knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
This document describes methods for quantitative analysis implemented within powdR via a range of reproducible examples that use open source data from the package.
One of the most powerful properties of XRPD data is that the intensities of crystalline (e.g., quartz, calcite and gypsum), disordered (e.g., clay minerals), and amorphous (e.g., volcanic glass and organic matter) signals within a diffractogram can be related to their concentrations within the mixture. This principal facilitates the quantification of phase concentrations from XRPD data.
Of the approaches available for quantitative XRPD analysis, the simple Reference Intensity Ratio (RIR) method has consistently proven accurate. A RIR is a measure of the diffracting power of a phase relative to that of a standard (most often corundum, Al~2~O~3~), usually measured in a 50:50 mixture by weight. The RIR of a detectable phase within a mixture is required for its quantification.
A given diffractogram can be modeled as the sum of pure diffractograms for all detectable phases, each scaled by different amounts (scaling factors). By combining these scaling factors with RIRs, phase concentrations can be calculated. Hereafter this approach is referred to as full pattern summation. Full pattern summation is particularly suitable for mixtures containing crystalline mineral components in combination with disordered and/or X-ray amorphous phases (e.g. soil), and further details on its implementation in powdR are provided in @Butler2021powdR.
A key component of the full pattern summation functions within powdR is the library of reference patterns. These are stored within a
powdRlib object created from two basic components using the
powdRlib() constructor function. The first component, specified via the
xrd_table argument of
powdRlib(), is a data frame of the count intensities of the reference patterns, with their 2θ axis as the first column. The column for a given reference pattern must be named using a unique identifier (a phase ID). An example of such a format is provided in the
library(powdR) data(minerals_xrd) head(minerals_xrd)
The second component required to build a
powdRlib object, specified via the
phases_table argument of
powdRlib(), is a data frame containing 3 columns in the following order.
phase_id: a string of unique IDs corresponding to the names of each reference pattern in the data provided to the
xrd_tableargument outlined above.
phase_name: the name of the phase group that this reference pattern belongs to (e.g. quartz, plagioclase, illite etc.).
rir: the reference intensity ratios of the reference patterns (relative to a known standard, usually corundum).
An example of the format required for the
phases_table argument of
powRlib() is provided in the
Crucially, when building the
powdRlib object, all phase IDs in the first column of the
phases_table must match the column names of the
xrd_table (excluding the name of the first column which is the 2θ axis), for example.
powdRlib objects can easily be visualised using the associated
plot() method (see
?plot.powdRlib), which accepts the
interactive arguments that are used to specify the X-ray wavelength, the reference patterns to plot, and the output format, respectively. In all cases where
plot() is used in this document, the use of
interactive = TRUE in the function call will produce an interactive html graph that can be viewed in RStudio or a web browser.
my_lib <- powdRlib(minerals_xrd, minerals_phases) plot(my_lib, wavelength = "Cu", refs = c("ALB", "DOL.1", "QUA.1", "GOE.2"), interactive = FALSE)
There are three
powdRlib objects provided as part of the powdR package:
data(minerals)], which is a simple and low resolution library designed to facilitate fast computation of basic examples.
data(rockjock)], which is a comprehensive library of 169 reference patterns covering most phases that might be encountered in geological and soil samples. The
rockjocklibrary in powdR uses data from the original RockJock program [@Eberl2003] thanks to the permission of Dennis Eberl. In
rockjock, each reference pattern from the original RockJock program has been scaled to a maximum intensity of 10000 counts, and the RIRs normalised relative to Corundum. All
rockjockdata were analysed using Cu Kα radiation.
data(afsis)], which contains 21 reference patterns measured on a Bruker D2 Phaser as part of the XRPD data analysis undertaken for the Africa Soil Information Service Sentinel Site programme. These are designed to supplement the
rockjocklibrary when analysing soil XRPD data.
To accompany the
rockjock reference library, a list of eight synthetic mixtures from the original RockJock program are also included in powdR in the
rockjock_mixtures data [accessed via
data(rockjock_mixtures)], and the known compositions of these mixtures provided in the
rockjock_weights data [accessed via
Occasionally it may be useful to subset a reference library to a smaller selection. This can be achieved using
subset(), which for
powdRlib objects accepts three arguments:
x argument specifies the
powdRlib object to be subset,
refs specifies the IDs and/or names of phases to select, and
mode specifies whether these phases are kept (
mode = "keep") or removed (
mode = "remove").
data(rockjock) #Have a look at the phase IDs in rockjock rockjock$phases$phase_id[1:10] #Remove reference patterns from rockjock rockjock_1 <- subset(rockjock, refs = c("ALUNITE", #phase ID "AMPHIBOLE", #phase ID "ANALCIME", #phase ID "Plagioclase"), #phase name mode = "remove") #Check number of reference patterns remaining in library nrow(rockjock_1$phases) #Keep certain reference patterns of rockjock rockjock_2 <- subset(rockjock, refs = c("ALUNITE", #phase ID "AMPHIBOLE", #phase ID "ANALCIME", #phase ID "Plagioclase"), #phase name mode = "keep") #Check number of reference patterns remaining nrow(rockjock_2$phases)
powdRlib objects from different instruments can be interpolated and then merged using the
merge methods (see
merge.powdRlib), respectively. For example, the
minerals library can be merged with the
rockjock library after interpolation using:
#Load the minerals library data(minerals) #Check the number of reference patterns nrow(minerals$phases) #Load the rockjock library data(rockjock) #Check the number of reference patterns nrow(rockjock$phases) #interpolate minerals library onto same 2theta as rockjock minerals_i <- interpolate(minerals, new_tth = rockjock$tth) #merge the libraries merged_lib <- merge(rockjock, minerals_i) #Check the number of reference patterns in the merged library nrow(merged_lib$phases)
In simpler cases where two libraries are already on the same 2θ axis and were measured using the same instrumental parameters, only the use of
merge() would be required.
#Load the afsis library data(afsis) identical(rockjock$tth, afsis$tth) rockjock_afsis <- merge(rockjock, afsis)
Once you have a
powdRlib reference library and diffractogram(s) loaded into R, you have everything needed for quantitative analysis via full pattern summation. Full pattern summation in powdR is provided via the
fps() function, whilst an automated version is provided in
afps(). Details on these functions are provided in @Butler2021rc and @Butler2021powdR.
fps() is specifically applied to
powdRlib objects, and accepts a wide range of arguments that are detailed in the package documentation (see
?fps.powdRlib). Here the
rockjock_mixtures data will be used to demonstrate the main features of
fps() and the various ways in which it can be used.
Often samples are prepared for XRPD analysis with an internal standard of known concentration. If this is the case, then the
std_conc arguments of
fps() can be used to define the internal standard and its concentration (in weight %), respectively, which is then used in combination with the reference intensity ratios to compute phase concentrations. For example, all samples in the
rockjock_mixtures data were prepared with 20 % corundum as the internal standard, thus this can be specified using
std = "CORUNDUM" and
std_conc = 20 in the call to
fps(). In addition, setting the
omit_std argument to
TRUE makes sure that the internal standard concentration will be omitted from the output and the phase concentrations recomputed accordingly. In such cases the phase specified as the internal standard can also be used in combination with the value specified in the
align argument to ensure that the measured diffractogram is appropriately aligned on the 2θ axis. These principles are used in the example below, which passes the following seven arguments to
libis used to define the
powdRlibobject containing the reference patterns and their RIRs.
smplis used to define the data frame or
XYobject containing the sample diffractogram.
refsis used to define a string of phase IDs (
lib$phases$phase_id) and/or phase names (
lib$phases$phase_names) of the reference patterns to be used in the fitting process.
stdis used to define the phase ID of the reference pattern to be used as the internal standard.
std_concis used to define the concentration of the internal standard in weight %.
omit_stdis used to define whether the internal standard is omitted from the output and phase concentrations recomputed accordingly.
alignis used to define the maximum positive or negative shift in 2θ that is permitted during alignment of the sample to the reference pattern that is specified in the
data(rockjock_mixtures) fit1 <- fps(lib = rockjock, smpl = rockjock_mixtures$Mix5, refs = c("ORDERED_MICROCLINE", "Plagioclase", "KAOLINITE_DRY_BRANCH", "MONTMORILLONITE_WYO", "CORUNDUM", "QUARTZ"), std = "CORUNDUM", std_conc = 20, omit_std = TRUE, align = 0.3)
Once computed, the
fps() function produces a
powdRfps object, which is a bundle of data in list format that contains the outputs (see
The phase concentrations can be accessed in the
phases_grouped data frames of the
#All phases fit1$phases #Phases grouped and summed by the phase name fit1$phases_grouped
Further, notice that when the concentration of the internal standard is specified then the phase concentrations do not necessarily sum to 100 %:
sum(fit1$phases$phase_percent, na.rm = TRUE)
It's also possible to "close" the mineral composition so that the weight percentages sum to 100. This can be achieved in two ways:
closed = TRUEin the
close_quant()function to the
For example, the phase composition in
fit2 created above can be closed using:
fit1c <- close_quant(fit1) sum(fit1c$phases$phase_percent, na.rm = TRUE)
In cases where an internal standard is not added to a sample, phase quantification can be achieved by assuming that all detectable phases can be identified and that they sum to 100 weight %. By setting the
std_conc argument of
NA, or leaving it out of the function call, it will be assumed that the sample has been prepared without an internal standard and the phase concentrations computed accordingly.
fit2 <- fps(lib = rockjock, smpl = rockjock_mixtures$Mix5, refs = c("ORDERED_MICROCLINE", "Plagioclase", "KAOLINITE_DRY_BRANCH", "MONTMORILLONITE_WYO", "CORUNDUM", "QUARTZ"), std = "CORUNDUM", align = 0.3)
In this case the phase specified in the
std argument is only used for 2θ alignment, and is always included in the computed phase concentrations.
Furthermore, the phase concentrations computed using this approach will always sum to 100 %.
The fitted patterns resulting from full pattern summation are most commonly derived by minimising an objective function. This process is computationally intensive and can therefore prove slow when a large number of scaling coefficients (i.e. a large number of reference patterns) are used. As a fast alternative to this approach, non-negative least squares [NNLS; @nnls] is also implemented in
fps() and can be defined using the
#Create a timestamp a <- Sys.time() fit2_n <- fps(lib = rockjock, smpl = rockjock_mixtures$Mix5, refs = c("ORDERED_MICROCLINE", "Plagioclase", "KAOLINITE_DRY_BRANCH", "MONTMORILLONITE_WYO", "CORUNDUM", "QUARTZ"), solver = "NNLS", std = "CORUNDUM", align = 0.3) #Calculate computation time Sys.time() - a
resulting in a computation time of less than half a second. Whilst the use of NNLS is fast, there is a small compromise in accuracy compared to the minimisation of an objective function [see Supplementary Material in @Butler2021powdR].
The selection of suitable reference patterns for full pattern summation can often be challenging and time consuming. An attempt to automate this process is provided in the
afps() function, which can select appropriate reference patterns from a reference library and subsequently exclude reference patterns based on limit of detection estimates. Such an approach is considered particularly advantageous when quantifying high-throughput XRPD datasets that display considerable mineralogical variation such as the Reynolds Cup [@Butler2021rc].
All of the principles and arguments outlined above for the
fps() function also apply to the use of
afps(). However, there are a few additional arguments for
afps() that need to be defined:
forceis used to specify phase IDs (
lib$phases$phase_id) or phase names (
lib$phases$phase_name) that must be retained in the output, even if their concentrations are estimated to be below the limit of detection or negative.
lodis used to define the limit of detection (LOD; in weight %) of the phase specified as the internal standard in the
stdargument. This limit of detection for the defined phase is then used in combination with the RIRs to estimate the LODs of all other phases [see @Butler2021rc and @Butler2021powdR for more details].
amorphousis used to specify which, if any, phases should be treated as amorphous. This is used because the assumptions used to estimate the LODs of crystalline and disordered phases are not appropriate for amorphous phases.
amorphous_lodis used to define the LOD (in weight %) of the phases specified in the
rockjock library, containing 169 reference patterns, will be used to quantify one of the samples in the
rockjock_mixtures data. Note that when using
afps(), omission of the
refs argument in the function call will automatically result in all phases from the reference library being used in the fitting process.
#Produce the fit a_fit1 <- afps(lib = rockjock, smpl = rockjock_mixtures$Mix5, std = "CORUNDUM", align = 0.3, lod = 1)
afps() accept a
shift argument, which when set to a value greater than zero results in optimisation of a small 2θ shift for each reference pattern in order to improve the quality of the fit. The value supplied to the
shift argument defines the maximum (either positive or negative) shift that can be applied to each reference pattern before the shift is reset to zero.
This shifting process is designed to correct for small linear differences in the peak positions of the standards relative to the sample, which may result from a combination of instrumental aberrations, mineralogical variation and/or uncorrected errors in the library patterns. Whilst this shifting routine provides more accurate results, the process can substantially increase computation time.
Occasionally it can be useful to apply a different grouping structure to the phases quantified within a
powdRafps object. This can be achieved using the
regroup function (see
#View the phases of the fit1 output fit1$phases #Load the rockjock regrouping structure data(rockjock_regroup) #View the first 6 rows head(rockjock_regroup) #Regroup the data in a_fit1 using the coarsest description fit1_rg <- regroup(fit1, rockjock_regroup[c(1,3)]) #Check the regrouped data fit1_rg$phases_grouped
powdRafps objects, derived from
afps(), respectively, is achieved using
plot(fit1, wavelength = "Cu", interactive = FALSE)
powdRafps objects the wavelength must be defined because it is required to compute d-spacings that are shown when
interactive = TRUE.
In addition to above, plotting for
powdRafps objects can be further adjusted by the
xlim arguments. When the
group argument is set to
TRUE, the patterns within the fit are grouped and summed according to phase names, which can help simplify the plot:
plot(fit1, wavelength = "Cu", group = TRUE, interactive = FALSE)
mode argument can be one of
"fit" (the default),
"both", for example:
plot(fit1, wavelength = "Cu", mode = "residuals", interactive = FALSE)
or alternatively both the fit and residuals can be plotted using
mode = "both" and the 2θ axis restricted using the
plot(fit1, wavelength = "Cu", mode = "both", xlim = c(20,30), interactive = FALSE)
The simplest way to quantify multiple samples via either
afps() is by wrapping either of the functions in
lapply() and supplying a list of diffractograms. The following example wraps the
fps() function in
lapply and applies the function to the first three items within the
multi_fit <- lapply(rockjock_mixtures[1:2], fps, lib = rockjock, std = "CORUNDUM", refs = c("ORDERED_MICROCLINE", "Plagioclase", "KAOLINITE_DRY_BRANCH", "MONTMORILLONITE_WYO", "ILLITE_1M_RM30", "CORUNDUM", "QUARTZ"), align = 0.3, std_conc = 20, omit_std = TRUE)
lapply in this way, the names of the items within the list or
multiXY object supplied to the function are inherited by the output:
lapply is a simple way to quantify multiple samples, the computation remains restricted to a single core. Computation time can be reduced many-fold by allowing different cores of your machine to process one sample at a time, which can be achieved using the
foreach packages, for example:
#Install the foreach and doParallel package install.packages(c("foreach", "doParallel")) #load the packages library(foreach) library(doParallel) #Detect number of cores on machine UseCores <- detectCores() #Register the cluster using n - 1 cores cl <- makeCluster(UseCores-1) registerDoParallel(cl) #Use foreach loop and %dopar% to compute in parallel multi_fit <- foreach(i = 1:2) %dopar% (powdR::fps(lib = rockjock, smpl = rockjock_mixtures[[i]], std = "CORUNDUM", refs = c("ORDERED_MICROCLINE", "LABRADORITE", "KAOLINITE_DRY_BRANCH", "MONTMORILLONITE_WYO", "ILLITE_1M_RM30", "CORUNDUM", "QUARTZ"), align = 0.3)) #name the items in the aquant_parallel list names(multi_fit) <- names(rockjock_mixtures)[1:2] #stop the cluster stopCluster(cl)
Note how the call to
fps uses the notation
powdR::fps(), which specifies the accessing of the
fps() function from the powdR package.
When multiple samples are quantified it is often useful to report the phase concentrations of all of the samples in a single table. For a given list of
powdRafps objects, the
summarise_mineralogy() function yields such summary tables, for example:
summarise_mineralogy(multi_fit, type = "grouped", order = TRUE)
type = "grouped" denotes that phases with the same
phase_name will be summed together, and
order = TRUE specifies that the columns will be ordered from most common to least common (assessed by the sum of each column). Using
type = "all" instead would result in tabulation of all phase IDs.
In addition to the quantitative mineral data, three objective parameters that summarise the quality of the fit can be appended to the table via the logical
summarise_mineralogy(multi_fit, type = "grouped", order = TRUE, rwp = TRUE, r = TRUE, delta = TRUE)
For each of these parameters, lower values represent a smaller difference between the measured and fitted patterns, and hence are indicative of a better fit.
All above examples showcase the use of R code to carry out full pattern summation. It is also possible to run much of this functionality of powdR via a Shiny web application. This Shiny app can be loaded in your default web browser by running
run_powdR(). The resulting application has six tabs:
powdRlibreference library from two '.csv' files: one for the XRPD measurements, and the other for the ID, name and reference intensity ratio of each pattern.
powdRlibreference library .
powdRafpsobjects to be viewed and edited via addition or removal of reference patterns.
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