Dirk Wissmann

For the elemental analysis of plant tissue materials, traditionally ICP-OES or ICP-MS are used. The advantages are clear, as the sample prep by digestion is standardized. The instruments deliver the required sensitivity and precision, and the measurement itself is very fast and can be automated. The same is true when analyzing fertilizer samples. However, the infrastructure necessary to operate an ICP instrument is not always available, or possible. An additional drawback of the traditional approaches is the chemical waste, resulting from the acids required for digestion. For fast on-site analysis of plant materials, portable energy-dispersive XRF (EDXRF) is a very suitable tool. For applications in the lab demanding high precision and lower detection limits, EDXRF in the past was limited, as the measurement times needed for the required precision were too long, and detection limits for some critical trace elements were not low enough. Based on new excitation concepts and advanced detector technology, modern EDXRF instruments are suitable to perform plant tissue analysis within a much shorter measurement time. Also, highly accurate analyses of major and minor elements in fertilizers can be done. For some critical trace elements in plant tissue and fertilizers, excellent detection limits can be achieved. This is combined with easy sample preparation and low investment into the infrastructure needed to operate the instrument.

	
	The determination of plant nutrients, trace elements, and heavy metals in plant tissue and in fertilizers is of great importance. High crop yields require a sufficient supply of plant nutrients, such as phosphorus and potassium, and of essential trace elements, such as copper and zinc. At the same time, the content of possibly harmful, heavy metals, such as lead, cadmium, nickel, arsenic, and uranium, must be kept low, in order to avoid accumulation in soil and their entrance into the food chain.

	In December 2018, the European Union decided to introduce limits for toxic contaminants in certain fertilizers, including a 60 mg/kg limit for cadmium. To limit the pick-up of uranium by plants, the German Environment Agency recommends a maximum concentration of 50 mg/kg of phosphate.

	XRF allows determination of potassium and phosphorus concentration in fertilizer. It also is suitable as a fast screening tool to test for trace element content. For example, important trace elements, are cadmium and uranium as possible contaminants of fertilizer.

	Improvements in detector and digital signal processing techniques have opened new possibilities for multi-element analysis using EDXRF instruments. In combination with optimized excitation, high sensitivities and low background levels can be achieved. The instrument setup is described in (1). For the application example described in this article, the following benefits can be achieved:

	The analysis of nutritional elements in plant tissue-this can accomplished with shorter measurement times.

	Lower detection limits for critical trace elements can be realized.

	Major element content in fertilizer material can be determined with high precision, which is a basic requirement for achieving high accuracy.

	The analysis of plant materials for toxic and nutritional elements requires little sample preparation. The sample material was dried and pulverized. From the powdered material, 8 g were taken and pressed into a pellet with a diameter of 40 mm, using a pressure of 15 tons.

	All measurements were made using a Spectro XEPOS simultaneous EDXRF spectrometer (Spectro Analytical Instruments). This instrument has an air-cooled, 50 W end-window X-ray tube. The tube anode consists of a thick, binary-alloy cobalt-palladium construction, combined with an adaptive excitation system. The analyzer includes a high-resolution silicon drift detector (SDD), plus the fixed excitation optics, and a filter changer in front of the X-ray tube. The resolution of the SDD is <130 eV (Mn Kα) at an input rate of 1 million counts per s (cps).

	Table I lists the measurement conditions of the EDXRF instrument.

	The analysis of fertilizer samples also requires drying and pulverization of the samples. The analysis of trace element content can be done from powdered samples. For sampling, 4 g of the fine powder is placed into a disposable sample cup with an outer diameter of 32 mm, which is typically closed with a polypropylene foil of 4 µm thickness at the analysis side. The main components, such as phosphorus pentoxide (P

O), and magnesium oxide (MgO), are ideally analyzed as fused beads. To prepare the fused beads, 1 g of the fine powdered material was mixed with 8 g of Lithium tetraborate and fused to a bead with 40 mm diameter using an automatic fluxer.

	Table II lists the measurement conditions of the EDXRF instrument.

Analysis of Plant Materials for Toxic and Nutritional Elements

	For the analysis of plant tissue samples, the analyzer was calibrated using various well-characterized plant material samples. The calibration is based on the fundamental parameters approach. The validation of the method was done using two reference samples from WEPAL (2) and from NIST (3).

	Table III lists the calibration ranges for some minor and trace elements as well as the correlation coefficients.

	The following example correlation graphs for P (Figure 1) and K (Figure 2) show good agreement between the reference values and the analyzed concentrations.

	To validate the method, two reference samples were analyzed (Analysis Results). These results are displayed in Tables IV and V. Information values from the certificate (Certified Concentration) are printed in italics. The printed error of the analysis results is the statistical error for a 1 sigma confidence interval.

	To test the precision of the EDXRF, the NIST 1570a reference sample was analyzed ten times. The results are displayed in Table VI.

Analysis of Fertilizers for Toxic and Nutritional Elements

	For the analysis of the fused beads, the calibration was done using various well-characterized fertilizer samples. The calibration is based on the fundamental parameters approach.

	Figures 3 and 4 show the correlation graphs for the analysis of P

	To test the precision of the analyzer, one of the fused bead samples was analyzed 10 times. The results are displayed in Table VII.

	For the analysis of the powder samples, a screening method for the analysis of various types of geological samples was applied. This screening method is based on a combination of fundamental parameters for fluorescence and scattering and on a Compton-normalization approach.

	Table VIII shows a summary of some typical fertilizer samples. The results for the major elements were taken from the analysis of the fused beads, those results for the trace elements are taken from the analysis of the powder samples. The printed error is the statistical error for a 1 sigma confidence interval.

	Since the sample preparation, especially when analyzing powder samples, involves only a little effort, an analysis using EDXRF can be done in a smaller lab, which is not equipped with a sample prep infrastructure required for ICP-OES or ICP-MS analyses.

	The determination of the content of major and minor elements can be done with good precision. For labs running batches of high numbers of samples, the relatively long measurement times of XRF are a drawback compared to those using ICP techniques. Therefore, EDXRF is very suitable for labs with low to medium sample throughput.

	The detection limits for some critical trace elements are sufficient for a quick screening analysis, but may not be sufficient for every application. Looking at the maximum tolerable limit of 50 milligrams uranium per kilogram of phosphate, as proposed by the German Federal Institute for Risk Assessment, this means a content of 2.5 to 5 mg/kg of uranium in the fertilizer. Reaching a limit of quantification of 2.5 mg/kg would mean reaching a limit of detection of <1 mg/kg, which is a challenge for XRF.

	The improvements in detector and digital signal processing techniques have opened new possibilities for multi-element analysis of plant tissue and fertilizer samples using energy dispersive XRF. As described the technique enables good precision and low detection limits and can be recommended to be used in labs with low to medium sample throughput.

Certificate of Analysis for Reference Material IPE Sample 154, Raye Grass /Lolium perenne 

(Wageningen Evaluation Programs for Analytical Laboratories, Wageningen, The Netherlands)

Certificate for SRM 1570a - Trace Elements in Spinach Leaves

 (National Institute of Standards and Technology, Washington, D.C., 2002)  .
		 

	R.E. Van Grieken and A.A. Markowicz, Eds., 

 (Marcel Dekker Inc., New York, New York, 2nd Ed.,1982).

	
	J. Willis, C. Feather, and K. Turner, 

 (James Willis Consultants Cape Town, South Africa, 2014).

	Dirk Wissmann is the Product Manager of XRF instruments with Spectro Analytical Instruments, in Kleve, Gemany. Direct correspondence to: 

Articles

New excitation developments and advanced detector technology enable the use of EDXRF for multielement analysis of plant material and fertilizers, with improved detection limits and reduced measurement time. These features are combined with easy sample preparation and low cost of investment.

XRF in Agronomy Applications—Analysis of Plant Tissues and Fertilizers

Fatty acid methyl esters (FAMEs) are usually derived from vegetable oil and are typically blended with regular diesel fuel in the 5–20% range. An elemental analysis of pure FAMEs, used vegetable oil, or the reprocessed vegetable oil as well as biofuel blends is required to show compliance with the relevant specifications. Several test methods are available that describe the use of energy-dispersive X-ray fluorescence (EDXRF) for the analysis of sulfur in fuel, such as International Organization for Standardization (ISO) 13032 and ASTM International D7220. The current test methods, using EDXRF, are limited to fuels with a discrete maximum amount of FAME or oxygen. This article describes a quick and economical procedure to check samples for their oxygen content to ensure that the material used for diesel fuel blending is really FAME-based and not mineral-oil based. The oxygen content determined by this approach is used to apply the required matrix correction without a priori knowledge (which is typically required using other approaches), improving sulfur content determination accuracy. In addition to the analysis of the sulfur content, this article describes EDXRF as a technique to monitor other elements, including P, Cl, Na, K, Mg, and Ca, among others, that can have detrimental effects on combustion engines.

	These days, renewable fuels and other biofuels play a growing role in the offer of automotive fuels. Currently, fatty acid methyl esters (FAMEs), mostly derived from vegetable oil, are typically blended with regular diesel fuel in the 5–10% or 20% range. An alternative for the fairly expensive fresh vegetable oil is recycled vegetable oil derived from the food industry. An elemental analysis of the pure FAMEs, the used vegetable oil, the reprocessed vegetable oil, and the biofuel blends is required to show compliance with actual specifications.

	Typically, this type of analysis is done using inductively coupled plasma–optical emission spectrometry (ICP-OES) using the following test methods:

	·      EN14538-2006 (1) describes the analysis of Ca, K, Mg, and Na in FAME, samples require a 1:1 dilution (50%) in kerosene (to reduce matrix effects) and no internal standard is used.

	·      EN16294-2012 (2) describes the analysis of P in FAME, samples require a 1:3 dilution (25%) in kerosene and the use of an internal standard is mandatory.

	Sulfur is not part of these methods and is determined using different technologies. The analysis requires some sample preparation and investment in purchase and operating cost of the analyzer.

	This report describes a quick and economic alternative to quantify the sulfur content in FAMEs using energy-dispersive X-ray fluorescence (EDXRF).

	Besides the analysis of the sulfur content, EDXRF equipment (depending on the analyzer performance), is capable of monitoring additional elements such as P, Cl, Na, K, Mg, Ca, and many more that can be harmful to car engines.

	Recycled vegetable oils can be added into diesel fuel and used as secondary burner fuel. An elemental analysis to show the absence of toxic elements is part of the quality control. This can also be done with the XRF equipment.

	In addition, it would be of interest to at least check the samples for their oxygen content to make sure that the material used for blending in the diesel fuel is really based on FAMEs and not based on mineral oil.

	Several test methods are available that describe the use of EDXRF for the analysis of sulfur in fuel. The International Organization for Standardization (ISO) method 13032 (3) and ASTM International method D7220 (4) are two of them.

	The currently available test methods using EDXRF are limited to fuels with a discrete maximum amount of FAME or oxygen. In the example of ISO 13032, the maximum content of FAME is 10% or 3.7% of oxygen. The reason is that the oxygen has a matrix effect on the sulfur signal measured by XRF. To be able to use XRF for this type of analysis, various approaches are possible:

	·      Correction using defined correction values depending on the known oxygen content, which requires a priori knowledge of the oxygen content.

	·      Correction using fundamental parameters based on the matrix, which has to be predefined, and which also requires a priori knowledge of the oxygen content.

	·      Correction using backscatter information from the sample. This approach does not require any a priori knowledge of the sample matrix, but the backscatter correction must be based on a signal, which should be close to the analyte’s X-ray fluorescence energy because otherwise additional effects occur.

	·      Additional references to test methods are also available (5–7).

	For other elements of interest, the same effect is visible. The influence on the results varies depending on the element.

	Our procedure used a Spectro Xepos analyzer in which samples were excited by a forced-air-cooled 50 W end-window X-ray tube combined with a doubly curved HAPG crystal for monochromatization and polarization of the primary tube spectrum. In addition, direct excitation using Pd-K and Co-K radiation can be used to optimize the excitation conditions for groups of elements. A high-resolution silicon drift detector (SDD) was used to collect fluorescence radiation from the sample. The highly stable spectral resolution of the SDD was <130 eV for Mn Kα.

	The measurement parameters are given in Table I.

	The sample preparation for the analysis is straightforward: 4 g of sample material is poured into a disposable XRF sample cup with an outer diameter of 32 mm. The analytical side of the sample cup is closed with a 4-µm-thick polypropylene film. For additional safety, the sample cups are placed into a sample holder, which is closed with a second polypropylene film of the same thickness.

	As explained above, biodiesel blends according to specification EN 590 typically can be analyzed with a calibration prepared for diesel fuel. Using the same calibration for the analysis of a biofuel sample with 12.5% oxygen would lead to a significant bias in the analysis result as can be seen in Figure 1.

					Figure 1: Calibration for S in mineral oil and biofuel with 12.5% oxygen.

	In many cases the oxygen content of the sample being analyzed is not known, and therefore the best option for an appropriate matrix correction is to determine the oxygen content of the sample. Using XRF, the direct determination of the oxygen content in fuel is not possible because the fluorescence energy of oxygen is very low and is not detectable.

	The second approach is to use the backscatter information of the sample to indirectly determine the content of oxygen. The so-called Compton scatter signal will decrease with increasing O-content. To be able to achieve this accurately, the energy of the Compton scatter signal should be at relatively low energy because otherwise volume effects will affect the determination.

	Since the EDXRF system used in our procedure is equipped with an X-ray tube with a thick binary Pd-Co alloy anode, in principle, there are two options to use backscatter information: one from the excitation using the characteristic radiation of Pd and the second using the characteristic radiation of Co.

	Because of the reasons listed above, the cobalt excitation of the EDXRF system can be used for this determination with good accuracy. Figure 2 shows the calibration of the measurements of samples with known oxygen concentration.

					Figure 2: Calibration for O in fuels.

	The figure demonstrates the sensitivity and accuracy of this indirect determination of the oxygen content. Based on this determination, the differentiation between materials based on FAMEs or on mineral oil is easily possible.

	For the accurate determination of the trace element content, the oxygen content determined with this procedure is used within a fundamental parameters program. Table II shows some examples.

	As previously listed, other trace elements can be determined simultaneously using EDXRF. Table III gives detection limits for additional trace elements.

	Although there is currently no specified test method using XRF for the determination of these elements in biofuels, the listed limits of detection (LODs) clearly demonstrate that at least a screening for the content of these elements is possible using the EDXRF, even at low levels.

	EDXRF clearly can be used for the analysis of S and other trace elements in biofuel. The determination of the oxygen content helps to compensate for matrix effects efficiently, and a priori knowledge of the oxygen content is not required.

		EN14538-2006: Fat and oil derivatives. Fatty acid methyl ester (FAME). Determination of Ca, K, Mg and Na content by optical emission spectral analysis with inductively coupled plasma (ICP-OES). 

		EN16294-2012: Petroleum products and fat and oil derivatives. Determination of phosphorus content in fatty acid methyl esters (FAME). Optical emission spectral analysis with inductively coupled plasma (ICP-OES). 

		International Organization for Standardization, ISO 13032, “Petroleum products – Determination of low concentration of sulfur in automotive fuels – Energy-dispersive X-ray fluorescence spectrometric method” (ISO, Geneva, Switzerland, 2012).

		ASTM International, D7220, “Standard Test Method for Sulfur in Automotive, Heating, and Jet Fuels by Monochromatic Energy Dispersive X-Ray Fluorescence Spectrometry” (ASTM International, West Conshohocken, Pennsylvania, 2017).

		European Committee for Standardization (CEN, French: Comité Européen de Normalisation) 

 is the senior product manager and product manager for XRF at Spectro Analytical Instruments GmbH in Kleve, Germany. Direct correspondence to: 

The EDXRF method described here is a quick and economical procedure to check samples for their oxygen content to ensure that the material used for diesel fuel blending is really FAME-based and not mineral-oil based.

EDXRF Analysis of Sulfur and Trace Element Content in FAMEs

This column installment describes a novel energy-dispersive X-ray fluorescence (ED-XRF) system that uses a combination of a Bragg polarizer, used simultaneously with a direct excitation source, together with a novel, highly annealed pyrolytic graphite (HAPG) crystal as a band-pass filter. By selection of the optimum configuration, this system allows for high analysis precision for minor and major elements across a wide wavelength range and lower detection capability for smaller groups of elements from potassium to manganese. To show the practical benefits of this technology, this study focuses on these performance metrics for the determination of titanium in polymer samples, together with the multielement analysis of high-purity graphite using the ashing sample preparation method. In particular, this installment of “Atomic Perspectives” demonstrates the improved performance for graphite that allows for lower sample weights to be used and results in significantly shorter ashing times, which is a requirement for process control applications and laboratories with a high sample workload.

	The improvements in detector and digital signal processing techniques have opened new possibilities for multielement analysis using energy-dispersive X-ray fluorescence (ED-XRF) instruments. The opportunity to process input count rates higher than 1 Mcps are a challenge for the development of new bright excitation optics to further improve elemental sensitivities and detection limits. Excitation of fluorescence lines must be efficient over a large energy range from about 0.5 keV to about 40 keV. The use of Bragg crystals as monochromators in ED-XRF in a selected energy range was first discussed in the early 1980s (1). However, today it is necessary to select crystals with a high integral reflectivity to cover large solid angles of >100 msr (millisteradian) and combine this technology with other excitation optics.

	Highly annealed pyrolytic graphite (HAPG) is a new mosaic crystal used for X-ray analysis because it has good bending properties, offers a significant increase of integral reflectivity compared to other crystals, and has other useful properties that are required in ED-XRF instrumentation. However, it cannot be used as a monochromator in the first order because of its weak mosaicity at the required bending radius. These properties make HAPG an optimal X-ray band-pass filter in the energy range of <10 keV, if they are placed between the X-ray tube and sample or between the sample and detector. In addition, such a filter can be combined simultaneously with the recognized benefits of a Bragg polarizer or other optics in combination with a binary alloy X-ray tube anode to provide even greater improvement in the sensitivity of the spectrometer in different energy ranges (2). Figure 1 shows the schematic of an XRF excitation system using a Bragg polarizer simultaneously with a direct excitation source.

					Figure 1: Schematic of an XRF excitation system using a Bragg polarizer simultaneously with a direct excitation source.

	Using the HAPG band-pass filter in this configuration allows for optimization of the excitation for different groups of elements. The radiation originating from the tube will be Bragg-reflected at the toroidal shaped crystal in an energy range and will be defined by the mosaic spread, d-spacing of HAPG and the beam guidance. Because of the large solid angle and the intensity of the excitation, radiation will also be increased, resulting in higher intensity of the fluorescence signal combined with lower background. The practical benefits are improved precision for major and minor elements together with an order of magnitude lower detection limits for trace elements. This type of band-pass filter excitation configuration is shown schematically in Figure 2.

					Figure 2: HAPG band-pass filter excitation configuration.

Determination of Titanium in Polymer Samples

	One of the most common types of analysis for ED-XRF is the determination of elemental impurities in polymer samples, and in particular trace levels of the transition elements, such as titanium. This is exemplified in Figure 3, which shows spectral measurements from 3.2–5.8 keV for four polymer standards containing Ti at concentration levels of 0, 0.3, 1.0, and 7.7 mg/kg (ppm) at a measurement time of 150 s. It can be clearly seen that the spectral peaks for Ti at about 4.5 keV can all be identified and quantified.

					Figure 3: Spectra from measurements of polymer samples containing trace levels of Ti.

	To determine the limit of detection and precision for this application, another low standard was analyzed 10 times. The net intensities, normalized to live time and tube current, as well as the analysis results are displayed in Table I.

	Where traditional ED-XRF instruments can achieve detection limits in the milligrams-per-kilogram (parts-per-million) range for this application, the band-pass-filter excitation approach can achieve a detection limit of about 66 µg/kg (ppb).

Multielement Analysis of Powdered Graphite

	Mosaic crystals like HAPG cannot be used for focusing the beam to the sample. The defocusing effects must be eliminated by beam guidance. By using optimized beam guidance, even smaller amounts of sample can be analyzed quantitatively without sacrificing precision. This approach can be used in applications where only small amounts of samples are available, such as the use of high-temperature ashing as a sample preparation technique. With this procedure, the elements are concentrated in the ashed sample. However, typically there is not enough sample to prepare a standard pressed pellet with a diameter of 32 or 40 mm, or to fill a standard XRF sample cup.

	One example is the analysis of impurities in high quality graphite with purity better than 99.5% or even 99.9%. To be able to ash enough material to prepare a standard pressed pellet, ~4 g of ash would be required, resulting in a long ashing time and making this procedure unrealistic for a process control application. The alternative is the microwave-assisted ashing procedure, where the sample material is reduced by a factor of 5–10 within 30 min. Using this approach, as little as 120–150 mg of the remaining ash in a 20-mm sample cup allows for an accurate determination of the elemental impurities in the graphite. This approach is exemplified in Figure 4, which shows spectra of a graphite sample with 99.5% purity analyzed directly in a standard 32-mm sample cup compared to the spectra obtained from analyzing the ashed sample in a 20-mm sample cup. It should be emphasized that the determination of volatile elements like S have to be carried out before the ashing procedure, but typically their concentrations are high enough to obtain precise results from the analysis of the powder directly.

					Figure 4: Spectra from the direct measurement of graphite (displayed in red) and after ashing (displayed in blue) of the sample.

	There is no question that it’s always going to be challenging to quantify all trace elements with different fluorescent energies up to 40 kV for ashed samples that cannot be prepared as a thin layer, or as a thick pressed pellet. Sherman (3) as well as Shiraiwa and Fujino (4) have developed mathematical algorithms for the computation of count rate values of fluorescent radiation emitted from a standard at a fixed detector position. Additionally, Weber (5) described effects caused by the instrumental and sample geometry that are neglected in the traditional fundamental parameter models for thick samples. As a result, modifications of the traditional fundamental parameter algorithms are therefore necessary to improve the accuracy of the analytical results, and for that reason, the density as well as the thickness of a sample must be known, independently of each other.

	This approach was used to analyze the graphite powder directly in a sample cup with an outer diameter of 32 mm, together with a smaller amount of ashed material in a 20-mm cup. The results for the direct graphite analysis and those determined from the ashed sample, but calculated back to the original substance, are displayed in Tables II and III, respectively. The results obtained demonstrate that there is a good agreement between both sampling methods for elements at concentration levels well above the limit of detection (LOD). However, it can be clearly seen that the standard deviation (SD) of the analytes in the ashed samples are lower, allowing for better precision (see SDs for MgO, Al

 as examples) and also better accuracy. In addition, the analysis of the ashed samples allows for the determination of some trace elements, which are not possible by directly analyzing the original material (see results for elements Sr, Y, and Zr as examples). The overall conclusion is that the quantitative analysis of ~150 mg of ashed sample is possible with a standard XRF instrument using the hardware configuration and calculation principles described in this study.

	The improvements in detector and digital signal processing techniques have opened new possibilities for multielement trace-element analysis with stationary and portable ED-XRF instruments. The development of new bright excitation optics improves elemental sensitivities as well as detection limits. The user benefits range from better precision to lower detection limits, which results in improved accuracy. By using optimized beam guidance even smaller amounts of sample can be analyzed quantitatively without sacrificing precision. This capability can be used where only small amounts of sample are available, such as in research and development applications, or when using rapid ashing procedures, which are a requirement for high sample workload laboratories and process control applications.

		T.C. Furnas, G.S. Kuntz, and R.E. Furnas, 

, Second edition (Marcel Dekker Inc., ISBN 0-8247-0600-5, 1982), pp. 618–626.

 (Wiley Heyden on-line library, 1983), pp. 12,1, 11–18.

 is the vice president of R&D at Spectro Analytical Instruments in Kleve, Germany, and is responsible for research and development of all products from Spectro.

 is the product manager for XRF instruments at Spectro Analytical Instruments and is responsible for the strategic development of the complete product line including application development and product support. Direct correspondence about this article to: 

This month’s column will describe a novel ED-XRF system, which utilizes a combination of a Bragg polarizer, used simultaneously with a direct excitation source, together with a novel, highly annealed pyrolytic graphite (HAPG) crystal as a band-pass filter. By selection of the optimum configuration, it will allow for high precision of minor and major elements across a wide wavelength range and/or lower detection capability for smaller groups of elements from potassium to manganese. To show the practical benefits of this technology, this study will focus on these performance metrics for the determination of titanium in polymer samples, together with the multielement analysis of high purity graphite using the ashing sample preparation method. In particular, it will be shown that the improved performance for graphite will allow for lower sample weights to be used resulting in significantly shorter ashing times, which is a requirement for high sample workload laboratories and process control applications.

Dirk Wissmann

Articles

XRF in Agronomy Applications—Analysis of Plant Tissues and Fertilizers

EDXRF Analysis of Sulfur and Trace Element Content in FAMEs

Optimization of ED-XRF Excitation Configuration Parameters to Determine Trace-Element Concentrations in Organic and Inorganic Sample Matrices