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".

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 Debye-Scherrer Camera provides a visual aid for understanding the X-ray diffraction powder method.

The essential features of the powder method includes a narrow beam of monochromatic X-ray radiation impinging upon a randomly oriented powder where all possible crystallographic planes are 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 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.

It essentially requires that distance from the source to the sample and the sample to the detector (i.e., R1 = R2) and that 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. Principle steps all rely on the fact that the sample is always equa-distant between the source and detector. (e.g., R-incident = R-reflected)

a. Axial alignment

b. Set take-off angle (6)

 

c. Zero alignment

 

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. Modern diffractometers radii range from 180 to 250 cm. Notice that for a 1 slit that a length of 2.5 cm is reached at 13 2 θ. 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 sample holder. At higher 2 θ angles less area 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 Kα1 radiation Early control choices of the operator are Kv and mA settings for the generator. Higher settings to increase peak (and background) intensity and counting statistics can be weighed against possible jeopardy to the tube.

2. Flat specimen error

 

3. Axial divergence

4. Penetration of the beam into the sample. (Sample transparency). The more intense the incident radiation, the farther the beam penetrates into the sample.

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.

Detector deadtime

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 deadtime. 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 quantitation. Perform the measurement a second time at half the tube current. If the second measurement is more than half the first then detector deadtime 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 chopper increment. For a chopper 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 +5%. 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 +4%.

 

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 Materials.

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 as 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 () 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
  • Incoherent (Compton) scattering which increases as when light elements are present
  • 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 are small.
  • Temperature diffuse scattering is general small unless soft materials are involved.
  • Peak fitting procedures typically need 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 all 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.

    It is possible to calculate the diffraction pattern (i.e., interference pattern) for any given crystal structure given:

    Under ideal conditions all the diffraction takes place at the Bragg angles of reflection.

    Diffraction effects (in one dimension) due to scattering from a grating can be described by an interference function:

    where:

  • D = separation of lines (e.g., d001)
  • N = the total number coherent scattering domains) Note: as N gets smaller, so does the breadth of line.
  • N x D is crystallite size
  • If N = 1 then F = 1 at all angles. Bragg reflection cannot occur from a single scattering center.
  • If N = 100, then F at the ideal Bragg angle is large. At the same time F is small away from the Bragg angle. In other words the peaks are very intense and narrow.
  • This relationship is shown graphically in the figure below.

    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 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.

    a. 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).

    b. Reduce the particle size of the material to about 5 to 10 m. This also minimized primary and secondary extinction effects. Using a wet grinding method reduces defects.

    c. 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.

    d. Slurry mounting with acetone can minimized preferred orientation. Water is a polar compound, where as acetone is less polar and it evaporates quickly. If particles are kept away from each other while drying, then van der waal forces will not let them to 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 (m) required for attenuation of a CuKa beam to 0.01 and 0.001 times incident intensity as a function of 2θ.

     

    Factors that influence the transparency of a specimen include:

    1. 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.
    2. 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.
    3. Porosity of the sample. This is minimized by good sample packing. If the sample is porous and thin, then the higher order (angle) reflections will be compromised.

    Data Processing and Interpretation

    Kα2 stripping.

    At low angles of 2θ, Kα1 and Kα2 peaks are closely overlapped. There are computer programs that will mathematically remove the Kα2 peak component. They use relationships between Kα1 and Kα2 radiation (1.54051 and 1.54433, respectively) via Braggs law and the assumption that the Kα1 peak intensity is double the Kα2 peak. The net intensity of stripped data is therefore one-third the observed intensity.


    Here's a list of commonly used radiation source wavelengths. Here is a list of all energies.


    Peak finding.

    There are various degrees of sophistication for determining peak properties. In increasing order of time and effort are as follows:

    1. Graphically picking peak intensity and positions with a cursor on the computer screen or print out the peak and use a ruler.

    2. Second-derivative peak finding routine.  The first derivative of a peak gives the peak position and the second derivative gives the peak width.

    3. Profile fitting. There are several models that can be used to fit peak data. A good reference is by Howard, S.A. and K.D. Preston (1989) Profile fitting of powder diffraction patterns. in Modern Powder Diffraction eds. D.L. Bish and J.E. Post, Reviews in Mineralogy, v. 20, p. 215-275.

    4. Rietveld refinement. This is a full pattern approach (beyond the scope of these course notes).  See Post, J.E.  and D.L. Bish (1989) Rietveld refinement of crystal structures using powder X-ray diffraction data. in Modern Powder Diffraction eds. D.L. Bish and J.E. Post, Reviews in Mineralogy, v. 20, p. 277-308.

     

    There is a tacit assumption often made by peak finding programs, which the method to determine peak shapes, for example  that the shape of the peak is Gaussian. This is not always the case, as noted above by the non-symmetrical instrument distortions that result from factors such as flat specimen, transparency, and displacement errors.  There are a variety of peak shape models to choose from that one can use to peak fit. Here's a table of commonly used functions.

    Your choice of peak fitting method depends on the use of your fitted parameters. Ask yourself any of the questions below.

    1. Are you doing a quick search and match to roughly identfy something you have no idea of what it is?
    2. Are you indexing a known mineral?
    3. Are you refining the lattice parameters of a mineral?
    4. Are you conducting a quantitative analysis?
    5. Are looking for solid-solution analysis?
    6. Are you evaluating order/disorder?
    7. Are you evaluating crystallinity using an index (e.g., ScherrerHinckely, Kubler, Opal)?
    8. Are you
    evaluating lattice strain (isotopic or anisotropic)?

    Calibration/correction.

    NIST SRM640b Silicon metal, SRM675 synthetic fluoro-phlogopite and more...

    Δ2θ = practical d-value - observed d-value

    Correction to the sample peak position is by simple linear interpolation.

    corrected = 2θ observed + ( Δ2θ left side - Δ2θ right side)


    Qualitative analysis - phase identification

  • Assertion - If you can enumerate the steps used to solve a problem, then a computer can do it for you.
  • Conversely - No computer can solve a problem for which an algorithm cannot be written.
  • Corollary - Don't expect a computer algorithm to produce results based information you do not supply to it!
  • What do you do once you have the d's and I's ?


    Search/match procedures

    Introduction to the Joint Committee Powder Diffraction Standards data base.

    Now called the International Centre for Diffraction Data - Powder Diffraction Files (PDF)

    Diffraction data is also now available from the Mineralogical Society of America Crystal Structure database


    Preconceived notions and the alphabetical listing

    Each phase in the data base is listed by chemical name and permutations of the name:

    Chemical Name: /mineral chemical formula Strongest 2nd 3rd PDF #

    iron carbonate:/ Siderite FeCO3 2.80x 1.734 1.743 29-696

    carbonate iron:/ Siderite FeCO3 2.80x 1.734 1.743 29-696

    What if the sample is a mixture?

    Example: Beach sand, you think it is composed of quartz.

    Observed Card #33-1161 Residual Card #5-628

    d Irel            d Irel             +Δd         d Irel        +Δd

    4.26 30           4.26 35           0.000

    3.345 100       3.343 100       0.002

    3.260 5           3.260 5 (11)                  3.258 13           0.002

    2.822 45         2.822 45 (100)              2.821 100         0.001

    2.459 12         2.458 12         0.001

    2.283 14         2.282 12         0.001     2.283 2(4)

    There are no rules when it comes to identifying an unknown! Use any and all information you can gather. Don't be to proud to ask the person who gave you the sample some questions about its origin. Remember to use some of the simple tests, such as dilute (5% HCl) acid fizzing, streak plates, specific gravity, magnetic properties, or hardness.  Independent methods such as XRF, ICP, EDS elemental analyses, FTIR, Raman, and thermal TGA or DSC data can always help constrain the identification process.

    In the case of beach sand above, there is a residual. Common sense tells you that shells are often found on the beach, so therefore why not go directly to the card for calcite or aragonite and consider them as a possibility? The residual turns out to be calcite.

    XRD


    Hanawalt Search method

    Hanawalt "grouping" is based upon the d-spacings of the strongest lines in the pattern.

    The lines are arranged as follows:

    where, A, B, C...H are the eight strongest lines in descending order.

    1. All patterns appear once with the first and second strongest lines in the following order: 2. The pattern will be listed twice if ratio of second to first strongest line (I2/I1) is in the range  > 0.75 to < 0.75 in the following order: 3. The pattern will be listed a third time if ratio of third to first strongest line (I3/I1) is   > 0.75 and  (I4/I1)< 0.75 in the following order:

    4. The pattern will be listed a forth time if ratio of forth to first strongest line (I4/I1) is   > 0.75 in the following order:

    The entire file is then arranged into 40 groups based upon ranges of d().

    999.99 - 10.00
    9.99 - 8.00
    7.99 - 6.00
    .
    .
    1.39 - 1.00
    1. Begin the search by arranging the observed d's from strongest ----> weakest.
    2. Select the strongest unused line. (example is quartz - 3.34)
    3. Proceed to the main group. The patterns are arranged into subgroups based upon descending d-values of the either the second or third strongest lines.
    4. If a match is suspected, then the other eight d-values and intensities area checked to see if they are present in the observed pattern.
    5. Note: There are errors ranges associated with each group. Peaks with d-values outside the range of the main group can be found within the group. This allows for errors from sample sensitive and specimen sensitive variations.
    6. If a match is likely, proceed to the actual PDF card file.
    7. Subtract the pattern and look for residual peaks.


    Fink Search method

    Fink "grouping" is based upon the eight strongest lines in the pattern.

    Works better than Hanawalt method when;

    There are four entries for each pattern, each starting with one of the four most intense lines.

    The remaining seven lines in each entry are arranged in descending order of d-value.

    Example: Lines ordered in terms of descending d-values (bold four most intense).

    d9 d3 d6 dx d7 d5 d8 d4

    The permutations then become:

    dx d7 d5 d8 d4d9 d3 d6
    d7 d5 d8 d4d9 d3 d6 dx
    d8 d4d9 d3 d6 dx d7 d5

     

    Like the Hanawalt system the entire file is then arranged into approximately equal size groups based upon ranges of d().

    999.99 - 10.00
    9.99 - 8.00
    7.99 - 6.00
    .
    .
    1.39 - 1.00

    Within each group entries are listed in descending order of the second d-value.

    1. Begin the search by arranging the observed d's from largest ----> smallest.
    2. Select the unused line with the largest d-spacing (one of the four strongest) (example is quartz 4.27)
    3. Proceed to the main group. The patterns are arranged into subgroups based upon descending d-values of the either the second d-value.
    4. If a match is suspected, then the other eight d-values and intensities area checked to see if they are present in the observed pattern.
    5. Note; errors ranges associated with each group. Peaks with d-values outside the range of the main group can be found within the group. This allows for errors from sample sensitive and specimen sensitive variations.
    6. If a match is likely, proceed to the actual PDF card file.
    7. Subtract the pattern and look for residual peaks.

    Criteria for a match.

    The tolerance for a possible window in d is related to the following factors:

    +Δd = precision + accuracy + chemical purity + order/disorder
     

    Accuracy is achieved by the appropriate calibrations and corrections (i.e., internal standard corrections).

    If you expect additional variations due to solid solution substitutions (i.e., a change in fn and fx, fy, fz).

    Error windows for d-values:

       
     102θ/minute
     12θ/minute
     d
     2θ(CuKa)
     +Δd
     +Δ2θ
     +Δd
      +Δ2θ
    20.0
    4.4 0.080
    0.1
    0.020
    0.04
    5.0  17.7  0.030  0.1  0.010  0.04
     3.0  29.8  0.010  0.1  0.003  0.04
     1.5  61.8  0.003  0.1  0.001  0.04

    Chen, P.Y., 1977, Table of key lines in X-ray powder diffraction patterns of minerals in clays and associated rocks: Geological Survey Occasional Paper 21, Indiana Geological Survey Report 21, 67 p.