Conadian Minerologist
Vol. 23, pp. 273-285(1985)
THE RELATIONSHIPBETWEENMINERAL HARDNESSAND COMPRESSIBILITY
(OR BULK MODULUS)
RONALD J. GOBLE
Departmentof Geolog/, Universityof Nebroska-Lincoln,433Morrill Holl, Lincoln, Nebraska68588-0340,
U.S,A.
STEVEN D. SCOTT
Deportrnentof Geologlt,Universityof Toronto, Toronto, OntarioMSSIAI
AssrRAcr
nir Ia duret6 H commedeuxidmed6riv6ede U par rapport i la distanceinteratomiquer, H: (&U/M)ro.
Hardness has been defined as "the resistanceof a
(Traduit par la R6daction)
mineral to scratching." Compressibility is the isothermal
changein volume of a compound with pressureand is
Mots'cl4s: duret6, compressibilitd, module de compresdependentupon many of the samefactorJthat determine
sion, 6nergiedu r6seau.
hardness. Bulk modulus K, the reciprocal of compressibility, has been defined as the second derivative
of lattice elergy U with change in volume,
INTR'DUCTI.N
K = V(d2UldV2)u^.Severalauthors have demonstrated
an empirical corr'elation between hardnessand comNumerous attempts have been made to derive an
pressibilityfor specificgroupsof minerals.Examination
empirical
relationship between hardness and various
bt data roi st soiidr tepiesentingI I structuretlpes shows
1926,
thiscorrelariontobeindirect.Thehardnessofiiubstancephysigal parameters (e.9., Friedrich
isdependentuponthesquareof theinteratomicdistance Goldschmidt 1926, Plendl&Gielisse 1963, Povarenwhereascompressibilityandbulkmodulusaredependent nykh 1964, Julg 1978). Most of these equations
upon the cube of the interatomic distance. Dependence incorporate several coefficients that are difficult to
upon valence is more complex, with the data examined determine for a particular
substance. Similar
falling into two groups, the Alkali Halide Group and the
attempts at empiricai correlations between compresSulfide-Oxide Group. Empirical equationsdevelopedfor
sibility or bulk modulus and crystal parameters have
the two groups suggestthat hardnessshould be defined
been
more successful
Bridgman
as the secondderivative of lattice energy with changein
-(e'g',
,1923, Ander-
son & Nafe 1965' D'L' Anderson 1967' Anderson
& Soga t967, O.L. Anderson 1970, 1972,Plendl &
Keywords:hardness,
compressibility,
bulk modulus,lat- Gielisse1969, 1970,Anderson & Anderson 1970).
tice energy.
Several investigators have demonstrated empirical
correlations between hardness and compressibility
(e.g.,Plendletal. 1965,Beckmannl9l, Scott1973),
S.MMATRE
suggesting that it might be possible to derive an
empirical equation relating hardnessto compressiLa duret6d,un mineralmesuresa rdsistance
d l'6ra- bility and' therefore' to other more readily deterflure. La compressibilite
d'un compos6,
le changement
The presentstudy was
isotherme
desonvolumelorsqu,ilestcomprim6,
d6pend 3i:Llp:q^parameters'
de plusieursdesfacteursqui ddterminent
la duret6.Le undertaken to test the empirical hardnessmodulede compression
K, qui estI'inversede la com- compressibility correlation of earlier workers, to
pressibilit6,correspond
autauxdechangement
del,6ner- determine its theoretical basis and the possibility of
gie U du r6seauavec le changement
en volume: usingitinapredictivefashion,andtodeterrninean
K : V(d2uldv2)yn,
Plusieurs
auteursont 6tabliunecor- empirical equation for hardness.
r6lationempiriqudentreduret€et compressibilit6
pour
certainsgroupesde mindraur.Un examendesdonndes CnySfaf-CfmnfcAl FACTORS
AFFECItr{c}IApONSSS
pouf 8l composds
solidesqui representent
ll typesde
structuremontrequecettecorrdlationestindirecte,La
Hardnessis defined as "the resistanceof a mineral
.
oureleq'une suDsBnceoepenq
interatomicdistance,H = aatur;t-'
ou c:lrrede la custance
rnteratomique,tandis que compressibiliteet module de compression d6pendent-du cubi de cette distance. La relation avec la valenceest plus compliquee;les donndes
ddfinissent deux groupes, celui des halog6nuresalcalins
et celui des sulfures et oxydes. Des relations empiriques
pour les deux groupesfont penserque I'on dewait d6fi-
to scratching"
@ates & Jackson 1980) or as
"resistance to plastic deformation, usually being
indentation" (Thrush 1968). It is a complex but
poorly defined property dependent upon such factors as the nature of the chemical bond, interatomic
distance, valence, atomic density and co-ordination
273
274
THE CANADIAN MINERALOGIST
nurrber. Several methods for approximating the
hardnessof solids are in common use. The Mohs
hardnessscaleranks minerals in the order of their
ability to scratch one another; numerical values of
hardness,r€mgrngfrom I to 10, havebeenassigned
to certain index-minerals.The technicalscaleis similar and uses many of the same index-minerals,
although the range is increasedto I to 15. The
Vickers and Knoop techniquesof measuringhardness€uemore precise;one measuresthe size of an
indentation produced by a diamond stylus under a
set load. Other methods include Shore hardness,
basedupon measurementof the height of rebound
of a steelball or hammer dropped upon the specimen, polishing hardness and Brinnell hardness.
Becausethe Mohs hardnessand Vickers hardnessare
the most commonly usedby mineralogists,we have
not consideredthe others.
Many hardnessdata are expressedas either Mohs
or Vickers hardnessbut not both, so that conversion from one scaleto the other is necessarv.Two
U/Y =24(Hn-2.7) for "hard-core" solids,
4<H-<9,
U/Y = 48(H--3.3) for "soft-core" solids,
H- > 4, and
U/V : % H^3 for all solids, H* ( 4.
at,
att
UJ
z.
o
OE
s
U'
TE
lrl
Y
c)
t
schemesof conversion,by Beckmann(1971)and by
Westbrook & Conrad (1973), are shown in Figure
l, togetherwith publishedexamplesfor which both
Vickers and Mohs hardnessare known. The curve
of Westbrook & Conrad fits the data better and was
used in our study. Whereasthere is good agreement
in Figure I for the Mohs index-mineralsupon which
the empiricalcurvesare based,thereis considerable
scatter of the data points (up to 3 H. units) for
non-indexminerals.This is at leastpartly dueto the
fact that the Viskers-hardnes data that we haveused
are averagein rangesof hardnesslisted in the source
fiterature and, overall, are for different loads (in
many casesnot included in the sourcedata). For this
reasonwe have used one type of hardnessdata (FI-,
Mohs) where possible.
Severalformulae have beenproposed for estimating hardnessfrom basiccrystal-chemicalparameters.
Most contain terms involving valence and mean
interatomic distance, as well as various constants
related to such factors as the crystal structure, the
covalencyof the bonds, the co-ordination number,
and atomic number. None of the formulae proposed
to date are particularly useful in predicting hardness
becausemost contain terms that are variable with
structure or bulk composition. In an attempt to circumventthis problem, Plendl & Gielisse(1963)proposedthat hardnessis equalto the volumetric cohesiveenergy,U/V. In the caseof Mohs hardness,they
found that:
fEsTBnooK 6co{MD
(|t7t)
(la)
(lb)
(1c)
U/V is given in kcal cm-'. They defined as "hardcore" those binary solids either of the AB type in
which A or B (or A and B) consist of elementsof
atomic number lessthan 10 or of any other type in
which one element has an atomic number lessthan
l0 and the other elementhas an atomic number less
than 18. "Soft-core" solids include all others.
Cnysr^q,L-CHEMTcAL
FACTonsAFrsc"uNc
CoMpnrssnrr,rrY AND Bun Monulus
Compressibilityis the isothermalchangein volume
of a compoundwith pressure,B = -(l/VXdV/dP)r
2468tO
(Birch 1966),and is dependentupon the samefacrcHS HARDNESS
tors that determinehardness.Compressibility can be
measureddirectly from the changein volume with
Ftc. 1. Comparison of Mohs and Vickers hardness data
( + ) of Tables l, 2 and 3. The correlationsof Beckmann confining pressureor can be computed from elastic
(1971),VHN: 86.3- 90.9Hn + 34.6H;2, and West- constants. The bulk modulus is the reciprocal of the
brook & Conrad (1973),VHN = 5.25 g.z.r3, for the compressibility,K = t/9, and is more useful than
Mohs index minerals (O) are showu as solid and dashed compressibility in the development of predictive
lines, respectively.
models.
275
THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY
The systernaticsof elasticconstantsweredeveloped (1965),D.L. Anderson (1967),Anderson& Soga
by Birch (1960),who demonstratedthat the density (1967), Anderson & Anderson (1970), and O.L.
and the mean atomic weight are the only variables Anderson (1972)haveevaluatedequations(2b) and
of importance in describingthe compressionalveloc- (2c) for a number of different elementsand types
ity of sound of rocks and minerals.Severalempiri- of compounds.
cal formulae have been developed,basedupon the
relationshipbetweenbulk modulus and the inverse
fourth power of the interatomic distance(e.g., Bridganp
BET'$T,EN
H-en-uwess
RELATToNSHIp
man 1923,Anderson & Nafe 1965,D.L. Anderson
CoMpREssrBrlrrt(oR But- Monulus)
1967,Anderson& Soga1967,O.L. Anderson 1970,
1972,Anderson& Anderson 1970,Plendl & Gielisse
Beckmann(1971)demonstratedempirical corre1969, 1970). A summary paper by Wang (1978) lations between hardness and compressibility for
showsthat:
specificgroupsof minerals.Scott(1973)useda similar empirical correlation for sulfide minerals to
K : V(d'zUld\p)v^:Y(dzU/df)(dr/dVE,^
Q-a) predict the compressibilityof troilite. In contrastto
"
: (AZJ,"*(n-r))/(9You\
Beckmann,who fitted straight linesto the data, Scott
eb)
fitted a curve. Plendl et al. (1965) expressedthe
where,4 is the Madelung constant, Z. the anionic empirical correlation betweenhardnessand compresvalence, Z" the cationic valence, e the electronic sibility as:
charge,n the repulsiveforce parameter,r the mean
(3)
atomic radius and Vo the specific volume. If
l/0 = p/V(Zm/q):H/(Zm/q)
AZuZ,"*(n-l),/ro in equation (2b) is assumedto be
constant for isostructural compounds,then:
whereB is the compressibility,U/V the volumetric
cohesiveenergy, H the hardness,Z the maximum
KVm: constant
Qc) valence,m the number of componentatoms, and q
the number of atomsper molecule.If the Mohs hardwhere V- is the molar volume and 16is the interanessfrom equations(la), (lb), (1c) is substitutedin
tomic distance at equilibrium. Anderson & Nafe
equation (3), bulk modulus is usedin placeof inverse
rngle i.
o U L KM o D U L u s ,
D A T Au s E Dr N D E T E R ! 4 r N tTNHGEc o N s r A N T _ E M p r R t R
p sM o N B
c AELL A T l o N s H r A
H A R D N E SM
S ,O L A R
MOHS
V O L U MAEN DV A L E N CFEO RT H ES U L F I D E - O X I DGER O U P
STRUCTURETYPE 'IZ
arc'I
ANDCoMPoUND
AgBr
AgCI
ZnS(c) Asl
CuBr
C U CI
Cul
N a CI
Ca0
Co0
Hgs
N a CI
Mso
Mn0
Pbs
PbTa
Z n S ( c ) CdT€
HgTe
ZnS
ZnSe
ZnT6
Z n S ( h ) B€0
cds
CdSs
ZnO
ZnS
Z n S ( c ) 1/zctFes2
TaC
N a CI
Tlc
TIN
uc
ZrC
ZnS(c) slc
2
2
2
2
2
2
2
Vm
(cn3/mote)
U
(kcal,/mole)
24.9
25.7
4 1. t
?7 .7
21.8
tt.4
16.8
l1 .5
24.5
11.2
lt.2
,1.4
40.4
20.'l
4 1. O
40.0
23.8
27.4
34.0
8,2
50.1
33.8
( 2 1 7'
(219'
(21t '
21.7
22.0
12.1
1t.5
14.4
15.4
12.4
(25't '
(900)
(987)
a 75 4 '
( 958)
(95J)
(810)
t763'
(866)
(655'
(662','
(788)
1727'
(686)
( 1074)
(7 46)
(705'
(96t)
(1A4t
( 2 7 6 7t
(2500)
(t1 92)
(2100,
K
(Mbar)
Hg
0.407
2.5
0.442. 1.75
!.5
0.24t
2.4
o.t45
0.398
2.25
0.t55
2.5
4.5
1.tJ6
1.452 (4.8)
0.230* 2.5
f;518
5.tt
5.5
1. 5 4 1
2.65
0.609
0.408
5.0
3.5
0.424
o.424
2.4
o.4t7
2,5
t,75
0.76t
o,595
1,75
0.510 (2.6',)
2.494
8.5
0.613
3.2
0.517
3.O
1.437
4.5
o.767
t.7 5
o.775
t.75
9.0
2.169
9.0
2.401
2.941
9.5
't.6t4 (6.1)
2 .23',t
8. '
?.464
9.5
vHN,
(kglnm',
90
,2
74
6t5
580
76
| 080
225
a7
49
164
60
28
195
90
r 150
72
fi0
254
'178
214
1 497
1200
724
2789
2480
rur/23/4 yrftlt tr'lt
11.76
il.16
10.04
9. 55
9.47
1 1. 1 2
11.J5
12.66
, .90r
10.24
12.11
1t.28
9. 80
10.14
| 0.f,4
t 0.39
t0.80
9.69
'10.t1
1 2. 1 6
10,97
'|
0.79
.l.l.79
10.81
10.t4
10.20
10,27
I1.96
10 . 6 3
1 2 . ' t5
t0.E0
23.54
15.24
17.92
2 ' t. 9 7
1I.6t
25.92
19.59
15.40
t4.69
I 6.81
19 . t 5
' |6
.61
22.24
16.62
20.97
18.42
19.54
2 1. 4 7
18.7t
21,77
19.50
't9.75
'16
.3|
't9.49
18.54
20.05
r 8.82
19.21
16.42
20.88
20.20
Hn
(ob-pr)
0.5
-0.5
-0.1
0.5
-0.|
0.7
-0. I
-1.2
-0.8
-0.7
0.I
-O.4
0.4
-0.5
0.2
-0.1
0. I
0.4
-0.1
| .0
0. I
0.1
-0.8
-0.1
0.1
0.4
-O,2
0.0
-0.8
0.7
0.5
Valres ln brackets () are ostlmatsd.
A n o m a l o u sv a l u e s a r E l n d l c a t e d b y a s t e r I s k s r .
Hm (ob-pr) ls the dlflerenca botrsen the observsd Mohs hardnees €nd fh6t prsdlcfsd uslng equatlon (10).
276
THE CANADIAN MINERALOGIST
a
o
a
A
+
z
:tr
clr
I
-o.5
log K
Ftc. 2. Bulk modulus and hardnessof solids. The curvesshow correlationsfor sulfide data by Beckmann(1971)and Scott (1973)as well as a solution to equations
(6b) and (6c) with tr = ozos.
T A B T E2 .
D A T A U S E DI N D E T E R M I N I NT
GH E C O N S T A NET M P T R I C A L
R E L A T I O N S H I PASM O N G
BULKMODULUS,
M O H SH A R D N E S SM, O L A RV O L U M E
AND VALENCE
F O RT H E A L K A L I H A L I D EO R O U P
S T R U o T UT
RyEp E l z - - l s t l z e l
u
K
H6
^vm
( c m t m o l e ) ( k c a l . / m o t e )( M b a r )
A N DC o M P o U N D
N a CI
KBr
KCI
KF
KI
LIBr
Ltcl
Ll I
NaBr
N a CI
NaF
Nal
RbBr
R b cI
RbF
Rbl
BaS
CaS
P bS e
5r5
NlAs
FaS
ZnS(c, AlSb
0aAb
'GaP
54
t6
28
72
38
20
12
56
46
28
20
64
72
54
46
90
.
0€Sb
I nAs
lnP
InSb
z
2
2
2
2
3
l.A4
l.4a
1.32
2,22
1.52
1.16
1,00
1. 8 8
|.68
1.32
t.t6
2.O4
2.20
1.84
t.68
2.55
41,t
2t.1
51.2
25.1
20.t
9.8
52.5
t2.0
26.9
14.8
40.8
48.4
42.9
27,0
59,5
t9.3
27 .7
34,5
32,9
1 a, 2
14.8
27.2
24,4
t4.5
3t.'l
t0 .4
40.9
15 9 . 3
1 6 5, 4
'I
89 .7
|50.8
I 88.5
199.2
240,1
174.1
t 74 . 5
t8r.1
213.4
165.9
'|
53 .5
I 60.7
18t.6
' t4 5 . t
l7 25'
(765)
il l7t)
t 1 2 0 1'
(1t07)
( t20t)
fi 170)
il196)
(1 lt2)
0.151
0.t80
0.tzt
0.12o
0.257
v H N ^ K v . , / z 3 1 4 p - u ? / 3t z 2 l 3
*
"'
(kg/ nnzt
1o 6 ' 1 rp)
2,0
2,0
2,8
2,2
2.5
0.663 5,75
0 . 1 8 8( 2 . 8 )
0.| 88 2.45
0.244 2.O
0. 4 9 0 2 . 2 5
0.t6t 2.3
o , 1 3 4( 2 . 7 '
0.160 (2.6,
0.27t (2.5'
0.tlt (2.7)
o.545 '.0
0.4rt 4.0
0.3t9 2.7
0.417 3,3
o.746 4.0
0.59t (4,9'
o.156 4.5
0 . 8 8 7 5. 0
0.56t 4.5
0.579 (4.7',r,
0.725 t5.l'
0.467 '.8
83
82
6,54
78
105
128
165
88
AA
95
t30
80
76
52
2t4
400
717
942
420
t57
412
220
Valuos In brackets () are ostlnated.
Hm (ob-pr) ls tho dlfference botroen the observed Mohs hardness and that
7 ,46
6.38
5,45
6.r9
6.50
6.tl
6.02
6,56
7 .25
6.57
6.68
6.86
7,37
6.60
5.97
5.65
6 .86
6.79
6.88
6.85
7 .21
6,47
6.t9
7.35
6.t7
predlcted
15,45
15.16
11.21
14 . 1 7
11.12
| 9.26
17.17
15.25
14.72
15.6t
11 . 7 0
15.58
la
?t
-0.5
0.0
o.f,
-0.2
-0.2
0.6
0.4
0.0
-0.1
-0,t
-0.7
-0.5
n
a
17.68
12.29
15.88
17.t3
18.31
0.5
-0.6
0.,l
0.J
0.6
16.94
13.44
't7.37
0.'
-0.4
0.6
-0.6
-0.5
0.1
0.2
0.4
-0.t
t4.r0
t5.56
14 . 0 2
15.89
| 6 . 1|
t6.59
t5.0t
-o.2
ustng equafton (12).
277
TI{E RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY
compressibility, and the units axeconverted,we find
that:
listed in Tables I , 2 and 3, and plotted againsthardnessin Figure 3. The empirical correlations of Plendl
& Gielisse(1963)(equationsla, b, c) are shownand
K : [0.s74(H^-2.1)]/ (Zm/ q)
fit the averagedata but, for specific structure-types
("hard-core",4<H.<9)
(4a) and ionic chargeswithin thesestructure types, more
K : [. 147(H^- 3.3)]/(Zm/ q)
precisecorrelations are evident. For example,the
("soft-core", H. > 4)
(4b) alkali halides conform to the expression
K = [0.012(H^)3)/(Zm/ q)
whereasaccordingto Plendl &
U/Y:2.2(Iln-2.5)
(all solids,H.<4)
(4c) Gielissethe relationship should be U/V =H^t".
Thus, the empirical correlations of Plendl & Gielisse
K is given in Mbar-l. Similar relationshipscan be seemto be oversimplified.
derivedby cornbining Zhdanov's (1965)equation for
Equation 3 demonstrates.acorrelation between
the cohesiveenergyU of an ionic crystal:
volumetric cohesiveenergyU/V and bulk modulus
K. Equation 3 is shown in Figure 4 (solid line) for
(5)
U: -(Z;ZceNA/r)(r-r/n)
in which N is Avogadro's number, with equations
(la,b,c) and (2b).Ndrops out becauseof differences
in definition of volume, giving:
(9Kln) = 0.574(H^-2.7)
("hard-core",4<Hm<9)
(9Kln): 1.147(H--3.3)
("soft-core", H- > 4)
(9Kln) : 0.012(HJ3
(all solids, H. < 4)
log VHN
f .o r.5 2.o 2.5 3.O 3.5
2.5
(6a)
(6b)
f t.to_ct-stt""tu;A
+ z"l
(6c)
s z-:e
0
/l
Equations (4) and (6) are identical if (9/n) equals
(Zm/q).
+ Z=l
f
.
Z=2
We havecollectedpublisheddata for 8l solids in
order to test the derived relationships betweenhardnessand bulk modulus or compressibility (Tables l,
2, 3). Values of hardness are from Palache et ol. :)
urv- qa(Ht3gJ +^
(1944, l95l), Deer et al. (1962),Beckmann(1971),
I
Ivan'ko (1971)and Uytenbogaardt& Burke (1971). C'I
Bulk moduli and compressibilitiesare taken largely I
o O I u/V=24(Hn-2.71
from the compilation of Simmons& Wang (1971),
o..
supplementedby data from Birch (1966),Anderson
I
& Anderson (1970), Beckmann (1971), Anderson
u\/v.H:N/z
uru= urHM-2.s]
(1972),Scott (1973)and King (1977).Bulk modulus
.[
and compressibilifydata are averagedvaluesat room
temperatureand one atmospherepressure.The data
set is plotted in Figure 2 togetherwith Beckmann's
(1971)empirical correlation-curvefor sulfides and
disulfides, Scott's (1973)curvefor sulfides,and equations (6b) and (6c) with the value of n as calculated
for sphalerite,rzns:5.1. Although Scott's curve
givesa better "average" fit for the sulfidesthan does
Beckmann's,as singlecurve satisfiesall of the data
in Figure 2. On the other hand, there do seemto be
o,5
l.o
o.o
trends in the bulk modulus - hardness plots for
log HM
different types of materials. In order to explorethese
trends, we have concentratedon the relationships
shown for the NaCl and ZnS structures before Frc. 3. Hardness and volumetric cohesiveenergy for the
attempting to expand these relationships to other
NaCl and ZnS structures. Correlations predicted by
structural types.
Plendl & Gielisse(1963)with equations(la, b, c) are
The volumetric cohesive energies U/V for
shown by solid lines; a curve fitted to the alkali halide
data is shown as a dashedline.
materials having the NaCl and ZnS structures are
278
TI{E CANADIAN MINERALOGIST
materials with the NaCl and ZnS structures. As
predicted,there is a good correlation (R : 0.99, using
a log scale)that is largely independent of structure
type. However, for the alkali halides and those
materialswith divalent cations, a more precisecorrelation would be (U/V) orKr.1et0.03
(R = 0.997, log
scaleused),suggestingthat the factor n in equation
(6) lor 9q/Zm in equation (3)] is approximately
inverselyproportional to K%, that is:
llidt-siro;mEl
+ z" I
.
:1".
O
z"z
0 z"s
t?;s sficr!@]
. z. I
a z-2
or
o
L
*
2"3
l.
n=9q/Zmqr/(ZuZa4)%.
A)
llno tlttod
=
l?,ouno,*--.-?'
/
tt!
-ot .
o.5
- o.5
log
o.o
o5
K
AB-type compounds
Ftc. 4. Bulk modulus and volumetric cohesive energy,
compensatedfor valence, for the NaCl and ZnS structures. The solid line representsthe fit of plendl et al.
(1965).The dashedlines representleast-squaresfits for
alkali halide, Z = 1, and non-alkali halide, Z : 2, data.
T A B L Ef .
CaQ.
BaFZ
l.4l
CaFt
l.4l
cdF,
l.4l
PbF2
1.41
SrF:
l .4 l
r l r * -.
CaCi2
l.4l
MgFrl.4t
S U L FI D E - O XI D E G R O U P
Cu2O
Cu20
1.41
MgAt2o
Afeor
slq
caF2
'll$
-
Asis
r.4r
Fel0 4
M g A| 2 0 4
M nF e 2 0 4
NtFezOa
Ateoi
2.31
2.tl
2.51
2.31
2.45
FeTt0r
sto2'
Thot
uoz
Sn02
rloi
2,45
2,A3
2.83
2.85
2,8t
2.83
crioi
reio\
Tables I and 2 and Figures 5 and 6 illustrate the
,relationshipbetweenbulk modulus and volume for
AB-type compounds in the halite, sphalerite, wurtzite and niccolite structures. The relationships are
D A T A U S E DI N D E T E R M T N T N
TO
H E C O N S T A NET U P I R I C A LR E L A T I O N S H T PASI , I O N B
6ULKMODULUS,
M o H SH A R D N E S SM
, o L A R V o L U M EA N D V A L E N C EF O R A x B y _ T I p E C O M P O U N D S
sTRUcruRE
rypE
lz"l
A N DC O M P O U N D
Asis
Equation (7) suggeststhat in order to correct for the
effect of ionic charge,K should be multiplied by Z%
rather than by Z as was done by Plendl et ol. (1965).
In order to evaluatetheseexpressionsproperly, we
must first know the exact correspondencebetween
n and both r andZ. This relationshipis also important in empirical expressionsused for deriving the
bulk modulus (e.9., equation 2) and hasbeenstudied
by Anderson (1970) and Anderson & Anderson
(1970).
z.4s
2.45
Ymt*
(cn37note)
KH.
( M b a r)
,.n)llr,
18.0
12.3
' |t . 8
1 5. 7
1 4. 7
25,6
9.8
0 .567
0.862
0.89'
0.611
0.699
0.229|
0.869
3 .0
4.O
4.O
t.2
t,5
5.0r
5.0
2t.4
54.4
ll.l
9.9
I t.6
10 . 9
0.515
0.198
1.872
2.020
1.851
1.82t
2.421
2.237
2,124
r .786
0.f,80r
1.9t1
z,l2a
2.O41
2.101
3.75
2.25
6.0
8.0
5.75
5.0
9.0
8.5
6.0
6.0
7.0
6.5
6.0
7 .0
6.5
9.7
l0.t
'I
0.6
I I .'
13.2
12.,
10.8
9.4
rv'rz!
127
272
'1
.22
7.50
t6J
317
'5t . 7 8
.27
4.45r
6.02
205
42
575
1 585
2265
95t
6Jl
11 2 0
t078
777
98t
10.68
I t.09
t0.68
11,46
10.6t
10.51
| 1.08
10.98
1.97'
1.69
12,00
l0.tl
9.06
'I
n,ufti3
rz$
14.47
r5.07
t4.66
14.19
14.85
18 . 4 2 r
t6.19
24,15
t9.03
t7.09
2l .'t I
t6.86
14.o7
20.61
zt .28
15.43
't
5 ,9t
1 7, 6 2
lE.l5
l 5.99
17,10
14,48
,o!!o"r
-o,2
-0. I
-0.2
-0 .5
-0. I
0.8
0.0
-o.7
-0 .8
-l.8
0.6
0.8
-l .2
-0 .6
-0.4
-0.8
-2.1
Anomalo!sval!ss ars lndlcatsd
ymrr lndlcaf6s molar volume for a formula,unlt
a slnglo asi6rlsk r.
b a s s d u p o n o n e a n l o n ( e . o , l / 4 !Fye ? O a ) . b a s a z
superscilpt h€s the valus I for the Atkail H€ilds Group
and 3/4 for fhe sulfld6-6xlde gro'upi c ,iea rn ilts iuy h€s'fhe vatue I for the Atkalt
Hsllds oroup and
2/3 lor ths Sulflde-Oxlds.Group.
i{m (obs-pr) ts the dtflerencE betteen the obs€rved Uohs h€rdn6ss and
lhat predlcted ustng equstlons il0) and fi2).
279
THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY
SULFI DE. OXIDE GROUP
+ Z. I
o Z=2
E 2"4
2A
30
MOLARVOLUME
FIo. 5. Bulk modulus - molar volume data for the Sulfide-Oxide Group' The curves
representthe equation K = 10.85Z /V n wlth Z = | (+), Z = 2 (O)' Z = 3 (no
symbol), arrdZ:4 (tr). The equationdeterminedby linear regressionis in logarithmic form: log (KV- ) = 1.025+ 0.781log Z; R : 0.9765'
A L K A L I H A L I D EG R O U P
.l.t.l.t.tl
+ Z= |
o 2.2
L z=3
@
f
J
D
o
o ns
Y
J
D
o
3
2
I
l,l,l
lrl,
o
to
20
30
40
50
60
MOI-AR VOLUME
Frc. 6. Bulk modulus - molar volume data for the Alkali Halide Group. The curves
representthe equationK = 6.65 Z/Y nwith Z : I ( + ), Z = 2 (O), Z = 3 (A), and
Z = 4 (no symbol). The equation determined by linear regressionis in logarithmic form: log (KVr) = 0.821+ 1.010log Z; R: 0.9919.
280
THE CANADIAN MINERALOGIST
S U L F I D EO
. X ID E
+ Zol
o 2.2
E 2.4
lo
30
2Q
40
50
MOLAR VOLUME
Ftc' 7. Mohs hardness- molar volume data for the sulfide-oxide Group. The curves
representthe equation Hm = 18.60Z%/V m%with Z = | (+ ), Z = 2 (O), Z = 3
(no symbol), arrdZ:4 (tr). The equation determinedby linear regressionis in
logarithmicforml log (H.V.%):1.289+0.6261og Z; R --0.902t.
essentiallythose shown by Anderson & Anderson
The same groups were used in determining the
(1970),Anderson (1972), Plendl & Gielisse(1969, empirical relationship anxongMohs hardness,molar
1970)and Plendl et al. (1965).We have chosento volume and valence.The data for the Sulfide-Oxide
usemolar volume in placeof the specificvolume or Group are shown in Fiirure 7. The correlation can
cube of the interatomic distanceused by some of
be describedby the equation:
theseinvestigators.We have divided our data into
two empiricalgroups, the Alkali Halide Group and H.V-% = 19,16 Z"%
(10)
the Sulfide-Oxide Group. The group namessimply
reflect the dominant compounds in eachgroup and Again, the proportionality of H. and V--% was
are not intended to be exclusive;other types of com- determined separately,and a fit of log (H.V.%)
pounds are included in each group. Figures 5 and against log (Z) then made. The correlation coeffi6 show the relationshipbetweenbulk modulus and cient, though not as high as for the KY^ versusZ"
molar volume for thesetwo groups, as well as the data, is still over 0.90. In the caseof the Alkali Halide
equation fitted to the data by linear and nonlinear
Group, no immediate correlation of H.V-% viith
regression;logarithmic scaleswereusedfor thesefits.
Z" is apparent, the data for the alkali halides
For the Alkali Halide Group the data are described (2": l) showing considerablescatter.However, if
by the equation:
thesedata are plotted separatelyagainst total number of electronsT, in the formula, a correlation is
KVm = 6.65 Z"
(8) apparent (Fie. 8), defined by the equation:
For the Sulfide-OxideGroup the appropriateequation is:
KVm: 10.84Z"
(9)
In practicethe proportionality of K and l,/V- was
determined separately,and a fit of log KV. against
log Z" was then made. Correlation coefficients of
greaterthan 0.97 confirm that thereis a good correlation among bulk modulus, molar volume and
valence.
H.V.* : ll.4l + 0.30 T"
(lla)
This relationship was used fo define an "effective
valence" Z" for the alkali halides. This "effective
valence" was basedupon a minimum of 12electrons
in the alkali halides. Rearrangingequation (lla)
based on this l2-electron minimum leads to the
equation:
H-V.*
: 14.99 Il - (Te- 12)/ 50.211
(llb)
281
THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY
ALKALI HALIDES
40
q30
(\lg
tE
E
20
o
loo
80
20
40
60
TOTAL NTJMBEROF ELECTRONS
Ftc. 8. Dependenceof H.V.% on total number of electronsin the structurefor the
alkali halides. The line determined by linear regression is:
H-V.% : 11.41+ 0.299 (total number of electrons);R = 0.9096.
+ Z. l
o 2"2
d 2"3
30
v'',(a42/3
Frc. 9, Mohs hardness- molar volume data for the Sulfide-Oxide Group. Molar
volume data for the alkali halideshavebeenadjusted to compensatefor the differencebetweenthe effectivevalence,2", alLdthe true valence,Z.For the remaining compoundsthe effective valenceis equal to the true valence, and Z/2" eqaals
l. The curvesrepresentthe equationHn = 15,01Z/Y ^% $/ith Ze : | (+), Z" = )
(a), Z"= 3 (A), aad Ze= 4 (no symbol). The equation determinedby linear
regressionis in logarithmic form: log (H*v.'z/; =1.176+ 1.023 log zri
R:0.9399.
282
THE CANADIAN MINERALOGIST
where:
l-(T"-L2)/50.21 = "effective valence":2"
(l lc)
Values of Z" for the alkali halides are tabulated in
Table 2; LiF, with 12 electrons, has an effective
valenceof l. H- and Vm data for the Alkali Halide
Group are plotted in Figure 9; values^ofV. have
been adjusted by a quantity (ZJZ)% to permit
inclusion of alkali halide data; for the non-alkali
halides,Z":2". The empirical relationship for all
data in the Alkali Halide Group is:
H.V.'/
(r2)
: L5,38Z"
The correlation coefficient, using logarithmic scales,
is again greater than 0.90.
The empirically derived equations for both the
,Sulfide-OxideGroup and the Alkali Halide Group
are summarizedin Table 4. Equations (8), (9), (10)
and (12) havebeenrearrangedto put them into common formatsin equations(8a),(9a),(l0a) and (l2a).
As can be seen,for the Alkali Halide Group the term
(n 2",J3) is common to both the bulk modulus
and hardnessequations; "effective valence" must
be usedin placeofvdlencein the hardnessequation.
For the Sulfide-Oxide Grodp the term (24 Z) is
conmon to both equations,but the exponenton this
term differs for the bulk modulus and hardnessequations.
Combination of the empirical equations (8)
through (12) and (8a)through (l2a) resultsin equations (13), (l3a), (14) and (l4a) relating hardnessto
T A B L E4 .
bulk modulus for the Alkali Halide and SulfideOxide Groups. The approximateterm (9Vnv"/4) is
common to the equationsfor both groups. The equations differ in that the Alkali-Halide-Group equations have a term compensating for the different
exponentson the valenceterm for the hardnessand
bulk-modulus equations.The latter term may or may
not havephysical significance;regressionanalysisof
(H. /KVnl6) against Z" shows a dependenceon
valence,but the correlation coefficient is low. Equations (13)through (l4a) showthat, asnotedby Beckmann (1971)and Scott (L973),thereis a relationship
betweenbulk modulus or compressibilityand hardness.However, as is shownby theseequations,this
correlation is not direct but involves other terms as
well.
Ap;type
compounds
Initially, we restricted ourselvesto a consideration
of data for AB-type compounds.Table 3 showsthe
relationship between bulk modulus and molar
volume for A*B;type compoundsin the fluorite,
rutile, quartz, corundum, spinel,cuprite and argentite structures;the ratio x:y is not equal to l. For
the volume term, we have chosento usethe molar
volume basedupon one anion; for valencewe have
usedthe squareroot of the product of the average
cation and anion valences,Zo. With these definitions, {Br-type compounds fall into two gxoups
corresponding to our original Alkali Halide and
Sulfide-Oxide Groups. Table 3 lists observedvalues
of (KY^/Z) and (H.Vr%/Zr) for the Sulfide-
S U M M A ROYF E M P I R I C A LC O R R E L A T I O N S
A L K A LI H A LI D E OROUP
gquallon
K . 6.65 Z"/V,
tu\{,3ttz"lz"tx
= l g l q t r y_Lml 3 ) ( z / z , K
e c
9a
n
H m : 1 9 . 1 6z l l 3 1 Y ^ 2 1 3
1Z
z (gl4, (.2olt, 2el\ nz/3
e
e ( 2 4 z c r 3 l 4/ v
8a
15.3s ze1yfil3
Hm- 2.3r
K - t0.84 tf;latt^
8
a . ( 2 0 / 3 14 / l n
a^ .
S U L F I D E - O X I DOER O U P
€qu€ilon
nunDer
l0
z'r 1\(24,"r'/3 ttrtl'
12a
I oa
Hm. 1.7s n!13ttt/z"tL/12*
8+l 2+l 5
'l5a
'ts/ 4,tv
{3ttttzaz"tL/12
e+to+14
r
t4a
G E N E R ARLE L A T I O N S H I P S
equailon
K-v(d2u/dv2tyo"r,r.:':ro"r'rrn_iirritiri"r^
-
l A Z u Z " e, . . , . , - , , .
,
Bm" (AZe4o (n-l))/(4Vm)
Hn .
H .
(AzaA
(z4z
Jl/12,
"2{n-,,)/t{4vr)
(dqJldrz)
.Hm
ro
m
2a
2b
lt6+2b'156
I 4a+2b' | 5b
I t€+l 5br | 5
THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY
283
e sulfide-oxlde group
A olkoll holide group
g
o
+ \ry
o
rl
compounds
4'
3
3
3o
UJ
z
o
E
r-l
o
I
o
= -2
MOHSHARDNESS
Frc. 10, The difference between observedand calculated Mohs hardnessas a function of observed Mohs hardness. Data are shown for the Sulfide-Oxide Group
(e ), Alkali Halide Group (A), and for A*\-type compoundsin both groups (+)'
The curve was constructedthrough points determinedby averagingdata over onehalf unit intervals in Mohs hardness.
Oxide Group. As can be seen,thesevaluesare close
to the coefficientsin equations(8), (9), (10)and (12)
and are within the ranges for AB-type compounds
shown in Tables I and 2, with the exceptionof two
anomalousmaterials, CaCl2and SiO2.The fact that
quartz, SiO2, is anomalousmay indicate that this
type of correlation will not be applicable to silicates.
CoNcr.usroNs
Equations(8), (9), (10)and (12) demonstratethat
empirical relationshipsexist by which bulk modulus and hardnesscan be modeled for certain groups
ofminerals.Equations(8a),(9a),(10a)and (12a)further demonstratethat theseempirical equationshave
a number of terms in common; combiningequation
(8a) with (l?a\ to derive (l3a) and equation (9a) with
(l0a) to derive (l4a) demonstratesthe basis for
empirical correlations between hardness and bulk
modulus (or compressibility)establishedby previous
investigators. Equations (2a) and (2b) show that for
bulk modulus these empirical equations can be
relatedto a theoreticalconcept,.thatis, bulk modulus is proportional to the second derivative of lattice energy with change of volume, (d2uldvlv
Equation (2b) can be combined with equations(133)
and (14a) to produce equations(l5a) and (l5b) as
shown in Table 4; comparison with equation (2a)
suggeststhat the Mohs hardnessis proportional
10 the second derivative of lattice energy with
changeof interatomic distance,(&U/df),o. "Abso-
lute hardness" could therefore be defined by the
equation:
H:
@2U/df)roa Hm
(15)
rather than being defined as equal to the volunetric
cohesive0attice) energy,U/Y, as proposedby Plendl
ofhardnessupon
& Gielisse(1963).The dependence
16rather than upon Ve is a reflection of the directional nature of hardness;i.e., we are stretchingand
breaking bonds in one direction rather than compressing the structure in three direclions.
One of the purposesof this study was to check
the possibility of using the empirical Mohs hardness
- bulk modulus correlations to estimate compressibilities. Examination of the scatterof data in the various figures and tables shows that, in most cases,
application of the empirical equations relating bulk
modulus to molar volume would be a more accurate
method for estimatingbulk modulus. However, the
use of the equations relating hardness to molar
volume may lie in determiningin which group a compound belongs prior to applying the appropriate
equation relating compressibility to molar volume.
It is possiblethat the equationsderived could also
be usedto estimatehardness.Tables l, 2 and 3 and
Figure 10 show the deviationof the valuesof Mohs
hardnessestimatedby useof equations(10)and (12)
from the observedvalues. The spreadis generally
about one-half unit about the observed value, but
the predicted value is consistently low in the range
284
THE CANADIAN MINERALOGIST
6 to 7 and consistentlyhigh in the range8 to 10. In
kristallinenSubstanzen.
len un&nichtmetallischen
the range6 to 7 this is probably due to inaccuracies
Kristall und Technik6, 109-117.
in the data (most of the data points are for A"B"
compressionalwaves
compounds);in the rangeS to l0 the discrepaicj BIncH,F. (1960):'Thevelocityof
l0 kilobars. J. Geophys'Res' 65'
may be real, sincemany iuthors havesuggestAth;
1i.1.:H^.
rudr-rruz'
the Mohs hardnessscaleshouldbe expandedbetween
9 and l0'
(1966):compressibility:elasticconstants..rz
AcrNowr_rocEMENrs
We gratefully acknowledgethe work of Z.L,
Mandziuk who, in a studentproject at the University of Toronto, laid the groundwork for our
research.We also thank H.E. King Jr. and C.T.
Prewitt of the StateUniversity of New York at Stony
Brook for helpful discussionsfollowing an oral
presentationof our work at the 1978Joint Annual
Meeting of GSA/GAC/MAC in Toronto (Scott &
Goble 1978).Our researchwas supportedby operating grant A7069 to Scott from the Natural Sciences
and EngineeringResearchCouncil of Canada and
by a PostdoctoralFellowshipto Goblefrom N.R.C.
NegotiatedDevelopmentGrant D56 to D.W. Strangway on behalf of the Department of Geology,
University of Toronto.
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ReceivedJuly 31, 1984, revisedmanuscript accepted
January8, 1985.