10.1.1.408.199

History and progress in the accurate determination of the Avogadro constant

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2001 Rep. Prog. Phys. 64 1945

(http://iopscience.iop.org/0034-4885/64/12/206)

Download details:IP Address: 132.248.103.229The article was downloaded on 14/10/2011 at 17:57

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 64 (2001) 19452008 PII: S0034-4885(01)12689-8

History and progress in the accurate determination ofthe Avogadro constant

Peter Becker

Physikalisch-Technische Bundesanstalt, Braunschweig, Germany

Received 24 September 2001Published 15 November 2001Online at stacks.iop.org/RoPP/64/1945

Abstract

The Avogadro constant, NA, is a fundamental physical constant that relates any quantity atthe atomic scale to its corresponding macroscopic scale. Inspired by the kinetic gas theoryAvogadro proposed his hypothesis in 1811, in order to describe chemical reactions as anatomic process between atoms or molecules. Starting from his pioneering findings, thedetermination of this large number has fascinated generations of scientists up to this day.The review of methods aimed at finding a value for NA starts with the calculations madeby Loschmidt (1865; NA 72 1023 mol1) who evaluated the number of molecules ina given gas volume, derived from estimates of molecular diameters and the mean free pathlength. Consideration of Brownian motion led to some more accurate determinations of NAaround the beginning of the 20th century (Perrin (1908); NA 6.7 1023 mol1). Othermethods developed in the following years are based on Millikans oil drop experiment (1917,NA 6.064(6) 1023 mol1), on the counting of alpha particles emitted from radium oruranium (Rutherford (1909); NA 6.16 1023 mol1) and on investigations of molecularmonolayers on liquids (Nuoy (1924); NA 6.004 1023 mol1).

A modern method to derive NA from the density, the relative atomic mass, and the unitcell length was introduced by Bragg in 1913. It makes use of the diffraction of x-rays bythe interatomic spacings of a crystal lattice and its periodic arrangement. The accuracyof this method is extremely affected by the fact that the lattice scale of the structurallyimperfect lattice can be calibrated only approximately in SI units. Data of NA were,therefore, found to be in disagreement with other fundamental constants (Bearden (1931);NA 6.019(3) 1023 mol1). A break though was achieved with perfect crystals of siliconand x-ray interferometry making available very precise data of atomic distances, expressed inSI units (Bonse and Hart 1965).

Today, metrology has re-discovered the Avogadro constant and uses it as one of severalpossible routes to a re-definition of the kilogram because the old platinum iridium artefactexhibits long-term stability problems. This application of the Avogadro constant presupposes afinal measurement uncertainty of about 1108, a challenge for the experimental determinationof the quantities involved, i.e. macroscopic density, isotopic composition, and unit cell volumeof a silicon crystal. Many years of research work were centred on the problem of how far theperfection of a real crystal is away from the ideal state. At present, it is widely accepted that,

0034-4885/01/121945+64$90.00 2001 IOP Publishing Ltd Printed in the UK 1945

1946 P Becker

in the limits of the desired uncertainty, the lattice parameter, and thus the unit cell volumeof silicon, can be seen as an invariant quantity when the influence of residual defects, forexample impurities, is taken into account. Up to a relative measurement uncertainty of a fewparts in 107 it has recently been shown that the molar volume, the ratio of molar mass todensity, is constant, too. The combination of data from several independent measurementsof the unit cell and the molar volumes has led to a value for the Avogadro constant ofNA = 6.022 1335(30) 1023 mol1 (De Bie`vre et al 2001) recommended by the nationalmetrology institutes involved in this research project (Becker 2001).

Prominent examples of the significance of the research work reviewed here are the use ofNA as an input independent of other data, for the adjustment of a consistent set of fundamentalconstants, and the accompanying outstanding experimental developments acting as spin-offsin the field of technology to make macroscopic dimensions traceable to the atomic scale.

History and progress in the accurate determination of the Avogadro constant 1947

Contents

Page1. Introduction 19482. Todays significance of the Avogadro constant 19483. Experimental determinations 1949

3.1. Avogadros hypothesis and the kinetic gas theory 19493.2. Early measurements on solid solutions in liquids 19503.3. Avogadros hypothesis and the Brownian motion 19503.4. The oil drop method 19513.5. The early x-ray/crystal density approach 19533.6. Discrepancies in x-ray data 1955

4. Establishing the link of the lattice parameter to the meter by x-ray interferometry 19554.1. Former XRCD key experiments 1956

5. Avogadros hypothesis: the XRCD method 19575.1. Material characterization 19585.2. The silicon lattice parameteran invariant quantity of nature? 19685.3. Determination of the molar mass 19795.4. Density of silicon volume standards 19835.5. Summary of NA results 19945.6. New developments 1996

6. Indirect determinations of the Avogadro constant from measurements of fundamentalphysical constants 1999

7. Spin-offs in the field of metrology and solid state physics 20018. The role of NA in the SI, present state and outlook 2003

Acknowledgments 2004References 2004

1948 P Becker

1. Introduction

The development of the kinetic gas theory in physics, chiefly due to Clausius, Maxwell andBoltzmann, and the development of a modern understanding of the atom or molecule inchemistry towards the end of the 18th century were the roots of the Avogadro constant. At thattime, chemistry stood on the verge of becoming an accurate science. John Dalton (17661844)declared in 1808 that the atoms of each element did not differ from one another, that they hadthe same properties and a characteristic atomic weight. These findings hold also for moleculesin a chemical compound. He also stated that the mass of an atom can be measured relative tothe mass of a hydrogen atom by their atomic weights. Around this time, in 1809, Joseph LouisGay-Lussac (17781850) studied the chemical reaction of gases. He found experimentally thatthe internal energy of an ideal gas was independent of its volume and that the ratios of volumesof the reacting gases were small integer numbers. Amedeo Avogadro (17761856) gave amore precise description of the reacting particles and distinguished between atoms as parts ofa molecule. Avogadro stated in 1811 that equal volumes of all gases at the same temperatureand pressure contain the same number of molecules, which is well known as Avogadrosprinciple. Avogadros work was almost completely neglected until it was forcefully presentedby Stanislao Cannizarro. In 1858 he published a consistent system of atomic weights andchemical formulae of all elements based upon the principles laid down by Avogadro.

In the following years, Avogadros hypothesis was taken as a starting point, from which,as logical as may be, the ideas of molecular weight, atomic weight, valency, radicals, etc weredeveloped, and from which, finally, the challenge was developed to estimate the number ofparticles in that volume and to comprehend the size of such a large number.

The term Avogadro constant was first used in a paper entitled Brownian Movement andMolecular Reality and published by Jean Perrin (18701942) in 1909. In his paper he wrote:The invariable number N is a universal constant, which may be appropriately designatedAvogadros constant and is reserved nowadays for the number of particles (entities) in amole, NA, even though Avogadro made no quantitative estimate of that constant. The termLoschmidts number is used then for the number of molecules in a cubic centimetre of a gasunder standard conditions, NL. Because the molar volume of an ideal gas is 22.4 litres, thevalue of the Avogadro constant is 22.4 times that of Loschmidts number.

Avogadro died in 1856 without his central hypothesis being recognized.The aim of this paper is to review and update the evolution of the determination of the

Avogadro constant from the early days of molecular theory to the present state of high-precisionevaluation derived from perfect silicon crystals. The structure of the paper is as follows:section 2 treats the significance of the Avogadro constant from the metrological and scientificpoints of view. Section 3 is devoted to early experimental methods for studying the value ofNA, from the kinetic gas theory to the x-ray method. Section 4 describes the basic concept ofx-ray interferometry as a break-through of todays highly precise evaluation of the Avogadroconstant, which is described in detail in section 5. The final chapters connect the Avogadroconstant with other fundamental constants and with spin-offs in science and technology.

Historical developments of this fundamental constant were also dealt with in the past byVirgo (1932), and in great detail by Deslattes (1980), Mathieu (1984) and, recently, by Manaand Zosi in 1995.

2. Todays significance of the Avogadro constant

The Avogadro constant has long been known as the scale factor between the macroscopic andthe atomic worlds. It helps to see the numerical value of a quantity in the macroscopic world


as being a multiple of the same quantity on the atomic scale. This is one of the reasons whythe Avogadro constant has been introduced into the International System of Units, the SI, interms of which physical and chemical quantities must be expressed. The concept of a physicalquantity was introduced by J C Maxwell in 1860: Every expression of a quantity consists oftwo factors or components. One of these is the quantity of the same kind as the quantity tobe expressed, which is taken as a standard of reference. The other component is the numberof times the standard is to be taken in order to make the required quantity. The chemicalquantity, called amount of substance, was introduced as a base quantity of the SI in 1971 andits assigned unit, the mole, is currently defined as: The amount of substance of a system whichcontains as many elementary entities as there are atoms in 0.012 kg of 12C. When the mole isused, the elementary entities must be specified and may be atoms, molecules, ions, electrons,or other particles or specified groups of particles. The number of entities in 12 g of 12C is theAvogadro number. A determination of the Avogadro constant, therefore, is a realization of thebase unit, mole.

There is yet another significance to the Avogadro constant: the present SI base unit formass is the prototype kilogram kept at Se`vres. However, the mass of that primary standardof mass is not stable. The estimated instability has been m/m 5 108 (Quinn 1991)over the past 100 years. The suggestion has, therefore, been made to change the definition ofthe kilogram following again the well known requirement by J C Maxwell that the physicalunits should not be founded on macroscopic quantities but on the properties of, for example,imperishable and unalterable molecules (Maxwell 1870). If a determination ofNA succeededto better than 5 108, in relative values, the definition could be changed from the mass ofthe prototype of the kilogram into the mass of 103{NA} atoms of 12C.

As a fundamental physical quantity, the Avogadro constant is related to other constants andcan be determined indirectly from the measurements of these. NA, then, appears to be, bothan input and an output of an overall least-squares adjustment of the fundamental constants.Through this approach, internal criticism among parts of physical theory is provided and aconsistent set of numerical values for the constants is obtained.

3. Experimental determinations

In chemistry, the concept of Avogadro and the mole emerged through a historical process.Over the years before 1860, the chemical community had been unable to come to agreements,neither on concepts nor on terminology. In physics, more than 50 years after a first calculationby Loschmidt of the number proposed by Avogadro, the numerical evaluation became reallyimportant.

3.1. Avogadros hypothesis and the kinetic gas theory

Concepts for atoms and molecules and their magnitudes can be found in the works of Newtondating back to 1704. The foundations of Daltons atomic theory (Dalton 1808) were definitelylaid on considerations arising from studies of chemical changes. It may be added that people atthat time were much concerned with relative weights of particles, but questions as to absolutedata of the particles of matter had no appeal for them.

One of the first investigations into this subject can be found in the work by Young publishedin 1807. He concluded that the diameter or distance of the particles of water is between0.125 108 and 0.025 108 cm.

Avogadro proposed his hypothesis in 1811. At that time there was no data at all on thenumber of particles in a mole, or an agreement on any atomic weights. The first estimates which

1950 P Becker

could give an approximative value for Avogadros number were deduced from observationsof the Brownian motion by Brown in 1828. Cannizarro (1856) used Avogadros principle todevelop a defensible set of atomic weights based on 1/16 of the atomic weight of oxygen, incontrast to the above-mentioned definition of the mole by a number of carbon atoms.

In 1850 and 1856, Clausius (182288) and Kronig (182279), respectively, developedthe idea that the motion of the gas molecules should be linear, except during collision withthemselves or with the container. They found that each molecule has its own hard elastic spherewith a certain radius and that the ratio of the free path length to the diameter of the moleculeand the ratio of the volume of the molecular sphere and the volume of the container are ofequal magnitude. Loschmidt gave first estimates of the size and the number density, deducedfrom the mean free path length (1865). He calculated the number of molecules in one cubiccentimetre of a gaseous substance under ordinary conditions of temperature and pressure tobe somewhere around 2.6 1019 molecules cm3. He was also able to obtain a moleculardiameter of the order of 1 nm using the new kinetic molecular theory which described themolecules as rigid spheres. This is usually known as Loschmidts constant.

The results of similar experiments performed by many authors were published withoutany measurement uncertainties being stated. This was due to an insuffcient development ofthe kinetic gas theory rather than to experimental difficulties. The gas theory has shown a waytowards the determination of the Avogadro constant and its order of magnitude but was notable to provide more precise data.

3.2. Early measurements on solid solutions in liquids

Many characteristics of the thermal motion in gases can also be found in liquids. As thenumber of molecules in a certain liquid volume is by a factor of a thousand larger than in a gasvolume, the free path lengths are much smaller and the mutual effects between the moleculescan no longer be neglected either. First attempts at establishing a quantitative analogy betweenfluids and gases were made by vant Hoff in 1877 and by van der Waals in 1881 when theyinvestigated osmotic phenomina. In 1890, Rontgen and Rayleigh investigated thin molecularfilms on water and obtained for molecules diameters of about 0.6 to 1 nm, resulting in a numberdensity of 67 1023 particles in a mole. Later on, by an improved analysis, DuNouy founda value of 6.003 1023 mol1 in 1924.

In 1890 a first attempt was made to introduce the concept of a volume related numberdensity into chemistry. In Ostwalds textbook Grundriss der allgemeinen Chemie (1889),the expression number in 1 l, N is still used. In the third, improved edition the term Molwith the meaning molecular weight in gram can be found for the first time (Oswald, 1899).At the end of the 19th century, a value of NL = 2.8 1019 cm3 was used.

3.3. Avogadros hypothesis and the Brownian motion

In early 1827, Brown found out that solid particles in a fluid are in continuous motion (Brown1928) which can be described as an irregular and independent lateral and tilt movement. Twomacroscopic phenomena characterized this movement, the diffusion as a drive to balanceparticle concentration and the particles vertical distribution due to gravity in the fluid. Asthe macroscopic objects were measurable and connections to the molecular entities could beestablished by a well-defined statistical analysis, properties of the molecules, e.g. their masses,could be inferred in a rather direct way.

Later on, a new significance to the Avogadro constant was given through the contributionsof four great physicists: two theoreticians, M Planck (1901) and A Einstein (1906), and twoexperimentalists, E Rutherford (1909) and J Perrin (1909).


First investigations were performed by Einstein in 1905 and Smoluchowski in 1906.Einsteins analysis started from the hypothesis of equipartition of energy among kinetic degreesof freedom of a system in equilibrium from which he deduced a new determination ofmolecular dimensions. After some correction, a revised value of 6.56 1023 particles/gat.wt. for the Avogadro constant was published by him in 1911. Plancks calculation (1901)of the three fundamental constants h, k, NL, was done in the context of the presentation of hisVerteilungsgesetz.

Einsteins deduction of an exponential density distribution with respect to height wasexperimentally proven by Perrin in 1909. Perrin demonstrated that, in an emulsion of particlesvisible under the microscope, the equilibrium repartition along a vertical axis was exponential,similar to an ideal gas under the influence of gravity. The repartition equation led to anAvogadro constant close to NA = 6.7 1023 particles/g at.wt. Perrin expressed his resultsin terms of a different unit, which he called Avogadros number or constant. This unit, whichsoon came into general use, is defined as the number of molecules in one gram-molecularvolume (namely 22.4 l) of a gas under normal conditions.

The work of Einstein and Perrin yielded some of the first specific evidence of theexistence of molecules. Subsequent to the work of Loschmidt and Perrin, many scientistscarried out numerous experiments using a variety of techniques to measure the value of theAvogadro constant as accurately as possible. As nearly all scientists paid more attention tothe determination of the molecular diameter than to the number density, it is difficult to assignto these measurements also an index of accuracy, ur . But, investigations of the movementof mercury droplets or spherical micro-organisms performed by Fletcher (1911), Nordlund(1914), Westgreen (1914), and by Shaxby in 1923 resulted in more precise values for theAvogadro constant. One of the last attempts to determine NA by movements of particles waspublished by Kappler in 1931, who observed the Brownian motion of a torsion balance fromwhich he obtained a value of NA = 6.059 1023 particles/g at.wt. (ur 1%).

In summary, improvements of the kinetic gas theory and the Brownian motion, andexperiments on the sedimentation equilibria of colloidal particles led to remarkable values:NA = (6.03 0.12) 1023 particles/g at.wt. At the beginning of the 20th century theAvogadro constant was increasingly derived no longer by investigations of gas molecules; theprinciple was applied more and more to fluids and solid samples, mainly due to the discoveryof x-rays.

3.4. The oil drop method

When the Brownian motion had been investigated also with charged particles, one was ableto obtain values for the product N e. These experiments, initiated by Townsend (1898) andcarried on by Thomson (1898) and Wilson (1903), were based on the condensation of clouds ofcharged droplets. In 1909 it was pointed out that these determinations were adversely affectedby two fundamental difficulties: one of them concerned the limitation of the law itself, derivedby Stokes in 1850, the other was due to evaporation from the droplets during the fall of thecloud. In 1908, Millikan began his long-lasting efforts to eliminate these two sources of errorsby his balanced-drop method holding the cloud in a stationary position by an electric fieldsufficiently strong to prevent its fall under gravity. The development of the oil drop methodwas extended and improved year by year. Finally, in 1917, Millikan was able to establish anaccurate value for the fundamental electric charge, and he proved that all electrons have exactlythe same charge. The experimental improvements directed to investigations of the Brownianmovements of individual droplets also provided the means for obtaining accurate values forthe Avogadro constant: 6.03 1023 mol1 (Schidlof 1915). In contrast to this, the indirect

1952 P Becker

Table 1. Values of NA and their uncertainties as determined from 1865. Data 117 are derivedmainly from atomic or molecular movement in gases or fluids, data 1824 mainly from x-raywavelengths, and data 2535 from the XRCD approach, together with least-squares adjustmentsof fundamental constants.

NAYear First author Method 1023 mol1 urel

1 1865 J Loschmidt Mean free path 72 1 1002 1873 J D van der Waals Kin. gas theory 11 5 1013 1890 W R Rontgen Atom. films on water 7 1 1004 1890 J W S Rayleigh Atom. films on water 6.08 1 1025 1901 M Planck R/k 6.16 1 1026 1903 H A Wilson Oil drop method 9.3 1 1007 1904 J J Thomson Oil drop method 8.7 1 1008 1908 A Einstein Diffusion theory 6 1 1009 1908 J Perrin Brownian movement 6.7 3 101

10 1909 E Rutherford -particle theory 6.16 6 10212 1914 T Fletcher Brownian movement 6.0 2 10113 1914 I Nordlund Diffusion in fluids 5.91 1 10214 1915 A Westgreen Diffusion in fluids 6.06 2 10115 1917 R A Millikan Oil drop method 6.064 6 10316 1923 T W Shaxby Diffusion in fluids 5.9 17 1924 P L du Nouy Thin films 6.004 8 10318 1929 R T Birge X-ray diffraction 6.064 4 6 10319 1931 J A Bearden X-ray gratings/plane 6.019 3 10320 1941 R T Birge Calcite, NaCl, KCl, . . . 6.022 83 1 10421 1945 R T Birge Diamond, LiF 6.023 38 2 10422 1948 J W M DuMond X-gratings/concave 6.023 2 1 10423 1949 M E Straumanis Calcite crystals 6.024 03 3 10424 1951 J W M DuMond 6.025 44 1 10425 1965 J A Bearden XRCD, Si 6.022 088 2 10526 1965 E R Cohen LS. adjustment 6.022 52 9 10527 1973 E R Cohen LS. adjustment 6.022 045 3 10528 1974 R D Deslattes XRCD, Si 6.022 094 3 1 10629 1987 R D Deslattes XRCD 6.022 134 1 10630 1987 E R Cohen LS. adjustment 6.022 136 7 6 10731 1992 P Seyfried XRCD 6.022 136 3 1 10632 1994 G Basile XRCD 6.022 137 9 4 10733 1995 P De Bie`vre XRCD 6.022 136 5 6 10734 1999 K Fujii XRCD 6.022 155 0 3 10735 2001 P De Bie`vre XRCD 6.022 133 9 4 107

determination from the electron charge, taking into account the actual value of the Faradayconstant, yielded NA = 6.064(6) 1023 mol1 (Birge 1935).

At the beginning of the 20th century, shortly after the discovery of radioactivity, twoof determinations of the Avogadro constant were published, based on the production ofhelium from radium (cf Rutherford (1909)), and derived from the half-life period of radium(cf Gleditsch (1919)). Selected values of NA, from 1865 until today, are presented in table 1,together with their uncertainty level. A graphical picture of the evolution of the uncertainty ofNA is also given by De Bie`vre and Peiser (1994) and De Bie`vre (1996). It shows a reductionof the uncertainty on NA by a factor of 10 every 15 years in the period from 1850 to 2000.


3.5. The early x-ray/crystal density approach

During the 18th century the atomic theory was introduced in crystallography by Hauy (1801).He described crystals as an entity of individual molecules of parallelepipedal shape. Later on,this picture was improved by Bravais (1848), Schonflies (1891) and Barlow (1898) who showedin detail the various arrangements of the lattices and their formation procedure. These findingsbecame of great importance through the discovery of x-rays by Rontgen in 1895. Since then,the following milestones for todays determinations of the Avogadro constant were found: in1900, the electro-magnetic nature of x-rays was postulated. A theory for x-ray diffraction bycrystals was developed and investigated by v Laue and co-workers (Friedrich et al 1912) andby W L Bragg in 1913. W H Bragg was the first to use the results published for the Avogadroconstant to calculate the atomic distances or lattice parameter of an NaCl crystal by means ofthe ansatz (Bragg 1914)

a0 =(n M NA

)1/3, (1)

where M is the mean molar mass of the atoms and n their number in the unit (cell) with avolume of a30 , and the macroscopic crystal density. At the same time, W L Bragg (1912)made a particular interpretation of v Laues theory, namely a relationship between the latticespacings and the wavelength of the diffracted x-rays:

dhkl = 2 sin , (2)with

dhkl = a0h2 + k2 + l2

, (3)

where h, k, l are Miller indices.The pioneering experimental work by Mosely in 1914 quickly made it evident that relative

magnitudes of x-ray wavelengths or lattice spacings could be much more precisely determinedthan the value of a0 in Braggs expression (1). For an NaCl crystal an atomic distance of2.814 1010 m was deduced with an uncertainty mainly due to the improper knowledge ofthe Avogadro constant. As a consequence, Siegbahn (1919) suggested defining a local x-rayunit scale, assuming that the numerical values available at that time were exact, in terms of anew unit, XU. The value recommended for the X unit was

1X = d(NaCl; 18 C)/2814 000 . . . . (4)From our knowledge it seems that it was adventurous to base the X scale on crystals with highlyimperfect lattice structures. This was obvious when absolute wavelength data corresponding tocertain x-ray spectral lines became available through experiments using gratings with a repeatdistance known in the unit of length, the meter, and the precise measurement of the diffractionangles (Bracklin 1928). Selected data forNA derived from such measurements were first foundby Birge in (1929), see Birge (1945), Bearden in (1931), and DuMond in (1948), see DuMond(1959). They reached measurement uncertainties in the order of 104. A comparison betweenwavelengths measured in the meter unit, m, and corresponding data derived from the X unit,X, showed a discrepancy of approximately 0.2%. This error was obvious, because the valuesof e from the Millikan oil drop experiment and, consequently, of NA, as well, were affectedby an error. To take this discrepancy into account, a conversion factor ,

= m/X (5)

1954 P Becker

was introduced. But it soon became clear that the conversion factor was not stable; itsvalue was affected by crystal imperfections such as impurities; more over, the measurementtechniques had meanwhile been improved.

Using , it was now possible to obtain new data for the Avogadro constant by invertingthe above-mentioned equation (1):

3 NA = n M a30

(6)

where a0 is measured in X units and is a geometrical quantity which depends on the slope ofthe crystal faces. In such a way, NA was derived from the lattice parameter, density and molarmass of an NaCl crystal by Birge in 1945: NA = (6.024 01 0.000 52) 1023 mol1.

However, since x-ray spectral lines have a relative width of more than 3 104, at theirnarrowest, and also internal structures, it was very unlikely that they could be used successfullyin any measurement technique at a precision level of the order of 1 106.

The determination ofNA using x-rays, once the lattice parameters are known, also requiresthe determination of the mean molar mass and the macroscopic density of the crystal. Early inthe 20th century, the first density measurements were carried out by hydrostatically weighingthe crystals in samples of water whose density had been precisely determined previously.Molar masses were first determined by chemical methods, and since about 1950 from physicalmeasurements of nuclear masses combined with abundance determinations. Since 1970,isotopic abundances have been directly established for the specimens used.

Almost all of the early work was carried out using calcite crystals, taking the latticeparameter a0 = 3029.040 XU of the purest calcite crystal as a basis. The trouble with thisdefinition was that there was no recipe for knowing how to define the purest calcite sample.The wide variation from one sample of natural crystals to the other was known, and calcitewas not an exception. In 1940, different samples were shown by Ievins and Straumanisto exhibit wide fluctations of a0 over a range exceeding 100 106. Birge also presentedin his paper measurements on diamond, LiF, KCl and calcite, showing an average value ofNA = 6.023 38 (23) mol1. Later measurements carried out by Straumanis in 1949 on calcitecrystals resulted in NA = 6.024 03 (18) mol1.

A vital source of uncertainty in these data was the wavelength of the characteristic x-ray line used to measure the crystal lattice spacing. One part was due to the uncertainty indefiniteness and reproducibility (approx. 3 104) of the centre of the x-ray lines limitingthe applicability of the XRCD method. Another source of error originated in the absolutedata for molybdenum and copper radiation, which were discrepant by about 40 106. Toovercome these difficulties, Bearden (1965) proposed to take the result that emerged for thecharacteristic tungsten x-ray line as a new fundamental standard: WK1 0.209 0100 A.The unit A* was numerically close to 1 , but it was operationally an x-ray, not an optical,unit. The problem of expressing the unit A in centimetres, however, did not differ from theproblem of relating X-units in centimetres. Density determinations by hydrostatic weighingalso introduced significant errors due to surface tension effects, corrosion etc, when the attemptwas made to approach a precision level of 1 106. The chemical methods for determiningmolar masses were not very accurate as well. Furthermore, when silicon or germanium cameinto use, material with a relatively low content of crystal defects, the measurement uncertaintycould not be reduced significantly. Work by Henins and Bearden in 1964, in which 17 siliconcrystal samples were studied yieldedNA3 = (6.059 770.000 15)1023 mol1. The majoruncertainty in this determination was contributed by the calculation of the atomic weight ofsilicon. So, in total, the level of precision reached at about 1960 converged at a value of about70 106, and, seemingly, did not improve further.


Figure 1. Schematic view of the x-ray interferometer for lattice parameter determination.

3.6. Discrepancies in x-ray data

The Avogadro constant belongs to the set of fundamental constants of physics, see section 6.Each of the values can be expressed in terms of a few constants serving as unknowns. Thisallowed experimental or theoretical results to be checked or improved. In 1955, a first attemptwas made by Cohen et al (published in 1957) to recommend a set of adjusted values of thefundamental constants. After 1965, revised versions were published (Cohen and DuMond1965, Cohen and Taylor 1973) mainly due to the fact that a new scale of atomic weights (mat(12C) = 12 g) had been adopted by international organizations in 1960. In this adjustment theauthors did not feel too well satisfied with some of the discrepancies which that least-squaresadjustment had revealed, particularly in the domain of x-ray measurements and thus for NA.At that time the general level of precision in other fields of physics had overtaken the precisionof the x-ray work. As a consequence, they decided to exclude completely from the calculationsall data derived from x-ray measurements.

The unsatisfactory situation as regards knowledge of the Avogadro constant washighlighted by the very large differences between values recommended by further least-squaresadjustments, where data derived from macroscopic quantum properties have been included.This problem could be solved only when, with the development of x-ray interferometry amethod was invented by which x-ray wavelength and lattice parameter could be measureddirectly in meters, the unit of length, and a separate x-ray unit thus became superfluous.

4. Establishing the link of the lattice parameter to the meter by x-ray interferometry

The most important characteristic of the x-ray interferometric method developed by Bonse andHart in 1965 is its ability to establish a lattice parameter without limitations to accuracy dueto the properties of the x-ray lines. The principle of measuring the lattice spacing of silicondirectly in terms of the meter is described in figure 1:

1956 P Becker

When x-rays pass through the first crystal S, they are diffracted, and this happens againin the second (M) and third crystal (A) which are mounted so that their crystal planes are veryaccurately parallel to one another. The x-rays diffracted by the first two crystals produce aregion of stationary wavefronts at the locus of the third crystal. This crystal also diffracts thex-rays, and the diffracted beams are monitored by a detector D. In a Gedankenexperiment,the third crystal is then moved through the stationary wave pattern, parallel to the diffractionvector. When the atomic planes of the third crystal are in line with those of the first two, amaximum of the counting rate of the detector occurs while at half a repeat distance a minimumis found. Attachment of a length ruler to the movable part of the crystal interferometer allowsthe displacement to be monitored very precisely. Simultaneously, the number of x-ray fringesper unit length can be counted and the crystal lattice period d determined in absolute valueswith extremely high precision when optical interferometric means are applied to measure thecrystal displacement (indicated by open arrows).

4.1. Former XRCD key experiments

Three technically distinct efforts towards such a combination of x-ray/optical interferometry(COXI) were first described at a conference on precision measurements and fundamentalconstants in 1970 by Bonse et al (1971), Hart (Curtis et al 1971) and Deslattes (1971).Work in this direction was also initiated by Tarbeyev (1984) at the D I Mendeleyev Institutefor Metrology (VNIIM) in St. Petersburg, Russian Federation. With the exception of thebasic principle, which all approaches had in common, there were differences in the technicalrealization of the design of the x-ray interferometer, of the type of the optical interferometerproviding the SI unit, the meter, and of the performance of the translation mechanism. Bonseand Deslattes used x-ray interferometers with the third crystal mounted separately on a specialtranslation stage in which the elastic deformation of spring elements facilitated the necessarysmooth translation. Hart used a monolithic interferometer which had an elastic spring stripdevice as part of the same silicon crystal. For the measurement of the displacement, the first twogroups used a multiple-beam interferometer of the FabryPerot type with extra optical mirrors,the third group a two-beam interferometer of the Michelson type with silicon mirrors polisheddirectly onto the interferometer crystal. The experiments were performed at the University ofMunster, at the National Physical Laboratory in Teddington and the former National Bureauof Standards (NBS) in Gaithersburg where, in 1973, a relative measurement accuracy in thedetermination of the lattice parameter of 0.15 106 was achieved (Deslattes and Henins1973).

While work at Teddington was stopped after a couple of years, the NBS continued its work,including the necessary experiments on molar mass and density for a determination of NA.Since 1970, the Physikalisch-Technische Bundesanstalt, PTB, in Braunschweig has carriedon the measurements started by the University of Munster. Later on, national metrologyinstitutes in Italy (Istituto di Metrologia G Colonnetti, IMGC, Turin), in Japan (NationalResearch Laboratory of Metrology, NRLM, Tsukuba), in Australia (CSIRO, Sydney) as wellas the Central Bureau for Nuclear Measurements (now the Institute for Reference Materialsand Measurements IRMM) joined PTB and NBS (now the National Institute of Standardsand Technology, NIST) first in scientific competition but today in a cooperative project underthe umbrella of the Bureau International des Poids et Mesures (BIPM) to concentrate andcoordinate their activities for a precise determination of the Avogadro constant.

In order to avoid the problems arising with naturally grown crystals, work was carriedout with silicon single-crystal material. The main reason was that Si can be grown in largequantities with a high degree of perfection. Any application of the dynamical theory of x-ray


diffraction (Laue and Ewald) to experimental results obtained on silicon at that time pointed toan almost ideal perfection of the crystal lattice. In further experiments it became evident thatthe degree of perfection is, of course, a question of the measurement uncertainty achieved.

Another candidate for the XRCD method, germanium, was also taken into consideration.It is a semiconductor material also used for electronic devices. Indeed, the first transistor wasfabricated from Ge. Soon after that, however, Si replaced Ge because it offered two mainadvantages: for metrological application, compared to Ge, silicon has a thermal expansioncoefficient smaller by a factor of three and a more stable surface oxide layer. Today, sincesilicon crystals are produced by the ton; the costs are therefore reasonably low and availabilityof the material is high. Thousands of scientific papers about silicon have been published todate, therefore, silicon is the material best investigated world-wide.

5. Avogadros hypothesis: the XRCD method

The Avogadro constant as a link between atomic and microscopic scales represents the numberof entities in a mole. To overcome the problems which accompany the direct counting of sucha great number with high precision, the XRCD method described here shows an alternativeway which makes use of the high perfection of a single crystal. For the determination of NA,equation (6) can be rewritten without any correction factors:

NA = n M a30

. (7)

NA is nothing else than the ratio of the molar volume Vmol and the atomic volume Vat or, inother words, the ratio of the molar mass Mmol and the atomic mass mat :

NA = VmolVat

, (8a)

and

NA = Mmolmat

, (8b)

respectively. The following quantities must be measured:

(1) The volume occupied by a single Si atom, derived from the knowledge of the structure andthe lattice spacing of a highly perfect, highly pure silicon crystal. These measurementstherefore include precise investigations into the content of impurity atoms and self-pointdefects,

(2) the macroscopic density of the same crystal, and(3) the molar mass and, thus, the isotopic composition of the Si crystal (Si has three stable

isotopes: 28Si, 29Si, 30Si).The last task is actively pursued by the Institute for Reference Materials and Measurements

(IRMM) of the European Union in Geel (Belgium).In various reports and at conferences, the national metrology laboratories and institutes

give information about their activities and the present state of their research work. Asymposium related to this work is held within the framework of the Conference onPrecision Electromagnetic Measurements (CPEM) every 2 years. A detailed summary ofthe experimental state around 1994 was published by Basile et al (1994) (special issue ofMetrologia) and, in a compact version, by Becker (1997).

1958 P Becker

5.1. Material characterization

The counting of silicon atoms by the XRCD method presupposes knowledge of the number ofimpurities and their influence on mass and volume. A precise determination of NA, therefore,requires by all means (1) the identification of the type and the content of the impurity and (2)the evaluation of the influence on mass and lattice parameter. These tasks are described in thefollowng sections.

All semiconductor-grade silicon is produced by converting silicon compounds to elementalsilicon by chemical vapour deposition (CVD). For approx. 75% of the world CVD production,trichlorosilane (Cl3SiH) is used as the silicon compound (for example by Wacker, MEMC,Tokuyama, Kyundo), and silane (SiH4) (for example by ASiMI, ethyl) is used for the remaining25%. The polycrystalline material shows significant differences in grain size and impurityconcentration: silicon from silane is purer and of finer grain size than that from trichlorosilane.As will be described later, in the case of silane the CVD process gives rise to a remarkableisotopic separation effect. For the production of silicon single crystals, two crystal growthmethods are commonly used: the Czochralski (CZ) method which yields about 85% of allsingle crystalline silicon, and the float-zone (FZ) method with a share of 15%.

Generally, all classes of crystal defects can be found in silicon as well. But if we focus ourdiscussion on selected undoped crystals grown under extreme metrological conditions, onlyzero-dimensional point defects occur in significant concentrations. In particular, in silicon thelow packing density of the diamond latticein contrast to the face-centred cubic lattice ofmetalsenables the incorporation of interstitial atoms. Parameters which can influence boththe silicon lattice parameter and the density are: temperature, pressure and lattice defects, suchas impurity atoms, self-point defects mainly incorporated into the lattice during the growthprocess, and the isotopic silicon composition. Only 1 108% of the atoms are foreignatoms (Zulehner et al 1993). This negligibly small amount is achieved by the highly efficientpurification process during the floating zone (FZ) refinement of the crystal growth. The keyparameter of this process is the distribution coefficient k, which is the ratio of the impurityconcentrations in the crystal and in the melt. For k < 1, the crystal will be purified duringthe growth process. In addition, a small percentage of some impurities, such as O, P, SiO, canbe eliminated by evaporation.

How perfect can a silicon crystal be? For the application discussed here, the following defectsmust be taken into account:

(1) impurity atoms occurring on regular lattice sites by the substitution of silicon atoms,(2) impurities on interstitial lattice sites, which increase the average number of atoms per unit

cell,(3) Si vacancies and Si self-interstitials favoured by the relatively small packing density of

the lattice (only 36% of the volume is filled up).In all cases, the defects can occur as point-like defects or as agglomerates such as swirls,

voids or precipitates with concentrations produced mostly during the crystal growth process.On these assumptions, the average number, n, of atoms per unit cell in equation (7) is no longeran integral number, and, consequently, the lattice parameters a and nmust be slightly modifiedby a and n = (No n), with No = 8 and n being a correction for the impurity content wellbelow 1 107.

At present, the concentration of the main residual impurities in undoped FZ-purified Si,either electrically active or inactive, can be measured by optical (IR) spectroscopy. For inactiveimpurities the method is not so sensitive, and the detection limits are close to the residual C andO concentrations in the best crystals (


electrically active impurities can be detected, for example, by electron spin or photothermalionization spectroscopy. Oxygen concentrations below 1013 cm3 were recently determined bymeans of phonon spectroscopy (Zeller et al 1999) with superconducting tunnelling junctions.Besides optical spectroscopy, a wide range of mass spectroscopy methods can also be used toobtain impurity concentrations.

Carbon and oxygen have already been introduced into the raw material, the polycrystallinerod, which is fabricated by CVD. Carbon with k = 0.07 can be reduced by multiple FZ-refinement. For oxygen, k is close to 1 and the amount of O cannot, therefore, be reduced bysegregation during crystal growth but effectively by evaporation. Oxygen can also penetrateinto the crystal when the growth atmosphere is not free of humidity. In the FZ-material usedin our experiments, the carbon and oxygen content is of the order of several times 1015 cm3and can be detected by IR spectroscopy with a detection limit of 5 1013 cm3 for C and2 1014 cm3 for O.

Nitrogen occupies interstitial lattice sites and is a doping material which prevents theagglomeration of self-point defects such as swirls, and forms N-vacancy or N-self-interstitialcomplexes. The mechanism of binding is not yet fully understood, but this model concept issupported by the fact that the N-concentration necessary for the suppression of swirls, and thevacancy concentration, are of the same order of magnitude. In general, this kind of N-dopingleads to a lattice expansion as is also found in the case of vacancies (Biernacki and Becker1999). The nitrogen is added to the growth atmosphere during the pulling process. Usually, adoping rate of several 1014 cm3 is needed (Wolf et al 1996). When IR spectroscopy is used,the detection limit is 1.5 1014 cm3.

The radial distribution of self-point defects is defined by the growth conditions and dependsstrongly on the growth rate (Chikawa et al 1986). For instance, a rate of 2.5 mm min1 forcrystals 100 mm in diameter leads to a concentric ring structure of point defects with analternative surplus of vacancies (D-swirls) or self-interstitials (A-swirls) separated by a neutralzone (0-zone) (Chikawa and Matsui 1993). An increase in the pulling speed is generallyaccompanied by a radial expansion of the vacancy-enriched central range. In contrast to this,sufficient speed reduction leads to a completely interstitial-dominated crystal. This behaviouris probably detectable in all FZ-materials and can be made visible in nitrogen-free crystalsby preferential etching or, in the case of nitrogen-doped crystals, for example by lithium iondrifting (Knowlton et al 1996).

In the case of vacancies and self-interstitials, investigations into their equilibriumconcentrations and diffusion are more difficult as their actual concentrations at roomtemperature are unknown. From calculations, values around 1011 cm3 are to be expectedbecause of the very strong covalent binding conditions, but these data are uncertain withinseveral orders of magnitude (Zulehner 1991a)! The following experimental methods wereapplied to determine and calibrate the in situ point defect density:

Positron annihilation: if a specimen contains vacancy-type defects, there is a finiteprobability of positrons being trapped by them. The lifetime of the positrons reflectsthe density of electrons in the places where positrons are annihilated. A brief introductionto this method can be found in Krause-Rehberg et al (1994).

Diffusion experiments, either with silicon isotopes 30Si, 31Si or with tracer elements suchas boron, phosphorus, or diffusion of metal atoms during growth: a comparison of theexperimental diffusion profiles with the simulation of the diffusion mechanism (Knowltonet al 1995) or the determination of the concentration of the incorporated atoms yields anestimate of the vacancy concentration (Lembke and Zulehner 1999).In an annihilation experiment, positron lifetime measurements in electron-irradiated

1960 P Becker

samples show an increase by 3 ps at low temperature. In native Si, only an increase of1 ps was found. From that, vacancy-like defects could be derived, with an upper limit ofa defect concentration of 1 1014 cm3 in the native silicon crystals (Gebauer et al 1999).Depending on doping with nitrogen, vacancies remained as single defects or formed voidsabout 200 nm in diameter and between 105 and 106 cm3 in density, with overall concentrationsbetween 6 1013 and 1 1014 cm3 . The calculations are in agreement with the positronannihilation measurements and density comparisons of samples cut from different swirlzones, cf section 5.1.3.

The concentration of interstitial Si at room temperature is unknown. But if we take intoaccount that vacancies and interstitials are almost in a state of equilibrium, the influence ofmass and lattice parameter changes is further reduced. Grown-in voids were also simulatedas spherical clusters of vacancies that nucleate homogeneously in a temperature range from1100 to 950 C (Vanhellemont et al 1997). The density and the size of the clusters areinfluenced by the cooling rate and the initial vacancy concentration. For the solidmeltinterface the simulation results predict concentrations of vacancies and self-interstitials inthe same order of 1015 cm3 and, with decreasing temperature, show a significant reductionby mutual recombination. The recombination process stops at temperatures below 1250 Cand self-interstitial lattice sites are almost completely depleted (von Ammon and Dornberger1999).

Hydrogen as an interstitial impurity diffuses very rapidly, and the maximum solubility israther low. At room temperature, diffusibility and solubility data show an H-density in argon-grown Si of between 105 and 2 107 cm3. This hydrogenation takes place during chemicaletching or when a semiconductor sample in water is simply boiled (Pearton et al 1992).Typically, these techniques lead to maximum hydrogen near-surface (a fewm) concentrationsof about 1016 cm3. In the case of 1 kg silicon spheres, the surface volume fraction of about3 104 would simulate an average hydrogen density of 3 1012 cm3. Crystals grownfrom silane poly-material show a high concentration of hydrogen, which could be qualitativelyidentified in the ambient argon gas. A percentage of hydrogen in the growth atmosphere, addedin the past to suppress the formation of swirls, can cause small voids filled with hydrogen underhigh pressure, which are perfectly enclosed by the host lattice.

In order to investigate the influence of hydrogen on density, crystals were grown in argonatmosphere containing a percentage of hydrogen (max. 2%), see section 5.1.3. A well-established method to detect voids larger than 20 nm is laser scattering tomography (LST),see for example Moriyama et al 1989. A typical result of hydrogen-induced voids in perfectsilicon is shown in figure 2 (Becker 1999).

Argon-related defects in silicon have been detected recently by Ulyashin et al (2000),stimulated by hydrogen implantation. The results obtained show that argon is present as anunintentional impurity in FZ-Si after material growth in the argon gas ambience. To investigatequantitatively this influence in our as-grown silicon, crystals were grown in atmospheres ofdifferent argon pressure and, for comparison, in vacuum. Density and lattice parameters ofthese crystals were measured and compared with one another. Within the experimental limits(2.5 108 rel.) no indication of the influence of argon on lattice parameter and density wasfound (Martin et al 1999), see also table 23.

Accurate measurements of the lattice parameter of a 28Si enriched (99.02%) single crystaland the results of first principle calculations of the dependence of the Si lattice parameteron temperature and on the isotopic composition were reported by Becker et al (1995). Theenriched crystals lattice parameter is larger than that of a naturally composed Si crystal byabout 2 106. Even though the isotopic composition varies from crystal to crystal bymore than 3 106, the influence of differences in the natural isotopic composition on the


(1)

(2)

Figure 2. LST images of bulk silicon voids, (1) CZ-grown by Wacker with a high number of voids,typical diameter of about 0.1 m. (2) FZ-grown by IKZ in Ar atmosphere with 2% of H2.

lattice parameter can, at present, be neglected when silicon material is used for metrologicalapplications.

A typical defect concentration in the best purified FZ-silicon material commerciallyavailable is summarized in table 2 (Zulehner 1999). The comparison with the defect densityin the polycrystalline material after the chemical vapour deposition process also shown givesan impression of the high degree of purification of the material by FZ-pulling.

Reduction of carbon and oxygen content in FZ-Si. Silicon material with an impurityconcentration of about 1 1015 cm3 is commercially available today and suitable formetrological applications. To our present knowledge, the carbon and oxygen content in almostpure silicon can be reduced by one order of magnitude. If this can infact be achieved, it wouldlead to a crystal material with a lattice parameter showing a relative stability of better than1108 (Martin et al 1998). This reduction is possible because the growing process for siliconmaterial can be improved to higher chemical purity and perfection. Then impurity correctionswould no longer be necessary.

1962 P Becker

Table 2. Typical concentrations of impurities in CVD and FZ-silicon. Source: WackerSiltronic.RT = room temperature.

CVD-poly-Si FZ-Si (argon)Point defect/impurity ratio ratio

Mono & di-vacancies 2 109 at RTSelf-interstitials 1 109 at RTC 4 108 4 108N


Baker et al (1968) who achieved a sensitivity of 51015 atoms cm3. Today, atomic fractionsof less than 108 in a cubic centimetre can be detected for elements which play an importantrole as dopants in silicon.

When impurity atoms enter regular lattice sites, the real silicon lattice parameter is affectedby the difference in atomic size between the atoms of the host lattice and the impurities. Inthe case of crystals with diamond structure, each atom has four close neighbours, their bondsare covalent, with some ionic character, and their interatomic distances are determined by thecorresponding covalent radii. In the case of a substitutional replacement of silicon atoms, thevalue of the covalent radius r compared with that of silicon, rSi, directly shows the trend ofthe volume change of the unit cell, and this changes linearly with the atomic concentration Niof the impurities, well known as Vegards law (Vegard 1921):

d/d = Ni (9)with:

= rrSi NSi (9a)

andr = ri rSi (9b)

with rSi = 0.117 nm and NSi = 5 1022 cm3 for the covalent radius and atomic density ofsilicon.

A similar discussion leads to density corrections for impurities

={

M

MSiNSi 3

}Nx = volNx, (10)

where M is equal either to MxMSi (substitutional atoms) or to Mx (interstitial atoms).If the impurities enter interstices between the host silicon atoms, prediction of quantitative

size factors is more complicated, because the development is affected by the electronicstructure, misfit strains and elastic coefficients.

The coefficient , which is a measure of the influence which impurities exert on thestructure of silicon crystals (Verges et al 1982), was quantitatively determined in the followingway: the data compiled in table 3 were derived from accurate comparisons between the latticespacing of doped crystals and that of a pure silicon reference material (Becker 1986). Thedoping rate was determined by IR spectroscopy against calibrated standards. The material wascarefully checked for a homogeneous distribution of the foreign atoms. The experimental datain column 3 are in excellent agreement with the concept of covalent radii of the crystals, derivedfrom the chemical bond theory (Pauling 1960) and displayed in column 2. The effective sizeparameters derived from these results (Zulehner 1991), including investigations of latticestrains by interstitial nitrogen and oxygen in silicon, are summarized in column 5 and compared(column 4) with data published by Pietsch and Unger in 1983.

The principle of the x-ray method is based on the 1st derivative of the Bragg equationwhere differences in lattice spacings are correlated with differences in the Bragg angles. Inpractice, a double-source double-crystal arrangement shown in figure 3 was chosen, whichoffers the following advantages.

When Bragg reflections from the front side and from the opposite side of the latticeplanes are simultaneously used (reflections by the reciprocal lattice vectors h andh), relativedifferences in lattice spacings are directly proportional to angular differences of their diffractingintensity profiles. The absolute determination of macroscopic angles is thus avoided. Rotationof the sample through 180 about the yaw axis cancels out the influence of lattice planebending.

1964 P Becker

Table 3. Effective radius r and lattice strain parameters of impurity atoms in FZ-Si. The datacalculated for an Si-vacancy have been taken from Scheffler et al (1985).1 2 3 4 5

r (nm) r (nm) (1024 cm3) (1024 cm3)Atom theor. exp. theor. exp.

C 0.077 0.077 6.9 6.9 0.2O (interst.) 0.142 +4.4 0.5N (interst.) 0.150 +5.7 1.0B 0.088 0.084 5.1 5.6 0.2P 0.110 0.109 1.4 1.3 0.2As 0.118 0.117 0 0.07 0.5Sb 0.136 0.133 +3.0 +2.8 0.2Vac 0.129 0.1274 +2 +1.7 0.5Si 0.117 0.1176

Figure 3. Schematic of ray tracing of the PTB lattice parameter comparator (Martin 1998).

Chromatic dispersion of the x-ray wavelength can be neglected if the relative differencein lattice spacing is no greater than 104.

The intensity profiles (rocking curves) are always symmetric: a detailed profile analysisis not necessary.

Basic principle. The two beams leaving the two x-ray sources (foci 1 and 2) are first Bragg-reflected in Laue geometry by the reference crystal, simultaneously by its h and h latticeplanes. The two reflected waves meet together at the entrance surface of the test sample andoverlap inside the crystal, where the difference in lattice parameters is measured. In contrastto this, defects in the reference crystal may influence the measured differences which mustbe treated as a constant offset in subsequent comparisons. This type of comparator was firstproposed and developed by Hart (1969). The advantage of the design is that a monochromatingsystem with well defined x-ray wavelength is not needed.

The reference crystal and the sample under investigation are mounted on the two axes of adouble-crystal diffractometer, and when the sample is rotated about the yaw axis, see figure 3,two rocking curves are obtained. If both crystals have the same lattice spacing d, both curveswill appear simultaneously at the same angular setting. When the crystals lattice spacingsdiffer byd, the rocking curves have an angular offset which is twice the difference inBragg angles, R and S , of the two crystals and linked to the differencesd in lattice spacings


Table 4. Assessment of errors, PTB approach.

urelDistribution (parts in 109)

Misalignment , Rectangular 2.5Temperature Rectangular 2.5Data processing Rectangular 2.5Angle measurement Rectangular 1Statistic Gaussian 2.5

5.1 Totala

a Root-sum-of-squares (rss).

by

= 2 (R S) (11)and

d

d= 1

2 cot*B. (11a)

In this way, the measurements of small differences in lattice spacings are converted into angularmeasurements. As double-crystal rocking curves are almost free of wavelength dispersion,they may be angularly narrow. Using high-order reflections combined with a short x-raywavelength, e.g. AgK radiation and Si(880) reflection, an angular range of 107 rad fullwidth at half height of the curve is easily obtained. If the peak position can be fixed to within1%, then relative differences of 109 rad in Bragg angle can be measured.

Misalignments. The relationship in (11a) holds only if the x-ray beams and the diffractionvectors for both crystals are coplanar, i.e. if they are in the same plane of diffraction. Whenthe sample is misorientated about the pitch axis (see figure 3) by an angle or/and when oneof the measuring x-ray beam paths, beams F1D1 and F2D2 in figure 3, are out of planecharacterized by the diffracting vectors h and hby an angle , then the rocking curves arealso separated by an angle :

= 2* 2 tan*B. (12)Combining equations (11) and (12) we obtain

d

d= 1

2cot B(2 2 tan B) (13)

(: angle in the vertical plane between the normals of the reflecting lattice planes; : departurefrom coplanarity between the two x-ray beams). Samples with different impurity contentswere investigated using the (880) reflection and AgK radiation. Differences in the latticespacing with respect to a reference crystal were detected with an uncertainty of 5 109.Error budget. The total measurement uncertainty of a lattice parameter comparison can bederived from the single uncertainties summarized in table 4. Measurement uncertaintiesbased on inhomogeneities of the lattice spacing due to impurities, isotopic composition ormacroscopic structure defects have not been included in the table. They must be taken intoaccount in the assessment of the individual comparison.

A similar set-up is currently used at NIST (Kessler et al 1994). Its special features arethe Bragg angle measurement by an optical angle interferometer, the easy exchange of crystalsamples and the comparison of four samples in a single set-up. Differences in lattice spacings

1966 P Becker

Table 5. Assessment of errors, NIST approach.

urelSystematic effect (parts in 109)Optical wavelength and 1st crystal orientation 7Temperature 3Crystal misalignment 2Nonlinearity of interferometry 5

9 Totala

a Root-sum-of-squares (rss).

Table 6. Impurities in crystals used for the Avogadro experiments (in 1015 cm3).Producer Dow chemical Wacker MEMC Shin-Etsu

Label Perfex WASO 4.2 WASO 17 WASO 04 MO SH1Laboratory NIST PTB PTB PTB, IMGC, CSIRO IMGC NRLM

Impurities in 1015 cm3

Carbon 39 4.3 1.2 4.5 1.2 2.5 0.5 13.7 0 5.6 0.9Oxygen 60 1.0 0.5 5.2 1.5 1.2 0.7 0.4 0.1 2.0 0.2Nitrogen 0.62 0.11


10 15 20 25 30 35 40 45 50

-1,0

-0,5

0,0

0,5

1,0

1,5

A zoneO zoneD zone

length position 121,5 cmlength position 73,5 cm

d/d

[10-

8 ]

radial position [mm]Figure 4. Radial variation of the lattice parameter investigated in WASO 04 silicon at two differentaxial positions and at three different radial positions D, O, and A according to the three differentswirl defects areas.

-0,3

0

0,3

D O A

( - (

Di))

/ in

10

-6

Figure 5. Density differences of three samples cutfrom different zones of the slightly nitrogen-dopedWASO 04 crystal.

investigation of the different swirl zones A, D, and 0 shows no significant variation of eitherthe lattice parameter (see figure 4), or the density (see figure 5).

From the total number density of the point defects in that material, the average numberof atoms in the unit cell n = (No n), with No = 8 and n = 1 107, a correction forthe interstitial impurity portion, can be derived for the WASO 04 crystal with a measurementuncertainty for n of about 10%.

Besides the crystals listed in table 6, other silicon materials were included in theexperiments. For comparison of data in this report, the corresponding values for density,molar mass and lattice spacing are labelled in the following by the same identification code.In most cases the location of the individual samples in the crystal boules are well supportedby documents, as has recently been shown by Fujii and coworkers (1999) in the case of theNRLM experiment.

Molar volume anomaly of sample SH1. For the sample SH1, a significantly lower density andthus a significantly higher molar volumetaking into account the isotopic compositionin

1968 P Becker

relation to other material were found, by about 3 106 (Becker 1999). These findings couldnot be confirmed by any material made available by the same supplier and by others, and theyare not related to changes in the lattice periods. Small voids are, therefore, suspected as thecause of this difference in this particular sample. Laser scattering tomography, secco-etching,electron spin resonance, small-angle scattering, and x-ray topography are being applied to theSH1 sample by different laboratories to detect voids or nano-voids. Concerning the detection ofvoids in sample SH1 results so far using x-ray topography are not convincing. This is becausethe reported size of the defects of about 50 m (Deslattes et al 1999) is not in agreementwith the null-result measurements carried out by laser tomography (Nakayama et al 2001).The observed defects can be related to defects at or close to the surface possibly generated byimproper sample preparation (Tuomi et al 2001). Since the density number and the size of thevoids are unknown parameters, the sensitivity of the IR method, with a resolution currently of20 nm, will be increased to a few nanometres to allow nano-voids to be observed.

To further investigate the density anomaly, PTB has collaborated with the IKZ in Berlin.Special FZ purified silicon single crystals were grown from SiHCl3 and SiH4 silicon compoundsto investigate the influence of growth parameters of the molar volume. These include the growthspeed, growth in different atmospheres (vacuum, argon at different pressures (1 and 3 bar), witha percentage of nitrogen and/or deuterium, as discussed in section 5.1), isotopic distribution,and grain size in the raw material. Recently, indications were found of molar mass separationsin the order of 105MSi in SiH4-deposited polycrystals 2 m in length. They are probably theconsequence of small mass differences of 3% between the three silicon compounds, in contrastto differences of 0.7% in the case of triclorosilane. Recent results of density comparisons of FZ-crystals made of two different poly-crystal materials showed a slight dependence of the molarvolume on the raw material, but they did not show any significant molar volume differencesdue to different growing conditions (Basile 2000), an indication that the overall perfection ofthe crystals will not suffer from variations of the growth conditions. The origin of the reportedmolar volume anomaly in SH1 is still unknown, and investigations and measurements are stillunder way.

5.2. The silicon lattice parameteran invariant quantity of nature?Four high-precision measurements of the (220) silicon lattice spacing have so far beencompleted, at NIST, PTB, IMGC and NRLM. The principles of these measurements are similarand the basic components are shown in figure 1.

As discussed before, up to a resolution of several femtometres (1 fm = 1015 m), thepurity of semiconductor-grade silicon is high enough so that no correction for imperfectionsmust be made. When the 1015 fm boundary is bridged, the actual content of impurityatoms, self-point defects and, less important, the actual isotopic composition must be takeninto account. Since the element-specific analysis of the impurities made by spectroscopicmeans (see section 3.1.2), the impurities influences on lattice homogeneity, and the averageinteratomic distance measured with special x-ray crystal arrangements are sufficiently exact,the general procedure is to correct for the measured lattice parameter in order to obtain datafor an ideal crystal structure. (This case is denoted by C, O 0 in the following tables forcarbon and oxygen reaching zero concentration.)

The first outstanding measurement made by Deslattes was a milestone on the way toreferring x-ray data to an absolute scale and showed the way to all other experimental groups.The NIST approach. A detailed description of the experimental arrangements and themeasurement procedure for the lattice parameter determination was given by Deslattes (1980a).An iodine-stabilized HeNe-laser was used in the optical interferometrical part of the apparatus


Figure 6. Diagram of the linear motion stage and of interferometers used in the NBS (NIST)experiment to determine the Si lattice parameter.

(figure 6). The wavelength of the laser stabilized to the inverse Lamb dip was 632 990.079 pm(0.004 106). The output power from the laser was 100 W. Radiation passed throughmode-matching optics, circular polarizers, and decoupling attenuators before being used inthe hemispherical FabryPerot interferometer (OI) for the displacement measurement. Highreflectivity coatings were used to produce narrow FabryPerot fringe signals of a finesse closeto 1500, i.e. the full width of a fringe corresponded to a change of about 316 nm in the plateseparation. Since the x-ray fringe period is approximately 0.2 nm a pointing precision of0.001 nm was achieved.

Precise and rectilinear motion of the movable component of the silicon crystalinterferometer, and precise relative orientation to ensure satisfactory x-ray fringe contrastwere achieved by a special mechanical device shown in figure 6. The structure also allowedfour rotational adjustments about two orthogonal axes to be made. The device was machinedfrom a solid block and, due to an arrangement of cuts and holes, it was equivalent to a seriesof levers and pivots. Translation motion was achieved by a fluid-filled bellows device drivinga pin which entered the structure at point O and actuated lever I through a cone-shaped socket.The pin was provided with a piezo-electric element permitting motions on the picometre scaleunder active feedback control. The motion of the micrometer screw thus was reduced by aanother factor of about 300. This device permitted displacement rates as small as 0.1 nm perminute. Excursions of the movable part of the interferometer by about 10 m were necessaryto achieve observations with an uncertainty of 0.2 106.

The requirement to align the three silicon crystals (SiI) very accurately with respect to

1970 P Becker

Table 7. Uncertainty budget for XROI experiment at NBS (NIST).urel

Parameter (108d220)Aligment XROI 4Temperature 2Fresnel phase shift 1Mirror radius 2Laser 13Bulk modulus 1Statistics 3Total (rss) 15

one another was met by fabricating them from a single block of silicon and mounting them ontheir respective supports. Fortunately, a narrow feature was found in the centre of the rockingcurve, when the two separated crystal parts were tilted around the Bragg position with respectto one another. It allowed a very high alignment sensitivity in the order of 0.01 arc sec tobe achieved. Later on, Bonse and Teworte (1980) were able to explain this very narrow finestructure as an interference effect inside the crystals due to Pendellosung fringes.

Essentially, the measurement consisted in counting x-ray and optical interference fringeswhile the system was made to traverse a common base line. The number of x-ray fringes n,corresponding to m optical half-wavelengths /2, is written as

d = m/n(m)/2 (14)with

n(m) = N(m) + f (m). (14a)First, the integer part of n(m), N(m), was determined by slow scans back and forth over a fewoptical wavelengths and by counting of the total number of fringes. Knowledge of the fractionalpart f (m) then was improved by tracing the progression of the x-ray phase as a function of theorder number of the optical interference fringes. In the experiment, N(1) 1648, dependingon the ambient experimental conditions (air; 22 C). The standard deviation due to statisticaleffects corresponded to 0.04 106.

Deslattes and Henins (1973) measured the d(220) lattice spacing of a vacuum float-zone-refined specimen of silicon (Deslattes 1980c) at temperatures between 22 and 23 C. A valueof 2.56 106 K1 was used as the appropriate coefficient of expansion of Si, and the finalresult wirth an uncertainty of 0.1 106 (Deslattes 1980b), was

d(220)NBS = 0.192 017 07(2) nm at 25 C and in air. (15)The contributions to the measurement uncertainty are listed in table 7. The total uncertainty wasmainly due to fixing of the order number and of the estimated first-derivative lock instability.

There were, however, systematic effects, attributable in particular to a problem with thetrajectory of the analyser crystal, which was only partially controlled. In a new version of theexperiment, specific steps were taken to ensure a more sensitive and precise determination ofthe x-ray phase and on-line measurements of trajectory errors, in particular so-called cosineand Abbe errors. First, a frequency-agile local oscillator laser was introduced, enabling thesystem to scan an entire x-ray period. Secondly, a four-beam interferometer of the polarization-encoded type was added whose signals could be processed to yield tilt angles independent ofdisplacement. Changes in the x-ray phase due to deliberate pitch and yaw motions were usedto measure the Abbe errors. Re-measurements reported by Deslattes et al (1987) produced


results of the lattice parameter, which were from 1.6 to 1.7106 smaller than those previouslyobtained:

d(220)NIST = 192 015.54(2) fm, (22.5 C, 0 Pa). (16)

The PTB approach. The PTB approach shown schematically in figure 7 differed completelyfrom the NIST set-up with respect to technical details. The crystal interferometer had thefollowing special features: the front faces of the crystals were polished to form optical mirrorsparallel to the measured lattice spacings, high enough to nullify the offset between the opticaland x-ray baselines. By this, the mirrors used for the optical measurement of the displacementwere part of the crystal lattice itself. The two entirely separated crystals were supported bythree steel spheres, each resting on supports Sa and Sb, without being glued to them in orderto facilitate exchanges of the crystals. A Lamb-dip-stabilized HeNe laser was used in theoptical interferometer. Its frequency was on-line controlled separately during the experimentby a 127I2 -stabilized HeNe laser in order to check at the same time the stability of the built-inmeter ruler. Optical feedback was avoided by a light trap (>30 dB attenuation). The two-beam interferometer of the Michelson type was developed after Curtis et al (1971) with thefollowing features: the interferometer signals were invariant with respect to a common rotationor translation of the whole x-ray interferometer. The two incident beams were polarizedperpendicular to each other by means of a half-wave plate. The phase difference of the beamscaused by the displacement was determined by electro-optical modulation in a Pockels cell andsynchronous detection. Thus a measuring uncertainty of 0.01 nm was obtained. The long-timestability of the interferometer signal was better than about 0.01 nm h1. The translation motionwas guided on a double parallel spring manufactured from one plate of high-quality steel. Apiezo-electric actuator was used as a drive for a 40 m displacement range. As the relativephase between optical and x-ray interferometer signals was measured as a function of differentalignments, the point of impact of the laser beam onto the crystal mirrors was optimized untilthe Abbe errors vanished. Residual guiding errors of the translation device were magneticallycompensated. Computer-aided on-line evaluation, as well as detailed data-processing allowedthe detection off (m) as small as 103 of an x-ray fringe. The measurement procedure followedthe NIST scheme. A detailed description of the experimental procedures was published in twoPTB-reports by Rademacher et al (1980) and Hanen et al (1981).

Measurements were carried out between 22.42 and 22.50 C in vacuum. Each result wasreduced to 22.50 C by correcting for the thermal expansion coefficient = (2.56 0.03)106 K1, measured separately on the same Si material. The results of 170 measurements areshown in figure 8. The mean value of n/m and the standard deviation m were obtained byfitting a Gaussian distribution n

m

=

2d= 1648.281 626 (17)

with

m = 0.000 007.The development of the mean values with time and the corresponding standard deviations werealso checked (Siegert and Becker 1984). No significant time-dependent systematic effects weredetected.

Several systematic corrections to the mean value of n/m had to be taken into account.(1) The laser wavelength was = (632 991 415 5) fm. The beam profile was Gaussian,

with a radius in the waist of w0 = (0.52 0.02) mm. The Fresnel phase shift in the beamled to a correction 1 = (3.8 0.2) 108n/m.

1972 P Becker

Figure 7. Principle of the PTB x-ray interferometer (1) with (a) movable, and (b) fixed siliconcrystals; optical interferometer (2) with /4-plates (c), /2-plate (d), compensation plate (e), phasemodulator (f) and polarizer (g).

(2) Misaligment during translation of the movable crystal was 2 = (0.10.2)108n/m.(3) Tilting of the crystal during motion combined with a lateral offset of the laser beam

impingement point on the mirror of the movable crystal with respect to the centre of thecrystal was 3 = 3.0 108n/m.


Figure 8. Histogram of the results of 170 measurements. The full curve is a least-squares fit to aGaussian curve.

(4) Uncertainty of crystal temperature (0.02 K) and thermal expansion coefficient (3 108 K1) led to 4 = 5.1 108n/m.The final result for the lattice spacing d(220) at 22.50 C and in vacuum (Becker et al

1981) wasd(220)PTB = 192 015.560(12) fm. (18)

In a second series of 336 single measurements the above value was confirmed with animproved accuracy of 0.02 106 (Becker et al 1982). In a third series of measurements,from the spacings of a whole set of equivalent (220) lattice planes related by symmetry, theaxial lengths, axial angles and the volume of the unit cell of a highly pure silicon samplewere obtained. Crystal-to-crystal comparisons (see section 5.1.2) led to the detection ofrelative deviations from the cubic shape, in particular in 022 and 202 directions, of theorder of +2 107 and 2 107 (relative values) respectively. Using an average valued(220) = (192 015.558 0.018) fm, a volume of the unit cell (at 22.50 C) of

V0 = (0.160 193 259 0.000 000 044) nm3 (19)under vacuum conditions was obtained (Siegert et al 1984). Within the uncertainty limits thevalue d(220) given in (18) and the average value d(220) were identical, an indication thatalthough the unit cell of highly pure silicon might be deformed by residual impurities; thevolume of the cell, however, seems to be unchanged.

For the determination of the Avogadro constant, different materials were used, namedWASO 17 in 1992, and WASO 04 in 2000. The individual lattice spacings were comparedwith the WASO 4.2 sample measured in absolute terms by the PTB lattice comparator describedabove. Former comparisons by Windisch and Becker (1990) were made with a transfer crystal,WASOref , current measurements with crystals cut from WASO 04. The results obtained andcompiled in table 8 are in excellent agreement with previous ones.

The IMGC approach. The IMGC adopted a similar experimental set-up as the PTB, but incontrast to PTB they used a classical set-up of the Michelson interferometer and a simple

1974 P Becker

Table 8. Relative differences in lattice parameter of PTB crystals used forNA. (dTest-dRef )/dRef d/d , compared with reference crystal WASO 04 (C = 3 1015 cm3, O = 1 1015 cm3);WASOREF (C < 0.7 1015 cm3, O < 0.3 1015 cm3) reference sample used before 1990.d/d[109] WASOREF WASO 4.2a WASO 17As grown 22 10 1 6 22 10C,O 0 11 10 10 7 15 10

Figure 9. Schematic drawing of the combined x-ray and optical interferometer used at IMGC.

parallel spring was used, see figure 9. The laser beam (127I2-stabilized HeNe laser) wasprovided by a single-mode preserving fibre polarization ensuring simple and stable remotealignment and positioning of the beam. Position-sensitive detectors, P and Q, were used inthe x-ray part and in the optical part in order to measure and compensate on-line parasitic tiltsof the translation stage. The fixed components of the x-ray and optical interferometer, C2 andM, were placed on a silicon base plate. The front faces of the crystals were optically polishedand metal-coated in order to enlarge the silicon reflectivity and to prevent laser heating ofthe crystal as well. The combination of finite-element analysis and active control allowedmillimetre displacements S to be achieved by constructing an elastic translation stage capableof scan velocities ranging from 1 pm s1 to 0.1 mm s1 and translations by up to 2 mm, smoothto within 1 pm, with yawing and pitching to within 1 nrad (Mana 1989). The crystal was grownand purified by FZ melting by MEMC electronic materials. The experiment was carried outin vacuum; the crystal temperature was measured to within 3 mK using a platinum resistancethermometer calibrated in accordance with ITS-90.

A detailed theoretical analysis of the achievable resolution and accuracy was undertaken byBasile et al (1991). Departures from an ideal experiment, such as limited machining accuracy,chemical etching and imperfect guiding, had been taken in account. Focusing conditions of thex-rays as well as thickness and alignment of the movable crystal changed with displacement.Accoto et al (1994) simulated these influences by calculating amplitude and phase of theinterfering x-rays using the dynamical theory of x-ray diffraction in a distorted mode.

High resolution and a brief observation time were achieved by a long scan, by the servo-control of crystal motion, and, in the final measurement, by data recording at the end pointsof translation only. The final data were collected in 196 runs, showing a 0.8 108 standarddeviation of the mean. Reproducibility was investigated by analysing 92 runs by re-assemblingpart of the experimental set-up.


Table 9. Corrections and uncertainties of the IMGC d(220) determination.Correction urel

Item (108d220) (108d220)Statistics 0 0.5Laser beam wavelength 0.2 0.2Laser beam diffraction 2.5 0.8Laser beam alignment 0.3 0.5Crystal temperature 0.5 0.8Crystal movement 0 1.8Crystal attitude 0 0.8Crystal geometry 0 0.8Crystal impurities 9.3 1.4Total (rss) 12.4 2.9

The final results for n/m and the lattice spacing d(220) at 22.50 C and in vacuum(Basile et al 1994) were n

m

=

2d= 1648.281 626 (20)

andd(220)IMGC = 192 015.569(6) fm. (21)

Going beyond the PTB approach the data were obtained by refining x-ray and opticalinterferometers to allow crystal position and attitude to be simultaneously measured, andby means of a detailed analysis of different portions of the interference pattern in bothinterferometers. Corrections and uncertainties of these measurements have been summarizedin table 9.

Based on these measurements, the IMGC researchers began work on improving themeasurement conditions to make measurements of a particular crystal with a relativeuncertainty at the 1 109 level. The issues addressed so far include the theory of scanningx-ray interferometers, beam astigmatism in laser interferometry, and the problem of how tomanage an analyser displacement by up to 2 mm corresponding to about 107 optical fringes. Anew translation stage was developed. Results of a new series of measurements of n/m carriedout with the same x-ray interferometer were reported by Bergamin et al (1999), who achieveda relative standard deviation of the mean n/m value of about 1 109, see figure 10.The NRLM approach. The improvements of the NRLM device in comparison with the devicespreviously used were the on-line correction of guiding errors during crystal motion and afurther miniaturization of the whole equipment (Nakayama et al 1992, Fujimoto et al 1995).The final experimental set-up shown in figure 11 was described by Nakayama and Fujimoto in1997. It featured the combination of two laser systems that measure simultaneously the angle,i.e. the lateral and the horizontal motion, between two crystals of the x-ray interferometer.The crystal shape and the alignment procedure were similar to those used at PTB and IMGC.The translation stage PAR consisted of a large and a small quadrilateral of links connected bya flexure-pivoted lever system. The stage was put in a rectangular frame that surrounded thebase of the stage. A PZT pushed the stage at lever L in a horizontal plane through the lowerpivots of the flexure hinges. Two additional hinge devices, M for pitch and B for yaw, weremounted on the stage to compensate pitch and yaw errors.

The trajectory interferometer measured pitch, yaw and roll angles as well as vertical andlateral shifts of the stage during translation. Pitch error signals of the stage measured on-line

1976 P Becker

Figure 10. Histogram of 45 consecutive measurement results of n, the number of x-ray fringesper optical period (reduced to 22.5 C). Solid curve is the best fit to a Gaussian distribution(measurement time 100 min).

Figure 11. Schematic diagram of the NRLM lattice spacing measurement device; M, B: tiltingdevices for pitch and yaw corrections, L: application point of driving system.

were fed into a computer which calculated the input voltage to an amplifier and fed the drivingvoltage into a PZT for compensation of the tilting stage M. The same procedure was usedfor yaw and roll errors. The displacement interferometer was of the Michelson type, too, andused polarization beam splitters to generate four parallel beams. The dimensions of the glass


Table 10. Corrections and uncertainties of the NRLM d220 experiment.

Correction urelItem (108d220) (108d220)Statistics 0 5Laser beam wavelength 0 0.2Abbe correction 0 1.0Cosine correction 0.8 0.2Crystal temperature 0 0.5Crystal movement 0.5 0Crystal impurities 3.0 0Fresnel diffraction 16.0 0.8Total (rss) 20.3 5.2

components were kept as small as possible in order to minimize the optical path length inthe interferometer. x-ray and optical interference signals were recorded by a computer. Afterscanning of several x-ray fringes, the feedback loop for yaw and pitch were switched off andthe crystal moved to the next optical order. This procedure was repeated at the 0th, 107th,and 217th or 0th, 100th and 200th optical order. The x-ray phase fraction at each optical nodewas calculated by fitting a sinusoidal function to the x-ray signal during displacement. Thehistogram over 829 accumulated n/m data showed a Gaussian distribution, and the value is

n/m = 1648.240 524 0.000 083 (22)in vacuum and at 22.50 C. Length measurements carried out with the optical interferometerwere affected by errors of different origin, such as laser misalignment with respect to the mirrorsurface of the x-ray interferometer, the deviation of the direction of movement from the surfacenormal of the mirror surface, the mode of the laser beam, and the Abbe offset. Compensationfor the change in the lattice parameter due to impurities was also included. The correctionsand uncertainties used in the deduction of the lattice spacing are summarized in table 10. Theresultant lattice spacing was (Nakayama and Fujimoto 1997)

d(220)NRLM = 192 015.593

Documents

10.1.1.408.199