X-Ray Powder Diffractometry
- Introduction to Scattering Techniques
- Important Aspects of X-Ray Powder Diffraction (XRD)
- Principles of the XRD Technique
- Bragg’s Law
- Scherrer equation
- Rigaku Multiflex
- Experimental Procedure
- Sample Preparation
- Interpretation of an XRD pattern
- Source of errors
- Quantitative Analysis
Scattering is a physical phenomenon wherein a beam characterized by a definite linear momentum is scattered from its original trajectory because of an object in its path. The beam can be constituted of visible light, X-rays and neutrons. The objects that intercept this beam could be electrons, atoms, molecules, or any nanometer to micrometer-sized structure.
For example, dynamic light scattering uses a laser in the visible light spectrum and the Brownian motion of suspended particles to determine their sizes. X-ray scattering uses the scattering of an X-ray beam by the electron cloud surrounding the atoms to extract information about the internal structure of the object under study. Neutrons are also used to study the internal structure of materials but in this case, a beam of neutrons is scattered by nuclei.
The main important aspect of the X-ray powder diffraction technique for a benchtop equipment are introduced in the following summary table.
Roetger showed for the first time in 1895 the existence of X-rays. It was not until 1912 that diffraction by crystals was discovered. Ever since 1912, diffraction with X-rays has been used to explain the internal structure of matter.
X-rays are electromagnetic radiation with energies ranging from ~120 eV to ~120 KeV, energies much higher than the energy of visible light, ~10 eV. X-rays can penetrate matter because of their high energy and corresponding short wavelengths. The wavelengths used in X-ray crystallographic diffraction experiments range from ~0.5 to ~25 Å.
The interaction between X-rays and matter can give rise to many physical phenomena. Fig. 1 illustrates those phenomena
From a material structure point of view, scattering is the relevant phenomena to use. X-ray scattering takes place when an X-ray photon, which is a discrete bundle or quantum of electromagnetic radiation, interacts with an electron belonging to the molecules of the sample. The photon can transfer energy to the electron, thereby undergoing inelastic scattering, or there could be no net loss or gain of energy, in which case there is elastic scattering. When elastic scattering occurs, there is only a change in the momentum of the X-ray. In this case, the energy of the incoming and scattered photon are equal which indicates that the incoming and outgoing wavelengths are necessarily equal based on the definition.
The discussion in this section is for elastic scattering only.
In order to elucidate the crystalline structure, which is in the order of few angstroms, the X-ray wavelength must be of the same order of magnitude as the size of the features that one is trying to characterize. X-rays of the order of 1 to 2 Å are suited to be used in revealing the atomic structure. A popular wavelength used is that of cooper, which is 1.54 Å.
Diffraction experiments are associated with the notion that X-rays scatter in certain preferential directions. Here, the concept of atomic or lattice planes is used. These planes are defined by the position of the atoms in the molecules making the sample. The atomic planes are viewed as arrangement of atoms in a 2D configuration. Identical planes are “stack” on top of each other. If the orientation of the atomic planes is such that the incoming X-ray gives rise to in–phase scattered X-rays, then a strong signal can be detected. If the atomic planes are not oriented “correctly” in relation to the incoming X-ray, then no signal is detected as the scattered X-rays are out of phase. The process of in-phase scattering is called diffraction.
In order to understand how the arrangement of these lattice planes gives rise to coherent scattering, a brief explanation of Bragg theory is provided here. In fats, these lattice planes can be used to identify the polymorphism as well as lipid bilayer or lamellae spacing.
Bragg’s theory uses the simple notion of mirror reflections, where the diffraction from a crystalline sample can be explained as a reflection of the incoming X-rays by a series of lattice planes, called crystallographic planes. A regular three dimensional array of points can be considered as the basis of a crystal. From these points the crystallographic planes can be defined according to the position of the atoms in the crystal, considering that parallel atomic-planes are similar in the atomic arrangement. The orientation of the planes in the lattice are defined in terms of the notation introduced by the English crystallographer Miller. Three numbers given in brackets specified the "Miller indices" of a family of atomic-planes (hkl), where h, k, l are three integers. (hkl) denotes the family of planes orthogonal to h c1 +k c2 + l c3, where ci are the basis of the reciprocal lattice vectors. Those parallel atomic planes are equidistant with a spacing denoted as dhkl between them, the value that one is interested in obtaining.
In Bragg's approach, each atomic-plane in the set (hkl) is considered a scattering object. Bragg proved that diffraction from the lattice-planes separated by an equal distance, is only possible at particular angles. Only the outgoing diffracted X-ray beams that are in phase will contribute to a constructive interference while all others will cancel among themselves.
Bragg’s law, can be geometrically derived with the aid of Fig. 2.
The incident X-ray beams 1 and 2 impinge at an angle onto a set of parallel atomic planes (hkl), P1 and P2, (Fig. 2). Resorting to wave-theory, one can use constructive interference. This phenomena happens when the difference in path length between the two waves is an integer multiple of the incoming wavelength. Since one is dealing with elastic scattering, the outgoing wavelength is the same as the incoming one. The difference in path between the two scattered waves directions is given in Eq. 1:
‘path length of X-Ray 1’ - ‘path length of X-Ray 2’ = AB + BC
where the segment is BC the shorter side of the rectangular triangle with vertices O, C, B; having 90 degrees between OC and CB as shown in Fig. 2. The same is valid for segment AB, for which the 90 degrees is between segments OA and AB . Coherent or constructive interference must satisfied the relation in Eq. 2
AB + BC = n λ
with n an integer but
AB = dhkl sin(θi); BC = dhkl sin(θs)
where θi is the angle between the direction of the incident X-ray and the atomic planes P1 and P2, θs is the angle between the direction of the scattered X-ray and the atomic planes P1 and P2, dhkl denotes the distance between the two atomic-planes. The subscript ‘hkl’ has been used to alert the reader to think about the Miller indices. It is obvious that AB = BC only when the incident and scattered angle θi = θs are the same. So under these condition, one can write
Where θ = θi = θs Eq. 2 and Eq. 4 are the equal. The integer n is called the “order” of the reflection. Its value is always taken as 1, since higher order can be represented as the relation shown in Eq. 5
Eq. 4 shows the relation required for parallel atomic-planes, separated by a distance d to generate constructive interference. It can be simplified to obtain Eq. 6, the Bragg Law
The typical output from an X-ray experiment is a plot of the intensity as a function of the diffracted angle θ that shows diffracted Bragg peaks. These peaks are the result of the in-phase or constructive interference of the scattered X-rays as explained above. The results of a diffraction experiment are usually plotted as values of intensities 2θ (rather than θ) and the plot is called a powder diffraction pattern.For sufficiently large and strain-free crystallites, the diffraction theory predicts that the lines of the powder pattern will be exceedingly sharp (Fig. 3A). In actual experiments, lines of such sharpness are never observed because of a combined number of instrumental and physical factors that broaden the “pure” diffraction line profile (Fig. 3B).
Interesting enough, once the instrumental factors are removed (not trivial for someone not familiar with the geometry, optic and detector) specific information on the crystal structure can be obtained from the shape of the peak.
When there is no strain or imperfection in the sample, the Scherrer (1906) Eq. 7 can be used to find the thickness of the crystallite,(
|thickness =||k λ|
B cos θ
where B is the half width at half maximum of the peak under study in radians (Fig 3B), λ is the wavelength (in nm) and K is a constant approximately equal to unity and related both to the crystallite shape and to the way that B and the thickness are defined. Scherrer’s original derivation (1906) was based on the assumptions of a Gaussian line profile and small cubic crystal of uniform size. This gave a value of K = 0.94. The derivation calculated by Klug and Alexander (1974) shows a value of K = 0.89 for no particular crystal shape. It has been quoted that this equation is valid only if the crystallites are smaller than 100 nm (West, 1984). The Scherrer equation shows that the thickness of the crystal increases as the width of the peak decreases. It is also known that broadening increases with the diffraction angle, and for that reason, this equation has been used in our laboratory for peaks that are in the region of 1 < 2θ < 10 degrees.
The Bragg Brentano geometry is one of the most popular geometry used in bench top diffractometer and will be briefly discussed here. A typical Bragg-Brentano setup is shown in Fig. 4. This figure shows that the arm containing the X-Ray tube and the arm containing the detector move simultaneously the same angle. Other set ups only move the detector while the X-ray tube stay fixed and some set ups only move the sample.
The X-ray source is a vacuum-sealed tube and requires continuous cooling for which an external chiller is used. Slits and monochromators are used to collimate (direct) the X-ray beam before and after it encounters the sample shown as “beam optic” in Fig. 4. The detector moves in a circle around the sample to cover all the angles of interest. The detector records the number of scattered X-Rays observed at each angle. The X-ray intensity is recorded either as counts or as counts per second.
It is important to mention that scattered X-rays appeared following cones with the opening corresponding to the various Bragg angles. These cones appear because of the nature of the sample: randomly oriented crystals, hence the name powder diffraction. The incident beam always finds a crystallite in an appropriate direction and the cone seems continuous but it is generated by different crystallites.
A point detector is usually employ in a Bragg-Brentano geometry. A point detector can not see the whole cone but rather a particular point and this is why manufactures must make the detector move in order to cover all scattered cones. If instead a 2 dimensional (2D) detector is used then it is kept stationary.
Another geometry used for x-ray instrument is the transmission one. In this case the incident x-ray beam goes through the sample instead of being reflected. The sample is positioned vertically and the incoming x-ray is directed at 90 degrees to the surface of the sample. The detector is mounted on the opposite side of the source. This kind of setup typically uses a 2-D detector to see all scattering cones stimultaneously since neither the X-ray source nor the sample are moved. Transmission geometry is used to observer-ray scattered at small angles (SAXS) otherwise, the 2D detectors must have a very large surface. Some manufactures use point detectors in transmission mode to reduce costs, but this implies that the detector has to be moved continuously in order to scan all the angles at which diffraction could had taken place.
Fig. 5 shows a picture of the goniometer of the Rigaku Powder X-ray Diffractomer, MultiFlex, theta/theta. At maximum power, this instrument operates at 2KW, 40 KV and 44 mA. One of the X-ray tubes that can be used in this equipment is a copper X-ray tube (wavelength of 1.54 Å). The denomination theta/theta indicates that the arms containing the X-ray tube and the detector move the same angle θ.
The system has four main parts/components: X-ray tube, sample holder, monochromator and detector. The X-ray tube and the detector are each attached to one of the two arms of the goniometer. The monochromator, attached to the same arm as the detector, is an optical device that transmits a mechanically selectable narrow band of wavelengths to work with. The arms move together and follow the measuring difractometer circle (Fig. 6) to maintain a θ/2θ relation, as described above, at all times.
The distance between the X-ray source and the sample is equal to the distance between the sample and the detector, which is provided by the factory. The sample position is fixed, in contrast with some instruments that make the sample spin on its axis. This instrument has a software-controlled Peltier system to control the sample holder temperature. A Peltier system is a device that increases or lowers the temperature by means of manipulating the voltage to a pre-programmed temperature. Sensors on the Peltier system tell the software if the temperature has been reached or not. The Peltier system requires continuous cooling provided by an external circulating water bath.
Fig. 6 is a schematic representation of the system in more detail showing the optics that collimate the X-ray beam, both before the sample and after it. It illustrates the following elements:
- DS : Goniometer divergence slit.
- SS : Soller slit assembly (SS1 on “tube” side, SS2 on detector side), a series of closely spaced parallel plates, parallel to the diffractometer circle (i.e., the plane of the paper), designed to limit the axial divergence of the beam.
- θ: incident angle of the X-ray beam onto the sample.
- 2θ: diffracted angle of the X-ray beam.
- SS: goniometer scatter slit. This system is designed in such a way that both DS and RS are located at the same distance from the sample and should have the same value (either ½ degree or 1 degree).
- RS: monochromator receiving slit.
- RSd: detector receiving slit.
- r’: primary focal circle radius.
- rg: goniometer radius.
- r: secondary focal circle.
The DS determines the irradiated area on the sample. It should be noted that, as the angle of incidence changes, the area that is irradiated changes. SS and DS must be the same number. The monochromator (possessing a graphite crystal with 2d = 6.708 Å) works like a filter, allowing only certain wavelengths to reach the detector.The angle θ can be varied from 1.0 degrees to about 80 degrees.
No daily check is necessary for the use of this machine. The machine gets serviced and checked once a year by a technician from the instrument manufacturer. The user can check the performance by using a standard material, such as silver behenate, to ensure that the scattering peaks of the standard have not shifted. If any differences are observed, then those changes should be reflected in the measured pattern.
A powder X-ray diffractometer is simple to operate. In addition to the x-ray instrument, an external refrigeration cooling system must be used to cool down the X-ray tube. The efficiency of the x-ray beam is only 10%, which means that there is a lot of heat to be dissipated. A water reservoir made of about 30% distilled water and the rest tap water is used for this purpose. The water must circulate whenever the x-ray tube is on. The x-ray tube has to be on at full power before it can be used for measurements. This is achieved by carrying out a procedure called “aging”, which consists of incrementing the voltage to the value the manufacture recommends for use. This procedure takes about 20 minutes. Aging of the x-ray tube is necessary, otherwise the X-ray tube life is shortened by taking it to full power in only one step. When the instrument is not in use for more than 24 hours, it is completely turned off. For periods of inactivity of less than 12 hours, the x-ray tube is left at a minimum power, again a value recommended by the manufacturer.
XRD instruments come with some built in features to prevent that the operator gets accidentally exposed to x-rays radiation. The idea is that there is no x-ray leakage and that it has safety features in place. For example, the Rigaku Multiflex door has a safety mechanism that shuts down the x-ray (no x-rays are produced at all) in case someone tries to open it without following the correct procedure. This is to avoid the door opening while the x-rays are exiting the x-ray tube.
The specimen is placed in a glass slide holder which is then placed on the sample holder inside the machine. The powder technique requires that no specific orientation is introduced and that the surface of the material is smooth and level with the top of the cavity in the sample holder (Pecharsky and Zavalij, 2009 Chapter 12). This is important in the Rigaku Multiflex as the top of the sample holder is considered the “0”. The goniometer is software-controlled. The user needs to specify the conditions for the analysis: starting angle, final angle, step size, scan speed and the DS and SS slit sizes. The slits help maximize the intensity as a function of θ, making sure that the incoming and scattered X-rays are directed (collimated) onto the sample and onto the detector. A typical value used in fats is ½ degree slits. The starting angle as well as the final one depends upon whether one is working in the wide or small angle region. The step size or sampling width represents the step that the detector is moved between two consecutive points where the data is collected. It is generally maintained at 0.02 degrees. The scan speed determines how long the detector will collect at each position. The faster the detector moves, the less resolution is achieved. A typical full scan for fats, covering both wide and small angle regions, will start at 1 degree and finish at 30 or 35 degrees, with a step size of 0.02 degrees and a scan speed of 1 deg/min. The Rigaku instrument allows the following types of scan: (1) continuous, where data is collected continuously as the X-ray tube moves; (2) fixed time, where the X-ray tube moves a specific interval of degrees, stops and collects data at this position for the duration of the selected period of time before moving to the next position and (3) fixed counts, where the detector moves a specific interval of degrees and collects a specified amount of counts.
A powder diffraction pattern is achieved when there are a sufficiently large number of small randomly oriented particles with dimensions between 10 and 50 μm. If the material is not in the form of small particles, particle size should be reduced. This is not a problem with fat samples, which are composed of crystals possessing a suitable size with no net orientation. The orientation of the crystals is eliminated by properly mounting the sample in the holder. The sample provided by the manufacture (Rigaku) is made from glass with a cavity that has the dimensions of 20 mm × 20 mm and 0.3 mm in depth. The sample holder can also be made from other materials, such as aluminium.
Filling the cavity with a powder-like material is done by using a flat spatula and making sure not to compress the sample in a particular direction. For samples that are paste-like, the same procedure is followed, resisting the temptation to squeeze down on the sample. Hard fat samples cannot be mounted on the cavity of the glass slide, since the process of trying to convert them to powder causes friction and melting. When the material needs to be melted prior to the analysis, it is simply poured into the cavity in liquid form. The only precaution is not to overfill or underfill the sample holder. The design of the glass slide and its cavity is such that it takes into account the slits and geometry of the goniometer. The user is not required to do any adjustments.
It is important to understand that the diffraction pattern obtain is the combination of all of the molecules that made the material. Having in mind that each phase or component has their unique diffraction pattern, the data collected is the sum of the diffracted patterns for each phase and it is the job of the researcher to make sure that she or he understand how to extract the information. Fig. 7 shows the XRD pattern of chocolate to the right. It can be seen that this pattern is the addition of 1) the coca butter fat (CB on Fig 7) 2) the sugar and 3) the oil.
If one has access to reference patterns, then one tries to match all of the Bragg peaks observed to different materials until a perfect match is obtained. Usually reference material are represented by sticks at the correct 2 theta position. The idea is to match all of the peak positions and intensities but a small mismatch in peak position and intensity is an acceptable experimental error.
Edible fats do not have references due to their complexities. Instead, researcher only match particular peaks. One looks for certain peaks in the WAXS and in the SAXS regions independently. Each region probes a different length scale of the material’s structure, and provides different information as shown in Table 1.
Bragg peak positions in the WAXS region characterize the lateral packing of the hydrocarbon chains in triacylglycerol (TAG) molecules. The XRD AOCS method (AOCS Official Method Cj 2- 95) states that if the sample shows a peak at d = 4.15 Å, then the polymorphic form is α, but if two peaks appear at positions d = 3.8 Å and d = 4.2 Å, then the polymorphic form is β’, while if the peak appears at position d = 4.6 Å, then the polymorphic form is β. These d-spacings obtained from the Bragg peaks define the subcell within the crystal lattice (Small, 1966), which can be hexagonal (α), orthorhombic (β’) or triclinic (β). Larsson (1966) introduced the criteria for the classification of the different crystal forms similar to the one stated by the AOCS method, but with the slight difference that the β polymorphic form should display three peaks at positions d = 4.6 Å, d = 3.8 Å and d = 3.7 Å.
Fig. 8 shows an example of fully hydrogenated canola oil (FHCO). Three polymorphs were obtained by manipulating the cooling rate. The α polymorph was obtained when the fat was subject to a fat cooling rate, achieved by a rapid cooling (in about 1 minute) from 80°C to 20°C and held at this temperature until complete crystallization was achieved. The α polymorph was obtained by setting the temperature on the sample holder to 55°C, and holding it there for 90 minutes. The β polymorph was obtained using a slow cooling rate, achieved by setting the temperature to 62°C for 30 minutes and then reducing it to 58°C in steps of 1°C at a time and with a waiting time of 15 minutes at each temperature.
Bragg peak position in the SAXS region characterize the longitudinal packing of TAGs, giving information about the size of the bilayer, or lamellae, formed by the stacking of the (001) planes. This (001) atomic plane can either be explained as formed by the methyl groups at the end of the hydrocarbon chains of the TAGs or as formed by the oxignes present in the glycerol backbone in the middle of the TAGs. Smaller peaks at angles that are exactly 2×, 3×, 4×, 5×, etc. relative to the first SAXS peak position correspond to higher order reflections of the (001) plane (see Eq. 4-6). Sometimes the position of the (001) peak is not well defined which force the researcher to choose a higher order reflection to find the right value of the bilayer. The (003) seems to show in most XRD patterns and it has been used with good results. The (003) d-position from Fig. 8 was multiplied by 3 to obtain the size of the bilayer: 49.4 Å for α, 45.6 Å for β’ and 45.0 Å for β. A 2L longitudinal packing of TAG molecules within a lamella indicates that the size of the lamellae is given by the length of 2 fatty acid chains. When the lamellar size corresponds to three fatty acid chains, this longitudinal packing is referred to as 3L. A simple calculation was performed to determine whether FHCO was crystallized in a 2L or 3L configuration. The number of carbons present in the fatty acid under study was used for this purpose. Carbons are joined in a zigzag form maintaining a 120° angle and three carbons are separated by 2.54 Å in the beta polymorphic form (Small, 1966). Since the predominant fatty acid present in FHCO contains 18 carbons, then the total carbon chain length is 2.54 Å × 8.5 = 21.54 Å. This result indicates that a 2L packing will give a value close to 43 Å, while a 3L packing will give a value close to 63 Å. The results presented in Fig. 8 are closer to 43 Å than 63 Å, which means that is safe to say that FHCO has crystallized in a 2L configuration.
The thickness of the crystal was also computed from the data obtained by the XRD pattern using the Scherrer equation (Eq. 7). The calculation was performed using the d-spacing for peak (003), the copper wavelength λ = 1.54 Å and K = 0.9. The three polymorphs were analyzed to obtain the crystal domain size as well as the number of lamellae contained in the crystal domain. The results are shown in Table 2.
Source of errors
1) Misalignment of the sample can lead to systematic peak position errors. This is why Rigaku emphasises having a perfectly flat sample’s surface that fills the sample holder cavity just right.
Fig. 9 shows the XRD patterns obtained on FHCO when the sample holder was overfilling and underfilling when compared with the sample holder filled up correctly.
2) Another problem is that the cooper anode in the X-ray tube generates a k-alpha1 and k-alpha 2 doublet that appears in the data. Many manufactures of XRD instruments are aware of this and recommend how to remove it using the software. The user should get familiar with the software as each one has their own way of removing it.
3) It is important to note that one cannot guess the relative amounts of phases based upon the relative intensities of the diffraction peaks when more than one phase are present.
4) The Scheerer equation has its limitation. Nanocrystallite size will produce peak broadening that can be quantified only when it is below certain value which depends on the diffractomer resolution. Most text quote an upper limit of 100 nm. It is not easy to separate all of the different potential causes of peak broadening, but it is necessary if an accurate number is desired.
XRD results are typically used to compute the unit cell dimensions. Food sciences hardly ever do this due to the complexity of the materials under study. The idea is that the interatomic distances can be correlated to define the unit cell dimension. Anything that might change this distances, like the temperature, or the presence of a dopping material or stress will be reflected on a change in the peak positions. Typically, the first step involves finding the interatomic atomic distance dhkl using the position of the peak and the Brag law.
Having in mind that hkl are unknown, the next step is to try to find them. Different software packages are available to try to match data from reference databases to the one collected. The software looks for the unit cell parameters that will allow the identification of the peaks. Iteration of the search is necessary and this process is called “refining”. The program has to be able to contemplate errors associated with the peak position slight differences due to specimen displacement, hence sophisticated programs had been developed. In many cases, the unit cell structure is not known and the researcher must use another specialise software that allows the design of unknown unit cell. This is a tedious and time-consuming process.
As mentioned above, the XRD technique is used in the study of edible fats to determine the characteristic spacings of the fat crystal structure, not the unit cell. Once the data is collected and displayed as intensity versus 2θ the process of identification can start.Data processing is done with a fitting analysis program. For example Jade 9.0 Plus 1995-2011 (MDI, Livermore, California USA) or Igor Pro 6.2 (WaveMetrics, Oregon, USA) are two popular software packages used by benchtop XRD users.Jade allows for the fitting of the Bragg peaks using different fitting equations like Gaussian, Pseudo-Voigt, Pearson-VII, and Lorentzian. The user needs to get familiar with the software in order to report correct values.
Every operator is then encouraged to spend some time familiarizing him- or herself with the equipment by thoroughly reading the manuals before performing any experiments. Investing time to understand the basic operation of the equipment, as well as the theory behind the technique, will guarantee the success of the measurements. Avoiding this step may lead to erroneous results that might put the validity of the results in jeopardy. Training sessions with experts in the area, specifically with the ones that built the equipment, might be a reasonable way to answer some of the inevitable questions. A good result is not equipment-dependent and its reproducibility should be universal, assuming that the correct technique has been used.This section covered the basic of X-ray diffraction. The polymorphic form present in an edible fat product has been used for years as a key indicator of how well the material will perform when used for a particular functionality. Whether the polymorph is the one responsible for the sensory attributes of the final product, such as texture and mouth-feel as not yet been proven. For example, margarine manufacturers know that a margarine in the β' polymorphic form will produce the desired functionality. On the contrary, margarine crystallized in the β polymorphic form showed a dull appearance, a mottled surface and the texture was brittle (deMan and deMan, 2001).It is however well known that when there is a polymorphic transformation, many characteristics of the product change. In some cases, the transformation renders the product useless, while I other cases it favours the development of certain attributes.Specific information about the structure like the lamellae or crystal thickness can be easily obtained from the X-ray diffraction data. Details of the internal structure in the length scale of up to few micrometres can be obtained using another X-ray technique: ultra small X-ray scattering (Peyronel et al., 2013).
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Update: January 26, 2017