National Institute of Standards & Technology

Certificate of Analysis
Standard Reference Material® 2000
Calibration Standard for High-Resolution X-Ray Diffraction
This Standard Reference Material (SRM) provides the high-resolution X-ray diffraction (HRXRD) community with
International System of Units (SI) [1] traceable Si (220) d-spacing in transmission, surface-to-crystal-plane wafer
miscut, and surface-to-Si (004) Bragg angle in reflection for our reference wavelength. A unit of SRM 2000 consists of
25 mm × 25 mm × 0.725 mm double-polished (100)-oriented, single-crystal Si specimens with a nominal 50 nm
Si0.85Ge0.15 epitaxial layer and 25 nm Si cap. These certified values can be used to calibrate HRXRD instrumentation.
Table 1. Certified Values for SRM 2000
Quantity(a)dSRM

Certified Value
0.192 016 1 nm (at 22.5 °C)

ξSRM

SRM

(a)
(b)

4.360
–360

Expanded Uncertainty(b), U for k = 20.000 000 9 nm

mrad

0.027

mrad

mrad

16

mrad

xSRM

–1.534

mrad

0.072

mrad

ySRM

4.0810

mrad

0.019

mrad

θsurface,SRM(004)0.604 785

rad0.000 076

rad

The identity of the quantity is defined in Table 2.
For uncertainty methods, see [2-4].

Expiration of Certification: The certification of SRM 2000 is valid indefinitely, within the uncertainty specified,provided the SRM is handled and stored in accordance with the instructions given in this certificate (see“Instructions for Handling, Storage, and Use”). Accordingly, periodic recalibration or recertification of this SRM is
not required. The certification is nullified if the SRM is damaged, contaminated, or otherwise modified.
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 (seeattached sheet) will facilitate this notification.
The overall coordination of the preparation and technical direction of the certification were performed by J.P. Cline of
the NIST Ceramics Division. Realization of the instrumental capability, development of the experimental design, andcollection of measurement data were performed by D. Windover of the NIST Ceramics Division. D.L. Gil, a guest
researcher for the NIST Ceramics Division, analyzed the data to estimate certified parameters. Hardware design andalignment procedures were facilitated by A. Henins of the NIST Ceramics Division.
Statistical analysis was provided by J. Filliben of the NIST Statistical Engineering Division.
Support aspects involved in the issuance of this SRM were coordinated through the NIST Measurement Services
Division.
Debra L. Kaiser, Chief
Ceramics Division
Gaithersburg, MD 20899
Certificate Issue Date: 05 March 2012

Robert L. Watters, Jr., Chief
Measurement Services Division

Certificate Revision History on Last Page

SRM 2000

Page 1 of 10

Supporting contributions to SRM certification: K. Bowen of Bede X-ray Metrology, Denver, CO, provided key
advice in the development of the HRXRD test structure. M. Wormington of Bede X-ray Metrology and
L. Bruegemann of Bruker AXS, Karlsruhe, Germany, provided supporting HRXRD measurements on the SRM
feedstock. D. Black of the Ceramics Division at NIST performed X-ray topographs at the XOR/UNICAT facility at theAdvanced Photon Source, Argonne, IL. R. Gates of the Ceramics Division at NIST performed SEM images at the
NanoFab Facility of the CNST at NIST. Applied Materials, Santa Clara, CA, deposited Si1-xGex epitaxial structures asa custom service for NIST.
Table 2. Definitions of the Certified Values for SRM 2000.
Quantity
dSRM
ξSRM

SRM
xSRM
ySRM
θsurface,SRM(004)

Description
Si (220) transmission case d-spacing of SRM 2000
Magnitude of wafer miscut (polar form)
Direction of wafer miscut (polar form)
Tilt of wafer miscut in x-plane (Cartesian form)
Tilt of wafer miscut in y-plane (Cartesian form)
Angle from wafer surface of Si (004) reflection peakMaterial Description(1): The specimens were diced from 9 sequential wafers out of a series of 25 type-p, B-doped,
200 mm Czochralski (CZ) Si wafers manufactured by ShinEtsu, SEH America Vancouver, WA. Each wafer was
deposited with a nominal 25 nm Si surface layer capping a strained, epitaxial, nominal 50 nm Si0.85Ge0.15 thin film on
one of the polished substrate surfaces (referred to as the “front” surface) using a Centura RP Epi deposition system by
the applications lab at Applied Materials. HRXRD measurements were performed on 24 areas (4 rows × 6 columns)
within a 150 mm × 100 mm rectangle on each wafer. Each measured sample area was then sectioned into a
25 mm × 25 mm square using a Disco 341 wafer dicing saw at the Center for Nanoscale Science and Technology(CNST) at NIST. Serial numbers (SNs) were then laser-etched onto the back surface of each specimen (see Figure 1
for SN convention). The SN is on the edge of the wafer closest to the wafer notch of the feedstock,  = 0 rad.

a)b)

Figure 1. (a) Schematic of specimen back showing the location of serial numbers. Serial number
convention: xx = wafer number, yy = specimen position on wafer. (b) Schematic of specimen front
showing azimuth, , convention for miscut orientation.

MEASUREMENT PROCEDURE
D-spacing and Bragg angle measurements: The NIST approach for establishing SI traceability in HRXRD
measurements uses a Si reference crystal of known d-spacing and compares it to the wafer feedstock values by adifference method to eliminate significant uncertainties arising from a number of systematic errors. The reference
crystal used in this study was cut from a piece of the WASO 04 boule of silicon grown for the Avogadro Project,(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 ofStandards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the
purpose.

SRM 2000

Page 2 of 10

and polished to a strain-free 450 μm thickness lamella. The WASO 04 boule is discussed in reference [5]; its(220) lattice spacing has been determined with a combined relative standard uncertainty ofuc(ΔdWASO04)/dWASO04 ≤ 610–8 by the d lattice comparator at NIST [6-9]. This reference crystal was measured
after system alignment changes so that wafer feedstock and reference crystal measurements could be paired only for
identical system configurations. The following iterative measurement and analysis approach was followed:
1. Align HRXRD instrument (see alignment methods [10-14])
2. Measure Si reference crystal with HRXRD instrument
3. Measure Si wafer feedstock with HRXRD instrument at all sample positions
4. Use reference data to provide SI traceability for wafer measurements.
Interpretation of a single diffraction measurement on an HRXRD instrument requires substantial knowledge about
instrument alignment and goniometer orientation. One way to eliminate some of the uncertainty resulting frommeasuring instrument-related parameters is to measure two diffraction features from the same family of planes,
ideally the non-dispersive and dispersive settings of the same planes and approximately same d-spacing of theplanes used in the instrument’s monochromator. (For a treatment of dynamical diffraction theory, see references 15
and 16). Our approach uses the (220) family to provide information on the Si substrate d-spacing. The Bragg angle,
θB(220), can be accurately determined by measuring the angles of the diffraction peak in each of the crystal’s two
settings. The angle between the diffracting conditions for the sample will be equal to 2θB(220), since a diffraction
peak is observed at both plus, (220)+, and minus, (220)–, the Bragg angle from a [100]-faceted Si crystal.

 B(220)  meas(220)  meas(220)

(1)

2

This approach to measuring d-spacing is commonly known as the Bond method [17-21]. This method is typicallyused in a reflection geometry, in which corrections for index of refraction and many other effects are necessary [22].
In contrast, the reference crystal and SRM wafer measurements have been taken in a transmission geometry toeliminate uncertainties related to these corrections. The impact of wafer radius-of-curvature and strain effects on
d-spacing determination is largely unexplored [23-25], but such effects should affect both certification
measurements and subsequent measurements similarly.
Each 200 mm wafer was measured at the 24 central points of the SRM specimens. At each point, or sample position, a
step-scan was taken for both the non-dispersive and dispersive Si (220) transmission peaks and for the Si (004)reflection peak from the Si substrate bulk. A longer angular range scan of the Si (004) and Si1-xGex (004) reflection
peaks was made to measure properties of the epitaxial surface structure. Table 3 provides the details of these scans.
This sequence of scans was performed at least twice at each of the 24 points.
Table 3. Scan Angular Ranges for SRM Measurements.
Scan Type
Non-dispersive Si (220)Range (rad)

steps
200

count time (s)2.0Non- dispersive Si (220) setup

4.3  10 3

50

1.0

Dispersive Si (220)4.3  10 4

100

2.0

Dispersive Si (220) setup

4.3  10 3

50

1.0

Reflection Si (004)

4.3  10 4

100

1.0

Reflection Si (004) setup

4.3  10 3

50

1.0

Reflection Si (004) long

2

500

5.0

4.3  10

2.6  10

4

Miscut Measurements: The measurements used for the certification of dSRM were collected with the wafer notch
facing up,  = 0 rad. The wafers were then rotated to  = /2 rad,  rad, and 3/2 rad orientations. At least two
reflection (004) scans were collected at each azimuthal orientation to determine θ. Equations 2 and 3 show thecalculations used for determining two miscut components, xSRM and ySRM, providing the angles between the polishedwafer surface and the Si crystal planes. The geometry of each of the miscut angles is provided in Figure 2.

SRM 2000

Page 3 of 10

a)b)
Figure 2. Schematic of wafer miscut from the Si crystal planes using measurements from two wafer
orientations. A miscut is calculated for both the horizontal xSRM and the vertical ySRM wafer orientations.
Figure 2(a) shows xSRM calculation parameters (see Equation 2). Figure 2(b) shows ySRM calculation
parameters (see Equation 3).

xSRM 

 / 2   3 / 2

ySRM 

(2)2

 0  (3)

2

By expressing the Cartesian miscut information, xSRM and ySRM, in polar coordinates, we can provide the miscut
information in azimuthal magnitude, ξSRM, and orientation, SRM (see Figure 3). Equations 4 and 5 are used to
calculate these quantities. Note that the miscut orientation is defined from the front of the sample, with the angle of
the wafer notch setting SRM = 0 rad.

Figure 3. Schematic of wafer miscut polar coordinates showing miscut orientation, SRM , andmiscut magnitude, ξSRM.

SRM  arctan tan 2 xSRM  tan 2 ySRMSRM  arctan

SRM 2000

tan xSRM
tan ySRM(4)(5)

Page 4 of 10

CERTIFICATION ANALYSIS
Calculating Si (220) transmission d-spacing: With the effective wavelength known for each alignment of thesystem, calculating the d-spacing simply requires applying Bragg’s law to pairs of dispersive and non-dispersive
peak positions. A small correction for temperature is also required with which a small uncertainty, uc(ΔTd) = 32 am,
is associated. These statistical calculations were performed in Python [26], R [27], and DATAPLOT [28] and thenindependently verified by the NIST Statistical Engineering Division (SED) using DATAPLOT.
Type A statistical uncertainty: The measured d-spacing for each SRM 2000 specimen varies because of statisticalvariation (from both instrument repeatability and sample d-spacing distribution) as well as systematic variation
arising from the sample position on the CDPBD wafer holder. Because of the systematic nature of the variation
with sample holder position, it was judged inappropriate to include this variation in the Type A statisticaluncertainty (k = 1), ui(dSRM). One-way ANOVA was therefore used to separate the statistical variation from thesystematic component by generating a correction to d-spacing based on sample position. The estimate of dSRM isthen the mean of the set of estimates, and ui(dSRM) = 0.24 fm is determined as the residual standard deviation of the
ANOVA model (that is, the variance not explained by the wafer holder position correction).
Type B systematic uncertainty related to misalignment: The only error relating to alignment that has a significant
impact on the uncertainties for dSRM is the relative misalignment of the SRM feedstock and the reference
crystal, Δδ2. Such misalignment may cause a systematic error because the wafers and the reference crystals aremounted differently. Therefore, based on a study of misalignment effects, we assign a large maximum offset tod-spacing of Δδ2d ≤ 0.62 fm and so a standard uncertainty (k = 1) of uj(Δδ2d) = 0.36 fm.
Best estimate and combined standard uncertainty: The best estimate of the Si (220) substrate d-spacing of the
SRM 2012 feedstock measured in transmission is dSRM = 0.192 016 1 nm. The combined standard uncertainty(k = 1) of the SRM 2012 d-spacing is the combination of the statistical uncertainty computed above, the uncertaintyof the misalignment difference between the SRM and the reference crystal, the temperature uncertainty in measuring
the specimens, and the uncertainty in the reference crystal d-spacing from reference 5, uc(dWASO04) = 12.1 am.



uc dSRM   ui dSRM   u j  2 d  uc  T d   uc d WASO04 
2

2

2

2

(6)

Thus the standard uncertainty (k = 1), uc(dSRM) = 0.43 fm. Expressed as a relative standard uncertainty (k = 1),
uc(dSRM)/dSRM = 2.2  10–6. The expanded uncertainty (k = 2) is U(dSRM) = 0.87 fm.
Calculating surface-to-crystal-plane miscut best estimates and combined standard uncertainty: The estimates used
for the certified values are the means over all of the wafers, and the type A standard uncertainties are their estimatedstandard deviations over that set. Their values are given in Table 5.
Calculating Bragg angle of Si (004) reflection best estimate and combined standard uncertainty: Because of theeffects of material index of refraction, wafer curvature, vertical divergence, long-range goniometer angle
uncertainties, and other systematic corrections to reflection Bragg angle, certification of the d-spacing of thesubstrate in reflection condition is not possible at this time. Instead, to provide a traceable parameter for a reflection
condition measurement of the SRM, we use the uncorrected Bragg angle of the Si (004) reflection diffraction peakfrom the surface of the wafer. This quantity can be calculated from the Bragg angle of the Si (004) reflection,
θB,SRM (004), corrected for the miscut of the surface relative to the crystal planes (see Equation 7).

surface,SRM(004)   B,SRM(004)  xSRM

(7)Since only one Si (004) reflection case peak could be measured, to determine the uncorrected Bragg angle of thereflection requires knowledge of the angle of the diffracting planes of the crystal, Δθ0 (see Equation 8).

 B,SRM(004)   meas(004)  0(8)This is known in this case from the positions of the dispersive and non-dispersive transmission peaks (used in thed-spacing calculation of Equation 1).

 0 

 meas(220)    meas(220) 
2




2

(9)

The Bragg angle of the Si (004) reflection was estimated for each sample position for each wafer; the mean andestimated standard deviation of this set are given in Table 6. Since temperature correction for d-spacing variation
SRM 2000

Page 5 of 10

cannot be applied directly to angle, an additional uncertainty (k = 1), ui(ΔTθB,SRM (004)) = 0.50 μrad, due totemperature variation over the course of the measurements must be incorporated. An uncertainty must be added due
to long-range error of the goniometer, uj(ωSRM) = 3.9 μrad at this angle. Finally, a correction for the uncertainty in
NIST monochromator wavelength, λSRM (see Table 7), and its corresponding contribution to the uncertainty of the
Bragg angle (k = 1), ui(λmonoθB,SRM (004)) = 7.3 rad, must be included. The resulting combined standard uncertainty(k = 1),uc ( B,SRM(004) )  [ui  B,SRM(004)   ui T B,SRM(004)   u j SRM   ui (mono B,SRM(004) ) 2 ]  11 μrad.
222

The best estimate and uncertainties of the certified angle of the (004) reflection from the surface are given in
Table 6.

Figure 4. Top-view schematic of Si (004) reflection Bragg angle showing determination of crystal
plane using Si (220) dispersive and non-dispersive peak positions and showing the miscut angle inthe specimen x direction relating wafer surface with crystal plane, xSRM.
SI TRACEABILITY
Uncertainty analyses performed for this SRM follow the ISO Guide to the Expression of Uncertainty in
Measurement [2]. Figure 5 shows the SI traceability pathway for Si (220) d-spacing determination using SI-definedunits of time, t, and distance, m, as input parameters [1,3,4]. This diagram illustrates the flow of SI units, theessential prior SI-traceable measurements, and the measured feedstock and material properties in an unbroken chain,
as required for computing the total uncertainty budget in a given measurement. Three different feedstock Si boules
and corresponding Si (220) d-spacings were used in the traceability chain for this SRM; (1) a purpose-commissioned
boule of Si known as the WASO 4.2 feedstock provided the direct link to the SI through optical frequency via its use
as the X-ray interferometer in the X-ray / optical interferometer at Physikalisch-Technische Bundesanstalt (PTB,
Braunschweig, Germany) [5]; (2) a purpose-commissioned boule of Si used in the Avogadro Project, known as
WASO 04 feedstock, provided reference artifacts for internal calibration of the NIST CDPBD instrument throughdirect comparison with WASO 4.2 on the NIST lattice comparator [29]; (3) a set of epitaxially-coated wafers
sectioned from a single commercial CZ boule of Si, known as SRM 2012 feedstock, are the certified artifacts
measured on the CDPBD by comparison with WASO 04.

SRM 2000

Page 6 of 10

Figure 5: SI traceability pathway for Si (220) d-spacing in transmission.

INSTRUCTIONS FOR HANDLING, STORAGE, AND USE
Storage and Handling: The epitaxial layer may be damaged or the wafer strained mechanically; therefore contact
with the front surface of the SRM should be avoided. To remove surface contamination, use filtered nitrogen gas.
Peak profiles: The measured diffraction peaks were significantly asymmetric. Supporting measurements suggestthat this asymmetry may not be entirely due to instrument effects, but in part due to properties of the SRM 2000
feedstock. Therefore, a split-Pearson VII profile was used for fitting and estimating peak positions and isrecommended for determining peak positions for calibration. Use of another profile function may result insystematic bias due to differing effects of asymmetry on profile position determination.
Use
Suggested use of d-spacing measured in Si (220) transmission: Calibration by direct use of the certified d-spacing,dSRM requires an instrument that can measure in transmission geometry. If this is possible, one can calibrate either
the instrument monochromator’s wavelength, λinst, or angle measurements near the Si (220) reflection (intransmission geometry), Δθinst. The following procedure provides an approach for this calibration (see Table 3 forscan ranges). Individual instrument manufacturers may have alternative suggested procedures.
1. Clean sample holder and SRM with filtered nitrogen prior to mounting.
2. Mount SRM in holder with serial number upright and facing away from source.
3. Locate non-dispersive Si (220) peak center, , using a setup scan.
4. Perform non-dispersive Si (220) normal range scan to measure peaks’ full width at half maximum(FWHM).
5. Re-align wafer azimuth and repeat (4) until minimum FWHM is achieved.
6. Locate dispersive Si (220) peak center, , using a setup scan.
7. Perform dispersive Si (220) normal range scan to measure FWHM.
8. Locate Si (004) reflection peak center, , using a setup scan.

SRM 2000

Page 7 of 10

9.

Perform a series of Si (004) reflection normal range scans at various wafer tilts, δ2 – defined as χ tilt on
most instruments.
10. Set δ2 for wafer based on Si (004) reflection FWHM minimum.
11. Measure non-dispersive Si (220) HRXRD data.
12. Measure dispersive Si (220) HRXRD data.
13. Repeat steps 11 and 12.
14. Use difference of step 13 to determine θinst(220).
15. Use θinst(220) and dSRM to calibrate either λinst or Δθinst.
Suggested use of surface-to-crystal-plane miscut: The miscut certification will allow calibration and repeatability
studies for rotating sample holders in commercial instruments in the field. By repeating miscut measurements on
the SRM, one can test the repeatability and stability of a sample holder during rotation and remounting. The
following procedure provides an approach for this sample holder calibration. Individual instrument manufacturers
may have alternative suggested procedures.
1. Clean wafer holder and SRM with filtered nitrogen prior to mounting.
2. Mount SRM in holder with wafer notch upright and scribe marks facing away from source.
3. Measure Si (004) reflection HRXRD data at  = 0 rad.
4. Rotate sample to  = π/2 rad and measure Si (004) HRXRD data.
5. Rotate sample to  = π rad and measure Si (004) HRXRD data.
6. Rotate sample to  = 3π/2 rad and measure Si (004) HRXRD data.
7. Calculate xinst using equation 2 and data from steps 4 and 6.
8. Calculate yinst using equation 3 and data from steps 3 and 5.
9. Calculate ξinst and inst using equations 4 and 5.
10. Compare values, ξinst and inst, with certified values ξSRM and SRM.
11. Use comparison data to align wafer holder.
12. Repeat steps 1 through 11 several times to reduce statistical uncertainty terms.
Suggested use of Bragg angle of Si (004) reflection: If transmission measurements are not possible, one can use theuncorrected Bragg angle of the Si (004) reflection, θB,inst(004), for calibration of either the instrumentmonochromator’s wavelength, λinst or angle measurements near the Si (004) reflection (in reflection geometry),Δθinst. The following relational formula uses Bragg’s law to relate the SRM and instrument parameters and may be
used to solve for either quantity of interest using equation 7:

inst

sin( B,inst(004) )



SRM

(10)

sin( B,SRM(004) )The following procedure provides an approach for this type of calibration. Additional uncertainties may result fromdifferences in beam conditioning optics. Individual instrument manufacturers may have alternative suggested
procedures.
1. Perform surface-to-crystal-plane misalignment calibration procedure.
2. Remount SRM in holder with serial number upright and facing sample holder.
3. Perform specular reflection alignment to determine specimen surface orientation with respect to the
instrument plane.
4. Reset instrument  zero to the wafer surface.
5. Measure Si (004) reflection HRXRD data at  = 0 rad.
6. Use measured θsurface,inst(004) and θsurface,SRM(004) to calibrate either λinst or Δθinst.
7. Repeat steps 2 through 5 several times to reduce the uncertainty in statistical mounting errors.
The procedures for aligning the specular reflection will introduce significant uncertainties that must be included inany uncertainty budget for use of the certified values.
Table 4. Uncertainty Budget for SRM 2000 Si (220) d-Spacing.
Quantity
dSRM
dSRM
Δδ2d
ΔTd
dWASO04

SRM 2000

Standard Uncertainty (k = 1)
0.43 fm
0.24 fm
0.36 fm
32
12

am
am

Uncertainty Type
Combined, uc(i)A, ui(i)B, uj(i)Combined, uc(i)Combined, uc(i)Page 8 of 10

Table 5. Uncertainty Budget for SRM 2000 Si (220) Miscut
QuantityξSRM

Standard Uncertainty (k = 1)
0.014 mrad
7.9
mrad
36
rad
rad
9.5
0.014 mrad
7.9
mrad
36
rad
rad
9.5

SRM
xSRM
ySRM
ξSRM

SRM
xSRM
ySRM

Uncertainty Type
Combined, uc(i)Combined, uc(i)Combined, uc(i)Combined, uc(i)A, ui(i)A, ui(i)A, ui(i)A, ui(i)Table 6. Uncertainty Budget for SRM 2000 Si (004) Surface Bragg Angle
Quantityθsurface,SRM(004)Standard Uncertainty (k = 1)
38
rad

xSRM
θB,SRM(004)θsurface,SRM(004)
ΔTθB,SRM(004)ωSRM
λSRMθB,SRM(004)

36
11
7.3
0.5
3.9
7.3

rad
rad
rad
rad
rad
rad

Uncertainty Type
Combined, uc (i )A, ui(i)Combined, uc(i)A, ui(i)A, ui(i)B, uj(i)B, uj(i)Table 7. Definitions used in SRM 2000 Uncertainty Analysis
Quantityui(dSRM)

Value
0.24 fmuj(Δδ2d)

0.36

uc(ΔTd)uc(dWASO04)
ui(ξSRM)
ui(SRM)
ui(xSRM)
ui(ySRM)
ui(xSRM)
uc(θB,SRM(004))

fm

0.032 fm
12
am
0.014 mrad
7.9
mrad
rad
36
rad
9.5
see aboverad
8.0

ui(θB,SRM(004))

7.3

rad

ui(ΔTθB,SRM(004))

0.5

rad

3.9
154 055.9
7.3

rad
fm
rad

ωSRM
λSRM
ui(λSRMθB,SRM(004))

Description
standard uncertainty (k = 1) (type A) of Si (220) transmission cased-spacingstandard uncertainty (type B) due to misalignment between reference
crystal and SRM feedstock
combined standard uncertainty of d-spacing due to temperature effects
combined standard uncertainty of WASO 04 d-spacing determination
standard uncertainty (type A) of magnitude of wafer miscut
standard uncertainty (type A) of direction of wafer miscut
standard uncertainty (type A) of tilt of wafer miscut in x-planestandard uncertainty (type A) of tilt of wafer miscut in y-plane
see above
combined standard uncertainty of Bragg angle from crystal planes for
Si (004) reflection peakstandard uncertainty (type A) of Bragg angle from crystal planes for Si(004) reflection peakstandard uncertainty (type A) due to temperature deviation overmeasurements
standard uncertainty (type B) of long range error in  axis at Si (004)reference monochromator wavelength for Si (004) measurements
standard uncertainty (type A) due to monochromator wavelength
deviation over Si (004) measurements

REFERENCES[1][2]

The International System of Units (SI); 8th ed.; Bureau International des Poids et Mesures, Paris (2006).
JCGM 100:2008; Evaluation of Measurement Data — Guide to the Expression of Uncertainty in
Measurement (ISO GUM 1995 with Minor Corrections); Joint Committee for Guides in Metrology (JCGM)(2008); available at http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf (accessed
Feb 2012); 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 http://www.nist.gov/pml/pubs/index.cfm (accessed Feb 2012).

SRM 2000

Page 9 of 10

[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]

Thompson, A.; Taylor, B.N.; Guide for the Use of the International System of Units (SI); NIST Special
Publication 811; U.S. Government Printing Office: Washington, DC (2008); available at
http://www.nist.gov/pml/pubs/index.cfm/ (accessed March 2012).
ISO; Quantities and Units; 3rd ed.; ISO: Geneva, Switzerland (1993).
Becker, P.; Bettin, H.; Danzebrink, H.U.; Glaser, M.; Kuetgens, U.; Nicolaus, A.; Schiel, D.; De Bievre, P.;
Valkiers, S.; Taylor, P.; Determination of the Avogadro Constant Via the Silicon Route; Metrologia,
Vol. 40(5), pp. 271–287 (2003).
Hanke, M.; Kessler, E.G.; Precise Lattice Parameter Comparison of Highly Perfect Silicon Crystals; J. Phys.
D: Appl. Phys., Vol. 38(10A), pp. A117–A120 (2005).
Deslattes, R.D.; Kessler, E.G.; Owens, S.; Black, D.; Henins, A.; Just How Perfect Can a Perfect Crystal
Be?; J. Phys. D: Appl. Phys., Vol. 32(10A), pp. A3–A7 (1999).
Becker, P.; The Determination of the Avogadro Constant - Not Simply a Metrological Problem; IEEE
Transactions on Instrumentation and Measurement, Vol. 48(2), pp. 225–229 (1999).
Becker, P.; Kuetgens, U.; Stumpel, J.; The Avogadro Problem: Summary of Tests on Crystal Imperfections;
IEEE Transactions on Instrumentation and Measurement, Vol. 50(2), pp. 612–615 (2001).
Bearden, J.A.; Thomsen, J.S.; Henins, A.; Sauder, W.C.; Marzolf, J.G.; Precision Redetermination ofStandard Reference Wavelengths for X-ray Spectroscopy; Physical Review A - General Physics,
Vol. 135(4A), p. A899 (1964).
Azároff, L.V.; X-ray Spectroscopy; McGraw-Hill Book Company, NY (1974).
Schnopper, H.W.; Spectral Measurements with Aligned and Misaligned 2-Crystal Spectrometers. I. Theory ofGeometrical Window; J. Appl. Phys., Vol. 36(4), p. 1415 (1965).
Schnopper, H.W.; Spectral Measurements with Aligned and Misaligned 2-Crystal Spectrometers. II.
Alignment; J. Appl. Phys., Vol. 36(4), p. 1423 (1965).
Bearden, J.A.; Thomsen, J.S.; Double-crystal X-ray Spectrometer - Corrections, Errors, and Alignment
Procedure; J. of Appl. Cryst., Vol. 4, p. 130 (1971).
Batterman, B.W.; Cole, H.; Dynamical Diffraction of X-rays by Perfect Crystals; Reviews of Modern
Physics, Vol. 36(3), p. 681 (1964).
Authier, A.; Dynamical Theory of X-Ray Diffraction, Oxford: Oxford University Press (2001).
Bond, W.L.; Precision Lattice Constant Determination; Acta Crystallographica, Vol. 13(10), pp. 814–818(1960).
Walder, V.; Burke, J.; Elimination of Specimen and Beam Tilt Errors in Bond Method of Precision LatticeParameter Determinations; J. Appl. Cryst., Vol. 4, p. 337 (1971).
Halliwell, M.A.G.; Measurement of Specimen Tilt and Beam Tilt in Bond Method; J. Appl. Cryst., Vol. 3,
p. 418 (1970).
Burke, J.; Tomkeieff, M.V.; Errors in Bond Method of Lattice Parameter Determinations-Further
Considerations; J. Appl. Cryst., Vol. 2, Part 6, p. 247 (1969).
Hubbard, C.R.; Mauer, F.A.; Precision and Accuracy of Bond Method as Applied to Small Spherical Crystals;
J. Appl Cryst.; Vol. 9, pp. 1–8 (1976).
Prince, E.; Mathematical, Physical and Chemical Tables; Vol. C, International Tables for Crystallography,
3rd ed., London: Kluwer Academic (2004).
Chantler, C.T.; X-ray-Diffraction of Bent Crystals in Bragg Geometry .1. Perfect-Crystal Modeling. J. Appl.
Cryst., Vol. 25, Part 6, pp. 674–693 (1992).
Chukhovskii, F.N.; Petrashen, P.V.; General Dynamical Theory of X-ray Laue Diffraction from a
Homogeneously Bent Crystal; Acta Crystallographica, Vol. 33, Section A, pp. 311–319 (1977).
Yan, H.; Kalenci, O.; Noyan, I.C.; Diffraction Profiles of Elastically Bent Single Crystals with Constant
Strain Gradients; J. Appl. Cryst., Vol. 40, Part 2, pp. 322–331 (2007).
Jones, E.; Oliphant, T.; Peterson, P.; and others. SciPy: Open source scientific tools for Python (2001).
R Development Core Team; A Language and Environment for Statistical Computing. R Foundation for
Statistical Computing; ISBN 3-900051-07-0, Vienna, Austria (2007).
Heckert, N.A.; Filliben, J.J.; NIST Handbook 148: DATAPLOT Reference Manual, Volume I: Commands,
National Institute of Standards and Technology Handbook Series, (2003);
http://www.itl.nist.gov/div898/software/dataplot.html/ (accessed March 2012).
Kessler, E.G.; Owens, S.M.; Henins, A.; Deslattes, R.D.; Silicon Lattice Comparisons Related to the
Avogadro Project: Uniformity of New Material and Surface Preparation Effects; IEEE Transactions on
Instrumentation and Measurement, Vol. 48(2), pp. 221–224 (1999).

Certificate Revision History: 05 March 2012 (Corrected uncertainty in Table 1 for dSRM; Editorial changes); 09 July 2009 (Original certificatedate).

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; fax (301) 926-4751;
e-mail srminfo@nist.gov; or via the Internet at http://www.nist.gov/srm.
SRM 2000

Page 10 of 10