Phosphorus fractionation diagram as a quantitative indicator of crystallization differentiation of basaltic liquids

0~~0

et Cozmoohlmlca Acta. 1969,

Vol. 83.

pp.4QS to

696. Pergmon

Preea. Print&

in Northern Ireland

Phosphorus fmctimtion diagram as a quantitative indicator of crgstallieation differentiation of basaltic liquids* A. T.

ANDERSON~ and L. P. GREENLBND U.S. Geologioal Survey, Washington, D.C.

(Received 18 September 1968; accepted in revised

form22 November 1968)

&&&-Distribution factors of phosphorus (P in mineral/P in liquid) between phenooryst 0.04 to 0.02; orthopyroxene minemls and ooexisting bssaltio groundmass are: olivine (F+,): (FE,): 0.01; au&e: O.OZto 0.01; plagioclase: 0.02; ilmenite: 0.04. Beoause of thesmsllness of these distribution factors the ratio of phosphorus in the initial liquid to that in the residual liquid (phosphorus ratio) ideally equals the mass fraction of residual liquid minus 0-00-O-04. The phosphorus ratio facilitates, therefore, quantitative oomparison of the variation of major and minor elements with crystallization of basaltio liquids. A &JS@XXM fractaonation diccgmmis a log-log graph of the wt. % of any ohemioal element or oxide vs. the phosphorus ratio. The slopes of variation ourves on suah a freotionation diagram approxim&ely equal unity minus the crystal aggregate/liquid distribution factor. Knowledge of the tiividpcal mineral/liquid distribution factors makes it possible to estimate the relative proportions of crystallizing minerals from the slopes of curves on a phosphorus freotionation diagram prior to the crystallization of apstite or other phosphorus&h mineral. This was done fairly suooessfully for the Alae Lava Leke, Hawaii.

General statemerbtof purpose Chemical differences between volcanic rocks can be understood largely in terms of crystallization differentiation, and qualitatively can be related to the crystallization of common igneous minerals. Chemical variation diagrams show the relationship between the elements and give an indication of the state of differentiation. Rocks from different volcanoes or rock series, however, generally show diverse variation diagrams. A quantitative mineralogical understanding of such differences depends upon the physical significance of the parameter of chemical differentiation. Given different initial compositions of liquids, the relationship between a particular differentiation index and the state of crystallization can be different. Any element which is largely excluded from the minerals and stored in the liquid can. serve as an approximate indicator of the amount of crystallization. Of several potentially useful elements (P, K, Ti), phosphorous is preferred (1) because it appears less susceptible to deuteric transport than potassium, and (2) because it enters less readily than titanium into pyroxene and than potassium into plagioclase (WAOER and DEER, 1939; WAGER, 1963; KORITNIO,1965; PECKet al., 1966; HENDERSON, 1968). * Publication authorized by Director, U.S. Geological Survey. t Present address: Department of Geophysioal Scienoes, University Illinois 60637. 493

of Chicego, Chicago,

494

A. T. ANDERSON and L. Y. GREENLAXI)

In this paper we present crystal/liquid distribution factors for phosphorus and show how phosphorus can be used to facilitate a quantitative, mineralogical interpretation of the chemical differentiation of basaltic liquids. This is done by reference to the fractionation law: &/A, = fjl-Q) (RAYLEIGH, 1896) wheref is the mass fraction of residual liquid in the total liquid + crystal system, &,/A, is the ratio of the mass concentration of substance A in the initial liquid (or in the whole system) to that in the residual liquid; and aA is the crystal aggregate distribution factor for A (i.e. the concentration of A in the crystal aggregate which is in equilibrium with the liquid divided by that in the liquid). In addition to the crystal aggregate distribution factor, arA,de&red in the foregoing sentence, we will speak of the individual mineral fractionation factor: uAa = concentration of A in mineral a divided by that in the liquid with which it is in equilibrium. In this paper we refer to A,/A, 5s the A ratio, in the case of phosphorus, the phosphorus ratio. ANALYTICAL

METHODS

Phenocryst minerals initially were concentrated from the 35 to 100 mesh size fraction (0.42 0.16 mm) of meohanioally ground lava speoimens. The mineral ooncentmtes were purified by hne grinding (0.070 to 0.045 mm) and by use of heavy liquids (diiodomethane, bromoform, and Clerici solution) and the Frantz magnetic separator. The purity of the final mineral separates ranges between 98 and 993 per cent. The principal impurity in sll mineral separates is attached groundmass or glass. Phosphorus occurring in attached groundmess (as apatite) or glass was removed chemioally. Mineral separates (except plagioclese) were leached with 10% HF for 15 min at 25%. After washing with 1 M NasCOs solution to remove amorphous silica, the mineral separates were washed with distilled water. Finally the separates were treated with 1.0 M HNOs for 12-20 hr, rewashed with distilled water and dried. Feldspar separates were leached with 1.6 M HNOs only. Portions of the minerals (particularly olivine) were dissolved by the lea&ing procedure. However, examination of the final separates in immersion oils revealed no new phases and indicated a mineral purity of 999 per cent or better. Sluggish diffusion in silicates and the faot that comparable results were obtained with the microprobe, indicate that it is unlikely that leaching preferentially removed phosphorus from the mineral lattices. For the analysis of small amounts of mineral separates, we have used a very sensitive neutron a&iv&ion procedure similar to that described in detail by GRXCH~NLAND (1987); it was convenient, but not neoessary, to use the same method for the bulk rook analyses ss well. In brief, 10 mg samples were irradiated for 2 hr in a thermal neutron flux of 7 x 10’s n/oms/sec. After about 10 days decay, samples and standards were dissolved in HF-HClO& in the presence of phosphorus oarrier and the PsB activity purifled by cation exchange and molybdate precipitation. The chemioal yield was determined by weighing the eunmonium phosphomolybdate. The phosphorus activity was determined by counting in a low-level 8_ oounter, and radiochemical purity was conthmed by following the deocty for at least 2 weeks. RESULTS

The analyses (Table 1) are limited to groundmass, olivine, plagioclaae, iron oxides, It is possible that hydrous ferromagnesian or other silicates can and pyroxenes. accept more phosphorus. Our results on the distribution of phosphorus between silioate minerals (Tables 1 and 2) are in fair agreement with the hypothesis of KORITNTCJ (1965) which states that the P/Si ratio of silicates is greater the less the degree of polymerization of SiO,4- tetrahedra. Thus, the P/Si ratio is highest in olivine which has unshared silica tetrahedra. Pyroxenes and feldspar with SiO,-* and (Al, Si)O, groupings have

A quantitative

indicator of crystallization

495

differentiation of basaltio liquids

Table 1. P,Os contents (ppm) in basaltic groundmass and phenocryst minerale

Cm(unleeehed)

Rook No.

Allgite Pl8gioehWe (leached with (leaohed with HF and HNO,) HNc,)

Bronzitehypemthene Olivine (lee&d with (leached with HF end HNO,) HF and HNO,)

Ilmenite

basalk,

Subalkaline

(MannaLa, Hawaii) HML1950SE HML1907

2100 f 2700 f

200 200’

50 & 10

20 f 50 f

5 50’

30 f 90 f

5 50+

90 f 10 309 & 80.

5500 f 5300 f

300 500

70 f 80 f

40* 60 f

10 10

40 f eo*

10 10

250 f 400 f

50 100

4800 f

200

so*

10

so*

10

Subalkaline

andssite,badi% Hawaii) HK1955KI HK1955AP (gihuee,

10 10

300 f

100*

Alkaline biwalt (H&&da, Maui) HM15

* Reeulta obtained on unleaehed materials with mioroprobe by A. T. Anderson using sn apetite standard, 0.2 x 10-e A specimen current, 15 KV eoeelerating potential, counting time 200 sec. PET oryetal. Errors estimatea are baeed on reproducibility and square root of the cum of the oounts on peek end background.

Table 2. Individual mineral distribution factors for phosphorus Distribution factor

Rock Type Pl

Subalkaline basalt Subalkaline Andesite basalt Alkaline basalt

AW aP*06

OPX aP*O5

0.024

0.009

0.014

0.043

0.014

0.009

0.009

o-055

the

ratios:

OLP*05

I1 aPP06

0.05

0.019

0.017

.

Pl

appo6, etc. are

01

aP2O5

.

PsOs m mmeral P,Os in groundmass

Superscripts Pl, Aug, Opx, 01 and 11 refer to plagioclase, augite, orthopyroxene, olivine, and ilmenite.

lower P/Si ratios. Contrary to Koritnig’s hypothesis, our results on pyroxene and plagioclase, like those of HENDERSON (1968), indicate that the P/Si ratio may be slightly lower for pyroxene than for plagioclase. The individual mineral distribution factors calculated from our analyses (Table 2) are all smaller than O-04. For a phenocryst assemblage of 60% plagioclase, 40 % pyroxene, and 10 % olivine the crystal aggregate distribution factor is about O-02. DISCUSSION

To facilitate a quantitative comparison of the fractionation trends of various basaltic liquids a chemical parameter of fractional crystallization must have two 5

496

A.

T. ANDEMO?;

and L. I’.

GHEEXLAND

attributes: (1) it must have a clear and exact physical significance, and (2) its physical meaning must be the same, quantitatively, for all magma series. The phosphorus ratio satisfies both of these requirements. The phosphorus ratio is practically equal to the mass fraction of residual liquid (Table 3). In Table 3 are tabulated the true phosphorus ratios which would result from ideal fractional and ideal equilibrium crystallization with the phosphorus distribution factor equal to 0.01 and 0.04. The phosphorus ratios are 0~000-0~040 larger than the fraction of residual liquid. Thus, the phosphorus ratio ideally exceeds the fraction of residual liquid by 0~080-0~040 depending upon the distribution factor and the extent of crystallization. If our measured distribution factors are typical of all basalt& then the phosphorus ratio gives a good measure of the proportion of ideal crystallization. Table 3. An evaluation of the phosphorus ratio as an approximation of the fraotion of residual liquid

I.000

1.000

l-000

I.000

1.000

0.900

0.901

0.904

0.901

0.904

0.800

0.802

0.807

0.802

0.808

0.700

0.703

0.710

0.703

0.712

0*600

0.603

0.612

0.604

0.616

0.500

0.603

0.514

0.505

0.520 0.424

O-400

0.404

0.416

0.406

0.300

0.304

0.315

0.307

0.328

0.200

0.203

0.213

0.208

0.232

0.100

0.102

0.110

0.109

0.136

0.010

0.010

0.012

0.020

0.050

f= the frection of residual liquid. PO/P,, and PO/ Pfe = the phosphorus ratio of the residual liquid according to ideal fracticmalcrystallization with the phosphorus distribution faotor equal to 0.01 e.g. pyroxene) and 0.04 e.g. olivine) respectively. PO/P,, and PO/P,, = the phosphorus ratio of the residual liquid according to ideal equilibrium crystallization with the phosphorus fractionation factor equal to 0.01 and 0.04 respectively.

With the onset of crystallization of a phosphorus-rich mineral, the crystal aggregate distribution factor for phosphorus will be large and the phosphorus ratio will lose quantitative significance. Most basalt8 have less than 0.3 wt. % of P,O, and commence to crystallize apatite if P,O, in the liquid exceeds about 1.0 wt. % (e.g. WAUER, 1960). Consequently, the phosphorus ratio ideally approximates the fraction of residual liquid for approximately the first 70 per cent of ideal crystalhzation of most b&X&c liquids. Crystalhzation of a liquid in nature is a complex process comp~ed to the ideal models. The sluggish difision of chemical elements and the low conductivity of heat through minerals and melts lead to chemical and thermal gradients within the system of crystallizing magma. Two deviations from the ideal models result: (1) Some liquid is trapped and isolated together with the crystals and (2) due to the thermal gradient, static, but not trapped, liquid can attain a different eompo&Gon

A quantitative indicator of crystallization differentiation of basaltic liquids

497

compared to that of the free liquid by means of diffusion. The composition of the trapped liquid generally is more highly fractionated than that of the free, residual liquid. (Our usage of the term : trapped liquid, is more general than that of WAGER et al. (1960), who requires the trapped liquid to have the same composition as the free, residual liquid). The free liquid can maintain a fairly uniform composition by means of convection. The real physical significance of the phosphorus ratio depends upon the proportion and composition of trapped liquid. In the following paragraphs we discuss the nature of this dependence and show that, for comparative purposes, it is the ideal, rather than the real, significance that is useful, because the ideal meaning is independent of the proportion of trapped liquid and trivially affected by its composition. The following general equation describes the behaviour of any element A during complex fractional crystallization which involves trapping of residual liquid

(&A,)

= fl’ - kc@+ @ - l)Ql

This expression is derived in Appendix 1. A, and A, are the weight concentrations of A in the original and residual liquid, respectively. The weight fraction of the residual liquid is f. ad and ad. are the distribution factors for the crystals and the trapped liquid. k is the ratio of trapped liquid to solid plus trapped liquid. The expression assumes that the trapped liquid behaves like an additional phase separating from the liquid. The volume in which diffusion takes place and in which the liquid has variable composition is neglected. The process is one of total fractionation, with no reaction between solids and liquid subsequent to crystallization. Although alternative assumptions and expressions are possible, we believe our simplifying assumptions are as realistic as are necessary in order to treat natural crystallization phenomena quantitatively. There are two special cases of the general expression: (1) if k = 0 (no liquid is trapped), it reduces to the simple, ideal fractionation law and (2) if k # 0, but ad. = 1 (trapped liquid and free residual liquid have the same composition), the resultant expression differs from the ideal law by only a multiplication constant. Consequently, if trapped and residual liquid are identical in composition the relative behaviour of two or more elements during crystallization is the same as in the ideal case and independent of the fraction of liquid trapped. The physical significance of the phosphorus ratio does depend upon the stage of fractionation of the trapped liquid. However, the extent of the dependence generally may be small. This fact is best illustrated by example. Let k = O-4, C(l-a~) = aA. and C = O-8. This means that the mass of the trapped liquid is 40 per cent of the total crystals plus trapped liquid and that the trapped liquid is 20 per cent more highly fractionated than the free residual liquid. With crystallization, two elements A and B with a, = 0, and aB = 2 (such as phosphorus and magnesium) are related to each other by an apparent ideal distribution factor of 2.04 rather than the true 2.00 value. This difference is negligible for all practical purposes. Although the significance of the phosphorus ratio is changed only trivially by different degrees of equilibrium attainment during crystallization, differentiation of other elements can be strongly influenced by the extent of equilibrium, because they may enter the lattices of the major igneous minerals. Therefore, the degree to which

498

A. !I’. ANDERSONand L. I’. CREESLSXD

equilibrium is attained during crystallization can affect the behaviour of other elements relative to that of phosphorus. The physical significance of the phosphorus ratio (in ideal terms) is approximately the same for both alkaline and subalkaline basalt. Although there may be a measurable dependence of the phosphorus distribution factors upon liquid composition, the absolute values of the factors are in every case too small to appreciably affect the accuracy of the estimate of the fraction of residual liquid. Hydrous silicates may accept more phosphorus than anhydrous silicates, but such minerals are unusual in basalts and KORITNIG’S (1965) amphibole analyses suggest that any effect would be small. In sum, the phosphorus ratio can be used to estimate the fraction of residual liquid within the same narrow limits of accuracy (0.04 to 0.00) for ideal crystallization of both subalkaline and alkaline basaltic liquids. In summary, the phosphorus ratio is an approximation of the fraction of free liquid which would remain, if the liquid completely separated from the crystals, as in the ideal models of crystallization. If trapping of liquid occurs, the phosphorus ratio is not an approximation of the real fraction of free residual liquid, but is, nevertheless numerically equal to the ideal fraction of liquid which would remain if no liquid were trapped. This idealized significance of the phosphorus ratio is the same, regardless of basalt magma composition. Consequently, relative to the phosphorus ratio, the behaviour of chemical elements with crystallization depends on only the crystal aggregate distribution factors and the extent of equilibrium. Practical use of the phospitorus ratio, the fractionation

diagram

Plotted logarithmically, the variation of the concentration of any element, A, vs. the logarithm of the phosphorus ratio as an abscissa has an especially useful physical significance. The expression d(log A,)/d(log f) = CI~ - 1 can be derived from the ideal fractionation law: (A,/A,) = f (l-uA) (see p. 501 for explanation of symbols). In the integration used to derive the fractionation law a, is considered a constant. The above differential equation, however, results from a pre-integration equation where we can consider aA to be a variable. The slope of a tangent to a variation curve of log A vs. the logarithm of the phosphorus ratio (wf) approximates (aA - 1). aA is always positive so that permissible slopes due to ideal fractional crystallization must be greater than - 1. Hereafter we will refer to a graph of the logarithm of the concentrations of substances in the liquid vs. the logarithm of the phosphorus ratio as a p?ws@orus fractionation diagram. A fractionation diagram shows the logarithmic variation of concentrations as a function of the logarithm of the amount of liquid which would remain, if crystallization were ideal. Changes of slope of curves on a fractionation diagram are independent of the proportion of liquid trapped and insensitive to its composition and reflect, quantitatively, changes of the crystal aggregate distribution factor, aA. A change in the slope of a fractionation curve can result from either or both of two causes : (1) An alteration of the individual mineral distribution factors and (2) a change in the relative abundances of the crystallizing minerals. If enough distribution factors for the separate minerals are known, then the relative proportions of the crystallizing minerals can be computed by solving simultaneous equations.

A quantitative indicator of cryetallizationdifferentiationof baaaltio liquids

499

Variation curves on fractionation diagrams can be affected by non-crystallization processes such as assimilation and gas transfer. In general, the fractionation diagram is not relevant to such cases. However, the restriction that slopes be greater than - 1 is a necessary condition of crystallization control of variations evident on a fractionation diagram. In constructing a fractionation diagram based on the phosphorus ratio, it is necessary to know the initial concentration of P,O,(P,) in the liquid, because P, has direct bearing on the estimation of the stage of fractionation. This choice of P, can be arbitrary but is best made on the basis of petrologic and field criteria. Since the slopes of the variation curves at any particular value of P, are independent of P,, it is not necessary to know P, in order to deduce either the chemical change with P, or the crystal paragenesis responsible for that change. Comparison

ofthephosphwus

ratio with

other indices

ofcy8tallization

di$!erentiation

Some of the most commonly used parameters of crystallization differentiation are plotted in Fig. 1 against the phosphorus ratio for Alae lava lake (Peck et al., 1966). Of those shown, the curve for the solidification index of KUNO et cd. (1957) most closely approaches the ideal curve; however, the curve for potash is even better. The curve for titania better parallels the ideal curve than either those of the ma& index (SIMPSON,1954; WAGER, 1956) or the differentiation index (THOBNTON and TUTTLE, 1960) throughout most of the range shown. It appears then that these commonly used parameters of crystallization differentation, although qualitatively adequate, are non-linear and thus do not allow quantitative comparisons. Most of these parameters of differentiation are not intended to be quantitative indicators of 0.3

0.4

0.9 0 0

0.7

0.6

0.5

0.4

C

(P,O,)dP,O,)f

Fig. 1. Indexes of crystallizationdifferentiationplotted against phosphorusratio, Alae lava lake, Hawaii. SI (circles) = solidification index (KUNO et al., 1967), DI (triangles) = differentiation index (THORNTONand TUTTLE, IQSO), MI (squares) = mafic index (SIIWSON, 1964). + = IL_,0 ratio, x = TiO, ratio. Data from I?ECKet al. (1966).

A. T. ANDERSON and

500

L. I’.

GREENLANII

crystallization but, rather, to be useful for classification purposes. The differentiation index, for example, indicates where a rock stands with respect bo the petrogeny residua system : SiO,-NaAlSiO,-KAlSiO,. The phosphorus ratio, on the other hand, tells where a rock stands with respect to the beginning of idea1 crystallization. Calculation Hawaii

of mineral paragenesis

from

a fractionation

diagram:

Alae

lava lake,

The crystallization of Alae lava lake, Hawaii (PECK et al., 1966) was studied by drilling through the crystallizing crust into the underlying melt. We have deduced a crystallization sequence from the published analyses of the initial lava and residual liquids for comparison with that observed by PECKet al. (1966) in thin sections of the cored lava specimens. Figure 2 is a fractionation diagram for Alae lava lake. Because the bulk distribution factors are related to slopes on this diagram by a = 1 + M (where Jf is the slope) and because a few mineral distribution factors are known approximately, it is possible to deduce the general outline of the crystallization of the lava lake. The alumina curve is quite even and nearly horizontal. Plagioclase is the principal host

‘--I------L--i 1" 0.9 0.8

07

0.5

0.4

0.3

(P~o&/(P*O,)f

Fig. 2. Fractionation

diagr:sm of bcsxslt and residue1 liquid from Alae lava lake, Hawaii. Data from l?I?XK et a.?. (1966).

A quantitative indioator of crystallizetion differentiation of basaltio liquids

601

mineral of alumina and generally contains roughly twice as much alumina as the liquid. Thus with iW = 0, a = I, and &plagioclase ~s~bution factor of 2, cryst4liz&ion of about 50 per cent plagioclase is indicated. For TiO, we see a steep rise followed by a sharp deoline with decreasing PJP,. The steep rise, H w 7 1, 01N 0, indicates little if any titanium crystallization. The turning point comes at about 60 per cent residual liquid and indicates the onset of crystallization of Fe-Ti oxides. Other, only slightly more complex, “thumbnail” observations can be made quickly as indicated on p. 503. The orystrtllizingassemblages which we deduced from the fractionation diagram (Fig. 2) are compared with those observed by PECK et al. (1966) in Table 4. It should be noted that these assemblages are the relative proportions of minerals which crystallize over an infinitesimally small interval of crystallization (d( 1 - f)). PECK et al. (1966) give a graph of the cumulative percentages of minerals and residual glass which we have transfo~ed

into the values quotc?d iu Table 4.

Table 4. Calculatedt and observedj relative abundances of crystallizing minerals, Alae lava lake, Hawaii Liquid Ml 60 40 30

Augite

Olivine

Plagiocliwe

Oxidea

Calc.

Obs.

c?&.

Obs.

Cslo.

Obs.

Calc.

Obs.

-0.08 0.006 -0.02 - 0.047

0.00 0.00 0.00 0.00

O-86 0.67 063 963

0.68 0.62 0.60 0.49

0.40 0.41 0.42 0.42

0.38 0.30 0.29 0.28

0.08 0.14 0.16

O-07 0.18 0.21 0.23

fh=§ 1.18 1.17 1.17 1.16

t Calculated by (a) assumingthat the pementof liquid equals (P,O&,/(PtO&f, (b) by using mineralfraetionstion fsotorabased on dsts given by &rats snd Richter, 1966; and (a) by using bulk frsctionetion fsotoradeduaed from a frautionationdiagram (Pig. 2). The arithmetiaapproxim~tiomrare expmizmdin Appendix 2. $ Observed abuodsnoesare deduoedfrom the d&a Of PEGKti al. (1966,Fig. 4) by fir& oonverting, to a weight b&s, seoondby plotting the changein the absolute abundanoeof uryst& ~ersuaper cent of glass rwn&ring, and third by normalizingthe total changein oryutalabundauw to 100Per cent. 6 The sum of the oaloulatedmintualfmotions should equal 190, if all mineral and bulk fractionationfactora are correot,and if the nrithmeticapproximationaare justified (sea Appendix 2).

The details whereby we calculated the crystallizing mineral assemblages are recorded in Appendix 2. Briefly, soda and lime, a;nd alumina and lime are used simultaneously to determine plagioclase ctnd augite. Magnesia is used to estimate olivine after allowing for augite. Titani& is used to estimate combined oxides, The degree of agreement between the calculated and observed crystallizing mineral assemblages is encouraging. Plagiocl;ase and augite are the dominant minerals in both cases and they crystallize in &pproxim~~ly equal proportions. Our ~tim&tions suggest that olivine primarily underwent resorption, whereas it is observed to undergo no change in a;mount throughout the range of fractionation stages covered by the residual liquids. The closest agreement is that for the oxides which is perfect within the limits of procedural errors. The over-all agreement is surprisingly good considering that the slopes of the fractionation curves can only be estimated to an accuracy of about &O-05, and that the ~ner&l/~quid fraction factors are assumed const~t (which they certainly are not). Since the individual mineral and crystal aggregate distribution factors are arrived at independently of each other, an internal summation test can be applied to the calculated estimates of fractions of crystallizing minerals. Their sum should equal

502

,4. T. ANDERSON and L. P. GREENLAND

unity. The last column of Table 4 gives these sums. Most are about 20 per cent larger than unity. Incorrect mineral distribution factors are the most probable cause of this discrepancy. The result of the foregoing exercise is an estimate of the relative proportions of crystal phases which crystallize as a function of crystallization. A similar result could be obtained by the more familiar method of subtraction of crystals. The advantage of the fractionation approach is that inspection of a fraction diagram gives a semiquantitative impression of the ideal crystallization sequence. Visual or “thumbnail” estimates can be made directly from the fractionation diagram using the following relationships : wt. y. phg.

m 0.5 a&(&, x 100

wt. o/oaugite M O-7 acaOx 100 - 0.8 (wt. % plag.) wt. o/oolivine =

0.15

aMso

x 100 - 0.4 (wt. % augite)

wt. o/0oxide w O-1 aTiOp x 100. Although the detailed algebraic expressions given in Appendix 2 may appear unwieldy, in practice they can be reduced to simple approximations such as the above which are easy to use and which are probably valid within error limits of lo-20 per cent for most basaltic magmas in which a low calcium pyroxene is rare or absent. CONCLUSIONS

The phosphorus ratio is a quantitative measure of ideal fractional or ideal equilibrium crystallization. It ceases to have quantitative signifioance with the crystallization of a phosphorus-rich mineral (e.g. apatite) which occurs well after about 70 per cent crystallization in most basalts. The fraction of liquid which ideally would remain may be up to about O-04 less than the phosphorus ratio, or up to 0.5 less if trapping of liquid occurs, However, trapping of liquid is of little consequence to comparisons between magma series which reveal differences in chemical fractionation due primarily to variations in original bulk composition (which controls the mineralogy and hence the crystal aggregate distribution factors) and degree of equilibrium attained during crystallization. Variation curves on a fractionation diagram have slopes which are approximately equal to (1 - Ed) where aA is the distribution factor. The fractionation diagram reveals the sequence of ideal crystallization of a liquid through knowledge of the individual distribution factors and the slopes of the variation curves for the chemical elements. The abundance of crystallizing minerals calculated from a fractionation diagram for Alae lava lake Hawaii, agree with those actually observed (PECK et al., 1966) within error limits from 0 to 20 per cent. The phosphorus ratio appears to serve as a useful quantitative yard&ok of crystallization differentiation. As knowledge is gained of the individual mineral distribution factors, the usefulness of fractionation diagrams to petrologic interpretation should increase. Acknowledgments-We record our debt to our colleaguea at the U.S. Geologioal Survey, DAVJD GO~FR~D, ROBERT I. TILLING, DALLAS L. PECK, and Z. S. A~~sowwwt for debate, oritieisms,

A quantitative indicator of crystallization differentiation of basaltic liquids

503

D who convinoed us of the need and helpful suggestions. Partioular thanks go to Dr. GOTTF&IE to improve upon the existing parameters of arystallization differentiation. Prompt reviews by the journal referees (B. W. EVANS, W. C. PHINNIEY,P. G. -IS and P. HENDERSON)are gratefully acknowledged. We are also grateful to P. HENDEE~ON, Glasgow University, for sending us a copy of his paper (1968) prior to its publication.

REFERENCES GREENLANDL. P. (1967) Determination of phosphorus in silicate rooks by neutron aotivation. U.S. Ueol. Surv. Prof. Papr 6%-C, C137-C140. HENDERSONP. (1968) The distribution of phosphorus in the early and middle stages of fraotionation of some basic layered intrusions. Beochim. Comochim. Acta 83, 897-911. KORITNIO S. (1966) Geoohemistry of phosphorus. I. The replaoement of Si’+ by Ps+ in rockforming silioate minerals. Beochim. Coemochim. Acta a@, 361-371. KUNO H., YABUSAKI K., IIDA C. and NA~ASHIMAK. (1957) Differentiation of Hawaiian magmas. Jap. J. Geol. Geogr. @@, 179-218. PECK D. L., WRIQHT T. L. and MOOREJ. G. (1966) Crystallization of tholeiitio basalt in Alae lava lake, Hawaii. Bull. VoZcanoZ.@@, 629-656. RAYLZIUH J. W. S. (1896) Theoretical considerations respecting the separation of gases by diffusion and similar processes. Phil. &fag. 42, 493498. SIMPSONE. S. W. (1954) On the graphical representation of differentiation trends in igneous rocks. &ol. Mag. 91, 238-244. THORNTONC. P. and TITLE 0. F. (1960) Chemistry of igneous rocks: I. Differentiation index. Amer. J. 6%. @5@, 664-684. WAUER L. R. (1956) A chemical definition of fraotionation stages as a basis for comparison of Hawaiian, Hebridean, and other basio lavas. Beoohim. Coemochim. Acta @,217-248. WAGER L. R. (1960) The major element variation of the layered series of the Skaergaard intrusion and a re-estimation of the average composition of the hidden layered series and of successive residual magmas. J. Petrol. 1, 364-398. WAGER L. R. (1963) The mechanism of adcumulus growth in the layered series of the Skaergaard intrusion. Min. Sot. Amer. Spec. Paper 1, Symposium on Layered Intruaiow, pp. l-9. WAQER L. R., BROWN G. M. and WADSWORTHW. J. (1960) Types of igneous cumulates. J. Petrol. 1,73-85. WAUER L. R. and DEER W. A. (1939) Geological investigations in East Greenland. Part III. The petrology of the Skaergsard intrusion, Kangerdlugssuch. Medd. chternland105, l-352.

APPENDIX

~-DERIVATION

OF A GENERAL EQUATION WHICH DESCRIBES

COMPLEX FRACTIONAL CRYSTALLIZATION Consider a system to oonsist of three parts which are considered as fractions of the whole. The mass fractions of these three parts are defined as follow: (1A) (1B) (1C)

f = the fraction of the free, residual liquid; f* = the fraction of the trapped liquid; l-f-f*

= the fraction of solids.

Let the distribution of substance A among these three parts be described thus: (2A) (2B) (2C)

A, = the weight conoentration of A in the free, residual liquid; a_pA, = the weight concentration of A in the liquid being trapped; aAA, = the weight concentration of A in the crystallizing aggregate.

504

and L. !?. ~REEHLAND

A. 'F.ANDERSON

Let an infinitesimal change in the fraction of trapped liquid be a constant proportion of an infinitesimalchange in the fraction of solids plus trapped liquid such that: dj* = k[d(l -f

- f*)

+ df*] = --kdf and, by substitution, d(1 - f --f*)

= (k - l)dj.

The differential equation for the conservation of the mass of ,4 with an infinit,esimalchange in the crystallinity of the system is: d(fA,)

(3)

- f - f*)

+ ora,Afdf* + aaA,d(l

Substitution, expansion of d(fAz),

Uf A,

and rearrangementgive: -7

df

((1 - k)a_‘j + kcca* - 1).

Taking aA, ag+ and k as constants, and taking the limits: definite integration yields: A = f (I_$

(54

= 0.

kaA* + (k -

f = 1, A, = A,, and f = f, A, = A,,

%A).

If k = 0 (i.e. there is no trapped liquid), the expression reduces to that which describes ideal fractional crystallization: A $

= f(l-Q).

The solutions of the general equation (5A) for k and xA9 are: (5A’)

k = [log (&/&)/log

fl +

aA -

aA -

1

CZA.

W”) Obviously aA* has no meaning if k = 0. There are two limiting C&BBS of ag*: (a) aA* = 1, (the compositions of the trapped and free liquids are identical); (b) aAl = Ic(l-“~), (the composition of the trapped liquid is ideally fractionated in proportion to its maas fraction of the total solids plus trapped liquid). If (a), then (5A) reduces to:

(W

A 2

= f Cl+k +

(k -

1)~)

Af

or (SC’)

k

=

log QLb%)llogf +

1

aA-

In general aA. will have some value between unity and k(l-=‘n) so that we can write: (6)

aA* = C(l-a~)

where k 2 C I 1.

C is thus a new constant which is inversely related to the extent of fractionationof the trapped liquid. If the trapped liquid is ideally fractionated, then G = b. If the trapped liquid is only 20 per cent fraotionated oompared to the free liquid then C = (1 - 0.2) = 0.8, eta. C is a constant characteristicof the process of liquid trapping and is the same for all elements only if all trapped liquids have oompositionswhich would result from ideal freotional orystallization.

A quantitative indioator of crystallization differentiationof basaltic liquids APPENDIX

%-ESTIMATION

OF CRYSTALLIZING ~KINERAL ASSEMBLAUES, ALAE LAVA LAKE

Ourestimations were arrived at by

using the following formulas:

(14

Aug’ =

(lb)

Pl’ =

(24

Aug’ =

aOaO

Pl

-

zgapOaCaO I4&0

*us Au(l %a0 - aN~BO

aNap - A’a$, Pl ‘Nr+O Pl /a” Alpo* %a0 - aAl*o*ac*o Aur Aus %a0 - ‘AIIOs _

Pln

505

=

aAl,O,

p1

A”aA”g 40.¶

a4O,

(34

Aug = (Aug’ + Aug”)/t

(3b)

Pl = (PI’ + P1”)/2 - Aa%% Ox = aT1o1oI aTIOl AW

(6)

o1

=

aMRo- “HE

-

Oxa%,

amgo

The symbols have the following meanings: Aug, Pl, Ox, 01 8re weight fraotions of au&e, plagioolase, oxides (1: 1 = m8gnetite:ilmenite) snd olivine. Aug’, Aug” and Pl’, Pl” are independent approximetions of Aug and Pl respectively whioh 8re averaged (equations 3a, b) to give Aug and Pl. ao,,o,ax.,0, a40*, ariO,, aMgo m-0 the distribution f8otors for OaO, N8s0, A.&O,, TiOs end MgO (arystaleggreg8te/liquid) whioh 8re reed graphioally from the fmotionation diagram, Fig. 2. Upper and lower o8se subsoripts indioate the minerel phase and oxide constituent to which the individual miner81distribution factors refer, e.g. a$$ is the distribution faator for CaO between augite end liquid. Equafions (la-3b) were used to 08k3ul8teto the fractions of augite and pl8giocl8se from combined oonsidertrtionsof (1) sod8 and lime and (2) alumina and lime. Equetions (4) and (6) were used to estimate the fractions of oxides and olivine from the TiO, and MgO distribution factors and the f?a&ion of augite estimated from equation (38). The individual miner81 distribution factors which we used are as follows: aa = 29, *= O-21, a&, = 2.4, a$3 = 10, a%0 e 6.9 aoX = l-0, a$3 = 1.4, a$0 = 1.1, ai20 = 0*;2, a?:0 = l-6. The source of these faotoma%e data for gl8ss (F-3), augid (F-l), and plagioolws of An 67.5 (MTJRATA and RICHTER,1966),and unpublished data of one of us on the oompositionsof iron titanium oxide phenoorysts and host glass in the 1956lava at the A vent (MACDON~~S and EATON, 1964). We ohose these d8t8 because we thought they best 8pphd to the orystallization of Alee lava lake on account of similaritiesin bulk chemiaal composition between the fractionated flank lavas of the 1960 Kilauea eruption and the least fraotionated residual liquids of Alas lava lake.