I. INTRODUCTION
1.9 I N - SITU CHEMICAL AND STRUCTURAL CHARACTERIZATION METHODS FOR
1.9.2 Pair Distribution Function
While traditional X-ray diffraction is useful for well crystallized, ordered materials, pair distribution functions are useful for understanding nanoscale, disordered, and even amorphous materials. Several different scattering events can occur when X-rays interact with matter including elastic, inelastic, coherent, and incoherent scattering. Elastic scattering events occur when there is no change in energy during an interaction between light and matter, while inelastic scattering is defined by a change in energy during a scattering event. Coherence describes phase relationships before and after a scattering event. During a coherent scattering event, two waves interfere, and the resulting intensity is equal to the square of the sum of the amplitudes of the individual waves. Incoherent scattering occurs when there is no interference between two waves and the resulting intensity is equal to the sum of the squares of the amplitudes of the individual waves.
Potential scattering events when light interacts with a material are summarized by:
i. Bragg scattering: elastic, coherent
ii. Laue monoatomic diffuse scattering: elastic, incoherent iii. Thermal diffuse scattering: inelastic, coherent
iv. Compton scattering: inelastic, incoherent
Pair distribution functions make use of both Bragg scattering and Laue monoatomic diffuse scattering, coherent and incoherent elastic scattering, to obtain information about the local environment of a material.64
To understand how the pair distribution function is obtained, one must first understand Q, the scattering vector. A schematic illustrating Q is shown in Figure 12.
Figure 12. Schematic of scattering vector, Q.
The scattering vector, Q, is related to the initial and final momentum of the X-ray beam, k0 and kf, respectively, through the following relation:
𝑄 = 𝑘0− 𝑘𝑓 (4)
where,
𝑘0 = |𝑘0| =2𝜋
𝜆0 (5)
𝑘𝑓 = |𝑘𝑓| =2𝜋
𝜆𝑓 (6)
In coherent scattering events
𝜆0 ≡ 𝜆𝑓 (7)
𝛹(𝑄, 𝑡) = 1
〈𝑏〉∑ 𝑏𝑖 𝑖𝑒𝑖𝑄∙𝑅𝑖(𝑡) (9)
The scattering amplitude cannot be measured directly. However, the intensity of the scattered beam, I(Q), is a measurable quantity, where I(Q) is related to the square of the scattering amplitude, such that
|𝛹(𝑄, 𝑡)|2 = 1
〈𝑏〉2∑ 𝑏𝑖𝑏𝑗𝑒𝑖𝑄∙(𝑅𝑖−𝑅𝑗)
𝑖,𝑗
(10)
Rearranging Equation 10 and multiplying by 𝑁1 where N is the number of atoms gives:
1
𝑁|𝛹(𝑄, 𝑡)|2〈𝑏〉2 = 1
𝑁∑ 𝑏𝑖𝑏𝑗𝑒𝑖𝑄∙(𝑅𝑖−𝑅𝑗)
𝑖,𝑗
(11)
Laue monotonic diffuse scattering contributions area also factored into this equation, where
𝐼(𝑄) = 𝑑𝜎𝑐(𝑄)
𝑑𝛺 + 〈𝑏〉2− 〈𝑏2〉 (12)
Normalized with respect to 〈𝑏〉2
𝑆(𝑄) =𝐼(𝑄)
〈𝑏〉2 (13)
The reduced structure function is defined by F(Q)
𝐹(𝑄) = 𝑄[𝑆(𝑄 − 1)] (14)
Equation 15 describes the general form of a sine Fourier transform where G(r) is the PDF.
𝐺(𝑟) = (2
𝜋) ∫ 𝐹(𝑄) sin(𝑄𝑟)
∞
0
𝑑𝑄 (15)
Realistically, PDFs do not achieve infinite limits. Reasons for this will be discussed later in this section. Therefore a Qmax and Qmin exist as limits to this integral calculation.
𝐺(𝑟) = (2
𝜋) ∫ 𝐹(𝑄) sin(𝑄𝑟)
𝑄𝑚𝑎𝑥
𝑄𝑚𝑖𝑛
𝑑𝑄 (16)
The PDF, G(r), is described by p(r), the atom-pair density function and p0, the atomic number density.
𝐺(𝑟) = 4𝜋𝑟[𝜌(𝑟) − 𝜌0] (17)
𝜌(𝑟) = 1
4𝜋𝑟2𝑁∑ ∑𝑏𝑖𝑏𝑗
〈𝑏〉2𝛿(𝑟 − 𝑟𝑖𝑗)
𝑖≠𝑗 𝑖
(18)
Acquiring a pair distribution function requires a wavelength of light that can probe a significant distance in Q space. Resolution of the PDF is dependent on Qmax where resolution =𝑄𝑚𝑎𝑥𝜋 . Laboratory PDF measurements typically use Ag or Mo sources which can achieve a Qmax resolution of ~22 Å-1 and ~17.5 Å-1, respectively.65 These measurements usually last on the order of tens to hundreds of hours due to the brilliance of laboratory X-ray sources of roughly 103 photons/s/mm2/mrad2/[1% band-width]. Beamline 11-ID-B at the Advanced Photon Source (Argonne National Lab, Argonne, IL) is one of the best PDF acquisition instruments to date. This beamline uses a wavelength of 0.2113 Å-1 which achieves a potential Qmax of ~60 Å-1. Rapid acquisition PDF measurements at synchrotron sources occur on the order of minutes with a brilliance of 1015 photons/s/mm2/mrad2/[1% band-width].
11-ID-B illuminates a transmission geometry with a 2D Si flat plate detector. The raw data acquired from 11-ID-B are 2D detector images containing Debye cones of the sample probed. The 2D detector images are integrated in software such as GSAS, GSASII, or FIT2D to obtain I(Q). The structure function, S(Q), is obtained by subtracting background and correcting for inelastic Compton scattering, shape of container, multiple scattering events, and other effects. Background subtraction is quite involved for in-situ studies and involves removing scattering contributions from components of the cell as well as air scattering. For measurements performed in capillaries, background subtraction is more straight forward and simply involves scaling background of the capillary container and contributions from air scattering. The PDF is then obtained by choosing Qmax and Qmin based on noise in the reduced structure function, F(Q), and S(Q), and performing a discrete Fourier transform on F(Q). The Fourier transform can be performed using software such as PDFgetX2 or PDFgetX3. Qmax and Qmin should be optimized where
Figure 13. Information gathered from PDFs include atomic distances, coordination number, disorder, and particle size.66
In a study performed in 2018, in-situ synchrotron XRD and PDF measurements were measured for layered NixMn1-xO2 during electrochemical cycling in NaSO4
electrolyte. Locations of Ni atoms were confirmed via neutron PDF refinements to be in the sheet as substitutions for the Mn ion rather than in the interlayer galleries. In-situ XRD showed a decrease in the interlayer spacing upon Na+ insertion caused by an increase in the electrostatic interactions between the sheet and interlayer, while the lateral direction experienced an expansion upon charging. Counteracting shifts between interlayer spacing and in-plane expansion and contraction are a common effect in layered energy storage materials to minimize the overall change in volume. Diffraction peaks returned to their original positions upon removal of the electrolyte ion. Reduction from Mn4+ to Mn3+ and Ni4+ to Ni3+ occurred at the charged condition, leading to an increase Ni/Mn – O and Ni/Mn – Ni/Mn bond length. Reduced ions have a larger ionic radii than oxidized ions, causing a longer bond length.67