Re-analysis of the uncertainty of the 0.895 [micro]m diameter (NIST SRM[R] 1690) and the 0.269 [micro]m diameter (NIST SRM[R] 1691) sphere standards.
The uncertainties of the mean diameters of the nominal 1.0 [micro]m SRM (1) (Storage Resource Management) The management of the storage resources in an organization in order to avoid duplication of files and to determine space utilization across all servers. [R] 1690 polystyrene polystyrene (pŏl'ēstī`rēn), widely used plastic; it is a polymer of styrene. Polystyrene is a colorless, transparent thermoplastic that softens slightly above 100°C; (212°F;) and becomes a viscous liquid at around 185°C; spheres and of the nominal 0.3 [micro]m SRM[R] 1691 polystyrene spheres are recomputed using the current NIST (National Institute of Standards & Technology, Washington, DC, www.nist.gov) The standards-defining agency of the U.S. government, formerly the National Bureau of Standards. It is one of three agencies that fall under the Technology Administration (www.technology. Guidelines for computing uncertainty. The revised expanded uncertainty (approximately 95% confidence level) for SRM[R] 1690 polystyrene spheres is equal to 0.005 [micro]m compared to previous value of 0.008 [micro]m. The revised expanded uncertainty for SRM[R] 1691 is equal to 0.004 [micro]m compared to the previous value of 0.007 [micro]m. The major cause of the reduction in the uncertainty for the 1.0 [micro]m spheres is from a decrease in the recomputed uncertainty of the refractive index A property of a material that changes the speed of light, computed as the ratio of the speed of light in a vacuum to the speed of light through the material. When light travels at an angle between two different materials, their refractive indices determine the angle of transmission of the polystyrene spheres. The 1.0 [micro]m spheres were used in calibrating the electron microscope electron microscope: see microscope. used to size the 0.3 [micro]m spheres, and the reduction in the uncertainty of 1.0 [micro]m SRM[R] uncertainty was the biggest factor in the decrease in the uncertainty of the 0.3 [micro]m spheres.
Key words: light scattering; Mie Scattering; polystyrene spheres; Standard Reference Materials; transmission electron microscopy “TEM” redirects here. For other uses, see TEM (disambiguation).
Transmission electron microscopy (TEM) is an imaging technique whereby a beam of electrons is transmitted through a specimen, then an image is formed, magnified and directed to appear either ; uncertainty analysis.
The NIST Standard Reference Material 1690 (NIST SRM[R] 1690) consists of a nearly monosize suspension of 0.895 [micro]m polystyrene spheres in water at a mass fraction of approximately 0.5%. This standard is used in the certification of secondary standards and also used directly in the calibration of electron microscopes, of scanning surface inspection systems in the semiconductor industry, and of other particle sizing instruments when the most accurate sizing standards are needed.
The certification of SRM[R] 1690 was based on the measurement of light scattering intensity versus scattering angle for a diluted di·lute
tr.v. di·lut·ed, di·lut·ing, di·lutes
1. To make thinner or less concentrated by adding a liquid such as water.
2. To lessen the force, strength, purity, or brilliance of, especially by admixture. suspension of the polystyrene spheres. Key features of the experiment  were the use of an intensity stabilized laser, an accurately indexed rotary table A rotary table is a precision work positioning device used in metalworking. It enables the operator to drill or cut work at exact intervals around a fixed (usually horizontal or vertical) axis. , and photon counting detection. The particle diameter was determined from a nonlinear A system in which the output is not a uniform relationship to the input.
nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. least squares fit of the predicted scattering based on Mie theory Mie theory, also called Lorenz-Mie theory or Lorenz-Mie-Debye theory, is a complete analytical solution of Maxwell's equations for the scattering of electromagnetic radiation by spherical particles (also called Mie scattering). and the measured data.
The uncertainty analyses used in this study  is not consistent with the current NIST policy [2,3] for reporting measurement uncertainty and result in overestimates of those uncertainties. The focus of this note is the recalculation re·cal·cu·late
tr.v. re·cal·cu·lat·ed, re·cal·cu·lat·ing, re·cal·cu·lates
To calculate again, especially in order to eliminate errors or to incorporate additional factors or data. of the uncertainty per the 1994 Guidelines . There is also an updated analysis of the refractive index component of the uncertainty.
NIST SRM[R] 1690 was also used in the measurement of SRM[R] 1691 (0.269 [micro]m diameter polystyrene spheres). The effect of the change in the uncertainty in SRM[R] 1690 on the uncertainty for SRM[R] 1691 is determined. The uncertainty in the SRM[R] 1691 is also recomputed per the 1994 guidelines, since the methodology used previously  is not consistent with current NIST policy.
2. Calculation of Expanded Uncertainty
The old procedure for computing the uncertainty is briefly reviewed and then the revised uncertainty analysis is presented based upon the 1994 Guidelines.
2.1 Old Method
The total uncertainty, [U.sub.T(old)], was computed by adding the random error, R, and the sum of the absolute values of the systematic errors, [u.sub.Bi] (1).
[U.sub.T(old)] = R + [summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (i)]|[u.sub.Bi]| (1)
The random component of the uncertainty, R, was computed as the product of a coverage factor, k, for a 95% confidence level times the uncertainty of the mean for 10 repeat measurements of the mean, [u.sub.r].
[u.sub.r] = [[1/[n(n-1)]][n.summation over (i=1)]([D.sub.n,i] - [bar.D.sub.n])[.sup.2]][.sup.1/2]. (2)
The quantity [D.sub.n,i] is the average diameter of the i-th sample, and [bar.D] is the average of the 10 samples. The computed value of [u.sub.r] equals 0.000229 [micro]m. The coverage factor k for 9 degrees of freedom based on Student's t-distribution In probability and statistics, the t-distribution or Student's t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. for "about 95%" confidence interval confidence interval,
n a statistical device used to determine the range within which an acceptable datum would fall. Confidence intervals are usually expressed in percentages, typically 95% or 99%. is 2.32. Thus the value of R is given by the following:
R = k[u.sub.r] = (2.32)(0.000229) =0.00053 i m. (3)
The systematic uncertainties are related to the particle properties and the optical system. The uncertainties are expressed in terms of the effect on the particle diameter and the values are given in Table 1. The particle related uncertainties include the refractive index of the spheres, [u.sub.B1], the presence of about 1% agglomerated agglomerated
of particles, compacted together into a mass.
particulated feeds compacted or extruded into pellets and similar forms. doublets dou·blet
1. A close-fitting jacket, with or without sleeves, worn by European men between the 15th and 17th centuries.
a. A pair of similar or identical things.
b. A member of such a pair. , [u.sub.B2], and multiple scattering from the particle suspension, [u.sub.B3]. The optical system related uncertainties include the reflection from the glass cell, [u.sub.B4], the finite acceptance angle of the detector of about [+ or -] 1[degrees], [u.sub.B5], and the slight optical misalignment mis·a·ligned
misa·lignment n. at zero angle, [u.sub.B6]. These uncertainties, which were referred to as systematic uncertainties in Mulholland et al. , are now classified as Type B uncertainties. These estimates are based on scientific judgment rather than based on statistical methods, as is done for Type A uncertainties.
The total uncertainty, [U.sub.T(old)] = 0.0074, is computed from Eqs. (1), (3), and the sum of the systematic uncertainties (see Table 1).
2.2 New Method
In 1994 the method for reporting uncertainties at NIST was unified  and aligned with the ISO (1) See ISO speed.
(2) (International Organization for Standardization, Geneva, Switzerland, www.iso.ch) An organization that sets international standards, founded in 1946. The U.S. member body is ANSI. Guide to the Expression of Uncertainty . In this approach each component of uncertainty of a measurement result is represented by an estimated standard deviation In statistics, the average amount a number varies from the average number in a series of numbers.
(statistics) standard deviation - (SD) A measure of the range of values in a set of numbers. , termed standard uncertainty with symbol u. There are two types of standard uncertainty. The first is computed by statistical means such as the standard deviation of the mean of several repeat measurements and is termed a Type A standard uncertainty. The second is often based on scientific judgment using all the relevant information available and is termed Type B standard uncertainty.
In the case of NIST SRM[R] 1690, the Type A standard uncertainty is the standard deviation of the mean of 10 repeat measurements of the number mean diameter, which is defined as [u.sub.r] in Eq. (2) and found to have a value of 0.000229 [micro]m.
The Type B standard uncertainties for NIST SRM[R] 1690 consist of six components of uncertainty given in Table 1 .
Following the NIST Guidelines, the combined uncertainty, [u.sub.c], is computed as the root-sum-of-squares of the Type A uncertainties and the Type B uncertainties. The basis of this approximation approximation /ap·prox·i·ma·tion/ (ah-prok?si-ma´shun)
1. the act or process of bringing into proximity or apposition.
2. a numerical value of limited accuracy. is that provided the variables are independent, the variance of a sum of independent variables is equal to the sum of the variances.
[u.sub.c] = [[u.sub.r.sup.2] + [summation over (i)][u.sub.Bi.sup.2]][.sup.1/2] = 0.0035. (4)
The expanded uncertainty, U, which defines an interval having a level of confidence of about 95% (95.4%), is computed as U = k[u.sub.c]. The quantity k is the coverage factor and its value is dependent on the number of degrees of freedom for [u.sub.c]. In the limit of infinite degrees of freedom, the value of k is 2.0. For a finite number of degrees of freedom, k is estimated as the t-factor from the Student's t-distribution based on the number of degrees of freedom and about a 95% confidence interval. For a combined uncertainty arising from several components each with degrees of freedom [[nu].sub.i], the effective number of degrees of freedom, [[nu].sub.eff], is estimated using the Welch-Satterthwaite formula :
[[nu].sub.eff] = [u.sub.c.sup.4]/[[summation over (i)][[[c.sub.i.sup.4][u.sub.i.sup.4]]/[[nu].sub.i]]]. (5)
It is assumed that the number of degrees of freedom for each of the Type B terms in Eq. (4) is infinity so that the only term in the sum is the Type A uncertainty given by Eq. (2). In this case the sensitivity factor [c.sub.i] is unity, the term [u.sub.i] = [u.sub.r] = 0.000229 [micro]m, and the degrees of freedom, [[nu].sub.i], is 9 (the number of repeat measurements minus one). The resulting value of [[nu].sub.eff] from Eq. (5) is 4.8 X [10.sup.5]. The corresponding value for k = 2.00. Given k, the value of the expanded uncertainty, U, is computed as 0.0070 [micro]m.
2.3 New Method With Revised Type B Uncertainty Estimates
Several of the systematic uncertainties estimated by Mulholland et al.  are over estimates. The term "at most" is used in describing both the uncertainty associated with the reflections from the glass cell, [u.sub.B4], and the finite acceptance angle of the detector, [u.sub.B5]. To convert these estimates to Type B standard uncertainties, we treat each of these quantities as having equal probability over the respective ranges of [+ or -] 0.001 [micro]m for [u.sub.B4] and [+ or -] 0.005 [micro]m for [u.sub.B5]. For this rectangular probability distribution Probability distribution
A function that describes all the values a random variable can take and the probability associated with each. Also called a probability function.
probability distribution , the standard deviation is [u.sub.Bi]/(3)[.sup.1/2]. So both of these uncertainties are reduced to 0.58 times their previous values, which corresponds to 0.0006 [micro]m for [u.sub.B4] and 0.0003 [micro]m for [u.sub.B5] (See corrected values in Table 1.)
The previous estimate of refractive index uncertainty corresponded to the range in the reported values from five studies . This provides an over-estimate and the estimate could be revised by the method used above for [u.sub.B4] and [u.sub.B5]. Instead, we compute the uncertainty based on the single particle refractive index measurements by Marx and Mulholland  for the SRM[R] 1690 particles. This is a more accurate approach because two of the other four studies involved particles at least a factor of three smaller than the SRM[R] and the other two studies, which used a method similar to , did not include a quantitative uncertainty analysis. The measurement of the refractive index was based on measuring the light scattering versus angle from 30[degrees] to 160[degrees] for a single, levitated SRM[R] sphere. The refractive index and particle size Particle size, also called grain size, refers to the diameter of individual grains of sediment, or the lithified particles in clastic rocks. The term may also be applied to other granular materials. were determined from best fits of Mie theory predictions to the scattering data for the incident laser polarization polarization
Property of certain types of electromagnetic radiation in which the direction and magnitude of the vibrating electric field are related in a specified way. direction both parallel to the scattering plane and perpendicular to the scattering plane . The best fit was based on the maximum in the harmonic mean har·mon·ic mean
The reciprocal of the arithmetic mean of the reciprocals of a specified set of numbers.
see harmonic mean. , 1/[Q.sub.1] + 1/[Q.sub.2], of the results for the two polarization directions. The quantities [Q.sub.1] and [Q.sub.2] are the sums-of-squares of the differences between the measured and predicted scattering for the laser polarization direction parallel and perpendicular to the scattering plane. The resulting mean and standard deviation of the mean for measurements on eight separate particles is 1.6121 [+ or -] 0.0013.
There are two sources of Type B uncertainty for the refractive index: uncertainty in the angle, [u.sub.[theta Theta
A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ]], and in the polarization direction, [u.sub.P]. There was a slight drift in the encoder A hardware device or software that assigns a code to represent data. See encode.
1. (algorithm, hardware) encoder - Any program, circuit or algorithm which encodes.
Example usages: "MPEG encoder", "NTSC encoder", "RealAudio encoder".
2. angle readout (1) A small display device that typically shows only a few digits or a couple of lines of data.
(2) Any display screen or panel. of 0.08[degrees] over the time that the measurements were made. Including this drift in the numerical simulation of the light scattering, it was found that the change in the refractive index was 0.005. We use this value as our estimate of [u.sub.[theta]]. It was found that for each polarization direction selected, there was about 0.5% of that intensity of light with the orthogonal At right angles. The term is used to describe electronic signals that appear at 90 degree angles to each other. It is also widely used to describe conditions that are contradictory, or opposite, rather than in parallel or in sync with each other. polarization direction. From numerical simulations, this effect was found to change the refractive index by 0.0032 (0.2%). This is our estimate of [u.sub.P].
The combined uncertainty in the refractive index, [u.sub.n], is obtained as the quadrature quadrature, in astronomy, arrangement of two celestial bodies at right angles to each other as viewed from a reference point. If the reference point is the earth and the sun is one of the bodies, a planet is in quadrature when its elongation is 90°. sum of the standard deviation of the mean and the two Type B uncertainties. The resulting value is 0.0061. The effect of this refractive index uncertainty on the uncertainty in the diameter of SRM[R] 1690, [u.sub.B1], is determined to be 0.0020 [micro]m based on the analysis on page 14 of the study by Mulholland et al. .
The combined uncertainty for the mean diameter is computed using Eq. (4) with the corrected values of [u.sub.B1], [u.sub.B4], and [u.sub.B5]. The resulting value is 0.0026 [micro]m. Equation (5) is used for computing the degrees of freedom with a resulting value of 1.47 X [10.sup.5]. For a 95% confidence level, the corresponding coverage factor is 2.00. Therefore the corrected expanded uncertainty is 0.0052 [micro]m. This value is about 1/3 less than the value of 0.008 [micro]m on the SRM[R] 1690 certificate.
3. Impact on the Certified See certification. Values for SRM[R] 1691
The uncertainty of the 0.3 [micro]m SRM[R] is recomputed to include the effect of the change in the uncertainty in the 1.0 [micro]m SRM[R]. The current NIST Guidelines for expressing uncertainty  are used in carrying out the analysis. The particle sizes for the 0.3 [micro]m spheres were measured by transmission electron microscopy (TEM TEM
1. transmission electron microscope.
3. transmissible encephalopathy of mink. ). Both the 1.0 [micro]m SRM[R] particles and the 0.3 [micro]m particles were deposited on five TEM grids. For each grid, at least 40 of both the 0.3 [micro]m particles and of the 1.0 [micro]m particles were sized. The 1.0 [micro]m SRM[R] served as the magnification Magnification
A measure of the effectiveness of an optical system in enlarging or reducing an image. For an optical system that forms a real image, such a measure is the lateral magnification m standard for the measurements. For each of the five TEM grids a mean size was computed. The average of these five mean sizes was found to be 0.269 [micro]m. The standard deviation of the means was found to be 0.00134 [micro]m, which is equal to the Type A uncertainty of the measurements, [u.sub.A].
One Type B uncertainty component is the uncertainty in the magnification. As shown in the study by Lettieri and Hembree , the magnification uncertainty is equal to the combined standard uncertainty in the 1.0 [micro]m SRM[R], 0.0026 [micro]m, multiplied times the ratio of the diameter of the 0.3 [micro]m SRM[R] to the 1.0 [micro]m SRM[R]. The resulting value of [u.sub.m] is equal to 0.00078 [micro]m. The second Type B component, [u.sub.e], is the uncertainty in the determination of the point in the particle image that corresponds to the actual edge of the particle. The estimated value  is 0.001 [micro]m.
The combined uncertainty, obtained from the quadrature sum of [u.sub.A], [u.sub.m], and [u.sub.e], is equal to 0.00184 [micro]m. The effective number of degrees of freedom computed using Eq. (5) is equal to 14. In this case the coverage factor is 2.20 and the expanded uncertainty is equal to 0.0040 [micro]m. This value is about 40% smaller than the value currently given on the SRM[R] 1691 Certificate.
* The revised expanded uncertainty (approximately 95% confidence level) for SRM[R] 1690 is equal to 0.005 [micro]m with number mean diameter of 0.895 [micro]m compared to 0.008 [micro]m on the SRM[R] certificates dated 2004 and earlier.
* The revised expanded uncertainty for SRM[R] 1691 is equal to 0.004 [micro]m with number mean diameter of 0.269 [micro]m compared to 0.007 [micro]m on SRM[R] certificates dated 2004 and earlier.
Table 1. Type B (systematic) Uncertainties for Measurement of SRM[R] 1690 Type B uncertainties Symbol Original value, Corrected value, [micro]m [micro]m Refractive index [u.sub.B1] 0.0030 0.0020 Particle doublets [u.sub.B2] 0.0010 0.0010 (a) Multiple scattering [u.sub.B3] 0.0010 0.0010 (a) Cell reflection [u.sub.B4] 0.0010 0.0006 Finite acceptance angle [u.sub.B5] 0.0005 0.0003 Optical misalignment [u.sub.B6] 0.0004 0.0004 (a) [summation over (i)] |[u.sub.Bi]| 0.0069 [[summation over (i)] ([u.sub.Bi])[.sup.2]] [.sup.1/2] 0.0035 0.0026 (a) These values are now considered to correspond to standard uncertainties.
Accepted: January 6, 2005
Available online: http://www.nist.gov/jres
(1) The symbol [[delta].sub.1], used in Mulholland et al. , has been replaced with [u.sub.Bi].
 G. W. Mulholland, A. W. Hartman, G. G. Hembree, E. Marx, and T. R. Lettieri, Development of a One-Micrometer-Diameter Particle Size Standard Reference Material, J. Res. Natl. Bur. Stand. (U.S.) 90, 3-26 (1984).
 B. N. Taylor and C. E. Kuyatt, Guidelines for evaluating and expressing uncertainty of NIST measurement results, NIST Technical Note 1297 1994 Edition, prepared under the auspices of the NIST Ad Hoc Committee ad hoc committee A committee formed with the purpose of addressing a specific issue or issues, which theoretically is disbanded once its raison d'etre is finished on Uncertainty Statements (U.S. Government Printing Office, Washington, DC 1994).
 International Organization for Standardization International Organization for Standardization (ISO)
Organization for determining standards in most technical and nontechnical fields. Founded in Geneva in 1947, its membership includes more than 100 countries. (1993), Guide to the Expression of Uncertainty in Measurement, Geneva Geneva, canton and city, Switzerland
Geneva (jənē`və), Fr. Genève, canton (1990 pop. 373,019), 109 sq mi (282 sq km), SW Switzerland, surrounding the southwest tip of the Lake of Geneva. , Switzerland (Corrected and reprinted, 1995).
 T. R. Lettieri and G. G. Hembree, Certification of NBS (National Bureau of Standards) See NIST.
NBS - National Bureau of Standards: part of the US Department of Commerce, now NIST. SRM[R] 1691: 0.3 [micro]m Diameter Polystyrene Spheres, NBSIR 88-3730, NIST, Gaithersburg, MD (1988).
 E. Marx and G. W. Mulholland, Size and Refractive Index Determination of Single Polystyrene Spheres, J. Res. Natl. Bur. Stand. (U.S.) 88, 321-338 (1983).
G. W. Mulholland
National Institute of Standards and Technology National Institute of Standards and Technology, governmental agency within the U.S. Dept. of Commerce with the mission of "working with industry to develop and apply technology, measurements, and standards" in the national interest. , Gaithersburg, MD 20899
About the authors: George W. Mulholland is a research chemist in the Fire Metrology Group in the NIST Building and Fire Research Laboratory. He conducts research in smoke particulate par·tic·u·late
Of or occurring in the form of fine particles.
A particulate substance.
composed of separate particles. phenomena and in particle size metrology. The National Institute of Standards and Technology is an agency of the Technology Administration, U.S. Department of Commerce.