Thickness of Epitaxial Films with Reflectometry

Non-destructive measurements of thin-film thicknesses can be challenging in cases where the film can not be scratched or masked for direct thickness measurements using AFM or profilometry.

X-ray reflectometry is a non-contact approach that involves measuring and modeling Kiessig fringes at very low angles. In this case an epitaxial thin-film of Ho₂Ti₂O₇ on yttria stabilized zirconia (YSZ) was scanned below 2° 2θ.

A coupled θ-2θ scan with very small divergence and receiving apertures (0.1° and 0.018° respectively) was performed with caution to not saturate the detector.  Constructive and destructive interference between reflection from the film surface and film-substrate interface result in periodic fringes that reflect the film thickness. These measurements are done in specular reflection, and rocking curves on primary substrate peak may be required to account for any substrate miscut angle or sample mounting error.

These fringes are generally modeled to obtain the film thickness, and in some cases interface and surface roughness parameters. There are freeware packages such as the NIST Ref1D or GenX. Modeling of this film was consistent with a 70 nm film thickness.

It should be noted that a Bragg-Brentano powder goniometer with fixed divergence apertures is generally not the appropriate tool for X-ray reflectometry. What makes it possible here is a combination of precise alignment, primarily of the zero-line and the divergence apertures, and careful sample mounting.

Thanks to Kevin Barry of the Beekman group, FSU Department of Physics.

Very Low Angle Calibration

High precision pXRD work is complicated by the presence of systematic errors in peak 2θ position. These aberrations are due to a variety of effects. At very low angles, the effects of sample transparency/displacement and axial divergence dominate. Standards exist for such work, such as NIST SRM 675 which is fluorophlogopite mica. This system has its lowest peak at 2θ =  8.853°.

Often it is important to characterize a goniometer well below the lowest peak of SRM 875. In this case silver behante, CH₃(CH₂)₂₀COOAg, which is a silver salt of the long chain fatty acid behenic acid, is a good standard. d(001) = 58.38 Å providing six strong diffraction peaks below 2θ = 10°.  The above coupled θ-2θ scan was taken with 0.1°, 0.3°, and 1.0° divergence apertures and 0.15° receiving aperture. Silver behenate was dissolved in toluene to form a super-saturated solution, and then drop-cast on a zero-background holder to provide large highly oriented domains.

From this data the error in 2θ as a function of 2θ can be determined and used as an external calibration or as a study of goniometer performance. This graph of δ2θ(2θ) was taken with 1.0° divergence apertures and no divergence Søller slit. In this angular range the aberrations are dominated by axial divergence and thus the increased error at low angles.

Powders can be mixed with Ag behenate in toluene and drop-cast or spun onto a sample support for a low angle internal standard for angles below those in SRM 675. The effects of the crystallization of Ag behenate on the preferred orientation of the power should be considered for quantitation. Also, Ag behenate can be combined with SRM 675 and either silicon or corundum to model aberrations through a wide angular range.

Simple Texture Analysis: Oriented Mica

Synthetic fluorphlogopite mica, NIST SRM 765, was dusted onto a zero background holder through a 100 mesh sieve. The SRM is ground to pass through a 200 mesh sieve, so this is a measure only to prevent aggregation of mica particles. A pXRD pattern was taken under normal conditions: 5-65° 2θ, at 2°/minute, 1° divergence apertures and 0.15° receiving aperture using the Siemens D500. After background subtraction and smoothing, the relative peak intensities were determined and then compared to ICDD PDF standard 16-0344. All peak intensities except for the (0 0 3) and (0 0 5) reflections were less than that of the presumably randomly oriented powder pattern presented in the PDF standard 16-0344, showing texture along the c-axis.

Since micas form plate-like crystal habits, the sample was then "wet prepped" by suspending the material in water and allowing it to dry on the zero background holder. A pXRD pattern was taken under the same conditions and compared to ICDD PDF standard 16-0344. Again, the (0 0 3) and (0 0 5) peaks are stronger than the randomly presented pattern in the PDF standard, but now most of the other reflections are gone, indicating even more highly enhanced texture. The only peaks that remain are the (0 0 1) and (0 0 2), consistent with c-axis texture, and (1 3 1) and (-1 3 5). What we're seeing is the plate-like crystals orienting in liquid and providing enhanced preferred orientation on drying.

Simple texture analysis is possibly by comparing coupled θ-2θ scans with a standard that is known to be without texture or preferred orientation. This probe of preferred orientation only addresses texture perpendicular to the sample surface, so a more in depth texture study will require a texture cradle.

in-situ Structural Decomposition

One use of high-temperature XRD is to correlate DSC or TGA data with structural information.  pXRD correlated with DSC can be useful to study structural phase transitions, while pXRD correlated with TGA can be used to study the decomposition of complex frameworks or the in-situ processing of reaction products.

In this series of pXRD data taken at temperatures ranging form ambient (run #7, back) to 325 C (run #1, front) a material with complex structure is found to decompose and collapse starting above 250 C (run #2). This data corroborated the interpretation of TGA data taken from 40 to 800 C indicating the decomposition of the material near 300 C.

In high-temperature XRD the sample is glued to a Pt-ribbon that acts as the sample holder as well as heater using an lacquer that decomposes at high temperatures. The Pt (111) and Pt (200) peaks are present throughout all scans and shift according to the Pt coefficient of expansion.

In the final run (run #1, front) it is noted that the decomposed products have a larger peak-width indicating a reduced correlation length. Note the narrower peaks just below 40° and just above 46°--these are the  (111) and (200) peaks from the platinum ribbon.

pXRD of Clays: Wet Preparation and Sedimentation Time Separation

Clays are phyllosilicate minerals that present lamellar crystal habits and complex diffraction patterns because of their low symmetry triclinic or monoclinic unit cells. They also tend to produce very low 2θ peaks because of their large unit cell parameters which are often over 8 Å. Clay systems also often include structural variations from ionic substitutions and varying degrees of hydration.

One key to the efficient study of these systems is to exploit the lamellar crystal habits. In the pXRD pattern shown, the red curve is the raw data prepared by dusting the sample through a mesh onto a greased zero-background holder. The PDF standard for quartz is shown. The sample is a naturally occurring clay sample.

The green  curve shows the same sample suspended in water and then dried on the zero-background holder. The vertical scale is normalized so that the strongest SiO₂ peaks have the same height. Throughout the range of the pXRD scan the signal from the phyllosilicates is enhanced. This is due to the lamellar crystal habits in solution drying on the zero-background holder and producing some preferred orientation or texture. The low-angle clay peaks are significantly enhanced.

An added step is to suspend the sample in a water column and use sedimentation time to separate the clay fraction. The sample material was suspended in water in a graduated cylinder and after the larger and heavier particles settled for 5-10 minute,s the water was decanted and filtered. The filtered material was placed on the zero-background holder with a little water and allowed to dry. This induced preferred orientation of the sample from the lamellar structure of the clay particles. This data is shown in the blue curve. The low angle clay peaks are greatly enhanced, indicating that the sedimentation time separation allowed for the preferential selection of the clay fraction for pXRD. Again the vertical scale is normalized so that the strongest SiO₂ peaks have the same height.

The purple curve is the same sample prepared after ~ 20 minutes of sedimentation in a water column, filtering and wet-preparation on the zero-background holder. The low angle clay peaks are even more enhanced. Again the vertical scale is normalized so that the strongest SiO₂ peaks have the same height.

RuO2: An Example of Particle Size in pXRD

To get the most out of powder XRD (pXRD) it is essential to be able to correlate aspects of the diffraction patterns with physical properties of the material being studied. The peak widths reflect the correlation length of the diffraction in the sample. The diffraction correlation length can be reduced in different ways. One is by the introduction of defects such as dislocations, or through the effect of strain. Another is the reduction of the scattering volume through a reduction of particle size. In general, reduced particle size broadens peaks in a Gaussian fashion, while increased strain broadens peaks in a Lorentzian fashion. This difference in the nature of the diffraction peak broadening is the basis of such techniques as the Warren-Averbach method of determining particle size and strain distributions.

The figure shows a RuO₂ powder sample synthesized under different conditions. The red curve was formed by annealing RuCl₃ at 400C, the blue curve through a sol-gel process that forms Ru(OH)₃. A huge difference in particle size is evident by the marked difference in peak width. The blue curve reflects a problem that arises in pXRD of nanoparticle systems: the peaks can be so broad as to almost be unidentifiable. Because of the reduction in peak intensity with reduced particle size, an area camera is often best suited for nanoparticle systems.

Thanks to Dr. Jim Zheng, FSU COE for the samples.

Phase Identification: An Isostructural Example

A MnO_x powder presumed to be α-MnO₂ was studied using XRD. In an initial study, two manganese phases matched nearly identically: α-MnO₂ and Mn(OH)₄. The automated search-match algorithm gave preference to the Mn(OH)₄ phase as reflected by its phase-ID figure of merit.

The MnOx powder was run again, this time mixing with a Si internal standard. It was sifted through a 200 mesh sieve onto a zero-background holder. 1° divergence and 0.15° receiving apertures were used. The scan shown is from 5° to 65° 2θ with 0.02° steps and a 2 second dwell time. The Si internal standard was used to calibrate the 2θ scale using a quadratic fit.

According to the ICDD PDF, α-MnO₂ (44-0141) and Mn(OH)₄ (15-0604) are seemingly isostructural. Both have I 4/m (87) space groups with nearly the same lattice parameters: a = 9.7847 Å and c = 2.863 Å for α-MnO₂ and a = 9.866 Å and c = 2.844 Å for Mn(OH)₄. Lattice parameters a & c differ by less than 1%. The data above shows the two phases to be nearly indistinguishable. The PDF standard for either phase can be used as a starting point for a metric refinement of the data-- and the results will be the same as the point and space groups are the same!

Interestingly-- the α-MnO₂ phase has two peaks the Mn(OH)₄ phase does not as indicated by arrows. These are the (220) and (330) peaks. How is that possible if they have the same space group? It's possible for a system to gain peaks if the symmetry of the system is reduced by strain or defects-- but the (220) and (330) lines are allowed for the Mn(OH)₄ phase.

Looking at the ICDD PDF editorial quality marks, α-MnO₂ 44-0141 is given a "*" which indicates the data is of the highest possible quality. "*" quality marks indicate very high precision peak positions, no serious systematic errors, and no unindexed or impurity reflections. In short, the data is high quality and consistent from a single phase sample.

The Mn(OH)₄ phase is not given any quality mark indicating the lowest possible quality for acceptance into the PDF. Such data is of low precision and from systems of uncertain chemistry that are either multiphase.

The different data qualities are reflected in the data shown in this example. Systems with the same space groups with the same lattice parameters should have peaks in the same positions. As such, there should be (220) and (330) peaks for the Mn(OH)₄ phase, and the presence of those peaks in the α-MnO₂ PDF standard should not, in itself be grounds for identifying this phase as α-MnO₂. What should be considered is the questionable quality of PDF card 15-0604 (Mn(OH)₄) and the high quality of PDF card 44-0141 (α-MnO₂).

There are several points to this example:

1. Do not blindly trust phase-ID algorithms.
2. Not all PDF cards are of the same quality.
3. Crystallographic information regarding phases should be considered.

D500 Goniometer Parts

This is a photo of the Siemens D500 θ-2θ goniometer. The 2θ angle is set to nearly 0.000° as would be found in alignment tests. The parts of the goniometer are labeled:

• A: Tube Stand. This contains the line focus Cu X-ray tube. Maximum power is 1200 W: 30 mA @ 40 kV.
• B: Ni Filter. A nickel foil is used to filter out continuum and Cu K_β radiation.
• I: This is the divergence aperture: 3°, 1°, 0.3° or 0.1°.
• SD: This is the divergence Soller slit to remove axial divergence.
• II: Anti-scatter slit to guarantee that only X-rays with the divergence defined by aperture I impinge upon the sample.
• ZBH: Zero background holder with the sample.
• III: Anti-scatter slit. Identical to anti-scatter slit II.
• SR: This is the receiving Soller slit to remove axial divergence.
• C: Optional Ni absorber. This allows one to operate without a diffracted beam monochromator (MC) by absorbing fluorescence in the sample.
• IV: Receiving aperture: 0.6°, 0.15°, 0.05° and 0.018°. Determines the resolution of diffractometer.
• MC: Graphite diffracted beam monochromator. This rejects any unfiltered primary beam-- continuum and Cu K_β, as well as any radiation fluoresced in the sample.

Agrillaceous Syenite

A sample of syenite was subject to SEM and EDS analysis. Domains of high density in BEI COMPO imaging were found to be iron rich, while the lower density material cementing these grains together were found to be a combination of Ca rich material and an Si-Al phase that was identified as not being tectosillicate due to the presence of Mg and thus thought to be phyllosillicate. Due to the loss of light element sensitivity due to EDS detector window absorption, it was thought that the Fe phase was FeO_x, the Ca phase calcite and the phyllosillicate phase a clay mineral. The image shows the BEI image.


The sample was powderized and drifted through a 200 mesh sieve onto a greased ZBH. Data was aquired from 5° to 75° 2θ with Δ2θ = 0.020° and a 1 second dwell time in step mode. 1° divergence and 0.15° receiving apertures were used with the diffracted beam monochromator.

The phase analysis confirmed this analysis obtained through EDS spectroscopy and EDS elemental mapping. Magnetite and calcite were present, as well as the phyllosilicate mineral chlorite-serpentine. Other phyllosilicate minerals are likely present and would be best identified from wet preparations to enhanced the low 2θ clay peaks. Thus this is agrillaceous or "clay containing" syenite.

The rare earth carbonate bastnaesite, LaCO₃F, was also present. This confirms the presence of the rare earth element lanthanum through EDS spectroscopy. This is significant as bastnaesite exists in multiple forms containing different rare earths, typically La, Ce and Y, and thus the most general bastnaesite is (La, Ce, Y)CO₃F. The atomic radii of the rare earths are very similar, and without such extensive studies as Rietveld refinement on single-phase material, differentiating the rare earth concentration in the bastnaesite by XRD is not possible. Previous EDS studies only shows the presence of La, suggesting that Ce and Y are absent or below the detection threshold of EDS-- a few 0.1 % weight.

A related mineral is parisite, Ca(Ce,La)₂(CO₃)₃F₂. Previous EDS mapping studies showed a negative correlation between La and Ca, either eliminating the presence of this mineral or placing it below a few 0.1 % weight given that neither La nor Ce were detected in the calcite regions.

Step Scans: The Effect of Step Size

This 3D graph shows the "five fingers" region of quartz between 67° 2θ and 69° 2θ using a 1° divergence aperture. The scans were taken with a 1 second dwell time in step scan mode. In step scan mode the goniometer moves by discrete angular steps, Δ2θ, but then stops and acquires data for the length of the dwell time. Then the goniometer steps another Δ2θ, acquires for the dwell time, and so on. The five finger region includes the quartz (212), (203) and (301) peaks and is in a range of 2θ that the K_α1-K_α2 doublet is resolvable. It is called the "five finger" region because the K_α2 (203) peak overlaps with the K_α1 (301) peak producing five instead of six peaks from three doublets.

The effect of step size, Δ2θ, was studied. The step sizes in the graph above are: 0.200° (front), 0.150°, 0.100°, 0.050°, 0.020°, 0.004° and 0.002° (back). In the case of continuous scans, the diffracted data is integrated over an angular range of 2θ. In the case of continuous scans, this integrated angular range is a function of step size, Δ2θ as the goniometer is continuously slewing at a constant angular velocity, and thus the intensity of the peaks is step-size dependent. In the case of step scans, the intensities are not step size dependent and are entirely determined by the dwell time. As such, the step size determines only the granularity of the data. While the diffraction patterns in the 3D graph above have different number of points-- all those points have the same intensity.

In the case of continuous scans, the step size, Δ2θ, has no effect on the scan time as it is determined by the total angular range and the slew rate in °/min. In the case of step scans, each step takes requires some "travel time" to accelerate and decelerate to the acquisition position as well as the dwell time, and thus scan time in step mode is step size dependent.

As in the study of continuous scans, the larger step sizes prevent even the identification of all of the peak positions, while below a step size of 0.100° there is only a minor effect on the resolution of peak positions and FWHM's. However, since extra steps cost extra time in step scan mode, a step size, Δ2θ, that is appropriate for the diffracted peak widths desirable as it saves time. For example, in the case of the 0.002° Δ2θ scan above, the position of the quartz (212) peak was found to be 2θ = 67.702 ± 0.001°, while the 0.020° Δ2θ scan yielded 67.699 ± 0.003°. That is not much of a change in measured certainty for a 8X increase in acquisition time.

SRM 1976: Post Alignment Performance Test

After aligning the θ-2θ goniometer, the NIST SRM 1976-- a plate of sintered corundum (Al₂O₃)-- was used as a performance test. The sample was scanned in step mode from 20° to 80° 2θ with 0.02° steps and a dwell time of 1 second. A 1° divergence aperture with no divergence Søller slit and a 0.15° receiving aperture was used. The data was smoothed with a 19-point quartic Savitsky-Golay filter and then the peaks were identified and then fit using pseudo-Voigts. K_α2 was assumed present only in peaks at high enough 2θ that the K_α1-K_α2 doublet was resolved as a split or asymmetry in the peak. The shape factors were fixed for all peaks. While not an entirely justified assumption given the variation in peak shape with axial divergence through the scan region, it did provide robustness of fit in regions with peak overlaps. ICDD PDF 46-1212 was used as an initial guess for metric cell refinement. All peaks were used in the refinement. The zero offset was fit and angular weighting was used.

The results were quite excellent. a = 4.7578 ± 0.0004 Å and c = 12.9907 ± 0.0005 Å. Δ2θ = 0.0029° and Δd = 0.00016 Å. The figure of merit: F(13) = 267.8(17). The smallest machine step, δθ, is on the order of 0.002°, so a Δ2θ < 2δθ is as close as one can get! The zero-offset was -0.037° ± 0.003°. While the alignment test with the alignment slit showed the zero-line to be -0.001° off, it should be noted that it is possible to introduce a θ-offset with the ZBH and that the effects of sample transparency correlate with a zero-offset in the metric cell refinement.

It should be noted that precisely determining peak positions is essential to metric cell refinement. The same data processed simply by smoothing and automatic peak ID produces a Δ2θ = 0.0249°, Δ2d = 0.00055 Å and a figure of merit of F(17)= 40.1(17).

Continuous Scans: The Effect of Step Size

This graph is a θ-2θ coupled scan of the "five finger" region of quartz which includes the (212), (203) and (301) peaks with 6%, 7% and 5% relative intensity respectively. This region is at high enough 2θ angle that the K_α1-K_α2 doublet is clearly resolved. The K_α2 (212) and (301) peaks are clearly visible. The K_α2 (203) peak overlaps with the K_α1 (301) peak. Thus-- five fingers.

This data was taken with a 1° divergence aperture and a 0.15° receiving aperture. The scan was configured in continuous scan mode in which there is a constant slew rate. The slew rate for this data set was 1°/min. As such, all of these scans took ~ 3 minutes to complete.

The effect of Δ2θ, the step size, was studied. The step sizes are: 0.200° (light green, top), 0.150°, 0.100°, 0.050°, 0.020°, 0.004° and 0.002° (dark green, bottom) respectively. In all continuous scans the reported intensity is integrated over a small range of 2θ. Because of this integration over a finite range of 2θ, the large Δ2θ scans have larger net intensity than the smaller Δ2θ scans because the intensity is integrated over a larger angular range-- Δ2θ.

While the 0.200°, 0.150° and 0.100° step sizes provide plenty of intensity, they also fail to define even the number of peaks in this region as the peak FWHM is on the order of ~ 0.150° given the divergence and receiving apertures used in this study. It is of interest that below a step size of 0.100° there is little effect on the peak widths and positions. Fitting to pseudo-Voigts with the peak FWHM's and shape parameters linked, there is only a variation of 0.004° in the peak width and a similar variation in the peak positions.

The bottom line: when running continuous scans, the step size needs to be chosen according to the peak FWHM for effective peak position determination-- but keep in mind that step size impacts net intensity.

Zero Background Holder

This image shows a zero background holder (ZBH). It consists of a single crystal of quartz that is cut and polished in an orientation such that it produces no diffraction peaks. For example, a metallic sample holder would produce diffraction peaks from the polycrystalline metal, and a glass slide would produce an amorphous background. The only background produced by the ZBH is that due to the choice of the divergence aperture.

The ZBH crystal is mounted in an aluminum frame. The dark region is where dislocations have been produced in the quartz by the X-rays. It shows the general region exposed to X-rays in the most common usage: a 1° divergence aperture with 2θ > 5°.

The ZBH is prone to certain types of damage. If the ZBH is scratched or marred, then there will no longer be zero diffraction background as diffraction peaks will arise from defects in the quartz. Also, nanoparticle materials can collect in the polishing marks on the ZBH surface resulting in a change in the ZBH background.

The graph shows the background from ZBH #2 using a 1° divergence aperture (with no divergence Søller slit) and a 0.15° receiving aperture with the diffracted beam monochromator. There is a little low angle background as expected for this set of apertures, but no presence of crystalline peaks through the region scanned.

Choice of Divergence Apertures

This is data taken from a zero background holder (ZBH) with no sample using various divergence apertures (with no divergence Søller slit) and a receiving aperture of 0.15°. The vertical scale is logarithmic, showing that at angles around 2θ = 5° the background can vary over four orders of magnitude depending upon the choice of divergence apertures. The reason for this is that at grazing incidence, larger divergence apertures allow more diffuse incoherent scattering and even direct incident beam into the receiving aperture.

The choice of divergence apertures depends largely upon the lowest d-spacing one needs to scan. For example, using Cu K_α, 2θ ~ 1° corresponds to a d-spacing of ~ 88 Å. There are very few crystalline systems in the PDF with a unit cell parameter in this range. There is a primitive hexagonal form of CdI₂; (PDF 87-0300) with c = 198.19 Å and the (001) peak at 2θ = 0.445°-- but such systems are rare. The protein tubulin has a (100) peak at 0.273°, but specialized low-angle systems are best suited to protein and polymer systems with very large d-spacings. These are available at the Florida State University in the Institute of Molecular Biophysics.

Using Cu K_α, 2θ ~ 4° corresponds to a d-spacing of ~ 22 Å. There are quite a few palmitate, stearate, and laurate systems with the longest d-spacing peak in this area. One could use either a 0.1° or 0.3° divergence aperture with these systems to achieve workable backgrounds. One of the largest d-spacing clay minerals, montmorillonite, has its longest d-spacing in the range of 13.5-15.0 Å for a range of 2θ angles of 5.8-6.2°. For such a system either a 0.3° or 1.0° divergence aperture is appropriate for workable backgrounds.

Kiessig Fringes

Even though the Siemens D500 is not well suited to very low angle work, with proper alignment beforehand much can be done.

In this graph a thin polymer film is scanned in coupled θ-2θ mode below 2θ = 1.500°. Very good aperture alignment is required so that the entire sample surface is exposed to X-rays with the smallest divergence aperture-- 0.1°. The sample is mounted to eliminate any θ-offset as well as any tilt in the axial direction.

Because of constructive and destructive interference between scattering from the film surface and the film-substrate interface, interference fringes called Kiessig fringes appear. This is an interference effect, not a diffraction effect, and thus Kiessig fringes can be seen from single layers as well as multilayers. Modeling of the reflectometry data can be used to determine film thickness as well as surface and interface roughness and electron density. As the interface and surface roughen, the fringes broaden and disappear as the interference between scattering from the surface and interface loses coherence.

Diffracted Beam Monochromators & Focusing Circles

This image (from Jenkins & Snyder's X-ray Powder Diffractometry) shows a schematic of a powder diffractometer with Bragg-Brentano parafocusing geometry. In such a geometry, the X-ray source (F), the sample (S) and the receiving aperture (RS) all fall on a circle of radius rf. The divergence aperture (DS) and divergence Søller slit (SS1) illuminate the sample with a diverging beam of X-rays; these X-rays are subsequently focused back to the receiving aperture on the focusing circle. The focusing circle radius, rf, is not constant. It is a function of the Bragg angle, θ, and the diffractometer radius, R: rf = R/(2sinθ). This is the most common diffractometer geometry.

The diffraction pattern shows data taken from quartz with a 1° divergence aperture and a 0.15° receiving aperture. The red curve is taken without the diffracted beam monochromator. The standard for quartz is below the diffraction patterns. Two things are immediately evident: the increased background and the presence of extra peaks due to the presence of Cu K_β radiation despite the initial filtering of the incident beam with Ni foil. The background is due to Bremstrahlung from the X-ray source as well as X-rays fluoresced in the sample itself.

The purpose of the X-ray monochromator is to remove these Cu K_β spectral artifacts and to reduce background. This is accomplished using another focusing circle of radius rm that contains the receiving aperture (RS), a crystal of HOPG (C) and the detector aperture (AS). The angle of incidence on the crystal, C, is set to the Bragg angle of Cu K_α for HOPG and fixed. The detector aperture, AS, subsequently functions to only pass Cu K_α radiation while rejecting Cu K_β and continuum radiation outside of the Cu K_α region. The blue data shows the same sample scanned with the same apertures and scan parameters using the diffracted beam monochromator. There are no diffraction peaks from Cu K_β, and the background is flat with an intensity of only a few Hz.

Diffracted Intensity: The Effect of Different Options

In powder diffractometry, different attachments increase performance-- but at the cost of intensity. The graphite diffracted beam monochromator removes background from fluorescence in the sample as well as artifacts introduced due to incomplete Ni filtering of the incident beam and spectral contamination of the X-ray tube. This spectral contamination is generally in the form of W and Fe from the tube electron source. The divergence Søller slits remove the effects of axial divergence. A good question is: at what cost to diffracted intensity?

In this figure the quartz (101) peak is scanned with a 1° divergence aperture and a 0.15° receiving aperture with and without different attachments. In summary:

• Yellow Curve: no diffracted mean monochromator, no divergence Søller slit
• Green Curve: no diffracted beam monochromator; divergence Søller slit
• Red Curve: diffracted beam monochromator; no divergence Søller slit
• Blue Curve: diffracted beam monochromator; divergence Søller slit

Without the monochromator and divergence Søller slit, the net diffracted intensity of the quartz (101) peak is ~ 145.1 kHz. This is at a loss of resolution, definition of peak shape and an increase in background. The background at 2θ = 27.600° is ~ 4.6 kHz. The background would be even larger for transition metal and rare earth samples which have a higher fluorescence yield than the Si and O in quartz. Adding the divergence Søller slit drops the net intensity to ~ 45.7 kHz. That is a reduction of ~ 3 X. The background is then ~ 1.3 kHz-- a reduction of ~ 3.5 X.

The "standard" configuration includes the diffracted beam monochromator but not the divergence Søller slit. The quartz (101) net intensity is shown to be ~ 29.8 kHz with a background of 49 Hz. Adding the divergence Soller slit drops the net intensity to ~ 7.3 kHz. That is a reduction of 4 X. The background is then ~ 17 Hz-- a reduction of ~ 3 X.

This gives us some general rules:

• Using the divergence Søller slit reduces the net diffracted intensity ~ 3-4 X
• Using the diffracted beam monochromator reduces the net diffracted intensity ~ 5-6 X
• Using both reduces the net diffracted intensity ~ 20 X

In applications where maximum diffracted intensity is required, e.g. phase ID of trace phases, and where spectral contamination and background is not a concern-- then it is useful to not install the divergence Søller slit and to remove the diffracted beam monochromator.

Axial Divergence

Axial divergence is one of the most important systematic aberrations in powder X-ray diffraction. While the divergence apertures and their corresponding anti-scatter apertures precisely define the angular divergence of the incidence beam in the plane of diffraction, the X-ray beam can also diverge in the perpendicular direction along the axis of the goniometer-- thus the term axial divergence.

In the top picture two rays are shown. One of them passes through the center of the sample plane. Another ray set at the same divergence angle hits the sample at its edge. Because of axial divergence its actual Bragg angle is smaller.

This is remedied by placing a Søller slit between the divergence aperture and its corresponding anti-scatter aperture. A Søller slit is a collimating optic consisting of parallel foils separated by spacers. This collimator limits the extent of the axial divergence. This reduction of axial divergence error is at the expense of transmitted intensity as the foils in the Søller slit do block some of the transmitted intensity. Not show is a second Søller slit that is placed between the receiving anti-scatter aperture and the receiving aperture.

The axial divergence error in peak position has the form: Δ2θ = -h²*(K1*cot2θ + K2*csc2θ)/3R². Here h is the axial width of the sample, R is the radius of the goniometer circle (501 mm for the D500 goniometer), and both K1 and K2 are constants determined by the two Søller slits (divergence and receiving). Notice that the axial divergence can change signs, but for small 2θ it is always negative.











One of the things that characterizes axial divergence is an asymmetry of the peaks that gives them a low-2θ tail. In the next two figures the quartz (101) peak is scanned with a 1° divergence aperture and 0.15° receiving aperture with and without the divergence Søller slit.

In the left picture the data reflects the loss of intensity caused by the divergence Søller slit. Without it the net intensity of the quartz (101) peak is ~ 29.8 kHz, which is reduced to 7.3 kHz with the Søller slit. That is a reduction of 4 x. Because most users are trying to do phase identification, we generally remove the divergence Søller slit for maximum diffracted intensity.

The right figure shows the same data scaled to equal net intensities. The asymmetry of the peak without the Søller slit is visible. Because of this asymmetry split Pearson-VI or split pseudo-Voigts are the best modeling line shapes. The small error in peak position is also evident. The data without the Søller slit yields a peak position of 2θ = 26.612° (0.001°): with the collimator it increases slightly to 2θ = 26.618° (0.001°). While this is a Δ2θ of only - 0.006°, this effect would be larger at lower angles.

Diffractometer Alignment

Aligning a powder diffractometer is simple if one understands the essential principles.

The most important part is establishing the zero-line. To accomplish this one installs an alignment slit in the position of the sample. This is a narrow channel etched between two pieces of glass which provides a high aspect ratio-- and thus collimating-- X-ray guide. The θ and 2θ angles are alternately and independently adjusted until the intensity passing through this alignment slit is a maximum. Ideally that would be at θ = 0.000° and 2θ = 0.000°, but in practice that is generally not true. If the 2θ for maximum intensity is not exactly 0.000°, then this is a sign that the zero-line does not pass through the center of the goniometer. The tube stand is moved up/down until the zero-line passes through the axis of the goniometer as indicated by the maximum intensity appearing at 2θ = 0.000°. While unrelated to the zero-line, at this time the tube stand can also be rocked sideways to maximize intensity. This aligns the line shaped X-ray source from the tube with the slit shape of the alignment slit.

After the adjustment of the zero-line, one mechanically adjusts the sample stage so that the intensity through the alignment slit is a maximum at exactly θ = 0.000 °. At this time the height of the sample stage is checked. For proper operation the sample plane must fall on the axis of the goniometer. The way this is checked is to independently adjust the θ and 2θ angles to maximize the intensity through the alignment slit. One makes special note of the 2θ angle. Then one flips the sample over by adding 180° to the θ angle and repeats this optimization of intensity through the alignment slit by adjusting the θ and 2θ angles. One again makes special note of the 2θ angle. If the difference between the 2θ angles is greater than zero, then the sample surface is above the goniometer axis by 9 µm per 0.010° of difference. In the current alignment it was determined that the sample plane is about 25.0 µm above the axis of the goniometer.

After this one addresses the alignment of apertures. With the θ and 2θ angles adjusted to maximize the intensity through the alignment slit, one systematically reduces the divergence and anti-scatter apertures. If a reduction of intensity through the alignment slit is noticed, then the height and/or the tilt of the divergence and receiving aperture assemblies are adjusted. In the current alignment with a 0.1° divergence aperture and a 0.018° receiving aperture, the various apertures were adjusted to pass an intensity of 156.2 kHz through the alignment aperture.

Finally one performs a θ-2θ coupled scan across the alignment aperture using the smallest divergence aperture-- 0.1° -- and the smallest receiving aperture-- 0.018°. The graph shows the results of that scan. These three peaks are fit to pseudo-Voigts with the assumption that no K_α2 component is present. The centroid of the large central peak indicates the calibration of the zero-line: in the current alignment it is - 0.001° off. This is acceptable given that the smallest 2θ step is 0.002° indicating the zero-line is dead-on. The ratio of the intensity of the two side peaks gives a measure of the 2:1 ratio-- the coupling ratio between θ and 2θ. In the current calibration they differ by 14 %, where a difference of > 30 % is considered problematic. Finally the intensity ratio between the central peak and the largest side peak should be about 3 X and is a measure of the alignment of apertures. In the current alignment the ratio is ~ 3.7 indicating good aperture alignment.

Thus, in the current alignment all aspects of the goniometer performance are adjusted and found to be acceptable.

Metric Cell Refinement

In metric cell refinement, an initial guess to the lattice parameters of a powder diffraction pattern of a single phase is used to refine those parameters to those that best define the pattern. The initial guess is generally a PDF standard. Data for metric cell refinement is generally corrected using an internal standard, though an external standard can be used-- or no standard at all.

In the data shown, Al₂O₃ was dusted onto a greased zero background holder through a 200 mesh sieve. No internal standard was used. The background was subtracted, the data was smoothed and peak locations identified. PDF standard 10-0173 was used as the starting point of the refinement.

The lattice parameters were refined to a = 4.7583 ± 0.0012 Å & c = 12.9864 ± 0.0029 Å. The average error in 2θ between the experimental data and refined model data was 0.006°. It should be noted that 0.020° steps were used in data acquisition and the smallest machine step of the goniometer is 0.002°. The average error in 2d spacing between the experimental data and refined model data was 0.0003 Å.

Metric cell refinement is particularly useful in the characterization of new phases and the study of doping on the lattice parameters of a system. In geological samples the refinement of the calcite lattice parameters can be used to infer the concentration of Mg due to Mg-substitutions.