Принципы нанометрологии, страница 9

It is one of a few cases where a name is given to the unit one,in order to facilitate the identification of the quantity involved. The namesgiven for the quantity angle are radian (plane angle) and steradian (solidangle). The radian is defined with respect to a circle and is the angle subtended by an arc of a circle equal to the radius (approximately 57.2958 ).For practical angle measurement, however, the sexagesimal (degrees,minutes, seconds) system of units, which date back to the Babylonian civilization, is used almost exclusively [19].

The centesimal system introducedby Lagrange towards the end of the eighteenth century is rarely used.Other units referred to in this section require either a material artefact(for example, mass) or a natural standard (for example, length). No ultimatestandard is required for angle measurement since any angle can be established by appropriate sub-division of the circle. A circle can only have 360 .In practice basic standards for angle measurement either depend on theaccurate division of a circle or the generation of an angle from two knownlengths.

Instruments that rely on the principle of sub-division includeprecision index tables, rotary tables, polygons and angular gratings [19].Instruments that rely on the ratio of two lengths include angular interferometers (see section 5.2.9), sine bars, sine tables and small angle generators.Small changes in angle are detected by an autocollimator [20] used inconjunction with a flat mirror mounted on the item under test, for examplea machine tool. Modern autocollimators give a direct digital readout of angularposition. The combination of a precision polygon and two autocollimators1314C H A P T ER 2 : Some basics of measurementenables the transfer of high accuracy in small angle measurement to thesame accuracy in large angles, using the closing principle that all anglesadd up to 360 .Sometimes angle measurement needs to be gravity-referenced and in thiscase use is made of levels. Levels can be based either on a liquid-filled vial oron a pendulum and ancillary sensing system.2.7 TraceabilityThe concept of traceability is one of the most fundamental in metrology andis the basis upon which all measurements can be claimed to be accurate.Traceability is defined as follows:Traceability is the property of the result of a measurement whereby itcan be related to stated references, usually national or internationalstandards, through a documented unbroken chain of comparisons allhaving stated uncertainties.

[21]To take an example, consider the measurement of surface profile usinga stylus instrument (see section 6.6.1). A basic stylus instrument measuresthe topography of a surface by measuring the displacement of a stylus as ittraverses the surface. So, it is important to ensure that the displacementmeasurement is ‘correct’. To ensure this, the displacement-measuringsystem must be checked or calibrated against a more accurate displacementmeasuring system. This calibration is carried out by measuring a calibratedstep height artefact (known as a transfer artefact).

Let us suppose that themore accurate instrument measures the displacement of the step using anoptical interferometer with a laser source. This laser source is calibratedagainst the iodine-stabilized laser that realises the definition of the metre,and an unbroken chain of comparisons has been ensured. As we move downthe chain from the definition of the metre to the stylus instrument that weare calibrating, the accuracy of the measurements usually decreases.It is important to note the last part of the definition of traceability thatstates all having stated uncertainties.

This is an essential part of traceabilityas it is impossible to usefully compare, and hence calibrate, instrumentswithout a statement of uncertainty. This should become obvious once theconcept of uncertainty has been explained in section 2.8.3. Uncertainty andtraceability are inseparable. Note that in practice the calibration of a stylusinstrument is more complex than a simple displacement measurement (seesection 6.10).Accuracy, precision, resolution, error and uncertaintyTraceability ensures that measurements are consistent and accurate.

Anyquality system in manufacturing will require that all measurements aretraceable and that there is documented evidence of this traceability (forexample ISO 17025 [22]). If component parts of a product are to be made bydifferent companies (or different parts of an organisation) it is essential thatmeasurements are traceable so that the components can be assembled andintegrated into a product.In the case of dimensional nanometrology, there are many exampleswhen it is not always possible to ensure traceability because there is a breakin the chain, often at the top of the chain.

There may not be national orinternational specification standards available and the necessary measurement infrastructure may not have been developed. This is the case formany complex three-dimensional MNT measurements. Also, sometimesan instrument may simply be too complex to ensure traceability of allmeasurements. An example of this is the CMM (see chapter 9). Whilst thescales on a CMM (macro- or micro-scale) can be calibrated traceably, theoverall instrument performance, or volumetric accuracy, is difficult and timeconsuming to determine and will be task-specific. In these cases it isimportant to verify the performance of the instrument against its specification by measuring well-chosen artefacts that have been traceably calibratedin an independent way.

Where there are no guidelines or where there is a newmeasurement instrument or technique to be used, the metrologist mustapply good practice and should consult other experts in the field.Traceability does not only apply to displacement (or length) measurements –all measurements should be traceable to their respective SI unit. In somecases, for example in a research environment or where a machiningprocess is stable and does not rely on any other process, it may only benecessary to have a reproducible measurement. In this case the resultsshould not be used where others may rely upon them and should certainlynot be published.2.8 Accuracy, precision, resolution, error and uncertaintyThere are many terms used in metrology that one must be aware of and it isimportant to be consistent in their use.

The ISO VIM [21] lays out formaldefinitions of the main terms used in metrology. Central to many metrologyterms and definitions is the concept of the ‘true value’. The true value ofa measurement is the hypothetical result that would be returned by an idealmeasuring instrument if there were no errors in the measurement. In practice the perfect scenario can never be achieved; there will always be some1516C H A P T ER 2 : Some basics of measurementdegree of error in the measurement and it may not always be possible tohave a stable, single-valued measurand.

Even if one had an ideal instrumentand measurement set-up, all measurements are ultimately subject toHeisenberg’s Uncertainty Principle; a consequence of quantum mechanicsthat puts a limit on measurement accuracy [23]. Often the true value isestimated using information about the measurement scenario. In manycases, where repeated measurements are taken, the estimate of the true valueis the mean of the measurements.2.8.1 Accuracy and precisionAccuracy and precision are the two terms in metrology that are mostfrequently mixed up or used indistinguishably. The accuracy of a measuringinstrument indicates how close the result is to the true value. The precisionof a measuring instrument refers to the dispersion of the results whenmaking repeated measurements (sometimes referred to as repeatability).

It is,therefore, possible to have a measurement that is highly precise (repeatable)but is not close to the true value, i.e. inaccurate. This highlights the fundamental difference between the two terms and one must be careful when usingthem. Accuracy is a term relating the mean of a set of repeat measurementsto the true value, whilst precision is representative of the spread of themeasurements.The VIM definition of accuracy is:closeness of agreement between a measured quantity value and a truequantity value of a measurandand the definition of precision is:closeness of agreement between indications or measured quantityvalues obtained by replicate measurements on the same or similarobjects under specified conditions.2.8.2 Resolution and errorThe resolution of a measuring instrument is a quantitative expression of theability of an indicating device to distinguish meaningfully between closelyadjacent values of the quantity indicated.

For example, for a simple dialindicator read by eye, the resolution is commonly given as half the distancebetween smallest, distinguishable indicating marks. It is not always eithereasy or obvious how to determine the resolution of an instrument. Considerfor example an optical instrument that is used to measure surface texture andfocuses light onto the surface. The lateral resolution is sometimes quoted inAccuracy, precision, resolution, error and uncertaintyterms of the Rayleigh or Abbe criteria [24] although, depending on thenumerical aperture of the focusing optics, the lateral resolution may bedetermined by the detector pixel spacing (see section 6.7.1).

The axialresolution will be a complex function of the detector electronics, the detection algorithm and the noise floor. This example highlights that resolution isnot a simple parameter to determine for a given instrument.It is also important to note that one should always consider resolutionhand in hand with other instrument performance indicators such as accuracyand precision.

Again using the example of the optical surface measuringinstrument, some surfaces can cause the instrument to produce errors thatcan be several hundred nanometres in magnitude despite the fact that theinstrument has an axial resolution of perhaps less than a nanometre (seesection 6.7.1).The error in a measuring instrument is the difference between the indicated value and the true value (or the calibrated value of a transfer artefact).Errors usually fall into two categories depending on their origin. Randomerrors give rise to random fluctuations in the measured value and arecommonly caused by environmental conditions, for example seismic noise orelectrical interference.

Systematic errors give rise to a constant differencefrom the true value, for example due to alignment error or because aninstrument has not been calibrated correctly. Most measurements containelements of both types of error and there are different methods for eithercorrecting errors or accounting for them in uncertainty analyses (see [25] fora more thorough discussion on errors).