955

VOL. 48, 1962 GEOLOGY: F. J. TURNER 955

search under grant NSF G-10236 administered by The University of Texas; to the Royal Depart-ment of Mines, Thailand, for their support of the research in Thailand, to Charles Mann andGaylord Walker of the United States Operations Mission to Cambodia, and to Edmond Saurinand Adrien Milli6s-Lacroix, University of Saigon, for cooperation in Cambodia and Viet Nam, andto others. too numerous to name here, who made this research possible.

Suess, F. E., K. geol. Reich. Wien, Jb., 50, 2, 193 (1900).2 Lacroix, A., C.R. Acad. Sci. (Paris), 200, 2129 (1935).Urey, H. C., Nature (London), 179, 556 (1957).

4Barnes, V. E., CERN Conference on Fission and Spallation Phenomena and Their Applicationto Cosmic Rays, September 26-29, 1961, abstract 37.

Barnes, V. E., Scientific American, 205, 5, 58 (1961).6 Urey, H. C., these PROCEEDINGS, 41, 27 (1955).

ROTATION OF THE CRYSTAL LATTICE IN KINK BANDS,DEFORMATION BANDS, AND TWIN LAMELLAE OF STRAINED

CRYSTALS*

BY FRANCIS J. TURNER

GEOLOGY DEPARTMENT, UNIVERSITY OF CALIFORNIA, BERKELEY

Communicated March 19, 1962

This paper is concerned with the geometry of rotation of the lattice of any highlystrained domain in a deformed crystal with respect to the lattice of the unstrainedhost crystal. In particular it is a revision of the geometry of external rotation ofnarrow kink bands, deformation bands, and twin lamellae in experimentally de-formed crystals of calcite and of dolomite. The degree and sense of externalrotation are correlated with strain and with the orientation and sense of glide in theactive glide system. The material studied is a series of single calcite crystals ex-perimentally deformed by D. T. Griggs and H. C. Heard over the period 1952-1961 at the Institute of Geophysics, University of California, Los Angeles.

Correlation of External Rotation and Strain in Deformed Metal Crystals.-Metal-lurgists have long recognized that in experimental deformation of single crystals,strain is concentrated in local domains such as kink bands, and that the crystal lat-tice within such a domain becomes progressively rotated with respect to the latticeof the undeformed ends of the specimen, in which a constant orientation is main-tained throughout deformation.1 The same effect has been observed in experimen-tally deformed crystals of calcite, dolomite, enstatite, and other minerals.2 Sincerotation is measured with reference to coordinates (e.g., lattice directions in the hostcrystal) external to the rotated domain, it is, in Sander's terminology, an externalrotation (to be distinguished from internal rotation of passive marker planes, suchas cleavage cracks, within the deformed domain).The magnitude and sense of external rotation depend upon (1) the magnitude of strain, (2)

the orientation of the active glide system in relation to the axes of principal stress a-, and 3 and(3) constraint of the undeformed ends of the specimen as determined by the mechanical condi-tions of loading. In all cases the active glide plane rotates toward the axis of minimal stress 93.(In conformity with nomenclature currently in use by students of experimental deformation ofrocks, compressive stresses are positive. tensile stresses negative; a, is algebraically greater than

956 GEOLOGY: F. J. TURNER PROC. N. A. S.

as). The axis of rotation lies within the glide plane T and is normal to the glide direction t. Inthe field of metallurgy' external rotation has been related to linear strain e by simple equations:[11 for extension sin xo/sinxi = li/lo = (1 + e); [2] for compression cos xi/cos xo = 14/lo = (1 -e); where xo and xi are the angles between the glide plane and the axis of the test cylinder re-spectively before and after the experiment, and lo and 11 are the respective lengths of the cylinderbefore and after the experiment. The angle of external rotation, cW, is (Xo - Xi).Anomalous Relations in Deformed Mineral Crystals.-The same expressions have

been applied to evaluate strain correlated with external rotation of kink bands anddeformation bands in experimentally deformed mineral crystals.2' 3 In these theaxis of observed external rotation is invariably normal to the glide direction tin the glide plane T as deduced from other data. But there is considerable dis-crepancy between the magnitudes of the calculated and of the observed strain, eventhough the latter is always in close agreement with strain deduced from internalrotation effects.There is good reason for such discrepancy. In experimentally and, presumably also, in naturally

deformed crystals, kink bands and deformation bands develop under conditions of loading thatdepart markedly from those upon which equations (1) and (2) are based.

1. Figure 1 illustrates external rotation of the lattice of a very broad kink band (between kk

ITk FIG. 1.-Sections normal to

glide plane T and parallel to/\\\| glide direction t to illustrate

j : |\ Pi 1external rotation of deformedsector of specimen in tensile test.ts\\\\\ 0 l l(a) Ends of specimen free; axis

l\ \ \\\ 19 of strained sector is rotatedN \\\< counterclockwise through co =l< \\\\\ 1(\xo- Xi = 380. (b) Ends of

K/\\ \\\ 1|1 specimen constrained to be co-6s, \\\\\ I; axial; lattice of strained sector

T (kink band between kk and k'k')is rotated clockwise through

SkAco = co' + o = 44'.

a

b

and k'k') in a conventional tensile test as carried out on a single crystal of a metal.1 4 xo = 530;xi = 15°; It/lo = 3.07; e = 2.07 (207W%). If the ends were free (diagram a) the axis of the strainedsection would be rotated counterclockwise through an angle c(380); the lattice, as indicated bythe glide plane T, would maintain a constant orientation throughout the whole specimen. Since,however, the conditions of loading constrain the ends to remain coaxial, the strained section as-sumes the form of a kink band within which the lattice is externally rotated clockwise through anangle w'. Because of the difference between their respective cross sections, the kink band and theunstrained ends are not quite coaxial. They diverge by an angle p, which is small (in this case 60)compared with w provided the value of e is high and the kink band occupies most of the lengthof the specimen. The observed angle of external rotation w' = co + o, whereas the ideal angle de-duced from equation (1) is w. In tensile experiments on calcite and other mineral crystals the kinkbands are usually narrow compared with the length of the cylinder. In consequence the straincalculated from co' is notably higher than the observed strain or the strain calculated from dataof internal rotation.

2. Equation (2) applies to external rotation of T (i.e., of the lattice) in short cylinders, homo-


/T~~~~~~~~~~/ -I

YiX,,,a. b. C.

FIG. 2.-Sections normal to T and parallel to t to illustrate external rotation of the lattice throughco, = (xI - xo), during compressive test in which end surfaces of specimen remain parallel but notcoaxial. (a) Before strain. (b) After strain. (c) Geometric relation lo/ = cos xo/cos xi for zerostrain in the plane T.

geneously strained in such a way that the end surfaces remain parallel. Figure 21, illustrates acase where l1/lo = 0.67; e = 0.33; xo = 300; xi = 54°. The axis of the cylinder is deflectedthrough a small angle which is the difference between the clockwise external rotation (xI - xo)and the counterclockwise internal rotation (d - ca) of the cylinder surface with respect to theglide plane T. Very different are the conditions familiar in compression experiments on mineralcrystals, where strain is far from homogeneous but is concentrated in narrow kink and deformationbands, and the ends of the cylinder are constrained to remain coaxial. Here, too, strain calculatedfrom observed external rotation commonly is greater than observed strain.

Clearly it is desirable to develop other relations between strain and rotation that hold good forkink and deformation bands observed in experimentally strained short cylinders cut from singlecrystals of minerals.

Revised Correlations of External Rotation and Strain in Kink and DeformationBands.-The closest approach to the condition illustrated in Figure lb is foundwhere, in extension experiments, strain is relatively large (e = 0.2 to 0.5) and islocalized in a single kink band whose width may be as much as half the length of thecylinder. This is illustrated in Figure 3, representing the outline of a thin sectionof a calcite crystal deformed by extension normal to the prism (1010), = ml, at300'C and 5,000 bars confining pressure.' The over-all strain e = 0.16; but thelocal strain in the kink band, calculated from reduction of one diameter from 0.49to 0.38 inches, is 0.29. The angle sp between the longitudinal trace of the kink bandboundary and the axis of extension (stress axis a73) = 60. Strain is due to glidingon (1011), = rl; xo = 450; xi has an average value of 280, so that the meanangle of external rotation of the kink band is 170.To take into account the angle up, the equation of strain is modified to eq. (3),

Is/lo = sin Xo/sin (xi + so). This gives a value of 1.24, and the correspondingstrain, E = 0.24, corresponds reasonably well with the value 0.29 calculated from thetransverse dimensions. The unmodified equation (2) on the other hand gives theimpossibly high value E = 0.47.

In compression experiments on short cylinders, strain is usually localized in asingle narrow oblique kink band (Fig. 4a). In some extension experiments, too,the kink band may be narrow compared with the length of the specimen, or strainmay be concentrated in several narrow lamellar or lenticular deformation bands sur-rounded on all sides by undeformed or slightly strained material (Fig. 5a). Clearlythere is no approach to the ideal conditions of Figures 1 and 2, and completely dif-ferent relations between strain and external rotation must therefore be explored.


t t^3r,

e0)~~~~~~~~~~~~~~~~~~~ ''

/ / " ~~~~~~~~Le.,,

4 ~ ~ ~~~~~~a.b.FIG. 3.-Broad kink band FIG. 4.--a) Kinkband (stippled) in section of calcite

(stippled) in section of calcite crystal deformed by compression normal to mi. Twincrystal deformed by extension gliding on el. The lightly stippled areas are transi-normal to mi. Translation glid- tional sectors of progressive strain. (b) Detail ofing on ri. secondary kink bands ki, k2, developed near D of dia-

gram a.

)A { b 11). /T . } ~~~~FIG. 5.-Glide plane T in host crys-a \b q tal (stippled) in and kink band (be-

ffii\ 1l\ t ~~~~low). (a) Ideal condition for equation|\/ / /\(4).(b) Condition for equation (5).

T<<_;T

7-~~~~~~~~.

FIG.: 3.-Broad kink band Fi. 4-a) KFIG. 6. Longitudinal sections, parallel to(stippled)in section o calce cstl d the deformation plane, of calcite cylinderscrystal deformed by extension gliding ondeformedby extensionparallel to[r2:ar3i .

normal toX in. Translation glid- tional sectNiols crossed Axis of m inimum streofing on rl. secondary k 1X-_axisof extension),dvp islongitudinal. (a)s

z > I .i 111t;Xg#~r~t fr _Experiment 563- 500'C, 4,700 bars. StrainAdz|gB' Am 10.3Strain rate 5 X /see.e

XM1 TJI4 4 ^fl -R&E w(b) Experiment 294; 400'C, 5,000 bars..*: ,i. FStrain G10%. Strain rate 3.3 XTh /se.

At least one boundary surface of a kink or deformation band) ticallyins planarthrough much of its length (Fig. 6b). Here extinction directions and cleavage ortwin traces are sharply deflected and the trace of the boundary surface can be

VOL. 48, 1962 GEOLOGY: F. J. TURAER 959

located to within a degree or so. For such a band a possible relation between ex-ternal rotation of the lattice and internal strain may be developed on two assump-tions: (1) the boundary surface is occupied throughout progressive strain by thesame mineral particles; (2) the initial angle between the active glide plane and theboundary surface approximates 900 (Fig. 6a). The first assumption can apply onlyto sharply defined boundaries such as A B of Figure 4a; it cannot be valid where theboundary is not a surface but rather a narrow zone of progressive strain merginginto the unstrained host crystal (B C of Fig. 4a). The second assumption is con-sistent with a large body of evidence relating to deformed metallic and ionic crystalswhose glide systems have been identified by other means.Any passive marker surface occupied by the same set of particles becomes in-

ternally rotated with reference to the lattice of the strained band2 according to theequation cot a - cot /3 = 6 sin y where a and /3 are the angles between the glideplane T and the rotated surface after and before rotation, s is the shear (= tan0, the angle of shear on T) and -y is the angle between the axis of rotation and theglide direction t. Since the kink-band boundary remains fixed in space as a surfacein common with the undeformed host crystal, we may consider the lattice of thekink band as being correspondingly rotated through the angle (/ - a) with ref-erence to the boundary surface. Thus the angle X of external rotation of the kink-band lattice is / - a; and according to our second assumption /8 is 900, and ais (900 - w). For slip on a single glide system, y = 900. Thus the relation ofexternal rotation w to strain e becomes equation (4), cot(900 - a) = s = E/So,where So is the coefficient of resolved shear stress on T, and the magnitude of strainis small. Equation (4) is an approximation; but for values of co up to 300 the erroris small even where the angle /3 departs by 100 or so from 900.Where the kink-band boundary is sharp and the identity of the glide plane is

known from other data, the relation between shear and external rotation can bedefined precisely. Let a and /3 be the angles between the glide plane T and thekink-band boundary in the kink band and in the host crystal, respectively. Usu-ally the geometric relations are as pictured in Figure 5b, where the normal to thekink boundary lies in the acute angle between the two orientations of T. Then (5),s = cot a + cot ,/. This relation is preferred to equation (4) because it dispenseswith the assumption that the glide plane in the host lattice is normal to the kink-bandboundary.

Illustrative Examples.-Figure 6a shows a longitudinal section, parallel to thedeformation plane, of a calcite cylinder deformed by extension parallel to the cleav-age edge [r2:r3] at 5000C and 4,700 bars confining pressure (Experiment 563 ofD. T. Griggs and H. C. Heard). The total strain e = 10.3%; but in the neck Eis of the order of 20%, and much of this is concentrated in oblique lensoid deforma-tion bands (dark in Fig. 6a) where the strain must be even higher. Within thedeformation band adjacent to X, el lamellae, now Lelti have been internally ro-tated counterclockwise along the arc e1 c ri (the zone circle of the a2 axis) as a resultof gliding on ri (Fig. 7a).

For this glide system a = 390, ,/ = 710, y = 900, So = 0.31. e = 0.31 (cot 39- cot 71) = 0.275 (27.5%). The lattice of the kink band has been externallyrotated clockwise through 400 (w) about the a2 axis, the path and sense of rotationbeing consistent with gliding on ri. From equation (4), s = cot (900-40°)


FIG. 7.-Projections (equal-area, lower hemisphere) of lattice directions, planes and lamellaemeasured in sections shown in Figure 6. Points in the undeformed sector are shown as crosses;those in the deformation or kink band are shown as circles. IR indicates internal rotation of eito Leirl; ER indicates external rotation of the lattice in the band. Sense of a gliding on r, inthe band is indicated by arrows. (a) Experiment 563. Deformation band near X, Figure 6a.Trend of band, dd. (b) Experiment 294. Diagonal kink band, Figure 6b. Trend of band, kk.

0.84. The corresponding strain, e = (0.31 X 0.84) = 0.26 (26%), is in reasonableagreement with the values calculated from internal rotation of Leirl and from cross-sectional dimensions of the neck. Onthemisfheot handifethe standard equation(1) were used, e =sl/lo -1 = (sin 71/sin 31) -1 = .83 (83%s). This s ssanim-possibly high value.

Figure 6b represents a longitudinal section through a similarly oriented calcitecylinder deformed by extension at a slow strain rate at 400sCand 5,000 bars con-fining pressure (Experiment 294, H. C. Heard). The over-all strain of 10%o is lo-calized in a single diagonal kink band inclined at 660 to r1 of the host lattice. Glid-ing on r1 within the kink band is established from clockwise internal rotation of eithrough 280 to LeIT' (Fig. 7b); and the corresponding local strain is calculated asee= 23%w. Counterclockwise external rotation of the kink band through 42°

about the a2 crystal axis is consistent with gliding on r1. Using equation (4),s = cot (90 - 42) = 0.90; e = 28%o. Since the orientation of the kink-bandboundary can be measured to within a degree or two, the preferred equation (5)can be employed; and from this s = (cot 66 + cot 72) = 0.77; e = 24%s. If,*however, the standard equation (1) were employed the calculated value of e wouldbe impossibly high:e= (sin 71/sin 29)-1 = 95%.

It is instructive to recompute strain on the basis of equations (4) and (5) usingpublished data on experimentally deformed crystals of calcite for which the value ofe has been determined independently from data of internal rotation or from the ge-ometry of e twinning. Figure 4a shows a longitudinal section through a calcite cylin-der deformed by compression normal to the prism (1010), = mi, at 2000 and 10,000bars.6 The over-all strain is a shortening of 15%. But this is concentrated in adiagonal kink band within which twinning oned is complete, so that the local strainis accurately known:t S = 0.694;i =26%e. The angle of external rotation of thekink-band lattice, co-shown by counterclockwise deflection oftk in Figure 4a-is37°. From equation (4), s = cot 53 = 0.75; the error is ± 8%. Contrast this


with the value of e calculated from equation (2). E = 1 - (cos 63/cos 26) = 0.50;the error is + 90%. At the edges of the main kink band of Figure 4a (e.g., nearpoint D) there are local microscopic, sharply defined kink bands showing differentdegrees of strain (el twinning) and of external rotation. Data for one of thesebands (k, in Fig. 4b) are as follows: Angle of internal rotation of e2 to Le, e = 180(clockwise). Angle of external rotation of lattice in ki = 280 (counterclockwise).Twinning on e1 in k1 is about 80% complete (s -0.56). From internal rotations = (cot 45 - cot 63)/sin 57 = 0.58. From external rotation, using equation(4), s = cot 62 = 0.53. All three values of s are mutually consistent.

Finally, Higgs and Handin7 have recorded rotational data for experimentallydeformed crystals of dolomite. In most of these experiments T = {0001 } and tis an a axis of the lattice. Values of the angle of external rotation are not large andare correct only to within one or two degrees. Nevertheless, as shown in Table 1,

TABLE 1VALUES OF e COMPUTED FROM ROTATIONAL DATA OF HIGGS AND HANDIN7 FOR

EXPERIMENTALLY DEFORMED DOLOMITE CRYSTALSComputed Values of e (%) -

Over-all, Local, from Local, fromSpecimen number from internal external rotation(page referencE) Orientation c dimensions rotation Eq 1 Eq 2 Eq 4

160 (p. 268)1 Extension 120 11.9 9.2-14.9 30.0 ... 9.0170 (p. 269)J normal to r 70 10.4 12.3 11.4 ... 5.3529 (p. 27-271)l Compression 140 6.5 9.0 ... 29.0 10.7156 (p. 269-270)f normal to r 100 7.8 12.0 ... 20.0 7.6161 (p. 271) Extension 100 7.5 7.9 or9 26 ... 7.6

parallel to[f2:f3I

622 (p. 266) (twin Extension 220 20 37 ... 12.7gliding on f = parallel to a102211)

the strain calculated from equation (4) commonly agrees satisfactorily with thatcomputed from internal rotation and from dimensional data, especially where (asin specimen 161) strain is concentrated in a sharply defined kink band. In onespecimen (170) the value of E computed from the standard equation (1) is more satis-factory than that obtained from equation (4); but here the kink band is ill defined.Experiment 622 was complicated by failure by shear fracture following twin glidingon {0221} = f.

Collectively the experimental data relating to calcite and dolomite confirm thegeneral validity of equations (4) and (5) as applied to narrow kink bands and de-formation bands.

Lattice Rotation in Complementary Twinning.-When twin gliding on el proceeds to completionin a calcite crystal, any pre-existing e2 twin lamella retains its identity and becomes internallyrotated through 220 (Fig. 8a). As an Le,' lamella it becomes reoriented so as to coincide with{1120} = a of the twinned host crystal.2 8 Dr. Iris Borg has drawn the author's attention to anaccompanying effect which she observed in completely rotated L.e21 lamellae: these themselvesbecome strained in sympathy with the enclosing host crystal, by internal twinning on e12 (thetwinned equivalent of el in the e2 lamella); and the orientation of the lattice so formed is identicalwith that of the completely twinned host. The boundary surfaces of the lamella are still recog-nizable in arrays of microscopically visible discontinuities.The geometry of the reorienting process may be followed on a projection upon the only recog-

nizable plane that survives the process of complementary twinning, namely Le1e (Fig. 8b). This


FIG. 8.-Relative rotation of lattices of host crystal and a rotated lamella Le2el in twinning of cal-cite. Crosses refer to the initial host lattice, circles to the host lattice twinned on e, (cl, a').Triangles (c') refer to lattice within the rotated lamella Lcele. Axis of rotation is [ei :e,]. (a)Plane of projection, e2. The L¢ele lamella and its lattice (as shown by c') is internally rotatedthrough 220 with reference to the fixed host lattice. (b) Plane of projection, Le~el All rotationsshown as rotations of the lattices with reference to fixed coordinates Le~el and [e : e,]. Solidcircles (c2/, e,') in upper part of figure refer to the lattice of the Leo~el lamella internally twinned one,' and rotated through 22° with respect to Le~el.

is held in a fixed position, and the two lattices-that of the host and that of the lamella-areshown as rotating in opposite senses as required by the geometry of twinning. The axis of rotationis the same for both lattices, the common edge R between e,, e, and e,'. In Fig. Sb what is conlven-tionally described as internal rotation of L~e1e within the host lattice is shown as external rotationof the twinned lattice in the opposite sense with reference to a fixed Level. In the upper part of thefigure the [00011 axis c2 of the lamella lattice is reoriented through 520 by complete twinning one,'. But in this process the lamella boundary surface e,' must become internally rotated to at 11201}orientation in the new lamella lattice. This involves relative rotation through 220 so that e,'comes to coincide with e,'; and [00011 in the twinned lamella (c2') now coincides with [00011 inthe host (cl). The lattices of the twinned host and the rotated twinned lamella completely coin-cide.To illustrate the general case, let a crystal lattice become completely twinned by gliding on a

plane h. And let a pre-existing twin lamella k (twinned on a plane k of the host lattice) therebybecome rotated to a new orientation Lkh. Lkh must be rationally oriented as a simple plane I ofthe twinned host lattice.9 If the lamella simultaneously twins on the complementary plane h'of its own lattice, the lamella boundary must assume the same orientation I in the lamella latticeas in the host. So the twinned host lattice and the twinned lamella have a plane I in common andalso a rotation axis r- the edge [h: k = [h': k] = [h:lJ-in common. This condition is satisfiedin one of two ways: (1) Where 1 is not a plane of symmetry the host and the lamella lattices arerelated by reflection twinning with 1 as twin plane; (2) Where I is a plane of lattice symmetry thehost andl the lamella lattices coincide.The second solution is exemplified by calcite, where I = L Ce,- {1120}-a plane of lattice

symmetry. The first solution is illustrated by complementary twinning onf1 = 102211} in dolomite.During fi twinning of the host lattice a pre-existing f2 lamella is rotated through 210 to LJwhich is f, of the host crystal.'0 This is not a plane of symmetry of the dolomite lattice. If theLFf' lamella simultaneously twins on its ownh f, plane, fd, the twinned lamella and the twinnedhost are finally related by twinning with f2 as twin plane. This may be demonstrated by geo-metric construction along the same lines as Figure Sb.Summary.-New equations are offered for correlating longitudinal strain e

or shear s (the tangent of the angle of shear on the glide plane) with external rota-tion of the crystal lattice in narrow kink bands and in deformation bands in ex-perimentally deformed crystals. Where xo and xl are the respective angles betweenthe single active glide plane and the longitudinal axis of the specimen(ael orlaof an axial stress system), (1) for extension experiments in which the kink band is

VOL. 48, 1962 MAT'HEMATICS: K. L. CHUNG 963

wide compared with the length of the specimen, and so is the angle between thelateral edge of the kink band and the axis of extension, E = [sin xo/sin (xi + so) ] - 1.(2) For narrow kink bands and deformation bands, (a) an approximate relation iss = cot (900 - w); (b) a more precise relation where a and ,3 are the respective anglesbetween the kink-band boundary and the active glide plane in the band and in theunstrained ends, s = cot a + cot fi. The validity of these equations has been testedagainst rotational and strain data for experimentally deformed crystals of calciteand dolomite.The above relations are based on the assumption that throughout progressive

strain the boundary surface of a kink or deformation band is occupied by the samematerial particles and serves as a datum with reference to which the crystal latticeof the band may be considered to rotate about the intersection of the kink-bandboundary surface and the glide plane T. By making a similar assumption with re-gard to a twin-lamella boundary in calcite, it is possible to explain complex ro-tational effects leading to simple geometric relations previously observed betweena completely twinned crystal lattice and the completely twinned lattice of an en-closed rotated lamella.

* This work was supported by the National Science Foundation (G 16316).1 Schmid, E., and W. Boas, Plasticity of Crystals (London: F. A. Hlighes & Co., Ltd., 1950),

57-63.2Turner, F. J., D. T. Griggs, and H. C. Heard, Bull. Geol. Soc. Am., 65, 898-901 (1954);

Higgs, D. V., and J. Handin, Bull. Geol. Soc. Am., 70, 247-249 (1959).3Handin, J., and D. T. Griggs, Bull. Geol. Soc. Am., 62, 863-886 (1951).4 Barrett, C. S., Structure of Metals (New York: McGraw-Hill Book Co., Inc., 1952). 369-370.5 Turner, F. J., D. T. Griggs, and H. C. Heard, Bull. Geol. Soc. Am., 65, 910-912, pl. 4A (1954).6 Ibid., 907, experiment 190.7 Higgs, D. V., and J. Handin, Bull. Geol. Soc. Am., 70, 245-278 (1959).8 Borg, I., and F. J. Turner, Bull. Geol. Soc. Am., 64, 1349 (1953).9 Pabst, A., Bull. Geol. Soc. Am., 66, 897-912 (1955).

10 Turner, F. J., D. T. Griggs, H. C. Heard, and L. E. Weiss, Am. J. Sci., 252, 485 (1954).

ON THE MARTIN BOUNDARY FOR MARKOV CHAINS*

BY K. L. CHUNG

DEPARTMENTS OF MATHEMATICS AND STATISTICS, STANFORD UNIVERSITY

Communicated by J. L. Doob, April 25, 1962

The Martin boundary for a discrete parameter Markov chain was first consideredby Doob,2 using the theory of R. S. Martin after whom the boundary is named.A direct and ingenious method was later found by Hunt,3 who also strengthenedDoob's results in several points. In this note, I shall sketch a natural approachto the theory in the continuous parameter case.t There, upon the introduction ofcertain intrinsic quantities (probabilities) which have no obvious discrete analogues,it is possible to derive the main results by familiar methods developed in reference1. They cover the discrete case through a simple transformation. Thus, in asense, what has been artificial becomes now the inevitable. It appears at the same

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