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Conversion of the magnetic susceptibility tensor into the orientation tensor in some rocks
Physics of the Earth and Planetary Interiors, 63 (1990) 71—77 Elsevier Science Publishers B.V., Amsterdam
71
Conversion of the magnetic susceptibility tensor into the orientation tensor in some rocks Franti~ekHrouda Geofyzika State Company, Brno (Czechoslovakia)
Karel Schulmann Central Geological Survey, Prague (Czechoslovakia) (Accepted for publication 1 November 1989)
ABSTRACr
Hrouda, F. and Schulmann, K., 1990. Conversion of the magnetic susceptibility tensor into the orientation tensor in some rocks. Phys. Earth Planet. Inter., 63: 71—77. The preferred orientation of minerals in a rock with a structural or sedimentary fabric can be referred to by the orientation tensor. This paper describes the determination of the orientation tensor of the crystallographic c’-axes from the rock magnetic anisotropy and the mineral anisotropy degree, in rocks in which the magnetic anisotropy is dominantly carried by one mineral with uniaxial magnetic anisotropy (e.g. phyllosilicates, pyrrhotite, hematite). The orientation tensor of biotite c’-axes in four samples of the Bite~orthogneiss determined in this way is similar to that determined independently through the universal stage measurement of biotite leaves in thin sections and is obtained much more rapidly by the magnetic method.
I. Introduction The preferred orientation of minerals in rocks is traditionally represented by means of illustrative contour diagrams presented on equal-area projections. However, for evaluating large datasets and for mathematical modelling of tectonic processes this method is not convenient. For this reason, the purely mathematical representation of the preferred orientation—the orientation tensor —has been developed (Watson, 1960, 1966; Scheidegger, 1965; Mardia, 1972; Woodcock, 1977; Cobbold and Gapais, 1979; Sanderson and Meneilly, 1981). Anisotropy of magnetic susceptibility (AMS) has been shown to be a very effective tool for investigating the preferred orientation of magnetic minerals in rocks. Originally, the AMS was be0031-9201/90/$03.50
© 1990
-
Elsevier Science Publishers B.V.
lieved to be mostly the result of the dimensional orientation of magnetite in rocks (and less frequently of the lattice orientation of pyrrhotite or hematite). Only recently has it been revealed that it can be controlled by the lattice preferred orientation of paramagnetic silicates in many rocks (Borradaile et al., 1986; Rochette, 1987). The parameters currently used to represent the AMS are different from those utilized in the X-ray pole figure goniometry and other non-magnetic methods. Obviously, it is desirable to represent the preferred orientation of minerals determined by AMS and by non-magnetic methods in the same terms. In our opinion, the best means to do this is to use the orientation tensor in both cases. The purpose of the present paper is to develop a method of converting the susceptibility tensor into the orientation tensor in rocks where the magnetic
72
F. HROUDA AND K. SCHULMANN
ariisotropy is carried by a mineral with uniaxial
magnetic anisotropy (phyllosilicates, hematite, pyrrhotite).
2. Theory The onentation tensor is defined as a 3 X 3 matrix of the sums of cross products of the direction cosines of the linear fabric elements evaluated (see, e.g. Scheidegger, 1965) ~ 1.2
o
~ ~l1rn1
=
~ 1,n1
K11=K(A sin2Ocos24)+B) K 2O sin~4)+ B) 22 = k~( A sin K 29 + B) K33 = K(A cos 29 cos 4)) 12 = K21 = K(A sin K 23 = K32 = k( A sin 9 cos 0 sin 4)) 1C33 = K3, = k( A ~m 9 cos 9 cos 4))
where K is the mean grain susceptibility and B are coefficients dependent on the degree of A,ant-
~ l.rn1
~
~
~rn1n1
~ mn1
individual grain. The components of the latter are (Owens, 1974; Hrouda et al., 1985)
(1)
n12
sotropy of an individual grain or the ‘grain anisotropywith degree’. are defined as follows. For grains oblateThey uniaxial anisotropy
where I,, rn, n are the direction cosines of the i th linear element represented by unit vector and n is the number of the linear elements evaluated. Concisely, this tensor can be expressed as
K= (2K,
o
and for grains with prolate uniaxial anisotropy
=
-~
~ o
(2)
where the tensor o can be named the elementary orientation tensor and its components, when using the conventional polar angles 9, ~ instead of the
+
K3)/3 —
A = (K3 K1)/K 3(1 P,,~)/(2P~ + 1) B = K1/K = 3P~/(2P~ + 1) —
—
K = (K1 + 2K3 )/3 A (K1 K3)/K= 3(P~ 1)/(P~+ 2) B = K3/K = 3/( ~, + 2) —
—
(6)
(7)
direction cosmes, are =
022
=
29 cos24) sin 2 2 ~ ~m 4)
033 = cos9 12 21 0 =0 = 023 = 032 = 013
=
031
=
2O
where K 1 and K3 are the maximum and minimum grain susceptibilities, respectively, and P~= KI/K3 is the grain anisotropy degree. (3\ ~
.
“ “
sin sm 9 cos 0 sin 4) sin 9 cos 9 cos 4)
From eqns. (3) and (5) it is clear that the rock susceptibility tensor can tensor be expressed the elementary orientation of thethrough revolution axis of the susceptibility ellipsoid, if B = 0 for the non-diagonal components
The magnetic susceptibility tensor of a rock containing n magnelic grains with uniaxial magnetic anisotropy can be in the first approximation expressed (see, e.g. Owens, 1974) as
—
k
pK ~ (Ao + IB)
(8)
(4)
where I is !he umty matnx. As p, K, A and B are scalars, it follows from eqn. (8) that the principal directions of the sus-
where p is the percentage of the magnetic mineral in a rock and K is the susceptibility tensor for an
ceptibility tensor are the same as those of the orientation tensor. Consequently, the following considerations will be made only in terms of
k
P~ K —
73
CONVERSION OF MAGNETIC SUSCEPTIBILITY TENSOR
principal values. The principal rock susceptibilities are k11=pK(A ~oi’+nB) n ~ 022 +
)
nB)
\/ a~i ,~ k33 =pK~A
~
033 +
(9)
nB
1=1
Dividing eqns. (9) by n and using the orientation tensor 0 instead of the elementary orientation tensor o yields k11 = npK(A011 + B) (10)
Using the ratios of principal susceptibilities eliminates n, p and k and yields two pairs of equations. The first is for grains with oblate anisotropy L=k1/k2=(A03+B)/(A02+B) P=k1/k3=(A03+B)/(A01+B)
L = (A01 + B)/(A02
+
B)
P=(A01+B)/(A03+B)
=
BL + 2BLP)/AJ
(AL + 2BL (AP + 2BP (ALP BP —
—
—
—
BLP
—
(12)
BP)/AJ
BLP BL)/AJ BL + 2BLP)/AJ —
(14)
the principal values of the orientation tensor can be determined knowing the rock anisotropy and the grain anisotropy degree. For grains with oblate anisotropy 03
=
[P~(2LP
—
L
—
P)/(1
—
[P~:2P— LP—L)/(1
F~)+ LP]/J
—Fe)
+
~1/J
(15)
01[P~(2L—LP—P)/(1—P~)+L]/J and for grains with prolate anisotropy 03 02 01
=
[(2L [(2P
=
R2LP
=
— —
LP F)/(F~ 1) + L]/J Li’ + F)/(P~ 1) + i’]/J —
— —
—
L
—
P)/(P~ 1) —
+
(16)
LP]/J
3. Experimental verification
where L and P are the convenient magnetic lineation and arnsotropy degree, k1 ~ k2 ~ k3 are the principal rock susceptibilities and 01 ~ °2 ~ 03 are the principal values of the orientation tensor. Solving the eqns. (11) or (12) together with the equation 01 + °2 + 03 = 1 (characteristic of the orientation tensor) gives two sets of equations enabling the principal values of the orientation tensor to be determined from magnetic am~. sotropy. For the grains with oblate anisotropy
02 03
—
(11)
and the second for grains with prolate anisotropy
=
BP
where J = (LP + P + L). Substituting for A and B from eqns. (6) and (7) into eqns. (13) and (14) yields equations in which
02
k22=npK(A022+B) k33 = npK( A 033 + B)
01
—
02=(AP+2BP—BLP—BL)/AJ 03=(AL+2BL—BLP—BP)/AJ
1=1 /
01= (ALP n
/
k22 =pK( A
and for grains with prolate anisotropy
(13)
In order to test the usefulness of eqns. (15) and (16) for the determination of the orientation tensor from the AMS, some specimens of the BIte~gneiss (S Moravia), in which the AMS is predominantly carried by biotite, were investigated. The AMS was measured by the KLY-2 Kappabridge (Jelinek, 1973, 1977, 1980) and the orientation tensor was calculated from eqns. (15) using a value of P~= 1.35 for the grain aiusotropy degree (Ballet, 1979; Borradaile et al., 1987; Zapletal, this issue). Then, the high-field magnetic anisotropy was measured in order to separate the paramagnetic and ferromagnetic anisotropy components (Jelinek, 1988; Panna, 1988; Hrouda and JelInek, 1985). Finally, the preferred orientation of biotite was measured in thin sections using a universal stage; poles to biotite cleavage planes, which are parallel to crystallographic pseudohexagonal c’-axes, were measured by the standard procedure described by
74
F. HROUDA AND K. SCHULMANN
TABLE 1
Table 1). The maximum values determined by the
Orientation tensor data of the Bitel gneiss determined from AMS and from universal stage measurement
universal stage are higher, the intermediate values are lower and the minimum values are similar.
Specimen number
Method
km
r
0,
02
03
BIT 40
AMS U-st
45
3.4
0.71 0.84
0.27 0.12
002 0.04
BIT 29
AMS U-st AMS
56
5.5 5.4
67
6.7
0.20 0.06 0.15 0.07 0.23 0.10
o.ii
43
0.69 0.91 0.79 0.88 0.71 0.87
Even though the differences are small, they are outside the range of experimental error. Their existence can be easily explained as follows. (1) The AMS determination comes from the whole specimen (cylinder 2.5 cm in diameter and 2.2 cm in height), whereas the universal stage determination comes from a thin section cut perpendicular to the metamorphic schistosity and to
BIT 36
U-st BIT 43
AMS U-st
0.03 0.06 0.05 0.05 003
a a: C
Turner and Weiss (1963, p. 200). The results are summarized in Table 1 and Figs. 1 and 2. In the table, the first column shows the specimen number, the second indicates the method used, the third gives values of the mean susceptibility (km in the order of 10-6 SI) and the fourth gives values of the parameter r = (k~1 k~3)/(kfl —
a
—
kf3) (indices p and f denote the paramagnetic and ferromagnetic principal susceptibilities). In the fifth to seventh columns the principal values of the orientation tensor are presented. It can be seen in Table 1 that the mean susceptibility of the specimens measured is low, whereas the r ratio of the paramagnetic to ferromagnetic anisotropy is high. This indicates that the AMS in these specimens is controlled mostly by the paramagnetic fraction. As biotite is the only representative of the paramagnetic minerals, the AMS is due to the preferred orientation of biotite. In Fig. 1 the orientations of the principal directions of the orientation tensor of c’-axes of biotite determined by universal stage measurement are presented, as well as the orientations of the principal directions of the whole-rock low-field magnetic susceptibility (determined by the KLY-2 Kappabridge) and of the paramagnetic susceptibility component (determined by high-field torque meter measurements). It can be seen in Fig. 1 that the principal directions of the orientation tensor determined by the universal stage and the grin.. cipal susceptibilities of the whole rock and of the paramagnetic component do not differ greatly. However, the principal magnitudes do differ (see
£a
•
-
~
b
£ A
•
u o Fig. 1. Orientations of principal directions of the orientation tensor of biotite leaf poles (c’-axes) determined by measurement using the universal stage, and orientations of principal directions of the whole-rock susceptibility tensor and the paramagnetlc susceptibility tensor in four samples of the Bite~ gneiss. Filled squares, triangles and circles denote the maximum, intermediate and minimum directions of the whole-rock susceptibilities, respectively. Squares, triangles and circles with dots denote thesquares, same directions paramagnetic susceptibilities. Open triangles of andthecircles denote the minimum, intermediate and maximum values of the orientation tensor. It should be noted that the maximum values of the orientation tensor are parallel to the minimum values of both whole-rock and paramagnetic susceptibilities and vice versa. This is because in biotite crystals the minimum susceptibility is parallel the cleavage coincide pole (c’-axis). Consequently, the minimumtosusceptibilities with the maximum concentration of the c’-axes. Structural cross co-ordinate system, equal-area projection on lower hemisphere.
75
CONVERSION OF MAGNETIC SUSCEPTIBILITY TENSOR
,//
A
2
m
/~BIT 36
20
/
10
,7i 5
BIT 43
-
,“‘
/
f
/
/
/
BIT 29
/ BIT 40
/
/
/ / / 7’ ,p’ /~ / A
,.‘
“I,’
00
0.2
0.4
0.6
ferromagnetic anisotropy components works only with deviatoric tensors and cannot provide the determination of the complete paramagnetic susceptibility tensor. (3) Our method for the determination of the orientation tensor from the AMS assumes magnetic grains of equal size. If the grain size is variable and direction dependent, the orientation tensors determined from the AMS and from the universal stage measurement can differ. (4) In calculating the orientation tensor from the AMS, the grain amsotropy degree of biotite was taken to be P~= 1.35, following the data of Ballet (1979), Borradaile et al. (1986) and Zapletal 85). However, although we have (1990; this issue p.with virtually unchioritized bioselected samples tite, weak chloritization cannot be entirely excluded. As the grain anisotropy degree of chlorites, though variable in general (Ballet et al., 1985), can be much lower (P~= 1.26; Borradaile et al., 1986), even weak chloritization of biotite can result in lowering of the anisotropy degree and in lowering the maximum value of the orientation tensor calculated from the AMS.
B2[T2J
4. Discussion and conclusion Fig. 2. Dependence of the amplitude of the second harmonic term (A 2) of the torque in the ac plane on the magnetic field squared for four samples of the Bitel gneiss with least-squares fit straight line. (The characteristic dependence for the paramagnetic anisotropy is the straight line passing through the origin (Jelinek, 1988).)
the mesoscopic lineation. The biotite leaves making a small angle with this section could not be measured. (2) Even though the AMS is predominantly due to the preferred orientation of biotite, it is not solely due to this phenomenon and the weak influence of a ferromagnetic mineral may also be possible. This can be seen in Fig. 2 where the dependence of the amplitude of the second harmonic term of the torque curve measured in the ac plane on the field squared is plotted. The points for some of the specimens do not lie on perfect straight lines passing through the origin, Our method for separating the paramagnetic and
.
A method of determination of the onentation tensor from the AMS was descnbed for the following conditions: when the rock AMS is carried by one mineral with known uniaxial grain anisotropy; when the grains of this mineral are of equal size; and when the grains do not interact magnetically. Consequently, the method cannot be used to determine the orientation tensor of rocks whose AMS is carried by magnetite, since magnetite grains in general show triaxial anisotropy (due to the shape effect) and the grain anisotropy
degree probably varies from grain to grain according to the shape variation. The situation is much better in rocks whose AMS is carried by pyrrhotite, hematite, mica or chlorite, because these minerals display magnetocrystalline anisotropy independent of the mineral shape which is reasonably constant from grain to grain (see Uyeda et al., 1963; Schwarz, 1974; Ballet, 1979; Beausoleil et al. 1983; Zapletal, 1990;
76
this issue p. 85). Moreover, the AMS of rocks with pyrrhotite or hematite is controlled only by the preferred orientation and not by their high grain anisotropy degree (Hrouda, 1980). As long as the rock is not a monomineralic pyrrhotite ore (if it was, the grains may interact magnetically) the magnetic interaction between the grains is negligible because of the low grain susceptibility of the other minerals concerned (for criteria of the interaction see Owens and Rutter, 1978). The non-equal grain size of the magnetic mineral can play a role only if the grain size is direction dependent. If the size is variable, but not direction dependent, the calculation of the orientation tensor from the AMS provides good results. The main problem of the application of the method to geological problems is to prove that the AMS is carried only by one mineral. Merely identifying this mineral is insufficient because a very
small amount of magnetite is present in many rock types and this can influence the results. To solve this problem, it is recommended that investigation of pilot specimens is undertaken in variable strong magnetic fields to separate the anisotropy components coming from individual minerals (see Owens and Bamford, 1976; Rochette et al., 1983; Hrouda and Jelinek, 1985; Hrouda et al., 1985).
If one is interested in the orientation tensor of phyllosilicates, one can assume that the AMS of rocks with a mean susceptibility <5 < 1O~~’ ~ mostly controlled by paramagnetic silicates
(Rochette, 1987). In this case the AMS can be a very powerful tool because the measurement of one specimen takes only a few minutes and no other method can compete with AMS in this respect. Note that the determination of the orientation tensor from the AMS does not necessarily provide definitive information if the AMS component analysis has not been made. It can be concluded that the orientation tensor can be determined from the AMS in rocks in which the carrier of AMS is pyrrhotite, hematite, micas or chlorites. If the AMS component analysis has been made and the carrier of AMS is wellknown, the determination of the orientation tensor is reasonably accurate. If the AMS is controlled only predominantly by one mineral, and not solely,
F. HROUDA AND K. SCHULMANN
the orientation tensor data calculated from the AMS are not necessarily definitive. Nevertheless, the high rapidity of AMS measurement suggests that the method developed can be an effective tool in the determination of the orientation tensor.
Acknowledgements The Director General of the State Company Geofyzika Brno, Ing. Jaroslav Ibnnajer DrSc., is thanked for permission to publish this paper. The research was supported by the Technical Development Foundation of Geofyzika Brno. References 2~ dans les silicates lamellaires: étude Ballet, 0., 1979. Fe magnétique et Mossbauer. These 3éme cycle, Grenoble, 120 PP. 0., Coey, J.M.D. and Burke, KJ., 1985. Magnetic Ballet, properties of sheet silicates, 2: 1 : 1 layer minerals. Phys. Chem. Mineral., 12: 370—378. Beausoleil, N., Lavallée, P. and Yelon, A., 1983. Magnetic properties of biotite micas. J. Appi. Phys., 54: 906—915. Borradaile, G.,magnetic Mothersill, J, Tarling, in D. aand Alford, 1986. Source of susceptibility slate. EarthC.,Planet. ~. ~., 76: 336-340. Borradaile, G., Keeler, W., Alford, C. and Sarvas, P., 1987. Anisotropy of magnetic susceptibility of some metamorphic minerals. Phys. Earth Planet. Inter., 48: 161—166. Cobbold, P. and Gapais, D., 1979. Specification of fabric shapes using eigenvalue method: discussion. Geol. Soc. Am. Bull., 90:an310—312. Hrouda, F., 1980. Magnetocrystalline anisotropy of rocks and massive ores: a mathematical model study and its fabric implications. J. Struct. GeoL, 2: 459—462. Hrouda, F. and Jeinek, V., 1985. Resolution of ferromagnetic and paramagnetic components, using low-field and highfield measurements. IAGA News, Scientific Session 1.14, Prague. Hrouda, F., Siemes, H., Herres, N. and Hennig-Michaei, C., 1985. The relationship between the magnetic anisotropy and the c-axis fabric in a massive hematite ore. J. Geophys., 56: 174—182. Jelinek, V., 1973. Precision A.C. bridge set for measuring magnetic susceptibility of rocks and its anisotropy. Stud. Geophys. Geod., 17: 36—48. Jelinek, V., 1977. The statistical theory of measuring anisotropy of magnetic susceptibility of rocks and its application. Geofyzika Brno, 88 pp. Jelinek, V., 1980. Kappabridge KLY-2. A precision laboratory bridge for measuring magnetic susceptibility of rocks (including anisotropy). Leaflet, Geofyzika Brno.
77
CONVERSION OF MAGNETIC SUSCEPTIBILITY TENSOR Jelinek, V., 1988. Potential energy density tensor and magnetic anisotropy problems. Phys. Earth Planet. Inter., 51: 361—
364. Mardia, K.V., 1972. Statistics of Directional Data. Academic Press, London. Owens, W.H., 1974. Mathematical model studies on factors
affecting the magnetic anisotropy of deformed rocks.
Tectonophysics, 24: 115—131. Owens, W.H. and Bamford, D., 1976. Magnetic, seismic and other anisotropic properties of rock fabrics. Philos. Trans. R. Soc. London, Ser. A, 283: 55—68. Owens, W.H. and Rutter, E.H., 1978. The development of magnetic susceptibility anisotropy through crystallographic preferred orientation in a calcite rock. Phys. Earth Planet. Inter., 16: 215—222. Parma, J., 1988. An automated torque meter for rapid measurement of high-field magnetic anisotropy of rocks. Phys. Earth Planet. Inter., 51: 387—389. Rochette, P., 1987. Magnetic susceptibility of the rocks matrix related to magnetic fabric studies. J. Struct. Geol., 9: 1015— 1020. Rochette, P., Fillion, G., Mollard, P. and Vergne, R., 1983. Utilisation d’un magnetometre a effet Josephson pour l’analyse de l’anisotropie magnetique des roches. CR. Acad. Sci. Paris, Ser. II, 296: 557—559.
Sanderson, DJ. and Meneilly, A.W., 1981. Analysis of threedimensional strain modified uniform distributions: andalusite fabrics from a granite aureole. J. Struct. Geol., 3: 109-116. Scheidegger, A.E., 1965. On the statistics of the orientation of bedding planes, grain axes, and similar sedimentological
data: U.S. Geol. Surv., Prof. Pap., 525-C: 164—167. Schwarz, EJ., 1974. Magnetic fabric in massive suiphide deposits. Can. J. Earth Sci., 11: 1669—1675. Turner, FJ. and Weiss, L.E., 1963. Structural Analysis of Metamorphic Tectonites. McGraw-Hill, New York, 545 pp. Uyeda, S., Fuller, M.D., BelshC, J.C. and Girdler, R.W., 1963. Anisotropy of magnetic susceptibility of rocks and mineral.
J. Geophys. Res., 68: 279—291. Watson, G.S., 1960. More significance tests on the sphere. Biometrica, 47: 87—91. Watson, G.S., 1966. The statistics of orientation data. J. Geol., 74: 786—797. Woodcock, N.H., 1977. Specification of fabric shapes using an eigenvalue method. Geol. Soc. Am. Bull., 88: 1231—1236. Zapletal, K., 1990. Low-field susceptibility anisotropy of some biotite crystals. Phys. Earth Planet. Inter., 63: 85—97.
71
Conversion of the magnetic susceptibility tensor into the orientation tensor in some rocks Franti~ekHrouda Geofyzika State Company, Brno (Czechoslovakia)
Karel Schulmann Central Geological Survey, Prague (Czechoslovakia) (Accepted for publication 1 November 1989)
ABSTRACr
Hrouda, F. and Schulmann, K., 1990. Conversion of the magnetic susceptibility tensor into the orientation tensor in some rocks. Phys. Earth Planet. Inter., 63: 71—77. The preferred orientation of minerals in a rock with a structural or sedimentary fabric can be referred to by the orientation tensor. This paper describes the determination of the orientation tensor of the crystallographic c’-axes from the rock magnetic anisotropy and the mineral anisotropy degree, in rocks in which the magnetic anisotropy is dominantly carried by one mineral with uniaxial magnetic anisotropy (e.g. phyllosilicates, pyrrhotite, hematite). The orientation tensor of biotite c’-axes in four samples of the Bite~orthogneiss determined in this way is similar to that determined independently through the universal stage measurement of biotite leaves in thin sections and is obtained much more rapidly by the magnetic method.
I. Introduction The preferred orientation of minerals in rocks is traditionally represented by means of illustrative contour diagrams presented on equal-area projections. However, for evaluating large datasets and for mathematical modelling of tectonic processes this method is not convenient. For this reason, the purely mathematical representation of the preferred orientation—the orientation tensor —has been developed (Watson, 1960, 1966; Scheidegger, 1965; Mardia, 1972; Woodcock, 1977; Cobbold and Gapais, 1979; Sanderson and Meneilly, 1981). Anisotropy of magnetic susceptibility (AMS) has been shown to be a very effective tool for investigating the preferred orientation of magnetic minerals in rocks. Originally, the AMS was be0031-9201/90/$03.50
© 1990
-
Elsevier Science Publishers B.V.
lieved to be mostly the result of the dimensional orientation of magnetite in rocks (and less frequently of the lattice orientation of pyrrhotite or hematite). Only recently has it been revealed that it can be controlled by the lattice preferred orientation of paramagnetic silicates in many rocks (Borradaile et al., 1986; Rochette, 1987). The parameters currently used to represent the AMS are different from those utilized in the X-ray pole figure goniometry and other non-magnetic methods. Obviously, it is desirable to represent the preferred orientation of minerals determined by AMS and by non-magnetic methods in the same terms. In our opinion, the best means to do this is to use the orientation tensor in both cases. The purpose of the present paper is to develop a method of converting the susceptibility tensor into the orientation tensor in rocks where the magnetic
72
F. HROUDA AND K. SCHULMANN
ariisotropy is carried by a mineral with uniaxial
magnetic anisotropy (phyllosilicates, hematite, pyrrhotite).
2. Theory The onentation tensor is defined as a 3 X 3 matrix of the sums of cross products of the direction cosines of the linear fabric elements evaluated (see, e.g. Scheidegger, 1965) ~ 1.2
o
~ ~l1rn1
=
~ 1,n1
K11=K(A sin2Ocos24)+B) K 2O sin~4)+ B) 22 = k~( A sin K 29 + B) K33 = K(A cos 29 cos 4)) 12 = K21 = K(A sin K 23 = K32 = k( A sin 9 cos 0 sin 4)) 1C33 = K3, = k( A ~m 9 cos 9 cos 4))
where K is the mean grain susceptibility and B are coefficients dependent on the degree of A,ant-
~ l.rn1
~
~
~rn1n1
~ mn1
individual grain. The components of the latter are (Owens, 1974; Hrouda et al., 1985)
(1)
n12
sotropy of an individual grain or the ‘grain anisotropywith degree’. are defined as follows. For grains oblateThey uniaxial anisotropy
where I,, rn, n are the direction cosines of the i th linear element represented by unit vector and n is the number of the linear elements evaluated. Concisely, this tensor can be expressed as
K= (2K,
o
and for grains with prolate uniaxial anisotropy
=
-~
~ o
(2)
where the tensor o can be named the elementary orientation tensor and its components, when using the conventional polar angles 9, ~ instead of the
+
K3)/3 —
A = (K3 K1)/K 3(1 P,,~)/(2P~ + 1) B = K1/K = 3P~/(2P~ + 1) —
—
K = (K1 + 2K3 )/3 A (K1 K3)/K= 3(P~ 1)/(P~+ 2) B = K3/K = 3/( ~, + 2) —
—
(6)
(7)
direction cosmes, are =
022
=
29 cos24) sin 2 2 ~ ~m 4)
033 = cos9 12 21 0 =0 = 023 = 032 = 013
=
031
=
2O
where K 1 and K3 are the maximum and minimum grain susceptibilities, respectively, and P~= KI/K3 is the grain anisotropy degree. (3\ ~
.
“ “
sin sm 9 cos 0 sin 4) sin 9 cos 9 cos 4)
From eqns. (3) and (5) it is clear that the rock susceptibility tensor can tensor be expressed the elementary orientation of thethrough revolution axis of the susceptibility ellipsoid, if B = 0 for the non-diagonal components
The magnetic susceptibility tensor of a rock containing n magnelic grains with uniaxial magnetic anisotropy can be in the first approximation expressed (see, e.g. Owens, 1974) as
—
k
pK ~ (Ao + IB)
(8)
(4)
where I is !he umty matnx. As p, K, A and B are scalars, it follows from eqn. (8) that the principal directions of the sus-
where p is the percentage of the magnetic mineral in a rock and K is the susceptibility tensor for an
ceptibility tensor are the same as those of the orientation tensor. Consequently, the following considerations will be made only in terms of
k
P~ K —
73
CONVERSION OF MAGNETIC SUSCEPTIBILITY TENSOR
principal values. The principal rock susceptibilities are k11=pK(A ~oi’+nB) n ~ 022 +
)
nB)
\/ a~i ,~ k33 =pK~A
~
033 +
(9)
nB
1=1
Dividing eqns. (9) by n and using the orientation tensor 0 instead of the elementary orientation tensor o yields k11 = npK(A011 + B) (10)
Using the ratios of principal susceptibilities eliminates n, p and k and yields two pairs of equations. The first is for grains with oblate anisotropy L=k1/k2=(A03+B)/(A02+B) P=k1/k3=(A03+B)/(A01+B)
L = (A01 + B)/(A02
+
B)
P=(A01+B)/(A03+B)
=
BL + 2BLP)/AJ
(AL + 2BL (AP + 2BP (ALP BP —
—
—
—
BLP
—
(12)
BP)/AJ
BLP BL)/AJ BL + 2BLP)/AJ —
(14)
the principal values of the orientation tensor can be determined knowing the rock anisotropy and the grain anisotropy degree. For grains with oblate anisotropy 03
=
[P~(2LP
—
L
—
P)/(1
—
[P~:2P— LP—L)/(1
F~)+ LP]/J
—Fe)
+
~1/J
(15)
01[P~(2L—LP—P)/(1—P~)+L]/J and for grains with prolate anisotropy 03 02 01
=
[(2L [(2P
=
R2LP
=
— —
LP F)/(F~ 1) + L]/J Li’ + F)/(P~ 1) + i’]/J —
— —
—
L
—
P)/(P~ 1) —
+
(16)
LP]/J
3. Experimental verification
where L and P are the convenient magnetic lineation and arnsotropy degree, k1 ~ k2 ~ k3 are the principal rock susceptibilities and 01 ~ °2 ~ 03 are the principal values of the orientation tensor. Solving the eqns. (11) or (12) together with the equation 01 + °2 + 03 = 1 (characteristic of the orientation tensor) gives two sets of equations enabling the principal values of the orientation tensor to be determined from magnetic am~. sotropy. For the grains with oblate anisotropy
02 03
—
(11)
and the second for grains with prolate anisotropy
=
BP
where J = (LP + P + L). Substituting for A and B from eqns. (6) and (7) into eqns. (13) and (14) yields equations in which
02
k22=npK(A022+B) k33 = npK( A 033 + B)
01
—
02=(AP+2BP—BLP—BL)/AJ 03=(AL+2BL—BLP—BP)/AJ
1=1 /
01= (ALP n
/
k22 =pK( A
and for grains with prolate anisotropy
(13)
In order to test the usefulness of eqns. (15) and (16) for the determination of the orientation tensor from the AMS, some specimens of the BIte~gneiss (S Moravia), in which the AMS is predominantly carried by biotite, were investigated. The AMS was measured by the KLY-2 Kappabridge (Jelinek, 1973, 1977, 1980) and the orientation tensor was calculated from eqns. (15) using a value of P~= 1.35 for the grain aiusotropy degree (Ballet, 1979; Borradaile et al., 1987; Zapletal, this issue). Then, the high-field magnetic anisotropy was measured in order to separate the paramagnetic and ferromagnetic anisotropy components (Jelinek, 1988; Panna, 1988; Hrouda and JelInek, 1985). Finally, the preferred orientation of biotite was measured in thin sections using a universal stage; poles to biotite cleavage planes, which are parallel to crystallographic pseudohexagonal c’-axes, were measured by the standard procedure described by
74
F. HROUDA AND K. SCHULMANN
TABLE 1
Table 1). The maximum values determined by the
Orientation tensor data of the Bitel gneiss determined from AMS and from universal stage measurement
universal stage are higher, the intermediate values are lower and the minimum values are similar.
Specimen number
Method
km
r
0,
02
03
BIT 40
AMS U-st
45
3.4
0.71 0.84
0.27 0.12
002 0.04
BIT 29
AMS U-st AMS
56
5.5 5.4
67
6.7
0.20 0.06 0.15 0.07 0.23 0.10
o.ii
43
0.69 0.91 0.79 0.88 0.71 0.87
Even though the differences are small, they are outside the range of experimental error. Their existence can be easily explained as follows. (1) The AMS determination comes from the whole specimen (cylinder 2.5 cm in diameter and 2.2 cm in height), whereas the universal stage determination comes from a thin section cut perpendicular to the metamorphic schistosity and to
BIT 36
U-st BIT 43
AMS U-st
0.03 0.06 0.05 0.05 003
a a: C
Turner and Weiss (1963, p. 200). The results are summarized in Table 1 and Figs. 1 and 2. In the table, the first column shows the specimen number, the second indicates the method used, the third gives values of the mean susceptibility (km in the order of 10-6 SI) and the fourth gives values of the parameter r = (k~1 k~3)/(kfl —
a
—
kf3) (indices p and f denote the paramagnetic and ferromagnetic principal susceptibilities). In the fifth to seventh columns the principal values of the orientation tensor are presented. It can be seen in Table 1 that the mean susceptibility of the specimens measured is low, whereas the r ratio of the paramagnetic to ferromagnetic anisotropy is high. This indicates that the AMS in these specimens is controlled mostly by the paramagnetic fraction. As biotite is the only representative of the paramagnetic minerals, the AMS is due to the preferred orientation of biotite. In Fig. 1 the orientations of the principal directions of the orientation tensor of c’-axes of biotite determined by universal stage measurement are presented, as well as the orientations of the principal directions of the whole-rock low-field magnetic susceptibility (determined by the KLY-2 Kappabridge) and of the paramagnetic susceptibility component (determined by high-field torque meter measurements). It can be seen in Fig. 1 that the principal directions of the orientation tensor determined by the universal stage and the grin.. cipal susceptibilities of the whole rock and of the paramagnetic component do not differ greatly. However, the principal magnitudes do differ (see
£a
•
-
~
b
£ A
•
u o Fig. 1. Orientations of principal directions of the orientation tensor of biotite leaf poles (c’-axes) determined by measurement using the universal stage, and orientations of principal directions of the whole-rock susceptibility tensor and the paramagnetlc susceptibility tensor in four samples of the Bite~ gneiss. Filled squares, triangles and circles denote the maximum, intermediate and minimum directions of the whole-rock susceptibilities, respectively. Squares, triangles and circles with dots denote thesquares, same directions paramagnetic susceptibilities. Open triangles of andthecircles denote the minimum, intermediate and maximum values of the orientation tensor. It should be noted that the maximum values of the orientation tensor are parallel to the minimum values of both whole-rock and paramagnetic susceptibilities and vice versa. This is because in biotite crystals the minimum susceptibility is parallel the cleavage coincide pole (c’-axis). Consequently, the minimumtosusceptibilities with the maximum concentration of the c’-axes. Structural cross co-ordinate system, equal-area projection on lower hemisphere.
75
CONVERSION OF MAGNETIC SUSCEPTIBILITY TENSOR
,//
A
2
m
/~BIT 36
20
/
10
,7i 5
BIT 43
-
,“‘
/
f
/
/
/
BIT 29
/ BIT 40
/
/
/ / / 7’ ,p’ /~ / A
,.‘
“I,’
00
0.2
0.4
0.6
ferromagnetic anisotropy components works only with deviatoric tensors and cannot provide the determination of the complete paramagnetic susceptibility tensor. (3) Our method for the determination of the orientation tensor from the AMS assumes magnetic grains of equal size. If the grain size is variable and direction dependent, the orientation tensors determined from the AMS and from the universal stage measurement can differ. (4) In calculating the orientation tensor from the AMS, the grain amsotropy degree of biotite was taken to be P~= 1.35, following the data of Ballet (1979), Borradaile et al. (1986) and Zapletal 85). However, although we have (1990; this issue p.with virtually unchioritized bioselected samples tite, weak chloritization cannot be entirely excluded. As the grain anisotropy degree of chlorites, though variable in general (Ballet et al., 1985), can be much lower (P~= 1.26; Borradaile et al., 1986), even weak chloritization of biotite can result in lowering of the anisotropy degree and in lowering the maximum value of the orientation tensor calculated from the AMS.
B2[T2J
4. Discussion and conclusion Fig. 2. Dependence of the amplitude of the second harmonic term (A 2) of the torque in the ac plane on the magnetic field squared for four samples of the Bitel gneiss with least-squares fit straight line. (The characteristic dependence for the paramagnetic anisotropy is the straight line passing through the origin (Jelinek, 1988).)
the mesoscopic lineation. The biotite leaves making a small angle with this section could not be measured. (2) Even though the AMS is predominantly due to the preferred orientation of biotite, it is not solely due to this phenomenon and the weak influence of a ferromagnetic mineral may also be possible. This can be seen in Fig. 2 where the dependence of the amplitude of the second harmonic term of the torque curve measured in the ac plane on the field squared is plotted. The points for some of the specimens do not lie on perfect straight lines passing through the origin, Our method for separating the paramagnetic and
.
A method of determination of the onentation tensor from the AMS was descnbed for the following conditions: when the rock AMS is carried by one mineral with known uniaxial grain anisotropy; when the grains of this mineral are of equal size; and when the grains do not interact magnetically. Consequently, the method cannot be used to determine the orientation tensor of rocks whose AMS is carried by magnetite, since magnetite grains in general show triaxial anisotropy (due to the shape effect) and the grain anisotropy
degree probably varies from grain to grain according to the shape variation. The situation is much better in rocks whose AMS is carried by pyrrhotite, hematite, mica or chlorite, because these minerals display magnetocrystalline anisotropy independent of the mineral shape which is reasonably constant from grain to grain (see Uyeda et al., 1963; Schwarz, 1974; Ballet, 1979; Beausoleil et al. 1983; Zapletal, 1990;
76
this issue p. 85). Moreover, the AMS of rocks with pyrrhotite or hematite is controlled only by the preferred orientation and not by their high grain anisotropy degree (Hrouda, 1980). As long as the rock is not a monomineralic pyrrhotite ore (if it was, the grains may interact magnetically) the magnetic interaction between the grains is negligible because of the low grain susceptibility of the other minerals concerned (for criteria of the interaction see Owens and Rutter, 1978). The non-equal grain size of the magnetic mineral can play a role only if the grain size is direction dependent. If the size is variable, but not direction dependent, the calculation of the orientation tensor from the AMS provides good results. The main problem of the application of the method to geological problems is to prove that the AMS is carried only by one mineral. Merely identifying this mineral is insufficient because a very
small amount of magnetite is present in many rock types and this can influence the results. To solve this problem, it is recommended that investigation of pilot specimens is undertaken in variable strong magnetic fields to separate the anisotropy components coming from individual minerals (see Owens and Bamford, 1976; Rochette et al., 1983; Hrouda and Jelinek, 1985; Hrouda et al., 1985).
If one is interested in the orientation tensor of phyllosilicates, one can assume that the AMS of rocks with a mean susceptibility <5 < 1O~~’ ~ mostly controlled by paramagnetic silicates
(Rochette, 1987). In this case the AMS can be a very powerful tool because the measurement of one specimen takes only a few minutes and no other method can compete with AMS in this respect. Note that the determination of the orientation tensor from the AMS does not necessarily provide definitive information if the AMS component analysis has not been made. It can be concluded that the orientation tensor can be determined from the AMS in rocks in which the carrier of AMS is pyrrhotite, hematite, micas or chlorites. If the AMS component analysis has been made and the carrier of AMS is wellknown, the determination of the orientation tensor is reasonably accurate. If the AMS is controlled only predominantly by one mineral, and not solely,
F. HROUDA AND K. SCHULMANN
the orientation tensor data calculated from the AMS are not necessarily definitive. Nevertheless, the high rapidity of AMS measurement suggests that the method developed can be an effective tool in the determination of the orientation tensor.
Acknowledgements The Director General of the State Company Geofyzika Brno, Ing. Jaroslav Ibnnajer DrSc., is thanked for permission to publish this paper. The research was supported by the Technical Development Foundation of Geofyzika Brno. References 2~ dans les silicates lamellaires: étude Ballet, 0., 1979. Fe magnétique et Mossbauer. These 3éme cycle, Grenoble, 120 PP. 0., Coey, J.M.D. and Burke, KJ., 1985. Magnetic Ballet, properties of sheet silicates, 2: 1 : 1 layer minerals. Phys. Chem. Mineral., 12: 370—378. Beausoleil, N., Lavallée, P. and Yelon, A., 1983. Magnetic properties of biotite micas. J. Appi. Phys., 54: 906—915. Borradaile, G.,magnetic Mothersill, J, Tarling, in D. aand Alford, 1986. Source of susceptibility slate. EarthC.,Planet. ~. ~., 76: 336-340. Borradaile, G., Keeler, W., Alford, C. and Sarvas, P., 1987. Anisotropy of magnetic susceptibility of some metamorphic minerals. Phys. Earth Planet. Inter., 48: 161—166. Cobbold, P. and Gapais, D., 1979. Specification of fabric shapes using eigenvalue method: discussion. Geol. Soc. Am. Bull., 90:an310—312. Hrouda, F., 1980. Magnetocrystalline anisotropy of rocks and massive ores: a mathematical model study and its fabric implications. J. Struct. GeoL, 2: 459—462. Hrouda, F. and Jeinek, V., 1985. Resolution of ferromagnetic and paramagnetic components, using low-field and highfield measurements. IAGA News, Scientific Session 1.14, Prague. Hrouda, F., Siemes, H., Herres, N. and Hennig-Michaei, C., 1985. The relationship between the magnetic anisotropy and the c-axis fabric in a massive hematite ore. J. Geophys., 56: 174—182. Jelinek, V., 1973. Precision A.C. bridge set for measuring magnetic susceptibility of rocks and its anisotropy. Stud. Geophys. Geod., 17: 36—48. Jelinek, V., 1977. The statistical theory of measuring anisotropy of magnetic susceptibility of rocks and its application. Geofyzika Brno, 88 pp. Jelinek, V., 1980. Kappabridge KLY-2. A precision laboratory bridge for measuring magnetic susceptibility of rocks (including anisotropy). Leaflet, Geofyzika Brno.
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CONVERSION OF MAGNETIC SUSCEPTIBILITY TENSOR Jelinek, V., 1988. Potential energy density tensor and magnetic anisotropy problems. Phys. Earth Planet. Inter., 51: 361—
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data: U.S. Geol. Surv., Prof. Pap., 525-C: 164—167. Schwarz, EJ., 1974. Magnetic fabric in massive suiphide deposits. Can. J. Earth Sci., 11: 1669—1675. Turner, FJ. and Weiss, L.E., 1963. Structural Analysis of Metamorphic Tectonites. McGraw-Hill, New York, 545 pp. Uyeda, S., Fuller, M.D., BelshC, J.C. and Girdler, R.W., 1963. Anisotropy of magnetic susceptibility of rocks and mineral.
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