GEOLOGICAL SURVEY: 

W» 2 1977

Weak-Field Magnetic

Susceptibility Anisotropy

and its Dynamic Measurement

GEOLOGICAL SURVEY BULLETIN 1418

G>£75"

Weak-Field Magnetic

Susceptibility Anisotropy

and its Dynamic Measurement

By WILLIAM F. HANNA

GEOLOGICAL SURVEY BULLETIN 141

Theory of directional properties of rock 

magnetic susceptibility and their measurement 

using a spinner magnetometer

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1977
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress Cataloging in Publication Data

Hanna, William F 1938-

Weak-field magnetic susceptibility anisotropy and its dynamic measurement. 

(Geological Survey bulletin ; 1418) 

Bibliography: p. 

Supt. of Docs, no.: I 19.3:1418

1. Rocks Magnetic properties. 2. Magnetic susceptibility Measurement. I. Title. 

II. Series: United States. Geological Survey. Bulletin ; 1418.

QE75.B9 no. 1418 [QE431.6.M3] 557.3'08s [552'.06] 76-608101

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402 

Stock Number 024-001-02934-1
CONTENTS

Page

Abstract _____________________________ 1

Introduction _______________..______________-_ 2

Acknowledgment _____________________ 3

Definitions of magnetic susceptibility __________-_ 3

General definition ______________________ 3

Definitions based on magnetization curve __ 3

The definition used in geophysics ___ - - - 6

Definitions based on other parameters _ 8

Units of magnetic susceptibility _____________ 8

Physical and theoretical framework of susceptibility anisotropy 9

Intrinsic vs. apparent susceptibility __ _____ 9

Intrinsic susceptibility of a magnetic domain 16

Magnetocrystalline anisotropy ______ _ -- 16

Intrinsic susceptibility of a multidomain grain 17

Apparent susceptibility of a spherical grain _ __ _ 18

Shape anisotropy of an ellipsoidal grain 22

Combinations of intrinsic susceptibility and shape anisotropies _ 24

Intrinsic susceptibility of a rock body _ 27

Minimizing anisotropy caused by rock specimen shape 30

Apparent susceptibility anisotropy of a rock formation _________ 31

Analysis of susceptibility anisotropy of a rock specimen _______ 31

Tensor components of linear susceptibilities ______ _ 31

Geometric representations ________________________ 35

Calculation of principal susceptibility components ________- 39

Nonlinear susceptibilities _________________________ 39

Anisotropy measurements using a spinner magnetometer _ _ ^1

Principles of measurement _ ______ ___ _____________ 41

Independent measurement of normal susceptibility component _ ^9

Methods of expressing anisotropy _________________ ^0

Mead-type spinner magnetometer _____________ __ *>2

Bhattacharya solution of equation of anisotropy ________ 55

Sample calculation _____________________________ ^2

Examples of anisotropy data ____________________ ^6

References cited _____________________________ __ 70

ILLUSTRATIONS

Page

FIGURE 1. Idealized plot of total magnetization vs. effective magnetic

field ______________________________ 4 

2. Magnetization curve with Rayleigh loop ___________ 6

in
IV CONTENTS

Page

FIGURE 3. Classification of metric units of intensity of magnetization,

magnetic field, and volume magnetic susceptibility __ 14

4. Diagram illustrating tensor combinations of K and N ___ 28

5. Sketch of materials possessing magnetic susceptibility

anisotropy __________________________ 32

6. Representations of susceptibility anisotropy _ _ 36

7. Diagram illustrating the relationship between rock sample

and pickup coil coordinates __________-_____- 42

8. Diagram illustrating the relationship among the coordinate 

axes of the components pickup coil, rotating rock sample, and principal susceptibility ________ _____ 43

9. Diagram showing sequential order of orientations of rock 

sample coordinate system during the measurement 

process ________________________-___- 47

10. Diagram illustrating a Mead-type spinner magnetometer

used for measuring susceptibility anisotropy ______ 53

11. Diagram illustrating the relationship of declination and 

inclination to direction cosines of a principal susceptibility axis ____________________________ 60

12. Plots of maximum susceptibility plane, minor susceptibility axis, and remanent magnetization of rock sample 66

TABLES

Page

TABLE 1. Criteria for definitions of various types of magnetic susceptibility __________________________ 10

2. Anisotropic susceptibility data for 42 volcanic rock specimens obtained by the dynamic method of measurement_ 68
WEAK-FIELD MAGNETIC SUSCEPTIBILITY

ANISOTROPY AND 

ITS DYNAMIC MEASUREMENT

By WILLIAM F. HANNA

ABSTRACT

The apparent magnetic susceptibility, KA, of a uniformly magnetized body, 

such as a magnetic domain, mineral grain, rock unit, or rock formation, depends upon both the intrinsic magnetic ^usceptbility, K, of the magnetic material and the demagnetizing factor, N, associated with the shape of the 

body. This relationship, in tensor form, is

which, for isotropic K and N, reduces to the scalar relation

K

KA =-

1+KN

Both K and N may be either isotropic or anisotropic. Anisotropic K of a 

single-domain magnetic jnineral grain is associated with magnetocrystalline 

anisotropy. Anisotropic K of a rock specimen reflects anisotropic K and (or) 

N of constituent magnetic mineral grains. Anisotropic N, associated with 

shape anisotropy of a body, becomes isotropic as the shape of the body becomes equidimensional. Laboratory measurements of the anisotropic KA of 

an equidimensional rock specimeji can be directly related to the anisotropic 

K of the rock. This amkiotropic K of the rock may, under certain conditions, 

give information about shape aniso,tropies assocated with alinements of 

magnetic mineral grains or magnetic domains within individual grains of the 

rock.

The weak-field apparent susceptibility anisotropy of a uniformly magnetized 

rock specimen can be dynamically measured by means of a spinner magnetometer if interfering second-harmonic signals due to iremanent dipole moments 

are cancelled and if effects due to electrical conductivity and remanent quadrupole moments are sufficiently small. Complete determination of the six 

susceptibility components, from which magnitudes and directions of principal 

axes can be computed, requires an additional measurement of one normal 

susceptibility component using a stationary-sample bridge. Measurements of 

106 miscellaneous volcanic rock specimens, by means of a Mead-type spinner 

magnetometer, and computations based on procedures of P. K. Bhattacharya, 

indicate that 42 specimens have measurable anisotropies and that 30 of these 

specimens have susceptibilities that are sufficiently linear to be represented 

by ellipsoids or spheroids.

1
2 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

INTRODUCTION

The anisotropy of magnetic susceptibility of a rock sample may 

be adequately measured by means of a spinner magnetometer provided that the induced magnetization is related linearly to the 

magnetic field within the rock sample and provided that unwanted 

signals and noise can be eliminated in the measurement process. 

This dynamic process of measuring anisotropy, introduced by 

Bhattacharya (1950), has received relatively little attention because of the general inaccessibility of this dissertation in the 

literature. The main attraction of the dynamic method is that the 

same apparatus routinely used to measure remanent magnetization of samples can, with minimal alteration, be used to measure 

susceptibility anisotropy. For rocks not satisfying the linearity 

condition between magnetization and magnetic field and for those 

generating unwanted signals or noise that cannot be eliminated, 

more sophisticated devices (Graham, 1967) l must be used to measure the associated relatively complex anisotropy.

After the work of Bhattacharya (1950), references to the dynamic method of measurement appeared in a report by Howell, 

Martinez, and Statham (1958) on magnetic properties of sedimentary and metamorphic rocks; in dissertations of Noltimier 

(1965; 1967) and Hanna (1965) on magnetic properties of redbeds and volcanic rocks, respectively; and in the instruction manuals by commercial manufacturers of spinner magnetometers, such 

as Princeton Applied Research Corporation and Schonstedt Instrument Company. However, detailed accounts of the measurement and computation of anisotropy by means of a spinner magnetometer are not generally available. This lack of a comprehensive published account has become evident during the past 10 

years from the numerous requests received for information about 

the dynamic method of measurement. This report is intended to 

provide an overview of the physical and mathematical framework 

of magnetic susceptibility and associated anisotropy, its measurement by means of a spinner magnetometer developed by Professor 

Judson Mead of Indiana University, its computation by the technique of the late Dr. P. K. Bhattacharya, and some examples of 

measurements and results for volcanic rocks.

1 The most highly refined technique for measuring susceptibility anisotropy of a rotating 

rock sample was developed over a 12-year period by the late John W. Graham, a pioneer 

in this type of geophysical investigation. Graham is'credited by Bhattacharya (1950) for his 

assistance in the original study involving the dynamic method.
DEFINITIONS 3

ACKNOWLEDGMENT

I first became interested in problems of weak-field susceptibility 

anisotropy in graduate research directed by Professor Judson 

Mead at Indiana University.

DEFINITIONS OF MAGNETIC SUSCEPTIBILITY 

GENERAL DEFINITION,

When a substance, such as a rock, is exposed to a magnetic field, 

it acquires magnetization, which is said to be caused by the field. 

The functional relationship between this magnetization and the 

effective magnetic field at a point within the substance is defined 

in terms of the magnetic susceptibility. If Heff is the magnitude of 

the effective magnetic field applied in a particular direction relative to coordinate axes of the material, K(Heff) is the magnitude 

of the magnetic susceptibility, and J(Heff ) is the magnitude of 

the magnetization caused by Hetf, the defining relation is

J (Heff ) = [K (He£f)] Heff. (1)

In equation 1, Heff is an independent variable and the component 

of an axial vector, J(Heff ) is a dependent variable of Heff and the 

component of an axial vector, and K(Heff ) is a multivalued function of Heff and the component of a tensor. The magnetic susceptibility is an aptly named quantity, for it expresses the extent to 

which a substance is susceptible of acquiring magnetization in the 

presence of a magnetic field. Because the general definition of 

magnetic susceptibility given by equation 1 contrasts slightly with 

a more restrictive definition ordinarily used in geophysics, we 

shall consider several susceptibility definitions in terms of a plot 

of magnetization versus effective magnetic field.

DEFINITIONS BASED ON MAGNETIZATION CURVE

The relationship between magnetization and effective magnetic 

field at a point within a ferromagnetic substance is shown in 

figure 1. In the diagram, we consider the magnetization to be 

total magnetization (combination of remanent and induced magnetization) and the substance to be homogeneous, at constant 

temperature and pressure, and in an initially demagnetized or 

neutral state. From its zero value at 0, J(H6ff) increases nonlinearly along the dashed line to its saturation value at B, where 

further increase in Heff results in negligibly small, if any, increase 

of J(Heff). If Heff at HB is decreased and reversed to its value at 

HF, J(Heff) decreases along the path BCDE'EF to its saturation
4 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

(Total magnetization)

Remanent magnetization, 

Rayleigh loop;

(Effective

magnetic

field)

FIGURE 1. Idealized plot of total magnetization vs. effective magnetic field 

showing initial magnetization curve OAB and hysteresis loop BCDEFGILB. 

Magnetization curve serves to define initial, differential, reversible, irreversible, total, and maximum magnetic susceptibility.

value in the opposite direction. Another reversal and increase of 

Heff causes J (Heff) to increase along the path FGILB to saturation 

again. Subsequent fluctuations of Heff between the values H« and 

HE drive J(Heff) repeatedly around the path BCDE'EFGILB, 

termed a hysteresis (from the Greek word meaning "to be behind") loop, because the decrease or increase of J(Hetf) to zero 

takes place after the decrease or increase of Heff to zero. The magnetization OD or OG is termed the remanent or permanent magnetization because it exists in the absence of a magnetic field, once 

the material is driven from its neutral state. The magnetic field 

OE or 01 is termed the coercive force or coercive field because it 

is the field required to coerce or force the magnetization from its 

remanent value to zero. The hysteresis loop serves as a basis for 

defining several magnetic-susceptibility terms used in the literature, such as initial, reversible, irreversible, total, and maximum magnetic susceptibility (see, for example, Stoner, 1934; 

Nagata, 1961; Chikazumi, 1964; Standley, 1972).
» DEFINITIONS 5

In terms of the hysteresis loop of figure 1, the initial susceptibility is the slope of the initial magnetization curve OAB at the

d origin 0. The differential susceptibility is defined as [J (Heff) ]

» at any point on the initial magnetization curve, assuming Heff is 

increasing. The reversible susceptibility is defined as the incremental change in J(H6ff) per incremental change in Heff as Heff is 

decreasing. The irreversible susceptibility is the algebraic difference between the differential and reversible susceptibilities at 

a point on the curve. The total magnetic susceptibility is defined 

as the ratio of J (Heff) to Heff at any point on the initial magnetization curve, and the maximum susceptibility is the maximum value 

of total susceptibility. The multivalued characteristic of K(Heff) 

is evident from the difference in slope of curves at points L, A, 

and C, corresponding to Heff=Ho. The vertical separation of these 

points manifests the multivalued characteristic of J(Heff ).

Of special interest in geophysical work are magnetic susceptibilities and magnetizations corresponding to weak magnetic fields 

on the hysteresis lopp. Because the effective magnetic fields in 

crustal rocks near the earth's surface are generally less than 1 

oersted in magnitude, the magnetic state of most rocks is close 

to point D on the hysteresis loop of figure 1. If, for example, 

the magnetic state of a rock is represented by point E' (fig. 

2), corresponding to a slightly negative effective magnetic field 

such as a naturally occurring demagnetizing field, an increase 

in Heff will not necessarily drive J(Hetf) along the path E'D to 

the point of remanent magnetization. Instead, J(H0ff) may be 

driven along the lower branch of an elliptical Rayleigh loop (Rayleigh, 1887) to point E", whereupon it returns along the upper 

branch to point E', provided that Heff is again reversed. This nonlinear change in magnetization for a small change in magnetic 

field results in a small change in remanent magnetization but no 

change in the overall magnetic state of the material given by 

point E'. The reversible susceptibility associated with the small 

Rayleigh loop is commonly taken to be the major axis of the loop, 

shown by a dashed line in figure 2.

If the change in magnetic field is kept sufficiently small, say, 

to less than 1 oersted, the magnetization of most crustal rocks 

undergoes a linear change. For example, if Heft fluctuates between 

the values HD , and H D,,in figure 2, J(Hpff ) will be repeatedly 

driven along the linear path segment D'DD". The slope of the 

linear segment D'DD" determines the reversible weak-field mag-
6 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

J(Heff)

(Total magnetization) r7"

Remanent magnetization < «^-;

Rayleigh loop

E/ Coercive force

/ HP HO,

 -^

H LJ LJ 

D" n£" "eff

(Effective 

magnetic field)

FIGURE 2. Part of magnetization curve shown in figure 1, showing Rayleigh 

loop. Dashed line E'E" is an approximate representation of a linear reversible magnetic susceptibility.

netic susceptibility, and the increment of magnetization associated 

with the nonzero magnetic field is linear-induced magnetization. 

It is this linear-induced magnetization that is ordinarily considered in the definition of magnetic susceptibility as used in solidearth geophysics.

THE DEFINITION USED IN GEOPHYSICS

In much geophysical work involving the interpretation of magnetic anomaly data in tectonic mapping, petroleum exploration, 

and ore prospecting, the magnetic susceptibility is considered to 

be not only a linear quantity but also a single constant. This assumption is justified in studies involving most sedimentary and 

metamorphic rocks and most plutonic and volcanic rocks of felsic
DEFINITIONS 7

to moderately mafic composition. If J is the magnitude of induced 

magnetization, Hew is the magnitude of effective magnetic field, 

and K is the magnitude of intrinsic magnetic susceptibilty of the 

rock, we may write

J = KHeff. (2)

In other geophysical work, such as studies of anomalies associated with highly magnetic mafic rocks or iron ores, analyses 

of the fabric of magnetic mineral grains within a rock, and investigations of the possible influence of induced magnetization on 

the acquisition of remanent magnetization, the magnetic susceptibility may be considered a linear quantity but not a single constant. Instead, the linear magnetic susceptibility in weak magnetic 

fields must generally be expressed by six constants. For these investigations, the magnetic susceptibility may be defined by the 

relation

-» A, -»

J = K.'Hett) (3)

~"^ **

where J is the induced magnetization vector, K is the intrinsic

-» 

magnetic susceptibility tensor, and Hett is the effective magnetic

W -» -» A<

field vector. Because K operates on Hett to produce J, K is known 

as a linear vector operator. This operator is also a second-rank 

tensor, a quantity having nine components, as shown by expanding equation 3 in terms of cartesian components

Jx Kxx (Hett) x + K-icy (Hett) y + K-xz (Heff ) a, 

Jy = Kyx(Hett) as + Kvv (Heft) y + Kyz ("eff) zt

and

Js =Kzx (Heft) a! +K,y(Hett) v+Ki:AHett) e. (4)

Thus, an effective magnetic field applied along, say, the y direction produces an induced magnetization not only along the y direction but also along the x and y directions.

The nine susceptibility components in equations 4 are reduced 

to six components as a consequence of the law of conservation of 

energy. This law requires that

KXV =KXV, Kxs=KzX, and Kyz = Kgy. (5)

Relations 5 indicate that, for example, the magnetization induced 

along the z direction by a magnetic field along the x direction is 

equal to the magnetization induced along the x direction by a field 

of the same strength along the z direction. Rocks for which
8 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

Kxy = K.Xz = KVz=Q (6) 

and

Ka!X = KyV = Ksz (7)

are said to have an isotropic magnetic susceptibility. Rocks for 

which relations 6 and 7 do not hold are said to possess an anisotropic magnetic susceptibility.

DEFINITIONS BASED ON OTHER PARAMETERS

In addition to susceptibilities conveniently defined in terms of 

the hysteresis loop and to those previously mentioned which have 

been adopted for use in geophysical research, many others commonly appear in the literature. A list of many of these special 

definitions of susceptibility, together with criteria for the definitions and comments about distinctive characteristics of each susceptibility, is presented in table 1. Susceptibility terms used in 

the present report include those described as ferromagnetic, volume, crystalline, domain, mineral grain, rock, rock formation, intrinsic, apparent, reversible, linear, constant, principal, weakfield, dynamic sample, stationary sample, A. C., and bulk, as noted 

in table 1. Anisotropic susceptibility treated throughout the report is considered to be linear, a characteristic which implies that 

the induced magnetization is related to only the first power of the 

effective magnetic field.

UNITS OF MAGNETIC SUSCEPTIBILITY

When considering various expressions of units that have been 

applied to magnetic susceptibility, it is important to note that 

metric systems of magnetic units are commonly grouped in three 

ways (Kennelly, 1936; Stratton, 1941; Smythe, 1968). First, they 

belong to either a centimetre-gram-second (CGS) system or to a 

metre-kilogram-second (MKS) system according to the mechanical units used. Second, they belong to either a classical or practical system depending upon the relative magnitudes of certain 

units, such as those of electric current or charge. Third, they belong to either the rationalized or unrationalized system depending 

upon whether factors of 4?r appear in certain expressions, such as 

those of current and charge. The CGS classical units may be subdivided into electromagnetic, electrostatic, or a mixture of these, 

according to arbitrarily chosen relationships of electrical permittivity and magnetic permeability to the speed of light in a vacuum.

The most commonly used systems of magnetic units are shown 

in the classification diagram of figure 3. The values given in figure
3 are all based on unit (value of one) magnitudes for these quantities in the SI metric system.

In rock-magnetic research, the Gaussian system has been used 

most widely. Volume magnetic susceptibility in this system has 

been traditionally expressed as emu/cm3 , an ambiguous designation identical with that used for volume intensity of magnetization in the same system of units. As an alternative to the emu/ 

cm3 Gaussian unit for susceptibility, the dimensionally equivalent 

gauss/oersted has been adopted by Collinson, Creer, and Runcorn (1967), even though it is 4?r times larger than the traditional 

Gaussian unit. The gauss/oersted is numerically equivalent to one 

SI unit.

As use of the internationally adopted SI units increases in 

geophysical work, the most important conversion factor for units 

of magnetic susceptibility will be

.emu 1 -=4* (SI) 

cm3

and

l(SI)=7.96xlO- 2 -^m^-. 

cm3

One SI unit of susceptibility (actually a dimensionless quantity) 

is representative of a susceptibility value characteristic of many 

types of highly magnetic coarse-grained rocks, such as gabbro, 

magnetite sandstone, and magnetite-rich iron ore.

PHYSICAL AND THEORETICAL FRAMEWORK OF 

SUSCEPTIBILITY ANISOTROPY

INTRINSIC VS. APPARENT SUSCEPTIBILITY

Because the induced magnetization of a finite magnetic body is 

measured upon application of a carefully measured magnetic field 

applied externally to the body, this magnetization can be conveniently referred to the external magnetic field rather than to the 

internal effective magnetic field. This convenience of expression 

leads to the definition of apparent magnetic susceptibility,

1 If .£7 (9l\ J **-A «ext> \&J

where J is the induced magnetization vector, Hext is the external 

magnetic field vector, and KA is the apparent magnetic susceptibility tensor. This definition contrasts with expression 3, defining 

intrinsic magnetic susceptibility,

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10 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

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 S* -ag

V AM 3 - O *

^i ffl ? -sl 4J 01

1? A§ C e *» f:-it^

iSlfi

§2"j»S 

 S^l § s gi

i 0* S K

§5-2 9

rt W = s^ 41

"P"" 1M-8 

gw^S .'J Op, ~ci

* 8s *< alls i »S o e 5 m

2 S>3 SJ S S

« o.

 rt t. 57 »£o E

rH"S S 1?

S a S5 .

OS rt-jJ 81^.

 H-g £ rH S 5 *<«D 

.- 2 « o <»

(Dresnei units (s matic" units 

QES the uds abamperes ng Cornelius, 113; 

 s^ss; ^.-2rt

Oa aj ^ ^ |i^? f I_J . s!^l

"coherent" metri selected only systems) CgfSand MkgfS, a ,

infrequen their ofbecause n

derived beMGS systems may 

Pidduck, 11947; andSas[947: 

ll^g^ .1 1 -?-i §

1^5* "w £"

Ofe n ^

0^ g
3 Excludes the Hartree and Quantum-electrodynamic natural systems of units, metrically derived fo

Ludovici system of units, based on values of fundamental constants of nature.

4 The MKSQ system (Sommerfield, 1952) is closely related to the MKSA system. 

5 International system of units, now almost universally adopted for many fields of study (Vigoureux

6 This system is identical with the four-quantity CGS electromagnetic (CGSm) system. The biot of cu

of current in the CGS-emu system. 

7 This system is identical with the four-quantity CGS electrostatic (CGSe) system. The franklin of 

coulomb of charge in the CGS-esu system. 

8 This magnetization is "loop" magnetization, defined, in terms of macroscopic electric current, as 

terms of fictitious magnetic charge (Smythe, 1968). 

9 In practice, the magnetic induction (units of weber per metre) is measured, rather than the mag

ton 1941; Parasnis, 1961) However, because the traditional definition of magnetic susceptibility relates 

than to magnetic induction, units of magnetic field are considered in the present classification scheme.

10 Dipole magnetization in the unrationalized MKS system is l/4ir of dipole magnetization in the rat

11 Susceotibility based on ratio of dipole magnetization to magnetic field in the unrationalized MKS 

rationalized MKS system (Kennelly, 1936). 

12 The unit of susceptibility in the SI system is dimensionless. In some references, the term "(SI)" i

1972). 

1:1 Magnetization units of gauss or "J-gauss" and susceptibility units of "gaues"/oersted adopted by 

units of "emu'Vcm3 in the traditional emu and Gaussian systems are equivalent to abampere/cm. Sus

are used merely to indicate that the intensity of magnetization has units of abampere/cm. 

KEY To ABBREVIATIONS: MKS=Metre-kilogram-second; CGS=centimetre-gram-3econd; MKSfl = metre-kilo

pere: SI=Systeme International; EMU=electromagnetic unit: ESU electrostatic unit; abampere=10 

mal = metre-tonne-second-thermie; CGS-thermal = centimetre-gram-second-calof ie; Mkgf S=metre- (kilo

second; MTS = metre-tonne-second; Mie (or CGSS) = centimetre-seventh gra'n (107 gram) -second; m

gram second; dm-kg-ds = decimeter-kilogram-decisecond; QES = quadrant (10T metres) -eleventh gram 

coulomb; CGSm=centimetre-gram-second-biot; CGSe=centimetre-gram-second-frankIin. 

FIGURE 3. Gassification of metric units of intensity of magnetization, magnetic f
PHYSICAL AND THEORETICAL FRAMEWORK 15

->

where, as previously noted, Heft is the effective magnetic field 

vector inside the body and K is the intrinsic magnetic susceptibility.

The apparent magnetic susceptibility is intimately related to 

the shape of the body and to an internal demagnetizing field 

which is produced within the body upon application of the external magnetic field. The intrinsic magnetic susceptibility, in 

contrast, does not depend upon the shape of the body. The intrinsic 

susceptibility is a property of the magnetic material itself which 

may be considered to extend continuously without bounds. The intrinsic susceptibility is ordinarily computed from the measured 

apparent susceptibility, given information about the shape of the 

magnetic body.

Both intrinsic and apparent magnetic susceptibility may be 

either isotropic or anisotropic. If the intrinsic susceptibility is 

isotropic, the apparent susceptibility can be isotropic only if the 

body is equidimensional. If the intrinsic susceptibility is anisotropic, the apparent susceptibility will in general be anisotropic 

also, whether or not the body is equidimensional. Under highly 

fortuitous circumstances, the apparent susceptibility of a nonequidimensional body possessing anisotropic intrinsic susceptibility may itself be isotropic, provided that the anisotropy associated 

with the shape of the body cancels the intrinsic anisotropy. In 

practice, measurements of susceptibility anisotropy are ordinarily 

restricted to equidimensional specimens so that observed apparent 

susceptibility anisotropy can be directly related to intrinsic susceptibility anisotropy.

The finite body to which the concepts of intrinsic and apparent 

susceptibility apply may be a mineral grain consisting of one or 

more magnetic domains,2 a rock body (a naturally occurring rock 

unit or a specimen cut from such a unit) composed of magnetic 

mineral grains, or a rock formation consisting of magnetic rock 

bodies. The intrinsic susceptibility anisotropy of any of these 

bodies depends upon the intrinsic susceptibilities and shapes of 

magnetic constituents that make up the body.

For example, if the finite body consists of a single magnetic 

domain, the intrinsic susceptibility of the domain is associated 

exclusively with the crystalline material of the domain. If the 

finite body is a multidomain magnetic mineral grain, the intrinsic 

susceptibility of the grain is associated with both the intrinsic

2 A magnetic domain is a spontaneously magnetized region generally having dimensions 

of a few hundredths of a micron, which forms all or part of a magnetic mineral grain (see, 

for example, Bates, 1961; Chikazumi, 1964).
16 WEAKnFIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

susceptibilities of the domains and the shapes of the domains. If 

the finite body is a rock unit, the intrinsic susceptibility of the 

rock unit is associated with both the intrinsic susceptibilities of 

the grains and the shapes of the grains. If the finite body is a rock 

formation composed of a group of rock units, the intrinsic susceptibility of the rock formation is associated with both the intrinsic susceptibilities of the rock units and the shapes of the 

rock units.

INTRINSIC SUSCEPTIBILITY OF A MAGNETIC DOMAIN

The smallest physical entity to which the concept of magnetic 

susceptibility is generally applied in rock magnetic studies is the 

magnetic domain.3 In the absence of an applied magnetic field, the 

spontaneous magnetization of a domain is alined parallel to one 

of several crystallographic directions called easy magnetization 

directions of the magnetic mineral. For example, in magnetite, 

the axis of easiest magnetization is perpendicular to the octahedral 

plane, whereas the axis of most difficult magnetization is perpendicular to the hexahedral plane (Nagata, 1961). If the crystalline material of a magnetic domain is subjected to an applied 

magnetic field, the spontaneous magnetization will tend to rotate 

toward the direction of the applied field. If the applied field is sufficiently weak, the magnetization will reversibly rotate to its initial direction upon removal of the applied field.

Because the crystalline material constituting a magnetic domain is more easily magnetized along some crystallographic directions than along others, the domain is said to possess a magnetocrystalline anisotropy of magnetic susceptibility. Thus, the 

magnetization of a domain composed of a particular magnetic 

mineral can be characterized by a separate curve, such as that 

shown in figure 1, for each direction of applied magnetic field 

relative to the crystallographic axes of the mineral. According to 

studies of Uyeda, Fuller, Belshe, and Girdler (1963), a weakfield magnetocrystalline anisotropy is negligibly small in cubic 

titanomagnetites but larger in rhombohedral ilmenohematites. 

These observations indicate that the initial magnetization curves, 

OA#(fig. 1), for various applied field directions in titanomagnetites coincide or overlap for small values of applied field but di3 Domains occur in a broad class of ferromagnetic materials, including true ferromagnetic, 

antiferromagnetic, and ferrimagnetic minerals referred to in this report as "magnetic min« 

erals." The atomic and subatomic origins of magnetic susceptibility of all materials, including those that are diamagnetic and paramagnetic, are subjects of quantum mechanics, beyond 

the intended scope of this report.
PHYSICAL AND THEORETICAL FRAMEWORK 17

verge or splay for larger values of applied field. The corresponding curves for ilmenohematites diverge or splay for relatively 

small values of applied field.

The geometry and relative mathematical complexity of magnetocrystallme anisotropy in a material is dependent upon the 

crystal class of the material. Studies of crystal symmetry using 

time- as well as space-coordinate transformations indicate the 

existence of 90 magnetocrystalline classes of materials having 

1,651 space groups, in contrast to the 32 crystal classes and 230 

space groups corresponding to space-coordinate transformations 

alone (Birss, 1964; Bhagavantam, 1966; Billings, 1969). Although 

the large number of magnetocrystalline classes places many restrictions on physical properties of a material not expressible by 

second-rank tensors, fewer restrictions are placed on second-rank 

tensor properties. The constraints posed by Neumann's Principle 

(see, for example, Nye, 1960, p. 20) merely require that the tensor geometry corresponding to magnetic susceptibility have a 

symmetry at least as high as the crystal symmetry of the material 

and that certain crystallographic axes in the material coincide 

with particular directions of the tensor property. The main crystallographic systems represented by naturally occurring magnetic 

minerals are the cubic system (magnetite-ulvospinel series and 

related inverse-spinel minerals) and the hexagonal system (ilmenite-hematite series and pyrrhotite).

INTRINSIC SUSCEPTIBILITY OF A MULTIDOMAIN GRAIN

In a typical magnetic mineral grain composed of two or more 

domains, distinct walls (called "Bloch walls" (Kittel, 1953, p. 

183)) separating domains of oppositely directed spontaneous 

magnetization (called 180° walls) tend to aline parallel to easy 

directions of magnetization. If a magnetic field is applied parallel to the easy direction, the 180° walls move laterally; domains 

having spontaneous magnetizations parallel to the applied field 

grow at the expense of domains having magnetizations antiparallel to the applied field. If a magnetic field is applied perpendicular to the easy direction, the 180° walls remain stationary, and 

the spontaneous magnetizations of the domains rotate toward the 

applied field direction (Stacey, 1961). Ordinarily, applied fields 

required to produce a given amount of magnetization by magnetization rotation are larger than those required to produce the 

same amount of magnetization by 180° wall displacement. In relatively weak applied fields of a few oersteds, 180° wall movements
18 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

and spontaneous magnetization rotations are found to be reversible upon cancellation of the applied field.

The development of 180° walls (and walls having other angular 

relationships between adjacent domain magnetizations) in a magnetic mineral grain depends upon the shapes of the domains, 

which, in turn, relate to amounts of magnetostatic energy developed at the margins of the domains. The ease of movements 

of domain walls upon application of a magnetic field are thus also 

related to the external shapes of the domains. The ease of rotations of magnetization upon application of a field are, however, 

not dependent upon the shapes of domains, but are dependent 

only upon the magnetocrystalline energy of the domain. The intrinsic susceptibility of a multidomain grain is therefore related 

to the intrinsic susceptibilities of domains having magnetocrystalline anisotropy and to the shapes of these domains. Elongations 

or alinements of domains can be detected in weak applied fields, 

provided that the grain has not been subjected to strong demagnetizing fields (Stacey, 1961) .

APPARENT SUSCEPTIBILITY OF A SPHERICAL GRAIN

When a magnetic body of finite dimensions, such as a magnetic 

mineral grain, is exposed to an external magnetic field, a demagnetizing field is established within the grain opposite in direction 

to the external field. The applied field within the grain, fundamental to the definition of intrinsic magnetic susceptibility given 

in equation 1, consists of the vector sum of external and demagnetizing fields. Because the external field is much easier to measure 

than the internal field, it is convenient to determine experimentally the apparent magnetic susceptibility, Kx , given by relation 8,

The apparent magnetic susceptibility satisfies all the second-rank 

tensoral properties of intrinsic magnetic susceptibility and is generally anisotropic.

The origin of the internal demagnetizing field of a mineral 

grain may be analytically demonstrated4 by using a method of 

potential field analysis credited to Poisson (Maxwell, 1892, art. 

437-438) . This method relates the magnetic potential, n, of a body 

having uniform magnetization to the gravitational potential, V,

*The internal demagnetizing field is classically pictured (Williams, 1931; Bitter, 1937) as 

the return lines of magnetic force associated with equatorial regions of magnetic dipoles 

within a body. Although these lines of force vectorially add to the external field in polar 

regions of each dipole, they subtract from the field in equatorial regions and are therefore 

equivalent to a demagnetizing field.
PHYSICAL AND THEORETICAL FRAMEWORK 19

of that body, modified by replacing a constant-density term by a 

magnetization term. Poisson's relation may be written

dV «=--rr, (9) 

d£

where £ is the direction of magnetization, the gravitational contant taken as unity. The internal demagnetizing field, HA, is obtained by taking the negative gradient of the magnetic potential,

The demagnetizing field component in the £ direction is therefore

//2V

(#d h= (10) de

The requirement of uniform magnetization for application of 

Poisson's relation is not in itself also a requirement that the internal demagnetizing field be uniform. However, if the external 

field is made uniform by experimental design, Poisson's relation 

will hold only if the demagnetizing field is also uniform. Further, 

if the demagnetizing field is uniform, the left-hand side of equation 9 is constant, and integration of equation 10 indicates that 

the modified gravitational potential, V, must be a quadratic function of coordinates within the body. As indicated by Maxwell 

(1892, art. 437), the only cases known for which this potential is 

a quadratic function of coordinates of a finite body are those in 

which the body is bounded by a complete surface of the second 

degree, namely, an ellipsoid. Therefore, a finite body, such as a 

mineral grain, exposed to a uniform external field will have a uniform magnetization and internal demagnetizing field only if it 

has the shape of a triaxial ellipsoid, spheroid (ellipsoid of revolution), or sphere.

As an illustration of applying Poisson's relation for determining the internal demagnetizing field of a body, we write the modified gravitational potential for a spherical mineral grain magnetized in the + x direction as

V = 2/3 7rJa! (3a2 -X2 -Y2 -Z2 ),

where J^ is the magnetization, a is the radius of the sphere, and 

X, Y, and Z are cartesian components of a point inside the sphere. 

Using Poisson's relation (equation 9), we have for the magnetic 

potential

n=- 

dX
20 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

The internal demagnetizing field, obtained by using equation 10 

is

If the spherical grain were magnetized along either the + y or + z 

direction instead of along the +x direction, similar equations result, namely

(Hd ), = -4/3 IT/, 

and

For the special case of a sphere, we may write

valid for any direction of magnetization. If we define the demagnetizing factor, N, of the spherical body by the relation

(11) 

we see by direct substitution that

 4

As alternatives to the method of Poisson used above, methods 

based on vector analysis (Stoner, 1945) and the application of 

Gauss' Theorum (see, for example, Chikazumi, 1964) may be 

used to derive the demagnetizing field and associated demagnetizing factor for a uniformly magnetized body.

If, in experimental work, the apparent magnetic susceptibility is 

measured, the intrinsic magnetic susceptibility must be obtained 

by computation. Considering a spherical grain having an isotropic 

intrinsic susceptibility, we may write for the applied field inside 

of the grain

H=H«t-N/,

and for the induced magnetization at that internal point

Because, from equation 3,

and by direct substitution, we have
PHYSICAL AND THEORETICAL FRAMEWORK 21

KA = ^-, (12)

1 + NK

or, alternatively,

K= KA , (13) 

1-NKA

provided that

KA <i,K>0,N>0, (14)

N

and that K is finite, conditions valid for magnetic minerals.

If we substitute the numerical value of N for the sphere into 

12,13, and 14, we have

OXT

(15)

3 + 47TK 

and

3KA K=- 

provided that

KA <-. (16)

47T

Relation 16 indicates that the maximum value of apparent susceptibility for a uniformly magnetized spherical grain is 2.39 x 

10 -1 emu/cm3 , no matter how high the intrinsic susceptibility 

value. Considering that naturally occurring titaniferous magnetite has an intrinsic susceptibility which ranges from 5 x 10~ 2 

to 5 X 10- 1 emu/cm3 (Nagata and Akimoto, 1961, p. 99), we see 

from equation 15 that spherical grains of this magnetite would 

have an apparent susceptibility in the range of 4.13 X 10~2 to 

1.62 x 10 -1 emu/cm3 . For a representative intrinsic susceptibility 

of 10 -1 emu/cm3 , the corresponding apparent susceptibility for 

a spherical grain is 7.05 x 10~2 emu/cm3 . It is of interest that 

the apparent and intrinsic susceptibilities of a uniformly magnetized spherical grain agree to within 1 percent if the intrinsic 

susceptibility is less than 2.41 x 10~3 emu/cm3 ; they agree to 

within .10 percent if the intrinsic susceptibility is less than 2.65 X 

10~ 2 emu/cm3 .

It may be noted that relations 12 and 13, written for a spherical 

grain having isotropic intrinsic susceptibility, 5 are also valid along

8 For example, magnetite has an isotropic intrinsic magnetic susceptibility in a sufficiently 

weak applied field.
22 WEAK-lFIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

each of three orthogonal axes in a spherical grain having anisotropic intrinsic susceptibility. In these expressions, K and KA are 

replaced by their corresponding values along a particular principal axis of the anisotropic grain. The anisotropy of apparent 

susceptibility of such an anisotropic spherical grain is associated 

entirely with its anisotropy of intrinsic susceptibility and not at 

all with its external shape.

SHAPE ANISOTROPY OF AN ELLIPSOIDAL GRAIN

Although any apparent susceptibility anisotropy possessed by a 

spherical grain must be associated entirely with its intrinsic 

anisotropy, the apparent anisotropy of an ellipsoidal grain is associated largely with its shape. For the ideal case of an intrinsically isotropic ellipsoidal grain, apparent anisotropy is associated 

entirely with grain shape. The relative contributions of intrinsic 

anisotropy and grain shape to apparent anisotropy depend upon 

the crystal symmetry, composition, and shape of the mineral grain. 

For example, experimental work of Uyeda, Fuller, Belshe, and 

Girdler (1963) indicates that apparent anisotropy of most naturally occurring grains of high-symmetry minerals, such as magnetite, is associated more with grain shape than intrinsic anisotropy (magnetocrystalline anisotropy) . Conversely, apparent 

anisotropy of most naturally occurring lower symmetry minerals, 

such as hematite, is associated more with intrinsic anisotropy 

than with grain shape.

It is convenient to discuss the concept of shape anisotropy by 

considering an ellipsoidal grain having isotropic intrinsic susceptibility. Just as intrinsic anisotropy derives from a tensoral relationship between two vectors in relation 3, shape anisotropy derives from a tensoral relationship between the demagnetizing field 

vector and the induced magnetization vector. If we extend the 

relation 11 to include cartesian coordinates of a vector, we may 

rewrite the expression by the three component equations

and

where the demagnetizing factor terms are components of a symmetrical second-rank tensor.

Because of the complexity of the analytical expression for the 

potential (gravitational or magnetic) of a triaxial ellipsoid, the
PHYSICAL AND THEORETICAL FRAMEWORK 23

components of demagnetization factor for a triaxial-ellipsoidal 

grain cannot be written in a simple form. Instead, these components are ordinarily expressed in terms of elliptic integrals 

(Stoner, 1945), and numerical values of the components may be 

tabulated. If two principal axes of the ellipsoid are equal, as in 

a spheroid or ellipsoid of revolution, the analytical expression for 

potential is simpler, and demagnetization factors are more easily 

tabulated. For example, the demagnetizing factors for prolate 

and oblate spheroids, first derived by Maxwell (1892) and later 

evaluated by Stoner (1945), may be expressed by equations given 

in simple form by Nagata and Kobayashi (1961, p. 70) as 

Prolate spheroid

where

e=-

Oblate spheroid

f 1 A/l^e5" 1

(19)

(20)

L\ e* I \ e* /J 

where

e= , b = c>a. 

b

In these expressions, a is the symmetry axis, or axis of revolution of the grain, and b and c are principal axes of equal length 

in the principal plane of symmetry. The coordinate x coincides 

with a, y with b, and z with c.

From expressions 17 through 20, graphs of anisotropy factor, 

P, defined by

1 + KV,

have been plotted as a function of the dimension ratio, ra, of the 

spheroidal grain, defined by
24 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

b m= , 

a

where K is the intrinsic susceptibility of the grain, which is assumed to be isotropic (Uyeda, Fuller, Belshe, and Girdler, 1963, 

p. 280-281). As an illustration of the use of such graphs, it may 

be quickly determined that the ratio of principal apparent susceptibilities of a prolate spheroidal grain having a major axis 

twice as long as its minor axis and an intrinsic susceptibility of 

5 x 10~2 emu/cm3 is 1.14 to 1. Computation of this result would 

be an extremely tedious task without the use of a .programmable 

calculator or a digital computer.

Expressions 17 through 20 may also be used to deduce the 

maximum and minimum values possible for an ellipsoid, and, 

therefore, for a uniformly magnetized grain. For a slender prolate 

spheroidal grain having the shape of a needle,

For a flat oblate spheroidal grain having the shape of a disc,

Nx = 4Tr&ndNv = Ns = Q. 

Thus, we see that the limits of values of N for a finite body are

These limits for values of N, in turn, set limits on maximum and 

minimum values of apparent susceptibility in an ellipsoidal grain 

as previously discussed.

Although the demagnetizing factor is strictly defined only for 

uniform magnetization, and, therefore, for ellipsoidal bodies, good 

approximations to N may be obtained for bodies having nonellipsoidal shapes (Bozorth and Chapin, 1942; Jahren, 1963; 

Olsen, 1966; Sharma, 1968). Thus, equations 17 through 20 can 

be applied to a wide variety of shapes of finite bodies, depending 

upon required precision of the measurement or calculation. For 

example, susceptibility anisotropy investigations of Uyeda, Fuller, 

Belshe, and Girdler (1963) have been successfully carried out using cylindrical specimens of rocks and minerals, approximated to 

spheroidal shapes having orthogonal axes of the same dimensions 

(lengths) as axes of the cylinders.

COMBINATIONS OF INTRINSIC SUSCEPTIBILITY AND SHAPE 

ANISOTROPIES

The apparent magnetic susceptibility tensor, KA , of a uniformly 

magnetized body measured in a weak magnetic field is a function 

of both the intrinsic susceptibility tensor, K, of the magnetic ma-
PHYSICAL AND THEORETICAL FRAMEWORK 25

terial, and the demagnetization factor tensor, N, associated with 

the shape of the body. The fundamental relationship among these 

three tensor quantities, one a dependent variable and the others 

*> independent variables, may be developed in the following way.

If J, KA , //ext, K, Heft, and N are the induced magnetization 

vector, apparent susceptibility tensor, external field vector, intrinsic susceptibility tensor, effective field vector, and demagnetization factor tensor, respectively, of a uniformly magnetized body, 

> we may write, using dyadic notation of Gibbs (Wilson, 1909)

t*> 

= K Hetf ,

If I is the idemfactor of Gibbs (Wilson, 1909, p. 288) ,

/ > -* -* 

I J=J,

and

[f+( 

then

}={[!+ 

The apparent susceptibility is therefore

KA =[I+(K-N)]- 1 -K. (21)

Tensor components of KA , referred to a coordinate system (o?i, 

x2 , #3 ) fixed to the uniformly magnetized body, are

(BF-CE)]/L, 

(KA ) 13 = [^lf (EI-FH)

(KA ) 32 = 

(KA ) 33 =
26 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

where

L=A(EI-FH) +B(FG-DI) +C(DH-EG),

A = 1 + K11N11 + K12N21 + K13N31 , 

E = KnNu + K12 Ng2 + K13NS2,

D=K21NU + K22N21 + K23NS1 , 

E = 1 + K21N1S + KS2 N22 + K23N32, 

F = K21N1S + K22N23 + K2SNSS, 

G = KS1NU + KS2N21 + KSSN31 , 

H = K31N19 + K32NS2 + K33N32 , 

1 = 1 + K»Nlt + K32N2S + K33N33. 

Using the symmetric properties

K.i 2 = K. 21) K.i 3 = K. 3 i, K. 23 ]\.s2, 

and

N12 =N21 , N13 =N31 ,N23 =NS2 ,

we may, by direct substitution, obtain the expected result 

(KA) is = (K-A)SI,(KA) is = (K^)si,(K^) 23 = (KA) as* 

If K is isotropic and N is anisotropic, relation 21 reduces to

(KA) 3 =

1 + KN2 

K

1 + KN3

where the subscripts 1, 2, 3, refer to the three orthogonal principal axes of N.

If N is isotropic and K is anisotropic, relation 21 reduces to

(KA ) 2 = K*

(KA ) 3 =

1 + NK2 

K3

1 + NK3

when referred to the principal axes of K.

If both K and N are isotropic, relation 21 reduces to the single 

expression
27

Although a combination of anisotropic K and anisotropic N 

generally gives rise tq^an anisotropic KA , this combination may 

produce an isotropic KA under fortuitous circumstances. If the 

principal axes of K happen to coincide with the principal axes of 

N, and if the relative magnitudes of principal components of K 

and N are in the right proportion, an isotropic K4 will result. For 

example, if Ki = 1 emu/cm3 , K2 = 2 emu/cm3 , K3 = 3 emu/cm3 , Ni = 

1, N2 = 3/2, and N3 = 5/3, an isotropic KA = l/2 emu/cm3 results. 

In naturally occurring materials, such combinations of alined K 

and N principal axes and relative magnitudes of principal components are not likely to occur.

The tensors K and N can combine in two ways to produce an 

isotropic K^ and in three ways to produce an anisotropic K^. These 

tensor combinations are summarized in figure 4. These combinations may be analyzed in terms of magnetic bodies of various 

scales, such as a single magnetic domain, combinations of magnetic domains composing mineral grains, combinations of mineral 

grains composing magnetic rock units (naturally occurring units 

or specimens cut from such units), and combinations of rock units 

composing magnetic rock formations. As an illustration of possible tensor combinations, figure 4 shows that 13 K and N combinations of single magnetic domains may give rise to an isotropic 

KA of a magnetic rock formation, depending upon how these domains are distributed in multidomain mineral grains and magnetic 

rock units. Twenty-one K and N^ combinations of single domains 

may give rise to an anisotropic K.4 of a magnetic rock formation. 

For a specimen cut from a magnetic rock unit, five K and N combinations of single domains can produce an isotropic KA , and eight 

combinations can produce an anisotropic K4 . As noted previously, 

laboratory investigations of Uyeda, Fuller, Belshe, and Girdler 

(1963) suggest that the effects of anisotropic K and N in rocks 

and minerals are often separable. In general, the ft term prej 

dominates in rocks composed of cubic titanomagnetites. The K 

term predominates in rocks composed of lower symmetry magnetic 

minerals, such as hematites, ilmenites, and pyrrhotites. In rocks 

for which K is very small in magnitude, anisotropy of K.4 will be 

small, regardless of the magnitude of N.

INTRINSIC SUSCEPTIBILITY OF A ROCK BODY

The intrinsic susceptibility of a body of magnetic rock, whether 

a naturally occurring rock unit or a specimen cut from such a
28 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

IJR

DO H-f- I N 1

rt «

*

«§ «

Hi

S.E 5-"

S3

3%

_« _O _«) 'S'S'S >M B 

o.-S
29

unit, depends upon the intrinsic susceptibilities and shapes of 

magnetic mineral grains and magnetic domains composing such 

grains within the rock unit. The relative contributions of intrinsic 

susceptibility and shape anisotropy of magnetic constituents of a 

rock can be determined only by a thorough microscopic analysis 

of grain and domain shapes, magnetizations, and compositions 

using techniques such as those described by Stoner (1934), Bates 

(1961), Chikazumi (1964), and Zijlstra (1967). In practice, however, measurements of intrinsic susceptibility anisotropy of rocks 

are often related to characteristics of magnetic constituents by 

using an assumed model. For example, a model of Nagata and 

Uyeda (1961) may be used to relate measured anisotropic K of 

rocks containing intrinsically isotropic titanomagnetite grains to 

the shapes of these grains.

Theoretical models of Stacey, Joplin, and Lindsay (1960) relate 

intrinsic susceptibilities of magnetic mineral grains to demagnetizing factors associated with shapes of the grains. In studies of 

igneous rocks, they suggest that a good approximation to average 

grain shape is the dimension ratio 1.65:1.45:1 of a triaxial ellipsoid. Where this double ratio for a triaxial ellipsoid was converted to a typical single ratio of 1.5:1 for an oblate spheroid, the 

authors computed that a randomly oriented distribution of such 

spheroidal particles in a rock would result in a mean isotropic 

demagnetizing factor of 3.91. This mean factor is within 7 percent of the equivalent demagnetizing factor of for a similar

3

distribution of spherical particles. Stacey (1961) further pointed 

out that an assembly of grains containing domains having random 

orientations has a mean isotropic intrinsic susceptibility of

K = l/3(Kpnr +2Kper),

where K is the intrinsic susceptibility, Kpnr is the weak-field susceptibility of the magnetic material measured parallel to its domain directions, and Kper is the susceptibility measured perpendicular to its domain directions.

In the model of Nagata and Uyeda (1961, p. 130), each magnetic 

mineral grain is identical, and each is assumed to be surrounded 

by a small hollow cavity in the continuous phase of the rock. If 

p, K, N, L, and M are, respectively, the volume percentage of 

magnetic mineral grains within the rock, the isotropic intrinsic 

susceptibility of a single mineral grain, the demagnetizing factor 

of the grain, the demagnetizing factor of the rock specimen, and
30 WEAKnFIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

the demagnetizing factor of the cavity, then the apparent susceptibility, KAt of the rock specimen is

l+[N+(L-M)p]K

If the shape of the cavity is assumed to be identical with the 

shape of the specimen,

L=M, 

and

1 + NK

Measurements of KA and estimates of p for rocks can be used to 

compute N, which is the mean demagnetizing factor of the rocks. 

Computed N values are ordinarily found to be less than the N 

value for a distribution of spherical particles. This fact suggests 

that the particles are elongate, a conclusion also reached in the investigations by Stacey, Joplin, and Lindsay (1960).

As noted by Graham (1954, 1967) and Balsley and Buddington 

(1960) in studies of rock susceptibilities, large numbers of irregularly shaped mineral grains within a rock can be expected to 

be dispersed into relatively simple population distributions. For 

example, grains that have flat or platy dimensions tend to aline 

parallel to planar features of rocks, such as bedding in sedimentary or volcanic rocks and foliation in metamorphic rocks. Grains 

that are elongate tend to aline parallel to linear structures of 

rocks, such as phenocryst alinements in volcanic flows and lineations in metamorphic rocks.

MINIMIZING ANISOTROPY CAUSED BY ROCK SPECIMEN SHAPE

The only requirement for eliminating the shape effect of rock 

specimens used in susceptibility anisotropy measurements is to 

know the geometries of the specimens to the precision of the 

measurement; however, it is convenient to use equidimensional 

specimens in practice to avoid excessive computation. In many 

rock-magnetic laboratories, specimens having the shape of a 

sphere, cube, cylinder, or volume formed by the orthogonal intersection of two or three cylinders are used. Calculations of Sharma 

(1968, p. 132) show that demagnetizing factors associated with 

orthogonal axes of a cube are identical with the demagnetizating 

factor of a sphere. As noted by Jahren (1963); calculations by 

Werner (1945) show that the demagnetizing factors associated 

with orthogonal axes of a cylinder are equal to that of a sphere, 

provided the length-to-diameter ratio of the cylinder is 0.89. A
PHYSICAL AND THEORETICAL FRAMEWORK 31

more recent computation by Noltimier (1971) indicates that the 

optimal length-to-diameter ratio for a uniformly magnetized cylinder is 0.865, if shape anisotropy is to be minimized. Experiments of Porath, Stacey, and Cheam (1966) on cylinders of iron 

and nickel indicate that the optimal length-to-diameter ratio in 

weak magnetic fields is 0.85 and in strong magnetic fields, 0.88. It 

may be noted that the length-to-diameter ratio of rock cores 

used by Doell and Cox (1965) to minimize nondipolar components of remanent magnetization was about 0.92. To one significant figure, the optimal length-to-diameter ratio may be taken 

to be 0.9.

APPARENT SUSCEPTIBILITY ANISOTROPY OF A ROCK FORMATION

The largest entity to which the concept of anisotropic susceptibility is ordinarily applied is. the magnetic rock formation, consisting of a number of magnetic rock bodies or units. If the magnitude of intrinsic susceptibility of the formation is very large, 

say, greater than 10~2 emu/cm3 , the susceptibility anisotropy will 

affect computation of magnetic anomalies associated with a model 

of the formation. In analyzing magnetic anomalies associated with 

strongly magnetized rocks, therefore, it is helpful either, to compute demagnetizing effects directly (Sharma, 1966) during computation of the magnetic field or to correct for shape anisotropy 

after uncompensated computations have been made. The gross 

geometric relationships among magnetic rock formation, rock 

unit, specimen, mineral grain, and magnetic domain are shown in 

figure 5.

ANALYSIS OF SUSCEPTIBILITY ANISOTROPY OF A 

ROCK SPECIMEN

TENSOR COMPONENTS OF LINEAR SUSCEPTIBILITIES

The quantity of direct interest in most studies of magnetic 

susceptibility anisotropy is the intrinsic susceptibility anisotropy 

of a rock. This quantity may, with certain assumptions, be related to mean shape anisotropies of magnetite grains within the 

rock, or magnetic domains composing individual grains within 

the rock. As this property of a rock is ordinarily determined by 

using finite specimens of the rock, the quantity actually measured 

is apparent magnetic susceptibility anisotropy. The apparent susceptibility anisotropy can be most easily related to desired intrinsic susceptibility anisotropy if the shape of the rock specimen
32 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

Magnetite grain containing Rock specimen of amphibolite 

5 METERS containing grains of magnetite magnetic domains

Magnetic

rock unit

(amphibolite)

Magnetic rock formation 

(Layer of amphibolite-quartzofeldspathic gneiss)

FIGURE 5. Idealized sketch of material entities possessing magnetic susceptibility anisotropy, including a magnetic rock formation, magnetic rock units, 

magnetic rock specimen, magnetic mineral grains, and magnetic domains.

is equidimensional. Specimens having the shape of a sphere, cube, 

cylinder having length-to-diameter ratio of about 0.9, or volume 

formed by the orthogonal intersection of two or three identical 

cylinders are ideal for use in laboratory work. The uniformly 

magnetized equidimensional rock specimen forms the practical 

basis here for quantitatively analyzing magnetic susceptibility 

anisotropy.

The linear relationship between the induced magnetization vec-

-» 

tor J of a uniformly magnetized equidimensional rock specimen -* 

and an external magnetic field vector, H, applied throughout the

volume of the rock specimen is expressed by relation 8.

 * ~ ~* 

«/ = K H,

where K is the apparent magnetic susceptibility of the rock specimen. As noted previously, K is a second-rank tensor6 having, in 

general, nine components which may be expressed in dyadic, 

matrix, or subscript form. The dyadic form has been used in a 

previous section to describe apparent magnetic susceptibility, and 

we shall have occasion to use the matrix and subscript forms to 

describe this quantity in various contexts. Discussions of these

6 The second-rank tensor property of apparent magnetic susceptibility has been previously 

noted by Nagata and Uyeda (1961) and Coe (1966).
ANALYSIS OF A ROCK SPECIMEN 33

forms of tensor expression may be found in Wilson (1909), Wills 

(1931), Nye (1960), Symon (1960), Tropper (1962), Post 

(1962),Sokolnikoff (1964), Hollingsworth (1967), Borisenko and 

Tarapov (1968), and Billings (1969). For a specimen character-

-»< -* -*

ized by anisotropic K, J is parallel to H only along three orthogonal axes of the specimen, which are called the principal axes

** ~* 

of the tensor. If a specimen is characterized by isotropic K, J

-* ~ is everywhere parallel to H, and K is reduced to a scalar quantity.

The nine components of the susceptibility tensor, Kti (i, j = l, 

2, 3), may be expressed in terms of three equations relating the 

induced magnetization vector components along three orthogonal 

coordinate axes of the specimen to the external magnetic field 

vector components along these axes. If the coordinate axes are 

numbered 1, 2, and 3, corresponding, for example, to Cartesian 

coordinates, Xa , X2 , and X3 , the three equations are

J i = KjjHj + K12H2 + K1SH3 

«/2 = KsiHj + KzuHz + K23HS 

J s = K-siHi + K3SH2 + K33HS 

or, in briefer notation,

3

/,= Y, K«Hi> i=1 > 2> 3' <22)

3 = 1

Considerations of the principle of conservation of energy (Nye, 

1960, p. 57-60; Smythe, 1968, p. 21) require that

Kij = Hji,i = l, 2, 3

indicating that the susceptibility is a symmetric tensor which 

may be expressed by six components rather than nine.

By use of the formulation of Bhattacharya (1950), we may 

express the induced magnetization intensity in any direction having direction cosines I,, 12 , and 13, by

3 

J,= £] JA (23)

i = l 

Combining equations 22 and 23, we write

/' = Z L WW (24) 

i=l j=l

-*

If h, 1 2 , and 1., are taken to be the direction cosines of H, we may 

write
34 WEAK-iFIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

Ht-l&.j-l.t.S (25)

Combining equations 24 and 25, we write for the component of 

induced magnetization parallel to the external field

3 3

The susceptibility component, KH, parallel to the external field 

may be defined as the induced magnetization parallel to the field, 

divided by the field. Thus,

Jn 3 3

** = 7T E E W^« < 26 > 

H i=lj=l

which, expanded becomes

This component of susceptibility is of primary importance in susceptibility measuring systems, which are arranged to detect the 

induced magnetization component along the external magnetic 

field direction.

Just as the general second-degree equation, with coefficients

3 3 

JT JTSyX^l (27)

i=l j = l

possesses principal axes such that a transformation of axes allows 

us to write equation 27 as

3

so too does equation 26 have principal axes that allow us to rewrite the equation as

3

V*4, (29)

or, expanded, as

K.JI'=- LI K-i ~\~ 1% K.z~r Ig J\.s>

Likewise, equation 22, which is the basic equation defining susceptibility, may be rewritten in terms of principal susceptibility 

components as

J^KJIi, i=l, 2, 3. (30)
ANALYSIS OF A ROCK SPECIMEN 35

GEOMETRIC REPRESENTATIONS

Tensor components of magnetic susceptibility may be represented in three-dimensional space by several mathematical surfaces. For example, one of the most direct representations of the

important component, KH, in equation 29 is the surface traced

-» 

by a radius vector, R, having a magnitude

R= =Ka,i = l,8,3 (31) 

h

where the Xt are coordinates and the li are directions cosines of

-»

R. If we substitute R of equation 31 into equation 29, we have

R*= 'XfKt.i-l.S.S. (32)

This equation is represented by the peanut-shaped surface shown 

in figure 6A. This surface may be referred to as the K& ovaloid. 

A second representation of the susceptibility tensor may be 

obtained by comparing equation 29 with equation 28, where the 

coefficients of Si are all positive in algebraic sign. Equation 28 is

therefore a triaxial ellipsoid having its center at the origin of the

-» 

Xi, X2 , and X3 coordinate system. If we take a radius vector, R,

-»

parallel to H, having direction cosines li, L, and ls, and extending 

from the origin to a point on the ellipsoidal surface, we may rewrite equation 28 as

3 

Jfc'ESiV-l. (33)

i = l 

If we rewrite equation 29 as

T^-£/W = l, (34) 

Ka i = l 

and take

Si=#i, (35)

and compare equations 33 and 34, we see that KI{ may be represented by an ellipsoid traced out by a radius vector of length

 ' *

This ellipsoidal surface is simpler in form than the KH ovaloid 

and is referred to as the representation quadric (Nye, 1960, p.
36 WE AIRFIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

FIGURE 6. Representations of susceptibility .anisotropy. A. Perspective view 

of KH ovaloid. B. Perspective view of representation quadric. C. Cross section of principal plane of representation quadric, showing radius-normal

-» 

property. If the external field vector, Ho, intersects the ellipse at point P,

-» 

the resulting induced magnetization vector, Jo, is normal to the tangent line

at P. D. Perspective view of magnitude ellipsoid.

-> 

17) . Along each coordinate axis, Xit R takes the value

Ri = ~, i = l, 2, 3

as shown in figure 6B.

The representation quadric may be used not only to express 

the magnitude of KH in any direction, but also to indicate the direction of induced magnetization for a given external field direc-
ANALYSIS OF A ROCK SPECIMEN 37

tion. This directional property may be demonstrated by considering the radius vector R, along the direction of H (shown as H0 in 

figure 6C), and having direction cosines I,, 12 , and ls, which extends 

from the origin to a point, P, with coordinates (Rli, R12 , Rls) on

the surface of the ellipsoid. The vector, Np , normal to the ellipsoidal surface at point P, is the gradient vector of the ellipsoidal 

function evaluated at point P (see, for example, Kaplan, 1952). 

From equation 32, the ellipsoidal function, F (Xi), is

/ 3 \

(36)

and the normal vector at P is

When we evaluate equation 36, we have

+KSX] -

= (2K1X1 )n1 + (2K2X2 )n2 + (2K3X3 )ns ,

= (2K1 lJR)n1 + (2K2 l2R)n2 + (2KslaR)ns, 

-* -» -» 

where nlf n2 , and n, are unit vectors along the X]f X2 , and X3 coordinates, and R= - We also note that the induced magnetization vector, «/ may be written from equation 22, referred to principal axes, as

} = (K1H1 )n1 +(K,H,)nt + (K3Hs )ns

= (K^H^n^ (K2 l2H)n2 + (K3 l3H)n,.

-* 

The induced magnetization vector, J0 , is seen to be parallel to the

-» 

normal vector Np , at point P, by taking the cross product of the

two vectors as follows

n, n2 ns

K2 12H Kslsti 

2K2 12R 2K3 13R

= 0.

The fact that J0 is parallel to Np is sometimes referred to as the 

radius-normal property of the representation quadric (Nye, 

1960, p. 28).
38 WEAKJFIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

A third representation of magnetic susceptibility may be expressed by the surface traced out by the induced magnetization

 » 

vector, J, for an external magnetic field of unit magnitude, or

H=I. (37) 

When we rewrite equation 30 as

Ji=Kili[I, 

square all terms

Jl=KlfH*

transpose terms, and sum i from 1 to 3, we have

By using relation 37, we may rewrite equation 38 as

(39)

where the h are direction cosines of J. We may compare equation 

39 with the general equation for an ellipsoid referred to its 

principal axes and containing coefficients, Si, which may be 

written

(40)

where the Si are the semiaxes of the ellipsoid. If the radius

.-» -» 

vector, R, is taken parallel to J, equation 40 may be expressed as

3 t li V

*' E(-T-) =1. (41) 

i = l\ s* /

When we compare equations 39 and 41 and use relation 35, we 

see that equation 39 is represented by an ellipsoid which has 

semiaxes of length Klf Kz, and Ks and which is traced out by a 

radius vector of length

R=J.

Because the radius vector is everywhere equal to J, assuming a 

unit external magnetic field, this surface is termed the magnitude 

ellipsoid (Nye, 1960, p. 47). It is shown in figure 6D.

The magnitude ellipsoid and representation quadric may be 

mutually derived from one another because the semiaxes of the
ANALYSIS OF A ROCK SPECIMEN 39

latter are the reciprocal square root of the semiaxes of the 

former. These two surfaces are sometimes drawn concentrically

to show in a single diagram both the magnitude and the direction

-» -» 

of J for a given H.

CALCULATION OF PRINCIPAL SUSCEPTIBILITY COMPONENTS

Magnetic susceptibility anisotropy is most conveniently expressed and geometrically represented by the magnitudes and 

directions of three principal susceptibility components rather 

than by the six components, Ky(i=l, 2, 3; Kij=Kji). The magnitudes and direction cosines (I,, lz, 13 ) of the principal susceptibilities may be obtained by using a generalized form of equation 30, 

in which Kt is taken to be K, and equation 22 combined as

3

/«=##<= £ J5T«ff,f i=.l,*f 5

j = l

or, by using relation 25,

3 

Kli= Y-K^, i = l,2,3. (42)

1 = 1 

When we expand equation 42, we have

(K,,-K)l, = 0. (43)

These homogeneous equations may be solved for I,, lz, and ls, provided

= 0. (44)

 " IS

KO.I

After expansion of the determinant, equation 44 is seen to be a 

cubic equation having three real roots, K = Ki, K2 , and K3 , which 

are the three principal susceptibilities. The directions of these 

principal susceptibilities, given by I,, 12 , and Is, may be determined 

by successively substituting the values Klt K2 , and Ks into equation 44.

NONLINEAR SUSCEPTIBILITIES

The assumption of linearity between magnetization and external field given by equation 22 is seldom, if ever, precisely observed in practice, either because the linear relationship is perturbed by experimental error or because the linear relationship
40 WEAK-JFIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

itself is not valid. Where departures from linearity are known 

to fall within the range of expected experimental error, and 

where sufficient measurements are obtained to produce a redundancy of data, the observed data may be systematically adjusted to satisfy the linear relation. Graphically, this adjustment 

is analogous to fitting dispersed points of an x-y plot to the 

best straight line defined in some sense, such as least squares. 

Analytically, the adjustment of data is equivalent to solving 

an overdetermined system of linear equations by minimizing 

residuals associated with experimental error, according to established procedures of numerical analysis (for comprehensive accounts, see Hamming, 1962, or Scheid, 1968). Analyses of errors 

involved in special types of anisotropic susceptibility measurements have been summarized by Stone (1967) and King (1967).

Where departures from linearity between magnetization and 

external field exceed expected experimental error, complex susceptibility anisotropies may be inferred. Nonlinear susceptibilities 

may theoretically occur as high-rank tensors that have components obeying special transformation laws (Saaty and Bram, 

1964; Bloembergen, 1965), but more commonly they occur as 

nontensor quantities having no well-defined transformation laws.

A formal representation of nonlinear susceptibilities may be 

expressed by expanding the induced magnetization of a specimen 

in a power series in the external field, using the polyadic notation of J. W. Gibbs (Wilson, 1909),

J=K-H+K:HH+K:HHH+ . (45)

The first term is a product of the linear susceptibility, a dyad, 

and the vector field. The second term is a product of the lowest 

order nonlinear susceptibility, a triad, and a dyad field term. 

The third term is a product of the next higher nonlinear susceptibility, a fourth-order polyad, and a triad field term. Rewritten 

in conventional subscript notation, equation 45 becomes

3 33 333

/,= £ *W+ E E KvtHjH* + E E E KijMHjHkHl + - 

J=l j=lk=l j=lk=ll=l

or, in briefer notation,

Ji KijHj+KukHjHk+KijkiHjHkHi + .

The susceptibility Ktj is the usual second-rank tensor, Kijk, a thirdrank tensor, and KijM, a fourth-rank tensor. As nonlinear tensoral 

susceptibilities become important only in relatively strong fields, 

they are not expected to contribute significantly to complex
MEASUREMENTS USING A SPINNER MAGNETOMETER 41

anisotropies observed in weak-field measurements. Nor, as previously noted, is the effect of magnetocrystalline anisotropy of 

magnetite expected to contribute to the complexity of weak-field 

anisotropies.

The most likely causes of complicated weak-field anisotropy in 

magnetite-bearing rocks appear to be highly irregular or uneven 

volume distributions of magnetite grains within rocks which contain a relatively small number of low-coercivity grains. If the 

number of grains per unit volume of rock is small, the statistical 

averaging effect of susceptibility anisotropies of dispersed grains 

is small. As the present report is limited to the measurement of 

linear susceptibilities, no attempt will be made to treat determination of nonlinear susceptibilities.

ANISOTROPY MEASUREMENTS USING A SPINNER 

MAGNETOMETER

PRINCIPLES OF MEASUREMENT

Because the susceptibility tensor may be represented by a surface such as the magnitude ellipsoid having three principal 

axes, it is reasonable to suppose that the tensor may be measured 

by determining the projections of these axes on three orthogonal 

planes. This procedure is analogous to measuring a remanent 

magnetization vector by determining its components projected on 

orthogonal planes. As remanent magnetization vector components may be effectively determined by rotating a rock sample in 

the vicinity of a pickup coil, this measurement procedure seems 

appropriate for measuring components of the anisotropic susceptibility tensor.

If a rock sample possessing anisotropic susceptibility is rotated about an axis near a pickup coil, the voltage induced in the 

coil is proportional to the time rate of change of the magnetic 

induction component parallel to the coil axis, integrated over 

the area of the coil loops (see, for example, Maxwell, 1892; 

Stratton, 1941). Further, the magnetic induction associated with 

the rock sample is proportional to the induced magnetization, and 

therefore to the susceptibility, of the rock sample. 7 If the susceptibility components, K^ (i, j = l, 2, 3), are referenced to the

7 According to Maxwell's equations for large-scale electromagnetic phenomena (Maxwell, 

1892; Stratton. 1941), the magnetostatic field giving rise to an electromotive force in the 

pickup coil may be generated by (1) a time rate of change of electric field intensity (electrostatic field) associated with electric charge or electric polarization, (2) electric current 

(conductive, rather than displacement), (3) a time rate of change of electric polarization, 

and (4) magnetization. In addition to the intrinsic magnetic susceptibility and the demagnetizing factor, second-rank tensors associated with the rock specimen (and, therefore the 

generated electromotive force) may include electric permittivitv (capacivity), magnetic 

permeability, electric conductivity or resistivity, electric susceptibility, electric depolarizing 

factor, and remanent magnetic or electric quadrupole moments.
42 WEAK-.FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

CUBICAL SAMPLE

PICKUP COIL

FIGURE 7. Relationship between the coordinates of the rotating rock sample, Xi, X2 , and Xa, and the stationary coordinates of the pickup coil, x, y, and z.

coordinate system of the sample (Xlf Xs, X3 ) , and the sample is 

rotated about the Xs axis relative to the pickup coil coordinate 

system, x, y, z, where z coincides with Xs, and Xt coincides with 

the coil axis (fig. 7) , we may write

At

or

_d

d£

(46)

where ex is the voltage or electromotive force induced in the coil. 

The susceptibility components, Kxx, Kxy, and Kxz, may be expressed 

in the terms of the Ky sample coordinates or a set of K^ coordinates which are referred to the principal susceptibilities in the 

plane normal to the rotation axis (fig. 8) by using transformation equations involving orthogonal matrices.

If the sample coordinate system is rotated clockwise by an 

angle, 0, about the Xs axis, which coincides with the z axis, the 

susceptibility components K{j (i, j = x, y, z) may be expressed in
MEASUREMENTS USING A SPINNER MAGNETOMETER 43

I '

I f Pickup coil coordinate axis 

 Rock sample coordinate axis 

I / Principle susceptibility coordinate axis

f/ For rotational speed, co, 8 = wf

Z,X3/ X3'

FIGURE 8. Relationship among the coordinate axes of the 

pickup coil, the rotating rock sample, and the principal 

susceptibility components in the Xi-Xs plane, normal to 

the rotation axis. Coordinate axes Z, Xs, and X3' are directed along the rotation axis, into the plane of the diagram.

terms of the K{i (i, j = l, 2, 3) components by the matrix equation

(

cos 8 sin 8 Q\/KnK12K13 \/cos 8 -sin 8 0\

- sin 8 cos 8 Oil K12K22K23 } I sin 8 cos 8 0 1

0 0 1/\K13K2SKJ\ 0 0 I/

IV V Tf \

/ n. a.xZ\. xv£±XZ\

= ( V K Tf I (&H\ I J\. aiyn.yVJ\yz I. \f± I )

\KxgKyzKgg /

If we take the matrix product on the left-hand side of equation 

47 and equate elements of the product to matrix elements on the 

right-hand side of the equation, we have

Kxas = K11 CQS2 0 + K22 sin2 8+2K12 sin 8 cos 8, 

Kyv =Kn sin2 8+K22 cos2 8-2K12 sin 8 cos 8,

and

Kxy= (K22 ~Kn ) sin 8 cos 8+ K12 (cos2 8-sm2 8), 

Kxe=Klscos 8 + K2Ssm 8,

K.yss = sin O+K,,, cos 8. (48)
44 WEAK+FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

When we use trigonometric identities for double angles, we may 

rewrite Kxa and Kxv as

Kxx = 

22 

(l + cos 26) + (1-cos 26) +K12 sin 26,

and

m 20+K12 cos 26.

2

If the Xs spin axis has a rotational velocity, o>, we may write 6 

as a function of time, t, as

8 = (at.

If we substitute Kxx, Kxy, and ^ into equation 46, we have

ex x. 

dt

+ [K15cos ut + K2Ssm wflHz}. (49) 

From equation 49, we see that the external field components, Hx 

and Hv, normal to the rotation axis, give rise to alternating voltages that have frequencies twice that of the rotational velocity, 

whereas the external field component, Hz, parallel to the rotation 

axis, gives rise to a voltage having the same frequency as the rotational velocity.

The signal generated by the anisotropic susceptibility of a rotating rock sample may be considered in greater detail by referencing the susceptibility components to coordinates X't , X'2 , and 

X\, of which X't and X' are the directions of principal susceptibilities in the plane normal to the rotation axis, and X\ is the direction of the rotation axis. From figure 8, we see that transformation of coordinate axes corresponding to the rock sample and 

pickup coil has the same form as the transformation of axes corresponding to the principal susceptibilities and pickup coil, except that the angle of rotation is increased from 0 = <at to 6' = <»t + 

fa, where fa is the angle between the principal susceptibility in 

the vertical plane and the corresponding pickup coil coordinate 

axis. Using orthogonal matrices similar to those of equation 47 

and noting that K']s = 0 when referred to the principal susceptibility coordinates, we have

cos((ot+fa) 

0

cos ( tat + <$> x ) - si 

sin ( a>t + fa) c

o

- sin ( tot + 4>r 

cos(ut+<j>x

o i/ /C/M^ (so)
MEASUREMENTS USING A SPINNER MAGNETOMETER 45

The equation for induced voltage, corresponding to equations 48 

and 49, is

22 ' -&' )cos 

+ [i/2 (K' -K' )sin 2 (

or

<*a M/^-tf;,) sin 2 ( 

+ Mff-/ ) cos 2 (<o£ +</> )]# (51)

If we combine the H^, and Hy terms, we may rewrite equation 51 

as

[' -K' )sin

22 11 '

(52)

The voltage associated with external field component H. may be 

cancelled by electrical filtering or by nulling the Hz component 

during the measurement process. Ordinarily, only the external 

field component, H^, is used during the measurement process, so 

that equation 52 becomes

(53)

The amplitude of the alternating emf is thus proportional to the 

principal susceptibility difference, (Kf -K'n ), the rotational 

velocity, <o, and the external field component, H.,.. The frequency 

of the sinusoidal voltage is 2<»t, that is, twice the rotational frequency of the rotation axis, and the observed phase angle is 

twice the angle between the principal susceptibility axis and the 

corresponding rock sample coordinate axis. The angular velocity, 

w, and the external field component, Hx, are known quantities in 

the measurement process, and they may be incorporated into a 

lumped coefficient giving the proportionality factor between the 

voltage amplitude, e*, and the susceptibility difference, (K'si - 

K'n ) This lumped coefficient may include design factors associated 

with coil construction (Collinson and Creer, 1960, p. 190-194) 

and constants obtained by calibration of the measuring apparatus. 

Thus, equation 53 may be rewritten in terms of the observed 

voltage amplitude
46 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

where £ is the lumped proportionality factor previously determined. If A is the adjusted observed voltage amplitude, such that

we may write

_ iff iff /i. /i

11 22

The observed phase angle, <f>x, for this axis orientation is called a, 

that is

Similarly, if the rock sample is rotated about the X, and X2 sample coordinate axes, we substitute for the left-hand side of equation 50 the matrix products

K'i, K'»,K',s

0 Kr K'

25 55,

and

If the phase angles are measured, in order, from y to x, x to z, 

and z to y (fig. 9) , we have

£=(#' -K' ),

v ' f

and

Given the measurements A, 5, C, a, ^8, and y, we now inquire 

about the relationship between the measured amplitudes and 

phase angles to the desired susceptibility components Ki} (i, j 1, 

2, 3). To obtain this relationship, we construct the following 

matrix equations
MEASUREMENTS USING A SPINNER MAGNETOMETER 47

FIGURE 9. Sequential order of orientations of rock sample coordinate system 

during the measurement process. (A) First spin position. (B) Second spin 

position. (C) Third spin position.

cos fa sin fa 0

- sin fa cos fa 0

001

cos fa 0 sin fa

010 

 sin fa 0 cos fa

K' K' 0

JI J2 K' K' K'

12 22 2S

0 K' K'

23 33

cos fa sin fa 0

sin fa cos fa 0

0 0 1

KisKzsK-33

cos 4>j, 0 sin </> 

01 0

sin fa 0 cos <£ 

(54)

(55)
48 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

and

'l 0 0 \/K',,K'u K',,\/l 0 0

0 cos fa sin fa II K'i2K'2z 0 II 0 cos 0« -sin fa

0 -sin fa cos fa l\ K'\ K' l\ 0 sin fa cos fa

\ / \ I* S3 / \

KJSK23K33 

From equation 54 we obtain

'n -K>22 )cos 20., 

' -K'u )cos 20., 

and

^lf = -%(«' -^22 )sm 20,. (57)

When we take the difference of the first two equations of syste'm 

57, we obtain

* -* -<*;,-*;,> a* 2*- (58) 

Thus, from relations 57 and 58

#ji--K2 2 = A COS 2a,

and

^2 =-i/2Asin2a. (59) 

Likewise, when using equations 55 and 56, we obtain

2/3,

KSS -KU = C cos 2y, 

and

^5 =- 1/2C'sin2y. (60)

As seen from equations 59 and 60, the cross susceptibility components, K12 , Kns, and K1S, are completely determined in the measurement process, whereas the normal susceptibility components, 

KU, K22 , and Kss, are determined only to within an unknown constant. Thus, in order to completely specify the susceptibility 

tensor, it is necessary to measure one of the normal susceptibility 

components independently. Without this measurement, the observed spinner magnetometer data are limited to determinations 

of the differences of principal susceptibilities rather than to determinations of the principal susceptibility values themselves. 

The orientations of principal susceptibility axes may be determined from the spinner magnetometer data alone.
MEASUREMENTS USING A SPINNER MAGNETOMETER 49

Although differences of principal susceptibilities may adequately describe susceptibility anisotropy in some types of rock magnetic investigations, determinations of individual principal susceptibility components are necessary where complete determinations of anisotropy are required. Complete determination of principal susceptibility values is possible only if the normal susceptibility components, Kn, K22 , and KS3, are known. Equations 59 

and 60 indicate that independent determination of one of these 

normal susceptibilities is sufficient to determine the other two.

A straightforward method of determining one of the normal 

susceptibilities is by orienting one of the rock sample coordinates 

parallel to an external magnetic field and measuring the induced 

magnetization of the sample in that direction. This measurement 

may be conveniently made by using a commercially available susceptibility bridge or a similar apparatus that uses an alternating 

external field. Consistency is achieved if the frequency of the 

external field is twice the rotational frequency of the spinner 

magnetometer used to determine the other susceptibility components.

If, for example, the measured induced magnetization along the 

X sample coordinate, «/*, is associated with an applied external

3 8

field, H*, in that same direction, the normal susceptibility component is

T

K =K* = (61)

S3 33 ff*

By substituting in equations 59 and 60, we have for values of 

normal susceptibilities

Kn =A cos 2a+B cos 2^ + K*s , 

K = Bcos2B+K*, 22 'S3'

and

K =K*. (62)

S3 S3 ^ '

These values, along with those for cross susceptibilities, K, 2 , K1S, 

and K2S, may be substituted into equation 44 to obtain the principal susceptibilities, K,, K2 , and K3 .

The direct measurement of K3S by using a bridge or similar apparatus suggests the possibility of directly measuring each of 

the normal susceptibilities by this method and reserving the
50 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

spinner magnetometer data for determinations of cross susceptibilities and orientations of principal susceptibility axes. However, for a detection system of given sensitivity, measurements 

of small differences of susceptibilities can be made more accurately than determinations of these differences by subtracting independently measured values of susceptibilities. Therefore, all of 

the signal observed in the spinner magnetometer measurement is 

directly related to anisotropy, whereas often less than 1 percent 

of the signal observed in the bridge measurement is associated 

with anisotropy.

METHODS OF EXPRESSING ANISOTROPY

The most direct way of expressing linear susceptibility anisotropy is by specifying the six components, Kllf K22 , Kss, K12, K13, 

and KZS, or, equivalently, by expressing the principal susceptibilities, Kj, K2 , and Ks in combination with the three angles associated with orientations of principal susceptibility axes relative 

to a fixed coordinate system of the rock specimen. 8 In addition to 

these direct expressions, other quantities have been defined by 

various authors to indicate the magnitude of anisotropy of a 

rock. Among the more common expressions are the following: 

(1} Magnetic lineation, L, where

j. "-max "-into L= I '

{2} Magnetic foliation, F, where

 *Mnt Kmin F =

K '

L {3} Prolateness= ,

F

{3} Oblateness= , 

L

where

j~_ Kmax +Kint +Kmin,

3

sometimes termed KB, the bulk susceptibility,

{5} Maximum anisotropy

v _v " max " min

8 A complete description of the anisotropy actually requires a determination of only five 

quantities, either five susceptibility tensor components, or a combination of three principal 

susceptibilities and two angles indicating the orientation of principal axes.
MEASUREMENTS USING A SPINNER MAGNETOMETER 51 

{6} Degree of anisotropy, P (max/min)

" mam

 " min

{7} Degree of anisotropy, P (max/int)

If __ "-mar

" int

{8} Degree of anisotropy, P (int/min)

{9} Anisotropy

£ "-mill

{10} Magnetic anisotropy ratio

tfper

where Kpar is measured for magnetic field parallel to rock layering and Kper is measured for magnetic field perpendicular to 

rock layering,

{11} Excess of susceptibility of the magnetic stratum plane, h, 

where

a K i^ffldfff + i K* .f xgfjg * __ _

h= Kmin, and 

2

{12} Azimuthal anisotropy quotient,

_ "-max "-mto

q h

Expressions 1, 2, 3, and 4 have been used by King and Rees 

(1962) and Khan (1962); they are similar to definitions by F. D. 

Stacey (Stacey, Joplin, and Lindsay, 1960) which were based on 

demagnetization factor components. These expressions convey information analogous to ratios of Kmax/Kint/Kmln , which have been 

rationalized by dividing by Kint . Ratios of Km(lx/Kint/Kmin , rationalized by dividing by Kmin, have been used by Girdler (1961a). 

Expression 5 was adopted by Howell, Martinez, and Statham 

(1958), and expressions 6, 7, and 8, by Uyeda, Fuller, Belshe, 

and Girdler (1963). Expression 9 was used by Balsley and Buddington (1960) and expression 10 by Jahren (1963). Expressions
52 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

11 and 12, defined by Granar (1959), are extensions of definitions first given by Isling (1943). Hamilton and Rees (1965) 

considered Kint to represent approximately the mean susceptibility, (Kmax Kmin ) /Kint, the intensity of foliation in terms of mean 

susceptibility, and q of expression 12, the intensity of lineation 

in terms of that of the foliation. If the susceptibility magnitude 

ellipsoid assumes the shape of an oblate or prolate spheroid, the 

anisotropy may be specified by the ratio of maximum-to-minimum susceptibility, equivalent to the anisotropy factor used by 

Uyeda, Fuller, Belshe, and Girdler (1963).

A convenient expression of anisotropy suitable for use with 

the spinner magnetometer, an apparatus by which differences 

of susceptibilities rather than susceptibilities themselves are 

measured, is expression 5. This definition is adopted in this report and modified to indicate a percent rather than a pure number. The term is given as

(

K -K \ 

max m^L ) 100 percent

where

= 1/3 (#,

MEAD-TYPE SPINNER MAGNETOMETER

A particular type of spinner apparatus used successfully in 

the early 1960's to measure susceptibility anisotropy was designed by Professor Judson Mead of Indiana University (Hanna, 

1965; Hanna and Mead, 1968) and is herein referred to as the 

Mead-type spinner magnetometer. The basic component of the 

apparatus, which is shown schematically in figure 10, is a motordriven spinner magnetometer (operated at a rotational speed of 

600 rpm) ordinarily used to measure the 10-Hz signal associated 

with the reman ent magnetization of a rock specimen. For purposes of measuring susceptibility anisotropy, an auxiliary horizontal shaft is geared to the main shaft in a 1:2 ratio so that 

the auxiliary shaft rotates at 300 rpm, or half the speed of the 

main shaft. A vertically oriented Helmholtz coil is placed in the 

region of the sample holder (which is attached to the auxiliary 

shaft) to provide a fixed uniform external magnetic field. When 

a rock sample is rotated at 300 rpm, the 10-Hz signal generated 

by the anisotropic susceptibility normal to the shaft is amplified, 

filtered, and detected by the pickup system of the basic spinner
MEASUREMENTS USING A SPINNER MAGNETOMETER 53

Helmholtz coil 

steady magnetic field

Specimen

Gear drive 

Auxiliary shaft

Reference coil 

^-^ Main shaft * . .f

^... bagaii;^«li i r " "t2a 

600 rpm

Illl

FIGURE 10. Mead-type spanner magnetometer used for measuring magnetic 

susceptibility anisotropy. The pickup and detection system, consisting of 

components shown below the rotating shafts in the diagram, is also used 

for measuring remanent magnetization of a rotating sample and magnetic 

susceptibility of a stationary sample.

magnetometer. For purposes of independently measuring the 

magnetic susceptibility along a particular rock specimen coordinate (to completely specify the susceptibility ellipsoid), the 

Helmholtz coil may be used to produce a 10-Hz alternating field 

surrounding the rock which is held stationary, so that the 10-Hz 

pickup system of the spinner magnetometer may again be used. 

An external field of two oersteds, which is sufficiently weak to 

insure linearity between magnetization intensity and magnetic 

field (see, for example, Nagata, 1961), is used for both the anisotropic susceptibility measurements of the rotating specimen and 

the apparent susceptibility measurement of the stationary specimen.

In addition to the usual sources of noise associated with the 

spinner magnetometer operation,9 unwanted 10-Hz signals may 

be introduced by the remanent quadrupole moment and anisotropic electrical conductivity of the rotating specimen as well as 

by second harmonics of various 5-Hz signals. Although these 5-Hz 

signals may be partly associated with effects of anisotropic susceptibility and conductivity, by far the strongest second harmonic 

is generated by the remanent dipole moment. Signals produced 

exclusively by external magnetic field components parallel to the

9 Noise sources include background electromagnetic fields which may be at least partially 

cancelled by external shielding or by budking coils, electrostatic charge on the surface of the 

plastic sample holder which may be effectively removed by commercially available phonograph 

record sprays or by steam (although the B-Hz fundamental frequency is removed by filtering, 

the 10-Hz second harmonic will be passed), mechanical vibrations associated with the rotating shaft or with the pickup coil, and many noise sources associated with electronic components of the pickup and detection system (for a comprehensive review of these latter 

sources, see Ficchi (1964) and King (1966)). The noise level of the entire system was such 

that the lower limit of precisely measurable susceptibility difference was approximately 5 X 10~7 

emu/cm3, and the lower limit of detectable susceptibility difference was about 5X1'0~3 emu/ 

cm3.
54 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

spinner shaft may be eliminated by cancelling these field components by using bucking coils or by orienting the spinner shaft 

perpendicular to the magnetic meridian of the laboratory. In 

previous studies of volcanic rocks (Hanna, 1965), the spinner 

shaft was oriented parallel to the magnetic meridian so that a 

single horizontally oriented coil or permanent magnet could be 

used to null all of the horizontal external field component, leaving only a vertical component of external field.

For volcanic rocks, the 10-Hz second harmonic signal associated with the remanent dipole moment is often many times 

larger than the desired signal associated with susceptibility 

anisotropy. In order to eliminate the effect of this second harmonic 

signal, a signal is introduced by the reference circuit component 

of the spinner magnetometer, ordinarily used for measuring remanent magnetization. This reference circuit component consists of 

a reference coil made up of two orthogonal square-section coils 

wound on a plastic cube matching the size of the cubical rock 

specimen and placed directly over a flat-lying pickup coil identical with the pickup coil directly beneath the specimen holder (fig. 

10). An external field produced by the vertically oriented Helmholtz coil is then adjusted until the field surrounding the specimen 

holder is cancelled. Rotation of the rock specmien in the resulting field-free space produces an output due to the second harmonic 

of the remanent dipole moment and to the remanent quadrupole 

moment. For the volcanic rocks studied, the second harmonic 

signal associated with the dipole moment was always at least an 

order of magnitude stronger than the signal associated with the 

quadrupole moment, and therefore the combined signal was almost 

purely sinusoidal in form. 10 This sinusoidal output is effectively 

nulled by introducing the correct proportions of direct current 

through each of the orthogonal coils of the rotating reference 

coil, that is, by adjusting the magnitude and phase of the reference circuit signal. After balance is achieved, a vertical external 

field is introduced within the Helmholtz coil, giving rise to the 

desired output associated with susceptibility anisotropy and to 

unwanted output associated with electrical conductivity.

Although there is no provision in the apparatus to eliminate 

possible signals associated with electrical conductivity, and although no attempt was made to measure the electric conductivity 

of rock specimens by using other equipment, there is reason to believe that the effects of conductivity were negligibly small for the 

rock specimens measured. For example, rocks possessing nearly

10 Of 106 specimens studied, the remanent quadrupole signal was detectable in only 16, that 

is, 15 percent of all measured specimens.
BHATTACHARYA SOLUTION OF EQUATION 55

isotropic susceptibility produced no observable signal, indicating 

an absence of conductivity effects. Also, rocks possessing anisotropic susceptibility generated an almost purely sinusoidal signal, 

suggesting that signals associated with conductivity were absent 

or were fortuitously in-phase or anti-phase with respect to the 

anisotropic susceptibility signal. Because the amplitudes of signals 

associated with electrical conductivity are directly proportional 

not only to the size and conductivity of the conducting material, 

but also to the time rate of change of the external magnetic field 

(see, for example, Chikazumi, 1964; Olsen, 1966), the relatively 

slow rotational speed chosen for the system tends to diminish 

conductivity effects.

BHATTACHARYA SOLUTION OF EQUATION OF 

ANISOTROPY

Having shown in equations 59, 60, and 61 that observed data 

from spinner magnetometer measurements and one bridge-type 

measurement directly determine values for K{j (i, j = l, 2, 3), we 

proceed to find the principal susceptibility axes by solving the 

cubic equation represented by equation 44. Using the terminology 

of Bhattacharya (1950) , we write

Q,=KU', b=K22 ', c = K3S

f=K23 ',g = Kls ;h = K12 . (65)

The cubic equation 44, after expansion of the determinant, is

K3 - (a + b + c)K2 + (bc + ca + ab-f*-g2 -h2 )K

fgh-af2 -bg2 -ch2 )=Q. (66)

The three real roots, K=K19 K2 , Ks may be routinely determined 

by numerical methods used in digital-computer programming. 

However, where digital computer access is not available, it is 

useful to have closed-form analytical expressions for Klf K2 , and 

K3 . Thus, we proceed with the solution of the equation of anisotropy as presented by Bhattacharya (1950, appendix 1, p. 94), 

using standard analytical techniques (see, for example, Dresden, 

1930; Uspensky, 1948). Setting

K=X+-(a + b + c) t (67) 

3

equation 66 then becomes

X*-qX-r = Q, (68)
56 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY 

where

q= - (a + b + c) 2 - (bc + ca + ab-f2 -g2 -h2 ) (69) 

3

2

r= (a + 6 + c) 3 

27

--(a + b + c) (bc + ca + ab-f2 -g2 -h2 ) 

3

+ (abc + 2fgh-af2 -bg~-ch2 ). (70)

Because equation 68 can be solved trigonometrically, we make 

the substitution

X=NY, (71)

and, from equation 68, we write

Y3 - Y- =0. (72)

N2 N3

If we use the trigonometric identity

3 1

cos 3 7? cos 17-- cos 377 = 0 (73)

4 4

and assume

7=cosT7 (74) 

in equation 72, we have, by comparing equations 72 and 73,

] 3 (75) 

and

4r r] / 21 cos3«= = -\ .

N3 2] q* (76)

The term, cos ?;, may be determined from equation 76 by trigonometric tabulations, the three roots of equation 72, expressed as

Y=cos r), cosf "n+~ )»cosf 17+ J. 

\ 3/ \ 3/ (77)

From equations 67, 71, and 75, the three values of susceptibility 

are

K=-(a + b + c) +2\/ -cosI 9+ 1 (78) 

3 Y 3 \ 3/

where

5=0,2,4.
BHATTACHARYA SOLUTION OF EQUATION 57

Because the first term of equation 78 is the bulk susceptibility, 

as defined by equation 63 or 64, we may consider each principal 

susceptibility component to consist of the algebraic sum of the 

bulk susceptibility of the sample and a term associated with 

anisotropy. Of further interest is that the sum of equations 78, 

with substituted values of S= 0, 2, and 4, is

(79) 

an invariant property of the second-rank tensor.

This invariant relationship between the sum of principal susceptibilities and the sum of normal susceptibilities may be used 

to demonstrate a practical procedure in the calculation of anisotropy. Because values of normal susceptibilities, Kn, K22 , and Kss, 

are ordinarily orders of magnitude larger than values of cross 

susceptibilities, computation of small percentages of anisotropy 

requires retention of large numbers of digits. If, however, the 

measured normal susceptibility, say Kt =K* , is set equal to 33 33

zero, corresponding normal susceptibilities have small values, 

similar to those of the cross susceptibilities, and fewer digits are 

required for computation. The corresponding computed principal 

susceptibilities of equation 78 are subsequently increased by addition of the constant, K* , to obtain the principal susceptibility 33

components. This procedure is evident from equations 62 and 79, 

which we combine as follows

(Kn + K,2 + Kss) = K*=A cos 2« + 2B cos 2(3 + 3K*

If Kss is set equal to zero, we have

(Kn+K8 , + K3S ) =0 = A COS 2a + 25 COS/?

K 33

-(f.-ij.) + <*.-*?,) + (*.-*:.).

The corresponding principal susceptibilities are deficient by the 

term K*s , and this term must be added to computed values K} , 

K2 , and Ks. The procedure of setting the measured normal susceptibility equal to zero and subsequently adding this constant 

to computed principal susceptibilities is most important where 

computation facilities are insufficient to accommodate large numbers of digits (significant figures) .

The direction cosines of each principal axis may be determined 

by using equation 44, following the procedure of Bhattacharya
58 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

(1950, p. 96). If we rewrite these equations in previously defined 

coefficients, we have

gl + fli + Col^O (80) 

where

a0 = a K 

b0 = b~K 

c0 = c-K. (81)

When we eliminate 12 between the first two equations of 80, we 

have

_/ b,g-fh\

''U-**./

and when we eliminate lt between the second two equations of 

80, we have

/ c0h-fg \

ls-\ - Ma- \o6)

\b0g-fh/

We may use the ratios, m and n, obtained from equations 82 

and 83, and write

m= ^= b.g-fh (84)

ls Ji2 -a0 b0 

n ..j.**-/g (gB)

ls b0g-fh 

to compute the direction cosines, or, alternatively the inclinations

and declinations of the principal susceptibility axes.

-» 

If a radius vector, R, passing through the the origin of the magnitude ellipsoid is taken parallel to a principal susceptibility axis,

-» 

then the unit vector R0 along this axis has direction cosines li, 12 ,

and ls, such that

-»-»-»-»

RO == IjX i + \<tX. 2 + lyX. 3

XlY X* Y ,X3y XQ/.V

= Xi + X2 + Xs (oo;

R R R 

-» -» -» 

where Xlt X2 , and X, are unit vectors along the specimen coordinate axes. From equation 86

R=£i=^f=£', (87)
BHATTACHARYA SOLUTION OF EQUATION 59

which is the equation of the line along which the principal axis 

lies. If we rewrite equation 87, we have

V V

(88) yi, yi,

or, using equations 84 and 85,

±=L=XS . (89) 

m n

When we note from equation 89 that

and

t = X2 (90) 

n

Xs= , (91) 

n

and if we use relation 87, we have

"A

n

(92)

&= , (93) 

R

and

(.-^ (94) 

R

where

D I / I 1 V"2 i v? . i - \ _ I / 21 ,x,2 i i /nir\ R = \l\ )X*+X* + [ } \ m*+n2 + l. (95)

n / \ n I n ] 

When we substitute equation 95 into equations 92, 93, and 94, we 

have for the direction cosines of the principal axis

l,= ^-^-, (96)

(97)

and

l.= 1 . (98)
60 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

For ease of plotting, we may express the direction of the principal susceptibility axis in terms of inclination and declination 

(Bhattacharya, 1950, p. 98). If /' is the inclination and D' is the 

declination measured positive from the +Xj to the +X2 axis (fig. 

11),

From diagram:

(1) cosD' = -£-, (2) sin D'=-£-, (3) cos l' = -^-t (4) sin /'=-£-, 

(5) /,= cos a'=-£-, (6) /2 =co*0' = -£-, (7) I3 =co*y' = ^-,

Therefore:

cos a'= cos D' cos /'; cos /3' = sin D' cos /'

tan D' = -

cos g' _ 1, 

cos a' /, /

sin 'I =

FIGURE 11. Relationship of declination, D', and inclination, /', to the direction 

cosines, t, k, and L of a principal susceptibility axis referred to the rock 

sample coordinate system.
BHATTACHARYA, SOLUTION OF EQUATION 61 

we have

(99)

l2 = sinD'cosI', (100) 

Z, = sin/'. (101)

If we divide equation 100 by equation 99 and use equation 95, 

we have

n> _ 's - n 

I, m

sin r = .

The direction cosines or inclination and declination of each principal axis may be determined by the above procedures, provided 

the appropriate principal susceptibility value is substituted into 

expressions for aot b0 , and c0 , in equations 81.

Estimates of uncertainties or errors in computed magnitudes 

and directions (orientations) of principal susceptibilities may be 

made in various ways, according to available information about 

errors in the magnetometer data. Standard expressions involving Taylor expansions have been developed for cases in which 

(1) the actual errors in data are assumed to be known and (2) 

some characteristic of errors in the data, such as the standard 

deviation of a set of measurements, is known (Beers, 1957; 

Parratt, 1961; Bevington, 1969). These expressions give either 

the error or error characteristic of the computed result by accounting for propagation of errors associated with the data. For 

purposes of illustration, we may consider the special case in which 

principal susceptibility axes coincide with coordinate axes of the 

specimen or specimen holder.

Considering the situation in which errors in measured values 

of A and a, given by AA and A«, are assumed to be known from 

previous experimental work, we may write for the error or uncertainty of the difference of principal susceptibilities, AA,
62 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY 

+ 3(AA) (Aa) 2 & A + (Acw)8v M33A 1

4(AA) 3 (Aa)-^- + 6(AA) 2 (Aa) 2-^- - + 4(AA) (Aa) 3- 

+ (Aa)

= [(AA) (cos 2a) -2(Aa)A sin 2a]

+ [~4(Aa) 2A COS 2a-4 (AA) (Aa)shl 2a] 

+ [-12(AA)(Aa) 2 COS 2a

+ 8(Aa) 2Asin2a]

+ [32(AA) (Aa) 3 Sin 2a 

+ 16(Aa) 4 AcOS2a] + .

If, in our example, a = 0° (corresponding to principal susceptibility axes coincident with axes of the specimen or specimen 

holder), the previously determined uncertainty in axis direction 

is ±2°, the measured susceptibility difference is 10~ 5 emu/cm3 , 

and the previously determined error in this difference is ±5 percent, or ±5xlQ- 7 emu/cm3 (corresponding to uncertainty in the 

measurement of electric current), the resulting error in susceptibility difference, AA, ranges from -5.41X10- 7 to +4.44X1Q- 7 

emu/cm3 . These figures are typical of propagated errors in the 

present study.

SAMPLE CALCULATION

As an example of the calculation of susceptibility anisotropy, 

data are presented which were obtained from measurements of 

a 2.54-cm cubical specimen of vesicular basalt from the Elkhorn 

Mountains Volcanics (Upper Cretaceous) northeast of Whitehall, 

Mont. The Xi X2 coordinate plane of the sample, as measured 

in the field, has a strike of N. 38° W. and a dip of 77° NE. The 

basaltic unit itself has a strike of N. 55° W. and a dip of 60° NE., 

as determined by the planar orientation of vesicles and the orientations of volcanic rock units stratigraphically above and below 

the basalt. The rock specimen, which was collected near the 

central part of the 82-m-thick flow unit, has a stable reversed 

remanent magnetization with declination, N. 54° E., inclination, 

-39°, and intensity, 1.04xlO- 3 emu/cm3 .

Apparently susceptibility " magnitudes and associated phase 

angles obtained by using the Mead-type spinner magnetometer 

are

11 Apparent susceptibility values may be converted to true susceptibility values using equation 3 and the demagnetization factor, N = 4ir/3.
BHATTACHARYA SOLUTION OF EQUATION 63

A = 2.13XlO- 5 emu/cm3 , 

5=7.81 xlO- 5 emu/cm3 . 

C=9.80XlO- 5 emu/cm3 , 

« = 173°, 

/?= 7°, 

y= 85°, 

and for the independent measurement of K.ri,

K*s = 2.30XlO- 3 emu/cm'.

In the computation process, we initially set K38 = 0 so that, 

from equations 59, 60, and 65, we write to three significant figures,

a=9.51 xlO~5 emu/cm3

6 = 7.58x10-° emu/cm3

/=0.945XlO- 5 emu/cm3 

g = 0.851 XlO- 5 emu/cm3 

and

ft= -0.258X10- 5 emu/cm3 .

Also, from equations 66 through 78, we have 

l/3(a+6 + c) =5.70X10- 5 emu/cm3 

l/3(a+6 + c) 2 = 97.3X!0- 5 units, 

2/27(a+6 + c) 3 = 370xlO- 5 units, 

bc + ac + ab f2 g2 h2 = 70.4x 10~ 5 units, 

abc + 2fgh-af2 -bg2 -ch2 =-14AxlQ- 5 units, 

q= 26.9 XlO~5 units, 

r= -45.7XlO- 5 units, 

cos 877 = -0.849 

377 = 148° 

r7=49.4° 

cos 77 = 0.651

 J = -0.983

/ 47r\ cosf 77+ 1 = 0.331

\ 3 /

N = 5.99xlO- 5 units,

1^=9.60xlO- 5 emu/cm3 , 

Ks = - 0.195 x 10- 5 emu/cm3
64 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

and

#S = 7.68X10- 5 emu/cm3 .

To find the actual values of principal susceptibilities, we must add 

the term K*y = 2.30xlO~3 emu/cm3 to each of the above terms, 

K 

123 

K , K . The actual principal susceptibility values, K* 

12

K*

K* , are

K* = 2.30 XlO- 3 + 9.60 xlO- 5 = 2.40 xlO~3 emu/cm3 , 

K* = 2.30XlO- 3 -0.195XlO- 5 = 2.30xlO~3 emu/cm3 ,

and

K* = 2.30 xlO- 3 + 7.68 XlO~5 = 2.38xlO~3 emu/cm3 .

The magnitude ellipsoid has the shape of an oblate spheroid with 

the maximum plane of susceptibility normal to /£*, the minor 

axis. The maximum percent anisotropy, as defined by equation 

74, is

 . /9.80xlO- 5 \ 

Maximum percent anisotropy = I 100 percent

\2.36X10- 3 /

= 4.15 percent.

The direction of the KI principal susceptibility axis is obtained 

from equations 81 through 103, as follows:

a0 = - 0.0911, 

60 =-2.02, 

c0 = -9.60, 

m = 12.5, 

n = 0.379,

with direction cosines li, 1 2 , and 1 3 ,

lt = 0.996 

lz = 0.0301 

ls = 0.0795,

and their associated angles 0Jf 02 , and 03 ,

^=4.88°,

V% == OO.« ,

and

0, = 85.4°.

The declination, D', and inclination, I', are obtained from

tan D' = 0.0303,

D'=1.73° 

sin/'=0.0795,
BHATTACHARYA SOLUTION OF EQUATION 65

and

7'=4.56°.

The direction of the K2 principal susceptibility axis is obtained 

from equivalent values, as follows:

a0 = 9.70,

b0 = 7.77,

c0 =-0.195,

w=-0.091,

n= -0.110,

Zj = -0.090,

lt = - 0.109,

13 = 0.990,

^=95.2°,

02 =96.2°,

0, = 8.12°, 

tan D' = 1.21,

£' = 50.4°, 

sin/'=0.990,

7'=81.9°. 

Similarly, the direction of the K3 principal axis is given by

a0 = 1.83,

b0 =- 0.104,

c0 =-7.68,

m = 0.604,

w = 7.59,

Z,= 0.0787,

le = 0.988,

Z,= 0.130,

0;=85.5°,

^ = 8.75°, 

^ = 82.5°, 

tan D' = 1.26, 

D'=51.5°, 

sin/'=0.130, 

and

/r =7.48°.

Projections of the maximum plane of susceptibility and the minor 

axis, K2 , are shown in figure 12 relative to the Xt-Xs coordinate
66 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

plane of the rock specimen, to geographic coordinates in the 

original outcrop position, and to the bedding plane. The direction 

of remanent magnetization, also shown for each of the three 

orientations in figure 12 is seen to be unrelated to the geometrical 

configuration of the susceptibility anisotropy.

EXAMPLES OF ANISOTROPY DATA

Additional examples of results from using the Mead-type 

spinner apparatus data on 42 volcanic rock specimens possessing anisotropic susceptibility are listed in table 2. Most of

NORTH

FIGURE 12. Plots of maximum susceptibility plane, minor susceptibility axis, 

and remanent magnetization vector of rock sample relative to (A) the Xi 

Xa coordinate plane of the sample, (B) geographic coordinates in the original outcrop position, and (C) the bedding plane. Solid lines and plots of 

the minor axis are on the lower hemisphere. Dashed lines and plots of the 

reversed remanent magnetization vector are on the upper hemisphere. Equalarea projections.
EXAMPLES OF DATA 67

these samples have relatively strong remanent magnetization intensities of 10 ~* to 10 ~ 2 emu/cm3 , but for all measurements, 

fundamental frequency signals associated with remanent quadrupole moments were undetected, and second-harmonic signals 

associated with remanent dipole moments were effectively cancelled. These 42 specimens are part of a collection of 106 samples 

tested for anisotropy from among 77 lithologic units in the Elkhorn Mountains Volcanics (Upper Cretaceous) of western Montana. Among the 64 specimens not listed in table 2, approximately two-thirds have anisotropies which are detectable but not precisely measurable; the remaining one-third have no detectable 

anisotropy. Because of the reconnaissance nature of the sampling, 

no firm conclusions about the relationships between anisotropy 

configuration and rock type or rock fabric can be confidently 

drawn. In general, most of these rocks have anisotropies which 

are triaxial ellipsoidal or nonellipsoidal, although those that possess oblate spheroidal anisotropy have maximum susceptibility 

planes which are oriented nearly parallel to bedding planes or 

planes of flow.

Available data from susceptibility anisotropy studies of other 

volcanic and igneous rocks (Girdler, 1961b; Khan, 1962; Stone, 

1963; Wing-Fatt and Stacey, 1966; Janak, 1972) suggest that relationships between susceptibility geometry and petrofabric elements of igneous rocks are not as well defined as these relationships in many sedimentary and highly foliated metamorphic rocks.

Intrinsic susceptibility components of the rock may be obtained directly from the measured apparent susceptibility components by use of the rock specimen demagnetizing factor N = 

47T/3. The principal intrinsic susceptibilities, KIR1 , KIR2 , and KIRS 

become

2.40 XlO~3 KIR1 = =2.42xlO~3 emu/cm3 , 

1-(4.19)(2.40X10- 3 )

o 30xlO~3

KTRS = : = 2.32xlO~3 emu/cm3 , 

1-(4.19)(2.30X10- 3 )

2 38xl0~3 KIR3= '- =2.40xlO~3 emu/cm3 .

1-(4.19)(2.38X10- 3 )

The intrinsic magnetic susceptibilities of the magnetic mineral 

grains within the rock specimen may be estimated using the 

measured volume percent of magnetite, 2.7 percent 12 and an estimated average mineral grain demagnetization factor of 3.6 obtained from various rock magnetic investigations noted by Nagata

12 Measured by point counts of one thin section and one polished section.
68 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

TABLE 2. Anisotropic susceptibility data for 4-2 volcanic rock

[nd not determined, because of inconsistency of data. All susceptibilities are apparent

demagnetizing

Rock type M. 1B *C ao 20 2 y

Andesite flow ..- -- .

Do ________ . .

Do - ____ . __ .

Vesicular flow _ .. _ _ .

Do ___ - _____ _

Welded tuff __________ _._..

Vitric tuff ___ _ __ ...

Vesicular tuff __ _ _ _ .____.

Basalt flow _ ______ _ .

Andesite flow _ __ - _____ . Ash flow __ _._ __ __ _ ___ . Do _____ . __________ . Andesite flow ----- _______

Do ... _________ __

Crystal tuff _____________ .

Basalt flow _ _ ___ ______ .

Tuff breccia ____ - _ _

Basalt flow __ . __ _-___ _ ___.

Do _______ . __ . __ ....

Crystal tuff _____________ .

Basalt flow ____ .. _____ _ . Do __._ _____ ______

Do _ . __ .. _ . ______

Crystal tuff ______________ .

Basalt flow __ .. _ ._ _ .. __ . Do ...-..._... _......._....

Do ..... _ .. ___ .... _ ...

Lithic tuff ______________ . Welded tuff _____ ___ _

Latite flow _ __.. _ _ _

Diabase intrusive .__... _ .. _ . Vitric tuff ..... _ . _ . ._.. . Hypabyssal intrusive _ . ___ ...

Do . _____ ..__...__. __ .

Do ..... _ ..._....__ _ .

Do ...__.._....._... ..

Latite tuff . _____

Crystal tuff __________ ____

Metamorphosed flow ...

Do ________________________

Metamorphosed intrusive .. __ _,

Do _._._.__..___._.___ ....

. ___ 33.0

4.04

. ___ .56

. ___ 87.2

. ___ 82.1

. ___ 66.6

_ ___ 2.89

. ___ 21.3

. ___ 23.6

. ___ 37.9

. ___ 49.6

. ___ 9.12

. ___ 22.5

. ___ 13.9

. ___ .12

._.__. 104

. ___ 19.9

. ___ 47.3

. ___ 281

. ___ 124

. ___ 109

. ___ 179

. _ 22.3

. ___ 124

. ___ 129

4.96

. ___ 11.1

. ___ 25.0

43.0

. __ 20.1

. ___ 310

61.6

. ___ 8.00

. _ 67.4

.. __ 48.0

. ___ 307

. ___ 45.7

._ . 17.9

. _ .. 1,190

. ___ 105

______ 744

55.1

14.8

2.02

.48

47.7

281

2.40

14.2

78.1

64.4

29.1

10.1

4.05

1.34

5.02

7.30

115

134

14.0

199

79.6

173

159

5.71

150

.50

2.92

26.9

12.4

16.6

.12

260

17.6

17.9

.88

76.1

333

7.64

3.62

626

81.6

320

104

4.34

10.7

.99

74.6

49.9

6.31

.06

98.0

28.3

40.1

54.3

16.6

2.32

7.40

7.82

135

39.9

13.5

27.0

15.8

112

161

5.42

94.1

19.1

17.1

9.90

.48

49.8

19.1

17.0

61.6

13.1

317

120

309

19.4

13.7

1,030

547

417

54.9

150

106

78

75

22

48

8

173

56

32

6

89

42

16

5

154

75

148

106

156

47

34

146

17

37

82

81

39

70

101

179

73

86

73

36

72

60

83

14

164

174

68

77

66

95

158

24

10

52

7

37

29

156

69

89

106

172

153

88

119

119

86

73

175

118

55

5

71

68

60

103

5

174

50

81

5

160

135

30

3

134

170

89

91

74

32

5

154

163

21

5

85

53

26

91

22

58

85

105

76

55

103

117

54

78

19

76

116

150

33

56

5

7

7

10

165

109

140

166

169

173

1

79

126

85

84

*A, B, C, magnitudes of anisptropic susceptibility signals in the first, second, and third 

spin orientations, respectively, adjusted to susceptibility units of 10"° emu/cm3.

2 o, |8, 7, phase angles measured in the first, second, and third spin orientations, respectively, 

in degrees.

3 K33-normal susceptibility component measured by static method parallel to Xs rock sample 

coordinate axis, in units of 10-° emu/cm3 .

and Uyeda (1961, p. 131). The estimated intrinsic susceptibilities, 

IMZ, and KIMS are

Kmi= 2>42Xl °~3 =1.32xlO- 1 emu/cm3 , 

0.027- (3.6) (2.42X10- 3 )

2 32x 10~3 

KIM2 = '- = 1.24xlO- 1 emu/cm3 ,

0.027- (3.6) (2.32X10- 3 )
EXAMPLES OF DATA 69

specimens obtained by the dynamic method of measurement

susceptibilities which may be converted to true rock susceptibilities using the rock sample 

factor of 47T/3]

«.

2,450

758

760

2,290

2,310

2,990

639

2,300

3.390

3,010

3.410

3,000

401

3,400

1,610

10,100

4,490

5,910

5.890

6,210

4,990

2,810

2,800

3,190

9110

509

930

3,490

4,810

1,590

10,100

6,790

620

6,210

2,790

10.000

2,800

1,490

51,100

41,900

50,500

2,540

*.

2,447

756.0

759.5

2,284

nd

2,989

638.0

2,367

3,398

nd

3,430

2,995

400.9

3,401

1,614

10,158

nd

6,911

nd

5,186

nd

nd

2,800

3,194

nd

506.0

nd

3,488

4.792

L584

nd

6,771

nd

6,192

nd

9,892

2,795

1,487

61.390

41,980

50.530

nd

«-«.

16.0

5.00

.493

55.6

nd

33.0

6.67

39.0

35.4

nd

34.8

7.62

11.3

7.42

3.07

98.1

nd

24.3

nd

60.1

nd

nd

.118

71.6

nd

8.30

nd

12.7

22.1

7.21

nd

26.2

nd

150

nd

279

20.5

7.26

751

276

389

nd

*-,.

 18.7

 5.98

 .625

 56.4

nd

 33.7

 7.68

 58.9

 27.9

nd

 22.2

 9.36

 11.4

 6.75

 5.28

 88.2

nd

 24.8

nd

 75.1

nd

nd

.116

 116

nd

 8.95

nd

 14.7

 32.9

 13.4

nd

 44.8

nd 169

nd 208

 25.8

 10.6 728

 274

 359

nd

*-*

3.62

.98

.033

.747

nd

.738

.905

19.8

 7.42

nd

 12.6

1.75

.083

 .673

2.21

 9.97

nd

.473

nd

25.0

nd

nd

 .234

44.8

nd

.650

nd

 1.92

10.8

6.16

nd

18.6

nd

18.9

nd

 71.7

5.21

3.37

 22.3

 2.07

 30.4

nd

Maximum 

percent 

anisotropy »

1.38

1.45

.13

4.90

nd

2.23

2.23

4.15

1.86

nd

1.66

.57

6.66

.42

.52

1.8-3

nd

.83

nd

2.41

nd

nd

.01

5.87

nd

3.41

nd

.78

1.15

1.30

nd

1.05

nd

5.14

nd

4.87

1.66

1.20

2.88

1.31

1.48

nd

Description 

of anisotropy

Triaxial ellipsoid.

Do.

Do.

Do.

Nonellipsoidal.

Triaxial ellipsoid.

Do.

Oblate spheroid.

Triaxial ellipsoid.

Nonellipsoidal.

Prolate spheroid.

Triaxial ellipsoid.

Do.

Do.

Oblate spheroid.

Triaxial ellipsoid.

Nonellipsoidal.

Triaxial ellipsoid.

Nonellipsoidal.

Oblate spheroid.

Nonellipsoidal.

Do.

Oblate spheroid.

Do.

Nonellipsoidal.

Triaxial ellipsoid.

Nonellipsoildal.

Triaxial ellipsoid.

Oblate spheroid.

Do.

Nonellipsoidal.

Oblate spheroid.

Nonellipsoidal.

Triaxial ellipsoid.

Nonellipsoidal.

Triaxial ellipsoid.

Do.

Oblate spheroid.

Triaxial ellipsoid.

Do.

Do.

Nonellipsoidal.

bulk susceptibility, denned as l/S(Kn+Kti+K»») = l/3(Ki+Kt+Ki), in units of 10~» 

emu/cm3.

*Ki KB), (Kt Ka), (Ks B), differences of principal susceptibility components and bulk 

susceptibility in units of 10-° emu/cm3.

T (Kmax Krnln) "I

6 Maximum percent anisotropy, defined as I I X 100 percent.

L KB J

KTMS 

2.38X10-'

 -1.29X10- 1 emu/cm3 .

0.027- (3.6) (2.38X10- 3 ) 

We may take as the estimated mean intrinsic susceptibility, 

Km, of the magnetic mineral grains the arithmetic mean of Knn , 

Kmt, and KIMS, which is

Km =l/3(KI1 +KI2 + IS ) =1.28X10- 1 emu/cm3 .
70 WEAK-FIELD MAGNETIC SUSCEPTIBILITY ANISOTROPY

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 £ U.S. GOVERNMENT PRINTING OFFICE: 1977 240-961/4