Принципы нанометрологии (1027506), страница 64
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The number of ions collectedis related to the current required to neutralize them supplied by an irradiated Josephson junction voltage source. The mass of ions, M, is then givenby the equationZn1 :n2 :ma tM ¼fðtÞdt(10.3)20where n1 and n2 are integers, ma is the atomic mass of gold, f(t) is thefrequency of the microwave radiation irradiated onto the Josephson junctionand ma ¼ 197 u, for gold isotope Au197, where u is the atomic mass unit(equal to 1/12 of the mass of C12).10.1.6.4 Levitated superconductor approachAs with the Watt balance approach, the levitated superconductor methodrelates the unit of the kilogram to electrical quantities defined from theJosephson and quantized Hall effects [23]. A superconducting body is levitated in a magnetic field generated by a superconducting coil.
The currentrequired in the superconducting coil is proportional to the load on thefloating element and defines a mass (for the floating element) in terms of thecurrent in the superconducting coil [24–26].10.1.7 Mass comparator technologyFrom the earliest days of mass calibration, the measurements have beenmade by comparison, each weight or quantity being compared witha standard of theoretically better accuracy. A series of comparisons wouldthus allow all measurements to be eventually related back to a primarystandard, whether it was a naturally occurring standard (such as a grain of295296C H A P T ER 1 0: Mass and force measurementwheat) or an artefact standard such as the current international prototypekilogram.Until recently these comparisons have been performed using two-panbalances. From the earliest incarnations to the present day the technologyhas relied on a balance beam swinging about a pivot normally at the centre ofthe beam.
The mechanical quality of the beam and in particular the pivot hasbeen refined until modern two-pan mechanical balances are capable ofresolutions of the order of one part in 109, equivalent to 1 mg on a 1 kg massstandard.10.1.7.1 The modern two-pan mechanical balanceTwo-pan balances consist of a symmetrical beam and three knife-edges. Thetwo terminal knife-edges support the pans and a central knife-edge acts asa pivot about which the beam swings.
Two-pan balances are generally undamped, with a rest point being calculated from a series of turning points.Some balances incorporate a damping mechanism (usually mechanical ormagnetic) to allow the direct reading of a rest point.Readings from two-pan balances tend to be made using a simplepointer and scale although some use more complicated optical displays.In all cases the reading in terms of scale units needs to be converted intoa measured mass difference. Capacities of such balances range from a fewgrams up to several tonnes.
The resolution of smaller balances is limitedto the order of 1 mg by the accuracy with which the central knife-edge canbe polished.10.1.7.2 Electronic balancesElectronic balances are usually top-loading balances with the applied loadbeing measured by an electro-magnetic force compensation unit or a straingauge load cell. Single-pan electronic balances give a direct reading of theweight applied whereas the other two mechanical balance types rely on thecomparison of two forces (an unknown weight with either an external orinternal weight). Despite the possibility of using these balances as directreading devices (applying an unknown weight and taking the balance readingas a measure of its mass) single-pan electronic balances will always performbetter when used as comparators, comparing a standard (A) and an unknown(B) in an ABA or ABBA sequence.Since the definition of the unit of mass is currently realised at the 1 kg level,the development of 1 kg electronic balances and mass comparators representsthe current state of the art and 1 kg mass standards can be compared toa resolution of one part in 1010 and with an accuracy approaching 1 mg.Low-force measurement10.2 Low-mass measurementAt loads less than 1 kg the sensing technology does not improve significantlyand resolution is limited to 0.1 mg.
Additionally the process of subdividingthe kilogram mass standard introduces significant uncertainties thatincrease as the mass is moved away from 1 kg. Traditionally there has notbeen a large demand for weighing quantities at the milligram level and belowto accuracies better than a few tenths of 1 %. This, coupled with uncertainties introduced by the sub-division process and the relative instability ofmilligram mass standards, has limited the development of weighing technology in this area.
Equally there has been no real drive to extend the massscale below its traditional limit of 1 mg as weights at this level become verydifficult to manufacture and handle (see section 2.4).Recently, however, demands from the aerospace, pharmaceutical, microfabrication, environmental monitoring and low force measurement areas haveled to increased research into the lower limits of the mass scale.
Traditionalmass standards of metal wire have been manufactured with values down toa few tens of micrograms. These have been calibrated using existing microbalance technology to relative accuracies of a few percent. Traceability is takenfrom kilogram mass standards by a process of subdivision.For mass standards below this level the physical size of wire weightsbecomes too small for easy handling. However, the use of particulates mayprovide a way forward for microgram and nanogram mass standards, withtraceability being provided by density and dimensional measurements.10.2.1 Weighing by sub-divisionSub-division is used for the most demanding mass calibration applications.
Itinvolves the use of standards of one or more values to assign values toweights across a wide range of mass values. A typical example of this wouldbe to use two or three 1 kg standards to calibrate a 20 kg to 1 mg weight set.Equally it would be possible to use a 1 kg and a 100 g standard for sucha calibration.Weighing by sub-division is most easily illustrated by considering howvalues would be assigned to a weight set using a single standard.
In reality theweighing scheme would be extended to involve at least two standards. Thestandard is compared with any weights from the set of the same nominalvalue and also with various combinations of weights from the set that sum tothe same nominal value. A check-weight, which is a standard treated in thesame manner as any of the test-weights, is added in each decade of thecalibration so that it is possible to verify the values assigned to the weight set.297298C H A P T ER 1 0: Mass and force measurement10.3 Low-force measurement10.3.1 Relative magnitude of low forcesA full derivation of the surface interaction forces significant at the MNTscaleis beyond the scope of this book, and indeed has been presented by variousgroups previously. Nevertheless the basic force separation dependenciesare worthy of consideration by the reader and a selection is presented inTable 10.1.
Equations obtained from referenced works have, where necessary,been adapted to use common nomenclature. To simplify comparison, theinteraction of a sphere and flat plate is considered where possible. Since thetips of most probes can be adequately modelled as a (hemi-) sphere, this isa suitable approach.
The sphere–plate separation is assumed to be much lessthan the sphere radius. Figure 10.1 is a comparative plot using typical valuesfor the given parameters. Section 7.3.7 also discusses surface forces in termsof the atomic force microscope.10.3.2 Traceability of low-force measurementsTraceability for force measurement is usually carried out by comparing toa calibrated mass in a known gravitational field (see section 2.4). However, asthe forces (and hence masses) being measured decrease below around 10 mN(approximately equivalent to 1 mg), the uncertainty in the mass measurement becomes too large and the masses become difficult to handle.
For thisTable 10.1Summary of surface interaction force equationsIn these equations F is a force component, U the work function difference between the materials, D thesphere-flat separation, g the free surface energies at state boundaries, H the Hamaker constant and q thecontact angle of in-interface liquid on the opposing solid surfaces. In the capillary force the step functionu(.) describes the breaking separation; e is the liquid layer thickness and r the radius of meniscuscurvature in the gapInteractionEquationElectrostaticF ¼ 30 U 2 pR 2 =D 2 [27]h 2eF ¼ 4pgR 1 uðh þ LÞ [27,28].2rHRF ¼ 2 for non-retarded,6DCapillaryVan der Waalsattractive forces [29]Casimir effectF ¼ Rp3 hc[30]360D 3Low-force measurementFIGURE 10.1 Comparative plot of described surface interaction forces, based on the following values: R ¼ 2 mm; U ¼0.5 V; g ¼ 72 mJ$m2; H ¼ 1018 J; e ¼ r ¼ 100 nm.
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