Second and Higher Order Elastic Constants

700 Pages·1992·31.9 MB·English

by  A. G. Every, A. K. McCurdy (auth.), D. F. Nelson (eds.)

1.1 Introduction 1
1 The elasticc onstantso f crystals
1.1 Introduction
1.1.1 Notation, units and abbreviations
a) List of symbols
CPU GPa elastics tiffnesse(sc ontracted)’)
1 GPa = 10gNm- ’ = 1010dyncm-2
c ’ GPa = fhl - 4
CL GPa = th + CIZ+ 22’ 4)
K GPa bulk modulusf or cubic crystals
K = fh + 2~12)
SP, (TPa)- ’ elasticc ompliances(c ontracted)’)
l(TPa)- ’ = 10-12m2N-1 = 10-13cm2dyn-1
SP strain components(c ontracted)’)
T K temperature
Tp GPa stressc omponents(c ontracted)’)
Tc,, 10-4K-1 temperaturec oefficientso f the elastics tiffnesses
Tcpd= &!!k (seet ext)
cp, dT
K Curie temperature
K melting temperature
K N6el temperature
K structuralt ransition temperature
K spin-flip temperature
K spin rotation temperature
K transition temperature
K martensitics tart temperature
GPa hydrostaticp ressure
(TPa)- ’ pressurec oefficientso f the elastics tiffnesses
Pcpo= Ldc,, (seet ext)
cw dp
V kms- ’ soundv elocity
VJ’ cp,Ip
n numbero f observations
s standardd eviation
v, cm3/mole molecularv olume
N cme3 carrier concentration
P kgmV3 density
b) Lettersu seda s superscripts
Letter Indication
S adiabatic( constante ntropy)
T isothermal( constantt emperature)
E constante lectricf ield
D constante lectricd isplacement
P constante lectricp olarization
*) For the tensonr otations, eeS ection2 .1.1
Land&-Bhstein
New Series III/29a
2 1.1 Introduction [Ref. p. 576
c) Other abbreviations
(piezoel.) After the nameo f a crystali n Tables3 . . .27 and Figs.3 .1. * .52.1 indicatesit is (or may be)
piezoelectric
at% atomic %
mole% mole %
wt % weight %
room temperature(Z 300K ; all resultsa re for RT unlesso therwises tated)
p: around a figure indicatet hat therei s somed oubt about it
BCC body-centerecdu bic
FCC face-centerecdu bic
1.1.2 Stiffnessa nd compliancec onstants
In the absenceo f electricala nd thermale ffects,t he stiffnessecs, , and the compliances pao f an anisotropic
materiala re definedb y the generalizedH ookes’ law
Tp= &%r Cpa= cop,
(p,o= 1,2,3,4,5 or 6) (1)
s, = sp,L spa= hp.
wheres ummationo ver indicesa ppearingt wice in any product is understood(E instein convention).
S,, S, are the six componentso f the strain matrix.’“) as definedb y Voigt [28vl].
Tp,T , are the six componentso f the stresst ensor”).
The cpaa nd spafo rm symmetrica6l x 6 arrays,a nd the numbero f independencto nstantsis therefore2 1i n the
most generalc aseo f a triclinic material”)‘ . The existence of symmetry elements in the material leads to
a reductioni n the numbero f independencto nstants[2 8vl, 57nl,61h l]. The schemeosf constantsa ppropriatet o
the variousc lasseosf crystals ymmetrya reg iveni n Tables1 and 2. Readingfr om the top downwardst,h eseta bles
show:n ameo f systemt;h e point groupsb elongingt o the system( Hermann-Mauguin otation”“)); the orienta-
tion of the principala xesw ith respecto the coordinatea xesx , , x2, and x3; the numbero f independencto nstants;
the columnn umber;a nd, in the body of the tables,t he schemeosf independenst tiffnessecs, ,,,,q uoting only the
suffixesp a. As an examplet,h e existenceof a simple4 -fold axiso r its equivalenpt arallelt o the x3 axisn ecessitates
the following relations:
Cl1= czz. Cl3 = c239 C44 = css. Cl6 = - C26.
Cl4 = Cl5 =C24 = C25 = C34 = C35 = C36 = C45 = C46 = C56 = 0,
and therea re 7 independenst tiffnesses”)‘ :
CllrC33,C12,C13.C44,C66.and c16.
Examinationo fcolumn6 , Table1 showst hat all theser elationsa rec ontainedin it, andi t thereforer epresenttsh e
schemeo f stiffnesseasp propriatet o point groups4 , 3, and 4/m. The other columnso f Tables1 and 2 similarly
representth e schemeosf stiffnesseasp plicablet o the point groupss howna t the heado f the particularc olumn.
The monoclinics ystemis listedf or threeo rientationsw, ith the 2-fold axisr espectivelpya rallelt o x1, x2, and x3;
the standardo rientationf or a monoclinicc rystali s with the 2-fold axis parallelt o x2.
I‘ The shear strains S,, Ss, S6 as defined by Voigt and most later writers must be halved to obtain true tensor components.
a1 The problem of tensor vs. matrix Voigt notation is discussedi n detail in Section 2.1.2.
Ir ’ In triclinic crystals the orientation of the coordinate system is not fixed by the crystal structure. Hence, by a suitable
rotation of axesi t is possiblet o reduce the number of independent elastic constants from 21 to 18 [65fl]. However a full
description of the elastic tensor now requires specification of the three Euler angles or equivalent parameters,a nd so the
information content is still the same.S imilarly in monoclinic crystalst he freedom to rotate the coordinate systema bout the
2-fold axis can be used to reduce the number of independent elastic constants from 13 to 12, with gain of one angular
parameter.I t is usual however fori’nvestigators to adopt the convention< coordinate systemf or a particular crystal and list
the higher number of elastic constants. For the tetragonal groups 4, 4 and 4/m, a suitable rotation about the xX axis
eliminates c,~ and the elastic constant tensor takes on the samef orm as for the remaining tetragonal groups, displaying
a higher degreeo f“acoustic symmetry” [61K2,65fl, 79B5,87E4,89Fl]. In a similar way c 14o r cl5c anb ee liminatefdo rt he
trigonal groups 3 and 3 yielding an elasticc onstant tensor which has the higher acoustics ymmetryo f the remaining trigonal
groups. Related questions that have attracted attention are the identification of material symmetry from given elastic
constants [87Cl] and the obtaining of invariants of anisotropic elastic constants [87T5].
- ’ For the corresponding Schoenfliesn otation, seeT able 4 in Section 2.1.3.
Land&-B6msrcin
New Swim 111,29a
2 1.1 Introduction [Ref. p. 576
c) Other abbreviations
(piezoel.) After the nameo f a crystali n Tables3 . . .27 and Figs.3 .1. * .52.1 indicatesit is (or may be)
piezoelectric
at% atomic %
mole% mole %
wt % weight %
room temperature(Z 300K ; all resultsa re for RT unlesso therwises tated)
p: around a figure indicatet hat therei s somed oubt about it
BCC body-centerecdu bic
FCC face-centerecdu bic
1.1.2 Stiffnessa nd compliancec onstants
In the absenceo f electricala nd thermale ffects,t he stiffnessecs, , and the compliances pao f an anisotropic
materiala re definedb y the generalizedH ookes’ law
Tp= &%r Cpa= cop,
(p,o= 1,2,3,4,5 or 6) (1)
s, = sp,L spa= hp.
wheres ummationo ver indicesa ppearingt wice in any product is understood(E instein convention).
S,, S, are the six componentso f the strain matrix.’“) as definedb y Voigt [28vl].
Tp,T , are the six componentso f the stresst ensor”).
The cpaa nd spafo rm symmetrica6l x 6 arrays,a nd the numbero f independencto nstantsis therefore2 1i n the
most generalc aseo f a triclinic material”)‘ . The existence of symmetry elements in the material leads to
a reductioni n the numbero f independencto nstants[2 8vl, 57nl,61h l]. The schemeosf constantsa ppropriatet o
the variousc lasseosf crystals ymmetrya reg iveni n Tables1 and 2. Readingfr om the top downwardst,h eseta bles
show:n ameo f systemt;h e point groupsb elongingt o the system( Hermann-Mauguin otation”“)); the orienta-
tion of the principala xesw ith respecto the coordinatea xesx , , x2, and x3; the numbero f independencto nstants;
the columnn umber;a nd, in the body of the tables,t he schemeosf independenst tiffnessecs, ,,,,q uoting only the
suffixesp a. As an examplet,h e existenceof a simple4 -fold axiso r its equivalenpt arallelt o the x3 axisn ecessitates
the following relations:
Cl1= czz. Cl3 = c239 C44 = css. Cl6 = - C26.
Cl4 = Cl5 =C24 = C25 = C34 = C35 = C36 = C45 = C46 = C56 = 0,
and therea re 7 independenst tiffnesses”)‘ :
CllrC33,C12,C13.C44,C66.and c16.
Examinationo fcolumn6 , Table1 showst hat all theser elationsa rec ontainedin it, andi t thereforer epresenttsh e
schemeo f stiffnesseasp propriatet o point groups4 , 3, and 4/m. The other columnso f Tables1 and 2 similarly
representth e schemeosf stiffnesseasp plicablet o the point groupss howna t the heado f the particularc olumn.
The monoclinics ystemis listedf or threeo rientationsw, ith the 2-fold axisr espectivelpya rallelt o x1, x2, and x3;
the standardo rientationf or a monoclinicc rystali s with the 2-fold axis parallelt o x2.
I‘ The shear strains S,, Ss, S6 as defined by Voigt and most later writers must be halved to obtain true tensor components.
a1 The problem of tensor vs. matrix Voigt notation is discussedi n detail in Section 2.1.2.
Ir ’ In triclinic crystals the orientation of the coordinate system is not fixed by the crystal structure. Hence, by a suitable
rotation of axesi t is possiblet o reduce the number of independent elastic constants from 21 to 18 [65fl]. However a full
description of the elastic tensor now requires specification of the three Euler angles or equivalent parameters,a nd so the
information content is still the same.S imilarly in monoclinic crystalst he freedom to rotate the coordinate systema bout the
2-fold axis can be used to reduce the number of independent elastic constants from 13 to 12, with gain of one angular
parameter.I t is usual however fori’nvestigators to adopt the convention< coordinate systemf or a particular crystal and list
the higher number of elastic constants. For the tetragonal groups 4, 4 and 4/m, a suitable rotation about the xX axis
eliminates c,~ and the elastic constant tensor takes on the samef orm as for the remaining tetragonal groups, displaying
a higher degreeo f“acoustic symmetry” [61K2,65fl, 79B5,87E4,89Fl]. In a similar way c 14o r cl5c anb ee liminatefdo rt he
trigonal groups 3 and 3 yielding an elasticc onstant tensor which has the higher acoustics ymmetryo f the remaining trigonal
groups. Related questions that have attracted attention are the identification of material symmetry from given elastic
constants [87Cl] and the obtaining of invariants of anisotropic elastic constants [87T5].
- ’ For the corresponding Schoenfliesn otation, seeT able 4 in Section 2.1.3.
Land&-B6msrcin
New Swim 111,29a
Ref. p. 5761 1.1 Introduction 3
Table 1. Elastics tiffnesseisn the triclinic, monoclinic,o rthorhombic,t etragonal,a nd cubic systems[6 1hl] I.’
Triclinic Monoclinic Orthorhombic Tetragonal Cubic
1 2 4 4mm 23
i m ii: 3 32rn m3
2/m mmm 4/m 42 23rn
4/mmm 43
m3m
2(x1) 2(x,) 2(x3) 4(x3) 4(x3)
or or or and
mh) mh) m(x3) 2(x2 or x1)
:6) ;3
11 11 11 11 11 11 11 11
12 12 12 12 12 12 12 12
13 13 13 13 13 13 13 12
14 14 0 0 0 0 0 0
15 0 15 0 0 0 0 0
16 0 0 16 0 16 0 0
22 22 22 22 22 11 11 11
23 23 23 23 23 13 13 12
24 24 0 0 '0 0 0 0
25 0 25 0 0 0 0 0
26 0 0 26 0 -16 0 0
33 33 33 33 33 33 33 11
34 34 0 0 0 0 0 0
35 0 35 0 0 0 0 0
36 0 0 36 0 0 0 0
44 44 44 44 44 44 44 44
45 0 0 45 0 0 0 0
46 0 46 0 0 0 0 0
55 55 55 55 55 44 44 44
56 56 0 0 0 0 0 0
66 66 66 66 66 66 66 44
l) Reproducedb y permissiono f Oxford University Press.
Landoh-Btmstein
New Series III 2‘ 98
4 1.1 Introduction [Ref. p. 576
Table 2. Elastics tiffnesseisn the triclinic, trigonal, hexagonala, nd isotropic systems[6 lhl] !‘
Triclinic Trigonal Hexagonal Isotropic
1 3 3m 6
i 3 3m 5
32 6/m
6mm ’
6m2
62
g//mm
3(x3) 3(x3) 6(x3)
2(x,)
11 11
12 12
13 13
14 14
15 15
16 0
22 11
23 13
24 -14
25 -15
26 0
33 33
34 0
35 0
36 0
44 44
45 0
46 -15
55 44 j(ll-12)
56 14 0
66 j(ll-12) f(ll-12)
” Reproducedb y permissiono f Oxford UniversityP ress
The schemeos f Table 1 apply to the compliancesa s well as to the stiffnessesT.a ble 2 is also valid for the
compliancess,u bjectt o the following modifications:
a) in the trigonal systemw here
C46 = - ClS, C56 = c14,
the correspondingre lationsf or the compliancesa re
s46 = - 2s,5, s56 = h4;
b) in the trigonal, hexagonala, nd isotropics ystemsw, here
C66 = fkl, - c,2),
the correspondingre lation for the compliancesis
S66 = as, I - s12).
It is evidentf rom equations( 1)t hat the stiffnessecsa n be convertedt o the complianceas nd vicev ersab y the
standardd eterminantp rocedurefo r solvings imultaneouse quationsT. he conversione quationsa, nd the special
caseso f them appropriatet o the variousc rystal systemsa re giveni n [61hl].
Landoh-BZmstein
New Series 11IR9.3
Ref. p. 5761 1.1 Introduction 5
1.1.3 Methodsf or the determinationo f the elasticc onstants
The methodsf or measuringt he elasticc onstantsa re describedin [46hl, 56h1,6 1h1,6 2wl,65zl, 67N2, 70m1,
71K7,71L4,72vl, 73sl,76C2,77Pl, 82cl,82dl, 83L9,84kl, 85n2,88sl]. The most widely usedm ethodsa re,o r
haveb een:
1 Ultrasonic wavet ransmissioni,n cluding the pulses uperpositionm ethod,
2 Resonancoef samplesin the shapeo f rods, bars,p arallelepipedas nd plates,
3 Light scatteringf rom ultrasonicw aves( Schaefer-Bergmann),
4 Static deformation,
5 Brillouin scattering,
6 Ultrasonic wedgem ethod,
7 Thermald iffuses catteringo f X-rays,
8 Neutron scattering.
Methods2 and 4 determinet he compliances ,, directly;t he remainderd eterminet he stiffnessecsp od irectly.
The accuracyo f the methodsv ariesw idely, and the abovel ist representsa n approximater anking in order of
decreasinga ccuracyM. ethods1 and 5 are the most widelyu seda t the present ime. Most completes etso f elastic
constantsh avei n fact beeno btainedb y ultrasonict ransmissionB, rillouin scatteringe nablesm easurementos be
madeo n smallc rystalsw ith sizeo f the order of a fewm illimeters,a nd hasp rovedi nvaluablein the studyo f effects
of p and Ton elasticc onstantsW. ith the aid of this methoda nd of the more traditional ones,t he behavioro f
manys ubstancehsa sb eenf ollowedt hroughp haset ransitionr egions;t he interpretationo f this behaviori n terms
of phonone ffectss, oft modes,o rder parametersa nd allied topicsh asd evelopedin to an extensives ubject,a nd is
not dealt with here.
Methodsh avea lso beenp roposeda nd usedf or the estimationo f singlec rystal elasticc onstantsf rom the
means quarea mplitudeso f atomic vibrations [78Sl, 80KlO], and from the elasticc onstantso f polycrystalline
aggregate[s7 8L2, 79B6,7 9H5, 81L11,8 3V3,8 7Bl], in somec asesc omplementedb y X-ray diffraction strain
measurement[s8 5H3]. Phononf ocusing[ 85nl, 86ml] yieldsd etailedi nformation on the elastica nisotropyo f
crystals,a nd henceth eir elasticc onstantr atios,b ut up to thep resenth asm ainly servedto confirme xistingv alues
of theser atios,a nd to explored ispersivea nd other effectsa t thermalp hononf requenciesR. ecentlyh, owever,a ll
three elasticc onstantso f an anisotropicc ubic crystal have beend eterminedf rom ultrasonicg roup velocities
measureda longo ff-symmetryd irectionsu singa point-source/poinrte ceivert echniqueT. he elasticc onstantsa re
varied so as to obtain a least-squarefsit betweent he measureda nd calculatedg roup velocities[ 90El]. There
have beena few reportedm easurementosf elasticc onstantsu sing surfacea cousticw aves[ 84Ml, 8115,8 5m1,
88B2],a nd Lamb modeso f orientedt hin films havey ieldedi nformationa bout elasticc onstants[ 87(35,88N7].
The vibrating reedm echanism[8 7B5] hasb eenu sedf or measuringY oungs’ moduluso f singlec rystal charge-
density-wavec onductorsa nd high-T, superconductors[8 9S3,89Xl], and a similar techniqueb asedo n the
bendingv ibrationso f singlec rystalw hiskersh asb eenu sed[ 8411].T he torsionalp endulumm ethod[ 82BS,8 4B10,
86B6,87Xl] has beenu sedf or measurings hearm oduli. There are other methodst hat have beenp roposed
[71C7,73N5,7323,7602,76SlO7, 786,88M4,89A2],b ut not widely useds o far.
Therei s an extensiveli teratureo n theorieso f elasticc onstantsm, ucho f whichc oncernsth e subjectso f phase
transitionsa nd lattice dynamics,a nd lies outsidet he scopeo f this compilation.F or recentr eviewsa nd other
generald iscussions ee[ 82bl, 8211,82sl8, 2S5,84M8,85fl,8 5Kll,86sl, 88F1,88sl].
Landolt-Bdmstein
New Series II1/29a
6 1.1 Introduction [Ref. p. 576
1.1.4S econdarye ffects:t hermal,e lectricala nd magneticc onditions
There are a number of secondarye ffectsw hich have to be taken into accountw hen discussingth e elastic
constantst,h e mosti mportant of which are thosea ssociatewd ith thermala nd electricacl onditions[ 57nl]. The
adiabatica nd isothermale lasticc onstantsd iffer from eacho ther, and are distinguishedfr om eacho ther when
necessarbyy the superscriptSs and T respectivel(ye .g.c i,, and c,o‘ ). Similarlyf or piezoelectricr ystals( identified
in Tables3 1. . 27 by (piezoel.a) ftert he nameo f the substancet)h, e elasticc onstantsa ppropriatet o conditionso f
constantf ield (c:~),c onstantp olarization( c:~),a nd constantd isplacemen(tc :~)a lso differ from eacho ther. All
thesed ifferenceasr e usuallyo f the sameo rder as,o r evens mallert han experimentael rrors,a nd aren ormallyn ot
distinguishedin the tables.H owever,f or a few substancesp,a rticularlyt hosef or which the differenceasr e large,
valueso f the differentt ypeso f constantsa re quotedw hena vailableb, ut no speciacl alculationsh aveb eenm ade
for presentp urposesF. or manyo f the materialsl abelled“p iezoel.“, the labellingi s basedo n the reportedp oint
group, and has not necessarilyb een confirmede xperimentallyI.n noncentrosymmetriccr ystals the linear
variation of the elasticm oduli with electricf ield definest he fifth-rank electroelasticte nsor [74P2,82R5].N o
attemptw ill be madeh eret o tabulatet he very limited amounto f data that is availableo n this tensor.
A varietyo f complicatede ffectso f magneticfi eld on elasticc onstantsh aveb eenr eported,p articularlyi n the
rare earth elementsD y, Er, Gd, Ho, Nd, Pr, and Tb [7OP3,70R4,72P27,3 14,73S9,74K2,74Pl,7 4R4,76Dl,
76P7,7 7J2,7 7P1,7 7P2,8 451,S SSfS].A cousticd e Haas-vanA lphen effectsh aveb eenr eportedi n rare earth
and other compounds[8 2N6,82T4,84L3,85N2,8586,87813,87El]a,n d other magneticfi eld effectsh aveb een
observedin heavy-fermionc ompounds[8 7K3,87N3,87W3]a nd in ferromagneticf,e rrimagnetica nd antiferro-
magneticm aterials[ 82B4,8311,8664,87M2],a nd in materialss howings pin reorientationp haset ransitions
[84D3]. A completes ummaryo f theser esultsh asn ot beena ttempteda; fewr esultsa req uotedf or illustration,b ut
references hould be madet o the original papersf or full details.U nlesso therwises tated,a ll resultsf or the
rare-earthsa nd materialsw ith similar behaviora re quoteda t zero magneticf ield.
Illumination hasb eenr eportedt o havea n influenceo n the elasticc onstantso f certainc rystals[ 81G6,85W4]
and the effectso f X-ray and y-ray irradiation havea lso beeni nvestigated[S SVl].
1.1.5 Secondarye ffectsf:r equency
Comparisono f elasticc onstantso btainedi n the ultrasonic,B rillouin scatteringa, nd neutrons catteringre gionsis
ofinteresti n connectionw ith the frequencyd ispersiono f the elasticc onstantsa, ndw ith soundp ropagationin the
first sounda nd zero soundr egimes[ 67C7]. The former correspondsto a collision-dominatedre gime( 07 < 1,
whereo is the frequencya, nd 7 is the phonon lifetime),e xploredb y ultrasonicp ropagationa nd, at higher
temperaturesb,y Brillouin scatteringte chniquesT. he secondc orrespondtso a collision-freere gime(~ 7 9 l), and
is exploredb y neutrons catteringt echniquesC. omparisonso f elasticc onstantso ver widef requencyr angesh ave
beenm adef or quartz [70B6,7389], sulfur [74Vl], copper[ 76Lll], potassiumb romide[ 68S4],a nd the nitrates
of Ca, Ba, and Sr [71M6]; the availabled ata for the rare-gass olids is summarizedin [72T2,77kl, 77R2].
However,t herei s conflicto ver the sizea nd event he signo f thesed ifferencesa,n d in order not to overburdenth e
tablesw ith detail, the differencesa re not normally distinguishedin the tables.
A lattice-dynamicaal pproachw hich takesi nto accountt he inertial effectso f optic modes[ 88N6] leadst o
couplingo f the strain and rotation fields,a nd therebyd ispersivec ontributionst o the elasticc onstantsI.n the
generalc aseo f triclinic symmetryt his extendede lastic constant matrix has 45 independentc omponents
(comparedw ith the usual2 1).I n [88P6] it is shownt hat only 36 of thesea reb ulk elasticc onstantst,h e remaining
9 beings urfacec onstantsO. bservable ffectsm ight be expectedin materialsw ith zone-centreso ft modesb, ut no
experimentacl onfirmationh as beenr eporteda s yet.
In certainc lasseosf crystalss, patiald ispersionis an important factora nd givesr iset o observable ffectse ven
at ultrasonicf requenciesIt. canb et reateda sa n extensionto continuume lasticityt heoryb y expandingth ee lastic
constantsi n powerso f the wavevector,a nd retaining only the leading terms of low order [68P2,80K17,
86D3,87E3].T he coefficientos f the lineart ermsc ompriseth e acousticg yrotropict ensor,a nd arer esponsiblefo r
the phenomenono f acousticaal ctivity.C ertainc omponentso f this tensorh aveb eenm easuredin a smalln umber
of crystals,b ut are not tabulatedh ere.
LandolbB6mswin
New Series 1112’ 9a
Ref. p. 5761 1.1 Introduction 7
1.1.6 Temperaturec oefficients
The temperaturec oefficientso f the elastics tiffnesseasr e definedf ormally as
Tc,, = a In cpolaT.
In practice,t hey haveb eenc alculatedf rom
Tc,, = Wc,,W,o/AT),
wherec pai s the stiffnessa t a particular temperatureI.f the temperaturec oefficienti s quoted at a particular
temperaturet,h e valueo f cpOa t that temperaturew as usedi n the calculationso, therwiset he referencete mper-
ature was usuallyt aken as 300K .
The above treatment of temperaturec oefficientsa ssumesa linear relationshipb etweenc and T. If the
relationshipi s not linear,a polynomialr elationshipt,r uncateda t the cubict erm,i s taken to representth e c vs. T
relationship[ 62B3,74S8]:
c-co AC
-=-= i Tc“’(‘ T - T,)“,
co co II=1
wheret he sufficesp a haveb eeno mitted, and the Td”) are generalizedte mperaturec oefficients:
This treatmento f temperaturec oefficientsh as beend evelopedw ith specialr eferenceto quartz, and some
generalizedte mperaturec oefficientsfo r this materiala reg iveni n Tables3 7 . . .39. Elsewheret,h e relationshipi s
takena sl inear,a nd the superscrip(t1 )i s omitted.G eneralr elationshipsc onnectingc with T haveb eend iscussed
in [7OV4] and [71Ll].
1.1.7 Pressurec oefficients
The hydrostaticp ressurec oefficientsa re definedf ormally as
kpa = a In cpoIap,
and haveb eenc alculatedf rom
&,a =
8 1.1 Introduction [Ref. p. 576
1.1.8 Accuracya nd selectiono f data
If the numbero f setso f observationso n any materiali s oneo r two, then all the data are given.I f the numbero f
setsi s threeo r more,t he average(2 ) is usuallyg iven,t ogetherw ith the standardd eviation( s).S incet he numbero f
sets( n)i s usuallys mall,s hasb eene stimatedfr om the rangea sd escribedin [56S2].H owever,fo r somem aterials,
n is too largef or this methodt o be applicable(e .gf.o r NaCI,n is at least3 5,a ndf or Cu at least2 0),a nd for n > 10,
the 10m ost recenta nd acceptables etsw ereg enerallyu sedf or the calculationo f Z?a nd s. Wheren ew data has
becomea vailableo n a materials incet he previouse ditionso f this tabulationi n Chaps.1 of Landolt-Bornstein,
New Series,V ol. III/II and Vol. III/l8, it has, wherea ppropriate,b eenc ombinedw ith existingd ata in the
tabulationst o calculaten ew averagesa nd standardd eviations.
The levelo f accuracya imeda t in the tablesi s threef igures,b ut in individual casesit may be greatero r less
than this.T hust hee lasticc onstantso fsomes etso f alloysa reg ivent o four figuresw, hereas omet emperaturea nd
pressurec oefficientsa re only given to two.
Most authorsr eport either the set of compliances pOo r the set of stiffnessecs, ,~b ut not both. All setsh ave
beenc onverteda s appropriatef or usei n the presentt ablesb y meanso f the equationsg ivene .g.i n [57nlJ or
[6lhl], but becauseo ferror propagationd uring the matrix inversion[ 81H2,81L5,81LlO],t he original accuracy
of the data is reducedd uring conversionI.n addition, becauseth e compliancesa nd the stiffnessehsa ve been
averageds eparatelys, omeo f the valuese nteredi n the tables may not obey exactly the relevantc onversion
relationsb etweent he compliancesa nd the stiffnesses.
In generalt he accuracyo f the off-diagonalc onstants( p # a) is lesst han that of the diagonalc onstants
(p = a).I n the monoclinics ystemt,h e off-diagonacl onstantsw ith suffices1 5,25,35a nd 46,w hicha rez eroi n all
other systemse xceptt he triclinic, are oftens malla nd havel arges tandardd eviationss , so it is doubtful whether
many of them differ significantlyf rom zero. In addition, there is ambiguitya bout the signso f someo f these
constants,e .g. the signs quoted for the materialsi n [70B4] differ in some instancesf rom those published
previously.
Large discrepancieesx ist for somes ubstancees. g.l ead nitrate and pentaerythritol.I n suchc asesg, reater
weighti s usuallyg ivent o the later resultso, n the assumptionth at the authort ook the earlierr esultsi nto account,
and would thereforeb e especiallyc arefula bout checkingh is own.
Somed iscrepancie(sio dic acid,l eadc hloride)e xistb ecaused ifferentw orkersu sedd ifferenta xial systemsI.n
suchc asest,h e axial systemg iven in [73dl] is accepteda, nd the resultsa re transformedt o this systemw here
necessaryH.o wever,if the axial differenceh asp reviouslyb eenn oticeda ndr esolvede .g.L iz SO4* HZ0 [52Bl], no
further alterationi s madeh ere.
Purelyt heoreticalc alculationso f the elasticc onstantsa ree xcludedb, ut somer esultso btainedb y combining
theoreticala nd experimentalr esultsa re included.T his appliest o elastic constantsd erived by interpreting
neutrons catteringe xperimentisn conjunctionw ith a particularm odelo f the substancuen deri nvestigation(o ften
oneo f the rare-gass olids),a nd to constantso btainedb y axial transformationb etweentw o modificationso f the
samem ateriale .g.w urtzite (hexagonal4) zincblende(c ubic)[ 72M8]; seea lso [60D2, 74F5,75W2].
A few sets of constants( KNb03 [72P5], triglycine sulfate [6162], 1,3,5triphenylbenzen[e7 3C12]) are
omitted becauseth ey violate the elastics tability conditionsf or crystals[ 57A2].
Incompletes etso f data are usuallyn ot includedi n the main tables,b ut partial setsf or materialso n which
completes etsa re not availablea re given in Tables1 0, 12, 16,2 0, 22,a nd 26.
Compositionsa reg iveni n atomico r mole% unlesso therwises tated,b ut uncertaintye xistsi n somei nstances
becausea uthorsd o not alwayss tateh ow their compositionsa re expressed.
Considerationh as beeng iven in the literature to factorsa ffectingt he accuracyo f measuringa nd deriving
elasticc onstantsE. rrorsa risingf rom crystalm isorientationin wavespeemd easuremenotsf elasticc onstantsh ave
beenc onsidered[ 59Wl, 88111t,h e effecto f phases hift of elasticw avesd ue to the transducera nd bond in
experimentaal rrangemenths as beene valuated[ 77D6], and the convergencien the calculationo f the elastic
stiffnessefsr om wave velocitiesi n cubic crystalsh as been examined[ 80NYJ.O ptimal methodsh ave been
discussedfo r determiningt he elasticc onstantsf rom wavespeemd easurementins symmetryd irections[ 89L2]
and in arbitrary non-principalp lanes[ 82C5,8588,89Cl, 89C2].A theory of the acousticm easuremenotf the
elasticc onstantso f a generala nisotropics olid has beenp ut forward [86V3].
Ref. p. 5761 1.1 Introduction 9
1.1.9 Arrangemenot f tablesa nd graphs
Tables3 . . . 27 contain the elasticc onstantso f the materialsc lassifiedb y the symmetrys ystemt o which they
belong at room temperatureu nlessi ndicated otherwise.W ithin each table, the substancesa re arranged
alphabeticallyb y their Englishn ame.B ecauseo f the largen umbero f substancerse presentedth, e cubics ystemis
subdividedI.n somec asesth e exactc lassificationis not certain,b ut any uncertaintyc anb e removedb y freeu seo f
the substancein dex (ch. 2.6):Unlesso therwises tated,a ll elasticc onstantsa re givena t room temperatureR, T,
(= 300K ). Tables2 8 . . .44 containt he temperaturec oefficientsa,n d Tables4 5 . . . 53t he pressurec oefficientisn
the sameo rder as in the respectiveT ables3 . . .27.
The compositionss, ynonymsa, ndp iezoelectrisct atuso f the materialsa reg iveni n Tables3 . . .27, but aren ot
repeatedin Tables2 8 . . .53, nor in generalo n the graphs,s o for full information,r eferencem ust be madet o the
relevante ntry in Tables3 . * .27. Most of the chemicafl ormulaef or mineralsa re takenf rom the list in [69H2];
variantso f someo f them can be found e.g.i n [73dl].
The graphs( Figs.3 .1. . .52.l ), follow the sameo rdero f arrangemenat st he substanceisn Tables3 . . 52,a nd
their numberingi ndicatest he tablest hey are associatedw ith. All graphsa re plotted to a left-hands caleu nless
a right-hands calei s specifiedm; ost of them refert o the stiffnessecsp av s.t emperatureT , but somea re included
showings tiffnessevss . pressurep , temperaturec oefficientsT c,, vs. T, compliances pav s. T, as well as a few
miscellaneourse lationshipsn ot includedu nder theseh eadings.
Additional informationr egardings omeo f the graphsa nd tabular entriesi s conveyeda sf ollows:p arentheses
( ) arounda figurei ndicatet hat therei s somed oubt abouti t; a questionm ark indicatesth at somen umericadl ata
are available,b ut that a definitev aluec annot be obtained;n o entry indicatesn o data available;a dashedli ne
(-----) on a graphi ndicatess omee lemenot f doubt or conjecturea boutt he relationshipc, ausede .g.b y errorsi n
readingf rom small-scaleg raphs,o r by error propagationi n convertingf rom stiffnessetso comphanceasn d vice
versa.
1.1.10N otes on bibliography
The bibliographyc ontainst wo setso f references“G: eneral references”,.identifiebdy a lower-casele tter,w hicha re
to sourceso f backgroundin formations ucha s textbooksa nd reviewa rticles,a nd “Special referencesi”,d entified
by a capital letter, which supply information on specifics ubstances.
Referenceqsu otedi n the tablesa red ividedi nto the two categories“M: ain referencesa”, nd“ Other references”.
The entriesi n the tablesa re basedo n the “Main references”a, nd the “Other referencesd” irect attention to
additionali nformationo n the material.O ther classeso f papersu nder“ Other referencesin” cludet hosec ontain-
ing doubtful data,a nd partial setso f constantsfo r materialsf or whichc ompletes etsa rea vailableT. his reference
classificationis also usedo n the graphs.
The data from Chaps.1 of Landolt-BornsteinN, ew Series,V ol. III/11 and Vol. III/l8 have largely been
retainedi n the presentc hapter.T o keepw ithin the ambit of “crystals” hash owevern ecessitateodm itting data
from thosee arlierc hapterso n anisotropicb ut non-crystallinem aterialss ucha s biologicalm aterials,p olymers,
and compositese tc.( poledc eramicsb einga notablee xception)T. he tablesa nd figuresh aveb eene xpandedw ith
newe xperimentadl ata that hasb ecomea vailables incet he last edition. Most of the “Other referencesf”r om the
earlier chaptersh ave beend eletedf rom the tables,f igures,a nd bibliography,a nd their referencen umbersl eft
vacant.N ew referencehsa veb eenn umberedc onsecutivelwy ith the earliero nes.M any papersp ublishedb efore
1956a re cited indirectly through the reviewa rticles[ 46hl, 52hl,56hl], but othersa re cited directly if required.
The literature was searchedto early 1990,b ut somel ater papersa re included.
Russianr eferenceas re to the original journal of publication.M ost of thesea re availablei n translationa s
showni n the list below.F or mostp apers,t he volumen umbera nd yearo f the translationa ret he samea st hoseo f
the original, but the pagen umbersd iffer.
Russianjo urnal Translationjo urnal
Akust. Zh. Sov.P hys-Acoust.
Defektoskopiva Sov.J . Nondestr.T est.
Dok. Akad. Nauk SSSR Sov.P hys.-Doklady
Fiz. Met. Metalloved. Phys.M et. Metallogr.( USSR)
Fiz. Nizk. Temp. Sov.J . Low Temp. Phys.
Fiz. Tekh. Poluprovodn. Sov. Phys-Semicond.
Fiz. Tverd. Tela Sov. Phys.-SolidS tate
Land&-Bhstein
New,Series II1/29a