Observation of Multipole Orderings in f-Electron Systems by Resonant X-ray Diffraction

Abstract

In f-electron systems, the spin and orbital degrees of freedom are coupled through strong spin–orbit interaction to form a J-multiplet. The J-multiplet is further split into several crystalline electric field levels, where each level has its own electric and magnetic characters. In such cases, it is appropriate to describe the electronic degrees of freedom in terms of multipole moments. Actually in f-electron systems, there are cases where higher order multipole moments order by themselves and play important roles in their physical properties. In this chapter, we start by explaining the basic concept and usage of multipole moments. It is especially important to understand electric and magnetic multipole moments are coupled in magnetic fields. After explaining how the multipole moments are involved in the formalism of resonant X-ray scattering, we describe some typical experimental cases, where resonant X-ray diffraction has played a conclusive role.

Appendix

1.1 Geometrical Factors

We summarize here the geometrical factors $P_{E2,\mu }^{(\nu )}$ for $z_{E2,\mu }^{(\nu )}$ at E2 resonance [4, 6, 7]. $\varepsilon _{j,j',j''}$ is the Levi–Civita tensor, where $\varepsilon _{j,j',j''}=1$ when $(j,j',j'')=(x,y,z)$, (y, z, x), or (z, x, y), and $\varepsilon _{j,j',j''}=-1$ when $(j,j',j'')=(x,z,y)$, (y, x, z), or (z, y, x).

$$\begin{aligned} P_{E2,1}^{(0)}= & {} \frac{3}{4\sqrt{5}} \bigl \{ (\mathbf {k}' \cdot \mathbf {k})(\mathbf {\varepsilon }' \cdot \mathbf {\varepsilon }) + (\mathbf {k}' \cdot \mathbf {\varepsilon })(\mathbf {\varepsilon }' \cdot \mathbf {k}) \bigr \} \end{aligned}$$

(39)

$$\begin{aligned} P_{E2,j}^{(1)}= & {} -i \frac{3}{4\sqrt{10}} \bigl \{ (\mathbf {\varepsilon }' \cdot \mathbf {\varepsilon })(\mathbf {k}' \times \mathbf {k}) + (\mathbf {k}' \cdot \mathbf {k} ) (\mathbf {\varepsilon }' \times \mathbf {\varepsilon }) \nonumber \\&+ (\mathbf {k}' \cdot \mathbf {\varepsilon }) (\mathbf {\varepsilon }' \times \mathbf {k}) + (\mathbf {\varepsilon }' \cdot \mathbf {k}) (\mathbf {k}' \times \mathbf {\varepsilon }) \bigr \}_{j} \;;\; (j=1,2,3) \end{aligned}$$

(40)

$$\begin{aligned} P_{E2,j}^{(2)}= & {} -\frac{3}{2\sqrt{14}} \bigl \{ (\mathbf {\varepsilon }' \cdot \mathbf {\varepsilon }) K_{j}(\mathbf {k},\mathbf {k}') + (\mathbf {k}' \cdot \mathbf {k}) K_{j}(\mathbf {\varepsilon } , \mathbf {\varepsilon }') \nonumber \\&+ K_{j}(\mathbf {k}' \times \mathbf {k} , \mathbf {\varepsilon }' \times \mathbf {\varepsilon }) \bigr \} \;;\; (j=1,2,3,4,5) \end{aligned}$$

(41)

$$\begin{aligned} P_{E2,1}^{(3)}= & {} i \frac{1}{4\sqrt{2}} \sum _{j=1}^{3} \bigl \{ (\mathbf {k}' \times \mathbf {k})_{j} K_{j+2}(\mathbf {\varepsilon }' , \mathbf {\varepsilon }) + (\mathbf {\varepsilon }' \times \mathbf {\varepsilon })_{j}K_{j+2}(\mathbf {k}' , \mathbf {k}) \nonumber \\&+ (\mathbf {k}' \times \mathbf {\varepsilon })_{j}K_{j+2}(\mathbf {\varepsilon }' , \mathbf {k}) + (\mathbf {\varepsilon }' \times \mathbf {k})_{j}K_{j+2}(\mathbf {k}' , \mathbf {\varepsilon }) \bigr \} \end{aligned}$$

(42)

$$\begin{aligned} P_{E2,j+1}^{(3)}= & {} i\frac{3}{8}\sqrt{\frac{5}{2}} \bigl \{ (\mathbf {k}' \times \mathbf {k})_{j} \varepsilon '_{j} \varepsilon _{j} + (\mathbf {\varepsilon }' \times \mathbf {\varepsilon })_{j} k'_{j}k_{j} \nonumber \\&+ (\mathbf {k}' \times \mathbf {\varepsilon })_{j} \varepsilon '_{j} k_{j} +(\mathbf {\varepsilon }' \times \mathbf {k})_{j} k'_{j} \varepsilon _{j} \bigr \} + \frac{1}{2} P_{E2,j}^{(1)} \;;\; (j=1,2,3)\end{aligned}$$

(43)

$$\begin{aligned} P_{E2,j+4}^{(3)}= & {} i\frac{3}{16}\sqrt{\frac{3}{2}} \bigl \{ (\mathbf {k}' \times \mathbf {k})_{j} \sum _{j',j''} \varepsilon _{j,j',j''} (\varepsilon '_{j'}\varepsilon _{j'} - \varepsilon '_{j''}\varepsilon _{j''}) \nonumber \\&+ (\mathbf {\varepsilon }' \times \mathbf {\varepsilon })_{j} \sum _{j',j''} \varepsilon _{j,j',j''} (k'_{j'}k_{j'} - k'_{j''}k_{j''})\nonumber \\&+ (\mathbf {k}' \times \mathbf {\varepsilon })_{j} \sum _{j',j''} \varepsilon _{j,j',j''} (\varepsilon '_{j'}k_{j'} - \varepsilon '_{j''}k_{j''})\nonumber \\&+ (\mathbf {\varepsilon }' \times \mathbf {k})_{j} \sum _{j',j''} \varepsilon _{j,j',j''} (k'_{j'}\varepsilon _{j'} - k'_{j''}\varepsilon _{j''}) \bigr \} \;;\; (j=1,2,3) \end{aligned}$$

(44)

$$\begin{aligned} P_{E2,1}^{(4)}= & {} \sqrt{\frac{2}{15}} \bigl \{ \frac{15}{4} (k'_{x} k_{x} \varepsilon '_{x}\varepsilon _{x} + k'_{y} k_{y} \varepsilon '_{y}\varepsilon _{y} + k'_{z} k_{z} \varepsilon '_{z}\varepsilon _{z}) - \sqrt{5} P_{E2,1}^{(0)} \bigr \} \end{aligned}$$

(45)

$$\begin{aligned} P_{E2,2}^{(4)}= & {} -{\frac{\sqrt{42}}{4}} (k'_{x} k_{x} \varepsilon '_{x}\varepsilon _{x} + k'_{y} k_{y} \varepsilon '_{y}\varepsilon _{y} -2 k'_{z} k_{z} \varepsilon '_{z}\varepsilon _{z}) - \frac{1}{2}\sqrt{\frac{6}{7}} \bigl \{(\mathbf {k}' \cdot \mathbf {k}) K_{1}(\mathbf {\varepsilon }' , \mathbf {\varepsilon }) \nonumber \\&+ (\mathbf {\varepsilon }' \cdot \mathbf {\varepsilon }) K_{1}(\mathbf {k}',\mathbf {k}) + (\mathbf {k}' \cdot \mathbf {\varepsilon }) K_{1}(\mathbf {\varepsilon }',\mathbf {k}) + (\mathbf {\varepsilon }' \cdot \mathbf {k}) K_{1}(\mathbf {k}',\mathbf {\varepsilon }) \bigr \} \end{aligned}$$

(46)

$$\begin{aligned} P_{E2,3}^{(4)}= & {} \frac{3 \sqrt{14}}{4} (k'_{x} k_{x} \varepsilon '_{x}\varepsilon _{x} - k'_{y} k_{y} \varepsilon '_{y}\varepsilon _{y}) - \frac{3}{2}\sqrt{\frac{2}{21}} \bigl \{(\mathbf {k}' \cdot \mathbf {k}) K_{2}(\mathbf {\varepsilon }' , \mathbf {\varepsilon }) \nonumber \\&+ (\mathbf {\varepsilon }' \cdot \mathbf {\varepsilon }) K_{2}(\mathbf {k}',\mathbf {k}) + (\mathbf {k}' \cdot \mathbf {\varepsilon }) K_{2}(\mathbf {\varepsilon }',\mathbf {k}) + (\mathbf {\varepsilon }' \cdot \mathbf {k}) K_{2}(\mathbf {k}',\mathbf {\varepsilon }) \bigr \} \end{aligned}$$

(47)

$$\begin{aligned} P_{E2,j+3}^{(4)}= & {} \frac{2}{8\sqrt{6}} \bigl \{ K_{j+2}(\mathbf {k}',\mathbf {k}) \sum _{j',j''} \varepsilon _{j,j',j''} (\varepsilon '_{j'}\varepsilon _{j'} - \varepsilon '_{j''}\varepsilon _{j''}) \nonumber \\&+ K_{j+2}(\mathbf {\varepsilon }',\mathbf {\varepsilon }) \sum _{j',j''} \varepsilon _{j,j',j''} (k'_{j'}k_{j'} - k'_{j''}k_{j''}) \nonumber \\&+ K_{j+2}(\mathbf {k}',\mathbf {\varepsilon }) \sum _{j',j''} \varepsilon _{j,j',j''} (\varepsilon '_{j'}k_{j'} - \varepsilon '_{j''}k_{j''})\nonumber \\&+ K_{j+2}(\mathbf {\varepsilon }',\mathbf {k}) \sum _{j',j''} \varepsilon _{j,j',j''} (k'_{j'}\varepsilon _{j'} - k'_{j''}\varepsilon _{j''}) \bigr \} \;;\; (j=1,2,3) \end{aligned}$$

(48)

$$\begin{aligned} P_{E2,j+6}^{(4)}= & {} \frac{3}{4\sqrt{42}} \bigl [ \{ 7 k'_{j} k_{j} - 3(\mathbf {k}' \cdot \mathbf {k}) \} K_{j+2}(\mathbf {\varepsilon }',\mathbf {\varepsilon }) \nonumber \\&+ \{ 7 \varepsilon '_{\mu } \varepsilon _{\mu } - 3(\mathbf {\varepsilon }' \cdot \mathbf {\varepsilon }) \} K_{j+2}(\mathbf {k}',\mathbf {k}) \nonumber \\&+ \{ 7 \varepsilon '_{j} k_{j} - 3(\mathbf {\varepsilon }' \cdot \mathbf {k}) \} K_{j+2}(\mathbf {k}',\mathbf {\varepsilon }) \nonumber \\&+ \{ 7 k'_{j} \varepsilon _{j} - 3(\mathbf {k}' \cdot \mathbf {\varepsilon }) \} K_{j+2}(\mathbf {\varepsilon }',\mathbf {k}) \bigr ] \;;\; (j=1,2,3) \end{aligned}$$

(49)

1.2 Scattering Intensity and Polarization Analysis

We use the scattering-amplitude-operator method to analyze the experimental results including polarization analysis [42]. This method is useful for describing the observed intensity at the detector in a general scattering geometry shown in Fig. 3, where the incident linear polarization is rotated to an arbitrary angle $\eta$ using phase plates and a crystal analyzer system is inserted to analyze the polarization state of the scattered X-ray.

All the information with respect to the target sample is summarized in a $2\times 2$ matrix $\hat{F}$ consisting of four elements of the scattering amplitude for $\sigma$–$\sigma '$, $\sigma$–$\pi '$, $\pi$–$\sigma '$, and $\pi$–$\pi '$ processes:

$$\begin{aligned} \hat{F} = \left( \begin{array}{cc} F_{\sigma \sigma '} &{} F_{\pi \sigma '} \\ F_{\sigma \pi '} &{} F_{\pi \pi '} \end{array}\right) . \end{aligned}$$

(50)

This determines the state of the target system. By using the identity matrix $\hat{I}$ and the Pauli matrix $\hat{\mathbf {\sigma }}$, $\hat{F}$ can generally be expressed as

$$\begin{aligned} \hat{F}= & {} \beta \hat{I} + \mathbf {\alpha }\cdot \hat{\mathbf {\sigma }} \nonumber \\= & {} \left( \begin{array}{cc} \beta + \alpha _3 &{} \alpha _1 - i \alpha _2 \\ \alpha _1 + i \alpha _2 &{} \beta - \alpha _3 \end{array}\right) , \end{aligned}$$

(51)

where the parameters $\beta$ and $\mathbf {\alpha }=(\alpha _1,\alpha _2,\alpha _3)$ are

$$\begin{aligned} \beta= & {} (F_{\sigma \sigma '} + F_{\pi \pi '}) /2 \nonumber ,\\ \alpha _1= & {} (F_{\pi \sigma '} + F_{\sigma \pi '}) /2 \nonumber , \\ \alpha _2= & {} i (F_{\pi \sigma '} - F_{\sigma \pi '}) /2 \nonumber ,\\ \alpha _3= & {} (F_{\sigma \sigma '} - F_{\pi \pi '}) /2 . \end{aligned}$$

(52)

Next, to calculate the scattering cross-section, information on the incident photon state is necessary. This is described by the density matrix

$$\begin{aligned} \hat{\mu }=(\hat{I}+\mathbf {P}\cdot \hat{\mathbf {\sigma }})/2 , \end{aligned}$$

(53)

where the Stokes vector $\mathbf {P}=(P_1, P_2, P_3)$ represents the polarization state of the incident photon. $P_1$, $P_2$, and $P_3$ represent ${\pm }45^{\circ }$, left or right handed circular, and $\sigma$ or $\pi$ polarization state, respectively [42]. For example, $\mathbf {P}=(0,0,1)$ and $(0,0,-1)$ mean the perfectly $\sigma$ and $\pi$ polarized state, i.e., $\eta =0^{\circ }$ and $90^{\circ }$ in Fig. 3, respectively. In general, since the beam is not perfectly polarized, we need to consider a situation with $P_1^{\;2}+P_2^{\;2} + P_3^{\;2} < 1$.

Once we know the matrix $\hat{F}$, the scattering cross-section $(d\sigma /d\varOmega )$ can be calculated by

$$\begin{aligned} \left( \frac{d\sigma }{d\varOmega } \right)= & {} \text {Tr} \{ \hat{\mu } \hat{F}^{\dag } \hat{F} \} \nonumber \\= & {} \beta ^{\dag }\beta + \mathbf {\alpha }^{\dag }\cdot \mathbf {\alpha } + \beta ^{\dag } (\mathbf {P} \cdot \mathbf {\alpha }) + (\mathbf {P} \cdot \mathbf {\alpha }^{\dag })\beta + i\mathbf {P} \cdot (\mathbf {\alpha }^{\dag } \times \mathbf {\alpha }) . \end{aligned}$$

(54)

The Stokes vector of the scattered X-ray, $\mathbf {P}'$, can be obtained from

$$\begin{aligned} \left( \frac{d\sigma }{d\varOmega } \right) \mathbf {P}'= & {} \text {Tr} \{\hat{\mu }\hat{F}^{\dag }\hat{\mathbf {\sigma }}\hat{F}\} \nonumber \\= & {} \beta ^{\dag }\mathbf {\alpha } + \mathbf {\alpha }^{\dag } \beta - i (\mathbf {\alpha }^{\dag } \times \mathbf {\alpha }) + \beta ^{\dag }\beta \mathbf {P} - i\beta ^{\dag } (\mathbf {P} \times \mathbf {\alpha }) \nonumber \\&\;\;\;\; +i (\mathbf {P} \times \mathbf {\alpha }^{\dag }) \beta + \mathbf {\alpha }^{\dag } (\mathbf {P}\cdot \mathbf {\alpha }) -\mathbf {\alpha }^{\dag } \times (\mathbf {P} \times \mathbf {\alpha }) . \end{aligned}$$

(55)

As an example, we show a case of fundamental Bragg reflection from the crystal lattice by non-resonant Thomson scattering. In this case, the scattering amplitude is written as

$$\begin{aligned} \hat{F} = -F_c \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} \cos 2\theta \end{array}\right) , \end{aligned}$$

(56)

where $F_c$ represents the structure factor of the reflection. From Eq. (52), $\beta =-F_c \cos ^2 \theta$, $\alpha _1=\alpha _2=0$, and $\alpha _3=- F_c \sin ^2 \theta$ are obtained. Then, from Eq. (54), the scattering cross-section becomes

$$\begin{aligned} \left( \frac{d\sigma }{d\varOmega } \right) = |F_c|^2 \left\{ 1 - \frac{1}{2}(1-P_3)\sin ^2 2\theta \right\} , \end{aligned}$$

(57)

and from Eq. (55), $\mathbf {P}'$ satisfies

$$\begin{aligned} \left( \frac{d\sigma }{d\varOmega } \right) P_1^{\;\prime }= & {} |F_c|^2 P_1 \cos 2\theta , \nonumber \\ \left( \frac{d\sigma }{d\varOmega } \right) P_2^{\;\prime }= & {} |F_c|^2 P_2 \cos 2\theta , \nonumber \\ \left( \frac{d\sigma }{d\varOmega } \right) P_3^{\;\prime }= & {} |F_c|^2 \left\{ P_3 + \frac{1}{2} (1-P_3) \sin ^2 2\theta \right\} . \end{aligned}$$

(58)

The Stokes vector of the incident X-ray, with the polarization angle $\eta$ in Fig. 3, is written as

$$\begin{aligned} \mathbf {P} = P_L (\sin 2\eta , \; 0, \; \cos 2\eta ) , \end{aligned}$$

(59)

where $P_L$ represents the degree of linear polarization of the incident X-ray.

The intensity after diffracted by the analyzer crystal is also described by the Thomson scattering, and Eq. (57) is applied. It is noted, however, that $P_3$ must be transformed to the value for the diffraction at the analyzer, which we write as $P_{3A}$:

$$\begin{aligned} P_{3A}= & {} P_1^{\;\prime } \sin 2\phi _{A} + P_3^{\;\prime } \cos 2\phi _{A} . \end{aligned}$$

(60)

Finally, the intensity at the detector is expressed as

$$\begin{aligned} I= & {} K \left( \frac{d\sigma }{d\varOmega } \right) \left\{ 1 - \frac{1}{2}(1 - P_{3A}) \sin ^2 2\theta _{A} \right\} , \end{aligned}$$

(61)

where $\left( d\sigma / d\varOmega \right)$ is the scattering cross-section of the sample expressed by Eq. (57) and K represents a constant factor.