National Institute of Standards & Technology

Certificate of Analysis
Standard Reference Material® 1976c
Instrument Response Standard for X-Ray Powder Diffraction
This Standard Reference Material (SRM) consists of a sintered alumina disc intended for the calibration of X-ray
powder diffraction equipment with respect to line position and intensity as a function of 2θ angle. The solid form of
the SRM eliminates any variability in intensity measurements introduced by the sample loading procedure. A unit of
SRM 1976c consists of a sintered alumina disc approximately 25.6 mm in diameter by 2.2 mm in thickness.
Material Description: The manufacturing process used to produce this SRM was developed for the production ofsubstrates for electronic components. The alumina powder used as the precursor for these substrates was of highphase purity (corundum structure) with a platelet particle morphology. The platelets were typically 5 μm to 10 μm in
diameter by 2 μm to 3 μm in thickness. The compaction procedure for the discs resulted in an axisymmetric texture
with the basal planes tending towards parallelism with the surface of the disc. This axisymmetric character of the
texture permits sample mounting in any orientation about the surface normal. The compacts were liquid-phase sinteredusing a small percentage of an anorthite glass phase. No crystalline impurities have been detected. The glass phase
involved in the liquid-phase sintering effectively prevents inter-particle contact and relaxes during the cooling of thepieces from the sintering temperature. This leads to a minimal development of micro-strain and its associated line
broadening; though some micro-strain is detectable as Gaussian broadening with a tanθ dependence. Given this, andthe essential absence of crystallite size broadening, SRM 1976c can be used to obtain an approximation of the
instrument profile function (IPF). Use of SRM 1976c is not recommended, however, for quantitative microstructure
analyses. The discs comprising the feedstock of this SRM were manufactured in a single, dedicated production run
to ensure consistency of microstructure with respect to grain size, shape, micro-strain, and texture.
Certified Values: A NIST certified value is a value for which NIST has the highest confidence in its accuracy in thatall known or suspected sources of bias have been investigated or taken into account. The measurands are the certified
values for relative intensity (dimensionless ratios) shown in Table 1, and the lattice parameters, shown in Table 2.
Metrological traceability is to the International System of Units (SI) units for the relative intensity values are thedimension of one, and for lattice parameters the dimension is length (expressed as nanometers). The certified values
and uncertainties were calculated according to the method described in the ISO/JCGM Guide [1].
Information Values: The analyses associated with certification of SRM 1976c included the computation of thediffraction line positions shown in Table 3. In order to use SRM 1976c on diffraction equipment of various opticalconfigurations, the effects of polarization must be taken into account. The values shown in Table 4 include a bias
applied to the certified values to account for this effect. The data of Tables 3 and 4 are presented as information
values. An information value is considered to be a value that will be of interest to the SRM user, but insufficient
information is available to assess the uncertainty associated with the value. Information values cannot be used toestablish metrological traceability.
Expiration of Certification: The certification of SRM 1976c is valid indefinitely, within the measurement
uncertainty specified, provided the SRM is handled and stored in accordance with instructions given in thiscertificate (see “Instructions for Storage and Use”). Periodic recertification of this SRM is not required. The
certification is nullified if the SRM is damaged, contaminated, or otherwise modified.
Overall coordination and technical direction of the certification were performed by J.P. Cline of the NIST Materials
Measurement Science Division.
R. David Holbrook, Chief
Materials Measurement Science Division
Gaithersburg, MD 20899
Certificate Issue Date: 05 April 2021

Certificate Revision History on Last Page

SRM 1976c

Steven J. Choquette, Director
Office of Reference Materials

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Preparation, measurements and data analyses were performed by J.P. Cline, M.H. Mendenhall and D. Black of the
NIST Materials Measurement Science Division and A. Henins of the NIST Quantum Measurement Division.
Statistical analysis was provided by J.J. Filliben of the NIST Statistical Engineering Division.
Support aspects involved in the issuance of this SRM were coordinated through the NIST Office of Reference
Materials.
Maintenance of SRM Certification: NIST will monitor this SRM over the period of its certification. If substantive
technical changes occur that affect the certification, NIST will notify the purchaser. Registration (see attached sheet
or register online) will facilitate notification.
INSTRUCTIONS FOR STORAGE AND USE
Storage: SRM 1976c consists of a sintered alumina substrate that is essentially identical to NIST SRMs that havebeen available since 1991. While no long-term stability studies have been performed, we have no evidence of any
degradation of the diffraction properties of this SRM when used under laboratory conditions. Furthermore, alumina
is known to be a highly stable oxide. Contamination of the surface with other crystalline materials may result in
impurity lines in the data. Discoloration may occur from exposure to X-rays; this does not affect the certification.
Use (Mounting of SRM 1976c): The disc format of the SRM was chosen to be amenable to many sample holder
geometries. Diffraction data should only be collected from the side opposite the label. The stamping operation used
to make the disc during manufacture resulted in the edge of the disc surface being depressed by approximately 10 μm
relative to the center. While this is not regarded as a significant difficulty due to the low attenuation of X-rays by
alumina, height justification during mounting should be with respect to the center of the disc (see also “Use of
SRM 1976c for Testing of Instrument Sensitivity”).
SOURCE, PREPARATION, AND ANALYSIS(1)Materials: The alumina discs used for this SRM were manufactured by International Business Machines Corporation,
East Fishkill, NY.
Certification Method: Certification was performed using data from a NIST-built diffractometer [2] that includes
several advanced design features. The optical layout is that of a conventional divergent-beam diffractometer of
Bragg-Brentano geometry, equipped with a Johansson incident beam monochromator (IBM), sample spinner, and asilicon strip position sensitive detector (PSD). Data analyses were via the fundamental parameters approach (FPA) [3]using the Pawley method [4]. These analyses were used to verify homogeneity, certify both the relative intensities of
13 lines and the lattice parameters. Use of the Johansson IBM simplified the IPF of the machine aiding in the accurate
fitting of the profiles; this was particularly important in the certification of the relative intensity values. Furthermore,
the residual errors to the fits were analyzed for systematic offsets in the intensity determination, with reported
intensities then being compensated for said offsets [5]. The linkage of the certified lattice parameter values to thefundamental unit of length, as defined by the SI [6], was established with use of the emission spectrum of Cu Kα
radiation as the basis for constructing the diffraction profiles. With the use of the FPA, diffraction profiles are modeled
as a convolution of functions that describe the wavelength spectrum, the contributions from the diffraction optics, andthe sample contributions resulting from microstructural features. Analysis of data from a divergent-beam instrument
requires knowledge of both the diffraction angle and the effective source-sample-detector distance. Two additionalmodels are therefore included in the FPA analyses to account for the effect of the sample height and attenuation.
Certification data were analyzed in the context of both Type A uncertainties, assigned by statistical analysis, andType B uncertainties, based on knowledge of the nature of errors in the measurements, to result in the establishmentof robust uncertainties for the certified values.
Certification Procedure: Twenty units of SRM 1976c were selected in a stratified random manner from the
population of units being certified. The 1.5 kW copper tube of standard fine-focus geometry was operated at a powerof 1.2 kW during certification measurements. The source size was approximately 8 mm × 0.04 mm. The variable
divergence incident slit was set to 0.9°. A 1.5° Soller slit was located in front of the PSD window to limit axial
divergence, no Soller slits were used in the incident beam. The goniometer radius was 217.5 mm. Samples were spunat 0.5 Hz during data collection. The PSD was scanned using a variable window length and a combination of coarse
and fine steps in θ and 2θ was used that allowed for timely data to be collected at high resolution [7]. The response(1) Certain commercial equipment, instruments or materials are identified in this certificate to adequately specify theexperimental procedure. Such identification does not imply recommendation or endorsement by the National Institute of Standards
and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

SRM 1976c

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of the PSD was measured with a scan of the attenuated direct beam to obtain a flat field correction; this ensured theuniformity in the measurement of X-ray intensity across the active area of the PSD. The machine was equipped with
an automated anti-scatter slit, located above the specimen, that blocked air scatter of the incident beam from entering
the PSD, where it would otherwise contribute to the low angle background level. The machine was located within atemperature-controlled laboratory space where the nominal short-range control of temperature was ± 0.1 K. The
temperature was monitored using two 10 kΩ thermistors with a Hart/Fluke BlackStack system that was calibrated at
the NIST temperature calibration facility [8] to ± 0.002 °C. The source equilibrated at operating conditions for at least
an hour prior to recording any certification data. The performance of the machine was qualified with the use of
SRM 660c Line Position and Line Shape Standard for Powder Diffraction (Lanthanum Hexaboride Powder) [9] andSRM 676a Alumina Powder (Quantitative Analysis Diffraction Standard) [10] using procedures discussed inreference 2.
Data Analysis: The certification data were analyzed using the FPA method with Pawley refinements as implemented
in TOPAS [11]. It was verified that TOPAS operated in accordance with published models for the FPA [12]. The
analysis used energies of the Cu Kα emission spectrum as characterized in reference 13. The optics of the Johansson
IBM were modeled using dynamical scattering from the monochromator in conjunction with the powder sample as
per the optics of a 2-crystal monochromator. The resulting “band pass” model provides a function that effectively
cuts off the Lorentzian tails of the native copper emission lines, providing good agreement with the shape of the tailsof the diffraction peaks. It also adds a dispersion term to the FPA emission model which adds to the width of themodeled lines, further improving the fit to the observation [14]. Parameters associated with the bandpass model, as
well as others of the IPF, the incident slit angle and the Soller slit angles of the “full” axial divergence model [15]were refined using scans from SRM 660c. They were then fixed at the SRM 660c values for the analyses of
SRM 1976c. Other refined parameters included the intensity values, 11 Chebyshev polynomial terms for modelingof the background, the lattice parameters, specimen displacement and attenuation terms and a term for Lorentzian size
and Gaussian strain broadening. The refined lattice parameters were adjusted using the CTE values found in Miyake
et al, [16] to values at 22.5 °C.
The relative intensity values were computed in a manner that avoided error correlation difficulties when the values
are based on the 100 % line alone. Each line was compared with the sum of the intensity values for all lines and thenthis ratio was re-scaled to yield the correct relative intensity values. Thus, the relative intensities were determinedwith a comparison of each given line intensity, x, to the total intensity of 14 lines measured from each sample. With
the number of samples being 20, n(s) = 20, and the number peaks being 14, n(p) = 14, we have for the jth peak of theith sample:
yij =

xij
n(p)
∑j=1 xij(1)

We then scale this value to the 100 % line by multiplying it by the average, over the 20 specimens, of the total line
intensity, relative to the average of the total intensity of the 100 % line:
1∑n(s) ∑n(p) xn(s) i=i j=1 ij
zij = 100 yij
1∑n(s) xn(s) i=1 i(100%)(2)Where z is the relative intensity value that is not corelated to an error in the measurement of the 100 % line alone.
The relative intensities of SRM 1976c and their expanded uncertainties, using the k = 2 factor, are shown in Table 1.
A statistical analysis of the data indicated that the means of the measurements were a = 0.475 909 20 nm andc = 1.299 336 50 nm with a k = 2 Type A expanded uncertainty of 0.000 000 79 nm and 0.000 000 99 nm for a and crespectively. However, a Type B uncertainty due to systematic error must be incorporated into the uncertainty boundsof the certified lattice parameter. Data were considered in the context of the uniformity in lattice parameter as afunction of 2θ angle; this, in turn, would reflect the functionality of the FPA model. This consideration leads to an
assignment of a Type B uncertainty that is roughly an order of magnitude larger than the Type A. The certified lattice
parameters and their expanded uncertainties, Type A compounded with Type B, are shown in Table 2. Peak positions
were computed from the certified lattice parameters for Cu Kα Radiation, λ = 0.154 059 29 nm, and are shown in
Table 3 as information values.

SRM 1976c

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Table 1. Certified Relative Intensity Data for SRM 1976c
Reflection, (hkl)

Relative Intensity

(012)
(104)
(113)
(024)
(116)
(300)
(1.0.10) & (119)(0.2.10)(226)(2.1.10)(324) & (0.1.14)(1.3.10)(146)(4.0.10)23.62
100.00
37.16
20.68
87.83
12.43
72.00
13.42
8.22
16.65
26.37
15.29
13.05
11.04

Expanded Uncertainty (k = 2)
± 0.21± 0.34± 0.23± 0.15± 0.22± 0.15± 0.49± 0.06± 0.05± 0.06± 0.16± 0.05± 0.07± 0.06

Table 2. Certified Lattice Parameters for SRM 1976c
Lattice Parameter
(nm)
a
c

Expanded Uncertainty(k = 2)0.475 909 2
1.299 337

± 0.000 008 0± 0.000 015

Table 3. Peak Position Information Values for SRM 1976c,
Lines Listed with a Relative Intensity >5 %, Computed Using Cu Kα Radiation, λ = 0.154 059 29 nm

SRM 1976c

Reflection, (hkl)

Peak Position(2θ, degrees)

(012)
(104)
(110)
(006)
(113)
(024)
(116)
(018)
(214)
(300)
(1.0.10)(119)(0.2.10)(0.0.12)(226)(2.1.10)(324)(0.1.14)(1.3.10)(3.0.12)(2.0.14)(146)(1.1.15)(4.0.10)(1.2.14)(1.0.16)(330)

25.57535.14837.77641.67343.35252.54957.49561.29766.51568.20776.86677.22988.98990.69995.243101.066
116.093116.588
127.670129.863131.083
136.066
142.292
145.154150.082
150.380
152.406
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Use of SRM 1976c for Testing of Instrument Sensitivity: The relative intensities reported for SRM 1976c are to
be used for the basis of a calibration of an instrument for diffraction intensity as a function of 2θ angle, or instrument
sensitivity. The k = 2 expanded uncertainties, shown in Table 1, represent our degree of confidence in the reported
relative intensity values; they do not play any role in a judgment concerning the qualification of a test instrument. It
is the responsibility of the end user, in conjunction with the instrument vendor, to determine the acceptance criteria
for their specific instrument. Such acceptance criteria may be based on a number of factors concerning random andsystematic influences on the measurement method. The “method” in this case refers to the entire suite of hardware,
software, and procedures used to obtain the results under consideration. In the determination of an acceptancecriterion, it may be appropriate to perform an extensive study of the measurement method itself; such a study would
have to include the sensitivity of the results on such environmental factors as power fluctuations, laboratory
temperature variations, etc.
The determination of accurate intensity values is, to say the least, complicated by the accurate determination of the
background. Complete scans of the entire 2θ range and the use of “whole pattern” data analysis methods arerecommended when the profile shape may include a significant Lorentzian component. Lorentzian profiles exhibit
considerable intensity far from the peak center. Given the Lorentzian nature of atomic emission lines, any machine
not equipped with an IBM would exhibit such profiles. Experiments comparing results from limited scans to those of“whole pattern” methods may be used to validate the more timely, limited region methods. Use of a sample spinner
will improve particle counting statistics. Graphical evaluation of the ratio of these test data to the certified values
vs. 2θ will allow for an appropriate judgment as to the condition of the test equipment. The desired result would
consist of unity values across the two-theta range. However, data should be considered as a whole and in the context
of testing for a slope in the data. A few outliers do not constitute a failure.
The use of SRM 1976c for the calibration of X-ray diffraction equipment of differing optical configurations may
require that a bias be applied to the certified values to render them appropriate for the machine to be qualified. Thisbias is needed to account for differences in the polarization effects from the presence, absence, and character of crystal
monochromators. The polarization factor for a diffractometer that is not equipped with a monochromator is, from
Guinier [17]:1 + cos2 2θ
2(3)

The polarization factor for a diffractometer equipped with only an incident beam monochromator is, from Mendenhall
et al. [18]:1 + cos 2θ𝑚𝑚 cos2 2θ
1 + cos 2θ𝑚𝑚

(4)

where 2θm is the 2θ angle of diffraction for the monochromator crystal. The polarization factor for a diffractometerequipped with only a diffracted beam, post monochromator is, from Yao and Jinno [19]:2

1 + cos2 2θ𝑚𝑚 cos 2θ
2(5)where, again, 2θm is the 2θ angle of the monochromator crystal. Equation 4 is considered appropriate for a “perfect”
monochromator crystal that would diffract in accordance with dynamical scattering theory. Equation 5 is consideredappropriate when the crystal is of an “ideal mosaic” structure, i.e., the diffracting domains are uniformly small and,
therefore, the crystal is diffracting in a kinematic context. The use of equations 3, 4 and 5 was evaluated with thediffractometer used in the certification of SRM 1976c. With the Johansson IBM, as per Equation 4, a Ge crystal (111)reflection (2θm = 27.3 degrees) was used. With respect to Equation 5, a pyrolytic graphite crystal (0002) basal planereflection (2θm = 26.6 degrees) as a diffracted beam, post monochromator was used. With respect to equation 3, themachine was configured with a conventional source and the PSD. Rietveld analyses of data from SRMs 660b, 1976c,
and 676a that included a refinement of the polarization factor, modeled as per Equations 3, 4 and 5 in TOPAS,
indicated that these equations were appropriate for these crystals and configurations. Given that the certification dataof SRM 1976c were collected using a Johansson IBM, the ratio of values computed from Equations 3, 4, and 5 wereused to bias the certified values to correspond to those of alternative configurations. The results are shown in Table 4.
The user may select the set of relative intensity values from Table 4 that are appropriate for the configuration of the
instrument to be qualified and proceed accordingly. Use of SRM 1976c for additional configurations may requirecomputation of biases alternative to those presented herein.

SRM 1976c

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Table 4. Information Values for Biased Relative Intensity Data for SRM 1976c
Reflection, (hkl)
(012)
(104)
(113)
(024)
(116)
(300)
(1.0.10) & (119)(0.2.10)(226)(2.1.10)(324) & (0.1.14)(1.3.10)(146)(4.0.10)SRM 1976c

Relative Intensity
No Monochromator

Graphite Post Monochromator

23.75
100.00
36.92
20.37
86.06
12.04
69.16
12.82
7.86
15.97
25.66
15.06
12.96
11.04

23.50
100.00
37.37
20.95
89.37
12.77
74.47
13.94
8.53
17.24
26.99
15.49
13.13
11.04

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REFERENCES[1]

JCGM 100:2008; Evaluation of Measurement Data — Guide to the Expression of Uncertainty in
Measurement; (GUM 1995 with Minor Corrections); Joint Committee for Guides in Metrology (JCGM) (2008);
availableathttps://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf (accessed
Apr 2021); see also Taylor, B.N.; Kuyatt, C.E.; Guidelines for Evaluating and Expressing the Uncertainty of
NIST Measurement Results; NIST Technical Note 1297; U.S. Government Printing Office:Washington, DC (1994); available at https://www.nist.gov/pml/nist-technical-note-1297 (accessed Apr 2021).
[2] Cline, J.P.; Mendenhall, M.H.; Black, D.; Windover, D.; Henins, A.; The Optics and Alignment of the Divergent
Beam Laboratory X-ray Powder Diffractometer and its Calibration Using NIST Standard Reference Materials;
J. Res. Natl. Inst. Stand. Technol., Vol. 120, pp. 173–222 (2015).
[3] Cheary, R.W.; Coelho, A..; A Fundamental Parameters Approach to X-Ray Line-Profile Fitting; J. Appl. Cryst.,
Vol 25, pp. 109–121 (1992).
[4] Pawley, G.S.; EDINP, the Edinburgh Powder Profile Refinement Program; J. Appl. Cryst., Vol. 13(6),pp. 630–633 (1980).
[5] Mendenhall, M.H.; Cline, J.P.; Model-independent Extraction of the Shapes and Fourier Transforms from
Patterns of Partially Overlapped Peaks With Extended Tails; Acta Cryst., Vol. A75, pp. 158–164, (2019).
[6] BIPM; International System of Units (SI), Bureau International des Poids et Mesures; 8th edition,
Sèvres, France (2006); available at https://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf (accessed
Apr 2021).
[7] Mendenhall, M.H.; Cline, J.P.; Efficient Collection of X-ray Powder Diffraction Data with a Solid State Position
Sensitive Detector; J. Appl. Cryst., submitted.
[8] Vaughn, C.D.; Strouse, G.F.; The NIST Industrial Thermometer Calibration Laboratory. In 8th Int’l Symp.
Temperature and Thermal Measurements in Industry and Science, Berlin, June 2001. available athttps://www.nist.gov/publications/nist-industrial-thermometer-calibration-laboratory (accessed Apr 2021).
[9] SRM 660c; Line Position and Line Shape Standard for Powder Diffraction (Lanthanum Hexaboride Powder);National
InstituteofStandards
andTechnology;
U.S.
Department
ofCommerce:
Gaithersburg, MD (29 October 2015).
[10] SRM 676a; Alumina Powder (Quantitative Analysis Powder Diffraction Standard); National Institute ofStandards and Technology; U.S. Department of Commerce: Gaithersburg, MD (04 November 2015).
[11] TOPAS, General Profile and Structure Analysis Software for Powder Diffraction Data, V5.0, Bruker AXS
GmbH, Karlsruhe, Germany.
[12] Mendenhall, M.H.; Mullen, K.; Cline, J.P.; An implementation of the Fundamental Parameters Approach for
Analysis of X-ray Powder Diffraction Line Profiles; J. Res. Natl. Inst. Stand. Technol. Vol. 120, pp. 223–251,(2015).
[13] Mendenhall, M.H.; Henins, A.; Hudson, L.T.; Szabo, C.I.; Windover, D.; Cline, J.P.; High-precision
Measurement of the X-ray Cu Kα Spectrum; J. Phys. B: At. Mol. Opt. Phys., Vol. 50, pp. 115004–115022 (2017).
[14] Mendenhall, M.H.; Black, D.; Cline, J.P.; The Optics of Focusing Bent-crystal Monochromators on X-ray
Powder Diffractometers with Application to Lattice Parameter Determination and Microstructure Analysis; J.
Applied Chryst., Vol. 52, pp. 1087–1094 (2019)..
[15] Cheary, R.W.; Coelho, A.A.; Axial Divergence in a Conventional X-Ray Powder Diffractometer. I. Theoretical
Foundations; J. Appl. Cryst., Vol. 31, pp. 851–861 (1998), and Cheary, R.W.; Coelho, A.A.; Axial Divergencein a Conventional X-Ray Powder Diffractometer II. Realization and Evaluation in a Fundamental Paramenter
Profile Fitting Procedure; J. Appl. Cryst., Vol. 31, pp. 862–868 (1998).
[16] Miyake, A.; Takaya, M.; Kodama, Y.; Ohi, S.; Thermal Expansion of Forsterite, Corundum, and Platinum To
1713 K; Photon Factory Activity Report 2010, Vol. 28, Part B (2011).
[17] Guinier, A.; X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies; Courier Dover
Publications: N. Chelmsford, MA (1994).
[18] Mendenhall, M.H.; Black, D.; Windover, D.; Cline, J.P.; Polarization Effects of X-ray Monochromators
Modeled Using Dynamical Scattering Theory, Acta Cryst., submitted.
[19] Yao, T.; Jinno, H.; Polarization Factor for the X-Ray Powder Diffraction Method with a Single-Crystal
Monochromator; Acta Cryst., Vol A38, pp 287–288 (1982).
Certificate Revision History: 05 April 2021 (Changes to equations 3 through 5 were used to update information values in Table 4; editorial changes);
02 April 2019 (Original certificate date).

Users of this SRM should ensure that the Certificate of Analysis in their possession is current. This can beaccomplished by contacting the SRM Program: telephone (301) 975-2200; e-mail srminfo@nist.gov; or via theInternet at https://www.nist.gov/srm.
SRM 1976c

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