This study explores the problem of fast estimation of soil particle content by visible–near-infrared (vis-NIR) spectroscopy. Four spectral pre-processing methods were used, and it was found that the multiplicative scatter correction (MSC) pre-processing transformation can maintain the original spectral characteristics and effectively eliminate the influence of scattering, with the best effect. Compared with the spectral index (SI) established by the original spectral data, the SI established by the MSC greatly increases its correlation and sensitivity with the soil particle content. The partial least squares (PLS) and random forest (RF) models were optimized using the successive projections algorithm (SPA), and it was found that the model complexity and calculation amount were greatly reduced, and the model accuracy was improved. The reserved samples were verified, and a good prediction effect was achieved. Our results show that the combination of spectral index and successive projection algorithm can be used as an effective means for rapid prediction of soil particle content.

Soil, composed of solid, liquid, and gas, plays a vital role in human life. As the global population increases and arable land decreases, our understanding of soil continues to deepen. Texture is one of the most basic physical properties of soil, and it has an important impact on soil permeability, water retention, and nutrient content. The soil texture can indicate the characteristics of the soil and provide a strong basis for the evaluation of soil fertility and crop suitability. Traditional soil texture measurement methods include hydrometry and laser particle sizing methods, which require large-scale sampling in the field and are low in efficiency and high in cost, making it difficult to measure large areas (1).

In recent years, hyperspectral technology has developed rapidly and has been widely used in the analysis of soil organic matter, heavy metals, and texture because the technology is fast and convenient. Decai Wang and his group established a soil texture prediction model using the original spectrum, differential processing, and orthogonal signal correction (OSC) pretreatment, discovering that the OSC algorithm could effectively eliminate irrelevant factors and improve the prediction accuracy of the model (2). Yanying Bai and his group established a soil texture prediction model based on the topsoil of farmland in river irrigation areas, and the results showed that the unitary regression model had the same accuracy as the back propagation (BP) neural network model (3). Na Zhang and his group studied the soil texture of the Hetao irrigation area and used unitary linear regression, stepwise multiple regression, and the BP neural network for modeling, discovering that the BP neural network was the most effective (4). Haiyan Song‘s group used a combination of the orthogonal signal and partial least square (PLS) models to effectively distinguish soil with different textures (5). Shimeng Li’s team explored the sensitive bands of the different particle-size distributions in the Yimin open-pit coal mine area using the thermal infrared spectrum, and found that the prediction accuracy for coarse sand grains using the support vector machine model was better than that of fine sand grains (6). Gmur’s group used regression tree algorithms in the 400–1000 nm spectrum to provide a powerful and fast method for evaluating nitrogen, carbon, and organics in the upper layers of soil (7). Gomez’s team calculated common soil properties using a PLS method, showing that CaCO

, iron, clay, and cation exchange capacity (CEC) can be obtained by hyperspectral data mapping (8).

This brief summary reveals that our predecessors have done a lot of research on spectral prediction and have achieved good results, but most of them concentrated on single-band applications. Hyperspectral data have a continuous narrow band, and a lot of useful information is hidden between the bands. Therefore, it becomes important to explore the powerful interpretation of soil texture by combining different bands. This study uses soil from central and southern Hebei, performing spectral preprocessing based on Matlab and Python programming platforms, to find the best spectral preprocessing method and then combine the spectral index to optimize the wave band. Band selection is performed by the successive projection algorithm (SPA) as an input variable for linear and nonlinear models. An effective prediction method for soil particle content is discussed to provide theoretical and methodological support for predicting the soil texture and other soil components using hyperspectral remote sensing technology.

The study area is located in the central and southern part of Hebei Province, North China, and is bounded by the 36°30′–40°30′ latitude to the north and the 114°–119°30′ longitude to the east. Located between a warm temperate zone and a wetland zone, four distinct seasons are clear, and there are about 2303 annual sunshine hours, which is a temperate and continental monsoon climate. The landform is complex and diverse, and the terrain is high in the northwest and low in the southeast. There are many soil types in the area, mainly brown soil and fluvo-aquic soil. The study area contains several types of plants and rich natural resources. Because of this, the study area has become an important grain and cotton producing area in China.

Figure 1 shows the technical process followed in the study. The soil samples were collected from the field. Randomly arranged grids were used to obtain 101 soil samples. The soil sampling depth was 0–20 cm, with the layout point as the center, and four more sampling points were collected around this center point using the plum pile method. During the collection process, stones, weeds, branches, and other debris were removed and uniformly mixed as the final soil sample. The soil sample was air-dried, ground through a 2 mm screen, and divided into two portions, each of 100 g. One portion was used to determine the soil particle size content, and the other was used for spectral measurement. The soil particle size content was determined by a particle size analyzer, and the international soil texture classification standard was adopted. The soil texture was divided according to the three-grain ratio of sand (2–0.02 mm), silt (0.02–0.002 mm), and clay (<0.002 mm).

The soil spectrum data were measured by a FieldSpec 4 spectrometer (Analytical Spectral Devices), and its wavelength range was 350–2500 nm. The spectrometer has different sampling intervals and resolution settings between different spectral regions. Within 350–1000 nm, the sampling interval was 1.4 nm and resolution was 3 nm; within 1000–2500 nm, the sampling interval was 2 nm and resolution was 10 nm. The light source was a 50 W halogen bulb, 50 cm away from the sample, the irradiation azimuth was 70°, with the signal collection probe 10 cm perpendicular to, and directly above, the sample. The sample was placed in a petri dish 10 cm in diameter and 2 cm in height, and the surface was scraped flat. Because the spectral measurement was susceptible to interference by external light, the spectral experiment was carried out in a dark room with no room light. To improve the optimization performance of the instrument and the accuracy of the spectral measurement, the instrument was on for 30 min, and then the instrument was calibrated according to the sequence of dark current acquisition, instrument optimization, and white reflectance correction. To ensure the accuracy of spectral data collection, white reflectance correction was carried out every 10 samples during the measurement. RS3 software (included with FieldSpec4 equipment) was used for spectral data collection, and ViewSpecPro software (also included with FieldSpec4 equipment) was used for data post- processing. Each sample was measured 10 times, and its average value was taken as the final sample spectrum.

Soil is a complex material composed of a variety of components, among which organic matter, heavy metals, and water content have a certain influence on the measured spectra. First, S–G convolution smoothing was performed on the 101 sample spectra to achieve a preliminary noise reduction. Second, to better highlight the relationship between the soil particle size and the spectral data, and to eliminate noise interference and to separate overlapping samples, the S–G smoothed spectral data were subject to first order differential (FDR) (9–11), reciprocal logarithm (RL) (12,13), and multivariate scatter correction (MSC) (14,15) pre-processing transformations.

The hyperspectral data has a continuous narrow band, which contains substantial spectral information. The spectral index method of random combination of two bands can make full use of the hidden information in the spectrum, extract the spectral bands sensitive to soil particle size, and improve the correlation between the spectrum and the soil particle size (16,17). This study uses four types of soil spectral indexes, including normalized differential soil index (NDSI), difference soil index (DSI), ratio soil index (RSI), and bare soil index (BSI). The calculation formula is as follows:

 are the spectral reflection values corresponding to the selected spectral wavelengths λ

The common linear regression methods used include multiple linear regression, logistic regression, and principal component regression. PLS, combining the advantages of principal component analysis (PCA), canonical correlation analysis (CCA), and linear regression, can handle multiple correlations between independent variables and allow modeling to be performed when the number of samples is less than the dimension (wavelengths) of the spectra (18,19).

Random forest (RF), first proposed by Leo Breiman and Adele Cutler, is an algorithm that uses multiple decision trees to train and predict samples (20,21). It can handle high-dimensional data and is not easy to fit. First, the bootstrap method is used to randomly extract 

 samples from the training set and these are used to build 

 decision trees. Assuming that there are 

 features in the training set, the best feature is selected for splitting, and each tree is split by feature until all training examples of the node belong to the same class. Second, each tree is allowed to grow to the maximum without pruning. Finally, multiple decision trees are formed into a random forest for classification or regression of new data. In the regression problem, the final prediction result is determined from the average of the predicted values of multiple decision trees.

The near-infrared reflection spectrum mainly indicates the absorption and scattering of organic substances. Because of the serious overlap of information in the multiple spectrum bands of different substances and the large number of spectral data bands, there is a lot of redundant information and noise in the full-band spectrum, which increases the workload and affects the model accuracy. Therefore, it is necessary to reduce the dimension of the data, reduce redundant information, and improve the running speed and stability of the model. The continuous projection algorithm is a method of selecting variables in a forward loop (22,23). It can extract valid information from severely overlapping spectral information, minimize the colinearity between spectral variables, and improves the modeling conditions of multiple linear regression models (24,25).

In this study, the successive projections algorithm (SPA) is used to select the spectral features. The main SPA algorithm principle is as follows (26,27): The initial iteration vector, the number of variables to be extracted and the spectral matrix are denoted as 

Step 2: Let the position of the unselected column vector be s, s={

 on the remaining column vectors in turn:

= 0,... ...,N − 1}, corresponding to each 

. Multiple linear regression analysis (MLR) is carried out after the cycle, and the internal cross-root mean square error (RMSE) of the correction set is obtained. The 

 corresponding to the minimum RMSE value are the optimal values.

Statistical Analysis of Soil Particle Content


To obtain descriptive statistical characteristics of the soil particle composition at the actual measurement points, the following analysis was performed using SPSS software (Table I). The soil texture in the study area was dominated by sandy loam and loam, with the highest content of sand and silt reaching 86.37 and 64.97%, while t e clay content was relatively low, ranging from 0.37 to 11.22%. The standard deviation of the contents of the three kinds of soil particles were <12%, and the degree of dispersion was small. The coefficient of variation (CV) was 18–46%, in which the coefficient of variation of the clay content reached >40%, belonging to the moderate variation intensity, which may be affected by different agricultural activities at various sampling points in the study area.

Spectral Feature Analysis and Data Preprocessing


A ground object spectrometer was used to obtain the soil spectral curve, and the original spectrum was subjected to S–G smoothing. Due to the large noise at both ends of the spectrum, the band data in the ranges of 350–399 nm and 2481–2500 nm were removed. Figure 2 shows the spectral data processed by S–G, FDR, MSC, and RL.

Soil contains many components, such as water and organic matter. A soil vis-NIR spectrum represents the combined signals of all soil components. Figure 1 shows that the spectral reflectance after S–G smoothing is between 0.1 and 0.5. Although the reflectance of each soil sample is different, the overall trend is roughly the same. The reflectance increases rapidly from 400 to 1000 nm in visible light, and the reflectance changes relatively smoothly in the near-infrared from 1000 to 2480 nm.

The spectral absorption band position and spectral reflectance changes are related to the chemical composition of the soil. It can be found that the characteristic bands of the spectrum were mainly located at 585 to 683 nm, 707 to 930 nm, 986 to 1061 nm, 1381 to 1467 nm, 1860 to 2032 nm, 2142 to 2217 nm, and 2315 to 2400 nm. It is generally believed that the perturbation of visible light and some near-infrared bands is caused by interactions of water, organic matter,, and metal ions—such as Fe

. Around 600 nm a reflection peak of organic matter is observed, and near 820 nm there is a sub-absorption peak of organic matter (28). The near-infrared spectrum after 980 nm is mainly the frequency doubling and harmonic frequencies generated by the bending vibration of group molecules formed by metal elements and hydroxide groups, such as Fe-OH, Al-OH, and Mg-OH. At 1400 to 1500 nm and 1850 to 1950 nm, there is a combined frequency band of the water molecule hydroxyl stretching vibration and Al-OH bending vibration in the soil silicate mineral, while the characteristic absorption band appearing at 2150 to 2250 nm is affected by the OH functional group in the organic matter (29). The absorption peak near 2455 nm is the band generated by the vibration of the CO

After FDR treatment, the overlapping samples were separated and the spectral characteristics peaks were more obvious, with different degrees of absorption bands around 1000, 1400, 1900, and 2200 nm. The reflectivity of the spectral curve after MSC treatment was between 0.1 and 0.5. By carrying out baseline translation and offset correction with the average spectrum data, the influence of scattering was effectively eliminated, and the changed characteristics of the original spectral data were retained, which were closer and smoother than the original spectrum. The difference between the spectral curve after LR treatment and the original spectral curve was the opposite. The overall reflectivity was between 1 and 2, and the reflectance decreased rapidly within the range of 400 to 800 nm, while the change of the spectral curve after 800 nm was relatively stable.

Correlation Analysis of the Soil Spectrum


Correlation analysis was performed between the smoothed original spectral data and the spectral data using different preprocessing transforms with the soil particle content (Table II). In the range of 400– 2480 nm, the spectral index of random pair-band combination under the original spectrum was calculated, and its correlation with the soil particle content was analyzed (Figure 3). The degree of correlation was determined using Pearson’s correlation coefficient, which is represented by 

It can be seen in Table II that the correlations were improved by varying degrees after spectral pretreatment. In the original spectrum, the absolute values of the maximum correlation coefficients of 0.477, 0.477, and 0.376 were obtained at 2478 nm for sand, silt and clay. The number of bands of sand and silt above 0.45 reached 133 and 142, and the number of bands of clay above 0.35 reached 176. After FDR treatment, the correlation coefficient was slightly improved. The maximum absolute values of the correlation coefficients of sand, silt, and clay were 0.519, 0.509, and 0.482, but there were fewer bands with absolute correlation coefficients above 0.4. After MSC processing, the correlation was significantly improved. The maximum absolute values of the correlation coefficients of sand, silt, and clay reached 0.69, 0.678, and 0.685, respectively. The maximum increase of the correlation coefficients was 0.309, and there were more bands with absolute values above 0.65. The RL pre-processing effect was general, and the absolute value of the correlation coefficients were all lower than 0.5, but the overall band correlation was improved compared to the original spectrum. Figure 3 shows the equipotential diagram of the correlation between the spectral index of sand and silt was in the opposite state, which may be because of the negative correlation between the content of sand and silt. The distribution trends of the NDSI, RSI, and DSI spectral indices on the same soil particle size were generally the same. The areas with high correlation of NDSI and RSI were mostly concentrated at the combination of bands of 700 to 1500 nm and 1850 to 2480 nm, while the areas with high correlation of DSI were slightly wider than the previous two, and mainly concentrated at 500 to 1600 nm and 1850 to 2480 nm. The effect of BSI on the three kinds of soil particles was not obvious, and the correlation coefficient reached only 0.4 at the band combination of 2150 to 2480 nm.

To further improve the correlation between the spectral index and the soil particle content, the MSC form with the best pretreatment effect was combined with four spectral index algorithms for band combination (Figure 4). Selected published domestic and foreign spectral index models, combined with the MSC processed spectral data in this paper, were substituted into each model in accordance with the corresponding combined bands, and the correlation analysis was conducted with different soil particle content to explore its applicability in determining soil texture (Table III).

Applying the spectral index calculation formulas published in previous studies, it was found that the correlation between different spectral indices and the soil particle content was significantly different, with r between -0.427 and 0.406. TCARI, TCARI/OSAVI, REIP, and TVI showed a slight correlation (|

| < 0.15), of which CRII had the highest correlation with sand content, with a maximum value of 0.427, while OSAVI had the highest correlation with silt content at 0.406. It can be seen that the correlation between the spectral index calculation formulas published by predecessors and the soil particle content is weak, which makes it difficult to estimate the soil particle content. Therefore, it is important to explore what spectral index is suitable for the estimation of soil particle content. Figure 4 shows that the spectral index established by the MSC transformation improved significantly, mainly indicated by the increase of the band combination range and overall correlation. The distribution trends of NDSI, RSI, and DSI are still roughly the same, and they have risen from the original near-infrared band combination of 750 to 1450 nm and 1900 to 2480 nm to the visible light and near-infrared band combination of 500 to 1800 nm and 1900 to 2480 nm. The maximum absolute value of the correlation of sand content reached 0.7360, 0.7361, and 0.7368, respectively, and the corresponding band combinations (in nm) were (990, 2480), (989, 2480), and (989, 2480). The maximum absolute values of silt content correlation reached 0.7107, 0.7128, and 0.7125, respectively, and the corresponding band combinations were (2479, 989), (989, 2479), and (2479, 989). The clay content achieved the absolute maximum correlation at the (2479, 994) band combination, with values of 0.7525, 0.7635, and 0.7472, respectively. The BSI spectral index established by MSC processing had the most significant improvement over the BSI spectral index established by the original spectral data. The absolute value of the correlation between the content of sand, silt, and clay increased from 0.4765, 0.4770, and 0.3751 to 0.7491, 0.7300, and 0.7452, respectively, with a maximum increase of 98.66%. This shows that the MSC can effectively reduce the effects of baseline translation and offset and enhance the correlation between the absorption spectrum of the measured component and the concentration data. The combination of MSC and the spectral index algorithm can reveal more hidden valuable spectral information, which plays a vital role in later modeling.

SPA to Extract Characteristic Wavelengths


The spectral data processed by MSC were used to construct a spectral index for regression analysis. Because the spectral index was combined through two bands, there are many independent variables, which made the data redundant and reduced modeling accuracy. Because of this, the dimensionality of the data should be reduced to decrease the number of independent variables (31). The distribution of NDSI-, RSI-, and DSI- correlated isopositive graphs were similar, and the effect of DSI was slightly better than the previous two.

Therefore, the following uses the SPA to select only BSI and DSI spectral index data. The SPA first calculates the root mean square error (RMSE) under different numbers of effective wavelengths, and then determines the number of effective wavelengths based on the minimum RMSE. Figure 5 and Table IV show the number of effective bands and specific band combinations determined by the SPA, respectively. It can be seen from the chart that the bands and number selected by the SPA are different for different soil particle content. For sand, the number of effective wavelengths determined by BSI and DSI spectral indices were 10 and 14, respectively, and the corresponding RMSE values were 6.6545 and 6.114, respectively. The corresponding band combinations were mainly concentrated in 400 to 420 nm, 580 to 1000 nm, 1320 to 1944 nm, and 1915 to 2480 nm. For silt, the number of effective wavelengths determined under the BSI and DSI spectral indices were 8 and 10, respectively, and the RMSE values were 5.603 and 5.2149, respectively. The main band combinations were concentrated at 900 to 1000 nm, 1680 to 1900 nm, and 2470 to 2480 nm. Based on the BSI and DSI spectral indices, the number of effective wavelengths of the clay were 22 and 12, respectively, and the RMSE values were 0.87297 and 0.84298, respectively. The corresponding band combinations were mainly at 550 to 1000 nm, 1350 to 1830 nm, and 2470 to 2480 nm. Generally, the sensitive combination bands all come from the positions affected by various chemical components in the soil, such as the absorption bands of organic matter, and the spectral responses of metal ions, the vibrational bands of water molecules, and the spectral bands generated by the vibration of the CO

 group in the soil carbonate. It has been proven that soil, as an organically complex material, has many component interactions. To achieve the prediction of soil particle content, the combination of these factors should be considered as a whole.

Articles

Peer-Reviewed Research

A rapid vis-NIR spectroscopy method for determining soil particle size and quality.

Optimization of a Soil Particle Content Prediction Model Based on a Combined Spectral Index and Successive Projections Algorithm Using Vis-NIR Spectroscopy

The need to calibrate and qualify analytical instruments and systems is well recognized in laboratories. However, calibrating and qualifying analytical instruments and systems are seen as discrete tasks. There is an increasing regulatory expectation that a risk-based life cycle approach is necessary to ensure “fitness for purpose” as part of an analytical procedure. The calibration and qualification lifecycle of analytical instruments and systems is a journey and not a sequence of disconnected events.

This article describes the principal stages of the lifecycle in both GXP or ISO/IEC 17025 laboratories that ensure that any analytical instrument and system is “fit for its intended use” as part of an analytical procedure, whether that is calibration or qualification. However, the tasks typically associated with these stages are often seen as discrete activities, but they should be integrated to form a continuum. In other words, the calibration and qualification lifecycle of analytical instruments and systems is a continuous journey, not a sequence of disconnected events. It is timely to consider a lifecycle approach because the United States Pharmacopoeia (USP) proposed a new General Chapter <1220> in September 2020 for analytical procedure lifecycle management (APLM) (1). There are no novel elements in this approach, but there is a structured process linking these activities that is described here.

Our starting point is the U.S. Good Manufacturing Practice (GMP) regulation for equipment design, size, and location in 21 

Equipment used in the manufacture, processing, packing, or holding of a drug product shall be of appropriate design, adequate size, and suitably located to facilitate operations for its intended use and for its cleaning and maintenance (2,3).

What does the phrase “intended use” mean and how should “fitness for intended use or purpose” be demonstrated in a regulated laboratory? Using our analytical instruments for analysis, we reviewed the high level process flow for the analytical procedure lifecycle from the proposed USP <1220> shown in Figure 1 (1). Notice that the key part of the process is stage 1 (Procedure Design, Development and Understanding). Method development is critical to the success of the whole process, and the draft USP <1220> covers an activity that the current International Council on Harmonization (ICH) Q2(R1) does not (4). Consequently, analysts developing methods must ensure that their analytical instruments are qualified and calibrated correctly and that the analytical systems are validated to ensure the smooth development and transfer of methods to the next stage of the lifecycle, which is called procedure performance qualification (PPQ). Qualification and calibration of analytical instruments is particularly important when analytical procedures are developed and validated in Research and Development (R&D) and later transferred to quality control laboratories.

Figure 1 shows that analytical procedure validation is no longer a discrete activity but a journey. As part of that journey, the first aspect of stage 2 PPQ requires qualification and calibration of instruments and systems as a prerequisite.

Analytical Process: Where Does Qualification and Calibration Fit? 

The analytical process, conducted within a pharmaceutical quality management system, provides reliable, traceable, and reportable values that are secure. The basic analytical process is shown in Figure 2 together with the relationship to stage 2 of the APLM highlighted in blue.

If we cannot demonstrate “fitness for purpose” of our instruments and systems, then outputs from our analytical process are unreliable. The current regulatory focus on data integrity and data governance requires us to put in place technical and procedure controls throughout the analytical process to ensure that data and records are trustworthy and reliable, and that quality decisions can be made in confidence.

The data governance and data integrity requirements and controls for a proposed analytical instrument and system (AIS) lifecycle that is defensible from a regulatory and technical perspective in Figure 2 can be mapped to the Foundation and Level 1 of a Data Integrity Model (5-8), shown in Figure S1, online. The Foundation ensures the right culture, ethos, and training for data governance and Level 1 ensures that instruments and computerized systems are calibrated, qualified and where appropriate validated for their intended use.

Analytical Instrument and System Lifecycle

The overall purpose of the AIS lifecycle is to demonstrate that not only is the instrument or system calibrated or qualified, but that any software is suitably validated and is operating in a controlled state. It is only then may we claim that the instrument or system is “fit for intended purpose” so that metrological data integrity is ensured. This article focuses on level 1 in this model concerning metrological integrity. The AIS lifecycle is shown in Figure 3. Data quality is only ensured if all components of data integrity and data governance are in place.

The levels of the AIS, with a specific focus on analytical instruments and systems, are presented at the bottom of Figure 3. Levels 1 and 2 are broken down into four elements each to provide metrological and procedural integrity, respectively. Levels 1 and 2 also overlay the culture and ethos of the foundation level. The quality oversight and some of the foundation level elements of the model are combined under data governance. Together these provide the essential requirements for the right reportable result at level 3 with associated data quality to ensure the correct decision is made.

The top portion of Figure 3 is based on the requirements of USP <1058>, analytical instrument qualification (9)that requires the following:

1) A laboratory user requirements specification (URS) to select the right instrument for the job (design qualification)

2) Following purchase of the instrument, a supplier IQ/OQ is per- formed to demonstrate that the requirements in the laboratory URS have been met. If this is not the case, the laboratory needs to conduct supplementary qualification tests to match the operating ranges specified.

3) Performance qualification (PQ) tests to demonstrate that the laboratory specifications are being met during operation of the instrument must be devised along with a plan for requalification of the instrument both after repair and maintenance but also periodically. 

4) The PQ test performance should be trended to detect any changes as well as maintaining deviation management and change control to demonstrate that the instrument is in a state of control.

The authors have published a risk assessment to help classify analytical instruments and systems into the USP <1058> groups and subtypes and suggest the extent of computerized system validation to be undertaken (10). Combining all the activities shown in Figure 3 ensures that analytical instruments and systems generate data and reportable results with appropriate integrity and quality.

Recently in Pharmacopoeial Forum, a stimulus to the revision process article was published discussing a lifecycle approach to both calibration and qualification (11). However, before we discuss this article, we need to consider the applicable FDA GMP regulations, by going back to the future.

Since the promulgation of the GMP regulations in 1978, approaches to ensuring that analytical instruments are “fit for intended use” have changed along with the terminology. Analytical instruments have changed and software, either in the form of firmware or a separate data system, now comprises a major component of modern analytical instrumentation. However, the wording of the regulations has not changed but the interpretation of them has. In 21 

 211.160(b) there are requirements that any laboratory control mechanisms must be scientifically sound and in 211.160(b)(4) we have the following requirement for calibration (3):

The calibration of instruments, apparatus, gauges, and recording devices at suitable intervals in accordance with an established written program containing specific directions, schedules, limits for accuracy and precision, and provisions for remedial action in the event accuracy and/or precision limits are not met. Instruments, apparatus, gauges, and recording devices not meeting established specifications shall not be used.

The records for such calibrations are defined in 211.194 (d) (3):

Complete records shall be maintained of the periodic calibration of laboratory instruments, apparatus, gauges, and recording devices required by 211.160(b)(4).

The problem is that the U.S. GMP regulations only mention calibration and therefore citations in 483 observations and warning letters will only cite deficiencies in instrument calibration when a qualification lapse could be the cause of the noncompliance, resulting in potential confusion. However, as our understanding of how to demonstrate fitness for intended use has evolved, so has the terminology.

The terms used within this section are one of the sources of great confusion between laboratories. For example, 

 can sometimes be used as a general “descriptive label” or have a specific meaning within the context of use. In addition to calibration, we now have the terms 

 also in common use. USP general chapter <1058> on analytical instrument qualification (9) provides an integrated approach to qualification of the analytical instrument and validation of the software. This general chapter mentions the terms calibration, qualification, verification, and validation. What do these terms mean? Are they linked? And where are we going in the future? All organizations, including regulatory authorities, need a technical glossary. There is no technical glossary that ensures a consistent meaning and use of terms throughout the pharmacopeia and other regulatory documents.

It is generally accepted that calibration forms part of the overall qualification process of an instrument or system. Calibration is all about metrology. Calibration is that part of qualification relating to measurement integrity of the ordinate response and abscissa functions, that is, all measurements from the instrument or system are within defined acceptance limits of a true or certified value.

The foundation for assurance of metrological “fitness for purpose” lies with the correct use of either a certified reference material (CRM) or a reference standard (RS). In the pharmacopeia, calibration is a comparison between a known or accepted value of a CRM or RS with its measured value on a specific laboratory analytical instrument or system being used for a pharmacopeial purpose the difference between these two values is compared with an acceptance criterion that may be an acceptable range or tolerance interval. For example, the measured value of a certified standard value of a wavelength standard must lie within the acceptance criteria of ±2 nm over the spectral range 200 to 400 nm. For some analytical instruments, calibration may all that is required (9).

Qualification is about overall “fitness for purpose” of an instrument or system, not only about good numbers. Hence the pharmacopeias require qualified instruments and systems. 

 <1058> is the only pharmacopoeial general chapter to describe the overall process of analytical instrument qualification (9).

 relates to the “fitness for purpose” of analytical procedures as required by the General Chapter on Verification of Analytical Procedures <1226> (12). However, verification is also used in the American National Standards Institute (ANSI) for software to refer to demonstrating that each individual step is mathematically or procedurally correct. GAMP 5 describes the testing phases of a computerized system lifecycle to demonstrate meeting user specifications (13). Therefore, some aspects of modular calibration (see later section) may be regarded as a verification process. The current version of 

 <1058> on analytical instrument qualification (AIQ) does not use 

Currently, validation relates either to the establishment of analytical procedures as required by General Chapter <1225>, Validation of Analytical Procedures (14) or particularly in General Chapter <1058>, Analytical Instrument Qualification for “fitness for purpose” (9). For example, in the “Instrument Control, Acquisition and Processing Software” section of <1058>, it states:

At the user site, integrated qualification of the instrument, in conjunction with validation of the software, involves the entire system. This is more efficient than separating instrument qualification from validation of the software (9).

However, there is a diversity of definitions for calibration, qualification, verification, and validation within the USP General Chapters that need to be harmonized to assist understanding and prevent user confusion. This is particularly true for the differentiation of holistic and modular calibration standards and the expression of acceptance criteria for establishing “fitness for purpose” (15,16).

The perennial question of “who does what and when” is answered in the lifecycle approach to calibration shown in Figure 4. The solid lines denote the primary process flows for establishing and confirming “fitness for purpose.” Dotted or dashed lines indicate either feedback loops triggered by data generated during the ongoing performance phase or, in the case of the URS, supported by the vendor.

The generic process of calibration and qualification should be designed to allow confirmation of “fitness for purpose” at all stages in the lifecycle including assurance of preventative maintenance and change control.

Users need to define their own control strategy to specify ongoing “fitness for purpose” by establishing the frequency of calibration, standards and acceptance criteria that will be employed and trend analysis of the ongoing performance data from the instrument.

One important aspect of the calibration and qualification lifecycle is the presence of change control feedback loops as part of the lifecycle. For example, if the qualified operational wavelength range for a UV spectrometer was originally 240 to 600 nm and the requirement is to extend this to 210 to 600 nm, then a new qualification standard is required to confirm “fitness for purpose” (17–19). In addition, the laboratory URS needs to be reviewed to ensure that the instrument was correctly specified originally to allow the revision of operational range.

Do You Specify Exactly What You Need the Instrument or System to Do?

 <1058> (9) is defined in a user specification. These are laboratory requirements for any analytical instrument with scientifically sound acceptance criteria. This does not mean simply copying the supplier’s specification, as discussed in an earlier column (18) but defining what is actually required. For example, if a UV-vis spectrometer has a supplier’s wavelength specification between 190 and 800 nm, what do you actually use? For example, if you only require measurement between 210 and 300 nm, then this should be your operating range. The requirement for wavelength accuracy is contained in 

 <857> (20) and this should be quoted in the specification document for the instrument. Testing wavelength accuracy using a CRM (for example, a holmium perchlorate solution or a glass filter) should demonstrate that this user requirement is met and that the instrument performs within scientifically sound acceptance criteria. Similar approaches need to be undertaken for the other technical requirements for the spectrometer.

Best Practices for Instrument Calibration

The best practice is to use the most stable artifact with the smallest uncertainty budget. For example, for routine monitoring of UV-vis and near infrared spectrometers, solid or liquid filters traceable to a national measurement institution (NMI), for both wavelength and absorbance accuracy qualification provide the most reliable option for the user.

For example, Figure 5 shows an overview of available standards for UV-vis-NIR over the overall spectral range available from one manufacturer. For example, to cover an operational wavelength range of 200 to 795 nm, two references would be required.

Focus on Quality

The calibration and qualification of analytical systems is a journey, not an event.

Are You Ready for a Life Cycle Approach for Analytical Instruments and Systems?

Group theory in conjunction with Raman polarization selection rules may be applied to the study and characterization of crystalline materials. The Polarization–Orientation (P-O) Raman method employs this relationship, and is useful for the identification of a material’s crystal class, orientation, degree of disorder, and determination of the symmetry species to which the Raman bands in a spectrum belong. Furthermore, it can be used to distinguish allotropes and polymorphs, differentiate single from polycrystalline materials, and determine whether materials at different locations in a fabricated device are crystallographically aligned. We discuss the theory and sensitivity of P-O Raman spectroscopy in the backscattering mode applied to crystals for which the nonzero Raman polarizability tensor elements of symmetry species are neither necessarily equal nor proportional. We demonstrate the effects of Raman polarizability tensor relative values and crystal orientation by calculating P-O Raman diagrams for the A and E symmetry species of the trigonal 

Raman crystallography is the application of group theory and Raman polarization selection rules to the study and characterization of crystalline materials. Characterizing crystalline materials by Raman spectroscopy is a good alternative when sample size or structure precludes one from doing so by X-ray diffraction. Where sample morphology or spatial variation of chemical composition or crystalline structure is on a scale that precludes or makes impractical the use of X-ray diffraction, polarized micro-Raman spectroscopy can often obtain crystallographic information. Applications of applied polarized Raman spectroscopy include characterization of minerals, integrated electronic and electro-optic devices, microelectromechanical systems (MEMS), and organic electronic devices including photovoltaics and organic light emitting diodes (OLEDs).

Most spectroscopists are familiar with Raman spectroscopy as a method of chemical analysis because of its sensitivity to the masses of a compound’s constituent atoms and their bond lengths and force constants. In addition, Raman spectroscopy can also be used for the characterization of the atomic structure of solids because Raman scattering depends on the polarization and direction of the incident light, the crystal symmetry and orientation of the solid sample, and the direction and polarization of the collected scattered light. Micro-Raman spectroscopic methods for determining local crystal orientation in silicon have been developed which take advantage of these properties (1,2). Raman crystallography can be applied to many materials, and the interested reader can find very elegant polarized Raman spectroscopic applications to characterize GaN (3), sapphire (4), and zirconia (5).

Polarization–Orientation Raman Spectroscopy

We discussed one method of Raman crystallography called Polarization–Orientation (P-O) Raman spectroscopy in a prior “Molecular Spectroscopy Workbench” installment (6). The P-O Raman method is useful for the identification of a material’s crystal class, orientation, degree of disorder, and determination of the symmetry species to which the Raman bands in a spectrum belong. To understand how this relationship can be applied to the characterization of crystals, we begin by considering the most fundamental expressions related to Raman scattering:

where α is the Raman polarizability tensor representing the vibrational modulation of the polarizability, 

 is the incident electric field vector, and 

 is the time varying dipole moment (also a vector) induced by the interaction of the incident electric field with the vibrationally modulated polarizability of the chemical bond. The Raman polarizability tensor is fixed relative to the positions of the atoms and the directions of the chemical bonds between them. Therefore, for a crystal fixed in space the relationship described by equations 1 and 2 is dependent upon the crystal symmetry and orientation of the sample relative to the direction and polarization of the incident and collected light. This relationship constitutes that part of the expression for Raman scattering intensity of interest to us in equation 3:

 is the intensity of the incident light, 

 is a factor incorporating the dependence on the frequency of the incident light, 

, the polarization of the incident light, and 

 and Ω are the polarization and solid angle, respectively, at which scattered light is collected. The significance of this expression for work with crystals is that Raman scattering intensity is proportional to the square of the dot product of the incident electric field vector, Raman polarizability tensor, and scattering vector.

The theory and general application of P-O Raman spectroscopy in the backscattering mode is based on the interdependence of incident electric field, Raman polarizability tensor and scattering vector. Most important is that the forms (whether elements are zero or nonzero) of the Raman polarizability tensors for crystals are dictated by the symmetry species of the crystal class to which the material belongs. The absolute magnitude of the tensor elements will be dictated by the specific chemical bonds between the atoms. However, whether a Raman tensor element is zero or nonzero is entirely dictated by the crystal symmetry. There are 32 crystal classes and the forms of the Raman polarizability tensors have been determined for all of them and tabulated in various publications (7–9).

In that previous introduction to P-O Raman spectroscopy, we demonstrated good agreement between the experimentally obtained results from single crystal Si and the calculated predictions for several crystallographic faces (6). The most common crystalline form of Si is cubic with two atoms in the primitive cell and belonging to crystal class 

. The first order optical phonon at 520 cm

 is triply degenerate and there are three 

 Raman polarizability tensors of the form:

where the letter d indicates a nonzero value. An experimental arrangement can be envisioned in which the laser beam is incident along the 

-axis with respect to the laboratory reference frame. This arrangement corresponds to the Raman microscope backscattering configuration. A cubic single crystal Si chip is positioned such that its crystallographic XY-plane is coplanar with the laboratory XY-plane. Backscattered light is collected in two configurations, with the analyzer either parallel or perpendicular to the polarization of the incident laser light. Raman scattering strengths (normalized) equal to the square of the dot product in equation 3 have been calculated for the analyzer parallel and perpendicular to the incident polarization as the crystal is rotated in the XY-plane. The calculated P-O diagram for Raman backscattering signal strength of the first order optical phonon at 520 cm

from the XY-face of cubic Si is shown in Figure 1.

The parallel and perpendicular responses are sinusoidal functions 45° out of phase, and, of course, because of the cubic symmetry the response is the same for the XZ- and YZ-faces. However, if the crystal is oriented out of these planes, one begins to see the effect of orientation on the P-O Raman plots. The method used to transform the Raman polarizability tensors from one crystal orientation to another is the transformation through Euler’s angles (10). The interested reader can refer to our previous publication to compare the P-O Raman results for the (100), (110) and (111) faces of cubic Si (6).

We have performed P-O Raman calculations for many other Si crystal orientations and have found the method to be very sensitive to crystal orientation. In fact, this sensitivity will be true for crystals of all symmetry classes because the P-O Raman diagrams depend only on the forms of the Raman polarizability tensors. Crystals of symmetry lower than that of the cubic system can have Raman polarizability tensors in which the nonzero elements are not equal, and in those cases, the P-O diagram will be affected by the relative magnitudes of the tensor elements. In fact, comparing calculated P-O Raman diagrams to the experimentally obtained P-O plots allows one to infer relative Raman scattering cross-sections of vibrational modes from crystals. The form or pattern of P-O Raman diagrams are dependent upon crystal symmetry and the relative magnitudes of the nonzero tensor elements, which depend upon the specific elemental, compositional, or molecular identity of the crystal. That is what makes the P-O Raman method a useful basis for Raman crystallography.

One can immediately recognize the application of this method for the characterization of crystal structure and orientation in either naturally occurring materials or fabricated devices. The P-O Raman method is complementary to X-ray diffraction, and should be considered when X-ray analysis is not practical or possible. P-O Raman spectroscopy may be used to identify vibrational modes, distinguish allotropes and polymorphs, distinguish single from polycrystalline materials, and determine orientation of the crystal and degree of disorder. Furthermore, one can determine whether materials at different locations in a fabricated device are crystallographically aligned based upon the phases of the experimentally obtained P-O Raman diagrams.

The nonzero elements of the triply degenerate 

 crystal class are all equal. Therefore, only the crystal orientation can affect the P-O Raman diagram for that symmetry species. The cubic system is a special case of high symmetry and all of the nonzero tensor elements of the various symmetry species of the cubic classes are either equal or proportional to each other. That is not the case for the other 27 crystal classes that do not belong to the cubic system.

Here, we discuss the theory and more general application of P-O Raman spectroscopy in the backscattering mode applied to crystals for which the nonzero Raman tensor elements of symmetry species are neither equal nor proportional. We demonstrate the effects of Raman tensor relative values and crystal orientation by calculating P-O Raman diagrams for the 

 crystal classes. The Raman tensors of these symmetry species are shown in equation 5. The 

 symmetry species is doubly degenerate and so there are two Raman polarizability matrices associated with that species.

 symmetry species in equation 5 and note that only the diagonal tensor elements are nonzero. Furthermore, α

, and, as we will see, the relative magnitudes of the tensor elements will affect the PO Raman diagram depending upon the sample’s crystal orientation. The P-O Raman experimental arrangement can be envisioned in which the laser beam is directed along the 

-axis, with respect to the sample’s crystallographic axes and at a crystal rotational angle of 0°. The XY-face of the trigonal 

 crystal is positioned coplanar with the XY-plane of the laboratory frame, and backscattered light is collected along the z-axis. Raman scattering strengths for α

 = 10 have been calculated for the analyzer parallel and perpendicular to the incident polarization as the crystal is rotated in the XY-plane. The calculated P-O diagram for Raman backscattering signal strength from the XY-face of the A symmetry species of a trigonal 

 crystal is shown in Figure 2. The P-O Raman responses for both parallel and perpendicular configurations are invariant with crystal orientation. The parallel Raman signal is allowed and always equal to the squared α

 values of 1, whereas the perpendicular response is forbidden, yielding a Raman signal of zero. Mathematically, the reason for these responses is that both diagonal tensor elements within the XY-plane of rotation are equal to 1, whereas α

 is perpendicular to the plane of rotation, and does not contribute to the signal in the backscattering configuration. In this experimental arrangement the specific value of α

, which in this example is 10, has no effect on the P-O Raman diagram.

Now, envision rotating the crystal 90° such that the laser is incident upon the YZ-face and is polarized along the 

-axis when the crystal rotation angle is at 0°. Relative Raman scattering strengths have been calculated for the analyzer parallel and perpendicular to the incident polarization as the crystal is rotated in the YZ-plane. In this configuration, the relative magnitudes of the α

 values directly affect the P-O Raman diagram, because the crystal rotation is in the YZ-plane and the dot product of equation 3 varies with crystal orientation. The calculated P-O diagrams for Raman backscattering signal strength from the YZ-face of the A symmetry species of a trigonal 

 values are shown in Figure 3. Note that the P-O Raman diagrams of the YZ- and XZ-faces are identical because the α

 values for the P-O Raman diagrams in Figure 3 are 0.01, 0.25, 0.5, and 0.75, all less than 1, the value we designated for α

. The Raman signal strengths for parallel and perpendicular configurations vary with crystal orientation, with the perpendicular response varying at twice the frequency as that of the parallel configuration. The P-O Raman diagrams of the A symmetry species obtained from the YZ-face are entirely different from that of the XY-plane, which was invariant with crystal angle. That is because the contributing Raman tensor elements are not equal when probing the YZ-face, whereas those contributing in the XY-plane are equal. The Raman signal reaches a maximum in the parallel configuration at 90° and 270°, whereas in the perpendicular configuration the Raman signal peaks at 45°, 135°, 225°, and 315°. The crystal angle separation between the maxima are 180° and 90° for the parallel and perpendicular polarization configurations, respectively. The P-O Raman response maxima remain at the same angles of crystal orientation for all α

. However, there are two noticeable variations in the P-O Raman diagrams, as the α

 values are changed relative to that of α

. The valley at 180° becomes less broad and more rounded with increasing α

 values. Also, the amplitudes of the parallel and perpendicular responses decrease, and the two functions separate, as the α

. These calculated P-O Raman diagrams demonstrate that experimentally obtained P-O Raman responses can be used to identify the crystalline face, assign the symmetry species of a particular Raman band, and determine the relative values of the α

 Raman polarizability tensor elements of the A symmetry species when α

Now, we examine those cases in which the Raman tensor values of the A symmetry species are α

 values for the P-O Raman diagrams in Figure 4 are 1, 1.5, 2.5, and 10, all greater than or equal to the value we designated for α

. The P-O Raman responses for both parallel and perpendicular configurations are invariant with crystal orientation when αxx = α

 = 1. The parallel Raman signal is allowed, and always equal to the squared α

 values of 1, whereas the perpendicular response is forbidden yielding a Raman signal of zero. This is the response that one would expect because, mathematically, the situation here is the same as for the XY-face P-O Raman diagram shown in Figure 2, where the contributing diagonal Raman tensor elements are equal. Looking back at Figure 3 with Figure 4a in mind, one can see the progression of the P-O Raman diagrams culminating in the invariant response of Figure 4a as the value of α

. The P-O Raman diagram of Figure 4b, with α

 = 1.5, appears similar to that of Figure 3d, with α

 = 0.75, with respect to the separation of the parallel and perpendicular responses. However, the phases of the parallel responses in Figures 4b–d are shifted 90° relative to those in Figure 3. Now the Raman signal reaches a maximum in the parallel configuration at 0°, 180°, and 360°, whereas in the perpendicular configuration the Raman signal peaks at 45°, 135°, 225°, and 315°, just as it did for α

. The phase shift of the A symmetry species P-O Raman parallel response informs us whether the α

 value is greater or less than that of α

, and the amplitudes and separations of the parallel and perpendicular responses allow us to determine the relative magnitudes of the contributing Raman polarizability tensor elements.

Thus far, we have discussed the P-O Raman response of only the A symmetry species. Now we turn our attention to the Raman active doubly degenerate E symmetry vibrational modes. The two 

 Raman polarizability tensors shown in equation 5 both apply when calculating the P-O Raman diagram for this symmetry species because of its double degeneracy. You can see from equation 5 that the 

 symmetry species is more complicated, because it has both diagonal and off-diagonal Raman tensor elements that are nonzero. Also, there are both positive and negative tensor element values. Nevertheless, there are only two absolute values or magnitudes in the two Raman tensors indicated by the letters 

 in equation 5. The differences of the magnitudes of tensor elements 

 can yield a wide variety of P-O Raman diagrams. We show just a sampling here so that you can get a sense of how sensitive the P-O Raman method is to crystal orientation and the relative values of the Raman tensor elements.

The P-O Raman experimental arrangement can be envisioned in which the laser beam is directed along the 

-axis with respect to the sample’s crystallographic axes and at a crystal rotational angle of 0°. The XY-face of the trigonal 

 crystal is positioned coplanar with the XY-plane of the laboratory frame, and backscattered light is collected along the z-axis. Raman scattering strengths for αxx = 2, α

 = -1 have been calculated for the analyzer parallel and perpendicular to the incident polarization as the crystal is rotated in the XY-plane. The calculated P-O diagram for Raman backscattering signal strength from the XY-face of the 

crystal is shown in Figure 5. The P-O Raman responses for both parallel and perpendicular configurations are invariant with crystal orientation and each is equal to a Raman signal of 4, the square of the tensor values α

 = -2. Mathematically, the reason for these responses is that only the Raman tensor 

elements within the XY-plane of rotation contribute to the Raman signal, and they all have the same absolute value of 2. The nonzero Raman tensor elements α

 are perpendicular to the plane of rotation, and do not contribute to the signal in the backscattering configuration. In this experimental arrangement the specific value of α

, which in this example is 1, has no effect on the P-O Raman diagram.

Note that the P-O Raman diagrams of both the 

symmetry species are invariant with crystal orientation within the XY-plane. Nevertheless, the two symmetry species can be differentiated and assigned to Raman bands from P-O Raman plots experimentally obtained from the XY-face of a crystal belonging to a trigonal 

 crystal class. That is because the perpendicular response of the 

 symmetry species is not allowed, whereas that of the 

 symmetry species is allowed and is equal to the parallel configuration response.

Now, envision rotating the crystal 90° such that the laser is incident upon the YZ-face and is polarized along the z-axis when the crystal rotation angle is at 0°. Relative 

Raman scattering strengths have been calculated for the analyzer parallel and perpendicular to the incident polarization as the crystal is rotated in the YZ-plane. Figure 6 shows the P-O Raman diagram for the YZ-face of a trigonal 

 = -1. Here, the P-O Raman diagram is no longer invariant with crystal orientation, even though the nonzero tensor absolute values contributing to the YZ-plane are 

all equal to 1. The P-O Raman diagram is oscillatory, and the parallel and perpendicular responses differ from each other in phase by 45°. The last two maxima of the parallel and perpendicular responses are labeled, and you can see that their separation within each configuration is 90°. Of course, the 

 nonzero tensor elements of equation 5 are not likely to be equal for a real uniaxial crystal. However, setting them equal demonstrates that the oscillatory response of the P-O diagram as shown in Figure 6 arises primarily from the contribution of the nonzero off-diagonal tensor elements to the YZ-plane, whereas equal values assigned to the diagonal and off-diagonal tensor elements within the XY-plane as in Figure 5 generate a P-O Raman diagram invariant with crystal rotation.

Having reviewed the P-O Raman response when the absolute values of the nonzero Raman tensor elements are equal, now let us analyze the effects when the relative magnitudes of the c and d nonzero tensor elements of equation 5 are varied. Figure 7a shows the P-O Raman 

 = -1; the absolute value of the Raman tensors in the XY-plane equals 1.5, and that in the XZ- and YZ-planes equals 1. We can see the effect of the increase in tensor magnitudes in the XY-plane by comparing this P-O Raman diagram with that of Figure 6. There are two noticeable differences between the two. The first is that the parallel and perpendicular responses in Figure 7a are shifted -5° relative to those in Figure 6 (the last two maxima of the parallel and perpendicular responses are labeled to assist in recognizing this shift). Also, the first and third maxima of the parallel configuration in Figure 7a peak at a lower signal strength than that of the corresponding maxima in Figure 6. Increasing the absolute value of the Raman tensors in the XY-plane to αxx = 5, α

 = -1 advances this progression, as shown in Figure 7c. Both parallel and perpendicular responses are shifted -15° relative to those in Figure 7a. Also, the first and third maxima of the parallel configurations in Figures 6 and 7a have now vanished.

It is useful to reverse the Raman tensor magnitudes of the XY-plane and the XZ- and YZ-planes, as shown in Figures 7b and 7d, to see the effects on the P-O Raman diagram. Compare the P-O Raman diagrams of Figures 7a and 7b in which the magnitudes of the Raman tensor elements 

have been maintained but simply reversed for the planes containing the ordinary axes (

). The parallel and perpendicular responses in Figure 7b appear similar to those in Figure 7a but are not identical. Several differences are clear. First, the phases of both the parallel and perpendicular responses of Figure 7b are shifted +10° relative to those in Figure 7a. Additionally, the first and third maxima of the parallel configurations in Figure 7b are more intense than the corresponding maxima in Figure 7a. Finally, the perpendicular response in Figure 7b is relatively stronger than that in Figure 7a. We see a similar progression when comparing the P-O Raman diagrams of Figures 7c and 7d. The phases of both the parallel 

and perpendicular responses of Figure 7d are shifted +32° relative to those in Figure 7c. Also, the perpendicular response in Figure 7d is relatively stronger than that in Figure 7c.

These subtle effects of the Raman tensor magnitudes on the phases and amplitudes of the P-O Raman diagrams are not intuitive. One might expect that such changes would occur if a crystal is rotated to some degree off of its crystallographic axis with respect to the direction of laser 

propagation, but not from changes in the Raman tensor magnitudes. P-O Raman calculations help one to anticipate and understand these effects. This observation is important, because more Raman spectroscopists are using experimental methods of laser polarization in conjunction with sample orientation to infer crystal phase and orientation along with the assignment of symmetry species to particular Raman bands. The calculated P-O Raman 

diagrams show how significantly the magnitudes of the Raman tensor elements affect the P-O Raman results and make clear that such calculations are necessary to properly interpret the experimentally obtained results of polarized Raman measurements obtained from crystals whose Raman tensor elements are not equal.

In this article, we discussed the application of group theory and Raman polarization selection rules to the study and characterization of crystalline materials. Specifically, we described one method of Raman crystallography called polarization–orientation Raman spectroscopy. The P-O Raman method has been shown to be useful for the identification of a material’s crystal class, orientation, degree of disorder, and determination of the symmetry species to which the Raman bands in a spectrum belong. Here, we discussed the theory and more general application of P-O Raman spectroscopy in the backscattering mode applied to crystals for which the nonzero Raman polarizability tensor elements of symmetry species are neither necessarily 

equal nor proportional. We demonstrated the effects of Raman polarizability tensor relative values and crystal orientation by calculating P-O Raman diagrams for the A and E symmetry species of the trigonal 

 crystal classes. The calculated P-O Raman diagrams show how significantly the magnitudes of the Raman polarizability tensor elements affect the P-O Raman results, and make clear that such calculations are necessary to properly interpret and understand the experimentally obtained results of polarized Raman measurements obtained from crystals whose Raman polarizability tensor elements are not equal.

(3) H.C. Lin, Z.C. Feng, M.S. Chen, Z.X. Shen, I.T. Ferguson, and W. Lu, 

(4) M.C. Munisso, W. Zhu, and G. Pezzotti, 

(5) K. Fukatsu, W. Zhu, and G. Pezzotti, 

 (Academic Press, New York, New York, 1972), pp. 358–361.

, D.J. Gardiner and P.R. Graves, Eds. (Springer-Verlag, New York, New York, 1989), pp. 22–24.

(10) E.B. Wilson, Jr., J.C. Decius, and P.C. Cross, 

Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra

 (McGraw-Hill, New York, New York, 1955), p. 285.

Raman spectroscopy is extremely useful for characterizing crystalline materials.

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