to ao(MrM'Xo):2a"(MX). -t ao(M'Xr) is +5.0 Io i6.5o/o for the pyrope-almandine ..... Doviations from the oxide additivity rule vs. bond strain. Bond strain values ...
American Mineralogist, Volume 77, pages 94-100, 1992
Dielectric constants of silicate garnets and the oxide additivity rule Ronrnr D. SruNNoN E.I. Du Pont de Nemours and Co., Inc., Central Researchand Development Department, ExperirnentalStation, Wilmington, Delaware I 9880-0356, U.S.A.
Gnoncr R. RossvraN Division of Geological and Planetary Sciences,California Institute of Technology,Pasadena,California 9 I 125, U.S.A.
Ansrn-c,cr The dielectric constantsand dielectric loss values of a seriesof garnetswere measured at I MHz using a two-terminal method and empirically determined edgecorrections.The results are five intermediate pyrope-almandine samples,x' : 11.96-12.35; spessartine,x' : I1.65; two andraditesamples,x' : 10.53-10.59;and threegrossularsamples,r' : 8.538 . 8l . The deviations of measureddielectric polarizabilities as determined using the ClausiusMosotti equation from those calculated using the sum of oxide polarizabilities according -t ao(M'Xr) is +5.0 Io i6.5o/o for the pyrope-almandine to ao(MrM'Xo):2a"(MX) -2.3o/ofor the andraditesamples,and -5.5 to -7.00/ofor samples,*1.9o/ofor spessartine, grossular.These deviations from additivity are believed to result from garnet structural constraints leading to "rattling" Mg ions and "compressed" Ca ions.
has beendiscussedby Heydweiller(1920),Cheng(1940), IrvrnooucrroN and Van Santen (1947), Roberts (1949, 1950, Jonker Dielectric polarizability, ao, is related to the measured Lasagaand Cygan(1982).This rule has been 1951), and dielectric constant, x', by the Clausius-Mosotti equation: applied to both electronic and dielectric polarizabilities (l) (Narayana Rao, 1949; Lasaga and Cygan, 1982; Arndt a": l/bl(V^)(x' - l)/(x' + 2)l paper we discussthe rule where Z* is the molar volume in At, D is assumedto be and Hummel, 1988),but in this polarizabilities. Earit to dielectric insofar as applies only part 4tr/3, and x', the real of the complex dielectric con(Jonker and Van Santen, 1947; Narayana Rao, work lier stant, is measuredin the range I KHz to l0 MHz (RobI 982) found the erts, 1950, l95l). The dielectric polarizability includes I 949; Roberts, 195 I ; I-asagaand Cygan, However, to 5-l0o/0. rule to be accurate additivity oxide both ionic and electronic components.The electronic pochrysoberyl, Y RE aluminates, on and our recent studies larizability, a", is related to the refractive index, n,by the phenacite, zircon, and a group lnrenz-Lorentz equation (Lorentz, 1880; Lorenz, 1880): spinel, olivine-type silicates, of beryllates, borates, and phosphates (Shannon et al., (2) 1989, 1990, l99la, l99lb, 1992;Shannonand Subraa": l/bl(V^)(n2 - l)/(n2 + 2)1. Assumption of point dipole ions and cubic symmetry leads manian, 1989; Shannon, l99l; Subramanianand Shanform a class of to b: 4zr/3(Tessmanet al., 1953;Kip, 19621' Anderson, non, 1989) show that these compounds polarizabilities folwhose dielectric well-behaved oxides 1974; Arndt and Hummel, 1988). Introduction of over+0.5-1.50/0. lapping electron distribution (covalency)leadsto b: 1.2- low the oxide additivity rule to paper purpose is to determine accurately of this The 1.3 for silicates(Arndt and Hummel, 1988;Marler, 1988). of a variety of silicate garconstants l-MHz dielectric the The Clausius-Mosotti equation is strictly valid only for validity of the oxide additivity the nets to evaluate and compounds in which the molecule or ion has cubic symminerals. rule in these metry (Szigeti, 1949;Bosman and Havinga,1963; Duffin, 1980; Kip, 1962; Megaw, 1957; Roberts, 1949, 1950, l95l; Dunmur, 1972) but is approximatelyvalid for a number of noncubic crystals(Roberts, 1949, l95l1,I-asaga and Cygan, 1982). The concept of additivity of molecular polarizabilities, whereby the molecular polarizability of a complex substancecan be broken up into the molecular polarizabilities of simpler substancesaccording to ao(Mrit'['X):2a"(MX) oo03-004x/ 92/0l 02-0094$02.00
* ao(M'Xr)
(3)
ExpnnrvrcNurThe samples were all gem-quality crystals. The grossular, andradite, and spessartineare relatively pure. We were unable to obtain end-member pyrope or almandine specimenssuitable for dielectric measurements.All samples labeledas pyrope contain significantquantities ofCa; sample Py5, in addition to Ca, contains considerableMn. The sourceand color of the samplesare listed in Table l.
94
SHANNON AND ROSSMAN: DIELECTRIC CONSTANTS
95
TleLE 1. Electron microprobe analyses,cell dimensions,colors, and sources of garnets Py2
Py1
Pv3
Py4
Pv5
Na.O 0.00 0.o2 0.00 0.00 0.00 0.01 0.00 0.01 0.00 0.00 MgO 0.03 o.28 0,04 0.18 0.17 19.57 18.08 15.25 10.30 11.18 At2o3 22.26 22.21 22.47 0.09 23.63 23.16 0.11 23.39 22.60 22.60 sio, 39.61 39.55 39.30 35.31 35.46 42.60 4 1. 1 1 41.14 40.75 40.32 CaO 36.07 37.50 37.24 33.09 33.20 1.08 5.73 2.43 2.56 4.12 Tio, 0.54 0.53 0.02 0.0s 0.01 0.08 0.00 0.02 0.05 0.03 VrO" 0.13 0.01 0.10 0.00 na 0.01 0.02 0.O2 0.28 Cr,O. 0.02 0.00 0.00 0.00 0.00 0.61 0.00 0.00 o.24 MnO 0.51 0.14 0.15 0.01 0.03 0.28 0.19 0.40 0.54 8.91 FeO 1.27 0.15 0.51 16.37 9.08 19.48 21.88 14.51 Fe,O" 31.24 31.55 Total 100.44 100.39 99.85 99.98 100.53 101.61 102.46 102.14 98.68 102.3 a,A11.8480(4)11.Bs0B(4)11.8507(s)12.0626(1)12.0626(1) 11.ss57(1)11.4926(3)11.5i49(1) 11.5309(1)11.s6 v^,4 207.89 208.04 208.04 219.40 219.40 191.88 189.74 190.85 192.04 193 color rose amber colorless l. green L green rose reddish red red red purple Source Asbestos, Umba Jeffrey Ala Valley, Val Northeast Madagas- Sri Lanka No.Creek, Tanzania Quebec River, Quarry, ltaly Malenco, AZ cat Ny Tanzania Quebec ltalv
Sampleswere oriented for cutting to obtain the maximum area for capacitance measurement. Although no systematiceffort was made to obtain information on possible chemical zonation, significant color variations were not observed in any crystal. Microprobe analysesof points on individual crystals did not reveal any significant chemical heterogeneities. Electron microprobe analyseswere made using a JEOL 733 electron microprobe. Data reduction methods are describedby Armstrong (1982, 1988).X-ray diffraction patterns were obtained using a Guinier-type focusing camero, CnKa, radiation, and Si SRM 640 as an internal standard.Cell dimensions were obtained by least-squares refinement. Rectangular-, triangular-, or trapezoidal-shapedsamples were cut from the bulk crystals using a low-speed diamond wheel saw. Sample thickness and area varied from 0.09 to 0.13 cm and 0.10 to 1.0 cm2,respectively. SputteredAu electrodeswere applied over the entire parallel surfacesof the sample using a Denton Vacuum Desk II sputtering unit. Sample preparation is describedin de-
tail by Subramanian et al. (1989). Dielectric constant measurementswere made with a parallel plate capacitance technique using Hewlett-Packard 4274A and,427 5A LCRbridges and fixture 160348 (Test Tweezers)according to the procedure described by Subramanian et al. (1989) and Hewlett-Packard(1984).Edgecorrectionswere made using the expression c, : (0.019 tn P/t - 0.043)P
(4)
where I : sample thickness and P: perimeter in centimeters. The overall accuracy of the dielectric constant measurementsusing the above techniquesis estimatedto be 1.0-1.50/0.Dielectric loss errors are estimatedto be 520o/oat levels of tan d : 0.002 and 50-100o/oat levels of 0.0004-0.0005.
Rnsur,rs Table I summarizes compositional data, unit-cell dimensions,colors, and sourcesofthe garnetsstudied here. Table 2lists all of the known garnet dielectric constants. There is good agreement among values found here and
TmLE2. Celldimensions, molarvolumes,anddielectricconstantsof garnetsinglecrystals Source Pyrop+dmandine
Spessartine Andradite Grossular
Northeastern AZ Madagascar Sri lanka? Tanzania North Creek, NY Warren, NY Mania, Madagascar Little 3 Mine, Ramona, CA Haddam. CT Val Malenco, ltaly Ala Valley, ltaly Graham Cty, AZ Umba River Jeffrey Quarry Asbestos, Quebec Transvaal Lake Jaco. Chihuahua
a, A
v-, A.
11 . 5 3 5 7 ( 1 ) 11.4926(3) 11.sl 48(1) 11.s6350) 11.5389(2)
191.88 189.74 190.85 193.28 192.O4
11.533 11 . 6 1 3 5 ( 1 )
191.75 195.79
12.0626(1) 12.0626(1)
219.40 219.40
11.8508(4) 11.8507(8) 11.8480(4)
208.04 208.04 207.89
11.844
207.68
x' 12.00 12.20 12.35 12.28 11.96 11.5 13 11.65 11.2 10.59 10.53 10.5 8.71 8.53 8.81 8.1 8.5
tan A
Frequency
Referencs
