X-ray magnetic circular dichroism: Difference between revisions

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X-ray magnetic circular dichroism (XMCD) is a difference spectrum of two X-ray absorption spectra (XAS) taken in a magnetic field, one taken with left circularly polarized light, and one with right circularly polarized light.^[1] By closely analyzing the difference in the XMCD spectrum, information can be obtained on the magnetic properties of the atom, such as its spin and orbital magnetic moment. Using XMCD magnetic moments below 10⁻⁵ µ_B can be observed.^[2]

In the case of transition metals such as iron, cobalt, and nickel, the absorption spectra for XMCD are usually measured at the L-edge. This corresponds to the process in the iron case: with iron, a 2p electron is excited to a 3d state by an X-ray of about 700 eV.^[3] Because the 3d electron states are the origin of the magnetic properties of the elements, the spectra contain information on the magnetic properties. In rare-earth elements usually, the M_4,5-edges are measured, corresponding to electron excitations from a 3d state to mostly 4f states.

Line intensities and selection rules

The line intensities and selection rules of XMCD can be understood by considering the transition matrix elements of an atomic state $\vert {njm}\rangle$ excited by circularly polarised light^[4]^[5]. Here $n$ is the principal, $j$ the angular momentum and $m$ the magnetic quantum numbers. The polarisation vector of left and right circular polarised light can be rewritten in terms of spherical harmonics

\mathbf {e} ={\frac {1}{\sqrt {2}}}\left(x\pm iy\right)={\sqrt {\frac {4\pi }{3}}}rY_{1}^{\pm 1}\left(\theta ,\varphi \right)

leading to an expression for the transition matrix element

\langle n^{\prime }j^{\prime }m^{\prime }\vert \mathbf {e} \cdot \mathbf {r} \vert njm\rangle

which can be simplified using the 3-j symbol:

\langle n^{\prime }j^{\prime }m^{\prime }\vert \mathbf {e} \cdot \mathbf {r} \vert njm\rangle ={\sqrt {\frac {4\pi }{3}}}\langle n^{\prime }j^{\prime }m^{\prime }\vert rY_{1}^{\pm 1}\left(\theta ,\varphi \right)\vert njm\rangle \propto \int _{0}^{\infty }dr~rR_{n^{\prime }j^{\prime }}(r)R_{nj}(r)\int _{\Omega }d\Omega ~{Y_{j^{\prime }}^{m^{\prime }}}^{*}\left(\theta ,\varphi \right)Y_{1}^{\pm 1}\left(\theta ,\varphi \right)Y_{j}^{m}\left(\theta ,\varphi \right)={\sqrt {\frac {(2j^{\prime }+1)(2j+1)}{4\pi }}}\langle {j^{\prime }~0~j~0}\vert {1~0}\rangle \langle {j^{\prime }~m^{\prime }~j~m}\vert {1~\pm 1}\rangle

The radial part is referred to as the line strength while the angular one contains symmetries from which selection rules can be deduced. Rewriting the product of three spherical harmonics with the the 3-j symbol finally leads to^[4]:

{\sqrt {\frac {(2j^{\prime }+1)(2j+1)}{4\pi }}}\langle {j^{\prime }~0~j~0}\vert {1~0}\rangle \langle {j^{\prime }~m^{\prime }~j~m}\vert {1~\pm 1}\rangle ={\sqrt {\frac {(2j^{\prime }+1)(2j+1)(2+1)}{4\pi }}}{\begin{pmatrix}{j^{\prime }}&j&1\\0&0&0\end{pmatrix}}{\begin{pmatrix}j^{\prime }&j&1\\m^{\prime }&m&\mp 1\end{pmatrix}}

The 3-j symbols are not zero only if

j,j^{\prime },m,m^{\prime }

satisfy the following conditions giving us the following selection rules for dipole transitions with circular polarised light^[4]:

$\Delta J=\pm 1$
$\Delta m=0,\pm 1$

References

^ Zhao, Jijun; Huang, Xiaoming; Jin, Peng; Chen, Zhongfang (April 2015). "Magnetic properties of atomic clusters and endohedral metallofullerenes". Coordination Chemistry Reviews. 289–290: 315–340. doi:10.1016/j.ccr.2014.12.013. ISSN 0010-8545.
^ Handbook of magnetism and advanced magnetic materials. Helmut Kronmüller, Stuart S. P. Parkin. Hoboken, NJ: John Wiley & Sons. 2007. ISBN 978-0-470-02217-7. OCLC 124165851.{{cite book}}: CS1 maint: others (link)
^ Stöhr, J. (1995-12-15). "X-ray magnetic circular dichroism spectroscopy of transition metal thin films". Journal of Electron Spectroscopy and Related Phenomena. Future Perspectives for Electron Spectroscopy with Synchrotron Radiation. 75: 253–272. doi:10.1016/0368-2048(95)02537-5. ISSN 0368-2048.
^ ^a ^b ^c de Groot, F. and Vogel, J. (2006). "Fundamentals of X-ray Absorption and Dichroism: The Multiplet Approach" in "Neutron and X-ray Spectroscopy". pp. 3–66. doi:10.1007/1-4020-3337-0_1. ISBN 978-1-4020-3337-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ J. Stöhr and Y. Wu (1994). "X-ray Magnetic Circular Dichroism: Basic concepts and theory for 3d transition metal atoms" in "New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources". pp. 221–250. doi:10.1007/978-94-011-0868-3. ISBN 978-94-010-4375-5.

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[1] Zhao, Jijun; Huang, Xiaoming; Jin, Peng; Chen, Zhongfang (April 2015). "Magnetic properties of atomic clusters and endohedral metallofullerenes". Coordination Chemistry Reviews. 289–290: 315–340. doi:10.1016/j.ccr.2014.12.013. ISSN 0010-8545.

[2] Handbook of magnetism and advanced magnetic materials. Helmut Kronmüller, Stuart S. P. Parkin. Hoboken, NJ: John Wiley & Sons. 2007. ISBN 978-0-470-02217-7. OCLC 124165851.{{cite book}}: CS1 maint: others (link)

[3] Stöhr, J. (1995-12-15). "X-ray magnetic circular dichroism spectroscopy of transition metal thin films". Journal of Electron Spectroscopy and Related Phenomena. Future Perspectives for Electron Spectroscopy with Synchrotron Radiation. 75: 253–272. doi:10.1016/0368-2048(95)02537-5. ISSN 0368-2048.

[:0-4] Groot, F. and Vogel, J. (2006). "Fundamentals of X-ray Absorption and Dichroism: The Multiplet Approach" in "Neutron and X-ray Spectroscopy". pp. 3–66. doi:10.1007/1-4020-3337-0_1. ISBN 978-1-4020-3337-7.{{cite book}}: CS1 maint: multiple names: authors list (link)

[5] J. Stöhr and Y. Wu (1994). "X-ray Magnetic Circular Dichroism: Basic concepts and theory for 3d transition metal atoms" in "New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources". pp. 221–250. doi:10.1007/978-94-011-0868-3. ISBN 978-94-010-4375-5.

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 In the case of [[transition metal]]s such as [[iron]], [[cobalt]], and [[nickel]], the absorption spectra for XMCD are usually measured at the [[X-ray notation#Use|L-edge]]. This corresponds to the process in the iron case: with iron, a [[Electron configuration|2p]] electron is excited to a [[Electron configuration|3d]] state by an [[X-ray]] of about 700 [[Electronvolt|eV]].<ref>{{Cite journal |last=Stöhr |first=J. |date=1995-12-15 |title=X-ray magnetic circular dichroism spectroscopy of transition metal thin films |url=https://dx.doi.org/10.1016/0368-2048%2895%2902537-5 |journal=Journal of Electron Spectroscopy and Related Phenomena |series=Future Perspectives for Electron Spectroscopy with Synchrotron Radiation |language=en |volume=75 |pages=253–272 |doi=10.1016/0368-2048(95)02537-5 |issn=0368-2048}}</ref> Because the 3d electron states are the origin of the magnetic properties of the elements, the spectra contain information on the magnetic properties. In [[rare-earth element]]s usually, the M<sub>4,5</sub>-edges are measured, corresponding to electron excitations from a 3d state to mostly 4f states.
+== Line intensities and selection rules ==
+The line intensities and selection rules of XMCD can be understood by considering the transition matrix elements of an atomic state <math>\vert{njm}\rangle</math> excited by circularly polarised light<ref name=":0">{{Cite book |last=de Groot, F. and Vogel, J. |title="Fundamentals of X-ray Absorption and Dichroism: The Multiplet Approach" in "Neutron and X-ray Spectroscopy" |year=2006 |isbn=978-1-4020-3337-7 |pages=3-66 |doi=10.1007/1-4020-3337-0_1}}</ref><ref>{{Cite book |last=J. Stöhr and Y. Wu |title="X-ray Magnetic Circular Dichroism: Basic concepts and theory for 3d transition metal atoms" in "New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources" |year=1994 |isbn=978-94-010-4375-5 |pages=221-250 |doi=10.1007/978-94-011-0868-3}}</ref>. Here <math>n</math> is the principal, <math>j</math> the angular momentum and <math>m</math> the magnetic quantum numbers. The polarisation vector of left and right circular polarised light can be rewritten in terms of spherical harmonics<math display="block">\mathbf{e} = \frac{1}{\sqrt{2}}\left(x \pm iy\right) = \sqrt{\frac{4\pi}{3}} r Y^{\pm 1}_{1}\left(\theta,\varphi\right)</math>leading to an expression for the transition matrix element <math>\langle n^{\prime}j^{\prime}m^{\prime}\vert \mathbf{e}\cdot\mathbf{r}\vert njm\rangle</math> which can be simplified using the 3-j symbol:<math display="block">\langle n^{\prime}j^{\prime}m^{\prime}\vert \mathbf{e}\cdot\mathbf{r}\vert njm\rangle =
+\sqrt{\frac{4\pi}{3}}\langle n^{\prime}j^{\prime}m^{\prime}\vert rY_{1}^{\pm 1}\left(\theta,\varphi\right)\vert njm\rangle\propto
+\int_{0}^{\infty}dr~rR_{n^{\prime}j^{\prime}}(r)R_{nj}(r)\int_{\Omega}d\Omega~{Y_{j^{\prime}}^{m^{\prime}}}^{*}\left(\theta,\varphi\right) Y_{1}^{\pm 1}\left(\theta,\varphi\right) Y_{ j}^{m}\left(\theta,\varphi\right) =
+\sqrt{\frac{(2j^{\prime}+1) (2j+1)}{4 \pi}} \langle{j^{\prime}~0~j~0}\vert {1~0}\rangle \langle {j^{\prime} ~ m^{\prime}~j~m}\vert {1 ~ \pm 1}\rangle</math>The radial part is referred to as the line strength while the angular one contains symmetries from which selection rules can be deduced. Rewriting the product of three spherical harmonics with the the 3-j symbol finally leads to<ref name=":0" />:<math display="block">\sqrt{\frac{(2j^{\prime}+1)(2j+1)}{4 \pi}} \langle {j^{\prime}~0~j~0}\vert {1~0}\rangle \langle {j^{\prime} ~ m^{\prime}~j~m}\vert {1 ~ \pm 1}\rangle = \sqrt{\frac{(2j^{\prime} + 1)(2j + 1)(2 + 1)}{4\pi}}
+\begin{pmatrix} {j^{\prime}} & j & 1 \\ 0 & 0 & 0 \end{pmatrix}
+\begin{pmatrix} j^{\prime} & j & 1 \\ m^{\prime} & m & \mp 1\end{pmatrix}</math>The 3-j symbols are not zero only if <math>j, j^{\prime},m,m^{\prime}</math> satisfy the following conditions giving us the following selection rules for dipole transitions with circular polarised light<ref name=":0" />:
+# <math>\Delta J = \pm 1</math>
+# <math> \Delta m = 0, \pm 1</math>
 ==See also==