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American Mineralogist, Volume 73, pages 855-860, 1988
Aluminum fluoride hydrates, yolcanogenic salts from Mount Erebus, Antarctica
Pnrr,rp E. RosnNnnnc
Department of Geology, Washington State University, Pullman, Washington 99164-2812,U.5.A.
AssrRAcr
Three aluminum fluoride hydrates, B-AlFj.HrO, AlF3.3HrO, and ralstonite (Na"Mg,Al, ,(F,OH)6'HrO) have been identified in volcanogenicincrustations from the summit area of Mount Erebus,Ross Island, Antarctica; anhydrous AlF, is probably also present. Associated salts are difficult to identify becauseof xno-peak overlap, compositional
variations, and, perhaps,postsamplingchangesin structural water.
The monohydratesand anhydrous AlF, coexist in sublimatescrystallized directly from
the Mount Erebus plume, whereas the trihydrate forms in aqueous solutions condensed
from the volcanic plume, possibly by hydration of the monohydrates or AlFr. Only AlFr.
3HrO could be isolated for characterization.It occurs as euhedral cubes l-6 pm on edge;
unit-cell (tetragonal)dimensionsareqo: 7.726(l) A and co: 3.657(l) A; mean refractive
index and calculateddensity are 1.413(l) and2.09 g/cm3,respectively.
Identification of aluminum trifluoride and its hydratesin volcanic incrustations suggests
that AlF, is one of the principal high-temperature salts present in the plumes of Mount
Erebus and many other volcanoes.
INrnooucTtoN
michael, l97l). Furthermore, water-soluble sublimates
Aluminum trifluoride (AtFr) is thought to be one of the destroyed or modified by hydrolysis at other volcanoes
principal high-temperature volcanogenic salts that form are more likely to be preservedat the summit of Mount
Erebus'
incrustations by fractional condensationofvolcanic gases
-posrA study of six small samplesof incrustations from the
(Oskarsson, 1981). Although AlF, has never been
tively identified as a mineral, the gaseousmolecule mat rygmit area, acquired in 1979 through the kindness of
be present as an initial constituent of the volcanic plume William Zoller of the University of Maryland, was unat Mount Hekla in Iceland(Oskarsson,198l) and at many dertaken in an effort to establish the presenceof AlF,
other volcanoes. Observation of AlF. as a sublimate in and/or its hydrates'
volcanic incrustations would serve to confirm this hybnNrrrrc,t:troN oF pHASES
pothesis.
Mount Erebus (elevation, 3794 m) is the most active
Positive identification of the crystalline phasesin involcano in Antarctica and the largest of four volcanic crustations from the summit area is difficult becauseof
conesthat form Ross Island. The predominant rock type their fine grain size and complex mineralogy. The preson Mount Erebus and the only one cropping out in the ence of compositional variations and postsampling
summit areais anorthoclasephonolite (Giggenbachet a1., changesin structural water are further obstaclesto minI 973; Kyle et al., 1976). The active crater containsa small eral identification.
persistent lava lake of phonolitic magma. Fumarolic acAttempts to fit observed X-ray patterns to those of
tivity has been continuous at the summit since the dis- known inorganic compounds have met with limited succovery of Mount Erebusin l84l (Joneset al., 1983).
cess(Joneset al. 1983). However, a list of crystalline
Incrustations of salts are widely distributed along the phasespositively or tentatively identified in samples from
coast of Ross Island and in the summit area. Many are the summit area(Keys and Williams, l98l) includesvaryellowish in color and so resemble native sulfur, which ious chlorides, sulfates,and fluorides; aluminum trifluohas not been reported on Mount Erebus (Jones et al., ride (AlFr) and ralstonite [Na"Mg"Al,
"(F,OH)6.HrO]
1983).Thesesalts appearto have been depositeddirectly were tentatively identified by means of X-ray
diffractofrom the plume of volcanic gasesand indirectly from it
metry.
by interaction ofthe acid gaseswith volcanic rocks.
X-ray diffraction data (Table l) for qualitative analysis
Mount Erebusis an ideal site for the study of F-bearing were collectedusing a monochromator-equippeddiffracvolcanic sublimates becauseit is an active, alkaline vol- tometer and CuKa radiation. The goniometer was calicano and becauseunique conditions (low temperature, brated using severalX-ray reflectionsfrom an external Si
high altitude) exist at its summit. F-rich volcanic gases standard.Data were obtained by averagingtwo measweare associatedwith alkaline volcanoes(Stormer and Car- ments per reflection at a scan rate of l'lmin. The preci0003404x/88/0708-085
5$02.00
855
856
ROSENBERG: ALUMINUM
FLUORIDE HYDRATES
TABLE1. X-ray powder-diffractiondata for salt sample E398
from the summitof Mount Erebus,Ross lsland,Antarctica,and for syntheticAlF3'3HrO
Saltsample
AlFs3HrO'
Saltsample
AlFo'3HrO-
d (A)
d (A)
d (A)
tlh
d (A)
tth
2.870
2.812
2.728
2.649
2.612
2.540
2.506
2.440
2.366
2325
2302
2.279
2.187
2.102
2.050
2.030
2.015
1.930
1.847
1.819
1,794
1.775
1726
1 668
5
5
20
15
5
5
20
30
5
5
5
3
10
5
10
20
8
10
10
I
10
20
25
5
2.73
2.65
30
I
2.51
2.44
12
55
2.19
211
c
2.03
12
1.933
1.850
1.824
l2
1.780
1.728
1.670
8
40
5
lh
ilh
Y O40
I 927
7 609
6.794
5 901
5 824
5.741
c.4cJ
4.897
4.03/
4.436
4.329
4.267
10
3
3
5
5
5
100
I
10
10
5 45
1R
20
4.UJJ
3.864
Fig. 1. Scanning-electron
microphotograph
of AlF3.3HrO.
Thelengthof thebaris 18.3pm.
sion of these measurementsis believed to be within
0.02 20 for all reflectionsand +0.01' 20 for t};lestronger
reflections.
Two mineralogically distinct sample goups were recognized by means of X-ray difractometry. The X-ray
powder-diffraction patterns of five samplesin one group,
representedby sample E68 (Table 4), are very similar to
that reported for a salt sample from the summit area of
Mount Erebusby Joneset al. (1983).The X-ray diffraction pattern of the remaining sample, E398 (Table l), is
unique. The location of sampling siteswithin the summit
area is unknown.
Salt sample E398
Phases present. A search of the Joint Committee on
Powder Diffraction Standards(JCPDS)listing of inorganic compounds using the three most intense X-ray diffraction maxima (Table l) yielded an acceptablematch with
AlFj.3HrO UCPDS card no. 9-1081basedon the positions and relative intensities of 18 X-ray reflections.The
trihydrate has not been reported in nature to date. The
identity of the other phasescomposing the sample could
not be established definitely because of the apparent
overlap ofX-ray reflections ofseveral coexisting phases.
In an attempt to identify additional phases,semiquantitative analysesfor Na, K, Ca, Al, Si, Fe, Mg, and S were
carried out using the analytical electron microscope(neu)
in the scanning mode. Five compositional groups were
recognizedon the basisof 54 random analysesofmineral
grains:(l) Al only (15 analyses),
identifiedas AlFr'3HrO
basedon morphology (seeFig. l). (2) Si only (7 analyses)
attributed to someform of SiO'. SeveralX-ray reflections
consistentwith the presenceof quartz were observed,but
owing to peak overlap, a positive identification cannot be
made by X-ray diffractometry. (3) S and Ca (5 analyses),
probably representingthe presenceof a small amount of
gypsum.ThreeunambiguousX-ray reflections7 .61,2.87,
3 531
3 504
3.477
3.398
3 336
3.299
3.213
3.031
2.915
50
20
3.86
3.65
OU
c
3
20
35
40
5
25
10
3.33
3.04
17
14
2
I
7
. JCPDS card no. 9-108.
and 2.07 A Gabte l) correspond to Sypsum, which has
been positively identified in the salts from the summit
areaof Mount Erebus(Keys and Williams, l98l). (4) Na
and Al predominant (15 analyses),possibly representing
a chloride or fluoride. (5) Si and K with lesseramounts
of Al and Na (7 analyses)indicating the presenceof a
silicate. The remaining five analysesshow the presence
of Si and Al with lesseramounts of Na, K, and Ca, but
elemental percentagesvary widely, suggestingthat these
analysesrepresentmixtures ofthe above groups.
Characterizationof AlFr'3HrO. Natural AlF.'3HrO was
characterizedby refined unit-cell dimensions as well as
by chemical composition and optical properties. X-ray
data are compared with those for synthetic AlF3'3HrO
in Table 2. Partial chemical analyseswere obtained using
an electron microprobe (eur).
AlF3.3HrO occurs as cubes l-6 pm on edge (average
-4.5 p.m)(Fie. 1) in yellowish, poorly crystalline to gellike masses.Isotropic material of low refractive index
(<1.40), yellowish needlelike crystals often in radiating
masses,and small rounded crystals with high birefringenceare also presentin this sample.The mean refractive
index of natural AlFr'3HrO, 1.413(l), is identical to the
measuredvalue for syntheticmaterial, 1.4130(6)(Ryss,
1956,p. 587);both have very low birefringence.
AlF3.3HrO was synthesizedby Ehret and Frere (1945)
who published Debye-SherrerX-ray powder data for the
stable form. Freeman (1956) obtained crystallographic
data for AlF3'3HrO basedon measurementsmade using
an X-ray diffractometer.He was able to index the powder
ROSENBERG: ALUMINUM
857
FLUORIDE HYDRATES
TnaLe3. Partialchemicalanalysisof salt sampleE68 (wt%).
39.0
10.3
0.43
1 24 0
1.9
0.81
19.5
0.43
0.21
1.29
0.35
0.06
86.68
F
Na
Mg
AI
5l
S
cl
K
Fig. 2. Electron-microprobe
analysesof AlFr.3HrO in the
systemAI-F-O. Compositionsof hydrousaluminumfluorides
havebeenprojectedinto theanhydrous
system.AlF, is a charged
product.
decomposition
FE
Mn
Ti
. W. H. Zoller.writtencomm.. 1979
pattern and calculate lattice parameters assuming a tetragonal unit cell. Although systematicabsenceswere observed,it was not possible to define the spacegroup unambiguously (Freeman, 1957).The density calculatedon
the basisof four formula weightsper unit cell, 2.09 g/cm3,
compares well with the measured density, 2.03 g/cm3,
considering that no correction was made for solubility
(Freeman,1956).
X-ray data for natural AlF3. 3HrO wererecollectedfrom
the bulk sample at a scan rate of Vz'/min; two data sets
were averagedto obtain d values that are compared to
those of Ehret and Frere (1945) and Freeman(1956) in
Table 2. The d values and relative intensities are in excellent agreementparticularly with the difractometer data
of Freeman(1956). The unit-cell dimensionsof natural
AlF3.3HrO calculatedby least-squares
refinementof 17
X-ray reflections are close to those reported for the synthetic end member (Table 2).
Partial chemical analyseswere obtained using a Cameca Cambax electron microprobe. Becausewater loss
was anticipated, samples were mounted on a gold substrate using ethanol as a dispersant and analyzed at an
acceleratingpotential of 8 kV. Al and F were the only
elementsdetected;oxygen was determined by difference.
The results are shown in Figure 2. Water has apparently
been lost in almost every analysis;many analysescluster
around AlF3 . 2HrO and AlF, .HrO. HF has also been lost
in analyses intermediate between AlF3.HrO and AlFrOH. A few analysesshow complete loss of water as
well as loss of HF and lie close to the composition AlFr.
These results are consistent with the identification of an
aluminum fluoride hydrate and, together with the X-ray
data, provide convincing evidencethat its chemical formula is AlF3'3HrO.
Salt sample E68
A partial chemical analysis(Table 3) by meansofatomic
absorption spectrophotometry and X-ray fluorescence
TABLE2. X-ray powder-diffraction
data for naturaland syntheticAlF3.3HrO
Natural
110
200
001
101
111
220
201
211
310
221
301
311
400
321
t002)
t330)
102
420
411
331
212
Synthetic
d.* (A)
d*. (A)
nh
5.468
3.862
5.463
3.863
100
J.OC/
3.305
3.040
2.732
2.656
2.512
2.186
2.103
2.030
1.931
1.848
1.821
1.779
1.728
1.670
1.631
1.616
. Freeman(1956,1957).
-'Ehret and Frere(1945)
3.305
3.039
2.732
z oco
2.511
2.443
2.188
2.106
2.O31
1.931
1.849
1.828
1.779
1.728
1.668
1.630
1. 6 1 6
a:7 .726(1)
A
c: 3.657(1)
A
OJ
24
60
40
ZJ
22
zo
49
14
28
13
'tl
I
31
Jb
4
7
4* (A)c.+c
3.86
3.65
3.33
3.04
2.73
z.b5
2.51
2.44
2.19
2.11
2.03
1.933
1.850
1.824
1.780
1.729
1.670
1.632
1. 6 1 8
NL
d* (A)-.
nh
100
62
5
17
14
30
I
12
5.47
3.86
3.66
3.29
3.01
2.72
2.64
2.50
2.43
2.18
2.08
2.O2
1.92
100
1.83
15
1.77
1.72
1.66
44
co
12
50
2
12
13
I
I
42
J
a : 7 . 7 3 4A
c: 3.665A
co
17
52
JC
27
25
27
58
10
4
23
858
ROSENBERG: ALUMINUM
FLUORIDE HYDRATES
TeeLe4. X-ray powder-diffraction
data for salt sample E68 from the summit of Mount Erebus,Ross lsland,Antarctica,and for
syntheticAlF3,B-A|F3HrO, A(OH).sFrr.HrO,and NaCl
E68
nL
8.588
100
TROA
d (A)
d (A)t
nL
d (A)tt
n6
63
100
6.2
100
d (A)+
tth
5.72
100
d (A)++ tlh
d (A)
n6
3.258
13
2.821
100
1.994
50
1. 6 2 8
15
O
6237
5717
s.025
4.541
4.260
3797
3.623
3.517
3.460
3.249
3220
3116
3069
2.923
2.831
2.821
2.757
2.550
26
6
30
26
48
3
34
45
3
3
6
36
7
44
11
15
50
14
2.515
2473
2.354
2.332
2.292
2.268
2.229
2.128
2.097
2 061
3
9
16
8
6
3
5
24
17
13
2.51
2 016
1.994
1. 9 8 1
1.872
1.802
1.757
1.722
7
20
15
3
6
19
3
1.658
1.643
1626
5
2
2
1.s80
1. s 5 8
2
3
lEea
7
1520
1.495
1.462
nh
NaCl'-
A(OH)o5Fr5 HrO.
B-A|F3HrO'
AtF3.
d (A)
4
5
10
352
100
3.68
63
3.63
52
3.58
27
3.55
24
3.44
3
21't9
20
2.074
3
2.019
1
1.600
1587
1. 5 6 0
1 460
25
3
15
7
253
5
2.3s
2.32
10
4
2.O4
2.O3
1. 8 1 7
1 765
1142
1691
18
16
32
45
I
10
100
7
42
3:13] 7s 3ll]
1.759
5.70
31
29
2.95
2.83
59
31
2.92
2.80
2.45
I
2.43
3
2.27
7
2.22
3
1. 9 9
21
28
31
2
2.53
2.30
4
2.20
4
2.02
1.932
13
13
1. 7 9 8
1.749
1.726
1.682
20
26
4
6
1.614
1.584
1. 5 6 0
6
8
16
1.598
1.573
1. 5 4 5
3
4
10
1. 5 2 9
22
1. 5 1 5
11
1.88
34
1.98
1.86
1.73
34
1.71
25
1 65
9
1.63
14
1.54
8
1.53
5
1. 4 9
13
1.47
9
1
. Grobelny(1977)
.'JCPDS card no. 5-0628.
t Synthesizedat 140'C.
it Heatedto 600'C
t Synthesizedat 140'C.
++ Heatedto 500'C.
techniqueswas provided with this sample(ZolIer, written
comm., 1979).The analysisindicatesthat over 80 wto/o
of this sample consists of Na, Al, Cl, and F and that
almost half of this percentageis F. Thus, the samplemust
contain largely chlorides and fluorides of Na and Al and
the fluorides should predominate.
ComparisonofX-rayditrractionmaxima(Table4)with
those known for chlorides and fluorides strongly suggests
the presenceof B-AIF..HrO based on the position and
relative intensities of 16 X-ray reflections.X-ray diffrac-
tion lines are listed (Table 4) for B-AlFr'HrO synthesized
fromanAlF.-bearingsolution al"l40"Candforthesame
material after heating at 600 'C. Although the crystal
structure is essentiallyunchanged,heating causesdehydration and contraction of cell parameters (Grobelny,
1977). The d spacingsof B-AlFr'H,O in the salt sample
may be expected to lie within or close to the range of
values for the fully and partially hydrated compounds
(Table 4). The monohydrate has not been reported in
nature to date. The presenceof AlF, is suggestedby six
ROSENBERG: ALUMINUM
reflectionsthat also correspondto the monohydrate and
three reflectionsunique to the anhydrouscompound. Only
two X-ray diffraction maxima of the monohydrate are
missing in the salt sample, rather weak reflections at approximately 1.69 and 1.60 A. The latter is the only reflection missing from the pattern for AlFr.
Further examination of the X-ray diffraction pattern
suggeststhat ralstonite may also be present in sample
E68. Natural ralstonites(Na,Mg,Al, ,(F,OH)..HrO) form
an isostructural solid-solution series between basic aluminum fluoride hydrate (x: 0) and a Na- and Mg-rich
end member (x: l) (Pauly, 1965).Despitedifferencesin
chemical composition that include the F/OH ratio and
the amount of water as well as the value of x in the structural formula, X-ray powder-difraction patternsof membersofthis seriesdiffer very little (Pauly, 1965;Rosenberg,
1972;Grobelny,1977).Thus, X-ray identificationof ralstonite is not difficult but estimatesof its chemical composition basedon X-ray powder-diffraction data alone are
not possible.
X-ray diffraction maxima for basic aluminum fluoride
hydrate, A(OH)05Frr.HrO, precipitated from an AlF,
solution at 140'C and for the same material after dehydration at 500 oC are compared with those for the salt
sample (Table 4). The d spacingsof ralstonite in sample
E68 should lie close to the range of values for hydrated
and dehydrated basic aluminum fluoride. All twelve X-ray
reflectionsfor the basic aluminum fluoride have reasonable equivalents in the salt sample; at least 10 are unambiguous.Despite the low intensity of the 5.72-A spacing in the salt sample as compared to the synthetic
compound, presumably causedby preferred orientation,
the presenceof ralstonite is highly probable. Unfortunately, B-AlFr.HrO, AlFr, and ralstonite could not be
isolated for purposesof characterization.
Inasmuch as Al and F were accountedfor, at least partially, the presenceof Na-bearing chlorides was suspected. In fact, four of the five X-ray diffraction maxima for
NaCl correspond closely to reflections in the diffractogram for sampleE68 (Table 4) including 3.258 A (l/lo:
r 3 ) , 2 . 8 2 rA l r O O ; ,r . 9 9 4A ( 5 5 ) ,a n d 1 . 6 2 8A ( r 5 ) ; o n t y
the weak reflectionat 1.701 A 1Z;is missing in the salt
sample. Thus, the presenceof halite, NaCl, which has
been positively identified in salt samplesfrom the summit area(Keys and Williams, l98l), may be inferred.
The identity of other phases composing the sample
could not be establishedwith any degreeofcertainty based
on X-ray diffraction data alone. Semiquantitative aru
analysesfor Na, K, Ca, A1, Si, Fe, Mg, S, and Cl were
carried out in an effort to identify additional phases.This
study proved to be difficult becauseof the extremely fine
grain size of the sample and becausethe .a,Euinstrumentation available is capableofanalysis only in the scanning
mode. However, three compositional groups were recognized based on 48 random analysesofapparent mineral grains:(l) Al greatly predominant (65-80 normalized
weight percent), attributed to aluminum fluorides; small
amounts of Mg are associatedwith Al in some of these
FLUORIDE HYDRATES
859
analysessuggestingthe presenceof ralstonite. (2) Na, Al,
and Cl with lesseramounts of Fe representinga chloride,
possibly NaAl4O4Cls,which has been tentatively identified
in saltsfrom the summit area(Keysand Williams, l98l).
Although the X-ray diffraction maxima of NaAloOoCl'
are similar to many of the unassignedreflections in the
salt sample,the correspondenceis not exact,perhapsowing to the efects of solid solution. (3) Fe and S with lesser
amounts of K, attributed to the presenceof jarosite that
has been tentatively identified in salts from the summit
area(Keys and Williams, l98l). X-ray diffraction maxima of the salt sample do not confirm the presenceof
jarosite, which may occur in minor amounts. Si is virtually absent in all analyses.
MrNrnLr- cENESrs
Although AlF3.3HrO is the only stablehydrate in temperature range 10-100 'C, higher hydrates (usually AlFr'
3.5HrO) precipitate from supersaturatedsolutions below
70 'C and gradually transform into AlFr'3HrO; between
70 and 100 "C the stable trihydrate crystallizes directly
from solution (Nielsen and Altintas, 1984). At higher
temperatures lower hydrates are more stable than the
trihydrate even in the presenceof aqueous solutions at
elevatedpressures;p-AlF3 .HrO crystallizesdirectly from
supersaturatedAlF3 solutions at temperaturesabove I l5
'C usually along with AI(OH,F)3.HrO (Grobelny, 1977).
These hydrates lose water on heating and are eventually
transformedinto a-AlFr, the most stableanhydrousform
of AlFr. However, B-AIF3 and AI(OH,F), persist to 640
and 550 'C, respectively.
Anhydrous AlF, and the monohydrates probably crystallized as sublimates directly from the volcanic plume.
Incrustrations containing thesecompounds are apparently common in the summit area of Mount Erebus. Keys
and Williams (198 l) report that AlF, hasbeen tentatively
identified in salts from the summit area. Furthermore an
X-ray powder diffraction pattern of a sample from the
summit area thaL is very similar to that of sample E68
(Table 4) has been published without any identification
of constituentminerals(Joneset al., 1983).
On the other hand, AlF3.3HrO in salt sampleE398 is
not a sublimate crystallizeddirectly from volcanic gasbut
a secondaryproduct formed in aqueous solutions condensedpartially or entirely from the vapor phase.AlFr.
3HrO may have precipitated directly from solutions
(Nielsenand Altintas, 1984;Sanjuanand Michard, 1987)
condensedfrom volcanic gasesor may have crystallized
by transformation of higher hydrates that precipitated directly from these solutions (Nielsen and Altintas, 1984).
Alternatively, AlF, or the lower hydrates, crystallized from
the vapor phase, may have hydrated in solution at or
below 100 "C to form the stable trihydrate. Inasmuch as
AlF, has been tentatively identified in the salts from the
summit areaof Mount Erebus(Keysand Williams, 1981,
and this study), hydration of AlF, seemsto be the more
likely mode of origin. The presenceof silicates in salt
860
ROSENBERG: ALUMINUM
sample 8398 suggeststhe reaction of acid gasesor condensateswith volcanic rock.
CoNcr-usIoNs
Three aluminum fluoride hydrates B-AIF3.HrO, AlF3'
3HrO and ralstonite (Na,Mg"Al,-,(F,OH)6.HrO) have
been identified in volcanogenic incrustations from the
summit area of Mount Erebus. Ross Island. Antarctica.
B-AIF3.HrO and AlFr.3HrO are, heretofore,unreported
in nature. Anhydrous AlF, is probably also present.Only
the trihydrate could be isolated for characterizalion.
The monohydratescoexist with anhydrousAlF, in five
samples that appear to represent assemblagesof sublimatescrystallizeddirectly from the Mount Erebusplume.
NaCl is also probably present, and NaAloOoCl, and
KFe3*(SO")dOH)u,jarosite, are possible additional constituents.
The trihydrate that has been identified and characterized in a sixth sample from the summit area probably
formed in aqueoussolutions condensedfrom the volcanic
plume, possibly by hydration of AlF. and/or the monohydrates. Quartz, gypsum, and a sodium-aluminum silicate are probably presentin this sample,but many X-ray
diffraction reflectionsremain unassigrred.
AlF3 with a sublimation temperature of 760 oC is one
of the principal high-temperature salts inferred to exist
in the volcanic plume of Mount Hekla, Iceland (Oskarsson, l98l). The identificationof aluminum fluoride hydrates and probably anhydrous aluminum fluoride from
the summit area of Mount Erebus confirms and extends
previous tentative reports of their existence (Keys and
Williams, l98l) and suggeststhat AlF3 may be present
along with many other speciesin the plumes of other
volcanoesas hypothesizedby Oskarsson(1981).
FLUORIDE HYDRATES
RnrpnnNcns crrED
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AcxNowr,nocMENTS
I am indebted to W. H. Zoller for the samplesfrom Mount Erebusand
to Gary Chandler for invaluable technicalassistance.The manuscript was
reviewed by F. F. Foit, Jr
M.q.Nuscnrpr REcEn'ED Ocronen 7, 1987
18, 1988
MeNuscmgr AccEPTED Mmcu