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1 Dec 1970

Volume 41, Issue 13, pp. 5043-5358

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back to top WELCOME AND INTRODUCTION

Special Section on Thermal Expansion of Solids

J. Appl. Phys. 41, 5043 (1970); http://dx.doi.org/10.1063/1.1658594 (1 page)

Online Publication Date: 16 December 2003

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Abstract Unavailable

Vibrational Effects in the Thermal Expansion of Noncubic Solids

T. H. K. Barron

J. Appl. Phys. 41, 5044 (1970); http://dx.doi.org/10.1063/1.1658595 (7 pages) | Cited 19 times

Online Publication Date: 16 December 2003

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The thermodynamics of thermal expansion and of the related Grüneisen function γ(T, V) is discussed first for cubic and isotropic solids. Vibrational effects are treated in the quasiharmonic approximation. For central force models two effects are distinguished‐that of the anharmonicity of the pair potential, normally contributing positively to the expansion, and that of the bond ``tensions,'' contributing negatively. The extension of the theory to noncubic solids is summarized, the strongly anisotropic example of zinc being used to illustrate the roles played by anisotropy in the elasticity and in the Grüneisen tensor. Recent work on central force lattice models, both ionic and nonionic, is reviewed. For ``internal expansion,'' in which the disposition of atoms within a unit cell varies with temperature, it is convenient to treat both internal and external strains on the same footing.

Sensitive Dilatometer for Low Temperatures and the Thermal Expansion of Copper below 10 K

F. N. D. D. Pereira, C. H. Barnes, and G. M. Graham

J. Appl. Phys. 41, 5050 (1970); http://dx.doi.org/10.1063/1.1658596 (5 pages) | Cited 14 times

Online Publication Date: 16 December 2003

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We describe a dilatometer which is an improved version of the one previously used in this laboratory. It consists of a doubly twisted strip of beryllium copper with a mirror attached to its central region. This system is immersed in liquid helium II. Dilations of the sample are sensed by this system via a thin diaphragm, causing rotations of the mirror, which is detected by an external optical lever. The system has a sensitivity to relative length changes Δl∕l of 10−11. Copper has been used as the test material to evaluate the performance of the instrument. It has been possible to make direct measurements of α, the coefficient of linear thermal expansion, down to 2 K for ΔT=0.2 deg. These results yield α=1.3×10−10 T+2.7×10−11 T3 deg−1. The ratio of these terms is much more accurately known than the absolute values because of uncertainty in the absolute calibration. The linear term leads to an electronic Grüneisen γ of 0.57. This value is compared with recent theoretical and experimental values.
back to top LOW TEMPERATURE

Theoretical Calculation of the Electronic Thermal Expansion of Simple Metals

Duane C. Wallace

J. Appl. Phys. 41, 5055 (1970); http://dx.doi.org/10.1063/1.1658597 (2 pages) | Cited 8 times

Online Publication Date: 16 December 2003

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At low temperatures the excitation of conduction electrons in metals gives rise to an electronic contribution C = mathT to the heat capacity, and also a contribution β∝T to the thermal expansion coefficient, where T is the absolute temperature. The electronic Grüneisen parameter is γ = VβB/C = (d lnmath/d lnV), where V and B are the crystal volume and bulk modulus, respectively, at T=0. We have used a local pseudopotential model to calculate all the contributions to Γ and its volume derivative, including the electron‐phonon interactions in first‐order perturbation. The calculated γ is 1.18 for Na, 1.01 for K, and 1.63 for Al, while the measured γ is 1.8 for Al. The free electron model predicts γ=⅔. From our calculations it appears that the Umklapp interactions of electrons with transverse phonons give important contributions to Γ.

Low‐Temperature Phase Transitions in Alpha Uranium

M. O. Steinitz, C. E. Burleson, and J. A. Marcus

J. Appl. Phys. 41, 5057 (1970); http://dx.doi.org/10.1063/1.1658598 (3 pages) | Cited 25 times

Online Publication Date: 16 December 2003

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Strain‐gauge thermal expansion and anisotropy of susceptibility measurements indicate the existence of three phase transitions below 50°K in alpha uranium. Two of the transitions are of first order and one is of second order.

Effect of Axial Ratio Changes on the Elastic Moduli and Grüneisen γ for Lower Symmetry Crystals

E. S. Fisher and M. H. Manghnani

J. Appl. Phys. 41, 5059 (1970); http://dx.doi.org/10.1063/1.1658599 (4 pages) | Cited 3 times

Online Publication Date: 16 December 2003

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Gerlich has shown that Sheard's model for calculating mode γ's from hydrostatic pressure derivatives of the elastic moduli of hcp Mg and Cd yields Gruneisen γ's at both high and low temperatures that are in good agreement with the γ's derived from thermal‐expansion measurements. For hcp Ti and Zr, however, large differences arise, primarily from very small values for dC44∕dP. It is proposed that these small values are caused by the changes in c∕a ratio with hydrostatic pressure because of a large dependence of C44 on the c∕a ratio. The disagreement with thermal‐expansion data can be removed by taking into account the difference in d(c∕a)∕dV between hydrostatic‐pressure and thermal‐expansion conditions. The effect of Δ(c∕a) is not found in tetragonal TiO2, rutile, where mathH is in excellent agreement with the thermal expansion γ∞.

Lattice parameters of ZnO from 4.2° to 296°K

Robert R. Reeber

J. Appl. Phys. 41, 5063 (1970); http://dx.doi.org/10.1063/1.1658600 (4 pages) | Cited 43 times

Online Publication Date: 16 December 2003

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See Also: Erratum

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The temperature dependence of the lattice parameters of ZnO was measured by x‐ray powder‐diffraction methods. Retrograde behavior was found to occur perpendicular to the c axis at 93°K. Lattice parameters at 4.2°K were a0=3.24826±0.00015Å, c0=5.2033±0.0018 Å. These results are compared with those for some other II‐VI compounds.

Thermoelastic Expansion and Creep of Polyethylene Terephthalate and Polypyromelitimide Film and Polyethylene Terephthalate Fibers from 20 to 295 K

P. M. McConnell, D. E. Daney, and J. B. Kirgis

J. Appl. Phys. 41, 5066 (1970); http://dx.doi.org/10.1063/1.1658601 (5 pages)

Online Publication Date: 16 December 2003

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See Also: Erratum

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A quartz tube dilatometer was used to measure the lineal thermal expansion and creep of single lengths of polyethylene terephthalate (PETP) film, polypyromelitimide (PPMI) film, and PETP multifiber yarn, while stressed under constant tension. Tensions below and above the conventionally defined yield strength were used and the sample temperature ranged from 20 to 295 K. Relative creep strain measurements, taken at the constant temperatures 77, 195, and 295 K were found to obey the equation
math
where y is a function of stress, time, and temperature and A′ is a constant depending on the material. This equation was used to correct the thermoelastic expansion measurements for creep at the higher stresses. PETP multifiber yarn subjected to a slight tension was found to elongate during cooldown from 293 to 20 K. Higher stresses caused less elongation; i.e., the coefficient of expansion increased with stress. This result is believed to be due to changes in crystallinity at the higher stresses. A similar stress effect was found with PETP film but not with PPMI film. The thermoelastic expansion of the film samples was also found to be sensitive to the thickness.
back to top DEFECTS, DIFFRACTION

Volume Expansion of Sodium near the Melting Point for Different Impurity Contents

M. Ritter, G. Fritsch, and E. Lüscher

J. Appl. Phys. 41, 5071 (1970); http://dx.doi.org/10.1063/1.1658602 (3 pages) | Cited 10 times

Online Publication Date: 16 December 2003

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The volume expansion of sodium with a potassium impurity content of 100, 200, or 500 ppm (mole) was measured by a capillary method. The temperature region covered extended from 60° up to 100°C. (Melting point of pure sodium 97.83°C.) In the temperature region 60° to about 90°C the results, when combined with x‐ray measurements of the lattice constant, agree well with published values and theoretical estimates. The formation energy of a single vacancy was found to be 0.40±0.05 eV, and 0.45±0.05 eV; the formation entropy, in units of Boltzmann's constant, to be 7.2±1.5, and 7.05±1.5 for the 100‐ and 200‐ppm specimens, respectively. Extrapolation of the single‐vacancy concentration up to the melting point yields 10.5±1.5×10−4 and 9.3±1.5×10−4, respectively, again for the two impurity contents, 100 and 200 ppm. The data taken between 90° and 97°C show an excess volume expansion over the extrapolated single‐vacancy values. Several possible mechanisms responsible for this behavior are discussed. The starting point of melting (Tc) was found by a careful analysis of the volume‐expansion curves versus temperature and time. A power law holds for the volume expansion in the region 10−6<(Tc−T)∕Tc<10−4. The exponential number is 0.66≈2∕3, independent of the potassium content of the sodium sample.

Low Thermal Gradient High‐Temperature Furnace for X‐Ray Diffractometers

H. A. McKinstry

J. Appl. Phys. 41, 5074 (1970); http://dx.doi.org/10.1063/1.1658603 (6 pages) | Cited 16 times

Online Publication Date: 16 December 2003

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The optimum conditions for a high‐temperature x‐ray diffractometer are considered. A furnace with a minimum of compromise with these optimum conditions is described in some detail. Comparisons of previously described furnaces are made. The thermal gradients in the furnace were measured by thermocouple probe at several temperatures, and a novel shielding technique (making use of the x‐ray diffraction beam) was used to establish that the thermal gradients in the furnace were indeed very low. A series of polymorphic transitions which have been studied are presented as a means of obtaining a good calibration for high‐temperature x‐ray studies.

Anisotropic Thermal Expansion of Refractory Carbides by High‐Temperature Neutron Diffraction

A. L. Bowman, G. P. Arnold, and N. H. Krikorian

J. Appl. Phys. 41, 5080 (1970); http://dx.doi.org/10.1063/1.1658604 (2 pages) | Cited 2 times

Online Publication Date: 16 December 2003

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The anisotropic thermal expansion of U2PtC2 (tetragonal), and UMoC2 (orthorhombic) has been measured using a high‐temperature neutron diffractometer. Coefficients of thermal expansion (units of 10−6∕°C) are U2PtC2, α11=9.3±0.1, α33=10.1±0.8; UMoC2, α11=8.6±0.2, α22=15.5±0.5, α33=10±3.

Thermal Expansion of Tungsten at Low Temperatures

Jayant S. Shah and M. E. Straumanis

J. Appl. Phys. 41, 5081 (1970); http://dx.doi.org/10.1063/1.1658605 (1 page)

Online Publication Date: 16 December 2003

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Abstract Unavailable

Thermal Expansion of NbC, HfC, and TaC at High Temperatures

Choll K. Jun

J. Appl. Phys. 41, 5081 (1970); http://dx.doi.org/10.1063/1.1658606 (1 page) | Cited 1 time

Online Publication Date: 16 December 2003

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The thermal expansion of niobium monocarbide, hafnium monocarbide, and tantalum monocarbide was determined from room temperature to 2700°C by dilatometric measurements. The apparatus and experimental procedure were described earlier by Miccioli and Shaffer.1 All carbides used in the present investigation were prepared at The Carborundum Company by hot pressing at 3150°C and 3000 psi. The bulk thermal expansion results of these three monocarbides are compared with other published dilatometric data1–3 and also with x‐ray lattice expansion data.4–6 The high‐temperature divergence between the bulk thermal expansion data and the lattice expansion data is due to the effect of nonstoichiometry, caused by noncongruent vaporization7–9 at high temperature in these cubic monocarbides. The previously reported inversion phenomena were observed in the vicinity of 2000°C on several refractory monocarbides which were prepared by hot pressing at about 2000°C. This inversion was not detected in the present investigation. The inversion is due to the densification at the temperature apparently at which the specimens were fabricated.
back to top MOLECULAR SOLIDS, GRAPHITE

Thermal Expansivity and Isothermal Compressibility Measurements on Rare Gas Solids by an Optical Method

H. J. Coufal, R. Veith, P. Korpiun, and E. Lüscher

J. Appl. Phys. 41, 5082 (1970); http://dx.doi.org/10.1063/1.1658607 (3 pages) | Cited 11 times

Online Publication Date: 16 December 2003

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The thermal expansivity and isothermal compressibility at p=0 of solid rare gases can be measured interferometrically in the temperature region between 2° and 150°K. The experimental errors are about 1% in the whole temperature region. First measurements have been done on solid Kr. We have found that the volume thermal expansion coefficient varies from (3.80±0.02)×10−5 °K−1 at 5°K to (1.82±0.012)×10−3 °K−1 at 115°K (triple point). The isothermal compressibility at p=0 varies from (2.93±0.022)×10−11 dyn−1 cm2 at 5°K to (8.83±0.10)×10−11 dyn−1 cm2 at 115°K.

Volume Dependence of the Thermal Expansion of Polymers

D. John Pastine

J. Appl. Phys. 41, 5085 (1970); http://dx.doi.org/10.1063/1.1658608 (3 pages) | Cited 17 times

Online Publication Date: 16 December 2003

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Recent theoretical developments dealing with the equation of state of polymers make it possible to obtain some estimate of the volume dependence of the thermal expansion α of these materials. It is clear at this point that α should depend not only on the volume V and temperature T, but also on the degree of crystallinity Fc of the polymer in question. The calculations presented take this latter parameter into account and theoretical values of α for polyethylene (PE) are given as a function of V, T, and Fc. The values of α for crystalline PE are converted into values for poly chloro tri fluoro ethylene (Kel F) by simple corresponding states relations. The efficacy of these relations is shown by accurately converting P‐V data on PE to P‐V data on Kel F.

Anisotropic Thermal Expansion of Pyrolytic Graphite at Low Temperatures

A. C. Bailey and B. Yates

J. Appl. Phys. 41, 5088 (1970); http://dx.doi.org/10.1063/1.1658609 (4 pages) | Cited 63 times

Online Publication Date: 16 December 2003

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Interferometric measurements of the linear thermal‐expansion coefficients α∥ and α⊥ in the ``c'' and ``a'' axial directions of well‐oriented pyrolytic graphite between approximately 20° and 270°K are reported. Grüneisen parameters γ∥ and γ⊥, defined for strain coordinates parallel and perpendicular to the ``c'' axis, respectively, have been calculated from the present thermal expansion coefficients and related data. Between 30° and 270°K γ∥ is positive, indicating the preponderance of longitudinal modes of vibration, while γ⊥ is negative, indicating the predominance of transverse modes. The quasiharmonic approximation has been applied to calculate the characteristic temperatures θ(n) corresponding to the maximum frequencies ωD(n) of the Debye distributions having the same nth moments 〈ωn〉 as the specimens, which vary between approximately 420° and 2270°K as n increases from −3 to +6. The dimensional dependence of the moments, defined by γ(n) =Σγjωjn ∕Σωjn corresponding to γ∥(n) and γ⊥(n), and their variations with n, are consistent with the existence of low‐frequency modes of vibration between planes of atoms perpendicular to the ``c'' axis, and high‐frequency modes within the planes, a result supported by calculations of the rms displacements of the atoms.

Thermal Expansion of Molecular Crystals and Its Graphical Display

J. L. Amoros

J. Appl. Phys. 41, 5091 (1970); http://dx.doi.org/10.1063/1.1658610 (1 page) | Cited 1 time

Online Publication Date: 16 December 2003

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Abstract Unavailable

Role of Thermal‐Expansion Measurements in the Experimental Graphite Program

Paul Wagner

J. Appl. Phys. 41, 5091 (1970); http://dx.doi.org/10.1063/1.1658611 (1 page)

Online Publication Date: 16 December 2003

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Abstract Unavailable

Lattice Dynamics of Argon at 4°, 40°, and 77°K

D. N. Batchelder, B. C. G. Haywood, and D. H. Saunderson

J. Appl. Phys. 41, 5091 (1970); http://dx.doi.org/10.1063/1.1658612 (1 page) | Cited 1 time

Online Publication Date: 16 December 2003

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Neutron scattering techniques have been used to measure phonon dispersion relations in a 1‐ml single crystal of 36Ar. Data were taken along the 100 crystallographic direction for the transverse mode at 4°, 40°, and 77°K. The measurements were performed on a free‐standing crystal in equilibrium with its own vapor pressure. Preliminary analysis of the results yields mean shifts of the frequency of the transverse mode of 7% at 40°K and 21% at 77°K, relative to the 4°K value. These frequency shifts appear to decrease slightly with increasing wave vector. The mean Grüneisen parameter for this mode between 4° and 77°K is then 3.2±0.4 if it is assumed that the frequency shifts are due to the change in volume alone. Good agreement is found between the experimental frequency shifts and those predicted using a classical lattice dynamical model.1

Thermal Expansion of Polycrystalline Graphite

Robert G. Naum and Choll K. Jun

J. Appl. Phys. 41, 5092 (1970); http://dx.doi.org/10.1063/1.1658613 (4 pages) | Cited 3 times

Online Publication Date: 16 December 2003

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Thermal expansion characteristics of a group of polycrystalline graphites have been measured to 2400°C by a method which permits acquisition of meaningful expansion data without resorting to reference standards. Direct measurements on specimens while in the furnace were accomplished by using two 8‐in. focal length Gaertner optical micrometers. Thermal expansion behavior is discussed in relation to orientation, impurity, degree of crystallinity, and grain size. Specimens were heated from room temperature to 2400°C, then cooled to room temperature, taking measurements of thermal expansion throughout both heating and cooling cycles. The existence of hysteresis is observed and discussed.
back to top STANDARDS, COPPER, METHODS

Thermal Expansion of Copper from 20 to 800 K—Standard Reference Material 736

Thomas A. Hahn

J. Appl. Phys. 41, 5096 (1970); http://dx.doi.org/10.1063/1.1658614 (6 pages) | Cited 36 times

Online Publication Date: 16 December 2003

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Copper is the first of a series of materials that will be certified as thermal‐expansion standards by the National Bureau of Standards. The results of tests on five specimens indicate the stock is of consistent quality so that it may be certified as Standard Reference Material 736. A Fizeau interferometer was used for the expansion measurements. Above room temperature a controlled‐atmosphere furnace using a calibrated Pt vs Pt‐10% Rh thermocouple was used. Below room temperature a cryostat capable of operation with both liquid nitrogen and helium was used with a calibrated platinum resistance thermometer. Values of expansivity were calculated between equilibrium temperatures. The expansivity was used in the analysis of the data. Third‐order polynomials were fitted to the data for each of the five specimens in the overlapping temperature ranges from 0 to 70 K, 50 to 270 K, and 210 to 800 K to test for variations between the specimens. The deviations between the five equations were well within the standard deviations of the data for each of the specimens in the respective temperature intervals. All the expansivity data were then pooled and used to obtain an equation for each of the temperature ranges given above. These equations and their integrals were used to calculate the final values of expansivity and expansion, respectively. The results of the statistical analysis of the expansion and expansivity data are presented. A comparison is made with the data in the literature.

Effects of High Pressure on the Thermal Expansion of Solids with a Linear Us‐up Relationship

David J. O'Keeffe

J. Appl. Phys. 41, 5101 (1970); http://dx.doi.org/10.1063/1.1658615 (2 pages) | Cited 2 times

Online Publication Date: 16 December 2003

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A theoretical technique for determining the volume dependence of the thermal expansion coefficient α along an isotherm is developed for solids which exhibit a linear Us‐up (i.e., shock velocity‐particle velocity) relationship. The method is founded on an accurate Grüneisen equation of state utilizing the temperature‐independent Grüneisen parameter γ which has been calculated from experimental shock data. Detailed results are presented for several isotherms at pressures as high as 1 Mbar. Based on the accuracy of the Grüneisen parameter, the accuracy of the calculated thermal coefficient α is estimated to be within ±10% down to 20% compression with greater accuracy for lower compressions.

Thermal Expansion: A Review of Test Methods, Standards, and Data Availability

C. D. Pears

J. Appl. Phys. 41, 5103 (1970); http://dx.doi.org/10.1063/1.1658616 (1 page)

Online Publication Date: 16 December 2003

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The purpose of this paper is to encourage agreement as to where we are and suggest a few ways that this symposium can contribute to some future requirements. Thermal expansion of solids is a subject of interest to a wide range of disciplines from those conducting basic studies of lattice behaviors to those analyzing temperature stresses on re‐entry. Temperatures of interest range from cryogenic to 6500°F. Pressures range from 10−12 Torr to several atmospheres. The chemistry of the environment is also concerned. Thus the total environment is important. As a result of this broad scope, many different test methods are involved, different precisions and accuracies are required, and data are difficult to relate. This paper describes the applicability of some of the major test methods in terms of some of the above requirements, the standards that are available, and some sources of data for the different regimes. Particular emphasis is placed on engineering behavior since that represents the bulk of the authors experience. The observations are made that we need some base line understanding as to where we now stand and as to the areas where the thermal expansion symposium could systematically attack the problems by (1) encouraging papers and discussions of the various test methods, their environments, and limitations, (2) resolving by discussion and by encouraged personal round robins the obtainable precision and accuracy of methods and standards, (3) making known sources of data through author references, (4) communicating needs to groups that develop standards, (5) providing a forum where the facts can emerge and (6) publishing informal bound proceedings for a few years to gather the existing information and then bound abstracts eventually to provide references of more detail. After this sorting of information, more sophisticated studies will be possible by more people.

Use of Critically Evaluated Data on Copper to Test the Mie‐Grüneisen Equation of State

Richard K. Kirby

J. Appl. Phys. 41, 5103 (1970); http://dx.doi.org/10.1063/1.1658617 (1 page)

Online Publication Date: 16 December 2003

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Thermal expansion data that is now available for copper is believed to be accurate within the range 0–1000 K. Using this data and other critically evaluated data on thermodynamics and elastic properties various approximations of the Mie‐Grüneisen equation of state
math
have been evaluated. These approximations, assuming zero pressure, are usually written as
math
.Both of these approximations fail to provide a reasonable fit to the data, especially in the low‐temperature region. If, contrary to the normal usage of these approximations, the Grüneisen parameter is allowed to be temperature dependent then a better fit, both qualitative and quantitative, to the data is obtained,
math
.This form of the Mie‐Grüneisen equation also provides a means of estimating the temperature dependence of the Grüneisen parameter when compressibility data is only available at room temperature
math
.
back to top COMPOSITES, CARBIDES

Thermal Expansion of Composite Materials

R. R. Tummala and A. L. Friedberg

J. Appl. Phys. 41, 5104 (1970); http://dx.doi.org/10.1063/1.1658618 (4 pages) | Cited 32 times

Online Publication Date: 16 December 2003

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An equation for predicting the thermal expansion coefficient of dilute binary composites is presented treating the dispersed particles as elastic spheres and taking into consideration the physical interactions between the dispersed phase and the matrix. Application of this particular equation to a variety of systems such as ceramic‐glass, glass‐metal, metal‐metal, and organic‐metal, is discussed, as well as the application of other equations to other material systems extant in the literature.
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