American Mineralogist, Volume 6E, pages 220-226, 1983
A Brillouin-zone model for compositionalvariation in tetrahedrite
Manv L. JouNsoN
Department of Geological Sciences
Harvard University
Cambridge, Massachusetts 02 138
nNo RavuoNo
JeINI-oz
Department of Geology and Geophysics
U niversity of California
Berkeley, Califurnia 94720
Abstract
A Brillouin-zone analysis based on a nearly-free electron model for tetrahedrite, an
important copper antimony sulfide, successfullyexplainsthe observedcompositionallimits
for synthetic and natural tetrahedrites.Tetrahedrite should occur with greaterthan 2M and
up to 208 valence electrons per unit cell (filling the 52nd Brillouin zone); it should be most
stable with 208 valence electrons. The classical formula CurzSblSlr is not expected to
occur. This model predicts that filling a zone(e.g ., by substitutingFe or Zn for Cu) should
cause electrical resistivity to increase. This is confirmed by experimental results which
show that a tetrahedrite with about 208 valence electrons per unit cell (approximate
composition Curo(Fe,Zn)z(Sb,As)aSs)has a resistivity 107times higher than that of a
The
tetrahedrite with approximately 205valenceelectrons (composition Cutz rz Sba.6eS13).
Brillouin-zone model also agreeswith observedlimits to tetrahedrite composition: natural
tetrahedrite samplesare stoichiometric with compositionsrestricted to two divalent cations
per formula unit, while synthetic samples,which are not stoichiometric, have compositions
bounded by the line
CureSbnSrr- Cu12Sba.67S13,
as expected.
sitional data fit within the range Cu12SbaS13Introduction
Tetrahedrite is one of a family of common sulfo- CulaSbaS
; none of theseendpoints
13-Cu12Sba.67S13
(Tatsuka
and Morimoto, 1977; Makosalt minerals,containingCu, Sb, S, and sometimes is reached
As, Ag, Fe, Zn, Hg, Bi, Te, Cd, Pb, and Se. Its vicky and Skinner, 1978;see Fig. 1).
These variations can be accounted for by a simmineralogical significancederives, in part, from the
fact that it is one ofthe silver-orephasesofgreatest ple, ionic (integer chargeper atom) model, and such
economic importance. Natural tetrahedritesare de- a model leads one to predict colTectly some of the
scribed approximately by the chemical formula apparent limits on the compositional variation of
A16B2CaD13,
where A : Cu, Ag; g : Cu, Fe, Zn, natural and synthetic tetrahedrites. The fact that
Hg, Cd, Pb; C : Sb, As, Bi, Te; D : S, Se sulfides and sulfosalts are not ionic in character is
(Springer, 1969;Charlat and Levy, 1974;Cech and well established.however. We are therefore motiHak, 1979;Patrick, 1978;Johanand Kvacek, l97l; vated to consider an alternative, and physically
Sandeckiand Amcoff, 1981;Basu et a/., 1981).This more plausible approach, such as that of the Brilformula, while complicated, is stoichiometric. By louin-zonemodeldiscussedby H. Jones(1975).Not
contrast, synthetic tetrahedrites in the Cu-Sb-S only is this model, as applied in the present study,
system show considerablevariation in the number as easily used as the ionic model, but more signifiof atoms per formula unit (i.e., the number of atoms cantly, the Brillouin-zone model is of substantially
per 13 sulfurs). All of the room temperaturecompo- greater predictive value: (1) It successfullypredicts
220
0003--(XX)083/0
I 02-0220$02.
00
JOHNSON AND JEANLOZ; BRILLOUIN-ZONE
Table l Ionic chargeand number of valence electronsper atom
fonic
Elenent
Charge
per Aton
Cu
Ag
Fe
Zn
H9
Fg
Sb
As
Bi
Te
+I
+2
+2
Number of
valence Electrons
per Aton
I
I
2
+2
it
1
+3
+3
+3
+4
3
3
MODEL FOR TETRAHEDRITE
equipoint (Makovicky and Skinner, 1979; Kalbskopf, 1971).Yet there is an apparentlimit of two,
not six, divalent cations per formula unit (Charlat
and Levy, 1974),which satisfies the ionic formula
Ai0B3*Ci*D?t. e. copper may be -rl or *2,
the "classical" formula Cu12Sb+Srr(: Cuio
Cu3*SU?*S?3
) is permitted.
Synthetic Cu-Sb-S tetrahedrites
The overstrict interpretation "two divalent catper formula unit" is not applicablefor synthetions
s-2
6
-2
Se
ic tetrahedrites, as the occurrence of mobile ions
(Makovicky and Skinner, 1979) permits the total
number of atoms per formula unit (per 13 sulfurs) to
be variable. There still must be 26 positive charges
the observed compositional limits which are in
accord with the ionic model; (2) lt gives additional per formula unit. This gives an upper limit to the
compositional limits not predicted by the ionic number of atoms in synthetic tetrahedrite along the
which
model, including an accountingfor the inaccessibil- compositionline CulaSbaSl3-Cu12Sbr.ozSrr,
is
confirmed
(Fig.
by
experimental
observations
1).
ity of the "classical" (Pauling and Neuman, 1934)
tetrahedriteformula Cul2SbaS13
(AroB2C4Dr:,A :
Compositional lower limits on numbers of atoms
B : Cu, C : Sb, D : S) in the pure Cu-SLS
There is nothing in the ionic model which resystem; and (3) It predicts variations of physical
stricts
the number of (nonsemimetal)cations to 12
properties, such as relative resitivities, for some
per formula unit. Kalbskopf (1974) has
or
more
tetrahedrites,which cannot be determinedfrom the
reported
copper
vacancies in synthetic tellurium
ionic model.
end-membertetrahedrite, CureTeaSl3,and Makovicky and Skinner (1979)report partial vacancy of
Ionic model
the 12d equipoint in a synthetic tetrahedrite with
Balancing chargesis easily performed for natural compositionnearCu12.3SbaS13.
One might therefore
and synthetic tetrahedrite formulae. In particular, a expect a hypothetical Cu2* end-member tetraheformal charge is assigned to every element, as drite, Cu?+Sb+Srr,
which has not beenseen;in fact,
shownin Table l, except that Cu may be either +l
even the "classical" tetrahedrite composition Cul6
or 12. The two choices for the charge on copper Cu3*SboS3is not found in synthesisexperiments
give one degree of freedom (or ambiguity) for this (Fig. 1). The simple ionic model offers no insight
model, and lead to the compositional limits dis- here.
cussedbelow.
4
D
Nataral tetrahedrites: two divalent cations per
formula unit
In the tetrahedrite structure (143m,Z : 2), there
are two metal equipoints (l2d and l2e, each with
six-fold multiplicity per formula unit, with tetrahedral and trigonal planar coordination, respectively),
one semimetal equipoint (8c, with fourfold multiplicity, with triangular pyramidal coordination),
thirteen sulfur sites (equipoints 24g and 2a, with
tetrahedral and octahedral coordination, respectively) (Weunsch, 1964), and twelve "mobile ion"
loci (equipoint 24g,roughly three-coordinated)(Makovicky and Skinner, 1979). Nowhere is there a
metal equipoint with multiplicity of two or less, and
divalent cations seem to congregate in the l2d
Zone model
History
The Brillouin zone concept was developed by
H. Jones from the discovery by Hume-Rothery
(1948)that the ratio of valence electrons to atoms
was constant for many alloy phaseswith the same
structures(see review of Jones, 1975).However,
Hume-Rothery has pointed out that "the number of
(valence) electrons per unit cell is more fundamental than the number per atom" (Hume-Rothery,
1963,p. 330), becauseit is the properties of the
fundamentalrepeat unit of the crystal (the Brillouin
zone in reciprocal space)that determinesthe bonding states.Furthermore, as the number of atoms per
222
JOHNSON AND JEANLOZ: BRILLOUIN.ZONE
MODEL FOR TETRAHEDRITE
TM 1977
Qrz$e.szsu
Cu14Sb4Sts
47
t?
Fig. L Tetrahedrite stability field at room temperature, data from synthesisexperiments. Portion of the Cu-Sb-S triangle (see
insert): Cu at lower left, Sb at lower right, S at top. Tie lines to other phasesnot shown. (a) Data of Tatsuka and Morimoto (1977);(b)
Data of Makovicky and Skinner (1978).
there is only one value for the number of valence
electrons for copper, as we assume that the filled
copper 3d shell contributes no electrons. (We have
thus stripped away the singledegreeoffreedom that
the ionic model contained; we will replace it with a
more restricted variation, that of the total number
of valence electrons per unit cell.) The number of
valence electrons per unit cell for some synthetic
and hypothetical tetrahedrite formulae zre shown in
Table 2, with the number of Cu+ and Cu2* atoms
Calculatio n fo r t et ra hedrit e
per formula unit for the ionic model listed for
The first Brillouin zone for a compound consists comparison. All of these formulae are allowed by
of the volume closer to the origin than to any other the ionic model.
point of the compound's reciprocal lattice. Since
On inspection, it is evident that formulae for
tetrahedrite has space grolp 143m, its lattice is body- which tetrahedrites are found correspond to over
centered, and the (non-primitive) first Brillouin zone 204 and up to 208 valence electrons per unit cell.
is bounded by the dodecahedron(110) (see Jones, 204 valence electrons completely fill the 51st Bril1975,p. 55-59).This volume is filled by 4 electrons, louin zone (51 times the volume of the first Brillouin
counting spin multiplicity.
zone), and 208 valence electrons completely fill the
The numbers of valence electrons per atom for
52nd Brillouin zone.
the elements in tetrahedrite corresponds in most
We recall that in order for a substanceto be an
(see
casesto the formal chargesof the ionic model
insulator or semiconductor, its number of electrons
Table 1); the differences are that sulfur, with a per unit cell must completely fill an integral number
charge of -2, has 6 valence electrons, and that of multiples of the first Brillouin zone. Otherwise,
formula unit is variable for tetrahedrite, we shall
follow his suggestionhere.
Subsequently, A. J. Frueh reported Brillouinzone calculations for chalcopyrite, bornite, digenite, chalcocite,covellite(1954),stromeyerite(1955),
and argentite (1961).Also, Hall (1972)contended
that a zone model could explain the limits of iron
and zinc substitution into tetrahedrite, but provided
no details.
TOHNSON AND JEANLOZ: BRILLOUIN-ZONE
Table 2. Composition and existenceof synthetic tetrahedrites
Zone Model:
Ionic Model:
Valence
A t o n s -DUenr i L
Found? Notes
Electrons
Fornula
per Unit Cel1 (Cu.Ag).r Cu-z
.
uonPosltron
rilaj2cj3o;l
204-20A
10
0-2
208
10
0
yes
b
CuI2.23Sb4.O8SI3
2O4.9
10.70
1.53
yes
c
cur2.43sb4,34sl3
206.9
11.88
0.55
yes
o
cu13.?75b4.O85l3 208.0
L3.77
0.0
yes
e
ru
z
no
culoFe2sb4sl3
cu12sb4sr3
204
cu?sb4st3
194
culoTe4sl3
2OB
a
t
OTno
t0
I
0
yes
n
Notes to Table 2:
a ceneral
b
"ionic"
nypothetical
fornula
for
endnenber,
natural
pauling
c cu-poor
tetrahedrite,
Tatsuka
d Sb-ri.h
tetrahedrite,
ibtd.
e Cu-rich
tetrahedrite.
ibid.
f-suggested
(1934)
I
tetrahedrite
Hypothetical
tetrahedrites
and Neunan (1934)
and Morinoto
fornula,
cu+2 tetrahedrite,
pauling
(19?7)
and
Neunan
MODEL FOR TETRAHEDRITE
223
lence electrons per unit cell (see Table 2), and is
thus a compositional limit for the Brillouin-zone
model as well. (In a semiconductor, electron states
are roughly continuous in energy up to the band
gap, where there is a Brillouin-zone boundary. We
assume a nearly-free electron model, which is to
say that azone fills completely until its boundary is
reached, and there is an energy barrier for tetrahedrite at the 52nd Brillouin zone boundary-208 valence electrons per unit cell.) Similarly, the line
CulaSbaS
limiting the composition
13-Cu12Sba.67S13
of synthetic tetrahedritesis the 208 valenceelectron
contour (Fig. 2); so that both models agree on this
(observed)limit as well.
As another example, synthetic tellurium endmember tetrahedrite CulsTeaSl3may be explained
as CuloTef*Sf3-, or again as having 208 valence
electrons per unit cell (Te contributes 4 valence
electrons due to the stabilization of the 5s shell.
analogousto Sb).
Re sistivity measurements
While resistivity measurementsare not directly
convertible to band gap values, it is possible to
predict relative resistivities of samples for which
the 52nd Brillouin zone is unfilled (metallic behavthere are accessible(conduction) states into which ior) and filled (semiconductor behavior). Samples
electrons may be excited by the addition of small with less than 208 valence electrons should have
increments of energy, and the compound is metal- lower electrical resistivities than samplesin which
lic.
the zone is filled, since electrons must cross the
band-gap energy barrier (a statistically unlikely
Large zone polyhedron
event at room temperature)in order to contribute to
While it has been common practice to determine conduction in the latter solid. The electrical resisthe shape of the polyhedron in reciprocal space tivities of a synthetic tetrahedrite (grown according
filled by the valence electrons of a compound (e.g. , to Tatsuka and Morimoto, 1973)with composition
Frueh, 1954,1955,1961;Jones,1975),sucha deter- Cu12.12Sba.seSrr,
and a natural tetrahedrite from
mination is based on the assumption that electrons Horhausen, Germany (Harvard specimen #82561,
and X-rays are Bragg-scatteredequivalently. This microprobe composition Cue.eaAg.srFes.BZn1.77
were measuredby using a twoassumption has been shown to be incorrect for Sb3.e7Ass.11St3.s),
diamond structure compounds, for which more point probe. For both samples, resistance was
exact calculations have been carried out (seeHarri- measuredwith a laboratory-built Cross-Correlator
son, 1979,p. 44G447). Hence the determination of (describedStratton et al., 1981),employing platia polyhedron is no longer widely employed (e.g., num-grid contacts attached with silver-epoxy paint
gamma brass structure, Pearson et al., 1976) and to freshly scraped surfaces on the sample. Sample
surfaceareasand lengths were measuredvisually at
will not be described here.
70x using a binocular microscopewith a calibrated
Discussion:predictions and comparisons
eyepiece.The first sample,with 204.8t0.1valence
electronsper unit cell, has a resistivity less than 3 x
Comparison with the ionic model
l0-s ohm-meters (indistinguishable from a short
The ionic model predicts a two-divalent-cation- circuit on the Cross-Correlator) compared with a
per-formula-unit limit for the compositions of natu- resistivity of 375 ohm-meters (i.e., at least 107
ral tetrahedrites; this limit corresponds to 208 va- higher)for the secondsample,which has 207.8-+0.3
h Synthetic
tellurian
tetrahedrite,
this
paper
xalbskopf (1974)
224
MODEL FOR TETRAHEDRITE
JOHNSON AND JEANLOZ: BRILLOUIN.ZONE
or16
Cupsb4.6?St3
J4l
l6
Fig. 2. Figure I contoured for number of valence electronsper unit cell, tetrahedritestructure. The number of electronsincreases
toward the lower right corner. (a) and (b) as in Figure l.
valence electrons per unit cell. This result confirms
the behavior predicted by the zone model.
Stability
It takes little energy to add electrons to tetrahedrite up to the 208 valence electron limit, where
there should be the greatest number of states per
unit energy(seeMott and Jones, 1958,p. 171).The
effect of this increasein the number of statesas the
band edge is approachedis to minimize the average
energy increment per valence electron added. This
is seen in Figure 3, where the average energy it
takes to add an electron to the 52nd Brillouin zone
of tetrahedrite is plotted as a function of the total
number of electrons. Since it takes less energy to
add an electron close to thd band edge, it becomes
energeticallyfavorable for synthetic tetrahedritesto
exsolve into high-electron (208 valence electrons)
and residual (nearly 204 valence electrons) tetrahedrite phases,consistent with the results of Tatsuka
and Morimoto (1977) and Makovicky and Skinner
(1e78).
Similarly, we would expect natural tetrahedrite
compositions to cluster around the 208 valence
electron value. This appears to be the case. For
Table 3, we have calculated the number of valence
electrons per unit cell from 121 microprobe analysesof natural tetrahedritesreported in the literature
(Springer, 1969;Charlat and Levy, 1974;Cech and
Hak, 1979, Pattrick, 1978), normalized to 4 semi-
c
o
o
ol
6 i'r
o
I
o
o
o
ct
o
o
o
o
o
c
CD
o
c
UJ
1,,
204
205
206
207
208
N u m b e ro f V o l e n c eE l e c t r o n sp e r U n i t C e l l
Fig. 3. Increase in the electronic contribution to the total
energy per valence electron (in units of the Fermi Energy, Er)
plotted againstthe number of valence electrons per unit cell for
the 52nd Brillouin zone. The solid curve is valid at 0 K (static
lattice) for an idealizedparabolic distribution of valenceelectron
statesas afunction ofenergy, given a single-valuedcutoffenergy
Ep (free-electronmodel). The dashedcurve (where ditrerent) is
calculatedfor kT : 0.1 Ep, or a temperatureof 472"C(near the
melting temperatureof tetrahedrite) for a 1.0 eV band gap. The
minimum energy for this curve occurs at slightly more than
207.7Evalence electrons per unit cell.
.IOHNSON AND JEANLOZ: BRILLOUIN-ZONE
Table 3. Natural tetrahedrites:number of valence electrons per
unit cell
Nunber of
Valence
Electrona
per unit
Ce11
Nunber of Analyses
Sprlnger
(1969)
< 204.0
204.0-204.5
2 0 4. 5 - 2 0 5 . 0
2 0 5 .0 - 2 0 5 . 5
205.5-206.0
206.0-206.5
206.5-207 .0
207.0-207 .5
2 0 7. 5 - 2 0 8 , O
208.0-208.5r
> 208.5t
I
4
o
)
I
2r
TotaI
*These
s train
I
0
1
0
conposltions
Charlat
a
Levy
(1974)
Cech
Sandecki
&
PattrickEet._Eak
Arcoff
(1979) (1978)
(1981)
Basu
Iocal
al.
(198r)
0000
001r
0000
0010
r1.20
5053
5046
L2017
23014
5000
10r0
I
J
t8
37
8
53L162l
also
violate
t2l
ionlc
nodel
con-
ts.
metals per formula unit (13 sulfurs per formula unit
assumed).Consideringthe variety of sources,the
clustering of tetrahedrite compositions about 208
valence electrons per unit cell is quite pronounced.
Again, this is in agreementwith the Brillouin zone
model, but is unexpected according to the ionic
model.
The classical (Pauling and Neuman, 1934)tetrahedrite composition is not reached in the synthetic
Cu-SLS system, and the simple ionic model used
here fails to explain this. In the Brillouin zone
model, the energy increment per valence electron
increases as we decrease the number of valence
electrons below 208 per unit cell (Fig. 3). Also,
Cu12SbaS13,
with 204 valence electrons which fill
the 5lst Brillouin zone, correspondsto the first
discontinuity in energy per valence electron that is
reachedby decreasingthe number of electrons. We
therefore expect that any change in the relative
stabilities of structuresfor the tetrahedrite composition is likeliest to occur at204 valence electronsper
unit cell, rather than at any marginally higher (or
lower) number of valence electrons. On this basis.
as multiphase assemblagesare stable for tetrahedrite compositions with less than 204 valence electrons per unit cell (see Tatsuka and Morimoto,
1977, and Makovicky and Skinner, 1978),tetrahedrite with the "classical" composition should not
occur, as is indeed observed.
Conclusions
The simple Brillouin zone model presented here
leads to the same compositional limits for tetrahedrite as the ionic model, and further predicts the
avoidanceof the composition Cu12SbaS13,
which
MODEL FOR TETRAHEDRITE
225
has been noted (Makovicky and Skinner, 1978),but
not explainedpreviously. In addition, the Brillouinzone model explains the clustering of natural tetrahedrite compositions at the limiting value of 208
valence electrons per unit cell, which is also seen.
However, the model's most obvious successlies in
the prediction of the relative resistivities of 52ndBrillouin-zone-filled and -unfilled tetrahedrites: a
prediction subsequentlyconfirmed, and completely
foreign to ionic models. Because the bonding in
sulfides and sulfosalts is far from being ionic, any
model invoking a balancing of formal ionic charges
is physically suspect.We demonstratethat an alternative model which may be more realistic for such
minerals, that based on the number of valence
states in the Brillouin zone, is indeed considerably
more successful in predicting the chemical and
physical properties of the tetrahedrite group of
minerals.
Acknowledgments
The authors would like to thank the following individuals for
their advice and help: Prof. H. Tuller, Dr. A. McHale,
H. Fujimoto (Department of Materials Science, M.LT.); Prof.
D. Tumbull, and Dr. C. Lobb (Division of Applied Sciences,
Harvard); M. Parisi (Intermetrics, Inc.); Prof. C. R. Rossman
and A. Hoffmeister (Department of Geological and Planetary
Sciences,C.LT.); Prof. B. J. Wuensch(Departmentof Materials
Science,M.I.T.); and Prof. C. W. Bumham (Departmentof
GeologicalSciences,Harvard). Helpful comments were provided by M. S. T. Bukowinski,T. J. Shankland,and the reviewers.
This work was supported by NSF grant EAR 8l-08592, the
Harvard Materials Research Laboratory, and the A.P. Sloan
Foundation. M.L.J. was also supported on NSF grant EAR 79-
20w5.
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IOHNSON
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