American Mineralogist, Volume 76, pages 1033-1037, 1991
Vectors, components,and minerals
DoN,c.Lo M. Bunr
Department of Geology, Arizona State University, Tempe, Arizona 85287-1404,U.S.A.
Ansrn-l.cr
Ionic substitutions in minerals are directed chemical displacementsand may be treated
as vector quantities. This approach, pioneered by J. V. Smith and J. B. Thompson, Jr.,
has many advantagesover the barycentric coordinates (such as Gibbs triangles) long familiar to mineralogists and petrologists. Among these are the convenience of plotting
recalculatedmineral formulas directly on x-y (cartesian)diagrams;a possiblereduction in
the number of end-members, e.g., in the spinel and garnet groups, resulting from the
valence insensitivity of exchangeoperatqrs; the identification of possible new end-members, illustrated with composition planesthat have up to five vertices;and the applicability
of the same set of vectors to different minerals, illustrated with regard to pyroxene, plagioclase,and melilite.
Planar vector diagrams for a given mineral group can have chemical limits, generally
dictated by ionic charge and bounded by lines representingzero contents of individual
ions, as well as narrower crystal-chemical limits, generally dictated by ionic radii and
bounded by lines representingconstraints such as all Si as tetrahedral and Mg as octahedral. A given vector doesnot necessarilyimply that a given mineral will be correspondingly
zoned chemically, unlessthe mineral's composition is very close to a chemical or crystalchemical limit.
INrnonucrroN
Geologistsconventionally plot mineral and rock compositions on triangles; the suggestionto plot compositions according to a center of gravity (barycentric coordinates) was made by J. W. Gibbs in 1878 (reprinted in
, . l l 8 ) . J . B . T h o m p s o nJ, r . ( 1 9 8 1 ,
G i b b s ,1 9 0 6 , 1 9 6 1 p
p. 159)cites W. L. Bragg(1937;'cf. Bragget al., 1965,p.
23) for first suggestingthat the compositions of complex
mineralogical solid solutions could be representedby the
extent of substitutions from a given end-member composition rather than by proportions of a bewildering variety of individual end-members.This approachwas used
by Smith (1959)for model amphibolecompositionsand,
much amplified, by Thompson (1981, 1982)for amphiboles and other minerals.
Ionic substitutions in minerals, generally represented
by an undirected notation such as Na + Si : Ca + Al
(e.g.,for plagioclase),can be representedmore succinctly
by directed exchange operators (Burt, 1974) such as
NaSi(CaAl)-,. This notation for the operation of ionic
exchangewas introduced in classlecturesin the late 1960s
by J. B. Thompson, Jr. My initial interest in exchange
operators (Burt, 1974, 1979) was prompted by the realization that they must be intrinsically acidic or basic in
the electronic or Lewis (1938) sense,becausedifferent
ions possessdiferent electronic environments. For many
years, exchangeoperators were used barycentrically (as
points on triangles) or algebraically or in terms of their
chemical potentials ("exchangepotentials"), rather than
0003-004x/9l /0506-1033$02.00
as composition vectors. My first use of composition vectors was in a systematization of lithium micas (in Cernj'
and Burt, 1984). Independent ionic substitutions in minerals, such as in this caseLiAlFe , and FeSiAl -2, are directed chemical displacements(i.e., they have both a direction and a magnitude) and are as much vectors as
physical displacement operations (cf. Hotrmann, 1966,
1975). The starting point for chemical displacement operations was termed the "additive component" by
T h o m p s o n( 1 9 8 1 ,1 9 8 2 ) .
The unique feature of my vector diagrams was that
they included a scaled vector inset that showed the algebraic relations among various ionic substitutions
graphically. Individual vectors on this scaledinset can be
labeled by ions that are not involved in the substitution
and are constant along their length. Their slopes correspond with the slopesof isocomposition lines on the associated composition diagram. This labeling of vectors
by noninvolved ions is somewhatanalogousto the labeling ofunivariant lines around an invariant point by noninvolved phases(e.g.,Zen, 1966), and it can be extended
into the third dimension.
This vector approach has also been applied to phyllosilicates in general (Burt, 1988), to tourmaline (Burt,
1989a),and to alarge variety of rare earth-bearing minerals (Burt, 1989b).The purpose of this article is to make
some generalizationsbasedon this previous work and to
give additional examplesof the application of vector diagramsto several mineral goups.
1033
1034
BURT: VECTORS, COMPONENTS, AND MINERALS
(Fig. 1A) that was derived following the method of Burt
(1974), and a subsequentexchangevector representation
/ rr\
(Fig. lB). In a sense,franklinite belongs to the ternary
z l' ,\ r
systemFerOo-Mn.Oo-Zn.Ou,
althoughthe last oxide composition is unstable (equivalent to a mixture of 3 ZnO
/
\ \ z n r t -t
andt/zOr), and the accessiblecompositions definea quadznFezooi\iziroo
rilateral with the vertices Fe.Oo,MnrOo, ZnMnrOo, and
ZnFerOo(e.g.,Mason, 1947).
For Figure lB, FerO. is the additive component
(Thompson, 1981, 1982); other additive components
s
MnFe-,
$ $
FeaOo
^
nn
could have been used instead, and each would have gen"o "^-erated the same diagram. Inasmuch as vector operations
+ €€
are commutative, the exchangeoperationscan be applied
=3
in any order. Vector operations are also associative,so
that
operations such as ZnMn , can be considered as
7an
ZnFeMn_2 ZnMn_1
ZnFe ,, then FeMn ,.
This diagram also illustrates two types of limits to such
diagrams, chemical and crystal-chemical.Strictly chemt\
ical limits are those applied by stoichiometry alone and
I
are, for this diagram, the limits defined by the three lines
''
\
Zn : 0 (horizontal), Mn : 0 (vertical), and Fe : 0 (ironKltntle
Hetcerolite
45"). These define a triangle with corners FerOo,MnrOo,
ZnFero4 ZnFe"*Md*o4 TnMnrOo
ZnrOo. The last corner, although chemically attainable,
must be unstable as a spinel becausetrivalent Zn does
not exist in minerals. The three points labeled ZnMn-,,
ZnFe-,, and MnFe ,, are only attainable mathematically;
they do not representobtainable chemical compositions.
Fe3Oa MnFe204 Fe-*/An
2Oo Mn3Oa
A second, more restrictive set of limits is provided by
M o g n e t i t eJ c c o b s i t e
H o u s m c n n i t e crystal-chemicalconsiderations.This is outlined in solid
lwokiite
lines and here definesa quadrilateral. Examplesare given
Fig. 1. Diagramsshowingthe compositionplaneof franklin- below for systemsin which the chemical limits alone dequadrilaterals and pentagonsrather than triangles.
ite spinel,(Zn,Mn,*,Fe,*)(Fe3*,Mn3*)rOo.
(A) Exchange
operator fine
vector representation such as Figure lB would be
A
or barycentricrepresentationof the system FerOu-MnFe
,ZnMn-r.(B)Exchange
vectorrepresentation
of compositions
de- especially useful for interpreting electron microprobe
rived from FerOoby the vectorsZnFe_,andMnFe_,.
analysesof franklinite. Although valencesof Mn and Fe
may not be known accurately, lines of constant Mn (of
any valence)are vertical and ofFe (ofany valence)have
a
slope of -1. Lines of constant Zn are horizontal, as
MrNnn c.LocrcAl ExAMpLEs
shown. These slopesare parallel to the slopesof the three
Franklinite and related spinels
vectors in the scaled inset labeled Mn, Fe, and Zn, rcExchangeoperators,a type of chemical component, are spectively,accordingto the noninvolved elementsor ions.
electrically neutral, and an operator such as MnFe , can
The basis vectors ZnFe-, and MnFe-r on Figure lB
representeither Mn2*(Fe,*), or Mn3*(Fe3*)_,.
Mn- and could have been drawn at an angle other than 90'(such
Fe-bearing minerals therefore can have fewer compo- as 60", forming an equilateral triangle) and could have
nents than one might infer from the number of named been scaleddifferently. The advantageofusing orthogoend-members,and certain "end-members" can be com- nal basis vectors ofequal length is that compositions can
positional intermediates in a solid solution series (al- be plotted on ordinary graph paper. Similarly, analyses
though this does not necessarilymake them invalid as or recalculatedmineral formulas, commonly recordedon
mineral species).
an electronic spreadsheet,can be plotted on ordinary x-y
As an example, consider spinels related to franklinite, diagrams (such as Zn vs. Mn in this case).
(Zn,Mn2*,Fe2+)(Fe3+,M1r+)rOo.
The five possible oxide
The scaledinset diagram also showsthe derived vector
componentsareZnO, MnO, FeO, FerOr,and MnrOr. This ZnFeMn ,, which representsa possible coupled substiis obviously too many components, becausefranklinite tution. Numerous other coupled substitutions could be
belongsto the four-component systemZn-Fe-Mn-O. One derived by linear combinations of the two basis vectors
could name six end-members: ZnFerOo, ZnMnrOo, MnFe-, and ZnFe'; ZnMnFe r, for example, would
MnFerOo,MnMnrOo, FeFerOo,and FeMnrOo.How many represent their sum. Note that solid solution mineral
components does this system really have? Only three, as compositions need not fall along such vectors unlessthe
shown on a barycentric exchangeoperator representation compositions approach a side line or some other crystalZnMn ,
- l
A
a
B
L || . ) v t
l 035
BURT: VECTORS, COMPONENTS, AND MINERALS
MnFe 1
MnCo,l
NoAl(CoMg)_,
A
"B l y t h i t e "
Jodeite
Mn2jMn3;si30r2
FeCo-1
CoAlrsiou
co/v\gsi
206No=o
Diopside
"C c t s "
Mn2;(Fe3.Mn3lsi3o
r2
o
CoLderlte
Mn2:Fe3;si3
or2
'\(., c
No2SiSirOt
B
(,*n';r"')r.t;r'.
o ,,
c o 3 / t n 3 i s i so t ;
C o 2 M g ( N o 2 S1i )
\,
{ N o c o ) { M 9.os l o 5 l s i 2 o Z
1- -\NoCo)A'S,2O7
'-r,| ..\
H
nnortn'i?3 Hi;,i.:';'i'"'"''
t
"'-?
planes
-\. | \'Sodo-melilrfe"
(Mn2.Fe2;)Fe3i
si3or2
co.corsi.o,,
co3Fe3;si3ot2
M n= 0
Fe2:Fe3;si3ol2
for
Fig. 3. Vectordiagramsshowingthe composition
plagioclase,
and melilite,usingas basisvectors
clinopyroxene,
Fig. 2. Vectordiagramshowingthe accessible
composition Al,(MgSi)-,and NaAl(CaMC),. (A) Clinopyroxeneplane.(B)
plane.(C) Melilite plane.Seetext for discussion.
plane of calciummanganese
garnet,
general
iron
of
formula Plagioclase
(Ca,Mn2*,Fe2*)r(Fe3*,Mn3*)rSirO,r.
Compositions
shownarederived from that of andraditebv the basisvectorsMnCa , and
of zero content of each component. Simply applying the
FeCa-,.
vectors MnCa-, and MnFe ' to the composition of andradite would have yielded the calderite and "skiagite"
chemical limit. Note also on Figure lB that the MnFerOo compositions at the corners of a triangle but would have
phases jacobsite (cubic) and iwakiite (tetragonal) are missed the entire left side of the diagram, including two
compositionally only intermediatesof a solid solution se- additional potential garnet end-members. The "tyranny
ries between Fe.Ooand MnrOo, although as ordered nor- of the triangle" would have won again.
mal spinels, they are valid species(Esseneand Peacor,
Clinopyroxene,plagioclase,and melilite
I 983).
The diagramsin Figures I and2 involve relatively simGarnet group
ple ionic substitutions, albeit for ions ofvariable valence.
Garnets can also contain both divalent and trivalent Much more complex coupled substitutions are also posFe and Mn. Accessiblegarnetcompositions derived from sible. For such cases,it is advantageousto condensethe
that of andradite using the substitutions FeCa-, and simple substitution vectors, such as FeMg r and AlFe ';
MnCa , (and the derived substitutions MnFe ,) are de- combining ions of similar ionic size and valence for
picted on Figure 2. The composition of calderite, derived graphical pu{poses has long been a familiar procedure.
from andradite by MnCa ,, is seento be intermediate in Most diagramsgiven below may be consideredto be sim'
a possible solid solution series between "skiagite" and ilarly condensed.
The samevector operations can be applied to different
"blythite" (both unknown in nature, but synthesizedat
high pressures,e.g.,Fursenko, 1986).The compositions mineral groups. Figures 3A, 38, and 3C show how the
within the crystal-chemical limits of garnet again define same exchangevectors can be applied to the formulas of
a quadrilateral becausethe theoretical, yet chemically at- diopside (clinopyroxene), albite (plagtoclase),and akertainable, end-member CarSirO,, must be unstable as gar- manite (melilite). The vector inset shows the horizontal
net (equivalent to CarSirO, plus CarSiOoplus YzOr), in- "Tschermak vector" Alr(MgSi)-' of constant Na and Ca
and the vertical vector NaAl(CaMg) , of constant Si. Deasmuch as trivalent Ca is unknown.
The dashed isocomposition lines inside the quadrilat- rived vectors of constant Mg and of constant Al, labeled
eral are not labeled on this and most following figures, on the diagram, are obtained by taking linear combinabut from the vector inset one can see that lines of con- tions of the basis vectors so as to eliminate Mg or A1.
stant Fe are vertical, lines of constant Mn are horizontal, Similar inset vector diagrams in subsequentfigures will
and lines of constantCa have a slope of - l. The chemical not be discussedin the text.
Figure 3,{ shows how the compositions of jadeite and
limits of the figure are provided by the lines Mn : 0, Fe
: 0, and Ca : 0; an additional limit at Ca:3 is provided of "CATS" (CaAlrSiOJ can be derived from that of diopside as the corners of a triangle. The chemical limits,
by the lack of trivalent Ca.
In general,in drawing such diagrams it is a good idea however, define a quadrilateral (unlike the triangles in
to seekall the chemical limits, which correspondto lines Figs. I and 2), with the lines Al : 0 and Ca : 0 conAndrodite
'Skicgite"
1036
BURT: VECTORS, COMPONENTS, AND MINERALS
MnS(NoCl)
M n 2 B e ( N q 2 S i )I
MnAl(NoS), V
.' /,
u"l//'
ffi
Be
AI
M n B e ( N o A) - ,
Be5iAl-2
MnB(Be2A loO24)Cl2
N o 2 M n 6 ( A l 1 2 O 2 a ) C2
No:o
Mnu(Bersiuoro)cl,
i,,',' ,'"i,i,',i 'i
li
t ,'t i j,';'2 N o
r M nu ( B eu S i u O r o ) C l ,
,(:
,'r',"',
,'.'i'i,,1 + 2 M n S ( N o C l ) _ r
l-,--2-i-
: Helv 1e
,ir'l'
"/'
NouMnr(BeoSiaOro)C,
Nou(BeuSisQ2iC2
/^n-o
N o B ( A l 6S i 6 O 2 4 ) C l 2
f
Nor(BerA 2SiBO24)Cl2
Sodclite
M n +, B e ^J S ( N o+, A l ^ C l' ), - .l
J
Tugtupite
Fig. 4. Vectordiagramfor possibleBe substitutionsin the
sodalitegroup(sodalite-tugtupite-helvite
relations).
Theprimary
interestof this diagramis that its chemicallimits definea penhgon.Seetextfor discussion.
vergingat a fourth vertex at composition Na(MgrSior)
SirOu. The presenceof I6lSiin pyroxene is improbable,
however, and the crystal-chemicallimit is defined by the
vertical line for Si : 2.
Figure 38 gives an analogousand congruent diagram
for the chemical limits of plagioclase.The main solid
solution here is betweenalbite and anorthite; substitution
of talMgin a feldspar is difficult, although Sclar and Benimofi 1980,have reportedthe synthesisof the end-member CaMgSirO8at 1200"C.
Figure 3C gives a similar diagram for melilite, with
gehleniteand "soda-melilite" compositions derived from
the akermanitecomposition. The chemicallimits are more
extensivethan for clinopyroxeneor plagioclaseand define
a sloping triangle with an apex at the composition
NarSiSirOr.Sucha composition with the largertetrahedral
site containing Si is somewhat improbable in melilite
(Goldsmith, 1948, failed to synthesizemelilites with any
solid solution toward it), and the vertical line of Si : 2
again provides a crystal-chemical limit that may, however, be tentative becauseoxynitride melilite, YrSi3O3N4,
doeshave three Si (Fukuhara, 1988).
Be in the sodalite group
Numerous natural and synthetic phasesisostructural
with sodaliteare listed by Hassanand Buseck(1989; cf.
Hassanand Grundy, I 984, I 985). Sucha long list provides
many opportunities for application of exchangevectors,
but one example involving Be is illustrative. The formula
oftugtupite can be derived from that ofsodalite by double
application of the vector BeSiAl ,, which only affectstetrahedral sites. Helvite, on the other hand, although iso-
? :6
Helvite
N o A l S i O ++ 2 M n S
N c u M n r ( A l u S6 a 2 l s 2
\
O
o
o
Mn, (BeuSiuOro)S,
\
t\
\t\\
\t\\
\t
\t
\t
\
\t
\
s:0
Ncs(Al6Si6O2a)Cl2
NorMnu(BeuSi6a2ic12
SOdOlite
? - 3 M n r s i o* o
3 B e 2 s o*42 N o c l
Fig. 5. Vectordiagramalongthe line Si : 6 in the previous
diagramshowingsodalite-helvite
relations.This is a reciprocal
ternarysubsystem.
structural with sodalite, only sharesSi in common with
it. Figure 4 representsan attempt to relate these two Bebearing sodalite structures.
On Figure 4, the horizontal vector is BeSiAl , and the
vertical vector is MnAl(NaSi) r, related to the plagioclase
substitution by the vector MnCa ,. The Ca analogue of
this vector diagram could be usedto model Be substitution
in plagioclase.When thesevectors are applied to the composition of sodalite, the pentagonal figure at the bottom
results. It is doubtful whether many of the compositions
shown are stable with the sodalite structure, although there
is no obvious reason other than dissimilarities between
Na and Mn why they should not be. Hassanand Buseck
(1989)list Si-freealuminate phaseswith the sodalitestructure. The chemical and crystal-chemical boundaries coincide in Figure 4, and they define a composition plane
with five sidesfor zero values of each of the five elements
Na, Be, Mn, Al, and Si. Such pentagonsare not uncommon in the vector world of mineral compositions; Burt
(1989b)givesexamplesinvolving the compositionsof apatite, pyrochlore, and chevkinite/perrierite.
On Figure 4 the Si : 6 line along the vector
MnBe(NaAl) , joins sodalite to a composition close to
that of helvite, Mnr(BeuSiuOro)Sr,
and related to it by the
vector 2 MnS(NaCl) ,, as labeled. This relation is also
indicated on Figure 5, which is a reciprocal ternary subsystem.The compositions at the upper left and lower right
of the rectangleare probably unstable as sodalite structures, as indicated. Helvite is related to sodalite by the
rather complex vector MnoBe.S(NaoAlrCl),, which is also
derived simply by subtractingthe formula of sodalitefrom
that of helvite. Working through this exercisein reverse
shows how such a complex vector can be resolved into
simpler ones.
Spacedoes not permit additional or more complex examples;others,reviewedby Cernj'and Burt (1984) and
Burt (1988, 1989a, 1989b),involve H gain or loss from
hydrous phases,vacancies,and interstitials and present
diagrams for which the number of anions is not constant.
BURT: VECTORS, COMPONENTS, AND MINERALS
SuNrNr.lnY oF FEATURES
Vector diagramscan be plotted on ordinary orthogonal
graph paper or using the standard x-y plotting routines
availablein electronicspreadsheetprograms.The formula
contentsofindividual phasesare plotted directly, without
the need for a barycentric recalculation. Lines of equal
contents of other ions can be determined from the scaled
vector inset that should accompany each vector diagram
(or x-y plot). Basisvectors can be varied accordingto the
data available.
Exchangevectors are electrically neutral and valence
insensitive; this fact can allow some apparently complex
composition spacesto be plotted on a plane. Examples
given above involve Mn and Fe-bearing spinels (franklinite-related compositions)and garnet (andradite-related
compositions).SeeBurt (1988)for a similar exampleinvolving annite. The valenceinsensitivity ofexchangevectors on mixed-valence diagrams may also be useful for
plotting microprobe analyses.
On the other hand, compositions conventionally plotted on a triangle may be more correctly plotted as a condensedvector space.Seethe vector section ofCernf and
Burt (1984) for an example involving the lithium iron
aluminum micas and Burt (1989b) for an example involving the isostructural alunite, beudantite, and crandallite groups.
Vectors are additive. A complex substitution vector can
therefore be expressedas the sum of simpler ones (as
above, for sodalite-helvite relations). Any convenient
combination of simple independentexchangevectors can
be applied in any order to arrive at a complex actual
mineral composition.
Vector diagrams commonly reveal hypothetical endmemberswhich can be soughtin nature or by experiment
in order to discover the compositional limits of a given
structure type. An example is given for the sodalite group
with up to five chemically determined verticesin a plane.
The boundaries of vector diagrams can be both chemical as determined by stoichiometry (i.e., lines of zero
content of individual elementsor ions) and crystal-chemical as determined by valence and ionic radius (i.e., lines
representingadditional constraints such as lack oftrivalent Zn or of octahedral Si or of tetrahedral Mg). The
chemical boundaries may reveal important parts of the
diagram, which otherwise might be overlooked.
Unless compositions lie very closeto a line determined
by chemical or crystal-chemicalconsiderations,the composition of a mineralogical solid solution is not necessarily
constrainedto vary along a singlevector, although it may
do so to a first approximation. Looking for a unique "substitution vector" in some minerals may be a pointless
exercise,unless account is taken of the compositions of
coexisting minerals and fluids.
AcrnowlnlcMENTS
A junior high school mathematics teacher, Mr. Stamer, enthusiastically
introduced me to vectors at about the level required to derive the diagrams
1037
in this paper. I am also grateful to J.B. Thompson, Jr. for fint suggesting
to me the algebraic concept ofexchange operators and for encouraging me
to use them. Finally, I thank John I-onghi and especially John Ferry for
useful reviews.
RnnnnnNcns cITED
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Exchangeoperators, acids, and bases.In V.A. Zharikov,
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Fennumv 5, 1990
MrNuscnrm RECEIvED
Mmcn 7, 1991
MnNuscmrr AcCEPTED