ArticlePublisher preview available

A low-uncertainty measurement of the Boltzmann constant

Metrologia

July 2013
50(4):354

DOI:10.1088/0026-1394/50/4/354

Authors:

Gavin Sutton

National Physical Laboratory

Paul Morantz

National Physical Laboratory

Show all 9 authorsHide

The Comité international des poids et mesures (CIPM) has projected a major revision of the International System of Units (SI) in which all of the base units will be defined by fixing the values of fundamental constants of nature. In preparation for this we have carried out a new, low-uncertainty determination of the Boltzmann constant, kB, in terms of which the SI unit of temperature, the kelvin, can be re-defined. We have evaluated kB from exceptionally accurate measurements of the speed of sound in argon gas which can be related directly to the mean molecular kinetic energy, . Our new estimate is kB = 1.380 651 56 (98) × 10−23 J K−1 with a relative standard uncertainty uR = 0.71 × 10−6.

Conceptual illustration of the operation of a combined acoustic and microwave resonator. Microwaves: the radius of the resonator is estimated from microwave measurements of up to eight TM1n resonance triplets. The frequencies must be corrected for the dielectric properties of argon gas. The uncertainty associated with this correction becomes smaller at low pressure. The radius at zero pressure is used in the estimate of the limiting low-pressure speed of sound. Acoustics: The speed of sound is deduced from the frequencies of several radial acoustic resonances and kB is deduced from the zero-pressure limit. In order to obtain this, several pressure-dependent frequency corrections must be applied.

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(a) Cross-section through the apparatus showing the NPL–Cranfield resonator NPLC-2 suspended within an isothermal enclosure within an outer 25 L pressure vessel. The pressure vessel was immersed in a stirred liquid bath at a temperature of approximately −0.2 °C. Also visible are some of the PRTs positioned in the neck, the equator and the south pole. In use, the argon gas flows into the resonator at a rate of 7.4 × 10⁻⁷ mol s⁻¹ (1 sccm) and then passes into the surrounding isothermal volume where the pressure is measured. The internal pressure is close to the external pressure, with the small difference being determined from microwave measurements of the dielectric constant of the argon gas. (b) Cut-away view of the resonator sphere showing the showing the coordinate system. Notice the –y-axis is shown. The outer cylindrical surface was cut at the same time as the inner surface so that when the hemispheres are assembled, good alignment between the two outer cylindrical surfaces ensures good alignment of the inner surfaces.

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Designs of the four types of plugs. (a) A blank plug. (b) Gas inlet and outlet ducts. (c) Microwave antenna with the central conductor adjusted to be flush with the spherical surface, and the surrounding volume filled with epoxy resin. (d) Acoustic transducer, in this case the microphone, including the pre-amplifier.

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(a) Magnitude of the TBL correction to cExp(P) (left-hand axis) and (right-hand axis) for the (0, 2) to (0, 9) resonances for a spherical resonator of volume 1 L. Note that the pressure axis is logarithmic. (b) The uncertainty in the pressure measurement (in pascal) required for 0.1 parts in 10⁶ concomitant uncertainty in . The uncertainty in arising from the estimation of the TBL is discussed in section 2.4.4. Note that both axes are logarithmic.

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The TM11 microwave resonance showing the triplet structure caused by the triaxial shape. The blue circles are data points and the red curve is a fit.

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IOP PUBLISHING METROLOGIA

Metrologia 50 (2013) 354–376 doi:10.1088/0026-1394/50/4/354

A low-uncertainty measurement of the

Boltzmann constant

Michael de Podesta1, Robin Underwood1, Gavin Sutton1, Paul Morantz2,

Peter Harris1, Darren F Mark3, Finlay M Stuart3, Gergely Vargha4and

Graham Machin1

1National Physical Laboratory, Teddington, Middlesex, TW11 0LW, UK

2School of Applied Sciences, Cranﬁeld University, Cranﬁeld, Bedfordshire MK43 0AL, UK

3Scottish Universities Environmental Research Centre, Scottish Ent. Tech. Park, Rankine Ave,

East Kilbride, G75 0QF, UK

4Effectech UK Ltd, Dove House, Dove Fields, Uttoxeter, Staffordshire, ST14 8HU, UK

E-mail: michael.depodesta@npl.co.uk

Received 19 April 2013, in ﬁnal form 4 June 2013

Published 11 July 2013

Online at stacks.iop.org/Met/50/354

Abstract

The Comit´

e international des poids et mesures (CIPM) has projected a major revision of the

International System of Units (SI) in which all of the base units will be deﬁned by ﬁxing the

values of fundamental constants of nature. In preparation for this we have carried out a new,

low-uncertainty determination of the Boltzmann constant, kB, in terms of which the SI unit of

temperature, the kelvin, can be re-deﬁned. We have evaluated kBfrom exceptionally accurate

measurements of the speed of sound in argon gas which can be related directly to the mean

molecular kinetic energy, 3

2kBT. Our new estimate is kB=1.380 651 56 (98)×10−23 JK

−1

with a relative standard uncertainty uR=0.71 ×10−6.

SOnline supplementary data available from stacks.iop.org/Met/50/354/mmedia

(Some ﬁgures may appear in colour only in the online journal)

1. Introduction

1.1. Introduction

1.1.1. Background. The current deﬁnition of the kelvin

[1,2] (the fraction 1/273.16 of the temperature of the triple-

point of water) has proved adequate for more than 50 years.

However, the nature of temperature as an intensive quantity

leads to difﬁculties in ‘scaling’ the unit to higher and lower

temperatures. In this sense the deﬁnition itself limits the

accuracy achievable at temperatures that differ signiﬁcantly

from the temperature of the triple-point of water (TTPW).

The CIPM now proposes [3] to introduce a new deﬁnition

of the kelvin, which will simply state that the kelvin has a value

consistent with a deﬁned value of the Boltzmann constant, kB.

This links the value of the unit of temperature, the kelvin, to the

value of the unit of energy, the joule (1 J =1kgm

2s−2)and is

independent of any particular temperature. In this conception,

the Boltzmann constant would be ﬁxed with no associated

measurement uncertainty. In order to make the transition from

one unit deﬁnition to another as seamless as possible it is

desirable to have a low-uncertainty estimate of the value of

kBin the current unit deﬁnition.

1.1.2. Overview of the acoustic resonance technique. Many

techniques can be used to estimate kB, but a recent review

[4] concluded that acoustic techniques were likely to achieve

the lowest uncertainty. One reason for this is the strikingly

simple relationship between the limiting low-pressure speed

of sound c0in a monatomic gas and the root-mean-squared

speed of the molecules, vRMS :c0=√5/9vRMS. In terms

of macroscopically measurable parameters this becomes c0=

√γ0RT/M where γ0is the ratio of the principal heat capacities

of the gas in the limit of low pressure, Tis the thermodynamic

temperature and Mis the molar mass of the gas. The molar gas

constant Ris deﬁned by R=NAkBwhere NAis the Avogadro

constant. Rearranging for kBwe ﬁnd

kB=Mc2

γ0TN

.(1)

Since γ0=5/3 exactly for monatomic gases, and NAis known

with a relative standard uncertainty uR=0.044 ×10−6[5]

Updated determination of the molar gas constant R by acoustic measurements in argon at UVa-CEM

Preprint

Full-text available

Sep 2024

A new determination of the molar gas constant was performed from measurements of the speed of sound in argon at the triple point of water and extrapolation to zero pressure. A new resonant cavity was used. This is a triaxial ellipsoid whose walls are gold-coated steel and which is divided into two identical halves that are bolted and sealed with an O-ring. Microwave and electroacoustic traducers are located in the northern and southern parts of the cavity, respectively, so that measurements of microwave and acoustic frequencies are carried out in the same experiment. Measurements were taken at pressures from 600 kPa to 60 kPa and at 273.16 K. The internal equivalent radius of the cavity was accurately determined by microwave measurements and the first four radial symmetric acoustic modes were simultaneously measured and used to calculate the speed of sound. The improvements made using the new cavity have reduced by half the main contributions to the uncertainty due to the radius determination using microwave measurements which amounts to 4.7 parts in

10^{6}

and the acoustic measurements, 4.4 parts in

10^{6}

, where the main contribution (3.7 parts in

10^{6}

) is the relative excess half-widths associated with the limit of our acoustic model, compared with our previous measurements. As a result of all the improvements with the new cavity and the measurements performed, we determined the molar gas constant R = (8.314 449

\pm

0.000 056) J/(K mol) which corresponds to a relative standard uncertainty of 6.7 parts in

10^{6}

. The value reported in this paper lies -1.3 parts in

10^{6}

below the recommended value of CODATA 2014, although still within the range consistent with it.

Second-order electromagnetic eigenfrequencies of a triaxial ellipsoid II

Article

Full-text available

Aug 2015
METROLOGIA

James B Mehl

Finite-element calculations of electromagnetic eigenvalues are known to converge to the exact solutions in the limit of vanishing element sizes. In an extension of previous work (Mehl 2009 Metrologia 46 554–9) the eigenfrequencies of the TM1n and TE1n (n = 1, 2, ··· 6) modes of triaxial ellipsoids were calculated as a function of mesh size. Higher-accuracy eigenvalues were obtained through a limiting process as the mesh size was reduced; the extrapolation was based on the theoretical convergence rate. The difference between the finite-element eigenfrequencies and the eigenfrequencies predicted by shape perturbation theory is found to be proportional to the cube of the fractional deformation parameter ϵ for all investigated modes. For ellipsoids with axes proportional to 1 : 1.0005 : 1.0010, the cubic term represents a fractional perturbation of the average TM16 eigenvalue k² by −0.16 × 10 − 6 and the average TE16 eigenvalue by −0.22 × 10 − 6. This work adds support to the correctness of the analytic second-order formula derived in the previous work, and also demonstrates the usefulness of finite-element methods for investigating the quasi-spherical resonators (QSRs) used in measurements of the Boltzmann constant. In principle, the method can be extended to QSRs whose shape differs from triaxial ellipsoids.

Influence of propellant injection directionality on the performance of an argon Hall thruster

Article

Full-text available

Dec 2024
J APPL PHYS

The performance characteristics of an argon propellant Hall thruster with two types of propellant injectors, the axial and swirl injectors, were investigated. In the swirl injector, the propellant is injected in the tangential direction. At a discharge voltage of 150 V, the swirl injector achieved a higher propellant utilization efficiency (30.3%) and anode efficiency (8.8%) compared to the axial injector (26.7% and 7%, respectively). A numerical simulation quantitatively explained the reason for these differences, which shows an increase in the neutral particle density of 32.6% near the injection region and 7.8% at the exit of the hollow anode with the swirl injector. Neutral particle accommodation on the anode wall was found to be the predominant mechanism, which reduces the injection effect.

Laser spectroscopy of cold molecules

Preprint

Jun 2016

This paper reviews the recent results in high-resolution spectroscopy on cold molecules. Laser spectroscopy of cold molecules addresses issues of symmetry violation, like in the search for the electric dipole moment of the electron and the studies on energy differences in enantiomers of chiral species; tries to improve the precision to which fundamental physical constants are known and tests for their possible variation in time and space; tests quantum electrodynamics, and searches for a fifth force. Further, we briefly review the recent technological progresses in the fields of cold molecules and mid-infrared lasers, which are the tools that mainly set the limits for the resolution that is currently attainable in the measurements.

Atomic spectroscopy for primary thermometry

Article

Full-text available

Aug 2015
METROLOGIA

Spectroscopy has been a key driver and motivator of new understanding at the heart of physics. Here we describe high-precision measurements of the absorption lineshape of an atomic gas with an aim towards primary thermometry. We describe our progress in pushing this type of spectroscopy to the ultimate limit, in particular in describing experimental work with Rubidium and Cesium, although we also consider the potential for other elements in expanding the precision, accuracy and range of the approach. We describe the important technical and theoretical limits which need to be overcome in order to obtain accurate and precise results—these challenges are not unique to atomic spectroscopy but are likely to afflict all high precision spectroscopy measurements. We obtain a value for kB=1.380 545(98)×10−23 > J K−1 > where the 71 ppm uncertainty arises with difficulties in defining the Lorentzian component of the lineshape.

Quantum Definition of New Kelvin and Way Forward

Chapter

Aug 2023

Calorimetry of a Bose-Einstein condensed photon gas

Preprint

Apr 2016

Phase transitions, as the condensation of a gas to a liquid, are often revealed by a discontinuous behavior of thermodynamic quantities. For liquid Helium, for example, a divergence of the specific heat signals the transition from the normal fluid to the superfluid state. Apart from liquid helium, determining the specific heat of a Bose gas has proven to be a challenging task, for example for ultracold atomic Bose gases. Here we examine the thermodynamic behavior of a trapped two-dimensional photon gas, a system that allows us to spectroscopically determine the specific heat and the entropy of a nearly ideal Bose gas from the classical high temperature to the Bose-condensed quantum regime. The critical behavior at the phase transition is clearly revealed by a cusp singularity of the specific heat. Regarded as a test of quantum statistical mechanics, our results demonstrate a quantitative agreement with its predictions at the microscopic level.

Thermophysical properties of argon gas from improved two-body interaction potential

Article

May 2024

Ab initio interaction potential for the electronic ground state of the argon dimer has been developed. The potential uses previously calculated accurate Born-Oppenheimer interaction energies while significantly improving the description of relativistic effects by including the two-electron Darwin and orbit-orbit corrections. Moreover, leading-order quantum electrodynamics corrections to the potential are calculated, and long-range retardation of the electromagnetic interactions is taken into account. Spectroscopic properties of the argon dimer are reported, such as the bond dissociation energy, positions of rovibrational levels, and rotational and centrifugal-distortion constants. Our potential supports eight rotationless vibrational states, and the existence of a ninth level can neither be confirmed nor ruled out at the current accuracy level. Finally, thermophysical properties of the argon gas, including pressure and acoustic virial coefficients, as well as transport properties—viscosity and thermal conductivity—are evaluated using the developed potential. For the virial coefficients, the obtained ab initio values are somewhat less accurate than the most recent experimental results. However, the opposite is true for the transport properties, where the theoretical results calculated in this work have significantly smaller uncertainties than the data derived from measurements.

Inter-laboratory re-determination of the atmospheric 22Ne/20Ne

Article

Dec 2023
CHEM GEOL

Cylindrical Acoustic Gas Thermometry

Article

Jul 2023

We review recent determinations of the Boltzmann constant kB and the differences T − T90 that used cylindrical acoustic gas thermometry (c-AGT). These determinations measured the acoustic resonance frequencies of argon gas enclosed by metal-walled, cylindrical cavities. (Here, T is the thermodynamic temperature and T90 is the temperature measured on the International Temperature Scale of 1990, ITS-90.) In the range 234–303 K, the standard uncertainty of c-AGT ranges from 1.9 × 10−6T to 2.6 × 10−6T. This uncertainty is much smaller than the errors in ITS-90; therefore, c-AGT can help improve ITS-90. Moreover, we are extending c-AGT up to 1358 K. With increasing temperatures, c-AGT becomes advantageous relative to AGT based on quasi-spherical cavities because long cylindrical cavities (1) naturally fit into cylindrical heat pipes or multi-shelled thermostats; (2) provide the immersion required by transfer temperature standards, such as long-stemmed platinum resistance thermometers; and (3) have more useful, low-frequency acoustic resonances. In preparation for high-temperature c-AGT, we identified suitable materials for fabricating cylindrical cavities and we developed techniques for measuring acoustic resonance frequencies using sources and detectors outside the high-temperature thermostat. We also considered alternative test gases and optimal dimensions of cavities.

He Thermophysical Properties: New Ab Initio Calculations

Article

Full-text available

Mar 2007

Since 2000, atomic physicists have reduced the uncertainty of the helium-helium "ab initio" potential; for example, from approximately 0.6 % to 0.1 % at 4 bohr, and from 0.8 % to 0.1 % at 5.6 bohr. These results led us to: (1) construct a new interatomic potential φ07, (2) recalculate values of the second virial coefficient, the viscosity, and the thermal conductivity of He from 1 K to 10,000 K, and (3), analyze the uncertainties of the thermophysical properties that propagate from the uncertainty of φ07 and from the Born-Oppcnheimer approximation of the electron-nucleon quantum mechanical system. We correct minor errors in a previous publication [J. J. Hurly and M. R. Moldover, J. Res. Nat. Inst. Standards Technol. 105, 667 (2000)] and compare our results with selected data published after 2000. The ab initio results tabulated here can serve as standards for the measurement of thermophysical properties.

Acoustic Eigenvalues of a Quasispherical Resonator: Second Order Shape Perturbation Theory for Arbitrary Modes

Article

Full-text available

May 2007

James B Mehl

The boundary-shape formalism of Morse and Ingard is applied to the acoustic modes of a deformed spherical resonator (quasisphere) with rigid boundaries. For boundary shapes described by r = a [1 - ∈ℱ(θ, φ)], where ∈ is a small scale parameter and ℱ is a function of order unity, the frequency perturbation is calculated to order ∈ 2. The formal results apply to acoustic modes whose angular dependence is designated by the indices ℓ and m. Specific examples are worked out for the radial (ℓ = 0) and triplet (ℓ = 1) modes, for prolate and oblate spheroids, and for triaxial ellipsoids. The exact eigenvalues for the spheroids, and eigenvalue determined with finite-element calculations, are shown to agree with perturbation theory through terms of order ∈ 2. This work is an extension of the author's previous papers on the acoustic eigenfrequencies of deformed spherical resonators, which were limited to the second-order perturbation for radial modes [J. Acoust. Soc. Am. 71, 1109-1113(1982)] and the first order-perturbation for arbitrary modes [J. Acoust. Soc. Am. 79, 278-285 (1986)].

Measurement of the Universal Gas Constant R Using a Spherical Acoustic Resonator

Article

Full-text available

Mar 1988
J Res Natl Bur Stand

We report a new determination of the Universal Gas Constant R: (8. 314 471 plus or minus 0. 000 014) J multiplied by (times) mol** minus **1K** minus **1. The uncertainty in the new value is 1. 7 ppm (standard error), a factor of 5 smaller than the uncertainty in the best previous value. The gas constant was determined from measurements of the speed of sound in argon as a function of pressure at the temperature of the triple point of water. The speed of sound was measured with a spherical resonator whose volume was determined by weighing the mercury required to fill it at the temperature of the triple point. The molar mass of the argon was determined by comparing the speed of sound in it to the speed of sound in a standard sample of argon of accurately known chemical and isotopic composition.

Article

Jun 2008

This paper gives the 2006 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA) for international use. Further, it describes in detail the adjustment of the values of the constants, including the selection of the final set of input data based on the results of least-squares analyses. The 2006 adjustment takes into account the data considered in the 2002 adjustment as well as the data that became available between 31 December 2002, the closing date of that adjustment, and 31 December 2006, the closing date of the new adjustment. The new data have led to a significant reduction in the uncertainties of many recommended values. The 2006 set replaces the previously recommended 2002 CODATA set and may also be found on the World Wide Web at physics.nist.gov/constants.

Internal Consistency in the Determination of the Boltzmann Constant using a Quasispherical Resonator

Conference Paper

Sep 2013

The use of a combined microwave and acoustic resonator to determine the Boltzmann constant, k B, permits several checks on the internal consistency of the data. Using measurements in argon gas in the NPL-Cranfield quasispherical copper resonator (NPLC-2), we describe four distinct types of internal consistency check. Firstly, we estimate k B using six distinct acoustic resonances varying in frequency from 3.55 kHz to 21.77 kHz. We thus span a wide range of systematic corrections, most notably in the effect of the thermal boundary layer (TBL), which varies strongly with mode. Secondly, the same theory which predicts the TBL corrections to the acoustic resonance frequencies also predicts the widths of the resonances. By comparing the measured and theoretically-expected widths we can place limits on the effect of any un-modeled physics. Thirdly, the equivalent radius of the resonator (∼62.03 mm) is inferred from analysis of 8 TM microwave resonances and the spread of the radius values inferred from each mode is a measure of how well the resonator has been modeled. Finally, the microwave data can be used to check the inferred density of gas within the resonator. Based on measurements of the dielectric permittivity of the argon gas, pressure discrepancies greater than ±6 Pa can be detected at all pressures up to 700 kPa. Taken together, these four checks improve confidence in the final estimate for k B and restrict the types of systematic error which may affect the result.

Gas-filled spherical resonator: Theory and experiment

Article

Feb 1986

Gas‐filled spherical resonators are excellent tools for routine measurement of thermophysical properties. The radially symmetric gas resonances are nondegenerate and have high Q’s (typically 2000–10 000). Thus they can be used with very simple instrumentation to measure the speed of sound in a gas with an accuracy of 0.02%. We have made a detailed study of a prototype resonator filled with argon (0.1–1.0 MPa) at 300 K, with the objective of discovering those phenomena which must be understood to use gas‐filled spherical resonators to measure the thermodynamictemperature and the universal gas constant R. The resonance frequencies f N and half‐widths g N were measured for nine radially symmetric modes and nine triply‐degenerate nonradial modes with a precision near 10− 7 f N . The data were used to develop and test theoretical models for this geometrically simple oscillating system. The basic model treats the following phenomena exactly for the case of a geometrically perfect sphere: (1) the thermal boundary layer near the resonator wall, (2) the viscous boundary layer (for nonradial modes), (3) bulk dissipation, and (4) the coupling of shell motion and gas motion. In addition, the following phenomena are included in the model through the use of perturbation theory: (5) ducts through the shell, (6) imperfect resonator geometry, and (7) the seam where the two hemispheres comprising the shell are joined. Some estimates of the effects of (8) roughness of the interior of the shell have also been made. Much of the lower pressure f N and g N data can be explained by our model of these phenomena to within ±5×10− 6 f N when a single parameter c 0/(V 0)1 / 3 is fit to a single resonance frequency at a single pressure. In this parameter, c 0 is the ideal‐gas speed of sound and V 0 is the resonator volume. If this volume were known, the prototype resonator could be used to measure the speed of sound of a gas with an accuracy approaching ±0.0005%. Improvements in resonator design which will circumvent difficulties discovered in this work are expected to lead to much better agreement between theory and the measuredf N and g N .

Second-order boundary corrections to the radial acoustic eigenvalues for a spherical cavity

Article

Aug 2012

Keith A Gillis

We calculated the eigenvalues of the radially symmetric acoustic modes of a gas-filled, spherical cavity to order (δt/a)2, where δt is the thickness of the thermal boundary layer and a is the radius of the cavity. Our results explain an anomaly revealed by high-precision acoustic measurements made to redetermine the Boltzmann constant.

New high-precision measurements of the isotopic composition of atmospheric argon

Article

Dec 2011
GEOCHIM COSMOCHIM AC

Earth’s atmosphere is used as a standard reference gas for mass spectrometric determinations of argon (Ar) isotopes used principally in geochronology. There are three published independent determinations of the Ar isotope composition of modern atmosphere that differ subtly. We have made new high-precision measurements of Ar isotope ratios of five different sources of air using a high-sensitivity multi-collector noble gas mass spectrometer in order to distinguish between them. The isotope ratios, corrected only for backgrounds, reside on a inverse square-root mass law fractionation line that passes through the air value proposed by Lee et al. [Lee J. Y., Marti K., Severinghaus J. P., Kawamura K., Yoo H. S., Lee J. B. and Kim J. S. (2006) A redetermination of the isotopic abundances of atmospheric Ar. Geochim. Cosmochim. Acta70, 4507–4512]. They are distinct from both the other proposed compositions and provide the first independent confirmation of the atmospheric Ar isotope composition. We suggest that the revised values should now be in routine employment.

Pyknometric volume measurement of a quasispherical resonator

Article

Jun 2012

We have measured the internal volume of a 1 litre, diamond-turned copper quasispherical resonator with a fractional uncertainty of approximately 1 part in 106 using two independent techniques. This is in response to the need for a uniquely accurate measurement of resonator volume, for the purpose of measuring the Boltzmann constant in pursuit of the redefinition of the kelvin. The first technique is a pyknometric measurement using water as a liquid of known density. We describe the development of a procedure that results in stable, reproducible volume measurements. We provide a detailed discussion of the factors that affect the water density, such as dissolved gases. The second technique is microwave resonance spectroscopy. Here, we measure the resonant frequencies of the TM1n modes and relate them to the dimensions of the resonator. We evaluate the frequency perturbations that arise from the coupling waveguides and the electrical resistivity of the copper surface. The results of the microwave measurements show evidence of a dielectric coating on the surface. We propose that this is an oxide layer and estimate its thickness from the microwave data. Finally, we compare the volume estimates from the two methods, and find that the difference is within the combined uncertainty.

Acoustic resonance frequencies of deformed spherical resonators. II

Article

Feb 1986

James B Mehl

The effects of imperfect spherical shape on the modes of a spherical acoustic resonator are considered. The resonator boundary is described by a series of spherical harmonics r=a+εa∑ c l k Y l k , where ε is a small parameter. It is convenient to summarize the results by considering deformations which do not change the resonator volume. Earlier work showed, for pure radial modes subject to axisymmetric shape perturbations, that the fractional shift of the resonance frequency is proportional to ε2. This work extends this result to arbitrary shape perturbations. Nonradial modes, for which the acoustic pressure is proportional to j n (k n s r) Y n m (θ, φ) with n≥1 are (2n+1)‐fold degenerate for a perfect spherical resonator. Deformation of the resonator normally shifts the resonance frequencies by an amount which is linear in ε. However, the average shift of a degenerate set of modes with indices {n s} is of order ε2. Axisymmetric shape perturbations normally reduce the degeneracy of the nonradial modes to (n+1)‐fold, and arbitrary deformations normally lift the degeneracy completely. The modes with indices {n s} are sensitive only to the terms c l k with even values of l in the range l≤z n. A formalism for quantitative estimates is developed. Numerical results are obtained for some n=1 and n=2 cases.