The diffractogram or diffraction pattern

The endproduct resulting from the interaction between monochromatic X-rays and crystalline material is a "diffractogram" or "diffraction pattern". It is incorrect to refer to a pattern as a spectrum. A spectrum infer a range of frequency domains or wavelengths, while recall that a diffractogram is generated using a single wavelength. The figures below are diffractograms showing the regions in angular space where constructive interference occurs as a result of Cu K_α radiation interacting with the atoms in the crystal lattice of calcite (CaCO₃). The x-axis shows the pattern is measured in 2θ and the y-axis is relative intensity of the diffracted beam. The figure below is the same diffractogram overlain with red lines marking the position of relative intensity of each peak. The positions are marked by the d-spacings (measured in Å) for the crystallographic planes responsible for the coherent scattering. Calcite's crystal symmetry is defined in the hexagonal class, where its axes are given by the unit lengths of  a = 4.99 Å, b = 4.99 Å, c = 17.06 Å and angles α = 90°, β = 90° and γ = 120°. The integers next to each "reflection" are the respective Miller indices (hkl) for the family of reflecting planes.


Each pattern can be represented by series of d-spacings (d's) and intensities (I's).  Ultimately the interpretation of XRD data requires an understanding of all the factors that can affect d's and I's.

This diagram illustrates the (-114) plane of the calcite structure.

Factors that affect d's and I's.

XRD data are sensitive to as many as 35 different factors. These factors can be grouped in to three general sources of "error". The word error is used here in the context of what causes deviations from theory. Sources of error include:

1. Instrument sensitive errors

2. Sample sensitive errors

3. Specimen sensitive errors

Theoretical d-value  (nλ = 2d sinθ)

vs.

Practical d-value (theoretical  + inherent aberrations)

vs.

Experimental d-value (practical + inherent sample aberrations + errors)

Geometry of the powder diffractometer

            The essential features of the powder method includes a narrow beam of monochromatic X-ray radiation impinging upon a randomly oriented powder, which has all possible crystallographic planes available for Bragg reflection.

Geometrical Principles of a Bragg-Brentano Parafocusing Diffractometer

1. X-rays diverge from source.

2. The "reflected" X-rays from the samples on the focusing circle are directed to their respective places back on the focusing circle.

3. The spots labeled G1, G2, G3 are the respective reflections of d-spacings d1, d2, d3.

 

The Bragg-Bentono geometry allows for a constant distance between the sample and the detector.

The geometry requires that the distance from the source to the sample and the sample to the detector  be equal (i.e., R1 = R2) and the sample is kept on the tangent of the focusing circle.

In order to keep the detector distance constant, the sample must rotate at 1/2 the angular velocity of the detector. As the angle of incidence (θ) changes, the detector must move 2θ . This is called 2:1 motion.

Instrument alignment.  All alignment steps rely on the fact that the sample is always equa-distant between the source and detector. (i.e., R1-incident = R2-reflected)

a. Axial alignment: The beam path must make a straight line from the source (F) to sample (S) to receiving slit (G) at the detector.

• F = source
• s1 = soller slit
• X = divergent or primary slit
• Y = primary scatter slit
• S = sample
• M = receiving scatter slit
• s2 = soller slit 2
• G = receiving slit.

2. Take-off angle (typically set to 6°). The effective size of the X-ray beam is determined by the shadow of the filament and the angle of view (i.e. take-off angle).  Manufacturers make different types of tubes with different focal breadths. These include broad focus (2 mm), normal focus (1 mm), and fine focus (0.5 mm).  Depending upon the take off angle (α) the resultant beam width will vary.

c. Zero alignment: The source, the sample, and the detector must all be perfectly in line at 0° θ.

d. Alignment of slits. The divergence slit limits the total irradiation area of the sample. The aperture of this slit hardly affects relative peak intensities if the slit is fixed and the specimen completely intercepts the X-ray beam. Sample mounts are typically hold material in 2.5 cm wide by 2.5 cm long holder.

 

            The figure above shows both the theoretical and measured the relationship between sample irradiation length versus 2θ for several different divergence slits. The raduis of modern diffractometers ranges from 180 to 250 cm. Notice that for a 1° slit that a length of 2.5 cm is reached at 13° 2θ. At lower angles the sample intercepts only part of the beam resulting in reduced beam intensity per unit area of sample surface and increases the likelihood of background scatter from the edges of the sample holder. At higher 2θ angles less area is irradiated, which would have an effect of decreasing diffraction intensity. The depth of penetration of the beam becomes commensurably deeper with higher angles. This effectively increases background as well as a sample displacement effect (see below). Theta compensating divergence slits on some diffractometers are designed to lessen high-angle intensity loss and displacement effects and low-angle background scattering. One must always recall however, that the standard form of reporting relative intensity is with fixed-slit conditions and that variable slit data must be appropriately corrected.

Five instrumental functions

1.  X-ray source - Gaussian distribution for K_α1 radiation. Control choices for the operator (you) are kV and mA settings of the generator. Higher settings to increase peak (and background) intensity and counting statistics can be weighed against possible jeopardy to the tube. The focal spot can be approximated by a Gaussian distribution function.

w = the half maximum breadth of the focal spot, and ε is the angular deviation of any point from the theoretical scattering angle 2θ.  Using the graph above, you can see that for a take-off angle of 6° and a fine-focus filament w = 0.021 mm.

An addition nuance, regarding beam voltage and current involves that fact that the tube filament may change location with different settings. The position of the filament at low voltage and current may not be the same at high voltage and current.  This can have consequences for alignment and other procedures that depend on accurate and precise intensity values from the tube.

2. Flat specimen error

Note that the diverging path lengths of the above beam are both longer and shorter than the center bram path (i.e., as it spreads out over the sample). Ideally, we would like the sample to sit as curved surface on the focusing circle.  It is impracticle to create such a curved surface (it's easy and more reproducible to make a flat surface), so we accept a small amount of distortion of the signal due to flat specimen that sits tangential to the circle.

3. Axial divergence

There is additional opportunity for the beam to diverge in the horizontal plane. The above figure shows that beam path can only be longer than the ideal centered beam. Closely spaced foil films oriented parallel to the center beam absorb the more widely diverging beam paths. These are call soller slits. Each instrument manufactuer has a varient for how to optimally minimize this affect.  Here is one example.

4. Penetration of the beam into the sample. (Sample transparency). The more intense the incident radiation, the farther the beam penetrates into the sample. This creates a situation similar to flat specimen error as well as penetration into the substrate holding the sample. In the latter case, diffraction from the substrate (off the focusing circle) will contribute to the pattern.

5. Receiving slit - Increasing the width of the receiving slit generally increases the peak height and width and decreases the ability to resolve peaks.

The net affect of instrument line profile modifications is to broaden and displace the theoretical line position to a lower theta angle. Kα₂ radiation will displace observed lines to higher angles.

6.  A sixth function can be considered as general misalignment. The figure below is reproduced from Klug and Alexander (1974). It shows the six functions and their weighting. Each is convoluted to form the final line profile. The sum of all sources of error results in the low angle bias to the peak position.  This is distortion can be measured using reference standards.

Detector dead time  

            When the measured count rate is not directly proportional to the photon rate entering the detector, the detector response is non-linear and said to have dead time. The effect is to increase the relative intensity of the weaker peaks. Most modern detectors (with calibration and computer correction) handle count rates up to 100,000 counts per second (cps). A quick check to determine if the detector not responding in a linear fashion is to measure the intensity of the strongest line with the tube current set for the normal quantification. Perform the measurement a second time at half the tube current. If the second measurement is more than half the first then detector dead time is the probable cause. In this latter case the detector correction routing must be recalibrated or the observed data must be corrected accordingly.

Counting Statistics 

 The precision of intensity data can be limited more by counting statistics than any other single parameter except preferred orientation. The figure below illustrates how the probable error of peak intensity measurements can vary with both the total number of counts and the ratio of the peak to background counting rate (R).

 Both the total number of counts and R depend upon scan rate and chop or step increment.  For a chop increment of 0.01° and scan rate of 2° 2θ min^-1, the count time per increment is only 0.3 s. If the counting rate at the peak top is 1000 cps, then the total number of counts is 300. If R is large, then the best precision attainable is about +4%. If R is 1.5, then probable error exceeds +10%. By decreasing the scan rate of 0.2° 2θ min^-1, probable error drops to +5%.

 

 Further improvement in peak intensity precision is possible by smoothing over the peak top in small increments and also averaging background over hundreds of increments on both sides of a peak.

Sample displacement 

            Displacement of the sample off the diffractometer focusing circle can be brought about in three ways. Firstly, there is instrumental misalignment. Often neglected is the tacit assumption that the goniometer is properly aligned before any experiment is run. This can be easily maintained with a proper alignment, using manufacturer provided tools and checking with a standard material such as the U.S. National Institute of Standards and Technology Reference Material 640b.

            If the sample holder is properly aligned, then the second potential source of displacement error comes from way the sample itself is packed into the sample holder (see next section).The figure below shows the changes in measured d-spacings for a series of reflections as a function of displacement from the goniometer focal plane.

Displacement error increases rapidly as 2θ falls below 20°  (see in the figure below).

Equally important is the decrease in diffraction intensity with increasing displacement. Finally, effective displacement or sample transparency can be a consequence of a low mass absorption coefficient (µ) or high sample porosity (see next section).

Background

Background can be produced from a number of sources. These include:

1. Fluorescent radiation emitted by the specimen
2. Diffraction of a continuous spectrum of wavelengths
3. Diffraction scattering from materials other than the specimen including soller slits, specimen binder, sample mount, and air.
4. Diffuse scattering from the specimen itself, including:          

    a. Incoherent (Compton) scattering, which increases when light elements are present.
    b. Coherent scattering:

• Diffuse scattering due to various crystallite imperfections. Extremely short-range ordered material, like glass, cause intense diffuse scatter.
• Low intensity maxima contribute to background when the number of unit repeats normal to the diffracting plane is small.
• Temperature (diffuse scattering is general small unless soft materials are involved)

Peak fitting procedures typically are needed to consider removal of background. Without correction for the background much of the trace is fit with nonsensical peaks that do not provide a unique solution Most XRD manufacturers provide peak fitting software to allow this procedure.

 

 

The figure above demonstrates a single peak fitting approach. The form of the fit is: 

Intensity = baseline + Kα₁ Gaussian (peak ht. and width) + Kα₂ Gaussian (peak ht. and width).

Sample Preparation 

           Specimen sensitive errors are most commonly introduced while preparing the sample for presentation to the X-ray beam. Itemized below are the most profound parameters related to preparation and mounting of powder.

1. Coherent scattering domain.

Excessive grinding of a sample, during preparation can induce defects in the crystal structure and reduce coherent scattering intensity. 

2. Preferred orientation

The XRD powder method relies on the principle that all possible crystallographic orientations are presented to the beam. This concept is known a random orientation. If there is a bias of orientations of one or more particular crystallographic plane, then this is known a preferred orientation. Preferred orientation is likely the most common cause of intensity variations in XRD powder experiments.

 Particles with perfect cleavage or acicular shapes, such as clay minerals, are the most prone to preferred orientation. Intensity variations up to 100% are possible. Here are some tips to minimize the effects of preferred orientation.

• Be consistent with sample mounting and packing methods.
• Sprinkling and backpacking may reduce preferred orientation.
• Apply a constant pressure (e.g. 200 psi for 20 s).

 

• Reduce the particle size of the material to the range of 2 to 10 µm.  This also minimizes primary and secondary extinction effects. Using a wet grinding method reduces defects.
• Mix the sample with 20% internal standard. Pick an internal standard that has equant particles and that does not interfere with the peaks of interest.
• Slurry mounting with acetone will minimized preferred orientation. Water is a polar compound, where as acetone is less polar. Acetone evaporates quickly.  If particles are kept away from each other while drying, then van der waal forces will not let them attract to each other and self align.  The pitfall of slurry mounting is that the sample may be too thin (see below).

3. Powder thickness and transparency    

            The thickness of the powder should be great enough to prevent the beam from passing through to the substrate below. The generally accepted reduction of the beam intensity is about 1/1000th of the initial beam intensity. This condition is termed "infinite thickness". 

Examples of powder thickness (µm) required for attenuation of a CuKα beam to 0.01 and 0.001 times incident intensity as a function of 2θ

Attenuation		factor			0.01					0.001
	Porosity			solid	0.9	0.8	0.7		solid	0.9	0.8	0.7
			4°	14.1	15.7	17.7	20.2		21.2	23.6	26.5	30.3
Gibbsite			20°	70.4	78.2	88.0	100.5		105.6	117.3	131.9	150.8
µ=56.8 cm-1			40°	138.6	154.0	173.3	198.0		207.9	231.0	259.9	297.0
µ*=24.2 cm2g-1			80°	261.0	289.0	326.0	372.0		391.0	434.0	488.0	559.0
			4°	8.3	9.3	10.4	11.9		12.5	13.9	15.6	17.8
Quartz			20°	41.4	46.0	51.8	59.2		62.2	69.1	77.7	88.8
µ=96.5 cm-1			40°	81.6	90.7	102.0	116.6		122.4	136.0	153.0	174.9
µ*=36.4 cm2g-1			80°	153.4	170.5	191.8	219.2		230.1	255.7	287.6	328.7
			4°	0.7	0.8	0.9	1.0		1.1	1.2	1.3	1.5
Hematite			20°	3.6	4.0	4.34	5.1		5.3	5.9	6.7	7.6
µ=1124.4 cm-1			40°	7.0	7.8	8.8	10.0		10.5	11.7	13.1	15.0
µ*=216.2 cm2g-1			80°	13.2	14.6	16.5	18.8		19.7	21.9	24.7	28.2

 

Factors that influence the transparency of a specimen include

• The mass absorption coefficient of the sample. The previous table gives requisite powder thickness for various materials with small, medium and large mass absorption coefficients and with different porosity. 
• Thickness of the sample. Sometimes you just don't have enough material to pack your holder. Using a "zero-background" plate minimizes scatter from the substrate. A common zero-background plate is a quartz crystal cut and polished 6° of the c-axis.
• Porosity of the sample. This is minimized via good sample packing. If the sample is porous and thin, then the higher order (angle) reflections will be compromised.