## Complex Refractive Index


Generally, optics based on the electromagnetics is described with the refractive index, while such index does not barely appear in X-ray diffraction (XRD) from the kinematical approach, presumably because of the different historical background; however, it is of great importance to see their relationship to understand the diffraction anomalous fine structure (DAFS) method. For example, the energy dependence of $$f\pprime$$ term is not completely equivalent to a linear absorption coefficient, $$\mu$$; $$f\pprime$$ should be divided by the photon energy in order to treat it as the absorption spectrum equivalent to $$\mu$$ in the theoretical framework and analysis of the XAFS field. Thus, the following section briefly describes the relationship between the complex atomic scattering factor derived before and the conventional complex refractive index.

The refractive index is defined as the ratio of the wave numbers in a material and a vacuum as follows:

\begin{align}
\tilde{n} \equiv k / K = 1 – \delta + i\beta = n + i\beta,
\label{Eq:refractive_index}
\tag{1}
\end{align}

where $$K$$ and $$k$$ are the wave numbers in the vacuum and the material, respectively, $$n$$ is the real part of the refractive index and $$\delta$$ corresponds to the its discrepancy from 1, and $$\beta$$ is the imaginary part of the refractive index. Note that the sign in the imaginary part depends on the definition of the wavefunction; the above description is based on the wavefunction of $$\exp \left\{ i \left( \vec{k}\cdot \vec{r} – \omega t \right) \right\}$$. Thus, the wave number in the materials is written as

\begin{align}
k = \tilde{n} K,
\label{Eq:wave_number_in_material}
\tag{2}
\end{align}

and consequently the electric field in the material is calculated as

\begin{align}
E &= E_{0} \exp \left\{ i (kx – \omega t)\right\} \notag \\
%&= E_{0} \exp \left\{ i(\tilde{n} K x – \omega t)\right\} \notag \\
%&= E_{0} \exp \left\{ i (nKx – \omega t) \right\} \exp (-\beta K x) \notag \\
&= E_{0} \exp \left\{ i(Kx-\omega t) \right\} \exp (-i \delta K x) \exp (- \beta K x) .
\label{Eq:wave_in_material}
\tag{3}
\end{align}

The second and the third terms in the last line of the above equation indicate the phase shift and the absorption, respectively. The absorption is further described with $$\beta$$ in the intensity (i.e., proportional to the square of the electric field ) as

\begin{align}
I(x) = I(0) \exp (- 2\beta K x).
\label{Eq:absorption_due_to_refractive_index}
\tag{4}
\end{align}

Since the absorption is also written as $$I(x) = I(0) \exp (- \mu x)$$ with the linear absorption coefficient, $$\mu$$, the relationship between $$\beta$$ and $$\mu$$ is obtained by comparing the exponential parts as

\begin{align}
\beta = \frac{\mu}{2K} = \frac{\lambda} {4\pi} \mu.
\label{Eq:absorption_coefficient}
\tag{5}
\end{align}

The complex refractive index is also written in the form of

\begin{align}
\tilde{n} = \sqrt{\frac{\epsilon \mu_{m}}{\epsilon_{0} \mu_{m0}}},
\label{Eq:full_refractive_index}
\tag{6}
\end{align}

where $$\epsilon$$ and $$\mu_{m}$$ are dielectric constants and magnetic permeability, respectively. A subscript 0 denotes the values in the vacuum. The material is magnetically equivalent to the vacuum. Thus, since $$\mu_{m} = \mu_{m0}$$, the above index can be reduced as

\begin{align}
\tilde{n} = \sqrt{\frac{\epsilon }{\epsilon_{0}}}.
\end{align}

The dielectric constant, $$\epsilon$$, is also related to an electric susceptibility, $$\chi$$, with an equation of

\begin{align}
\epsilon = \epsilon_{0} (1 + \chi_{e}).
\end{align}

### Connection between the refractive index and the scattering factors

The refractive index is related to the atomic scattering factor as follows. Electric dipole moment, $$P_{e}$$, is written as $$P_{e}= \epsilon_{0} \chi_{e} E$$. At the same time, $$P_{e}$$ is also described as $$-n_{s} ex$$, where $$n_{s}$$ is the volume density of the dipoles. Then,

\begin{align}
P_{e} = – n_{s} e x_{0} = \epsilon_{0} \chi_{e} E_{0}.
\end{align}

Therefore, $$\chi_{e}$$ is further calculated with the amplitude of the forced oscillator described in Resonant Scattering as follows:

\begin{align}
\chi_{e} &= \frac{-n_{s} e}{\epsilon_{0} E_{0}} x_{0} \notag \\
&= \frac{-n_{s} e}{\epsilon_{0} E_{0}} \left(- \frac{eE_{0}}{m} \frac{1}{(\omega_{s}^{2} – \omega^{2} -i\omega\Gamma)} \right) \notag \\
&=\frac{n_{s} e^{2} }{\epsilon_{0} m} \frac{1}{(\omega_{s}^{2} – \omega^{2} -i\omega\Gamma)} \notag \\
& = \frac{n_{s} e^{2} }{\epsilon_{0} m \omega^{2}} \frac{- \omega^{2}}{(\omega_{s}^{2} – \omega^{2} -i\omega\Gamma)}.
\end{align}

Furthermore, the last term can be replaced by the atomic scattering factor of the single oscillator. Then,

\begin{align}
\chi_{e} & = – \frac{n_{s} e^{2} \lambda^{2}}{\epsilon_{0} m (2\pi c)^{2}} f_{s} \notag \\
& = – \frac{r_{0} }{\pi} n_{s} \lambda^{2} f_{s}. \notag \\
\end{align}

Thus, we obtain the relationship between the electric susceptibility and the dispersion correction term of the atomic scattering factor.

On the other hand, the complex refractive index is written with the electric susceptibility, $$\chi_{e}$$, by assuming $$\chi_{e} \ll 1$$ and consequently with the dispersion correction term as follows:

\begin{align}
\tilde{n} &= \left( \frac{\epsilon}{\epsilon_{0}} \right)^{\frac{1}{2}} \notag \\
&= \left( 1 + \chi_{e} \right)^{\frac{1}{2}} \notag \\
&\sim 1 + \frac{1}{2} \chi_{e} \notag \\
&= 1- \frac{r_0}{2\pi} \lambda^{2} n_{s} f_{s}.
\label{Eq:refractive_index_with_fs}
\tag{7}
\end{align}

In the scope of the single forced oscillator mode, this refractive index is expressed in the form of

\begin{align}
\tilde{n} = 1- \frac{2\pi r_{0} n_{s} c^{2}}{\omega^{2} – \omega_{s}^{2} + i\Gamma \omega}.
\label{Eq:refractive_index_of_oscillator_model}
\tag{8}
\end{align}

Again, the refractive index is also affected by the bound of the electron to the nucleus as seen in the atomic scattering factor.

Practically, $$f_{s}$$ is replaced by the complex atomic scattering factor determined from experiments, i.e., $$f_j (\vec{Q}, E) = f^{0}_{j}(\vec{Q}) + f’_{j}(E) + i f\pprime_{j}(E)$$. When assuming the forward scattering, i.e., $$\vec{Q} = 0$$, $$f^{0}$$ value is identical to the atomic number, $$Z$$, and then

\begin{align}
\tilde{n}= 1- \frac{r_0}{2\pi} \lambda^{2} \sum_j n_{j} \left( Z_{j} + f’_{j} + if”_{j} \right),
\label{Eq:refractive_index_with_scattering_factor}
\tag{9}
\end{align}

where $$n_{j}$$ denotes the number of atoms of element $$j$$ in a unit volume. If the material is a crystal, Eq. \eqref{Eq:refractive_index_with_scattering_factor} is also rewritten with the unit cell volume, $$v_{c}$$ as

\begin{align}
\tilde{n}= 1- \frac{r_0}{2\pi v_{c}} \lambda^{2} F(Q=0, E) =1- \frac{r_0}{2\pi v_{c}} \lambda^{2} \sum_j \left( Z_{j} + f’_{j} + if\pprime_{j} \right).
\label{Eq:refractive_index_with_structure_factor}
\tag{10}
\end{align}

Thus, the real and imaginary pars of the complex refractive index are obtained by comparing each pat in the above equation and Eq. \eqref{Eq:refractive_index} as follows

\begin{align}
\delta = \frac{r_0}{2\pi v_{c}} \lambda^{2} \sum_j \left( Z_{j} + f’_{j}\right),\ \ \beta = -\frac{r_0}{2\pi v_{c}} \lambda^{2} \sum_j f\pprime_{j}.
\label{Eq:each_parts_of_refractive_index_with_structure_factor}
\tag{11}
\end{align}

Furthermore, $$\mu$$ is linked with $$f\pprime$$ by Eq. \eqref{Eq:absorption_coefficient} by

\begin{align}
\mu = \frac{4\pi}{\lambda} \beta = \frac{2\lambda r_{0}}{v_{c}} \sum_j \left( -f\pprime_{j} \right).
\label{Eq:mu_and_f_double_prime}
\tag{12}
\end{align}

Therefore, $$f\pprime$$ obtained from the DAFS method is completely equivalent to $$\mu$$ by multiplying wavelength, $$\lambda$$, to $$f\pprime$$ (or divided by the photon energy). Importantly, $$f\pprime$$ is a negative value because $$\mu$$ is positive from Eq. \eqref{Eq:mu_and_f_double_prime}. Conventionally, $$f\pprime$$ appears as a positive value in many textbooks and/or tables. It causes no problem as long as we discuss the diffraction intensity, where only the square of $$f\pprime$$ appears in the intensity; however, it is of importance for the DAFS method to distinguish the sign of $$f\pprime$$ values because we need to extract the site- and/or phase dependent $$f\pprime$$ value itself by directly solving the simultaneous equation of the weighted $$f\pprime$$ values.

###### For further study…(This article was written based on the following books)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
• 菊田惺志(S. Kikuta), X線散乱と放射光科学 基礎編 (X-ray scattering and synchrotron radiation science -basics- (English title was translated by T.K.)), 東京大学出版 (University of Tokyo Press), 2011.

## Resonant Scattering


### Resonant scattering term

In the previous article “scattering by one electron”, the classical Thomson scattering from an extended distribution of free electrons is derived as $$-r_{0}f^{0}(\vec{Q})$$, where $$-r_{0}$$ is the Thomson scattering length of a single element and $$f^{0}(\vec{Q})$$ is the atomic form factor. The atomic form factor is a Fourier transform of the electron distribution in an atom; therefore, it is a real number and independent of photon energy. In contrast, there exists an absorption edge and a fine structure in an absorption spectrum in the x-ray region. Thus, the absorption term should be included into the scattering length as an imaginary part, which is proportional to the absorption cross-section, by assuming a more elaborated model rather than that of a cloud of free electrons.

As energy dependent terms from the forced oscillator model will be derived in the following discussion, an atomic scattering factor consists of real and imaginary energy-dependent terms as well as the conventional atomic form factor as follows:

\begin{align}
f(\vec{Q}, E) = f^{0} (\vec{Q}) + f'(E) + i f\pprime(E),
\label{Eq:complex_scattering_factor}
\tag{1}
\end{align}

where $$f’$$ and $$f\pprime$$ are the real and imaginary parts of the dispersion corrections. These terms are called resonant scattering terms, while at one time it was conventionally referred to as anomalous scattering factors. Usually $$\vec{Q}$$ dependence of the resonant scattering terms are negligible, because the dispersion corrections are dominated by electrons in the core shell such as $$K$$ shell, which is spatially confined around an atomic nucleus.

Fig. 1 shows the theoretical curve of the resonant scattering terms of a bare Ni atom. $$f’$$ shows local minimum cusps at each absorption edge, while $$f\pprime$$, which should be , strictly speaking, negative values, has absorption edges as described in the XAFS section. These features of the resonant terms enable us to carry out various sophisticated x-ray scattering techniques, e.g., Multi-wavelength Anomalous Diffraction (MAD) for the determination of the unique crystalline structure without suffering from the phase problem [1]. Furthermore, by using the polarization and azimuthal dependences of the resonant terms, the resonant scattering techniques also contribute to the determination of spin and orbital orders seen in a strongly-correlated electron system and enantiomeric materials such as quartz [2, 3]. The structural analysis is usually carried out by hard x-ray (typically, >5 keV), where $$K$$ absorption edges of third-period elements locates as shown in Fig. 2.

Thus, we can evaluate the occupations of the similar elements such as Fe, Co and Ni in a crystal by the resonant scattering technique thanks to the characteristic of steep decrease in $$f’$$ at the each absorption edge, while the nonresonant x-ray diffraction technique hardly distinguishes the contributions from similar elements to a certain crystallographic site. This kind of approach to extract the structural information at a specific element is also applicable to the structural analysis of the amorphous materials by total scattering measurements [4, 5, 6]. Importantly, the diffraction anomalou fine structure (DAFS) method is also one of measurement techniques utilizing this resonant feature. In this method, we observe the $$f\pprime$$ and $$f’$$ spectra, which also reflect the fine structure as seen in a X-ray absorption fine structure (XAFS) spectrum of $$f\pprime$$, through the scattering channel.

In this article, a forced charged oscillator model is introduced to explain the basic principles behind how $$f’$$ and $$f\pprime$$ appear in the atomic scattering factor. This model is obviously classical and a crude approximation; however, it can help us to understand the relationship between $$f’$$ and $$f\pprime$$, i.e., scattering and absorption.

### The forced charged oscillator model

Suppose that an electron bound in an atom is subjected to the electric field of an incident x-ray beam, $$\vec{E}_{in} = \hat{\vec{x}} E_{0} \e^{-i\omega t}$$, which is linearly polarized along the $$x$$ axis with amplitude $$E_{0}$$ and frequency $$\omega$$. The motion equation for this electron is

\begin{align}
\ddot{x} + \Gamma \dot{x} + \omega_{s}^{2}x = – \left( \frac{eE_{0}}{m}\right) \e^{-i\omega t},
\label{Eq:forced_motion_electron}
\tag{2}
\end{align}

where $$\Gamma \dot{x}$$ is the velocity-dependent damping term corresponding to the dissipation of energy from the applied electric field due to the re-radiation, $$\omega_{s}$$ is the resonant frequency usually much larger than the damping constant, $$\Gamma$$. The solution for the differential equation is described as $$x(t) = x_{0}\e^{-i\omega t}$$ and consequently the amplitude of the forced oscillation is

\begin{align}
x_{0} = – \left( \frac{eE_{0}}{m}\right) \frac{1}{\omega_{s}^{2} – \omega^{2} -i\omega\Gamma}.
\label{Eq:coefficient_of_solution}
\tag{3}
\end{align}

The radiated field for an observer at a distance $$R$$ and at time $$t$$ is proportional to the $$\ddot{x}(t-R/c)$$ at the earlier time $$t’ = t -R/c$$; therefore,

\begin{align}
E_{\mathrm{rad}}(R,t) = \left( \frac{e^{2}}{4\pi \epsilon_{0} m c^{2} }\right) \ddot{x}(t-R/c),
\end{align}

where the polarization factor $$\hat{\vec{\epsilon}}\cdot \hat{\vec{\epsilon}}’$$ is assumed to be 1. By inserting the specific value of $$\ddot{x}(t-R/c)$$ calculated from $$x(t) = x_{0}\e^{-i\omega t}$$ and Eq. \eqref{Eq:coefficient_of_solution}, the above equation is expanded to be

\begin{align}
E_{\mathrm{rad}}(R,t) = \frac{\omega^{2}}{\omega_{s}^{2} – \omega^{2} -i\omega\Gamma} \left( \frac{e}{4\pi \epsilon_{0} R c^{2} }\right) E_{0} \e^{-i\omega t} \left( \frac{\e^{ikR}}{R} \right)
\end{align}

or equivalently

\begin{align}
\frac{E_{\mathrm{rad}}(R,t)}{E_{\mathrm{in}}} = -r_{0}\frac{\omega^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma} \left( \frac{\e^{ikR}}{R} \right).
\end{align}

The atomic scattering length, $$f_{s}$$, is defined to be the amplitude of the outgoing spherical wave, $$(\e^{ikR}/R)$$. (cf. scattering by one electron) Thus, $$f_{s}$$ in units of $$-r_{0}$$ is

\begin{align}
f_{s} = \frac{\omega^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma},
\label{Eq:resonant_terms_of_single_oscillator}
\tag{4}
\end{align}

where a subscript, $$s$$, denotes the “single oscillator”. For frequencies greatly larger than the resonant frequency, i.e., $$\omega \gg \omega_{s}$$, the value of $$f_{s}$$ should be approximated to be the Thomson scattering length of 1. Thus, the following reduction makes the equation clearer to understand the resonant scattering terms:

\begin{align}
f_{s} &= \frac{\omega^{2} – \omega_{s}^{2} + i\omega \Gamma + \omega_{s}^{2} – i\omega \Gamma}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma}
= 1 + \frac{ \omega_{s}^{2} – i\omega \Gamma}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma} \notag \\
&\sim 1 + \frac{ \omega_{s}^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma},
\label{Eq:fs_reduction}
\tag{5}
\end{align}

where the last line is derived based on the fact that $$\Gamma$$ is usually much less than $$\omega_{s}$$. Eq. \eqref{Eq:fs_reduction} clearly shows that the second term corresponds to the dispersion correction to the scattering factor. When the dispersion correction is written as $$\Delta f (\omega)$$, the value is described as

\begin{align}
\Delta f ( \omega) = f’_{s} + if\pprime_{s} = \frac{ \omega_{s}^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma}
\end{align}

with the real part given by

\begin{align}
f’_{s} = \frac{ \omega_{s}^{2} ( \omega^{2} – \omega_{s}^{2} )}{(\omega^{2} – \omega_{s}^{2} )^{2} + (\omega\Gamma)^{2}}
\label{Eq:dispersion_correction_real}
\tag{6}
\end{align}

and the imaginary part also given by

\begin{align}
f\pprime_{s} = – \frac{ \omega_{s}^{2} \omega \Gamma}{(\omega^{2} – \omega_{s}^{2})^{2} + (\omega\Gamma)^{2}}.
\label{Eq:dispersion_correction_imaginary}
\tag{7}
\end{align}

The dispersion correction terms calculated from the forced oscillator model are shown in Fig. 3. The imaginary part of the dispersion correction, $$f\pprime$$, corresponds to the absorption, showing the peak profile at $$\omega = \omega_{s}$$. In contrast, the absorption spectrum of a real material is like an edge rather than the peak; the discrepancy between two spectra shows the limitation of the single forced oscillator model. In order to model this behavior, we need to take into account so-called oscillator strength, $$g_{o}(\omega_{s})$$, to compensate the gap between the model and a real material, which gives the population of the single oscillators dependent on the photon energy; however, it still gives no explanation of XAFS (oscillation characteristic observed in the condensed material). Eventually, though the single oscillator model is not adequate for the quantitative understanding of the resonant scattering terms, it is helpful to understand the emergence of the dispersion correction terms due to the bound of the electron to the nucleus. The quantitative explanation and its evaluation require more sophisticated approach of quantum mechanics, where $$f’$$ and $$f\pprime$$ are derived from the 1st- and 2nd-order perturbation theory of the interaction Hamiltonian of $$(e \vec{A} \cdot \vec{p} / m)$$.

###### For further study…(This article was written based on the following books)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
• 菊田惺志(S. Kikuta), X線散乱と放射光科学 基礎編 (X-ray scattering and synchrotron radiation science -basics- (English title was translated by T.K.)), 東京大学出版 (University of Tokyo Press), 2011.
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## Calculation of a structure factor

A structure factor is calculated by summing up scattering factors of each atom with multiplying the phases at each atomic position in a unit cell as described previously in an equation. It is a facile approach to calculate the structure factor; however, the calculation becomes complicated when the unit cell includes the large number of atoms. Furthermore, it is frequently difficult to distinguish the equivalent and non-equivalent sites in this approach.

For example in body-centered cubic (BCC) metal, atom occupy (0, 0, 0), (1/2, 1/2, 1/2) sites, and all of them are equivalent. Thus, these atoms must have exactly the same local/electronic structure. In contrast, in cesium chloride structure, (0, 0, 0) and (1/2, 1/2, 1/2) are occupied by chloride and cesium atoms, respectively, being inequivalent each other; therefore, the local/electronic structure must be different. The “site-distinguished” analysis of the diffraction anomalous fine structure (DAFS) method provides individual XAFS spectra at the inequivalent sites in the material as seen in the latter case. These relations and concepts are comprehensively understood in the framework of Space Group, which is well-described in the International Tables for Crystallography vol. A[1]. Any crystalline materials except for a quasicrystal belong to a certain space group, and the structure factor of each space group is calculated and included in the international table [2].1 Thus, the calculation of the structure factor should be carried out based on the crystallographic site in space group, which is more versatile and convenient.

The table provides the structure factor as values of “$$A(\vec{G}\cdot \vec{r}_{m})$$” and “$$B(\vec{G}\cdot \vec{r}_{m})$$”, whose definition is as follows:

\begin{align}
A(\vec{G}\cdot \vec{r}_{m}) &= \sum_{e} \cos (\vec{G}\cdot \vec{r}_{m}) \\
B(\vec{G}\cdot \vec{r}_{m}) &= \sum_{e} \sin (\vec{G}\cdot \vec{r}_{m}),
\label{Eq:def_A_and_B}
\tag{1}
\end{align}

where $$\sum_{e}$$ denotes the summation of equivalent positions belonging the site in a unit cell. Then, the structure factor is described with these $$A$$ and $$B$$ values as

\begin{align}
F(\vec{G}) = \sum_{j} f_{j}A_{j} + i \sum_{j} f_{j}B_{j},
\label{Eq:A_B_based_structure_factor}
\tag{2}
\end{align}

where $$\sum_{j}$$ is the summation of independent sites, and $$f_{j}$$ is the atomic scattering factor of the atom at site $$j$$. Note that both $$A$$ and $$B$$ values should be multiplied by the ratio of the numbers of atoms at general and special positions. For example, when calculating the structure factor of $$32e$$ site (Wyckoff letter for a special position) in space group $$F d \bar{3} m$$ (No. 227), the factor of 32/192 ($$192i$$ is for general position in this space group) should be multiplied in order to take into account the overlap of the atoms at the special position. Also note that the value of $$B$$ disappears in the space group with centrosymmetry, because $$\sum_{e} \sin (\vec{G}\cdot \vec{r}_{m})$$ becomes 0 when the same element locates at $$\vec{r}_{m}$$ and $$-\vec{r}_{m}$$ due to the nature of $$\sin$$ function.

###### References
[1] International Tables for Crystallography Volume A, 5th ed., T. Hahn, Ed., Springer, 2006.
[Bibtex]
@book{Hahn2006,
edition = {5th},
editor = {Hahn, Th},
file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Unknown/Unknown\_Unknown\_2006.pdf:pdf},
isbn = {0792365909},
publisher = {Springer},
title = {{International Tables for Crystallography Volume A}},
year = {2006}
}
[2] International Tables for X-ray Crystallography Volume I, N. F. M. Henry and K. Lonsdale, Eds., Birmingham: Kynoch Press, 1952.
[Bibtex]
@book{Henry1952,
editor = {Henry, Norman F. M and Lonsdale, Kathleen},
file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Unknown/Unknown\_Unknown\_1952.pdf:pdf},
pages = {367},
publisher = {Kynoch Press},
title = {{International Tables for X-ray Crystallography Volume I}},
year = {1952}
}

## Lorentz factor

In the previous articles, the scattered intensity and related factors are derived. In an actual experiment of diffraction anomalous fine structure (DAFS) as well as the conventional X-ray diffraction (XRD), we need to “scan” a sample and/or a finite-size detector to obtain the whole shape of the diffraction peak, whose area is proportional to the structure factor described above. Thus, the factor concerning $$\vec{Q}$$ step2 and/or the fraction of detective diffraction, which are dependent on scattering angle and also energy, should be introduced for the evaluation of the scattered intensity, being called the Lorentz factor. This factor is essentially quite different from the other factors described above, i.e., atomic scattering factor, structure factor and Debye-Waller factor, because the Lorentz factor gives no structural information about the crystalline sample, being derived only from the experimental aspect. Thus, the effect of the Lorentz factor should be corrected for the subsequent structural and spectroscopic analyses using X-ray diffraction.

In the Lorentz factor appearing in a textbook and/or articles regarding the conventional XRD, only the angular dependence is focused and discussed, presumably because such a measurement is carried out by a single monochromatic x-ray. In contrast, the correction of the energy-dependence in the Lorentz factor is necessary in the spectroscopic analysis like the DAFS method. Thus, the following article briefly describes the derivation of the Lorentz factor including its energy dependence as well as that of the scattering angle.

### Single crystal diffraction

Let’s start from the case of a single crystal. A schematic of the experimental setup to measure the integrated intensity of diffraction from a single crystal particle is shown in Fig. 1. In the calculation, it is assumed that the incident and scatted beam are monochromatic, and that the incident is perfectly collimated while the scatted beam necessarily is not perfectly collimated, because the number of the lattice $$N$$ is finite and the beam will have some divergence. Left-hand side of Fig. 1 shows the schematic of the reciprocal space around a reciprocal lattice point, where only the portion of the point on the Ewald’s sphere, which is a sphere depicted by the possible terminal points of the outgoing wave number vector $$\vec{k’}$$, is observed at a certain incident angle (purple lines). Thus, the crystal has to be rotated (i.e., rocking scan) to obtain the integrated intensity from the reciprocal lattice points, which is drawn with watery, purple, pink lines, on an axis of $$\theta$$.

As shown in the previous article, the Laue function becomes the Dirac’s delta function when the number of the lattices is sufficiently large; therefore, the integrate intensity is described as follows:

\begin{align}
\int \diff \hat{\vec{k}}’ \delta (\vec{Q}-\vec{G} ) = \int \diff \hat{\vec{k}}’ \delta(\vec{k} – \vec{k}’ – \vec{G}),
\label{Eq:integration_of_k_LF}
\tag{1}
\end{align}

where $$\hat{\vec{k}}’$$ is a unit vector along $$\vec{k}’$$. The element of the solid angle $$\diff \hat{\vec{k}}’$$ is two-dimensional vector, which means the integration of the all scattered angles under a fixed incident angle. For the calculation, the vector, $$\vec{s} = k’ \hat{\vec{s}}$$, is introduced instead of $$\vec{k}’$$, where $$\hat{\vec{s}}$$ is a unit vector (see Fig.2). Then, the integration is transformed by adding the integration of the unit value2 into

\begin{align}
\tag{2}
\int \diff \hat{\vec{k}}’ \delta (\vec{k} – \vec{k}’ -\vec{G} ) = \overbrace{\frac{2}{k’} \int s^{2} \delta(s^{2} – k’^{2}) \diff s}^{1}\int \delta(\vec{k} – \vec{s} – \vec{G} ) \diff \hat{\vec{s}},
\end{align}

where $$\vec{k}’$$ is replaced by $$\vec{s}$$ in the second integration. The trick of adding the integration and the change of variable is to transform the integration of two-dimension into that of three-dimension.

Based on the schematic of the integration parameter in Fig. 2, the above equation is furthered transformed into

\begin{align}
\int \diff \hat{\vec{k}}’ \delta (\vec{k} – \vec{k}’ -\vec{G} ) &= \frac{2}{k’} \int \delta(s^{2} – k’^{2}) \delta(\vec{k} – \vec{s} – \vec{G} ) \diff \hat{\vec{s}}\diff s \notag \\
&= \frac{2}{k’} \int \delta(s^{2} – k’^{2}) \delta(\vec{k} – \vec{s} – \vec{G} ) \diff \vec{s}.
\label{Eq:int_para_change_LF}
\tag{3}
\end{align}

When $$\vec{s} = \vec{k} – \vec{G}$$, $$\delta(\vec{k} -\vec{s} – \vec{G}) = \delta (0)$$ and the integration is reduced to be

\begin{align}
\int \diff \hat{\vec{k}}’ \delta (\vec{k} – \vec{k}’ -\vec{G} ) &= \frac{2}{k’} \delta((\vec{k} – \vec{G})\cdot (\vec{k} – \vec{G}) – k’^{2}) \notag \\
&= \frac{2}{k} \delta(G^2 – 2kG \sin \theta ).
\label{Eq:int_result_LF}
\tag{4}
\end{align}

Finally, the preparation for the integration of $$\theta$$, which means the evaluation of the integrated intensity under the rocking scan, is completed.

The differential cross section of the diffraction is described by using the result above as

\begin{align}
\label{Eq:cross_section_LF}
\tag{5}
\left( \frac{\diff \sigma}{\diff \Omega}\right)_{\mathrm{int. \ over\ } \vec{k}’} = r_{0}^{2}P |F(\vec{Q})|^{2} N v_{c}^{*} \frac{2}{k}\delta(G^{2} -2kG \sin \theta).
\end{align}

Because the integration of the delta function by $$\theta$$ gives the following value

\begin{align}
\label{Eq:int_delta_func_LF}
\tag{6}
\int \delta(G^{2} -2kG \sin \theta) \diff \theta = \left[ \frac{-1}{2kG \cos \theta} \right] _{t = 0} = \frac{-1}{2k^{2} \sin 2 \theta},
\end{align}

The cross-section is further derived as

\begin{align}
\left( \frac{\diff \sigma}{\diff \Omega}\right)_{\mathrm{int. \ over}\ \vec{k}’, \theta} &= r_{0}^{2}P |F(\vec{Q})|^{2} N v_{c}^{*} \frac{2}{k} \frac{1}{2 k^{2} \sin 2 \theta} \notag \\
&= r_{0}^{2}P |F(\vec{Q})|^{2} N \frac{\lambda^{3}}{v_{c}} \frac{1}{\sin 2 \theta}
\label{Eq:cross_section_result_LF}
\tag{7}
\end{align}

and detective intensity is

\begin{align}
I_{\mathrm{SC}}\left( \mathrm{ \frac{photons}{sec}} \right) = \Phi_{0} \left( \mathrm{ \frac{photons}{unit\ area \times sec}} \right) r_{0}^{2}P |F(\vec{Q})|^{2} N \frac{\lambda^{3}}{v_{c}} \frac{1}{\sin 2 \theta}.
\label{Eq:scat_intensity_result_LF}
\tag{8}
\end{align}

Therefore, when we discuss the energy dependency of the DAFS spectrum, we need to correct the factor, $$\lambda^{3}/\sin 2 \theta$$ or $$1/(E^{3} \sin 2 \theta)$$, in a single crystal.

### Powder diffraction

In the powder diffraction, the Lorentz factor is decomposed into three parts as follows:

\begin{align}
\label{Eq:three_LF_in_XRPD}
\tag{9}
L(\theta, E) = L_{1}L_{2}L_{3},
\end{align}

where $$L_{1} = 1/ ( E^{3} \sin 2\theta )$$ same as that in single crystal, and both $$L_{2}$$ and $$L_{3}$$ are additional Lorentz factors for the powder diffraction, which will be subsequently introduced.

$$L_{2}$$ is derived from the angle dependence of the number of the observable crystalline particles. A sample for X-ray powder diffraction (XRPD) is randomly oriented crystallites, whose scattered intensity is described as a simple summation of the scattering intensity from each small crystalline particle. Thus, the scattered intensity is dependent on the number of the observable reciprocal lattice points at the same time. The sphere of the numerous reciprocal lattice point, whose radius is $$|\vec{G}| = 2\pi / d \equiv G$$, and the observable area on the “reciprocal sphere” drown as a ribbon are shown in Figure 3. When it is supposed that the number of the crystalline particle is $$N$$ and their reciprocal lattice points homogeneously distribute on the sphere, the angle dependence of the area of the ribbon corresponds to the $$L_{2}$$ value.

Straight line CP is a perpendicular to the lattice plane of a crystalline particle we observe, and $$\Delta \theta$$ is an acceptable angle of the diffraction derived from the divergence and the energy width of the incident beam. Namely, the particles whose reciprocal points locate among the range of $$\Delta \theta$$ satisfy the diffraction condition. Thus, from the geometric consideration of Fig. 3, the number, $$\Delta N$$, is described as

\begin{align}
\Delta N &= G \Delta \theta 2 \pi G \sin (\pi – \theta) \notag \\
&= 2\pi G^{2} \Delta \theta \cos \theta .
\label{Eq:number_of_particle_on_the_ribbon}
\tag{10}
\end{align}

Since the area of the whole sphere is $$4\pi G^{2}$$, the fraction of the observable number of the particles is

\begin{align}
\label{Eq:ratio_of_the_number_of_particles}
\tag{11}
\frac{\Delta N}{N} = \frac{\Delta \theta \cos \theta}{2}.
\end{align}

Therefore, the integrated intensity of XRPD is proportional to the factor:

\begin{align}
\label{Eq:L2_factor}
\tag{12}
L_{2} = \cos \theta.
\end{align}

The factor of $$L_{3}$$ is from the observation technique of XRPD, where we usually observe a portion of the Debye-Scherrer ring, by scanning a detector with a finite-size sensitive area. When the camera length, i.e., the distance between the detector and the sample, is $$R$$, the radius of the Debye-Sherrer ring at the detector position is $$R \sin 2 \theta$$, and consequently that of the length is $$2 \pi R \sin 2 \theta$$ as shown in Fig. 4.

If we observe the portion of the Debye-Scherrer ring of $$\delta R$$, the ratio of these lengths, i.e., $$\delta R/ 2\pi R \sin \theta$$, corresponds to the observable scattered intensity.
Thus, the integrated intensity is proportional to the factor:

\begin{align}
\label{Eq:L3_factor}
\tag{13}
L_{3} = \frac{1}{\sin 2 \theta}.
\end{align}

Finally, we obtain the complete Lorentz factor for XRPD and powder-DAFS as follows:
\begin{align}
L(\theta, E) &= L_{1} L_{2} L_{3} \notag \\
&= \frac{1}{E^{3}\sin 2\theta} \cos \theta \frac{1}{\sin 2\theta} \notag \\
&= \frac{1}{4 E^{3}\sin ^{2} \theta \cos \theta}.
\label{Eq:full_lorentz_factor_for_XRPD}
\tag{13}
\end{align}

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}

## Debye-Waller factor


The lattice has been assumed to be “perfectly rigid” in the evaluation of the scattering amplitude from the crystal in the previous article; however, the atoms vibrate due to two distinct causes in a real material. The first is from the uncertainty principle of the quantum mechanics, which is independent on temperature and observed even at 0 K, being called zero-point fluctuation. The second is from the elastic wave and/or phonon in the crystal, depending on the temperature. Whether the vibration is caused by above two mechanisms, the atomic vibration reduces the magnitude of the interference of the scattering wave from the different atoms due to the “ambiguity” of the atomic position, eventually decreasing the scattering amplitude. This attenuation factor is known as the Debye-Waller factor in x-ray diffraction.

The Debye-Waller factor is affected by some factors. The magnitude of the attenuation basically depends on the element; a heavier element shows small attenuation at a certain temperature. Furthermore, the attenuation magnitude is also dependent on the crystallographic site even when the same element is occupied. In addition, this vibration effect is enhanced in the higher scattering angle, i.e., small lattice spacing, because the scattering at the higher angle is more sensitive to the phase difference than that at the lower angle. Usually, the Debye-Waller factor is implemented into the structure factor by multiplying an exponential attenuation term, whose derivation will be given in the subsequent section. The site-selectivity of the diffraction anomalous fine structure (DAFS) originates from the difference in the contribution of the atoms to the a certain diffraction as described in the derivation of the structure factor. Thus, this factor should be included into the DAFS analysis to accurately separate each contribution of the atom in the different crystallographic sites.

For the sake of simplicity of the derivation, the scattering amplitude of a crystal consisting of a single element with some displacement from the average position is evaluated as follows:

\begin{align}
\label{Eq:DW_cal_displacement}
\tag{1}
F^{\mathrm{crystal}}(\vec{Q}) = \sum_{n} f(\vec{Q})\e^{i\vec{Q}\cdot (\vec{R}_{n} + \vec{u}_{n})},
\end{align}

where $$\vec{R}_{n} + \vec{u}_{n}$$ is the instantaneous position of the atom, $$\vec{R}_{n}$$ is the time-averaged mean position, and $$\vec{u}_{n}$$ is the displacement, which temporal average value, $$\left< \vec{u}_{n} \right>$$, is zero from the definition. Since the scattering intensity is calculated by taking the product of the scattering amplitude and its complex conjugate, the time-average scattering intensity is

\begin{align}
\label{Eq:time_average_intensity}
\tag{2}
I &= \left\langle \sum_{m} f(\vec{Q})\e^{i\vec{Q}\cdot (\vec{R}_{m} + \vec{u}_{m})} \sum_{n} f^{*}(\vec{Q})\e^{-i\vec{Q}\cdot (\vec{R}_{n} + \vec{u}_{n})} \right\rangle \notag \\
&= \sum_{m} \sum_{n} f(\vec{Q}) f^{*}(\vec{Q}) \e^{i\vec{Q}\cdot (\vec{R}_{m} -\vec{R}_{n})} \left\langle \e^{i\vec{Q}\cdot (\vec{u}_{m} -\vec{u}_{n})} \right\rangle.
\end{align}

For the further calculation, the last term of the second row in the equation is rewritten as

\begin{align}
\label{Eq:DW_dimension_reduction}
\tag{3}
\left\langle \e^{i\vec{Q}\cdot (\vec{u}_{m} -\vec{u}_{n})} \right\rangle = \left\langle \e^{iQ (u_{Qm} – u_{Qn})} \right\rangle,
\end{align}

where $$u_{Qn}$$ is the component of the $$\vec{u}_{n}$$ parallel to the vector, $$\vec{Q}$$, for the $$n$$-th atom. By using the Baker-Hausdorff theorem expressed as

\begin{align}
\label{Eq:BH_theory}
\tag{4}
\left\langle \e^{ix} \right\rangle = \e^{-\frac{1}{2} \left\langle x^{2} \right\rangle},
\end{align}

the right hand side in Eq. \eqref{Eq:DW_dimension_reduction} is reduced to be

\begin{align}
\left\langle \e^{iQ (u_{Qm} – u_{Qn})} \right\rangle &= \e^{-\frac{1}{2} \left\langle Q^{2}(u_{Qm}-u_{Qn})^{2} \right\rangle} \notag \\
&= \e^{-\frac{1}{2} Q^{2} \left\langle (u_{Qm}-u_{Qn})^{2} \right\rangle} \notag \\
&= \e^{-\frac{1}{2} Q^{2} \left\langle u_{Qm}^{2} \right\rangle} \e^{-\frac{1}{2} Q^{2} \left\langle u_{Qn}^{2} \right\rangle} \e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle}.
\tag{5}
\label{Eq:reduction_of_u}
\end{align}

Because of the translation symmetry, $$u_{Qn}^{2} = u_{Qm}^{2}$$ and its value will be simply expressed as $$u_{Q}^{2}$$. In addition $$\e^{-Q^{2} \left\langle u_{Q}^{2} \right\rangle /2}$$ is also expressed as $$\e^{-M}$$ in the following derivation. In order to separate the scattering intensity into two terms, the last term of Eq. \eqref{Eq:reduction_of_u} is written as

\begin{align}
\label{Eq:correlated_vibration}
\tag{6}
\e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle} = 1 + \left\{ \e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle} – 1 \right\}.
\end{align}

Then, the scattered intensity is decomposed into two terms as

\begin{align}
I &= \sum_{m} \sum_{n} f(\vec{Q}) \e^{-M} \e^{i\vec{Q}\cdot \vec{R}_{m}} f^{*}(\vec{Q}) \e^{-M} \e^{-i\vec{Q}\cdot \vec{R}_{n}} \notag \\
&+ \sum_{m} \sum_{n} f(\vec{Q}) \e^{-M} \e^{i\vec{Q}\cdot \vec{R}_{m}} f^{*}(\vec{Q}) \e^{-M} \e^{-i\vec{Q}\cdot \vec{R}_{n}} \left\{ \e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle} – 1 \right\}.
\label{Eq:decomposed_scattered_intensity}
\tag{7}
\end{align}

The first term is the elastic scattering from a lattice, i.e., x-ray diffraction; however the scatted intensity is weaken by the factor of $$\e^{-M} ( < 1)$$, which is known as the Debye-Waller factor. This factor can be generally introduced by replacing the atomic scattering factor by

\begin{align}
\label{Eq:introduction_of_DW_factor}
\tag{8}
f^{\mathrm{atom}} = f^{0}(\vec{Q}) \e^{- \frac{1}{2} Q^{2} \langle u_{Q}^{2} \rangle} \equiv f^{0}(\vec{Q}) \e^{-M}.
\end{align}

Conventionally, the magnitude of the Debye-Waller factor is given and discussed in the form of $$B_{T}$$ as

\begin{align}
\label{Eq:introduction_of_B}
\tag{9}
M = \frac{1}{2}Q^{2}\langle u_{Q}^{2} \rangle = \frac{1}{2} \left( \frac{4\pi \sin \theta}{\lambda} \right)^{2} \langle u_{Q}^{2} \rangle = B_{T} \left( \frac{\sin \theta}{\lambda} \right)^{2},
\end{align}

with

\begin{align}
\label{Eq:definition_of_B}
\tag{10}
B_{T} \equiv 8\pi^{2} \langle u_{Q}^{2} \rangle,
\end{align}

because of the traditional reason of the XRD description, where the angle dependence of a parameter is favorably expressed as a function of $$(\sin \theta / \lambda)$$ rather than $$Q$$ (for example, the atomic form factor is also given in the above form in equation in the previous article. If the atoms vibrate isotropically, $$\langle u^{2} \rangle = \langle u^{2}_{x} + u^{2}_{y} + u^{2}_{z} \rangle = 3 \langle u^{2}_{x} \rangle = 3 \langle u_{Q}^{2} \rangle$$, then

\begin{align}
\label{Eq:isotropic_DW_factor}
\tag{11}
B_{T, \mathrm{isotropic}} = \frac{8 \pi^{2}}{3}\langle u^{2} \rangle.
\end{align}

Though the deviation above proceeded on the assumption of the single element, the structure factor of plural elements is analogically derived as

\begin{align}
\label{Eq:structure_factor_with_DW_factor}
\tag{12}
F &= \sum_{m} f_{m} \exp \left( -M_{m}\right) \exp \left( i\vec{Q}\cdot \vec{r}_{m}\right) \\
&= \sum_{m} f_{m} \exp \left\{ -B_{T, m} \left( \frac{\sin \theta}{\lambda}\right)^{2} \right\} \exp \left( i\vec{Q}\cdot \vec{r}_{m}\right).
\end{align}

The magnitude of the Debye-Waller factor of each element can be evaluated by a preliminary XRD analysis such as the Rietveld analysis based on the XRPD. Typical values are available on International tables for x-ray crystallography vol. II, ranging from 0 to 2. The refined value should be used for the site-separation of the absorption spectrum obtained from the DAFS method.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}

## Scattering from a crystal

In a crystal, atoms or molecules form a periodic structure with the translational symmetry, which is frequently called long range order. Thus, scatterings from each atom are interfered and eventually create a diffraction pattern, which reflects the crystalline structure. Since each diffraction has a different contribution from each atom, the diffraction anomalous fine structure (DAFS) method is capable to distinguish the crystalline site-specific spectroscopic information by measuring an energy dependence of the diffraction intensity. This article will provide a brief review about the description of the conventional x-ray diffraction.

As seen in the discussion of the scattering from an atom, the scattering amplitude from plural atoms, i.e., a crystal, are similarly calculated from the summation of phases multiplied by atomic scattering factors of each atom. First, the position of an atom in the crystal is described by taking the translational symmetry into account as

\begin{align}
\label{Eq:position_vector}
\tag{1}
\vec{r}_{l}= \vec{R}_{n} + \vec{r}_{m},
\end{align}

with

\begin{align}
\label{Eq:atomic_position_vector_definition}
\tag{2}
\vec{R}_{n} &\equiv n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3} \\
\vec{r}_{m} &\equiv x_{m}\vec{a}_{1}+y_{m}\vec{a}_{2}+z_{m}\vec{a}_{3},
\end{align}

where $$\vec{R}_{n}$$ is a vector specifying the number $$n$$-th unit cell, $$\vec{r}_{m}$$ is a vector indicating $$m$$-th atomic position in a unit cell, $$\vec{a}_{1}$$, $$\vec{a}_{2}$$ and $$\vec{a}_{3}$$ are lattice vectors along with $$a$$, $$b$$ and $$c$$ axes, and $$x_{m}$$, $$y_{m}$$ and $$z_{m}$$ are the fractional coordinate of $$m$$-th atom. Thus, the scattering amplitude from the crystal can be described as

\begin{align}
\label{Eq:scat_amp_all_atom}
\tag{3}
F^{\mathrm{crystal}}(\vec{Q}) = \sum^{\mathrm{All\ atoms}}_{l}f_{l}(\vec{Q})\e^{i\vec{Q}\cdot \vec{r}_{l}},
\end{align}

where $$f_{l}(\vec{Q})$$ is the atomic form factor3 of the atom placed at position $$\vec{r}_{l}$$. The scattering amplitude is readily decomposed into two contributions from the lattice and that from the inside of the unit cell with Eq. \eqref{Eq:position_vector} as

\begin{align}
\label{Eq:decomposition_of_scattering_amplitude}
\tag{4}
F^{\mathrm{crystal}}(\vec{Q}) = \sum^{\mathrm{All\ atoms}}_{\vec{R}_{n}+\vec{r}_{j}} f_{j}(\vec{Q})\e^{i\vec{Q}\cdot(\vec{R}_{n}+\vec{r}_{j})} = \overbrace{\sum_{n}\e^{i\vec{Q}\cdot\vec{R}_{n}}}^{\mathrm{Lattice}} \overbrace{\sum_{m}f_{j}(\vec{Q})\e^{i\vec{Q}\cdot\vec{r}_{m}}}^{\mathrm{Unit\ cell}}
\end{align}

The first term of Eq. \eqref{Eq:decomposition_of_scattering_amplitude} is the sum of the scattering from the lattice, while the second term is the sum of atoms in the unit cell, which is known as structure factor.

The diffraction from a crystal is observed under the diffraction condition, which is conventionally described as $$2d\sin \theta = \lambda$$. The equivalent diffraction condition in the vector form is

\begin{align}
\label{Eq:laue_condition}
\tag{5}
\vec{Q} = \vec{G},
\end{align}

where $$\vec{G}$$ is a vector pointing reciprocal lattice defined as

\begin{align}
\label{Eq:def_G}
\tag{6}
\vec{G} = h\vec{a}^{*}_{1}+k\vec{a}^{*}_{2}+l\vec{a}^{*}_{3}.
\end{align}

$$h$$, $$k$$ and $$l$$ are all integers and called “Miller indices”. $$\vec{a}^{*}_{1}$$, $$\vec{a}^{*}_{2}$$ and $$\vec{a}^{*}_{3}$$ are basis vectors of reciprocal space defined as2

\begin{align}
\tag{7}
\vec{a}^{*}_{1} &= 2\pi \frac{\vec{a}_{2}\times \vec{a}_{3}}{v^{*}_{c}}\\
\vec{a}^{*}_{2} &= 2\pi \frac{\vec{a}_{3}\times \vec{a}_{1}}{v^{*}_{c}}\\
\vec{a}^{*}_{3} &= 2\pi \frac{\vec{a}_{1}\times \vec{a}_{2}}{v^{*}_{c}},
\end{align}

where the volume of the unit cell in the reciprocal space, $$v^{*}_{c}$$, is calculated as $$v^{*}_{c} = \vec{a}_{1}\cdot (\vec{a}_{2}\times \vec{a}_{3}) = \vec{a}_{2}\cdot (\vec{a}_{3}\times \vec{a}_{1}) = \vec{a}_{3}\cdot (\vec{a}_{1}\times \vec{a}_{2})$$.
These vectors fulfill a condition that

\begin{align}
\label{Eq:relation_between_real_and_reciprocal_vector}
\tag{8}
\vec{a}_{i} \cdot \vec{a}_{j} = 2\pi \delta_{ij},
\end{align}

where $$\delta_{ij}$$ is the Kronecker’s delta function. This diffraction condition is derived from the feature of the first term of right hand side in Eq. \eqref{Eq:decomposition_of_scattering_amplitude}, i.e., the scattering from the lattice. That is

\begin{align}
\label{Eq:laue_function}
\tag{9}
\left| \sum_{n}\e^{i\vec{Q}\cdot\vec{R}_{n}} \right|^{2} \to Nv^{*}_c \sum_{\vec{G}}\delta (\vec{Q}-\vec{G}) \qquad \text{as} \quad N \to \infty
\end{align}

where $$\delta$$ is the Dirac’s delta function, $$N$$ is the total number of unit cell, i.e., $$N_{1}\times N_{2}\times N_{3}$$. $$N_{1}$$, $$N_{2}$$ and $$N_{3}$$ are the number of unit cell along with $$\vec{a}_{1}$$, $$\vec{a}_{2}$$ and $$\vec{a}_{3}$$, respectively. Therefore, the diffraction intensity, $$\left| F^{\mathrm{crystal}}(\vec{Q}) \right|^{2}$$, is observed only when $$\vec{Q} = \vec{G}$$ as described in Eq. \eqref{Eq:laue_condition}, since the Dirac’s delta function, $$\delta(x)$$, shows the non-zero value only at $$x=0$$.

Under the diffraction condition, the structure factor is reduced as follows:

\begin{align}
\sum_{m}f_{m}(\vec{Q})\exp \left( i\vec{Q}\cdot\vec{r}_{m} \right)
&= \sum_{m}f_{m}(\vec{G})\exp \left( i\vec{G}\cdot\vec{r}_{m} \right) \notag \\
&= \sum_{m}f_{m}(\vec{G})\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\},
\end{align}

where $$\sum_{m}$$ is a summation of all atoms in a unit cell3 Therefore, the scattering amplitude from a crystal under diffraction condition is finally derived to be

\begin{align}
\label{Eq:derived_scattering_amplitude}
\tag{10}
I(\vec{Q} = \vec{G}) \propto \left| \sum_{m}f_{m}(\vec{G})\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\} \right|^{2}.
\end{align}

The essence of the site-distinguished analysis by the DAFS method is this structure factor, where the atomic scattering factor is multiplied by the phase factor, $$\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\}$$. This factor causes the difference in the contributions from a certain element at each diffraction line, consequently enabling us to site-selectively analyze the energy dependence of the atomic scattering factor4 of the same element at the different sites.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}

## Scattering by an atom


In the description of scattering by an atom, we need to take into account the interference of x-rays radiated from different positions in the atom since an electron spreads around the atomic nucleus in quantum-mechanical picture even when the number of electrons is one.

Figure 1 shows the configuration of the scattering process by an atom, whose atomic number is $$Z$$. $$\vec{k}$$ and $$\vec{k’}$$ are the wave number vectors, which lengths are the same, i.e., $$|\vec{k}| = |\vec{k’}| =k = 2\pi/\lambda$$, $$\vec{r}$$ is a position where we evaluate the interference of x-ray, $$\rho(\vec{r})$$ is an electron density at $$\vec{r}$$, and $$\diff V$$ is a volume element at $$\vec{r}$$. The phase difference, $$\Delta \phi(\vec{r})$$ between x-rays radiated at the origin and position $$\vec{r}$$ is described to be

\begin{align}
\label{Eq:phase_difference_in_an_atom}
\tag{1}
\Delta \phi(\vec{r}) = \left( \vec{k} – \vec{k}’ \right) \cdot \vec{r} = \vec{Q} \cdot \vec{r},
\end{align}

where

\begin{align}
\label{Eq:dif_Q_vector}
\tag{2}
\vec{Q} = \vec{k} – \vec{k}’.
\end{align}
This interference occurs in the whole atom; therefore, the scattering amplitude is as follows:

\begin{align}
\label{Eq:integral_of_phases}
\tag{3}
-r_{0}\int \rho(\vec{r}) \e^{i \Delta \phi (\vec{r})} \diff V =
-r_{0}\int \rho(\vec{r}) \e^{i \vec{Q}\cdot \vec{r}} \diff V
\equiv -r_{0}f^{0}(\vec{Q}),
\end{align}

where the integration is carried out in the whole atom and $$f^{0}(\vec{Q})$$ is known as the “atomic form factor“. From the definition, it is definitely expected that $$f^{0}(\vec{Q} = 0)$$ is identical to the atomic number, i.e., the number of electrons, $$Z$$. Thus, $$f^{0}(\vec{Q})$$ value is scattering power described with a unit of the number of electrons called “electron unit (eu), which is frequently used to discuss the absorption amplitude in the DAFS method as well as the conventional diffraction technique.

When we assume the spherical electron density, i.e., $$\rho(\vec{r}) \to \rho(r)$$, the atomic form factor is described and evaluated in a simpler form because the electric density is just a function of distance $$r$$. With the absolute value5 of $$|\vec{Q}| = 4\pi \sin \theta / \lambda$$, the atomic form factor can be reduced2 into

\begin{align}
\label{Eq:integration_f_zero}
\tag{4}
f^{0}(Q) = \int_{0}^{\infty} 4\pi r^{2} \rho(r) \frac{\sin Qr}{Qr} \diff r.
\end{align}

Thus, if one knows the electron density, $$\rho(r)$$, by some theoretical calculations, the atomic form factors of each element and ions can be evaluated and used for the structure analysis by x-ray diffraction technique. As seen in Eq. \eqref{Eq:integration_f_zero}, the atomic factor is a function of only $$Q$$; therefore, the values determined from the electron density calculated by the quantum mechanical approaches such as Hartree-Fock or Fermi-Thomas-Dirac are available as a function of $$(\sin \theta/ \lambda)$$ in International tables for crystallography; vol. C [1] as in the following form:

\begin{align}
\label{Eq:model_atomic_form_factor}
\tag{5}
f^{0}\left(\frac{\sin \theta}{\lambda}\right) = \sum_{i =1}^{4 \mathrm{\ or\ } 5} a_{i} \exp \left\{ -b_{i} \left( \frac{\sin \theta}{\lambda} \right)^{2} \right\} + c,
\end{align}

where $$a_{i}$$, $$b_{i}$$ and $$c$$ values of each element and ion are given in the book.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}
###### References
[1] International Tables For Crystallography Volume C, 3rd ed., E. Prince, Ed., Wiley, 2004.
[Bibtex]
@book{Prince2004,
doi = {10.1107/97809553602060000103},
edition = {3rd},
editor = {Prince, E},
publisher = {Wiley},
title = {{International Tables For Crystallography Volume C}},
year = {2004}
}

## Scattering by one electron


The ability of an electron to scatter an x-ray is described in terms of differential scattering length defined as follows:

\begin{align}
\label{Eq:dif_scattering_length}
\tag{1}
\left( \frac{\diff \sigma}{\diff \Omega}\right) \equiv \frac{I_{\rm{SC}}}{\Phi_0 \Delta \Omega},
\end{align}

where $$\Phi_0$$ is the strength of the incident beam (the number of photons passing through a unit area per second), $$I_{\rm{SC}}$$ is the number of scattered photons recorded per second in a detector positioned at a distance $$R$$ away from the object, $$\Delta \Omega$$ is a solid angle of the detector.

The values of the right-hand side in Eq. $$\eqref{Eq:dif_scattering_length}$$is also described by the electric fields of incoming and scattered x-ray with $$\Phi_0 = c \left| \vec{E}_{\rm{in}}\right|^2 /\hbar \omega$$ and $$I_{\rm{SC}} = cR^2\Delta \Omega \left| \vec{E}_{\rm{rad}}\right|^2 /\hbar \omega$$
as follows3:

\begin{align}
\label{Eq:dif_scattering_length_electric_field}
\tag{2}
\left( \frac{\diff \sigma}{\diff \Omega}\right) = \frac{\left| \vec{E}_{\rm{rad}}\right|^2 R^2}{\left| \vec{E}_{\rm{in}}\right|^2}.
\end{align}

In a classical model of the elastic scattering of the x-ray, the scattered x-ray is generated by the electron forcedly vibrated by the electric field of the incoming x-ray 2. The radiated field is proportional to the charge of the electron $$-e$$, and to the acceleration, $$a_X(t’)$$, cause by the electric filed of the incident x-ray, which is a linearly-polarized in $$x-z$$ plane, evaluated at a time $$t’$$ earlier than the observation time $$t$$ since the speed of light is finite value of $$c$$. Thus, the electron field of the radiated x-ray is written as

\begin{align}
\tag{3}
\end{align}

where $$t’ = t – R/c$$. The full acceleration from the force on the electron is evaluated with Newton’s equation of motion as

\begin{align}
\label{Eq:acceleration}
\tag{4}
a_{X}(t’) =\frac{-e E_{0}\e^{-i\omega t’}}{m}
= \frac{-e}{m}E_{\rm{in}}\e^{i\omega (R/c)}
= \frac{-e}{m}E_{\rm{in}}\e^{ikR},
\end{align}

where $$E_{\mathrm{in}} = E_{0}\e^{-i\omega t}$$ is the electric field of the incoming x-ray. Therefore, Eq. $$\eqref{Eq:rad_electric_field}$$ can be rearranged to be

\begin{align}
\label{Eq:ratio_of_electric_field}
\tag{5}
\frac{E_{\mathrm{rad}}(R,t)}{E_{\mathrm{in}}} \propto \left( \frac{e^2}{m} \right)\frac{\e^{ikR}}{R} \sin \Psi.
\end{align}

In order to complete the derivation of the differential cross section of the electron, it is necessary to check the dimension of both members of Eq. $$\eqref{Eq:ratio_of_electric_field}$$. First, the left-hand side is definitely dimensionless. On the other hand, the dimension of $$\e^{ikR}/R$$ is the inverse of the length. Therefore, the proportionality coefficient of Eq. $$\eqref{Eq:ratio_of_electric_field}$$ must have units of length. By noting that in SI units the Coulomb energy at distance $$r$$ from a point charge $$-e$$ is $$e^2/(4\pi \epsilon_{0}r)$$ while the dimensionally the energy is also described as the form of $$mc^2$$, the proportionality coefficient $$r_{0}$$ is then written as

\begin{align}
\tag{6}
r_{0} = \left( \frac{e^2}{4\pi \epsilon_{0} m c^{2} } \right) = 2.82 \times 10^{-5} \ \A.
\end{align}

This value is known as the Thomson scattering length or classical radius of the electron. By generalizing the relationship of the electric fields of incident and radiated x-rays, the ratio of the radiated electric field to the incident electric field described in Eq. $$\eqref{Eq:ratio_of_electric_field}$$ is reduced to

\begin{align}
\label{Eq:complete_ratio_of_electric_field}
\tag{7}
\frac{E_{\mathrm{rad}}(R,t)}{E_{\mathrm{in}}} = -r_{0} \frac{\e^{ikR}}{R} \left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|,
\end{align}

where “$$-$$” indicates the radiated x-ray has a different phase from that of the incident x-ray by 180$$\deg$$ because the charge of the electron is negative, $$\hat{\vec{\epsilon}}, \hat{\vec{\epsilon}}’$$ are unit vectors for the electric field of the incident and radiated x-rays.
Therefore, the differential cross-section becomes

\begin{align}
\label{Eq:derived_cross_section}
\tag{8}
\left( \frac{\diff\sigma}{\diff\Omega} \right) = r_{0}^{2}\left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|^{2}.
\end{align}

The factor of $$\left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|^{2}$$ is called the Polarization factor and its value is dependent on the polarization of the incoming x-ray and experimental geometry:

\begin{align}
\label{Eq:polarization_factor}
\tag{9}
P = \left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|^{2} =
\begin{cases}
1 & \text{horizontal linear polarization: vertical scattering plane} \\
\cos^{2} \Psi & \text{horizontal linear polarization: horizontal scattering plane}\\
\left( 1 + \cos^{2} \Psi \right)/2 & \text{unpolarized source: x-ray tube}
\end{cases}
\end{align}

The resultant equation predicts that the scattering intensity becomes very weak if $$\Psi$$ is around 90$$\deg$$ when a detector is scanned in the horizontal plane with the light source of horizontal linear polarization, which is definitely unfavorable to the usual scattering measurement3. This is the reason why a detector is vertically scanned in a synchrotron radiation facility, whose polarization is generally linear in the horizontal plane due to the electron orbit in a storage ring.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}
###### References
[1] J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}

## Lithium-ion Battery

Cure of our addiction to oil and stabilization of the human-induced climate change are overriding issues of the energy policy in the world. Electric vehicles (EVs) are one of the promising technologies for tackling these problems thanks to their higher energy efficiency and compatibility to the green energies in comparison with a conventional gasoline car. A key technology of the EV is a rechargeable battery such as a lithium ion battery (LIB), which has been utilized in our mobile devices; however, further developments of the battery are still necessary for the dissemination of EVs.

A LIB is a kind of rechargeable (secondary) batteries, firstly released by a Japanese company, Sony, by using LiCoO2 proposed by Mizushima et al. (Prof. Goodenough’s group) [1] as a positive electrode (PE) and graphite as a negative electrode (NE) in 1990’s. Since then, LIBs has contributed to developments of mobile devices thanks to their high energy-density and power, and long cycle life in comparison with the other popularized secondary batteries such as lead-acid storage battery, NiCd battery and nickel-metal hydride battery [2, 3]. Recently, polyvalent rechargeable batteries [4], where the polyvalent cations, e.g., Mg2+, Ca2+ and Al3+ etc., are used as a “guest ion” instead of Li ions, and a dual-salt battery [5] also have attracted attention for their relatively higher energy density and safety than those of LIBs; however, the science of LIBs are still of great importance for the wide applications and as a model system of intercalation chemistry.

In LIBs, an essential electrochemical reaction is Li insertion/extraction (frequently called, lithiation/delithiation) between solid-state electrodes and liquid electrolyte in a typical cell system. As for discharge of the battery, Li atoms are extracted from a NE, e.g., conventionally graphite, to the electrolyte. On the other hand, Li ions in the electrolyte are inserted to a PE.
At the same time, the electrons are also extracted from the NE and inserted into the PE through an outer circuit, which is connected to some devices such as a bulb and a motor in EVs etc. In the PE, on the basis of conventional understanding of solid-state electrochemistry, the valence state of the transition metal, i.e., Co ion in the figure, changes by receiving Li and electron as the charge compensation, which is one of the important electrochemical reaction in the PE.

The above electrochemical reactions are summarized as follows:

\begin{align}
\mathrm{Co(IV)O_{2} + Li^{+} + e^{-} } &\to \mathrm{LiCo(III)O_{2}} \\
\mathrm{LiC_{6}} &\to \mathrm{ C_{6} + Li^{+} + e^{-} }
\end{align}

Thus, the reduction reaction proceeds on the PE, while the oxidation reaction does on the NE in the discharging process. When using a Li metal as a reference electrode in a three-electrode cell in the same electrolyte, the open circuit voltage (OCV) of the positive and negative electrodes corresponds to electrode potential (V vs. Li+/Li), which directly corresponds the chemical potential of Li atoms in each electrode by defining chemical potential of Li in the Li metal as a standard condition. Since the electromotive force is the difference of electrode potentials of the PE and the NE, a material of high redox potential is suitable for the PE, whereas that of low redox potential for the NE. Such systems are often called “Rocking chair type battery” because Li ions move between the NE and the PE through the electrolyte; the total amount of Li ions in the electrolyte is always constant during the battery operation. Thus, any combinations of the electrode materials for the PE and NE are allowed in LIBs as long as a candidate material accommodates the considerable amount of Li at a reasonable potential, which may makes the LIB science fascinating. In the current LIBs, since the capabilities of the PE in terms of the energy density, which is the product of potential and capacity, is the bottleneck of the whole battery system and required to be further developed, I would like to focus on structural chemistry of the PEs.

In the era of “Li (primary) battery”, MnO2 is one of the conventional electrode material, where Li ions are inserted into the MnO2 structure, whereas the re-extraction of the inserted Li ions is difficult. In contrast, TiS2 and MoS2 are capable of the reversible lithiaion/delithiaion; therefore, they were used as the PE in “Li (secondary, rechargeable) battery”, where the Li metal used to be used as the NE. Unfortunately, this kind of battery did not become common, because the use of Li metal in the rechargeable battery was risky due to the short circuit caused by the dendritc plating nature of the Li pure metal. Through the development period of Li batteries, the emergence of a “Li Ion Battery” from Sony started a new era of Li secondary batteries, where graphite and Li transition-metal complex oxides are used as the electrodes.

The most conventional PE material is LiCoO2 (LCO) [1]. Its crystalline structure is categorized into α-NaFeO2 type layered rock-salt structure, where Li and Co atoms occupy the octahedral sites in the face centered cubic (FCC) lattice of oxygen atoms. The name of “layered” derives from the layer-by-layer cation ordering of [111] direction in the cubic lattice, which reduces the space symmetry from cubic to hexagonal. Consequently, the structure of LiCoO2 is understood as a layered compound, where Li occupies the interlayer gallery inbetween CoO2 sheets (see the figure). LiNiO2 belongs to this family of electrode, and various related electrodes were reported, e.g., LiNi1/2Mn1/2O2 [6, 7] and LiNi1/3Mn1/3Co1/3O2[8]. In these modified materials, the available amount of Li in the layered structure was significantly improved, whereas only 0.6Li can be extracted from LCO with keeping the good electrode charge and discharge cycle life, i.e., cyclability.

LiMn2O4 (LMO) and LiFePO4 (LFP) are also important materials, which were reported from the same group of LCO [9, 10]. LMO has the normal spinel structure, where Li and Mn occupy the tetrahedral and octahedral sites in the oxygen FCC framework, respectively. The cyclability of LMO is known to be relatively low because the Jahn-Teller Mn3+ ion is unstable in the crystalline and dissolves into the electrolyte during the cycling. Thus, the partial exchange of Mn with the other transition element of Cr, Ni etc., was conducted in order to keep the valence state of Mn to IV, which effectively improved the cyclability and potential of the spinel-type electrode[11].Today it is known as high-voltage electrode materials.Both in LCO and MCO, only the half of the possible valence change of the transition metals is used for the charge compensation accompanied with lithiation/delithiation, because of low availability of the amount of Li atoms in the layered rock-salt structure and high atomic ration of Mn/Li in the spinel electrode, respectively. Thus, the structure of LiFePO4, which was also reported by Goodenough’s group [10], is understood as follows; the half of the transition metal in the composition of LiCoO2, is exchanged by phosphorous in order to enhance the structural stability and the electrode potential, which is called ordered-olivine structure and categorized to polyanion compounds. In recent years, LFP is one of the most promising electrodes thanks to high thermal stability, low cost and relatively high capacity. Furthermore, LFP is a favorable model material for the analyses of the electrochemically-driven structural phase transition
[12, 13, 14, 15, 16, 17, 18, 19] owing to its simple two-phase reaction behavior between the wide Li composition. Since the phase transition of LFP causes the considerable volume change about 5%-8%, the effects of the strain energy on the electrochemical capabilities of the electrode have been intensively studied with the phase-field micromechanical simulation [20, 21] and the in-situ simultaneous XRD and XAFS measurement [22] by our group.

As the further developed electrode materials based on polyanionic LFP, high capacity electrodes such as Li2FeSiO4 [23, 24, 25, 26, 27], Li2FePO4F [28, 29, 30], Li2MP2O7 (M = Fe, Mn) [31] and Li4NiTeO6 [32] were reported, where two Li reaction are available and would contribute to the significant improvement of the energy density of the battery. On the other hand, the electrode of the layered rock-salt structure was also developed to be Li-rich layered electrode material, in which LiMO2 ( M = Mn, Co, Ni ) is stabilized by adding electrochemically inactive Li2MnO3 and shows the capacity of > 200 mAh g-1 by the almost one Li extraction from the structure[33, 34], while the conventional LiMO2 is unstabilized by only 0.6~0.7 Li extraction. As described above, Li2MnO3 is electrochemically inactive because the valence state of Mn in this materials is Mn4+ and further oxidation by the electrochemical delithiation is generally difficult; however, substitution of Mn atom by other transition metal in forth-period element such as Ru, Mo opens a new way to utilize these Li-rich layered electrodes [35, 36, 37, 38]. Furthermore, the redox reaction of oxygen is found to be available in this family of electrodes, which was demonstrated by substituting Sn in Li2RuO3 [39], whereas the importance of the redox contribution of oxygen has been pointed out from the ab initio calculations[40, 41]. Ab initio calculations also greatly contributes to the material design for the electrodes of LIBs [42, 43, 44, 45].

The cation mixing is also very common phenomena in the above electrode materials as well as LNO, which affects diffusion and electrode potential, phase transition, electric and magnetic properties, and eventually the whole electrode properties.Furthermore, the cation mixing is driven by the repetitive cycling, which degrades the energy density of the electrode in the long-term perspective. In some cases, some complex electrode materials have plural sites for a certain transition metal, where the electronic/local structural changes accompanied by lithiation/delithiation would be different each other. Thus, the demonstration of the site-selective analyses by the DAFS method in LNO is of great importance to show how to understand the relationship between the cation-mixing and the electrochemical properties in the electrode materials for LIBs.

###### References
[1] K. Mizushima, P. C. Jones, P. J. Wiseman, and J. B. Goodenough, “A NEW CATHODE MATERIAL FOR BATTERIES OF HIGH ENERGY DENSITY,” Mater. res. bull., vol. 15, p. 783–789, 1980.
[Bibtex]
@article{Mizushima1980a,
abstract = {A new system LixCoO 2 (0
 
 [2] M. Armand and J. -M. Tarascon, "Building better batteries.," Nature, vol. 451, iss. 7179, p. 652–657, 2008. [Bibtex] @article{Armand2008, author = {Armand, M and Tarascon, J.-M.}, doi = {10.1038/451652a}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nature/Armand, Tarascon\_Nature\_2008.pdf:pdf}, issn = {1476-4687}, journal = {Nature}, keywords = {19th Century,20th Century,21st Century,Air,Automobiles,Automobiles: history,Bioelectric Energy Sources,Bioelectric Energy Sources: economics,Bioelectric Energy Sources: history,Bioelectric Energy Sources: trends,Biomass,Biomimetics,Cellular Phone,Cellular Phone: history,Conservation of Energy Resources,Conservation of Energy Resources: economics,Conservation of Energy Resources: history,Conservation of Energy Resources: methods,Conservation of Energy Resources: trends,Electrochemistry,Electrochemistry: economics,Electrochemistry: history,Electronics,Electronics: economics,Electronics: history,Electronics: trends,History,Lithium,Lithium: chemistry,Nanotechnology,Nanotechnology: trends,Oxygen,Oxygen: chemistry}, month = feb, number = {7179}, pages = {652--657}, pmid = {18256660}, title = {{Building better batteries.}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/18256660}, volume = {451}, year = {2008} } [3] J. -M. Tarascon and M. Armand, "Issues and challenges facing rechargeable lithium batteries.," Nature, vol. 414, iss. 6861, p. 359–67, 2001. [Bibtex] @article{Tarascon2001, abstract = {Technological improvements in rechargeable solid-state batteries are being driven by an ever-increasing demand for portable electronic devices. Lithium-ion batteries are the systems of choice, offering high energy density, flexible and lightweight design, and longer lifespan than comparable battery technologies. We present a brief historical review of the development of lithium-based rechargeable batteries, highlight ongoing research strategies, and discuss the challenges that remain regarding the synthesis, characterization, electrochemical performance and safety of these systems.}, author = {Tarascon, J.-M. and Armand, M}, doi = {10.1038/35104644}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nature/Tarascon, Armand\_Nature\_2001(2).pdf:pdf}, issn = {0028-0836}, journal = {Nature}, month = nov, number = {6861}, pages = {359--67}, pmid = {11713543}, title = {{Issues and challenges facing rechargeable lithium batteries.}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/11713543}, volume = {414}, year = {2001} } [4] D. Aurbach, Z. Lu, A. Schechter, Y. Gofer, H. Gizbar, R. Turgeman, Y. Cohen, M. Moshkovich, and E. Levi, "Prototype systems for rechargeable magnesium batteries.," Nature, vol. 407, iss. 6805, p. 724–727, 2000. [Bibtex] @article{Aurbach2000, abstract = {The thermodynamic properties of magnesium make it a natural choice for use as an anode material in rechargeable batteries, because it may provide a considerably higher energy density than the commonly used lead-acid and nickel-cadmium systems. Moreover, in contrast to lead and cadmium, magnesium is inexpensive, environmentally friendly and safe to handle. But the development of Mg batteries has been hindered by two problems. First, owing to the chemical activity of Mg, only solutions that neither donate nor accept protons are suitable as electrolytes; but most of these solutions allow the growth of passivating surface films, which inhibit any electrochemical reaction. Second, the choice of cathode materials has been limited by the difficulty of intercalating Mg ions in many hosts. Following previous studies of the electrochemistry of Mg electrodes in various non-aqueous solutions, and of a variety of intercalation electrodes, we have now developed rechargeable Mg battery systems that show promise for applications. The systems comprise electrolyte solutions based on Mg organohaloaluminate salts, and Mg(x)Mo3S4 cathodes, into which Mg ions can be intercalated reversibly, and with relatively fast kinetics. We expect that further improvements in the energy density will make these batteries a viable alternative to existing systems.}, author = {Aurbach, D and Lu, Z and Schechter, A and Gofer, Y and Gizbar, H and Turgeman, R and Cohen, Y and Moshkovich, M and Levi, E}, doi = {10.1038/35037553}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nature/Aurbach et al.\_Nature\_2000.pdf:pdf}, issn = {0028-0836}, journal = {Nature}, month = oct, number = {6805}, pages = {724--727}, pmid = {11048714}, publisher = {Macmillian Magazines Ltd.}, shorttitle = {Nature}, title = {{Prototype systems for rechargeable magnesium batteries.}}, url = {http://dx.doi.org/10.1038/35037553}, volume = {407}, year = {2000} } [5] S. Yagi, T. Ichitsubo, Y. Shirai, S. Yanai, T. Doi, K. Murase, and E. Matsubara, "A concept of dual-salt polyvalent-metal storage battery," J. mater. chem. a, vol. 2, iss. 4, p. 1144–1149, 2014. [Bibtex] @article{Yagi2014, abstract = {In this work, we propose and examine a battery system with a new design concept. The battery consists of a non-noble polyvalent metal (such as Ca, Mg, Al) combined with a positive electrode already well-established for lithium ion batteries (LIBs). The prototype demonstrated here is composed of a Mg negative electrode, LiFePO4 positive electrode, and tetrahydrofuran solution of two kinds of salts (LiBF4 and phenylmagnesium chloride) as an electrolyte. The LIB positive-electrode materials such as LiFePO4 can preferentially accommodate Li+ ions; i.e., they work as a “Li pass filter”. This characteristic enables us to construct a septum-free, Daniel-battery type dual-salt polyvalent-metal storage battery (PSB). The presented dual-salt PSB combines many advantages, e.g., fast diffusion of Li+ ions in the positive electrode, high cyclability, and a high specific capacity of lightweight polyvalent metals. The concept is expected to allow the design of many combinations of dual-salt PSBs having a high energy density and high rate capability.}, author = {Yagi, Shunsuke and Ichitsubo, Tetsu and Shirai, Yoshimasa and Yanai, Shingo and Doi, Takayuki and Murase, Kuniaki and Matsubara, Eiichiro}, doi = {10.1039/c3ta13668j}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Journal of Materials Chemistry A/Yagi et al.\_Journal of Materials Chemistry A\_2014.pdf:pdf}, isbn = {10.1039/C3TA13668J}, issn = {2050-7488}, journal = {J. Mater. Chem. A}, language = {en}, month = dec, number = {4}, pages = {1144--1149}, publisher = {The Royal Society of Chemistry}, title = {{A concept of dual-salt polyvalent-metal storage battery}}, url = {http://pubs.rsc.org/en/content/articlehtml/2014/ta/c3ta13668j}, volume = {2}, year = {2014} } [6] K. Kang, C. Chen, B. J. Hwang, and G. Ceder, "Synthesis, Electrochemical Properties, and Phase Stability of Li 2 NiO 2 with the Immm Structure," Chem. mater., vol. 16, iss. 13, p. 2685–2690, 2004. [Bibtex] @article{Kang2004, abstract = {The electrochemical properties and phase stability of the orthorhombic Immm structure of composition Li2NiO2 are studied experimentally and with first principles calculations. The material shows a high specific charge capacity of about 320 mAh/g and discharge capacity of about 240 mAh/g at the first cycle. The experimental results and first principles calculations all indicate that the orthorhombic Immm structure is rather prone to phase transformation to a close-packed layered structure during the electrochemical cycling. The possibility of stabilizing the orthorhombic Immm structure during the electrochemical cycling by partial substitution of Ni is also evaluated. A detailed analysis of the crystal field energy difference between octahedral and square-planar coordinated Ni2+ indicates that crystal field effects may not be large enough to stabilize Ni2+ in a square planar environment when the cost of electron pairing is taken into account. Rather, we attribute the stability of Li2NiO2 in the Immm structure to the more favorable Li arrangement as compared to a possible Li2NiO2 structure with octahedral Ni. The electrochemical properties and phase stability of the orthorhombic Immm structure of composition Li2NiO2 are studied experimentally and with first principles calculations. The material shows a high specific charge capacity of about 320 mAh/g and discharge capacity of about 240 mAh/g at the first cycle. The experimental results and first principles calculations all indicate that the orthorhombic Immm structure is rather prone to phase transformation to a close-packed layered structure during the electrochemical cycling. The possibility of stabilizing the orthorhombic Immm structure during the electrochemical cycling by partial substitution of Ni is also evaluated. A detailed analysis of the crystal field energy difference between octahedral and square-planar coordinated Ni2+ indicates that crystal field effects may not be large enough to stabilize Ni2+ in a square planar environment when the cost of electron pairing is taken into account. Rather, we attribute the stability of Li2NiO2 in the Immm structure to the more favorable Li arrangement as compared to a possible Li2NiO2 structure with octahedral Ni.}, author = {Kang, Kisuk and Chen, Ching-Hsiang and Hwang, Bing Joe and Ceder, Gerbrand}, doi = {10.1021/cm049922h}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Chemistry of Materials/Kang et al.\_Chemistry of Materials\_2004.pdf:pdf}, issn = {0897-4756}, journal = {Chem. Mater.}, month = jun, number = {13}, pages = {2685--2690}, publisher = {American Chemical Society}, title = {{Synthesis, Electrochemical Properties, and Phase Stability of Li 2 NiO 2 with the Immm Structure}}, url = {http://dx.doi.org/10.1021/cm049922h}, volume = {16}, year = {2004} } [7] Z. Lu, Z. Chen, and J. R. Dahn, "Lack of Cation Clustering in Li[Ni x Li 1/3 - 2 x /3 Mn 2/3 - x / 3 ]O 2 (0 < x ≤ 1 / 2 ) and Li[Cr x Li (1 - x )/3 Mn (2 - 2 x )/3 ]O 2 (0 < x < 1)," Chem. mater., vol. 15, iss. 16, p. 3214–3220, 2003. [Bibtex] @article{Lu2003, abstract = {Recent papers by Ammundsen et al. and Pan et al. give evidence for the formation of local regions high in Mn content and other local regions high in Cr or Ni content in Li[Li0.2Cr0.4Mn0.4]O2 and Li[Ni0.5Mn0.5]O2 by EXAFS and NMR methods, respectively. These observations are surprising for the following reasons:? (1) each of these materials is a part of a solid solution series, Li[CrxLi(1-x)/3Mn(2-2x)/3]O2 (0 < x < 1) or Li[NixLi1/3-2x/3Mn2/3-x/3]O2 (0 < x < 1/2); (2) the materials are made at high temperature, and entropy considerations suggest that like transition-metal atoms should not cluster; and (3) the electrochemical and structural properties of the materials vary smoothly with composition. Here, using careful X-ray diffraction on many samples from each solid solution, we show that it is very unlikely that such local regions high in Mn, Ni, or Cr exist. We show that long-ranged lithium ordering on the 31/2 a ? 31/2 a superstructure occurs as expected based on the work of Schick et al., however, this does not imply local regions of Li2MnO3. Instead, the diffraction angles of the superstructure peaks shift with composition suggesting that the Mn, Cr, or Ni are uniformly mixed on the transition-metal sites. In addition, we show that the electrochemical behavior of Li[NixLi1/3-2x/3Mn2/3-x/3]O2 heated to 1000 °C is improved compared to that of samples made at 900 °C. Recent papers by Ammundsen et al. and Pan et al. give evidence for the formation of local regions high in Mn content and other local regions high in Cr or Ni content in Li[Li0.2Cr0.4Mn0.4]O2 and Li[Ni0.5Mn0.5]O2 by EXAFS and NMR methods, respectively. These observations are surprising for the following reasons:? (1) each of these materials is a part of a solid solution series, Li[CrxLi(1-x)/3Mn(2-2x)/3]O2 (0 < x < 1) or Li[NixLi1/3-2x/3Mn2/3-x/3]O2 (0 < x < 1/2); (2) the materials are made at high temperature, and entropy considerations suggest that like transition-metal atoms should not cluster; and (3) the electrochemical and structural properties of the materials vary smoothly with composition. Here, using careful X-ray diffraction on many samples from each solid solution, we show that it is very unlikely that such local regions high in Mn, Ni, or Cr exist. We show that long-ranged lithium ordering on the 31/2 a ? 31/2 a superstructure occurs as expected based on the work of Schick et al., however, this does not imply local regions of Li2MnO3. Instead, the diffraction angles of the superstructure peaks shift with composition suggesting that the Mn, Cr, or Ni are uniformly mixed on the transition-metal sites. In addition, we show that the electrochemical behavior of Li[NixLi1/3-2x/3Mn2/3-x/3]O2 heated to 1000 °C is improved compared to that of samples made at 900 °C.}, author = {Lu, Zhonghua and Chen, Zhaohui and Dahn, J. R.}, doi = {10.1021/cm030194s}, issn = {0897-4756}, journal = {Chem. Mater.}, month = aug, number = {16}, pages = {3214--3220}, publisher = {American Chemical Society}, title = {{Lack of Cation Clustering in Li[Ni x Li 1/3 - 2 x /3 Mn 2/3 - x / 3 ]O 2 (0 < x ≤ 1 / 2 ) and Li[Cr x Li (1 - x )/3 Mn (2 - 2 x )/3 ]O 2 (0 < x < 1)}}, url = {http://dx.doi.org/10.1021/cm030194s}, volume = {15}, year = {2003} } [8] T. Ohzuku and Y. Makimura, "Layered Lithium Insertion Material of LiCo1/3Ni1/3Mn1/3O2 for Lithium-Ion Batteries," Chem. lett., vol. 30, iss. 7, p. 642–643, 2001. [Bibtex] @article{Ohzuku2001, author = {Ohzuku, T. and Makimura, Yoshinari}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Chem. Lett/Ohzuku, Makimura\_Chem. Lett.\_2001.pdf:pdf}, journal = {Chem. Lett.}, keywords = {LCNMO}, number = {7}, pages = {642--643}, publisher = {J-STAGE}, title = {{Layered Lithium Insertion Material of LiCo1/3Ni1/3Mn1/3O2 for Lithium-Ion Batteries}}, url = {http://joi.jlc.jst.go.jp/JST.JSTAGE/cl/2001.642?from=Google}, volume = {30}, year = {2001} } [9] M. M. Thackeray, P. J. Johnson, L. A. de Picciotto, P. G. Bruce, and J. B. Goodenough, "Electrochemical extraction of lithium from LiMn2O4," Mater. res. bull., vol. 19, iss. 2, p. 179–187, 1984. [Bibtex] @article{Thackeray1984, abstract = {Lithium has been removed electrochemically at 15 $\mu$A/cm2 from LiMn2O4 (spinel) to yield single phase Li1−xMn2O4 for 0 < × ⩽ 0.60. The electrochemical curve suggests that beyond x = 0.60 an electrochemical process other than lithium extraction occurs. Powder X-ray-diffraction spectra indicate that during the extraction process the [Mn2]O4 framework of the spinel structure remains intact. Previous results have shown that 1.2 Li+ ions can also be inserted into LiMn2O4, which suggests that lithium may be cycled in and out of the [Mn2]O4 framework of the spinel structure over a wide range of x, at least from Li0.4Mn2O4 to Li2Mn2O4. Discussion of the mechanism of formation of $\lambda$-MnO2 in an acidic environment is extended.}, author = {Thackeray, M.M. and Johnson, P.J. and de Picciotto, L.A. and Bruce, P.G. and Goodenough, J.B.}, doi = {10.1016/0025-5408(84)90088-6}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Materials Research Bulletin/Thackeray et al.\_Materials Research Bulletin\_1984.pdf:pdf}, issn = {00255408}, journal = {Mater. Res. Bull.}, month = feb, number = {2}, pages = {179--187}, title = {{Electrochemical extraction of lithium from LiMn2O4}}, url = {http://www.sciencedirect.com/science/article/pii/0025540884900886}, volume = {19}, year = {1984} } [10] A. K. Padhi, K. S. Nanjundaswamy, and J. B. Goodenough, "Phospho‐olivines as Positive‐Electrode Materials for Rechargeable Lithium Batteries," J. electrochem. soc., vol. 144, iss. 4, p. 1–7, 1997. [Bibtex] @article{Padhi1997, author = {Padhi, A.K and Nanjundaswamy, K.S and Goodenough, J.B}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/J. Electrochem. Soc/Padhi, Nanjundaswamy, Goodenough\_J. Electrochem. Soc.\_1997.pdf:pdf}, journal = {J. Electrochem. Soc.}, number = {4}, pages = {1--7}, title = {{Phospho‐olivines as Positive‐Electrode Materials for Rechargeable Lithium Batteries}}, url = {http://link.aip.org/link/?JESOAN/144/1188/1}, volume = {144}, year = {1997} } [11] C. Sigala, D. Guyomard, A. Verbare, Y. Piffard, and M. Tournoux, "Positive electrode materials with high operating voltage for lithium batteries: LiCryMn2 − yO4 (0 ≤ y ≤ 1)," Solid state ionics, vol. 81, iss. 3-4, p. 167–170, 1995. [Bibtex] @article{SIGALA1995, abstract = {Reversible lithium deintercalation of chromium-substituted spinel manganese oxides LiCryMn2 − yO4 (0 ≤ y ≤ 1) in the voltage range 3.4–5.4 V versus Li, occurs in two main steps for 0 < y < 1: one at about 4.9 V and the other at about 4 V. The 4.9 V process capacity increases with the chromium content while the 4 V process capacity decreases at the same time. Excellent cyclability was observed for y ≤ 0.5 while materials with y ≥ 0.75 were loosing capacity rapidly upon cycling. Changing the chromium composition of these materials enables the control of the average intercalation voltage in the range 4.05–4.5 V versus Li, a voltage range where no material was known before. A low manganese to chromium substitution rate in LiMn2O4 was found to be beneficial to the specific capacity and energy and to the cyclability of the spinel materials. Due to the selected electrolyte composition with high stability against oxidation, extra capacity due to electrolyte oxidation at each cycle remained very low even though the charge voltage was highly oxidative.}, author = {Sigala, C and Guyomard, D and Verbare, A and Piffard, Y and Tournoux, M}, doi = {10.1016/0167-2738(95)00163-Z}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Solid State Ionics/Sigala et al.\_Solid State Ionics\_1995.pdf:pdf}, issn = {01672738}, journal = {Solid State Ionics}, keywords = {Cathode material,High voltage material,Li battery,Li intercalation,Lithium chromium manganese oxide,Spinel oxides}, month = nov, number = {3-4}, pages = {167--170}, title = {{Positive electrode materials with high operating voltage for lithium batteries: LiCryMn2 − yO4 (0 ≤ y ≤ 1)}}, url = {http://www.sciencedirect.com/science/article/pii/016727389500163Z}, volume = {81}, year = {1995} } [12] H. Liu, F. C. Strobridge, O. J. Borkiewicz, K. M. Wiaderek, K. W. Chapman, P. J. Chupas, and C. P. Grey, "Batteries. Capturing metastable structures during high-rate cycling of LiFePO₄ nanoparticle electrodes.," Science, vol. 344, iss. 6191, p. 1252817, 2014. [Bibtex] @article{Liu2014, abstract = {The absence of a phase transformation involving substantial structural rearrangements and large volume changes is generally considered to be a key characteristic underpinning the high-rate capability of any battery electrode material. In apparent contradiction, nanoparticulate LiFePO4, a commercially important cathode material, displays exceptionally high rates, whereas its lithium-composition phase diagram indicates that it should react via a kinetically limited, two-phase nucleation and growth process. Knowledge concerning the equilibrium phases is therefore insufficient, and direct investigation of the dynamic process is required. Using time-resolved in situ x-ray powder diffraction, we reveal the existence of a continuous metastable solid solution phase during rapid lithium extraction and insertion. This nonequilibrium facile phase transformation route provides a mechanism for realizing high-rate capability of electrode materials that operate via two-phase reactions.}, author = {Liu, Hao and Strobridge, Fiona C and Borkiewicz, Olaf J and Wiaderek, Kamila M and Chapman, Karena W and Chupas, Peter J and Grey, Clare P}, doi = {10.1126/science.1252817}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Science (New York, N.Y.)/Liu et al.\_Science (New York, N.Y.)\_2014.pdf:pdf}, issn = {1095-9203}, journal = {Science}, month = jun, number = {6191}, pages = {1252817}, pmid = {24970091}, title = {{Batteries. Capturing metastable structures during high-rate cycling of LiFePO₄ nanoparticle electrodes.}}, url = {http://www.sciencemag.org/content/344/6191/1252817}, volume = {344}, year = {2014} } [13] R. Malik, F. Zhou, and G. Ceder, "Kinetics of non-equilibrium lithium incorporation in LiFePO4," Nat. mater., vol. 10, iss. 8, p. 587–590, 2011. [Bibtex] @article{Malik2011, author = {Malik, Rahul and Zhou, Fei and Ceder, G.}, doi = {10.1038/nmat3065}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nat. Mater/Malik, Zhou, Ceder\_Nat. Mater.\_2011.pdf:pdf}, issn = {1476-1122}, journal = {Nat. Mater.}, month = jul, number = {8}, pages = {587--590}, title = {{Kinetics of non-equilibrium lithium incorporation in LiFePO4}}, url = {http://www.nature.com/doifinder/10.1038/nmat3065}, volume = {10}, year = {2011} } [14] C. Delmas, M. Maccario, L. Croguennec, F. {Le Cras}, and F. Weill, "Lithium deintercalation in LiFePO4 nanoparticles via a domino-cascade model.," Nat. mater., vol. 7, p. 665–671, 2008. [Bibtex] @article{Delmas2008a, abstract = {Lithium iron phosphate is one of the most promising positive-electrode materials for the next generation of lithium-ion batteries that will be used in electric and plug-in hybrid vehicles. Lithium deintercalation (intercalation) proceeds through a two-phase reaction between compositions very close to LiFePO(4) and FePO(4). As both endmember phases are very poor ionic and electronic conductors, it is difficult to understand the intercalation mechanism at the microscopic scale. Here, we report a characterization of electrochemically deintercalated nanomaterials by X-ray diffraction and electron microscopy that shows the coexistence of fully intercalated and fully deintercalated individual particles. This result indicates that the growth reaction is considerably faster than its nucleation. The reaction mechanism is described by a 'domino-cascade model' and is explained by the existence of structural constraints occurring just at the reaction interface: the minimization of the elastic energy enhances the deintercalation (intercalation) process that occurs as a wave moving through the entire crystal. This model opens new perspectives in the search for new electrode materials even with poor ionic and electronic conductivities.}, author = {Delmas, C and Maccario, M and Croguennec, L and {Le Cras}, F and Weill, F}, doi = {10.1038/nmat2230}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nat. Mater/Delmas et al.\_Nat. Mater.\_2008.pdf:pdf}, issn = {1476-1122}, journal = {Nat. Mater.}, month = aug, pages = {665--671}, pmid = {18641656}, title = {{Lithium deintercalation in LiFePO4 nanoparticles via a domino-cascade model.}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/18641656}, volume = {7}, year = {2008} } [15] C. Delacourt, P. Poizot, J. Tarascon, and C. Masquelier, "The existence of a temperature-driven solid solution in LixFePO4 for 0 ≤ x ≤ 1," Nat. mater., vol. 4, iss. 3, p. 254–260, 2005. [Bibtex] @article{Delacourt2005, author = {Delacourt, Charles and Poizot, Philippe and Tarascon, Jean-Marie and Masquelier, Christian}, doi = {10.1038/nmat1335}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nature Materials/Delacourt et al.\_Nature Materials\_2005.pdf:pdf}, issn = {1476-1122}, journal = {Nat. Mater.}, month = feb, number = {3}, pages = {254--260}, title = {{The existence of a temperature-driven solid solution in LixFePO4 for 0 ≤ x ≤ 1}}, url = {http://www.nature.com/doifinder/10.1038/nmat1335}, volume = {4}, year = {2005} } [16] B. Kang and G. Ceder, "Battery materials for ultrafast charging and discharging," Nature, vol. 458, iss. 7235, p. 190–193, 2009. [Bibtex] @article{Kang2009, author = {Kang, Byoungwoo and Ceder, Gerbrand}, doi = {10.1038/nature07853}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nature/Kang, Ceder\_Nature\_2009.pdf:pdf}, issn = {0028-0836}, journal = {Nature}, number = {7235}, pages = {190--193}, publisher = {Macmillan Magazines Ltd, Brunel Rd, Houndsmills, Basingstoke, Hants, RG 21 2 XS, UK}, title = {{Battery materials for ultrafast charging and discharging}}, url = {http://dx.doi.org/10.1038/nature07853 http://burgaz.mit.edu/PUBLICATIONS/nature07853.pdf}, volume = {458}, year = {2009} } [17] W. Dreyer, J. Jamnik, C. Guhlke, R. Huth, J. Moškon, and M. Gaberšček, "The thermodynamic origin of hysteresis in insertion batteries," Nat. mater., vol. 9, iss. 5, p. 448–453, 2010. [Bibtex] @article{Dreyer2010, abstract = {Lithium batteries are considered the key storage devices for most emerging green technologies such as wind and solar technologies or hybrid and plug-in electric vehicles. Despite the tremendous recent advances in battery research, surprisingly, several fundamental issues of increasing practical importance have not been adequately tackled. One such issue concerns the energy efficiency. Generally, charging of 10(10)-10(17) electrode particles constituting a modern battery electrode proceeds at (much) higher voltages than discharging. Most importantly, the hysteresis between the charge and discharge voltage seems not to disappear as the charging/discharging current vanishes. Herein we present, for the first time, a general explanation of the occurrence of inherent hysteretic behaviour in insertion storage systems containing multiple particles. In a broader sense, the model also predicts the existence of apparent equilibria in battery electrodes, the sequential particle-by-particle charging/discharging mechanism and the disappearance of two-phase behaviour at special experimental conditions.}, annote = {From Duplicate 1 ( The thermodynamic origin of hysteresis in insertion batteries. - Dreyer, Wolfgang; Jamnik, Janko; Guhlke, Clemens; Huth, Robert; Mo\v{s}kon, Jo\v{z}e; Gaber\v{s}\v{c}ek, Miran )}, author = {Dreyer, Wolfgang and Jamnik, Janko and Guhlke, Clemens and Huth, Robert and Mo\v{s}kon, Jo\v{z}e and Gaber\v{s}\v{c}ek, Miran}, doi = {10.1038/nmat2730}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nat. Mater/Dreyer et al.\_Nat. Mater.\_2010.pdf:pdf}, issn = {1476-1122}, journal = {Nat. Mater.}, month = apr, number = {5}, pages = {448--453}, pmid = {20383130}, publisher = {Nature Publishing Group}, title = {{The thermodynamic origin of hysteresis in insertion batteries}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/20383130 http://www.nature.com/nmat/journal/vaop/ncurrent/full/nmat2730.html}, volume = {9}, year = {2010} } [18] Y. Asari, Y. Suwa, and T. Hamada, "Formation and diffusion of vacancy-polaron complex in olivine-type LiMnPO_\4\ and LiFePO_\4\," Phys. rev. b, vol. 84, iss. 13, p. 134113, 2011. [Bibtex] @article{Asari2011, author = {Asari, Yusuke and Suwa, Yuji and Hamada, Tomoyuki}, doi = {10.1103/PhysRevB.84.134113}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Phys. Rev. B/Asari, Suwa, Hamada\_Phys. Rev. B\_2011.pdf:pdf}, issn = {1098-0121}, journal = {Phys. Rev. B}, month = oct, number = {13}, pages = {134113}, title = {{Formation and diffusion of vacancy-polaron complex in olivine-type LiMnPO\_\{4\} and LiFePO\_\{4\}}}, url = {http://link.aps.org/doi/10.1103/PhysRevB.84.134113}, volume = {84}, year = {2011} } [19] Y. Orikasa, T. Maeda, Y. Koyama, H. Murayama, K. Fukuda, H. Tanida, H. Arai, E. Matsubara, Y. Uchimoto, and Z. Ogumi, "Direct observation of a metastable crystal phase of Li(x)FePO4 under electrochemical phase transition.," J. am. chem. soc., vol. 135, iss. 15, p. 5497–500, 2013. [Bibtex] @article{Orikasa2013a, abstract = {The phase transition between LiFePO4 and FePO4 during nonequilibrium battery operation was tracked in real time using time-resolved X-ray diffraction. In conjunction with increasing current density, a metastable crystal phase appears in addition to the thermodynamically stable LiFePO4 and FePO4 phases. The metastable phase gradually diminishes under open-circuit conditions following electrochemical cycling. We propose a phase transition path that passes through the metastable phase and posit the new phase's role in decreasing the nucleation energy, accounting for the excellent rate capability of LiFePO4. This study is the first to report the measurement of a metastable crystal phase during the electrochemical phase transition of LixFePO4.}, author = {Orikasa, Yuki and Maeda, Takehiro and Koyama, Yukinori and Murayama, Haruno and Fukuda, Katsutoshi and Tanida, Hajime and Arai, Hajime and Matsubara, Eiichiro and Uchimoto, Yoshiharu and Ogumi, Zempachi}, doi = {10.1021/ja312527x}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/J. Am. Chem. Soc/Orikasa et al.\_J. Am. Chem. Soc.\_2013.pdf:pdf}, issn = {1520-5126}, journal = {J. Am. Chem. Soc.}, month = apr, number = {15}, pages = {5497--500}, pmid = {23544671}, publisher = {American Chemical Society}, title = {{Direct observation of a metastable crystal phase of Li(x)FePO4 under electrochemical phase transition.}}, url = {http://dx.doi.org/10.1021/ja312527x}, volume = {135}, year = {2013} } [20] T. Ichitsubo, K. Tokuda, S. Yagi, M. Kawamori, T. Kawaguchi, T. Doi, M. Oishi, and E. Matsubara, "Elastically constrained phase-separation dynamics competing with charge process in LiFePO4/FePO4 system," J. mater. chem. a, vol. 1, p. 2567–2577, 2013. [Bibtex] @article{Ichitsubo2013, author = {Ichitsubo, Tetsu and Tokuda, Kazuya and Yagi, Shunsuke and Kawamori, Makoto and Kawaguchi, Tomoya and Doi, Takayuki and Oishi, Masatsugu and Matsubara, Eiichiro}, doi = {10.1039/c2ta01102f}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Journal of Materials Chemistry A/Ichitsubo et al.\_Journal of Materials Chemistry A\_2013(3).pdf:pdf;:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Journal of Materials Chemistry A/Ichitsubo et al.\_Journal of Materials Chemistry A\_2013.pdf:pdf}, issn = {2050-7488}, journal = {J. Mater. Chem. A}, language = {en}, pages = {2567--2577}, publisher = {The Royal Society of Chemistry}, title = {{Elastically constrained phase-separation dynamics competing with charge process in LiFePO4/FePO4 system}}, url = {http://pubs.rsc.org/en/content/articlehtml/2013/ta/c2ta01102f}, volume = {1}, year = {2013} } [21] T. Ichitsubo, T. Doi, K. Tokuda, E. Matsubara, T. Kida, T. Kawaguchi, S. Yagi, S. Okada, and J. Yamaki, "What determines the critical size for phase separation in LiFePO4 in lithium ion batteries?," J. mater. chem. a, vol. 1, iss. 46, p. 14532–14537, 2013. [Bibtex] @article{Ichitsubo2013c, author = {Ichitsubo, Tetsu and Doi, Takayuki and Tokuda, Kazuya and Matsubara, Eiichiro and Kida, Tetsuya and Kawaguchi, Tomoya and Yagi, Shunsuke and Okada, Shigeto and Yamaki, Jun-ichi}, doi = {10.1039/c3ta13122j}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Journal of Materials Chemistry A/Ichitsubo et al.\_Journal of Materials Chemistry A\_2013.pdf:pdf}, issn = {2050-7488}, journal = {J. Mater. Chem. A}, language = {en}, month = nov, number = {46}, pages = {14532--14537}, publisher = {Royal Society of Chemistry}, title = {{What determines the critical size for phase separation in LiFePO4 in lithium ion batteries?}}, url = {http://pubs.rsc.org/en/Content/ArticleHTML/2013/TA/C3TA13122J}, volume = {1}, year = {2013} } [22] K. Tokuda, T. Kawaguchi, K. Fukuda, T. Ichitsubo, and E. Matsubara, "Retardation and acceleration of phase separation evaluated from observation of imbalance between structure and valence in LiFePO4/FePO4 electrode," Apl mater., vol. 2, iss. 7, p. 70701, 2014. [Bibtex] @article{Tokuda2014, author = {Tokuda, Kazuya and Kawaguchi, Tomoya and Fukuda, Katsutoshi and Ichitsubo, Tetsu and Matsubara, Eiichiro}, doi = {10.1063/1.4886555}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/APL Materials/Tokuda et al.\_APL Materials\_2014.pdf:pdf}, issn = {2166-532X}, journal = {APL Mater.}, month = jul, number = {7}, pages = {070701}, title = {{Retardation and acceleration of phase separation evaluated from observation of imbalance between structure and valence in LiFePO4/FePO4 electrode}}, url = {http://scitation.aip.org/content/aip/journal/aplmater/2/7/10.1063/1.4886555}, volume = {2}, year = {2014} } [23] S. Nishimura, S. Hayase, R. Kanno, M. Yashima, N. Nakayama, and A. Yamada, "Structure of Li2FeSiO4.," J. am. chem. soc., vol. 130, iss. 40, p. 13212–13213, 2008. [Bibtex] @article{Nishimura2008a, abstract = {A large-scale lithium-ion battery is the key technology toward a greener society. A lithium iron silicate system is rapidly attracting much attention as the new important developmental platform of cathode material with abundant elements and possible multielectron reactions. The hitherto unsolved crystal structure of the typical composition Li2FeSiO4 has now been determined using high-resolution synchrotron X-ray diffraction and electron diffraction experiments. The structure has a 2 times larger superlattice compared to the previous beta-Li3PO4-based model, and its origin is the periodic modulation of coordination tetrahedra.}, author = {Nishimura, Shinchi and Hayase, Shogo and Kanno, Ryoji and Yashima, Masatomo and Nakayama, Noriaki and Yamada, Atsuo}, doi = {10.1021/ja805543p}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/J. Am. Chem. Soc/Nishimura et al.\_J. Am. Chem. Soc.\_2008.pdf:pdf}, issn = {1520-5126}, journal = {J. Am. Chem. Soc.}, month = oct, number = {40}, pages = {13212--13213}, pmid = {18788804}, publisher = {American Chemical Society}, title = {{Structure of Li2FeSiO4.}}, url = {http://dx.doi.org/10.1021/ja805543p}, volume = {130}, year = {2008} } [24] D. Rangappa, K. D. Murukanahally, T. Tomai, A. Unemoto, and I. Honma, "Ultrathin nanosheets of Li2MSiO4 (M = Fe, Mn) as high-capacity Li-ion battery electrode.," Nano lett., vol. 12, iss. 3, p. 1146–1151, 2012. [Bibtex] @article{Rangappa2012, abstract = {Novel ultrathin Li(2)MnSiO(4) nanosheets have been prepared in a rapid one pot supercritical fluid synthesis method. Nanosheets structured cathode material exhibits a discharge capacity of \~{}340 mAh/g at 45 ± 5 °C. This result shows two lithium extraction/insertion performances with good cycle ability without any structural instability up to 20 cycles. The two-dimensional nanosheets structure enables us to overcome structural instability problem in the lithium metal silicate based cathode materials and allows successful insertion/extraction of two complete lithium ions.}, author = {Rangappa, Dinesh and Murukanahally, Kempaiah Devaraju and Tomai, Takaaki and Unemoto, Atsushi and Honma, Itaru}, doi = {10.1021/nl202681b}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nano letters/Rangappa et al.\_Nano letters\_2012.pdf:pdf}, issn = {1530-6992}, journal = {Nano Lett.}, keywords = {Artificial,Electric Power Supplies,Electrodes,Energy Transfer,Equipment Design,Equipment Failure Analysis,Ions,Lithium,Lithium Compounds,Lithium Compounds: chemistry,Lithium: chemistry,Manganese,Manganese: chemistry,Membranes,Nanostructures,Nanostructures: chemistry,Nanostructures: ultrastructure,Oxides,Oxides: chemistry,Particle Size,Sulfates,Sulfates: chemistry}, month = mar, number = {3}, pages = {1146--1151}, pmid = {22332722}, title = {{Ultrathin nanosheets of Li2MSiO4 (M = Fe, Mn) as high-capacity Li-ion battery electrode.}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/22332722}, volume = {12}, year = {2012} } [25] R. Dominko, M. Bele, M. Gaberšček, a. Meden, M. Remškar, and J. Jamnik, "Structure and electrochemical performance of Li2MnSiO4 and Li2FeSiO4 as potential Li-battery cathode materials," Electrochem. commun., vol. 8, iss. 2, p. 217–222, 2006. [Bibtex] @article{Dominko2006, author = {Dominko, R. and Bele, M. and Gaber\v{s}\v{c}ek, M. and Meden, a. and Rem\v{s}kar, M. and Jamnik, J.}, doi = {10.1016/j.elecom.2005.11.010}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Electrochem. Commun/Dominko et al.\_Electrochem. Commun.\_2006.pdf:pdf}, issn = {13882481}, journal = {Electrochem. Commun.}, keywords = {Li2FeSiO4,cathode material,crystal structure,iron silicate,lithium-ion battery,manganese silicate}, mendeley-tags = {Li2FeSiO4}, month = feb, number = {2}, pages = {217--222}, title = {{Structure and electrochemical performance of Li2MnSiO4 and Li2FeSiO4 as potential Li-battery cathode materials}}, url = {http://linkinghub.elsevier.com/retrieve/pii/S1388248105003607}, volume = {8}, year = {2006} } [26] T. Masese, Y. Orikasa, T. Mori, K. Yamamoto, T. Ina, T. Minato, K. Nakanishi, T. Ohta, C. Tassel, Y. Kobayashi, H. Kageyama, H. Arai, Z. Ogumi, and Y. Uchimoto, "Local structural change in Li2FeSiO4 polyanion cathode material during initial cycling," Solid state ionics, vol. 262, p. 110–114, 2014. [Bibtex] @article{Masese2014a, abstract = {To elucidate the Li+ extraction and insertion mechanism for Li2FeSiO4 nanoparticles, at the atomic scale, X-ray absorption spectroscopy (XAS) measurements at Fe and Si K-edges were performed. Fe K-edge XAS spectra suggest irreversible changes occurring in the local and electronic environment of iron which can be attributable to the characteristic shift in potential plateau during initial cycling of Li2−xFeSiO4 system. While the local environment around Fe atoms significantly changes upon initial cycling, the local SiO environment is mostly maintained.}, author = {Masese, Titus and Orikasa, Yuki and Mori, Takuya and Yamamoto, Kentaro and Ina, Toshiaki and Minato, Taketoshi and Nakanishi, Koji and Ohta, Toshiaki and Tassel, C\'{e}dric and Kobayashi, Yoji and Kageyama, Hiroshi and Arai, Hajime and Ogumi, Zempachi and Uchimoto, Yoshiharu}, doi = {10.1016/j.ssi.2013.11.018}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Solid State Ionics/Masese et al.\_Solid State Ionics\_2014.pdf:pdf}, issn = {01672738}, journal = {Solid State Ionics}, keywords = {Li2FeSiO4,Lithium-ion battery,X-ray absorption spectroscopy (XAS)}, month = sep, pages = {110--114}, title = {{Local structural change in Li2FeSiO4 polyanion cathode material during initial cycling}}, url = {http://www.sciencedirect.com/science/article/pii/S0167273813005973}, volume = {262}, year = {2014} } [27] T. Masese, Y. Orikasa, C. Tassel, J. Kim, T. Minato, H. Arai, T. Mori, K. Yamamoto, Y. Kobayashi, H. Kageyama, Z. Ogumi, and Y. Uchimoto, "Relationship between Phase Transition Involving Cationic Exchange and Charge–Discharge Rate in Li 2 FeSiO 4," Chem. mater., vol. 26, iss. 3, p. 1380–1384, 2014. [Bibtex] @article{Masese2014, abstract = {Li2FeSiO4 is considered a promising cathode material for the next-generation Li-ion battery systems owing to its high theoretical capacity and low cost. Li2FeSiO4 exhibits complex polymorphism and undergoes significant phase transformations during charge and discharge reaction. To elucidate the phase transformation mechanism, crystal structural changes during charge and discharge processes of Li2FeSiO4 at different rates were investigated by X-ray diffraction measurements. The C/50 rate of lithium extraction upon initial cycling leads to a complete transformation from a monoclinic Li2FeSiO4 to a thermodynamically stable orthorhombic LiFeSiO4, concomitant with the occurrence of significant Li/Fe antisite mixing. The C/10 rate of lithium extraction and insertion, however, leads to retention of the parent Li2FeSiO4 (with the monoclinic structure as a metastable phase) with little cationic mixing. Here, we experimentally show the presence of metastable and stable LiFeSiO4 polymorphic phases caused by lithium extraction and insertion. Li2FeSiO4 is considered a promising cathode material for the next-generation Li-ion battery systems owing to its high theoretical capacity and low cost. Li2FeSiO4 exhibits complex polymorphism and undergoes significant phase transformations during charge and discharge reaction. To elucidate the phase transformation mechanism, crystal structural changes during charge and discharge processes of Li2FeSiO4 at different rates were investigated by X-ray diffraction measurements. The C/50 rate of lithium extraction upon initial cycling leads to a complete transformation from a monoclinic Li2FeSiO4 to a thermodynamically stable orthorhombic LiFeSiO4, concomitant with the occurrence of significant Li/Fe antisite mixing. The C/10 rate of lithium extraction and insertion, however, leads to retention of the parent Li2FeSiO4 (with the monoclinic structure as a metastable phase) with little cationic mixing. Here, we experimentally show the presence of metastable and stable LiFeSiO4 polymorphic phases caused by lithium extraction and insertion.}, author = {Masese, Titus and Orikasa, Yuki and Tassel, C\'{e}dric and Kim, Jungeun and Minato, Taketoshi and Arai, Hajime and Mori, Takuya and Yamamoto, Kentaro and Kobayashi, Yoji and Kageyama, Hiroshi and Ogumi, Zempachi and Uchimoto, Yoshiharu}, doi = {10.1021/cm403134q}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Chemistry of Materials/Masese et al.\_Chemistry of Materials\_2014.pdf:pdf}, issn = {0897-4756}, journal = {Chem. Mater.}, month = feb, number = {3}, pages = {1380--1384}, publisher = {American Chemical Society}, title = {{Relationship between Phase Transition Involving Cationic Exchange and Charge–Discharge Rate in Li 2 FeSiO 4}}, url = {http://dx.doi.org/10.1021/cm403134q}, volume = {26}, year = {2014} } [28] B. L. Ellis, W. R. M. Makahnouk, Y. Makimura, K. Toghill, and L. F. Nazar, "A multifunctional 3.5 V iron-based phosphate cathode for rechargeable batteries," Nat. mater., vol. 6, iss. 10, p. 749–753, 2007. [Bibtex] @article{Ellis2007a, abstract = {In the search for new positive-electrode materials for lithium-ion batteries, recent research has focused on nanostructured lithium transition-metal phosphates that exhibit desirable properties such as high energy storage capacity combined with electrochemical stability. Only one member of this class--the olivine LiFePO(4) (ref. 3)--has risen to prominence so far, owing to its other characteristics, which include low cost, low environmental impact and safety. These are critical for large-capacity systems such as plug-in hybrid electric vehicles. Nonetheless, olivine has some inherent shortcomings, including one-dimensional lithium-ion transport and a two-phase redox reaction that together limit the mobility of the phase boundary. Thus, nanocrystallites are key to enable fast rate behaviour. It has also been suggested that the long-term economic viability of large-scale Li-ion energy storage systems could be ultimately limited by global lithium reserves, although this remains speculative at present. (Current proven world reserves should be sufficient for the hybrid electric vehicle market, although plug-in hybrid electric vehicle and electric vehicle expansion would put considerable strain on resources and hence cost effectiveness.) Here, we report on a sodium/lithium iron phosphate, A(2)FePO(4)F (A=Na, Li), that could serve as a cathode in either Li-ion or Na-ion cells. Furthermore, it possesses facile two-dimensional pathways for Li+ transport, and the structural changes on reduction-oxidation are minimal. This results in a volume change of only 3.7\% that--unlike the olivine--contributes to the absence of distinct two-phase behaviour during redox, and a reversible capacity that is 85\% of theoretical.}, author = {Ellis, B L and Makahnouk, W R M and Makimura, Y and Toghill, K and Nazar, L F}, doi = {10.1038/nmat2007}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nat. Mater/Ellis et al.\_Nat. Mater.\_2007.pdf:pdf}, issn = {1476-1122}, journal = {Nat. Mater.}, month = oct, number = {10}, pages = {749--753}, pmid = {17828278}, publisher = {Nature Publishing Group}, shorttitle = {Nat Mater}, title = {{A multifunctional 3.5 V iron-based phosphate cathode for rechargeable batteries}}, url = {http://dx.doi.org/10.1038/nmat2007}, volume = {6}, year = {2007} } [29] B. L. Ellis, T. N. Ramesh, W. N. Rowan-Weetaluktuk, D. H. Ryan, and L. F. Nazar, "Solvothermal synthesis of electroactive lithium iron tavorites and structure of Li2FePO4F," J. mater. chem., vol. 22, iss. 11, p. 4759–4766, 2012. [Bibtex] @article{Ellis2012, author = {Ellis, B. L. and Ramesh, T. N. and Rowan-Weetaluktuk, W. N. and Ryan, D. H. and Nazar, L. F.}, doi = {10.1039/c2jm15273h}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Journal of Materials Chemistry/Ellis et al.\_Journal of Materials Chemistry\_2012.pdf:pdf}, issn = {0959-9428}, journal = {J. Mater. Chem.}, number = {11}, pages = {4759--4766}, title = {{Solvothermal synthesis of electroactive lithium iron tavorites and structure of Li2FePO4F}}, url = {http://xlink.rsc.org/?DOI=c2jm15273h}, volume = {22}, year = {2012} } [30] N. Recham, J-N. Chotard, L. Dupont, C. Delacourt, W. Walker, M. Armand, and J. -M. Tarascon, "A 3.6 V lithium-based fluorosulphate insertion positive electrode for lithium-ion batteries," Nat. mater., vol. 9, iss. 1, p. 68–74, 2010. [Bibtex] @article{Recham2010a, abstract = {Li-ion batteries have contributed to the commercial success of portable electronics, and are now in a position to influence higher-volume applications such as plug-in hybrid electric vehicles. Most commercial Li-ion batteries use positive electrodes based on lithium cobalt oxides. Despite showing a lower voltage than cobalt-based systems (3.45 V versus 4 V) and a lower energy density, LiFePO(4) has emerged as a promising contender owing to the cost sensitivity of higher-volume markets. LiFePO(4) also shows intrinsically low ionic and electronic transport, necessitating nanosizing and/or carbon coating. Clearly, there is a need for inexpensive materials with higher energy densities. Although this could in principle be achieved by introducing fluorine and by replacing phosphate groups with more electron-withdrawing sulphate groups, this avenue has remained unexplored. Herein, we synthesize and show promising electrode performance for LiFeSO(4)F. This material shows a slightly higher voltage (3.6 V versus Li) than LiFePO(4) and suppresses the need for nanosizing or carbon coating while sharing the same cost advantage. This work not only provides a positive-electrode contender to rival LiFePO(4), but also suggests that broad classes of fluoro-oxyanion materials could be discovered.}, author = {Recham, N and Chotard, J-N and Dupont, L and Delacourt, C and Walker, W and Armand, M and Tarascon, J.-M.}, doi = {10.1038/nmat2590}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nat. Mater/Recham et al.\_Nat. Mater.\_2010.pdf:pdf}, issn = {1476-1122}, journal = {Nat. Mater.}, month = jan, number = {1}, pages = {68--74}, pmid = {19946280}, title = {{A 3.6 V lithium-based fluorosulphate insertion positive electrode for lithium-ion batteries}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/19946280}, volume = {9}, year = {2010} } [31] S. Nishimura, M. Nakamura, R. Natsui, and A. Yamada, "New lithium iron pyrophosphate as 3.5 V class cathode material for lithium ion battery," J. am. chem. soc., vol. 132, iss. 39, p. 13596–13597, 2010. [Bibtex] @article{Nishimura2010a, abstract = {A new pyrophosphate compound Li(2)FeP(2)O(7) was synthesized by a conventional solid-state reaction, and its crystal structure was determined. Its reversible electrode operation at ca. 3.5 V vs Li was identified with the capacity of a one-electron theoretical value of 110 mAh g(-1) even for ca. 1 $\mu$m particles without any special efforts such as nanosizing or carbon coating. Li(2)FeP(2)O(7) and its derivatives should provide a new platform for related lithium battery electrode research and could be potential competitors to commercial olivine LiFePO(4), which has been recognized as the most promising positive cathode for a lithium-ion battery system for large-scale applications, such as plug-in hybrid electric vehicles.}, author = {Nishimura, S and Nakamura, M and Natsui, R and Yamada, A}, doi = {10.1021/ja106297a}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/J. Am. Chem. Soc/Nishimura et al.\_J. Am. Chem. Soc.\_2010.pdf:pdf}, issn = {1520-5126}, journal = {J. Am. Chem. Soc.}, keywords = {Diphosphates,Diphosphates: chemical synthesis,Diphosphates: chemistry,Electric Power Supplies,Electrochemistry,Electrodes,Iron,Iron: chemistry,Lithium,Lithium: chemistry,Models,Molecular}, month = oct, number = {39}, pages = {13596--13597}, pmid = {20831186}, title = {{New lithium iron pyrophosphate as 3.5 V class cathode material for lithium ion battery}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/20831186}, volume = {132}, year = {2010} } [32] M. Sathiya, K. Ramesha, G. Rousse, D. Foix, D. Gonbeau, K. Guruprakash, A. S. Prakash, M. L. Doublet, and J-M. Tarascon, "Li4NiTeO6 as a positive electrode for Li-ion batteries.," Chem. commun., vol. 49, p. 11376–11378, 2013. [Bibtex] @article{Sathiya2013d, abstract = {Layered Li4NiTeO6 was shown to reversibly release/uptake ∼2 lithium ions per formula unit with fair capacity retention upon long cycling. The Li electrochemical reactivity mechanism differs from that of Li2MO3 and is rooted in the Ni(4+)/Ni(2+) redox couple, that takes place at a higher potential than conventional LiNi1-xMnxO2 compounds. We explain this in terms of inductive effect due to Te(6+) ions (or the TeO6(6-) moiety).}, author = {Sathiya, M and Ramesha, K and Rousse, G and Foix, D and Gonbeau, D and Guruprakash, K and Prakash, A S and Doublet, M L and Tarascon, J-M}, doi = {10.1039/c3cc46842a}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Chemical communications/Sathiya et al.\_Chemical communications\_2013.pdf:pdf}, issn = {1364-548X}, journal = {Chem. Commun.}, language = {en}, month = oct, pages = {11376--11378}, pmid = {24165856}, publisher = {The Royal Society of Chemistry}, title = {{Li4NiTeO6 as a positive electrode for Li-ion batteries.}}, url = {http://pubs.rsc.org/en/content/articlehtml/2013/cc/c3cc46842a}, volume = {49}, year = {2013} } [33] M. M. Thacheray, C. S. Johnson, J. T. Vaughey, N. Li, and S. A. Hackney, "Advances in manganese-oxide ‘composite’ electrodes for lithium-ion batteries," J. mater. chem., vol. 15, iss. 23, p. 2257–2267, 2005. [Bibtex] @article{Thacheray2005, abstract = {Recent advances to develop manganese-rich electrodes derived from ‘composite’ structures in which a Li2MnO3 (layered) component is structurally integrated with either a layered LiMO2 component or a spinel LiM2O4 component, in which M is predominantly Mn and Ni, are reviewed. The electrodes, which can be represented in two-component notation as xLi2MnO3·(1 − x)LiMO2 and xLi2MnO3·(1 − x)LiM2O4, are activated by lithia (Li2O) and/or lithium removal from the Li2MnO3, LiMO2 and LiM2O4 components. The electrodes provide an initial capacity >250 mAh g−1 when discharged between 5 and 2.0 V vs. Li0 and a rechargeable capacity up to 250 mAh g−1 over the same potential window. Electrochemical charge and discharge reactions are followed on compositional phase diagrams. The data bode well for the development and exploitation of high capacity electrodes for the next generation of lithium-ion batteries.}, annote = {とりあえずcomposite electrodeと認識しているLi-rich系のレビュー}, author = {Thacheray, M. M. and Johnson, Christopher S. and Vaughey, John T. and Li, N. and Hackney, Stephen A.}, doi = {10.1039/b417616m}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Journal of Materials Chemistry/Thacheray et al.\_Journal of Materials Chemistry\_2005.pdf:pdf}, issn = {0959-9428}, journal = {J. Mater. Chem.}, language = {en}, month = jun, number = {23}, pages = {2257--2267}, publisher = {The Royal Society of Chemistry}, title = {{Advances in manganese-oxide ‘composite’ electrodes for lithium-ion batteries}}, url = {http://pubs.rsc.org/en/content/articlehtml/2005/jm/b417616m}, volume = {15}, year = {2005} } [34] M. M. Thackeray, S. Kang, C. S. Johnson, J. T. Vaughey, R. Benedek, and S. A. Hackney, "Li2MnO3-stabilized LiMO2 (M = Mn, Ni, Co) electrodes for lithium-ion batteries," J. mater. chem., vol. 17, iss. 30, p. 3112, 2007. [Bibtex] @article{Thackeray2007b, author = {Thackeray, Michael M. and Kang, Sun-Ho and Johnson, Christopher S. and Vaughey, John T. and Benedek, Roy and Hackney, S. A.}, doi = {10.1039/b702425h}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Journal of Materials Chemistry/Thacheray et al.\_Journal of Materials Chemistry\_2007.pdf:pdf}, issn = {0959-9428}, journal = {J. Mater. Chem.}, number = {30}, pages = {3112}, title = {{Li2MnO3-stabilized LiMO2 (M = Mn, Ni, Co) electrodes for lithium-ion batteries}}, url = {http://xlink.rsc.org/?DOI=b702425h}, volume = {17}, year = {2007} } [35] J. Lee, A. Urban, X. Li, D. Su, G. Hautier, and G. Ceder, "Unlocking the potential of cation-disordered oxides for rechargeable lithium batteries.," Science, vol. 343, iss. 6170, p. 519–22, 2014. [Bibtex] @article{Lee2014a, abstract = {Nearly all high-energy density cathodes for rechargeable lithium batteries are well-ordered materials in which lithium and other cations occupy distinct sites. Cation-disordered materials are generally disregarded as cathodes because lithium diffusion tends to be limited by their structures. The performance of Li1.211Mo0.467Cr0.3O2 shows that lithium diffusion can be facile in disordered materials. Using ab initio computations, we demonstrate that this unexpected behavior is due to percolation of a certain type of active diffusion channels in disordered Li-excess materials. A unified understanding of high performance in both layered and Li-excess materials may enable the design of disordered-electrode materials with high capacity and high energy density.}, author = {Lee, Jinhyuk and Urban, Alexander and Li, Xin and Su, Dong and Hautier, Geoffroy and Ceder, Gerbrand}, doi = {10.1126/science.1246432}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Science (New York, N.Y.)/Lee et al.\_Science (New York, N.Y.)\_2014.pdf:pdf}, issn = {1095-9203}, journal = {Science}, month = jan, number = {6170}, pages = {519--22}, pmid = {24407480}, title = {{Unlocking the potential of cation-disordered oxides for rechargeable lithium batteries.}}, url = {http://www.sciencemag.org/content/343/6170/519}, volume = {343}, year = {2014} } [36] H. Kobayashi, "Structure and lithium deintercalation of Li2−xRuO3," Solid state ionics, vol. 82, iss. 1-2, p. 25–31, 1995. [Bibtex] @article{Kobayashi1995, abstract = {Lithium deintercalation process of lithium ruthenium oxide, Li2RuO3, was characterized by X-ray diffraction and electrochemical measurements. The deintercalation proceeded from x = 0.0 to 1.3 with two-phasic reactions for 0 < x ≤ 0.5 and 0.7 ≤ x ≤ 1.0. Monophasic properties were observed for the compositions, Li1.4RuO3 and Li0.9RuO3; Li1.4RuO3 has a monoclinic cell isostructural to Li2RuO3, and Li0.9RuO3 has rhombohedral symmetry with the ilmenite-related structure. The lithium deintercalation from the lithium layer caused the rearrangement of the oxide-ion array from a cubic close packed (ccp) to a hexagonal close packed (hcp) structure. Further, electrical and magnetic properties were discussed on the basis of electrical and magnetic measurements.}, author = {Kobayashi, H}, doi = {10.1016/0167-2738(95)00135-S}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Solid State Ionics/Kobayashi\_Solid State Ionics\_1995.pdf:pdf}, issn = {01672738}, journal = {Solid State Ionics}, keywords = {li2−xruo3,lithium deintercalation,ruthenium oxide}, month = nov, number = {1-2}, pages = {25--31}, title = {{Structure and lithium deintercalation of Li2−xRuO3}}, url = {http://dx.doi.org/10.1016/0167-2738(95)00135-S}, volume = {82}, year = {1995} } [37] J. Ma, Y. Zhou, Y. Gao, X. Yu, Q. Kong, L. Gu, Z. Wang, X. Yang, and L. Chen, "Feasibility of Using Li 2 MoO 3 in Constructing Li-Rich High Energy Density Cathode Materials," Chem. mater., vol. 26, iss. 10, p. 3256–3262, 2014. [Bibtex] @article{Ma2014, abstract = {Layer-structured xLi2MnO3·(1 ? x)LiMO2 are promising cathode materials for high energy-density Li-ion batteries because they deliver high capacities due to the stabilizing effect of Li2MnO3. However, the inherent disadvantages of Li2MnO3 make these materials suffer from drawbacks such as fast energy-density decay, poor rate performance and safety hazard. In this paper, we propose to replace Li2MnO3 with Li2MoO3 for constructing novel Li-rich cathode materials and evaluate its feasibility. Comprehensive studies by X-ray diffraction, X-ray absorption spectroscopy, and spherical-aberration-corrected scanning transmission electron microscopy clarify its lithium extraction/insertion mechanism and shows that the Mo4+/Mo6+ redox couple in Li2MoO3 can accomplish the task of charge compensation upon Li removal. Other properties of Li2MoO3 such as the nearly reversible Mo-ion migration to/from the Li vacancies, absence of oxygen evolution, and reversible phase transition during initial (de)lithiation indicate that Li2MoO3 meets the requirements to an ideal replacement of Li2MnO3 in constructing Li2MoO3-based Li-rich cathode materials with superior performances. Layer-structured xLi2MnO3·(1 ? x)LiMO2 are promising cathode materials for high energy-density Li-ion batteries because they deliver high capacities due to the stabilizing effect of Li2MnO3. However, the inherent disadvantages of Li2MnO3 make these materials suffer from drawbacks such as fast energy-density decay, poor rate performance and safety hazard. In this paper, we propose to replace Li2MnO3 with Li2MoO3 for constructing novel Li-rich cathode materials and evaluate its feasibility. Comprehensive studies by X-ray diffraction, X-ray absorption spectroscopy, and spherical-aberration-corrected scanning transmission electron microscopy clarify its lithium extraction/insertion mechanism and shows that the Mo4+/Mo6+ redox couple in Li2MoO3 can accomplish the task of charge compensation upon Li removal. Other properties of Li2MoO3 such as the nearly reversible Mo-ion migration to/from the Li vacancies, absence of oxygen evolution, and reversible phase transition during initial (de)lithiation indicate that Li2MoO3 meets the requirements to an ideal replacement of Li2MnO3 in constructing Li2MoO3-based Li-rich cathode materials with superior performances.}, author = {Ma, Jun and Zhou, Yong-Ning and Gao, Yurui and Yu, Xiqian and Kong, Qingyu and Gu, Lin and Wang, Zhaoxiang and Yang, Xiao-Qing and Chen, Liquan}, doi = {10.1021/cm501025r}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Chemistry of Materials/Ma et al.\_Chemistry of Materials\_2014.pdf:pdf}, issn = {0897-4756}, journal = {Chem. Mater.}, month = may, number = {10}, pages = {3256--3262}, publisher = {American Chemical Society}, title = {{Feasibility of Using Li 2 MoO 3 in Constructing Li-Rich High Energy Density Cathode Materials}}, url = {http://dx.doi.org/10.1021/cm501025r}, volume = {26}, year = {2014} } [38] M. Sathiya, K. Ramesha, G. Rousse, D. Foix, D. Gonbeau, A. S. Prakash, M. L. Doublet, K. Hemalatha, and J. -M. Tarascon, "High Performance Li2Ru1–yMnyO3 (0.2 ≤ y ≤ 0.8) Cathode Materials for Rechargeable Lithium-Ion Batteries: Their Understanding," Chem. mater., vol. 25, p. 1121–1131, 2013. [Bibtex] @article{Sathiya2013a, author = {Sathiya, M and Ramesha, K and Rousse, G. and Foix, D and Gonbeau, D and Prakash, A.S. and Doublet, M. L. and Hemalatha, K. and Tarascon, J.-M.}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Chemistry of Materials/Sathiya et al.\_Chemistry of Materials\_2013.pdf:pdf}, journal = {Chem. Mater.}, pages = {1121--1131}, title = {{High Performance Li2Ru1–yMnyO3 (0.2 ≤ y ≤ 0.8) Cathode Materials for Rechargeable Lithium-Ion Batteries: Their Understanding}}, volume = {25}, year = {2013} } [39] M. Sathiya, G. Rousse, K. Ramesha, C. P. Laisa, H. Vezin, M. T. Sougrati, M-L. Doublet, D. Foix, D. Gonbeau, W. Walker, A. S. Prakash, M. {Ben Hassine}, L. Dupont, and J-M. Tarascon, "Reversible anionic redox chemistry in high-capacity layered-oxide electrodes," Nat. mater., vol. advance on, 2013. [Bibtex] @article{Sathiya2013, author = {Sathiya, M. and Rousse, G. and Ramesha, K. and Laisa, C. P. and Vezin, H. and Sougrati, M. T. and Doublet, M-L. and Foix, D. and Gonbeau, D. and Walker, W. and Prakash, A. S. and {Ben Hassine}, M. and Dupont, L. and Tarascon, J-M.}, doi = {10.1038/nmat3699}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nat. Mater/Sathiya et al.\_Nat. Mater.\_2013.pdf:pdf}, issn = {1476-1122}, journal = {Nat. Mater.}, month = jul, publisher = {Nature Publishing Group}, shorttitle = {Nat Mater}, title = {{Reversible anionic redox chemistry in high-capacity layered-oxide electrodes}}, url = {http://dx.doi.org/10.1038/nmat3699}, volume = {advance on}, year = {2013} } [40] M. Aydinol, A. Kohan, G. Ceder, K. Cho, and J. Joannopoulos, "Ab initio study of lithium intercalation in metal oxides and metal dichalcogenides," Phys. rev. b, vol. 56, iss. 3, p. 1354–1365, 1997. [Bibtex] @article{Aydinol1997b, author = {Aydinol, M. and Kohan, A. and Ceder, Gerbrand and Cho, K. and Joannopoulos, J.}, doi = {10.1103/PhysRevB.56.1354}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Physical Review B/Aydinol et al.\_Physical Review B\_1997.pdf:pdf}, issn = {0163-1829}, journal = {Phys. Rev. B}, month = jul, number = {3}, pages = {1354--1365}, title = {{Ab initio study of lithium intercalation in metal oxides and metal dichalcogenides}}, url = {http://link.aps.org/doi/10.1103/PhysRevB.56.1354}, volume = {56}, year = {1997} } [41] G. Ceder, Y. -M. Chiang, D. R. Sadoway, M. K. Aydinol, Y. -I. Jang, and B. Huang, "Identification of cathode materials for lithium batteries guided by first-principles calculations," Nature, vol. 392, iss. 6677, p. 694–696, 1998. [Bibtex] @article{Ceder1998c, abstract = {Lithium batteries have the highest energy density of all rechargeable batteries and are favoured in applications where low weight or small volume are desired — for example, laptop computers, cellular telephones and electric vehicles1. One of the limitations of present commercial lithium batteries is the high cost of the LiCoO2 cathode material. Searches for a replacement material that, like LiCoO2, intercalates lithium ions reversibly have covered most of the known lithium/transition-metal oxides, but the number of possible mixtures of these2, 3, 4, 5 is almost limitless, making an empirical search labourious and expensive. Here we show that first-principles calculations can instead direct the search for possible cathode materials. Through such calculations we identify a large class of new candidate materials in which non-transition metals are substituted for transition metals. The replacement with non-transition metals is driven by the realization that oxygen, rather than transition-metal ions, function as the electron acceptor upon insertion of Li. For one such material, Li(Co,Al)O2, we predict and verify experimentally that aluminium substitution raises the cell voltage while decreasing both the density of the material and its cost.}, author = {Ceder, Gerbrand and Chiang, Y.-M. and Sadoway, D. R. and Aydinol, M. K. and Jang, Y.-I. and Huang, B.}, doi = {10.1038/33647}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Nature/Ceder et al.\_Nature\_1998.pdf:pdf}, issn = {0028-0836}, journal = {Nature}, month = apr, number = {6677}, pages = {694--696}, shorttitle = {Nature}, title = {{Identification of cathode materials for lithium batteries guided by first-principles calculations}}, url = {http://dx.doi.org/10.1038/33647}, volume = {392}, year = {1998} } [42] L. Wang, T. Maxisch, and G. Ceder, "Oxidation energies of transition metal oxides within the GGA+U framework," Phys. rev. b, vol. 73, p. 195107, 2006. [Bibtex] @article{Wang2006, author = {Wang, Lei and Maxisch, Thomas and Ceder, Gerbrand}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Phys. Rev. B/Wang, Maxisch, Ceder\_Phys. Rev. B\_2006.pdf:pdf}, journal = {Phys. Rev. B}, keywords = {calculation}, mendeley-tags = {calculation}, pages = {195107}, title = {{Oxidation energies of transition metal oxides within the GGA+U framework}}, volume = {73}, year = {2006} } [43] Y. Koyama, Y. Makimura, I. Tanaka, H. Adachi, and T. Ohzuku, "Systematic Research on Insertion Materials Based on Superlattice Models in a Phase Triangle of LiCoO[sub 2]-LiNiO[sub 2]-LiMnO[sub 2]," J. electrochem. soc., vol. 151, iss. 9, p. A1499, 2004. [Bibtex] @article{Koyama2004, author = {Koyama, Yukinori and Makimura, Yoshinari and Tanaka, Isao and Adachi, Hirohiko and Ohzuku, Tsutomu}, doi = {10.1149/1.1783908}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/J. Electrochem. Soc/Koyama et al.\_J. Electrochem. Soc.\_2004.pdf:pdf}, issn = {00134651}, journal = {J. Electrochem. Soc.}, language = {en}, month = sep, number = {9}, pages = {A1499}, publisher = {The Electrochemical Society}, title = {{Systematic Research on Insertion Materials Based on Superlattice Models in a Phase Triangle of LiCoO[sub 2]-LiNiO[sub 2]-LiMnO[sub 2]}}, url = {http://jes.ecsdl.org/content/151/9/A1499.full}, volume = {151}, year = {2004} } [44] S. Mishra and G. Ceder, "Structural stability of lithium manganese oxides," Phys. rev. b, vol. 59, iss. 9, p. 6120–6130, 1999. [Bibtex] @article{Mishra1999, abstract = {We have studied stability of lithium-manganese oxides using density functional theory in the local density and generalized gradient approximation (GGA). In particular, the effect of spin-polarization and magnetic ordering on the relative stability of various structures is investigated. At all lithium compositions the effect of spin polarization is large, although it does not affect different structures to the same extent. At composition LiMnO2, globally stable Jahn-Teller distortions could only be obtained in the spin-polarized GGA approximation, and antiferromagnetic spin ordering was critical to reproduce the orthorhombic LiMnO2 structure as ground state. We also investigate the effect of magnetism on the Li intercalation potential, an important property for rechargeable Li batteries.}, author = {Mishra, S. and Ceder, Gerbrand}, doi = {10.1103/PhysRevB.59.6120}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Physical Review B/Mishra, Ceder\_Physical Review B\_1999.pdf:pdf}, issn = {0163-1829}, journal = {Phys. Rev. B}, month = mar, number = {9}, pages = {6120--6130}, publisher = {American Physical Society}, shorttitle = {Phys. Rev. B}, title = {{Structural stability of lithium manganese oxides}}, url = {http://link.aps.org/doi/10.1103/PhysRevB.59.6120}, volume = {59}, year = {1999} } [45] J. Reed and G. Ceder, "Role of Electronic Structure in the Susceptibility of Metastable Transition-Metal Oxide Structures to Transformation," Chem. rev., vol. 104, iss. 10, p. 4513–4534, 2004. [Bibtex] @article{Reed2004, author = {Reed, John and Ceder, Gerbrand}, doi = {10.1021/cr020733x}, file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Chemical Reviews/Reed, Ceder\_Chemical Reviews\_2004.pdf:pdf}, issn = {0009-2665}, journal = {Chem. Rev.}, month = oct, number = {10}, pages = {4513--4534}, publisher = {American Chemical Society}, title = {{Role of Electronic Structure in the Susceptibility of Metastable Transition-Metal Oxide Structures to Transformation}}, url = {http://dx.doi.org/10.1021/cr020733x}, volume = {104}, year = {2004} } 08/29/201806/29/2019 by tkawaguchi