While Fourier transform–infrared (FT-IR) spectroscopy is commonly thought of as a laboratory analytical technique for which annual demand tops $200 million worldwide, it is also becoming an important technique in the post-9/11 era of heightened security measures. FT-IR has several advantages over dispersive infrared that make it highly advantageous for most security applications. There are a fair number of application areas for FT-IR within the security instrumentation market. 

An FT-IR instrument has only a single moving part, which makes it an inherently simpler and more reliable instrument, which comes in particularly useful for in-field, handheld, and vehicle-mounted instrumentation. The configuration of an FT-IR instrument also allows for internal calibration with each measurement and provides for a much more rapid analysis with higher sensitivity than a typical dispersive infrared instrument. 

2007 FTâIR Security Application Demand ($26M)			

Military applications are by far the largest area of application in the security market for FT-IR instrumentation. The ruggedness of the systems and the ability to provide open path detection of chemical warfare agents (CWAs) and other toxic chemicals hundreds or even thousands of meters away from the instrument make them ideal for military applications, and they are now regularly found installed on vehicles, including tanks. FT-IR systems that can identify hundreds or even thousands of hazardous or toxic gases can now be built into portable, briefcase-sized systems that are useful for first responders including police, firefighters, and paramedics, as well as for various transportation and shipping security applications. 

The market for FT-IR–based security instrumentation is estimated to be worth $25 million to $30 million in 2007. Smiths Detection and Bruker Daltonics are the two leading vendors in this market, although General Dynamics, Edo, and Thermo Electron are also significant competitors. 

The foregoing data were extracted from SDi's market analysis and perspectives report entitled 

Analytical Instrumentation for Security Applications, May 2006

. For more information, contact Stuart Press, Senior Consultant, Strategic Directions International, Inc., 6242 Westchester Parkway Suite 100, Los Angeles, CA  90045, (310) 641-4982, fax: (310) 641-8851, 

Articles

Departments

While Fourier transform–infrared (FT-IR) spectroscopy is commonly thought of as a laboratory analytical technique for which annual demand tops $200 million worldwide, it is also becoming an important technique in the post-9/11 era of heightened security measures. FT-IR has several advantages over dispersive infrared that make it highly advantageous for most security applications. There are a fair number of application areas for FT-IR within the security instrumentation market.

Market Profile: FT-IR for Security Applications

Middle-infrared (mid-IR) laser sources operating over a "molecular fingerprint" 2–15 μm spectral range are in great demand for a variety of applications including molecular spectroscopy, noninvasive medical diagnostics, industrial process control, environmental monitoring, atmospheric sensing and free space communication, oil prospecting, tracking of naturally occurring gas emitters — methane seeps and volcanoes — and numerous defense-related applications such as infrared countermeasures, monitoring of munitions disposal, and standoff detection of explosion hazards. Mid-IR wavelengths usually are generated using relatively complex nonlinear optical conversion techniques or by means of direct generation in heterojunction lead-salt, antimonite, or quantum cascade semiconductor lasers featuring limited output power and tuning range.  

Impurity-doped crystalline vibronic lasers having a gain bandwidth up to 50% of the central wavelength represent another viable route for broadly tunable mid-IR coherent sources. Among them, transition metal (TM) (TM

)–doped binary and ternary chalcogenide crystals represent a relatively new class of solid-state gain media with strong and ultrabroad absorption and emission bands in the mid-IR range of optical spectra. In 1996, scientists from the Lawrence Livermore National Laboratory (1) were the first to show that among different types of crystalline gain materials, TM

-doped wide bandgap II–VI semiconductor crystals could be very special for mid-IR lasing. Active interest in TM-doped II–VI compounds inspired by this research was explained by the fact that these media are close mid-IR analogs of the titanium-doped sapphire (Ti-S) in terms of spectroscopic and laser characteristics, and it was anticipated that TM

-doped chalcogenides will lase in the mid-IR with a great variety of possible modes of oscillation, similar to the Ti-S laser. Indeed, optically pumped room-temperature lasers based upon Cr:ZnS, Cr:ZnSe, Cr:Cd

Te, Cr:CdSe, and Fe:ZnSe crystals providing access to the 2–5 μm spectral region with high (up to 70%) efficiency, multi-watt-level output powers, tunability in excess of 1000 nm, and narrow spectral linewidth (<20 MHz), have been reported by several groups (2,3). Arguably these lasers represent the simplest and most cost-effective light sources in this 2–5 μm wavelength range.  

In addition to effective room-temperature mid-IR lasing, TM-doped II–VI media, being wide-band semiconductors, hold potential for direct electrical excitation. Two promising routes for achieving TM:II–VI mid-IR lasing under electrical excitation were envisioned: utilization of TM-doped quantum well II–VI heterostructures, and utilization of thin films based upon TM-doped nanocrystals, or nanocrystals dispersed in a conductive matrix (4). In both cases, quantum confinement enables an efficient energy transfer from the host to the active transition metal (Cr or Fe) impurity. It was verified experimentally for Mn

:ZnS nanocrystals (5–7) and qualitatively explained by the increase of the oscillator strength of the exciton bound to the impurity center. The idea of achieving the goal of electrically pumped mid-IR semiconductor lasers based upon TM-doped quantum confined structures was reported (4,8,9), and first steps were performed by studying bulk mid-IR electroluminescence of Cr (4,10,11) and Cr excitation and lasing via photoionization transitions (4,12). Here, we describe our first steps toward fabrication and spectroscopic characterization of TM-doped nanocrystals. 

Preparation of II–VI Semiconductor Nanocrystals Doped with TM Ions 

The majority of methods of metal and semiconductor nanocrystal synthesis are based upon chemical approaches in which a variety of functional groups are used to stabilize nanocrystals and serve as capping agents. Various byproducts of chemical reactions can also interact with nanocrystals, and their removal from the surface is tricky. Even small amounts of residual organic molecules used in nanocrystal synthesis might reabsorb or quench nanocrystal luminescence in the IR region.  

An entirely physical approach for the fabrication of metal and semiconductor nanocrystals, such as laser ablation, enables formation of pure uncapped nanocrystals. Laser ablation in a liquid environment has been used for the fabrication of semiconductor and metal nanoparticles (13,14). However, formation of semiconductor nanocrystals doped with TM ions under laser ablation has not been documented.  

TM- (Cr, Co, and Fe) doped II–VI (ZnS, ZnSe) semiconductor nanocrystals were prepared by laser ablation in a liquid environment in several steps. Initially, chemical vapor deposition (CVD)–grown polycrystalline ZnS and ZnSe samples placed in evacuated and sealed quartz ampoules were subjected to 7–10 days of postgrowth thermo-diffusion of TM ions from the metal phase at 970 °C. The detailed description of the process can be found elsewhere (15). The average TM ions concentration was 10

Nanocrystals were prepared by ablation of bulk polycrystalline TM-doped ZnS and ZnSe samples in double-distilled, deionized water with a radiation of the fundamental harmonic of Nd:YAG laser with 30-ps pulse duration, 10-Hz repetition rate, and pulse energy of 10 mJ. The laser radiation was focused to a spot size of approximately 1 mm. Uncapped nanocrystals quickly aggregated and formed precipitate. Precipitated nanocrystals were collected and dried under ambient conditions for X-ray diffraction (XRD) and spectroscopic studies.  

At the last stage of nanocrystal preparation, their suspension in a quartz spectroscopic cell was illuminated additionally by the radiation of the third harmonic (355 nm) of a picosecond Nd:YAG laser with pulse energy of 15 mJ. The laser beam was not focused and had ~1 cm diameter. Initially cloudy colloidal suspension became clear after 10 min of irradiation, which indicates the formation of smaller particles. The prepared nanocrystals tend to form aggregates and precipitate in several hours. Precipitated aggregates were collected and dried under ambient conditions.  

Dry nanocrystals were evenly dispersed on microscope slides for XRD measurements or kept in a quartz cell for spectroscopic studies.  

XRD Characterization of II–VI Nanocrystals Doped with TM Ions

The crystalline grain size and structure of nanocrystals was studied by XRD using θ–2θ angle mode of diffractometer with a Cu K-alpha anode. The XRD spectra depicted in Figure 1 correspond to Cr-doped ZnS and ZnSe polycrystalline samples as well as nanocrystals prepared from the bulk samples by laser ablation. Initial bulk polycrystalline ZnS and ZnSe samples had a zinc-blend structure, as seen in Figure 1a. The most prominent ZnS diffraction peaks located at 2θ = 28.6, 33.2, 47.6, and 56.5° coincide with (111), (200), (220), and (311) planes reflections of the zinc-blend structure. Similarly, the ZnSe polycrystalline target had XRD peaks at 2θ = 27.5, 45.5, and 53.3° corresponding to (111), (220), and (311) planes reflections of cubic ZnSe.  

As shown in Figure 1b, the nanocrystals fabricated by 1064-nm laser ablation also featured several additional peaks characteristic to wurtzite structure. Specifically, Cr

:ZnS nanocrystals exhibit XRD pattern (Figure 1b) with several wurtzite ZnS (100), (101), (102), and (103) peaks. The other peaks of the wurtzite structure (002), (110), and (112) overlap with main peaks of the zinc-blend structure (111), (220), and (311). Similar results were obtained for Cr

:ZnSe nanocrystals. This indicates that the ZnS and ZnSe nanocrystals prepared by laser ablation had mixed wurtzite and zinc-blend structures.  

XRD peaks of nanocrystals obtained after additional ultraviolet irradiation exhibit significant broadening, indicating the particle's size reduction. The mean nanocrystal size can be calculated from the width of XRD peaks using Scherrer's equation, 

 is a measured full width half maximum (FWHM) of the peaks, and 

:ZnS nanocrystals was 11 nm before and 3 nm after the UV treatment. Similarly, Cr

:ZnSe nanocrystals have a size of 13 nm before the UV treatment. However, a prolonged UV treatment of ZnSe nanocrystals results in sample degradation. The size of nanocrystals fabricated by a single-stage 1064-nm laser ablation and estimated from the XRD measurements was consistent with the transmission electron microscope images of nanocrystals depicted in Figure 1.  

Spectroscopic Characterization of II–VI Nanocrystals Doped with TM Ions

) ions. This state is split in a weak tetragonal T

 crystal field of the II–VI crystals into the ground 

-doped chalcogenides spans over 1.5–3.5 μm. The broad absorption bands of the Cr

  ions in ZnS and ZnSe are centered at 1680 and 1780 nm, respectively, and feature FWHM of ~400 nm (Figure 2).  

Room-temperature photoluminescence spectra of Cr

:ZnSe nanocrystals are demonstrated in Figure 2. The photoluminescence was excited by the 5 ns radiation of an optical parametric oscillator operating near the absorption maxima, and photoluminescence spectra were measured by IR spectrometer-InSb detector-boxcar integrator combination. The photoluminescence spectra of Cr-doped bulk crystals obtained at the same experimental conditions are also shown for comparison in Figure 2. The photoluminescence bands of Cr-doped ZnS and ZnSe nanocrystals are not only blue-shifted with respect to the photoluminescence of bulk polycrystalline samples, but they also have a different shape with a strong photoluminescence signal suppression at wavelengths longer than 2600 nm. The blue shift of the Cr-doped nanocrystals photoluminescence maxima can be attributed to the increase of the strength of the crystal field resulting in larger splitting between the ground 

E states of Cr in nanocrystalline environment, while the most probable factor affecting the shape of Cr

:ZnS and ZnSe nanocrystals photoluminescence spectra is emission reabsorption by OH groups possibly remaining on the nanocrystal surface.  

:ZnSe nanocrystals are depicted in Figure 2 as inserts. The room-temperature luminescence lifetime of Cr

:ZnS nanocrystals was 2.6 μs, which is noticeably shorter than the 6 μs lifetime typical of Cr

:ZnS nanocrystals' lifetime was 5 μs at 77 K, which is close to that of the bulk samples. It is probable that the shortening of the Cr

  excited-state lifetime in ZnS nanocrystals at room temperature with respect to the bulk samples is due to Cr

  photoluminescence quenching by the surface states and OH groups. On the other hand, a smaller spectral overlap of Cr

  photoluminescence and OH absorption at 77 K mitigates this quenching. Similar results were obtained for Cr

:ZnSe nanocrystals photoluminescence lifetime behavior.  

The absorption and photoluminescence spectra of Co

:ZnSe polycrystalline bulk samples as well as photoluminescence of Co

:ZnSe nanocrystals are shown in Figure 3. The ground level 

) ion is split by the crystal field into three levels: 

-doped II-VI bulk crystals typically have two bands associated with 

 transition is relatively weak because it is symmetry-forbidden (1). 

 room-temperature luminescence spectra of Co

:ZnSe nanocrystal and bulk samples excited via 

 transition (1600 nm) are shown in Figure 3. One can see from Figure 3 that the photoluminescence signal of Co

:ZnSe nanocrystals is strongly suppressed at 2800 nm compared with the bulk sample emission, which can be attributed to light reabsorption by OH groups remaining at the surface of the nanocrystals. Indeed, the insert in Figure 3 demonstrates that the room-temperature luminescence lifetime of Co

:ZnSe nanocrystals is 160 μs, which is practically twice smaller than the 290 μs photoluminescence lifetime featured by the bulk sample. It is noteworthy that Co

:ZnSe nanocrystals' photoluminescence lifetime increased and became comparable with that of a bulk sample when nanocrystals were dried using anhydrous alcohol. This fact is in agreement with the hypothesis of OH groups affecting emission properties of Cr- and Co-doped ZnS and ZnSe nanocrystals. It also indicates that the number of OH groups can be reduced and thus, parasitic nonradiative relaxation of excited Cr and Co ions can be mitigated by the application of an appropriate nanocrystal drying procedure.  

D is a ground term. The ground state in the tetrahedral crystal field of the II–VI crystals is split into doublet 

. However, in contrast to the chromium ions, the lowest level of the iron is doublet 

  ions is smaller compared to chromium ions, and as a result, the iron absorption and emission bands usually are shifted to the longer wavelength. Recently, a room-temperature lasing of Fe:ZnSe tunable over the 3.77–5.05 μm spectral range was demonstrated (16,17). For these low-energy mid-IR transitions, the major factor limiting photoluminescence efficiency is nonradiative quenching of the excitation. Remarkable differences in thermal quenching of Mn

  ion photoluminescence in nanocrystal and bulk ZnS material were revealed by Tanaka (7), where an increased quantum efficiency of Mn

:ZnS nanocrystal photoluminescence was explained by a weaker electron-phonon coupling in the nanocrystals. Hence, it is of interest to study whether, similarly to Mn in ZnS, the thermal quenching of Fe

:ZnSe nanocrystals can be mitigated and whether one can obtain broadband room-temperature mid-IR photoluminescence with wavelengths longer than 4 μm. The absorption and photoluminescence spectra of iron-doped ZnSe bulk and nanocrystal samples are depicted in Figure 3. The Fe

:ZnSe room-temperature absorption spectrum features an ultrabroad band spanning over 2.5–4.5 μm. Room-temperature luminescence of Fe

:ZnSe nanocrystals and bulk material under excitation by the 2.8-μm radiation of Er:Cr:YSGG laser is also depicted in Figure 3. As one can see, at room temperature, Fe

:ZnSe nanocrystals have a broadband (3.5–5.2 μm) luminescence practically identical to Fe

:ZnSe bulk photoluminescence with a characteristic dip at 4.25 μm caused by absorption of atmospheric CO

TM-doped II–VI media hold the potential for lasing at room temperature under direct electrical excitation. These studies demonstrate the initial effort toward achieving this goal by studying Cr

-doped nanocrystals. We have demonstrated a novel method of TM-doped II–VI nanocrystal production by laser ablation in a liquid environment. This method enables fabrication of II–VI nanocrystals doped with a variety of TM ions using laser ablation of thermodiffusion-doped polycrystalline targets. This method has a significant advantage over chemical methods because of its simplicity and the absence of residual organic molecules on the nanoparticles' surface. For the first time, to our knowledge, TM-doped II–VI nanocrystals demonstrated strong mid-IR luminescence. This technology opens new pathways for future development of optically and electrically pumped mid-IR lasers based upon TM-doped quantum confined structures.  

This material is based upon work supported by the National Science Foundation under Grants No. ECS-0424310, EPS-0447675, and BES-0521036. 

C. Kim, D.V. Martyshkin, V.V. Fedorov, I.S. Moskalev

are with the Department of Physics, University of Alabama at Birmingham, Birmingham, AL.

(1) L.D. DeLoach, R.H. Page, G.D. Wilke, S.A. Payne, and W.F. Krupke, 

(3) S. Mirov, V. Fedorov, I. Moskalev, and D. Martyshkin, 

(4) V.V. Fedorov, A. Gallian, I. Moskalev, and S.B. Mirov, 

(5) R.N. Bhargava, D. Gallagher, X. Hong, and A. Nurmikko, 

(8) S.B. Mirov, V.V. Fedorov, K. Graham, I.S. Moskalev, I.T. Sorokina, E. Sorokin, V. Gapontsev, D. Gapontsev, V.V. Badikov, and V. Panyutin, in 

5th International Conference on Mid-Infrared Optoelectronic Materials and Devices

, Annapolis, MD, Sept. 8–11, 2002, Abstract Book pp. 143-144. 

(9) A. Gallian, V.V. Fedorov, J. Kernal, J. Allman, S.B. Mirov, E.M. Dianov, A.O. Zabezhaylov, and I.P. Kazakov, 

(10) L. Luke, V.V. Fedorov, I. Moskalev, A. Gallian, and S.B. Mirov, in 

Solid State Lasers XV: Technology and Devices

, H.J. Hoffman, R.K. Shori, Eds., Proc. Of SPIE Vol. 

 (SPIE, Bellingham, Washington, 2006), 61000Y-1-8. 

(11) J. Jaeck, R. Haidar, E. Rosencher, M. Caes, M. Tauvy, S. Collin, N. Bardou, J.L. Pelouard, F. Pardo, and P. Lemasson, 

(12) A. Gallian, V.V. Fedorov, J. Kernal, S.B. Mirov, and V.V. Badikov, in 

Advanced Solid State Photonics 2005 Technical Digest on CD-ROM 

(The Optical Society of America, Washington, DC, 2005), MB12. 

(13) K.V. Anikin, N.N. Melnik, A.V. Simakin, G.A. Shafeev, V.V. Voronov, and A.G. Vitukhnovsky, 

(14) T. Sadaki, Y. Shimizu, and N. Koshizaki, 

(15) S.B. Mirov, V.V. Fedorov, K. Graham, I.S. Moskalev, I.T. Sorokina, E. Sorokin, V. Gapontsev, D. Gapontsev, V.V. Badikov, and V. Panyutin, 

(16) J. Kernal, V.V. Fedorov, A. Gallian, S.B. Mirov, and V. Badikov, 

(17) V.V. Fedorov, S.B. Mirov, A. Gallian, D.V. Badikov, M.P. Frolov, Y.V. Korostelin, V.I. Koslovsky, A.I. Landman, Y.P. Podmar'kov, V.A. Akimov, and A.A. Voronov, 

A novel method of transition metal (TM) (Cr, Co, and Fe)-doped nanocrystals fabrication based upon laser ablation was demonstrated. For the first time, mid-IR luminescence from TM:II-VI nanocrystals was reported.

Mid-IR Luminescence of Nanocrystalline II-VI Semiconductors Doped with Transition Metal Ions

Diamonds are unique among gemstones because they are composed of a single element (carbon), while virtually all other gems contain multiple elements, including significant amounts of oxides. The infrared spectrum of a diamond is equally unique and can be used as an easy way to confirm that a stone is authentic. Figure 1 compares the spectrum from a diamond with several of the most common diamond simulants. The three materials are cubic zirconia (CZ: ZrO

), and clear corundum (white sapphire: Al

). The strong vibrational modes from the Al-O, Zr-O, and Si-O chemical bonds absorb the entire infrared signal below 1000 cm

A diamond consists of a crystalline lattice of carbon atoms; the extreme conditions required to create diamonds also provide a way for trace amounts of other elements to be trapped in the crystal matrix. The most important trace element is nitrogen, which can be found in different forms in the diamond crystal and reveals a great deal about the diamond's character and how it was created. These nitrogen aggregates create unique features in the infrared spectrum and are a key element in classifying diamonds. The spectra from three different types of nitrogen aggregates are compared with a nitrogen-free diamond (Type IIa) in Figure 2. 

Hydrogen, boron, and carbonates are other important trace materials that are found in diamonds, all of which have identifiable features in the infrared spectral region. Although it is easy to confirm that a stone is a diamond, it is much more difficult to determine if it is synthetic or treated. Laboratory-grown high pressure–high temperature diamonds are now available commercially, and their identification is one of the major challenges for the gem industry today. In certain cases, a similar process can be used to treat lower quality diamonds and significantly improve their appearance. The infrared peaks corresponding to the nitrogen aggregates or the presence of other elements can be used to provide valuable evidence that a stone might not be natural. In this article, several techniques based upon Fourier transform–infrared (FT-IR) spectroscopy that are commonly employed in gemological laboratories will be discussed.  

The instrument used in this work consisted of a Nicolet 6700 FT-IR spectrometer (Thermo Fisher Scientific, Madison, Wisconsin), equipped with an optimized beam condenser or reflection accessory. TQ Analyst (Thermo Fisher Scientific) multivariate analysis software was employed to calculate the results. The infrared spectra were acquired at 1-cm

  spectral resolution, and a baseline correction was applied before further processing. 

Rapid Confirmation That a Stone Is a Diamond

					Automatic material confirmation is a technique that has been used extensively in the pharmaceutical industry to ensure that incoming material is labeled correctly. Reference spectra are acquired from all of the materials that are used in the products, and then a similarity-matching algorithm is employed to calculate the similarity of the incoming material spectrum to the specified referenced material spectrum. For diamonds, a reference spectrum from a Type IIa diamond was used to develop a similarity-match method. A match value is then computed. The value will be 100 if the spectrum from the sample is identical to the reference and zero if there is no similarity between the two spectra. Because the spectra for all diamonds are slightly different, the match value will always be less than 100. In this experiment, setting a threshold for a pass of 80 gave no false positives and yet accepted a certain amount of variance in the sample spectra. The primary features observed in a spectrum of pure diamond are the infrared phonon bands resulting from the vibrational modes of the crystal lattice structure. These peaks are observed in the 1500–4000 cm

  spectral region. The major exception is Type IIb diamonds with high boron levels, which result in severely distorted features in the phonon region. An example of analyzing a typical diamond is shown in Figure 3. 

 A technique used extensively in quantitative analysis is classical least squares (CLS). The basis of CLS is the assumption that the spectrum from a sample that is a mixture of known materials can be modeled by a linear combination of the spectra from the pure reference materials. In this example, spectra from diamonds with well-defined nitrogen aggregates have been used: Type IaA, Type IaB, and Type Ib. Because we do not know the actual concentrations of nitrogen in the spectra, we have arbitrarily assigned the value to be 100 ppm, so the actual values reported here are relative to the arbitrary value for illustrative purposes. 

The results of this example CLS method are shown in Figure 4. Although the calculated standard error terms indicate that all three nitrogen types are present, adding more standards to the method might resolve any questions about the presence of the Type Ib aggregates. In many cases, an infrared spectrum is acquired from a stone before it undergoes the high temperature–high pressure process, because the nitrogen aggregates behave differently during the process. 

Many diamond spectra have a peak in the 1360 cm

  region, which has been attributed to the formation of platelets in the crystal. This peak is interesting in that it both shifts and broadens depending upon the overall makeup of the sample. Curve resolution, or peak fitting, is a technique that can be very effective in this type of situation. This technique synthesizes peaks based upon a predefined line shape and minimizes the difference between the actual spectrum and a linear combination of the synthetic peaks. Figure 5 shows two platelet peaks that are shifted by several wave numbers. 

The results of the Peak Resolve curve fit reveal that the full width at half maximum (fwhm) has gone from 3.2 cm

.  A closer examination of the curve fit results also shows that the peaks are asymmetric and can be fitted better with two synthetic peaks. 

Assisting in the Detection of Synthetic Diamonds

 In addition to the platelet peak, several other peaks in the spectrum can be helpful when trying to identify synthetic or treated stones. In many of these cases, the amount of nitrogen in the diamond is very low, so it can be difficult to confirm the presence of particular peaks. In this example, a simple CLS method was developed to confirm the presence of trace-level nitrogen and hydrogen by measuring the intensity of peaks at 3107 cm

.  A key advantage to the CLS approach is that a standard error term is reported with the analysis. Although it is not a rigorous statistical confidence level, if the peak intensity is three to five times the reported error term, the presence of a peak has been confirmed. An example for a sample containing very low levels of nitrogen is shown in Figure 6. In this example, all three peaks are present at detectable levels. 

Numerous gemological laboratories employ FT-IR on a daily basis to characterize diamonds and other gemstones. As with most natural materials, there is a broad range of trace component concentrations found in diamonds. As more sophisticated methods are developed to help ensure that a stone is natural and untreated, the developers of synthetic diamonds and high pressure–high temperature treatments also learn how to better mimic the natural processes. The combination of an FT-IR spectrometer and an optimized accessory can be used with multivariate statistical analysis techniques to provide a rapid, reliable information source of great importance to classifying diamonds. By automating these methods, the results are consistent and supported by a quality, or confidence, metric. In many cases, FT-IR is used by an operator to rapidly screen out the samples that clearly pass or fail the test, allowing the outliers to be examined by an expert. While it is straightforward to detect simulants, the proliferation of high-quality synthetic and treated stones makes advanced analysis techniques such as FT-IR a key part of the modern gemological laboratory. 

 is with Thermo Fisher Scientific, Madison, Wisconsin. 

(2) S.R. Boyd, I. Kiflawi, and G.S. Woods, 

(7) C.P. Smith, G. Bosshart, J. Ponahlo, V.M.F. Hammer, H. Klapper, and K. Schmetzer, 

(8) J.E. Shigley, S.F. McClure, C.M. Breeding, A.H. Shen, and S.M. Muhlmeister, 

(9) S. Fernandes, M. Kahn, and G. Choudhary, 

Diamonds are by far the world's most popular gemstone. Because of this popularity and the high prices of quality gemstones, there is a large market for cheaper counterfeit stones that resemble diamonds. Detecting these counterfeit stones can be a significant problem for diamond buyers when the stones are represented as the genuine article by unscrupulous sellers. Fourier transform-infrared (FT-IR) spectroscopy can be a useful tool for buyers and sellers to determine authenticity.

Analysis of Diamonds by FT-IR Spectroscopy

Principal component analysis (PCA) is arguably one of the more, if not actually the most, common methods of multivariate analysis applied to molecular spectroscopy nowadays. Therefore, we decided to present the long, complicated, tedious, and difficult approach for reasons that are good and sufficient, and that we will explain here and now. 

In addition to our own previous column, there are, of course, multiple discussions and presentations of the topic. There are books devoted entirely to the subject (2) and innumerable books whose main topic is statistics, chemometrics, or one or another field of spectroscopy or a related field that include one or more chapters on the topic. A good example is reference 3. All of those publications inevitably present the discussion using the formalism of matrix notation, and for good reason: writing out all the necessary equations otherwise is rather unwieldy. However, unless someone is really, really facile at slinging the matrix equations, matrix algebra loses the intuitive feel that the reader might like to develop for the subject. Those who are really, really facile at slinging matrix equations know who they are and don't need what we're about to do here; this is for the rest of us. 

We ran into a similar situation recently in some previous columns when we found that the presentation of multiple regression using the matrix formulation also was not a very intuitively understandable method of presenting the derivation of the equations needed to formalize the mathematical least squares definition of the multiple linear regression (MLR) algorithm. We found that going "back to the basics" of applying fundamental concepts from algebra, aided by some relatively elementary operations from calculus, was necessary to achieve intuitive clarity (4,5). 

So it is with principal components. The presentation of the subject in terms of matrix operations is rigorous and concise, but it also tends to form a barrier that separates the reader from an intuitive understanding of what is going on "under the hood" of the algorithm. After writing these columns for close to 30 years now (starting with "Statistics in Spectroscopy" before moving on to "Chemometrics in Spectroscopy"), one of the points we have come to appreciate is that while we are limited in what we can include in any single column, we can treat any topic at whatever length is necessary by spreading it out over several columns, which is what we plan to do with the explanation of principal component analysis. As described earlier, so many treatments simplify the algorithm past the point of comprehension that it's long past time for the long, complicated, but comprehensible treatment that can be achieved by doing it all (well, mostly, anyway) with a lot of algebra and a little calculus, just as we did with multiple regression. 

Therefore, over the next several columns, we will present the derivation of the principal component algorithm using elementary algebra. Now, strictly speaking, our presentation will not be completely algebraic. As we found necessary to do with derivation of MLR, we also will need to introduce some concepts from elementary calculus, in particular the use of derivatives, and the concept that the maximum or minimum of a function occurs at the point where the derivative is zero (but you remember all that, don't you?). For the most part, however, we eschew the use of matrix operations for the main derivations.  

We will include some matrix notation for some purposes and goals, however: 					

At main junctures, we will include the matrix equation corresponding to the derived result. This will allow readers to compare the result with other, more standard presentations, because the matrix equation will act as a "landmark" in the derivation. This also will allow readers to see algebraically how a result in the standard presentations came about.

We will use matrix notation for setting up some of the data arrays upon which we will operate. This will provide us with the advantages of clarity and conciseness of expression that matrices provide. For those who are not familiar with basic matrix notation, we have included the basics in Appendix A of this column.

Much of the analysis can be performed by treating the equations involved one at a time, as though they were univariate. Toward the end of the process, however, we will reach a point at which the analysis can continue and reach the desired goal only by recognizing and dealing with the inherent multivariate nature of the situation. At this point, it will become necessary to use inherently multivariate methodology, which is defined in terms of matrix behavior; thus, we will be forced to deal with matrices as such. That won't happen for a long time, though.

So we will stick primarily to the algebra for the derivations, including portions that seem complex. The apparent complexity, however, is not due to any use of advanced concepts; it is due more to the fact that we have all forgotten most (or, at least, much) of the elementary algebra we once knew rather than because the concepts are so advanced, difficult, or inherently complex. So the actual operations and derivations will be done algebraically to enhance the intuitive clarity that we are striving for. 

In the later stages of the derivations, it will become necessary to introduce some results based upon theorems that are inherently multivariate as described earlier, and then we will, of necessity, have to use matrix representations. We've done this sort of thing before, too. For example, in the previously discussed column about MLR (5), we had to introduce the concept of a matrix inversion. Sometimes there is no help for this. If and when it becomes necessary, we will try to minimize the pain and maximize the intuitive comprehension with our descriptions. We'll do our best, but there's only so much we can do. 

Principal components have been defined in many ways. One definition that captures the essential characteristics of principal component analysis was presented in reference 6. To paraphrase it slightly: 

Principal component analysis is a mathematical technique that generates eigenvectors, corresponding to the eigenvalues of a variance–covariance matrix, that are orthogonal, and unique linear combinations of which account for the largest possible amount of variance of the data with the fewest such linear combinations. 

The underlying basis of principal components is that they are a way of reconstructing the data matrix (for example, the spectra) from which they were derived. In this respect they are similar to Taylor series or Fourier series. In all three methods of reconstructing the data, the reconstruction is approximate; using the first term from a Fourier series or the first term from a Taylor series gives some degree of approximation. Using two terms gives a better approximation, and so forth, until sufficiently many terms are included (usually this means as many terms as there are data points in the function, or spectra in a data set), at which point the reconstruction becomes exact. Principal components have this property also. The first principal component gives some degree of approximation, and the approximation gets better and better as more principal components are included in the approximation. Eventually, the reconstruction becomes exact here, also. 

There is also a fundamental difference between Taylor (and Fourier) series as compared with principal components, however. The functions used when calculating a Taylor series (polynomials) or Fourier series (trigonometric functions: sines and cosines) involve approximations to the data that are defined by 

 mathematics. When performing this task using principal components, the functions used are defined by, or at least derived from, the data. In this sense, principal component analysis is more nearly akin to spectral subtraction or to Gauss–Jordan curve fitting; both of these are ways of fitting empirical functions (the spectra of the components of a mixture) to the spectra making up the data matrix (normally, the spectra of mixtures).  

Principal component analysis is a way to generate empirical functions that similarly can be used to reconstruct a data matrix, except that the functions so derived are abstract (in the sense that they don't necessarily represent any particular feature of the spectra of the components of the mixture comprising the samples whose spectra were measured) and have certain predefined mathematical properties. There are three key properties of the functions resulting from the calculation of principal components: 					

Within a set, they are all mutually orthogonal.

They capture the correlation structure of the data from which they were derived (that is, each principal component "explains" those parts of the data that are correlated with each other).

They are maximum-variance functions (the first principal component includes more of the variance of the data that any other single function can; the first two principal components include more of the variance of the data than any other two functions can, and so forth).

These properties are not entirely independent. As we'll see when examining the mathematics of the situation, the first two properties are, to some extent, consequences of the third, although they also can exist independently (in other types of data analysis). For example, the sines and cosines comprising a set of Fourier functions are also all mutually orthogonal, although they do not share the property of being maximum-variance functions. 

It is the third property that is unique to principal components and that distinguishes it from all other methods of approximating the data. Other functions, such as sines and cosines (Fourier components, generating this term by analogy), are orthogonal, and can be used to "explain," or reconstruct, the data. Other sets of functions, such as Taylor components (again using an analogy to represent the polynomials), also can be used for the same purpose, even though the underlying functions (polynomials in this case) are not necessarily even orthogonal. But principal components will reconstruct the data set from which they were derived better than any other set of functions can, and that is the key defining property of this method of data analysis (although we note that we have not yet defined what "better" means, in this sense). 

One further note about principal components being a "maximum variance estimator": As we will see once we get well into it, trying to derive the properties of principal components tends to lead to trivial solutions. Sometimes the trivial solution reached is one in which all the values making up the principal components are zero. This is obviously a minimum variance solution, because the variance of anything multiplied by zero is itself zero, and a variance cannot be less than that. 

Another type of trivial solution is one in which the various coefficients making up the principal component become larger and larger without bound (they approach being "infinitely" large). This is certainly one type of maximum variance solution, because the variance also would become "infinitely" large. Certainly infinity is the largest possible amount of variance that anything can have, just as zero is the smallest possible value for variance. This is not a useful solution, nor is it the solution we are seeking. In order to understand the intended meaning of "maximum variance estimators," we need to develop some preliminary results. 

We begin with a review, and some further development, of some of the concepts we discussed earlier, including the analysis of variance of data. Let us start with a data array, as shown in equation 1 (we remind our readers about Appendix A, for those who need it). 

Note: While our basic approach will avoid the use of matrices in the explanations, we will include the matrix expression corresponding to several of the results we derive. There are several reasons for this. One is that upon reading the literature, you will see the matrix formulations, and our including them here will allow you to relate the standard presentations to the underlying mathematical derivations. Another is that toward the end of our presentations, some of the global aspects will become easier to comprehend by avoiding some of the more confusing details (it will be easier to "see the forest without the trees," although as we just said, because you will have both presentations, you can relate them to each other). 

Thus, we see that the elements of the data matrix [

 columns. Generally, in our usage, the columns represent different variables (for example, the absorbance at different wavelengths) and the rows represent different samples. For the sake of keeping our intuition satisfied, we will consider that the rows of matrix [

] represent spectra of different samples, and the columns represent the absorbance values at the various wavelengths in each of the spectra. 

We showed previously (7) that the following relationship holds, for any given variable: 

 (TSS), for obvious reasons. The terms on the right-hand side of equation 2 are called the 

sum of squares due to the differences from the mean

 (SSE). We also had shown, a long time ago, that the mean itself is a least-square estimator of the data from which it was calculated (8). If we take the point of view that the mean is an approximation to the data, or a "prediction" of the data, then the difference 

mean is the "error" of the mean as an approximation, and the sum of squares corresponding to that is then the SSE.  

The first two terms of equation 2a are not of particular interest at this time, but the third term, the second one on the right-hand side of equation 2a, can be seen to be the variance of 

, the quantity whose square root is the standard deviation of 

. On the other hand, except for the division (or multiplication by the constant (

 – 1)), there is no difference in the basic structure or behavior between equations 2 and 2a.  

So for a given dataset, equation 2 shows that the TSS (the left-hand side of equation 2) can be split up into two other sums of squares, shown on the right-hand side of equation 2: the SSM and the SSE that add up to the original, TSS. This operation is known in statistical parlance as the 

. Similarly, equation 2a shows that a quantity akin to the variance of the 

 also can be split up, into terms that also have a correspondence to quantities that correspond to variances, of the pieces into which the total is split. Therefore, this operation also is called 

. We will see below how actual variances can be used to perform the same operation. 

If the deviations from the mean can be further separated into two meaningful subsets of data, in which the data points from each subset can be put into a one-to-one correspondence, then we can do a further partitioning of the sums of squares. We showed this previously also; if we designate the subsets of the initial partitioning representing Σ (

, then we show that the following relationship holds between two variables (5): 

It simplifies the conceptual picture of the situation (and also the notation) if we consider the data after the mean already has been subtracted from the data in the dataset, wavelength by wavelength, and along with it, the SSM. Because for any given data set the mean is constant and therefore the variance of the mean is zero, the subtraction of the mean from the rest of the data makes no change to the further analysis of the variance components of the data set and simplifies the equations by eliminating a useless term. We will therefore perform this simplification, doing it in two steps. 

Then we note that the left-hand side of equation 3 is the variance of (

), and the first two terms on the right-hand side of equation 3 are the variances of the 

 values separately. The third term is closely related to the correlation coefficient between 

. Therefore, if the two variables are uncorrelated (the correlation coefficient = 0), then equation 3 simplifies to 

That is, the variance of the sum equals the sum of the variances — we've seen this before, too (8). As before, subtracting the means of the data before doing the partitioning gives us the simplified expression: 

Similarly, as we did with equation 2, we can (in this case) multiply both sides of the equation by 

to demonstrate the corresponding relationship between the sums of squares. As in equation 2, there is no fundamental difference between using the sums of squares or using the variances. 

Applying this method to the relationship between the spectral data and the predicted values from the approximation made by principal components, we can identify these two pieces of data with the 

 of equation 6, and so begin our discussion. We will concentrate our attention on the data at one of the wavelengths. Let's consider some arbitrary wavelength, which we'll call wavelength 1, for the sake of simplicity. Let 

 subscript standing for "prediction," here) represent the value of a particular principal component at the first wavelength (let's consider only the first principal component). For any value of 

], in particular a value from the first wavelength (that is, 

 represents the approximation to the data at the first wavelength represented by the principal component.  

As mentioned earlier, we will consider the mean of the data set to have been subtracted from the data. This also has another consequence: in addition to making the mean of the resulting data set zero, it also makes the mean of the data set formed by the approximating functions zero. While it is not critical for the means to be zero, it is critical that they be the same, and making them zero is the easiest way to demonstrate that in fact they are the same, in addition to simplifying the notation. 

Making this change of notation, we can write the following identity: 

Chemometrics in Spectrosocpy

In this month's installment, columnists Howard Mark and Jerome Workman, Jr. present the derivation of the principal component algorithm using elementary algebra.

The Long, Complicated, Tedious, and Difficult Route to Principal Components: Part I

The 34th Annual Federation of Analytical Chemistry and Spectroscopy Societies (FACSS) meeting will be held from October 14–18, 2007, at the Memphis Cook Convention Center in Memphis, Tennessee. The conference is an annual meeting that covers the whole of analytical chemistry with an emphasis on spectroscopy. 

The Federation was founded in order to combine many small meetings, previously organized by individual societies, into an annual meeting that covers the entirety of analytical chemistry. The affiliate organizations of FACSS are ACS-Analytical Division, ANACHEM, American Society for Mass Spectrometry, Coblentz, ISA, Royal Society of Chemistry, and the Society for Applied Spectroscopy. The FACSS conference has also been established as the National Meeting for the Society for Applied Spectroscopy. 

The FACSS conference begins each day with plenary presentations given either by internationally recognized speakers or annual award recipients. After the daily plenary lectures, there are eight concurrent sessions within each half-day period, and each of those sessions is composed of poster and oral presentations. Prizes are awarded daily for outstanding poster presentations. Original research papers are presented at the conference from all areas of analytical chemistry and spectroscopy. Symposia topics will include atomic spectroscopy, chemometrics, forensics, mass spectrometry, nanotechnology, process control, surface plasmon resonance, bioanalytical, fluorescence, imaging science, molecular spectroscopy, pharmaceutical, Raman spectroscopy, and terahertz. 

Ira W. Levin, Deputy Director of the Division of Intramural Research and Chief of the Section on Molecular Biophysics, National Institute of Diabetes and Digestive and Kidney Diseases, NIH (Bethesda, Maryland), will open the 2007 meeting with a lecture titled 

Interdisciplinary Biophotonics: Molecular Domains to Organelles to Organs

. Bruce E. Bursten, president-elect of the American Chemical Society (Washington, DC), will present a plenary lecture titled 

Workshops and short courses are conducted by leading experts in their respective fields. The FACSS workshop program covers a wide range of subjects that span basic state-of-the-art topics in analytical chemistry and spectroscopy. 

The instrument exhibit is one of the focal points of FACSS. Ultimately, the realization of the research discussed during the FACSS symposia is exhibited at FACSS in the form of modern innovative instrumentation, software, and supplies. Exhibitors will also have the chance to participate in the "What's Hot" Vendor Symposium, where they will be given 7–10 minutes to present information about their products and services. 

Monday exhibit opening reception: 4:30 PM–6:30 PM

Each year, FACSS presents the Tomas A. Hirschfeld Scholar Award and FACSS Student Award to outstanding graduate students. The 2007 Tomas A. Hirschfeld award will be presented to Matthew V. Schulmerich, University of Michigan (Ann Arbor, Michigan), and Junrong Zheng, Stanford University (Stanford, California). The 2007 FACSS Student Award will be given to Sen Li, Purdue University (West Lafayette, Louisiana). 

Spectroscopy previews the 34th Annual Meeting of the FACSS, which will be held in Memphis, Tennessee.

Spectroscopy-09-01-2007

Market Profile: FT-IR for Security Applications

Mid-IR Luminescence of Nanocrystalline II-VI Semiconductors Doped with Transition Metal Ions

Analysis of Diamonds by FT-IR Spectroscopy

The Long, Complicated, Tedious, and Difficult Route to Principal Components: Part I

34th Annual FACSS Meeting

Light-Emitting Diodes

Products