the experimental XA spectrum at the N edge^{35} (circles).

empty partial DOSs (in arbitrary units) for the surface layer in the 9/3 MLS (full line) and the bulk structure (dashed line)^{36} in comparison with the experimental XA spectrum at the N edge^{35} (circles).

The LSDA partial and density of states in .^{72}

The LSDA partial and density of states in .^{72}

Isotropic absorption and XMCD spectra of at the uranium edges calculated in the LSDA (solid lines) and (OP) (dashed lines) approximations.^{72} The experimental spectra^{66} (circles) were measured at and at magnetic field (the spectrum is shifted by to include it in the figure). The upper panel also shows the background spectra (dashed line) due to the transitions from inner levels to the continuum of unoccupied levels.

Isotropic absorption and XMCD spectra of at the uranium edges calculated in the LSDA (solid lines) and (OP) (dashed lines) approximations.^{72} The experimental spectra^{66} (circles) were measured at and at magnetic field (the spectrum is shifted by to include it in the figure). The upper panel also shows the background spectra (dashed line) due to the transitions from inner levels to the continuum of unoccupied levels.

XMCD spectra of at the uranium , , , and edges, calculated in the LSDA approximation.^{72} All the XMCD spectra are multiplied by a factor (the and spectra have been shifted by and , respectively, to include them in the figure).

XMCD spectra of at the uranium , , , and edges, calculated in the LSDA approximation.^{72} All the XMCD spectra are multiplied by a factor (the and spectra have been shifted by and , respectively, to include them in the figure).

XMCD spectra of at the Fe , , and edges in bcc Fe and Fe in , calculated in the LSDA approximation.^{72} The XMCD spectrum at the edge has been multiplied by a factor .

XMCD spectra of at the Fe , , and edges in bcc Fe and Fe in , calculated in the LSDA approximation.^{72} The XMCD spectrum at the edge has been multiplied by a factor .

The LSDA self-consistent fully relativistic, spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) of , , and .^{86}

The LSDA self-consistent fully relativistic, spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) of , , and .^{86}

The partial density of states in calculated in the LSDA and (OP) approximations.^{86}

The partial density of states in calculated in the LSDA and (OP) approximations.^{86}

Isotropic absorption and XMCD spectra of at the uranium edges calculated in the LSDA (dotted lines) and (OP) (solid lines) approximations.^{86} The experimental spectra^{71} (circles) were measured at and at a magnetic field of (the spectrum is shifted by to include it in the figure).

Isotropic absorption and XMCD spectra of at the uranium edges calculated in the LSDA (dotted lines) and (OP) (solid lines) approximations.^{86} The experimental spectra^{71} (circles) were measured at and at a magnetic field of (the spectrum is shifted by to include it in the figure).

The XMCD spectra of and at the uranium edges calculated in the LSDA (dashed lines) and (OP) (solid lines) approximations.^{86} The experimental spectra for UCoAl^{71} (circles) were measured at a magnetic field of . The experimental data for is from Ref. 70 (the spectra are shifted by to include them in the figure).

The XMCD spectra of and at the uranium edges calculated in the LSDA (dashed lines) and (OP) (solid lines) approximations.^{86} The experimental spectra for UCoAl^{71} (circles) were measured at a magnetic field of . The experimental data for is from Ref. 70 (the spectra are shifted by to include them in the figure).

Uranium XMCD spectrum of calculated in the ) approximation (full line) and spectra calculated with the and DOSs artificially shifted by (dashed line) and (dotted line).^{86}

Uranium XMCD spectrum of calculated in the ) approximation (full line) and spectra calculated with the and DOSs artificially shifted by (dashed line) and (dotted line).^{86}

Self-consistent fully relativistic energy band structure and total DOS (in states/(unit cell⋅eV)) of calculated within the LSDA and approximations with and .^{116}

Self-consistent fully relativistic energy band structure and total DOS (in states/(unit cell⋅eV)) of calculated within the LSDA and approximations with and .^{116}

The partial and densities of states in , , and , calculated in the LSDA and approximations.^{116}

The partial and densities of states in , , and , calculated in the LSDA and approximations.^{116}

The partial density of states in and , calculated within the approximation.^{116}

The partial density of states in and , calculated within the approximation.^{116}

The XMCD spectra of , , and at the uranium edges calculated within the LSDA (dashed lines), LSDA + U(OP) (dotted lines), and (solid lines) approximations.^{116} The experimental spectra of US^{71} (circles) were measured at a magnetic field of (the spectra are shifted by to include them in the figure).

The XMCD spectra of , , and at the uranium edges calculated within the LSDA (dashed lines), LSDA + U(OP) (dotted lines), and (solid lines) approximations.^{116} The experimental spectra of US^{71} (circles) were measured at a magnetic field of (the spectra are shifted by to include them in the figure).

The self-consistent fully relativistic, spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) of calculated in the LSDA and approximations.^{140}

The self-consistent fully relativistic, spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) of calculated in the LSDA and approximations.^{140}

The partial density of states in and , calculated in the approximation.

The partial density of states in and , calculated in the approximation.

The model representation of the (a) and (b) XMCD of for two solutions with occupied states (full lines) and (dashed lines): (a) presents the partial densities of states ; (b) (full line) and (dashed lines)^{140} (see the explanation in the text).

The model representation of the (a) and (b) XMCD of for two solutions with occupied states (full lines) and (dashed lines): (a) presents the partial densities of states ; (b) (full line) and (dashed lines)^{140} (see the explanation in the text).

The XMCD spectra of and at the uranium edges, calculated in the LSDA, (OP), and approximations.^{140} The experimental spectra for ^{65} (circles) were measured in a magnetic field of at (the spectra are shifted by to include them in the figure).

The XMCD spectra of and at the uranium edges, calculated in the LSDA, (OP), and approximations.^{140} The experimental spectra for ^{65} (circles) were measured in a magnetic field of at (the spectra are shifted by to include them in the figure).

The self-consistent fully relativistic, spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) of calculated in the LSDA and approximations.^{140}

^{140}

The partial density of states in calculated in the LSDA approximation^{140} (the partial DOS has been multiplied by a factor of 3 for clarity).

The partial density of states in calculated in the LSDA approximation^{140} (the partial DOS has been multiplied by a factor of 3 for clarity).

Isotopic absorption and XMCD spectra of at the uranium edges calculated in the LSDA (dashed lines) and (full lines) approximations.^{140} The experimental spectra^{69} (circles) were measured at and in a magnetic field of (the spectra are shifted by to include them in the figure).

Isotopic absorption and XMCD spectra of at the uranium edges calculated in the LSDA (dashed lines) and (full lines) approximations.^{140} The experimental spectra^{69} (circles) were measured at and in a magnetic field of (the spectra are shifted by to include them in the figure).

The self-consistent fully relativistic, spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) of and , calculated in the LSDA approximation.^{140}

The self-consistent fully relativistic, spin-polarized energy band structure and total DOS (in states/(unit cell⋅eV)) of and , calculated in the LSDA approximation.^{140}

The partial density of states in .^{140}

The partial density of states in .^{140}

The XMCD spectra of and at the uranium edges calculated in LSDA and approximations.^{140} The experimental spectra^{69} for were measured in a magnetic field of and . The experimental data for the XMCD spectrum of is from Ref. 68 (the spectra are shifted by to include them in the figure).

The XMCD spectra of and at the uranium edges calculated in LSDA and approximations.^{140} The experimental spectra^{69} for were measured in a magnetic field of and . The experimental data for the XMCD spectrum of is from Ref. 68 (the spectra are shifted by to include them in the figure).

Crystal structure of .

Crystal structure of .

The energy band structure and total density of states (in states/(unit cell⋅eV)) in , calculated in the LSDA and approximations.^{140}

The energy band structure and total density of states (in states/(unit cell⋅eV)) in , calculated in the LSDA and approximations.^{140}

The -projected and total density of states in , calculated in the LSDA and approximations.^{140}

The -projected and total density of states in , calculated in the LSDA and approximations.^{140}

Comparison of the calculated partial DOS in the LSDA (dotted line) and approximations with the experimental BIS spectrum (circles) of .^{184} The dashed line shows the DOS calculated with the nonspherical correction to the Coulomb matrix elements, whereas the full line is calculated with averaged and .^{140}

Comparison of the calculated partial DOS in the LSDA (dotted line) and approximations with the experimental BIS spectrum (circles) of .^{184} The dashed line shows the DOS calculated with the nonspherical correction to the Coulomb matrix elements, whereas the full line is calculated with averaged and .^{140}

Isotropic absorption and XMCD spectra of at the uranium edges, calculated in the LSDA (dotted lines), and approximations. The dashed line presents XMCD spectra calculated with nonspherical corrections to Coulomb matrix elements, whereas the full line results are calculated with averaged and .^{140} The experimental spectra^{65} (circles) were measured at and in a magnetic field of (the spectrum is shifted by to include it in the figure).

Isotropic absorption and XMCD spectra of at the uranium edges, calculated in the LSDA (dotted lines), and approximations. The dashed line presents XMCD spectra calculated with nonspherical corrections to Coulomb matrix elements, whereas the full line results are calculated with averaged and .^{140} The experimental spectra^{65} (circles) were measured at and in a magnetic field of (the spectrum is shifted by to include it in the figure).

(a) Theoretically calculated^{193} (dashed line) and experimental^{191} (circles) isotropic absorption spectra of at the edges. The experimental spectra were measured with external magnetic field at . The dotted lines show the theoretically calculated background spectra; the full thick lines are the sum of the theoretical XAS and background spectra. (b) The experimental^{191} (circles) XMCD spectra of at the edges in comparison with theoretically calculated ones in the LSDA (dotted lines) and (full lines) approximations.

(a) Theoretically calculated^{193} (dashed line) and experimental^{191} (circles) isotropic absorption spectra of at the edges. The experimental spectra were measured with external magnetic field at . The dotted lines show the theoretically calculated background spectra; the full thick lines are the sum of the theoretical XAS and background spectra. (b) The experimental^{191} (circles) XMCD spectra of at the edges in comparison with theoretically calculated ones in the LSDA (dotted lines) and (full lines) approximations.

(a) Theoretically calculated^{193} (dashed line) and experimental^{191} (circles) isotropic absorption spectra of at the U *N* _{3} edge. The dotted lines show the theoretically calculated background spectrum; the full thick line is the sum of the theoretical XAS and background spectrum. (b) Experimental^{191} (circles) XMCD spectrum of at the edge in comparison with theoretically calculated ones in the approximations (full line).

(a) Theoretically calculated^{193} (dashed line) and experimental^{191} (circles) isotropic absorption spectra of at the U *N* _{3} edge. The dotted lines show the theoretically calculated background spectrum; the full thick line is the sum of the theoretical XAS and background spectrum. (b) Experimental^{191} (circles) XMCD spectrum of at the edge in comparison with theoretically calculated ones in the approximations (full line).

(a) Theoretically calculated^{193} XMCD spectra of at the and Ge edges at different Ge sites; (b) experimental^{191} (circles) XMCD spectra of at the Ge and edges in comparison with the theoretically calculated one in the approximation (full line).

(a) Theoretically calculated^{193} XMCD spectra of at the and Ge edges at different Ge sites; (b) experimental^{191} (circles) XMCD spectra of at the Ge and edges in comparison with the theoretically calculated one in the approximation (full line).

(a) Theoretically calculated^{193} XMCD spectra of at the edge at different Ge sites; (b) theoretically calculated XMCD spectrum of at the Ge edge in the approximation (full line) in comparison with the experimental one^{192} (circles). The experimental spectrum was measured at in an external magnetic field applied along the axis.

(a) Theoretically calculated^{193} XMCD spectra of at the edge at different Ge sites; (b) theoretically calculated XMCD spectrum of at the Ge edge in the approximation (full line) in comparison with the experimental one^{192} (circles). The experimental spectrum was measured at in an external magnetic field applied along the axis.

## Tables

The dipole allowed transitions from core levels to the unoccupied valence states for left- and right- polarized x rays.

The dipole allowed transitions from core levels to the unoccupied valence states for left- and right- polarized x rays.

The dipole allowed transitions from core levels to the unoccupied valence states for left- and right- polarized x rays.

The experimental and calculated spin , orbital , and total magnetic moments at the uranium site (in ) for , , and .

The experimental and calculated spin , orbital , and total magnetic moments at the uranium site (in ) for , , and .

The experimental and calculated spin , orbital , and total magnetic moments at the uranium site (in ) for , , and .^{116}

The experimental and calculated spin , orbital , and total magnetic moments at the uranium site (in ) for , , and .^{116}

The experimental and calculated spin , orbital , and total magnetic moments at the uranium site (in ) for and . The magnetic moments are calculated for the easy magnetic axes, namely, the hexagonal plane in and the axis in .^{140}

The experimental and calculated spin , orbital , and total magnetic moments at the uranium site (in ) for and . The magnetic moments are calculated for the easy magnetic axes, namely, the hexagonal plane in and the axis in .^{140}

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