Clays and Clay Minerals, Vol. 46, No. 1, 38-50, 1998.

XRD MEASUREMENT OF MEAN THICKNESS, THICKNESS DISTRIBUTION AND STRAIN FOR ILLITE AND ILLITE-SMECTITE

CRYSTALLITES BY THE BERTAUT-WARREN-AVERBACH TECHNIQUE

V. A. DRITS, 1 D. D. EBERL 2 AND J. SRODOI~I 3

Institute of Geology RAN, Pyzhevsky 7, 109017 Moscow, Russia

2 U.S. Geological Survey, 3215 Marine St., Boulder, Colorado 80303-1066

3 Institute of Geological Sciences PAN, Senacka 1, 31002 Krakow, Poland

Abstrac t A modified version of the Bertaut-Warren-Averbach (BWA) technique (Bertaut 1949, 1950; Warren and Averbach 1950) has been developed to measure coherent scattering domain (CSD) sizes and strains in minerals by analysis of X-ray diffraction (XRD) data. This method is used to measure CSD thickness distributions for calculated and experimental XRD patterns of illites and illite-smectites (I-S). The method almost exactly recovers CSD thickness distributions for calculated illite XRD patterns. Nat- ural I-S samples contain swelling layers that lead to nonperiodic structures in the c* direction and to XRD peaks that are broadened and made asymmetric by mixed layering. Therefore, these peaks cannot be analyzed by the BWA method. These difficulties are overcome by K-saturation and heating prior to X-ray analysis in order to form 10-,~ periodic structures. BWA analysis yields the thickness distribution of mixed-layer crystals (coherently diffracting stacks of fundamental illite particles). For most I-S sam- pies, CSD thickness distributions can be approximated by lognormal functions. Mixed-layer crystal mean thickness and expandability then can be used to calculate fundamental illite particle mean thickness. Analyses of the dehydrated, K-saturated samples indicate that basal XRD reflections are broadened by symmetrical strain that may be related to local variations in smectite interlayers caused by dehydration, and that the standard deviation of the strain increases regularly with expandability. The 001 and 002 reflections are affected only slightly by this strain and therefore are suited for CSD thickness analysis. Mean mixed-layer crystal thicknesses for dehydrated I-S measured by the BWA method are very close to those measured by an integral peak width method.

Key Words--Crystall i te, Fundamental Illite Particles, Illite, Illite-Smectite, MacEwan Crystallites, Mixed-Layer Clay, Thickness Distributions, Warren-Averbach Method.

I N T R O D U C T I O N

The B W A techn ique (Ber taut 1949, 1950; Warren and A v e r b a c h 1950) ana lyzes profi les of X R D reflec- t ions and permi ts de t e rmina t ion of the a rea -weigh ted m e a n size of CSDs (also refer red to as " c rys t a l l i t e s " ) , the d is t r ibut ion of these sizes and f luctuat ions in the d - spac ings (micros t ra lns) of the CSDs. This method, based on Four ie r analysis , has been appl ied wide ly to meta l s and alloys, bu t unt i l recent ly (Eberl and Srodofi 1988; ,~rkai et al. 1996) has not b e e n appl ied to min- erals, except for the work of K o d a m a (1965) and Ko- d a m a et al. (1971), devo ted ma in ly to the analysis of mic ros t ra in in f ine-gra ined d ioc tahedra l micas. The CSD size and size d is t r ibut ion are essent ia l character- ist ics of the real c rys ta l s t ructure of a mineral . For example , the m e a n th ickness and th ickness d is t r ibut ion of ill i te and I - S C S D s could serve as indicators of the degree of d iagenes is and of low-grade me tamorph i sm.

Eber l and Srodofi (1988) were the first to apply the B W A approach to the s tudy o f mixed- l aye r I - S sam- ples, u t i l iz ing the S iemens Corpora t ion D5000 soft- ware (S iemens 1990). Th i s appl ica t ion was not com- ple te ly appropr ia te for I - S studies, because the B W A m e t h o d was deve loped for per iodic crystals . Eber l and B l u m (1993) t r ied to solve this p r ob l em by saturat ing

Copyright �9 1998, The Clay Minerals Society 38

the smect i te in ter layers wi th e i ther Ca or Sr ca t ions to obta in 15-,~ water--clay complexes . Al t e rna t ion of 15-

smect i te and 10-,~ illi te layers p rese rves the coher- ency of the waves scat tered in the d i rec t ion of the di f f ract ion m a x i m u m , wi th d = 5.0 ,~. Therefore , Fou- r ier analys is of this ref lect ion should yie ld the pa ram- eters of interest . However , subsequen t resul ts (unpub- l i shed data) have indica ted that the 15-,~ complex was not comple te ly deve loped for m a n y of the i r samples .

Recently, Lanson and Kubler (1994) used the same technique and the same software to analyze profiles of basal reflections for a large collection of illites and I S . For some samples they found a strong disagreement among mean thicknesses of CSDs obtained by Fourier analysis and by other techniques. These authors conclud- ed that, in its standard form, the method cannot be ap- plied to the study of illite and I~S.

In the course of fur ther exper iments with the com- mercial B W A software, we real ized that some of the informat ion conta ined in the peak profile was lost when mathemat ica l funct ions were fitted to the exper imenta l data pr ior to Fourier analysis. We then decided to de- velop our own program (Eberl et al. 1996) to avoid this and other simplifications. In the present paper, we de- scribe the theory on which the p rogram is based, and

Vol. 46, No. 1, 1998 XRD measurement of illite crystallite thickness distributions 39

Inc ident waves D i f f rac ted waves

m ' + I

rTI'

m ' - I

o

m + l

m B

r n - 1

Figure 1. A schematic illustrating the difference in paths traveled by diffracted waves scattered by m' and m layers separated by n interlayer spacings (n = m' - m), so that the distance between the layers Z, = m'd(O01) - md(00l) = nd(O01).

applicat ion of this p rogram to the study of clay min- erals, wi th part icular reference to illite and I -S .

T H E O R Y OF T H E B W A T E C H N I Q U E

The t rea tment p resen ted be low fol lows the or ig inal works of Ber tau t (1949, 1950) and War ren and Aver- b a c h (1950), bu t it is s implif ied for the analys is of basa l ref lect ion profiles. We h a v e s impli f ied the equa- t ions by e l imina t ing cons tan ts which do not affect the shape of 00I reflections. We use the t e rm " C S D " for a s tack o f layers paral lel to each other.

Exper imen ta l De t e rmina t i on of the In te r fe rence Func t ion

The in tens i ty d i s t r ibu t ion I(20) for a 001 ref lect ion of a layered mineral , cons is t ing o f CSDs wi th a strict per iodic i ty a long the c* axis, can be wr i t ten as:

1(20) = Lp(20)GZ(20)~b(20) [1]

Lp(20) is a c o m b i n a t i o n o f the Loren tz and polar iza- t ion factors wh ich accounts for in tens i ty con t r ibu t ions due to X- ray b e a m polar izat ion, and for geomet r ica l factors re la ted to the v o l u m e and or ien ta t ion of crys- tals in the X- ray b e a m (Reynolds 1986; M o o r e and Reyno lds 1989). G2(20) is the s t ructure factor tha t rep- resents d i f f ract ion f rom the a r r angemen t o f a toms wi th in the uni t cell. The symbol +(20) is the interfer- ence funct ion, w h i c h depends on the th ickness of CSDs , the th ickness d i s t r ibu t ion and the micros t ra ins or layer spacing f luctuations.

The values of Lp(20) and G2(20) can be ca lcula ted a p r i o r i i f expe r imen ta l condi t ions for d i f f rac t ion and chemica l and structural da ta for the mine ra l under s tudy are wel l known. These funct ions are no t sensi-

n=!

M - n = 5 4

n=2

A

H=6

n=3 n=4 n=5

5 2

d(O0

, f

H-n is the number of l aye r pairs separa ted by n i n t e r l a y e r spacings d(O01 ).

Figure 2. Relation between total number of layers in a crys- tal (M) and amount of pairs of layers (M - n) separated by n spacings of d(001). An increase in n from 1 to 5 decreases the number of nth neighbors from 5 to 1.

t ive to CSD sizes and micros t ra ins . Therefore , we can l imit cons idera t ion to the in te r fe rence funct ion:

1(20) qb(20) [2]

Lp(20)G2(20)

It is c o n v e n i e n t to replace the 20 var iab le of X R D pa t te rns wi th a new con t inuous var iable , Z*, a long the c* axis in rec iprocal space. The va lues of Z* are re- la ted to the co r re spond ing 20 va lues by the equat ion:

Z* = 2 s in 0/k [3]

where k is the w a v e l e n g t h of the rad ia t ion and 0 is a con t inuous var iable .

Phys ica l M e a n i n g of the In te r fe rence Func t i on

The interference function represents effects of the phase differences that appear during wave scattering by all the nth nearest layer pairs that exist in CSDs. To make this clear, let us consider first a sample where CSDs have the same thickness and consist of M identical layers (Fig- ure 1). Let us consider 2 layers, m ' and rn, separated by n = m' - m interlayer spacings d(001). The paths trav- eled by waves scattered by these 2 layers differ by the length A = (AB + BC) = 2Z n sin 0, where Z,, = nd(001) and 0 is the angle be tween the initial X-ray beam and (001) planes (Figure 1). The phase difference (p) be- tween the waves is related to A:

21r A = 2 ~ r Z 2 sin 0 2~rZ, Z * [4] P = k ~ -

This phase d i f fe rence con t r ibu t ion to d i f f rac t ion is ex- p ressed by a cos 27rZ, Z * t e rm (James 1965).

The va lue of (M - n) g ives the n u m b e r of layer pairs separated by n in ter layer spac ings tha t exis t in the CSD. This re la t ionship is i l lus t ra ted in F igure 2. The product :

(M - n)cos 2~rZ, Z* [5]

40 Drits, Eberl and Srodofi Clays and Clay Minerals

Two- theta

6 10 14 18 22 26 30 i , i , J , i , a , , , i

001 002 003

. J Z* /C*

- i ~=2 g~3

Figtare 3. 001 peaks of the interference function located along the Z' /c* axis. For the 002 peak. the origin is chosen at its maximum so that a new variable s* may vary from - � 8 9 to +�89 Z' /e* = l + s*.

represents the effect o f the d i f fe rence in phase of waves scat tered by all the n th neares t layer pairs in a CSD of a g iven M. The s u m m a t i o n of these produc ts for all Inl = I n ' - m[ ( that is, for n = m ' - m and - n = rn - m ' ) , no rma l i zed for the uni t cell ( that is, di- v ided by M), g ives the in te r fe rence funct ion, d~(Z*), wh ich descr ibes the total effect of the phase d i f ference on the in tens i ty d is t r ibut ion a long the Z* axis:

~b(Z*) = ~ (M - ]n[)co s 2~rZ.Z* [6] n = - M M

It is phys ica l ly unreal is t ic to a s sume that samples cons is t of CSDs h a v i n g the same n u m b e r o f layers. Therefore , let us cons ider a sample that consis ts of CSDs wi th a d is t r ibut ion of th icknesses def ined by

f (M) , so that:

M2 M2

f ( M ) = 1 and ~ M f ( M ) = M [7] MI MI

where M~ and M 2 co r respond to CSDs hav ing the smal les t and the largest n u m b e r of layers respec t ive ly and M is the m e a n n u m b e r of layers per CSD.

The n u m b e r o f n th neares t layer pairs in all CSDs cons i s t ing o f M layers is equal to f ( M ) ( M - n). Since C S D s have di f ferent th icknesses , the m e a n n u m b e r of layer pai rs separa ted by n in ter layer spac ings is:

M2

N(n) = ~ (M - n ) f ( M ) [8] MI

I f w e r e p l a c e (M - Inl) and M in Equa t ion [6] by N(n) and M, respect ively , we will ob ta in the in te r fe rence

func t ion for samples hav ing a d is t r ibut ion of CSD th icknesses :

M~ N(n) M2 ~b(Z*) = ~ - - cos 2~rZ.Z* = ~ H(n)cos 2~rZ.Z*

n= M 2 m n = - M 2

[9]

where:

1 M2 n ( n ) = ~ ~ ( M - n ) f ( M ) [10]

MI

The in te r fe rence func t ion represen ts a series of bel l - l ike peaks o f ident ical shape, loca ted a long the Z* axis at pos i t ions co r respond ing to 001 ref lect ions (Figure 3).

The In te r fe rence Func t ion as a Four ie r Series

Le t us replace the absolu te Z* coordina te by the f ract ional coord ina te Z*/c*, that is, express it in t e rms of a f rac t ion o f the rec iprocal uni t cel l pa r ame te r c* = I /d(001) . T h e n pos i t ions of in tensi t ies a long the Z* axis for each 00l ref lect ion (defined as each 00l inter- ference func t ion peak) m a y be expressed as:

Z*lc* = ~ / c * + s* = 1 + s* where 1 --> s* --> 0 [11]

where Z~ = 2 sin 0j/X = I/d(O01) is the coord ina te co r respond ing to the Bragg law pos i t ion for a 001 re- f lect ion and s* is a new var iable parameter .

Le t us now treat each peak separate ly and choose the or igin o f the coord ina te for each ind iv idua l 00l ref lect ion at the pos i t ion of its m a x i m u m , that is, set- t ing l = 0. In this case, s* values wi th in a 001 reflec- t ion are l imi ted by an in te rva l f rom - � 8 9 to �89 (Figure 3) and:

Z,Z* = nd(OO1)c*s* = ns* [12]

Thus:

M2

+(Z*) = d~(s*) = ~ H(n)cos 2~rns* [13] n ~ - M 2

Because n is an integer, the in te r fe rence func t ion rep- resents a Four ie r series. The Four ie r coeff ic ients H(n) are ob ta ined by the back Four ie r t r ans fo rmat ion of +(s*) , that is:

1/2

H(n) = ~ ~b(s*)cos 2~rns* [14] s * = - I/2

m

Dete rmina t ion o f f ( M ) and M

H(n) is a func t ion dependen t on the m e a n th ickness and the th ickness d is t r ibut ion of CSDs. It fo l lows f rom Equa t ion [10] that:

02H(n) f ( M ) OH(n) 1 and - On .-~o ~ ~ ' On2 .-~o ~. [15]

The th ickness T o f a CSD is T = Md(001) . In the case

Vol. 46, No. 1, 1998 XRD measurement of illite crystallite thickness distributions 41

of illite, d(001) = 1 rim, so T = M nm. Thus, the first derivat ive o f H(_n) at n ~ 0 determines the mean thick- ness o f CSDs (7) and the second derivat ive gives the C S D thickness distribution.

Fluctuations in the Posit ion o f Layers

A periodic structure o f CSDs may be disturbed due to slight fluctuations in the layer thicknesses. To any pair of the nearest layers (n = 1) separated by the distance Zl, one can associate a variat ion ~1 with re- spect to the ideal value d(001). This % variable may vary f rom one pair of layers to another and the vari- ation o f Z~ may be described by a statistical distribu- tion function, ~/I(Z). In the general case of nth nearest neighbors, we have the general relation:

Z. = nd(001) + ~. = (n + 5.)d(001) [16]

where ~. is the difference be tween the ideal distance and the true distance and 5. is the same parameter expressed as a fraction o f d(001). The variable ~. may vary for each pair of the nth nearest neighbors and the variation of Z. may be described by a statistical dis- tribution function %(Z.) (Guinier 1964; Drits and Tchoubar 1990). Because of the spacing fluctuations, the cosine terms in Equat ion [9] must be averaged over all possible Z. values for each given n. Thus, the mean phase difference o f waves diffracted by all layer pairs separated by different interlayer spacings may be written as:

M2 ~b(Z*) = ~ H(n)cos 21rZ.Z* [17]

n = - - M 2

Using Equations [11] and [16], the product Z~Z* may be modif ied as:

Z.Z* = (n + 5.)d(001)(/ + s*)c* = nl + ns* + 15. [18]

because c* = l /d(001) and we have neglected the 5.s* term, since 5. , ~ 1 and s* < 1. Then:

cos 27rZ.Z* = cos 2~r(ns* + 15.)

= cos 2~rns*cos 2~r15,,

- sin 2~rns*sin 2wt5. [19]

because the mean applies only to terms containing 5. and cos(n/ + ns*) = cos(ns*), nl being an integer. Equat ion [17] then can be presented as:

~b(s*) = ~ A(n)cos 2~rns* - ~ B(n)sin 2~ns* [201

where:

A(n) = H(n)cos 2~rl5. and B(n) = H(n)s in 2~rl5. [211

As was mentioned, every pair o f layers, m ' and m, enters into summat ion twice as n = m ' - m and - n = m - m' . Since 5. = 5~. - 5., and - 5 _ . = 5,. - 5 m , ,

it fol lows that 5. = 5_.; hence, after summation, the sine terms in Equat ion [20] vanish:

~b(s*) = 2 A(n)cos 2~rns* (22)

Again ~b(s*) has a form of the Fourier series and the A(n) coefficients may be determined f rom the ex- per imental profile o f a 001 reflection by the equation:

1 /2

A(n) = ~ ~b(s*)cos 2~rns* [23] s * = 1 /2

As can be seen f rom Equat ion [10], the size term. H(n), is independent o f L whereas the cos 2~r15. term depends upon the reflection order. Therefore, using A(n) one may determine H(n) and fluctuations in layer positions if at least two 001 reflections are analyzed.

In order to obtain the precise mean values o f the cosine terms in Equat ion [21] we must know the dis- tribution law for the deviation. % or 5.. The simplest assumption is that this distribution fol lows the Gauss- Jan function:

~/(~) = ~ - - ~ e x p - ~ = ~ - ~ e x p - [ - ~ l

[24]

in which er = (crY) 1/2 = (~)u2 = (5--7)uz d(001) is the standard (or root mean square) deviat ion equal to the half-width at ha l f -max imum of the strain distribution, y(e). d iv ided by (2 in 2) I/z. First one must take into account the fluctuations in layer positions of all nearest layer pairs. In order to do this, one has to sum ~/(51) cos 2~rl51 for any possible values of 51, where ~/(St) is the probabili ty to find 51. Thus:

cos 2~rl51

- X/~--~1 cos 2~rlSlexp(-5~dz(O01)12~) d5 l

= exp [d2(001) ] [251

where ~r~ = (8~)d2(001) is a var iance o f ~ or 51 or a mean square deviat ion o f these values. As fol lows f rom Equat ion [21], we are interested in calculat ing cos 2~rl5, for any n. The standard deviat ion of the dis- tribution of e, is related to the standard deviat ion of nearest layer pairs as follows: cr~ = ncr~ (Maire and Met ing 1960; Reynolds 1989; Drits and Tchoubar 1990). Hence:

A(n) = H(n)cos 2~r15.

and

In A(n) = In H(n)

- - = H(n)exp [ - 2~rZlZn~r~l]d2(O01 ) ]

m

2~r212ntr~ d2(001)

[26]

[27]

42 Drits, Eberl and Srodofi Clays and Clay Minerals

If o n l y 2 basal reflections, 00ll and 0012, are avail- able, the tr 2 value can be found analytically:

- - dz(OO1)in[At,(n)/Al2(n)] tr~ -- [281

2~rZn(l~ - l~)

If ~j follows the Gaussian function, then tr~ does not depend on n.

Using Equation [23], the A(n) values are found for each 00l reflection. Then, for a fixed value of n, a plot of the values for In A(n) versus l 2 is constructed. The extrapolation to l_= 0 gives H(n) and the slo_pe of the plot gives -2~r2ntr~/d2(001). The value of (tr2) l/z is the root mean square fluctuation of adjacent layer thick- nesses in, for example, ,~ units.

In the general case, when the distribution law of e, is unknown, one may use an approach which is valid for small l and n values:

cos 2~Ign ~ 1 - 4~r2/2

~ 1 - 2~2/ag~ ~ exp -2~r/28~ [29]

The plot In A(n) versus F will be represented by straight lines only in the small n region, but this is just a region most suitable for a linear extrapolation to l = 0. This method does not require any assumptio__n about the nature of layer thickness fluctuations, If 82 values determined by Equation [29] and divided by n lead to a constant value, it means that the distribution law cor- responds to a Gaussian function (compare Equations [26] and [29]).

EXTRACTION OF THE INTERFERENCE FUNCTION FROM AN XRD PATTERN

The theoretical treatment presented above ignores experimental conditions that may affect the shapes of XRD peaks. These effects include background, over- lapping of K~I and KoL 2 peaks and instrumental broad- ening (Klug and Alexander 1974). We attempted to recover the interference function (which contains the particle size and strain information) from XRD peaks first by smoothing the XRD peak and removing the background, then by extracting the K ~ peak and de- convolving instrumental broadening and finally by ex- tracting the interference function according to Equa- tion [21.

This order of calculation proved unsuccessful: smooth- hag noise from the intensity data by using a moving av- erage broadened the XRD peaks; the background inten- sity function could not be removed because its shape was unknown; insmmaental broadening cannot be evaluated theoretically; and finding an appropriate instrumental standard for illite (a powdered mineral completely free from crystal size and strain broadening with a d close to the XRD peak to be analyzed) proved difficult. Finally, removing Lp(20)G2(20) (hereafter referred to as LpG 2) by division according to Equation [2] often distorted the

interference function so badly that it could not be used for crystallite size and strain analysis.

To circumvent these problems, a computer program (MudMaster; Eberl et al. 1996) was written using a calculation sequence that was optimized by trial and error to circumvent these problems. XRD patterns were recorded from 4.5-cm-long oriented clay prepa- rations using a diffracted beam monochromator and relatively long count times (at least 5 s/0.02 ~ step, with a tube current of 40 kV and 30 mA) to minimize background and noise, so that the intensity data did not require smoothing. An analytical range was chosen that was broad enough to encompass the complete range of the interference function peak (that is, half- way between the peaks of subsequent orders), so that little intensity was lost during background removal. A Siemens D500 diffractometer using incident and dif- fracted beam S611er slits, a 1 ~ divergence slit and a relatively narrow receiving slit (0.15 ~ was employed to minimize instrumental broadening. An attempt was made to use the >20 txm size fraction of NITS SRM 675 synthetic fluorophlogopite as an instrumental stan- dard. Deconvolution of instrumental broadening by the Stokes (1948) technique (Klug and Alexander 1974) yielded acceptable mean thicknesses for natural illites and I-S, but distorted crystallite thickness distributions for samples having mean thicknesses greater than about 15 nm. By comparison between samples run with and without the instrumental standard, instrumen- tal broadening was found to be insignificant under our instrumental conditions for crystallite thicknesses <20 r i m .

The values of G: for the illite unit cell were calcu- lated using a program written by the authors (CALCLPG2) and atomic coordinates given by Moore and Reynolds (1989). To analyze NEWMODr - lated illite XRD pattems, potassium content (K) was determined as follows:

equivalents K per half-unit cell = 0.89[(T - 1)/-/] [30]

where T is the mean illite crystallite thickness in nm and 0.89 the charge of an illite interlayer per half-unit cell. This equation corrects K-content for the presence of crystal edges which, for NEWMOD�9 patterns, contain no K. For natural I -S, K-content was determined from the percentage of smectite interlayers (expandability, that is, %S) measured from glycolated samples (Srodofi 1980; Srodofi and Elsass 1994) prior to K-saturation and heating, assuming that the typical smectite layer charge is - 0 . 4 equivalents/Ol0(OH)2. The relation is:

equivalents K

= 0 .01 (T- 1)[0.40(%S ) + 0.89(100 - %S)] + 0.4

T [311

Vol. 46, No. 1, 1998 XRD measurement of illite crystallite thickness distributions 43

Q e -

e-

0 0 1 I I I I te

0 0 3

0 0 2

1 0 2 0

0 0 5

3 0 4 ' 0 5 0

Two-theta

Figure 4. A NEWMOD�9 illite XRD pattern (thin line) and its LpG z (thicker line).

A n Lp factor was used that c o m b i n e s the r a n d o m and s ingle-crys ta l Lp factors (Equa t ion 3.13 in M o o r e and Reyno lds 1989). However , use o f the r a n d o m Lp fac- tor, of the s ingle-crys ta l Lp fac tor or o f Lp factors that were ca lcula ted for a d i f ferent sample or ien ta t ion and ins t rumenta l slit sys tem did not s ignif icant ly change the results.

L p G 2 o f layer si l icates is modu la t ed and has ap- p rox ima te ly zero values at several 20 angles (Figure 4). The in te r fe rence func t ion p roduced by d iv i s ion o f the expe r imen ta l X R D in tens i t ies (Figure 5A) by L p G 2 may be de fo rmed nea r such angles (Figures 4 and 5B), because smal l inaccurac ies in the es t imat ion o f L p G 2 or in the de t e rmina t ion of b a c k g r o u n d have a s t rong effect on the shape of the r ecovered in te r fe rence func- t ion in these regions . For example values of L p G 2 are ca lcula ted for per iodic crys ta ls and therefore canno t take into accoun t crysta l end-ef fec ts w h i c h cause N E W M O D r and natura l i l l i te c rys ta ls to be sl ightly aperiodic. There fo re a t echn ique was devel- oped that rel ies on the fact that the in te r fe rence func- t ion peaks are str ict ly symmet r i c (Figure 3). The un- d is tor ted ha l f o f the in te r fe rence func t ion peak is " ' f l ipped" over a ver t ical p lane pass ing th rough the peak m a x i m u m (Figure 5C). For illites, 001, 002 and 004 ref lect ions are f l ipped f rom low-20 to high-20. T he 003 and the 005 can be f l ipped in e i ther d i rec t ion i f one side o f a peak is de fo rmed b y minera l admixtures . Af te r fl ipping, or i f f l ipping is no t applied, the res idual b a c k g r o u n d is r e m o v e d by set t ing to zero the m i n i m a on e i ther side of the in te r fe rence func t ion peak. Pr ior to fl ipping, the K,x 2 c o m p o n e n t to the in te r fe rence func t ion can be r e m o v e d by a Four ie r m e t h o d (Gan- gulee 1970; Eber l et al. 1996).

The in te r fe rence func t ion peak (Figure 5C), so ex- t racted f rom the X R D in tens i ty data (Figure 5A) , then was ana lyzed by the B W A techn ique as was descr ibed

I i I I i i i i I i i

A XRD peak

B Deformed interference

function

C Flipped Interference

function

4 6 8 10 12 14

Two theta

Figure 5. Patterns for K-saturated and dehydrated Zempleni sample, a 15% expandable R > 1 ordered I~S prior to de- hydration. A = 001 XRD peak; B = interference function calculated by dividing the XRD intensities in A by LpG 2 for an illite with the appropriate chemical composition; C = final interference function to be analyzed by MudMaster, formed by flipping the left half of the peak in B from left to right.

44 Drits, Eberl and Srodofi Clays and Clay Minerals

Table 1. Parameters of crystal thickness distributions measured by the BWA and by the integral peak width (Drits et al. 1997) methods from XRD patterns of illites that were calculated using NEWMOD, with lognormal distributions of crystallite thickness having means of 5 and 20 layers.

Pattern & reflection (nm) (nm) (nm) a /32 (A)

Lognor 5 input = 1.51 input = 0.20 input = 9.98 .~ 001 5.2 5.3 5.0 1.55 0.22 10.00 002 5.4 5.4 5.2 1.61 0.17 9.98 003 5.3 5.3 4.9 1.57 0.18 9.98 004 4.6 4.6 4.7 1.42 0.21 9.99 005 4.9 4.9 4.7 1.48 0.20 9.98

Lognor 20 input = 2.83 input = 0.34 input = 9.98 ,~ 001 19.7 20.2 20.2 2.84 0.33 9.98 002 20.4 20.5 18.6 2.87 0.30 9.98 003 20.1 20.0 18.5 2.83 0.35 9.98 004 19.9 19.9 17.9 2.83 0.32 9.98 005 19.7 19.7 18.4 2.81 0.34 9.98

Key: Te, Td, and ~, are mean thicknesses calculated from the first derivative of the Fourier coefficients plot, from the crystal thickness distribution (without smoothing, except for the Lognor 20, 001 reflection for which a smoothing power of 1 was used) and from the integral peak width method of Drits et al. 1997, respectively. The symbols et and 62 are the lognormal parameters for the input distribution and for distributions calculated from the BWA method. The representation d(001) is the position of the maximum of the interference function.

in the " T h e o r y " section. Four ie r coeff ic ients A(n) were ca lcu la ted for each ref lect ion f rom n = 0 to a m a x i m u m n equal to 5T where T the m e a n n u m b e r of ill i te layers in the crystal l i tes . A t this m a x i m u m n the f requency genera l ly is less than 0.001 of the m a x i m u m f requency and therefore larger n ' s can be ignored. Fou- r ier coeff ic ients at smal l n ' s f requent ly are d is tor ted due to loss o f in tens i ty dur ing b a c k g r o u n d r em ova l and therefore mus t be cor rec ted for the hook effect (Klug and A lexande r 1974). Ana lys i s o f N E W M O D e- ca lcu la ted il l i te X R D pat te rns indica tes that the hook cor rec t ion for the Four ier coeff ic ients is bes t per- fo rmed by ex t rapola t ing the s teepest slope of the c u r v e of H(n) versus n to H(n) = 0 to find the m e a n thick- ness , and that the size d is t r ibut ion is bes t de t e rmined by tak ing the second der iva t ive of the same curve not cor rec ted for the hook effect (Equa t ion [15]). T he first and second der iva t ives o f the Four ie r coeff ic ients can be smoothed , us ing a m o v i n g average of 3 or 5 ( smoo th ing p o w e r = 1 or 2 in the M u d M a s t e r pro- g ram) to decrease noise in the th ickness dis t r ibut ions . The th ickness d is t r ibut ions can be cut off at a desig- na ted th ickness to e l iminate r ipples that m ay appear in the d is t r ibut ion at large th icknesses . I f such r ipples are present , the d is t r ibu t ion is cut o f f at a size tha t makes the d is t r ibut ion m e a n approx imate ly equal to the m e a n de t e rmined by extrapolat ion.

V E R I F I C A T I O N U S I N G N E W M O D C- C A L C U L A T E D X R D P A T T E R N S

To tes t the B W A techn ique incorpora ted in the pro- g ram MudMas te r , we ana lyzed X R D pat te rns tha t were ca lcula ted for a r ange of crysta l l i te size d is t r ibut ions u s i n g N E W M O D S K Y , a v e r s i o n o f N E W M O D ~ (Reynolds 1985) modi f ied by R. C. Reynolds , Jr., to

pe rmi t input of crys ta l l i te size d is t r ibut ions f rom an externa l file. Opt imal input pa ramete r s for M u d M a s t e r for the h o o k correct ion, for the 20 in te rva l of the anal- ysis, for smoo th ing condi t ions and for the m a x i m u m n to be used in the ca lcu la t ion were the reby were es- tabl ished, and were then used to s tudy the more com- p l ica ted X R D pat terns o f na tura l i l l i tes and I -S .

Mos t N E W M O D S K Y ill i te pa t te rns were ca lcu la ted us ing lognorma l crys ta l l i te th ickness d is t r ibut ions , be- cause this d is t r ibut ion law is c o m m o n to m a n y min- erals, inc lud ing il l i te (Eber l et al. 1990; Dri ts et al. 1997). Two parameters , ct and 132, wh ich cor respond, respect ively, to the m e a n va lue and the var iance of In T, where T is the crysta l l i te th ickness and f (T) is the f r equency of tha t th ickness , charac te r ize the dis t r ibu- t ion comple te ly , where:

et = "~ In Tf(T), and [32]

132 = ~ [In T - ot]2f(T) [33]

Values of et, 132 and T for 2 crys ta l l i te th ickness dis- t r ibut ions used to ca lcula te N E W M O D �9 X R D pat te rns are compared wi th values ext rac ted f rom the calcula t - ed X R D pat te rns by M u d M a s t e r (Table 1). The agree- m e n t is sat isfactory. The bes t data are ob ta ined i f the in te r fe rence func t ion peaks can be r ecove red over the comple te 20 range, s tart ing m i d w a y be tween ne igh- bor ing _001 reflections. I f the analy t ica l r ange is nar- rower, T tends to be overes t ima ted and 132 underes t i - mated.

N E W M O D �9 pa t te rns h a v i n g CSDs wi th a var ie ty o f shapes were ca lcula ted and then ana lyzed by M u d - Master. F igure 6 i l lustrates that N E W M O D ~ inputs hav ing equal (Figure 6A) and lognorma l (Figure 6B) d is t r ibut ions can be r ecovered a lmos t perfectly. A

Vol. 46, No. 1, 1998 XRD measurement of illite crystallite thickness distributions 45

1.0

|

0.5

0.0 ~ ; 0

. . . . , . . . . , . . . . , . . . . , . . . . , . . . .

I I N E W M O D mean = 12.0 nm

0 � 9 . � 9

, | , ' . . . . 5 10 15 20

~ A A A A ~ A .

2 5 30

1.00" B

0.80

0.60"

0.40"

0.20" ; O

0.00 - - - - 0

. . . . . . . . . . . . . . . . . . . . . . . . .

,r i BW'm--n=10:n I

5 10 15 20 25 30

0.03

0 . 0 2

0.01

0.00

C

m _ v 20 40

. , . , .

BW,~ mean" = 48.5 nm - NEWMOD mean = 47.1 nm

I_f ;I 6 0 80 100

Thickness (nm)

Figure 6. Crystallite size distributions used in NEWMOD �9 calculations of 003 illite XRD peaks (solid lines) compared with distributions recovered by MudMaster analysis of those peaks (dots). A = even distribution, from 4 to 20 layers (no smoothing); B = lognormal distribution having a mean of 10 layers (no smoothing); C = distribution spelling the word "'MUM" (smoothing power of 1).

more compl ica ted distribution that spells the word " M U M " was tested to see i f the program can recover mul t imodal distributions. The Ka~ - K o s 2 doublet was included in the pattern by adding N E W M O D ~ patterns calculated with Ka~ and Ka2 radiation in the propor- tion 2:1. The mean and the distribution were recovered almost perfectly (Figure 6C) f rom each of first 5 basal reflections, provided that the K o t 2 correction was ap- plied to reflection orders greater than 1.

The effect of the Ket I - K o t 2 doublet on the B W A calculation was invest igated for calculated illite pat- terns for illites having mean thicknesses ranging f rom 7 to 45 nm. The results indicate that Kcz 2 need not be r emoved from the_patterns, even for the 005 peak, i f

< 20 nm; but i f T = 45, distributions were distorted and the mean values were lowered for reflection orders greater than the 001 i f Kct 2 was not removed.

These studies of synthetic X R D patterns indicate that, i f applied correctly, the B W A method is capable of reproducing almost perfectly the original crystall i te thickness distributions used in calculated patterns.

A P P L I C A T I O N OF T H E B W A T E C H N I Q U E T O E X P E R I M E N T A L X R D PATTERNS

Fifteen samples representing the full range of I - S expandabil i ty were studied by the B W A technique. Expandabil i ty o f the samples was determined accord- ing to Srodofi (1980, 1984)�9 The same patterns were used to develop the modif ied integral width method (Drits et al. 1997)�9 K-saturated samples were dehy- drated by heating overnight at 300 ~ and then were X-rayed in a dry atmosphere. This procedure, per- formed under the low relat ive humidi ty condit ions o f Boulder, Colorado, was found sufficient to r e m o v e X R D peak broadening caused by swell ing interlayers. Dehydrat ion was confirmed by observing that the 002 peak was narrower than the 003 (002 broadening is most sensit ive to the presence of 1-water layer smec- tite) and that the mean d(001) values calculated f rom the 002, 003 and 005 were nearly identical (the mean deviat ion f rom the mean thickness value did not ex- ceed --_0.008 ,~ for 13 samples and ---0.012 ,~ for the other 2).

The progression in a BWA-MudMas t e r analysis for K-satura ted, dehydra ted Z e m p l e n i c lay (Viczifin 1997), which prior to dehydrat ion was a 15% expand- able, R > 1 ordered mixed- layer I--S, is presented in Figures 5, 7 and 9 for the 001 reflection. The flipped, undistorted interference function (Figure 5C) was sub- jec ted to Fourier analysis (Equation [14]), and the Fou- rier coefficients, H(n), were plotted against CSD thick- ness (Figure 7A). Extrapolat ion of the steepest slope of this curve yields and extrapolated mean (T~) o f 10.5 nm. The second derivat ive o f this curve (Equation [15]) yields the thickness distribution (Figure 7B) f rom which the distribution mean (Td) was calculated and found also to equal 10.5 nm. I f there is no exper imen-

46 Drits, Eberl and Srodofi Clays and Clay Minerals

1.0

O

O 0.5

A e, X

1 5rim A

�9 �9

. . . . . . . . . . , ' , ; . 2 4 6 8 10 12 14 t6 18 20

>, g

O"

u.

0.12 I

0.06

0 . 0 0

Distribution mean = 10.5 nm B

1 0 2 0 a-0 . . . . . . . go

0.14

Smoothed strain C corrected distribution

)

o~" 0.07

0.00 - - - ~ . ~ . = ~ . = ~ . = ~ . = ~ . ~ . 7 . ~ . = ~ .

0 10 20 30 40

Thickness (nm)

Figure 7. BWA analysis of K-saturated, dehydrated Zem- pleni sample. See Figure 5 for the XRD data_. A = BWA method for determining the extrapolated mean (Te); B = crys- tallite thickness distribution calculated from the curve in A (no smoothing); C = crystallite thickness distribution calcu- lated from Fourier coefficients that have been corrected for strain (smoothing power of 3).

= ==

=.

25

20

15

10

0 0 5 10 15 20 25

Te thickness (nm)

Figure 8. Mean thicknesses for the samples listed in Table 3 determined by extrapolation of Fourier coefficients (Te; see Figure 7A) compared with those determined from the distri- butions (Td; see Figure 7B) for the 001 reflection.

tal error, the means ca lcu la ted by b o t h approaches should be equal to each other, as is genera l ly the case for the 001 ref lect ion (Figure 8). However , the extrap- olated m e a n is the more accurate and s table mean , because the d is t r ibut ion m e a n is sens i t ive to the max- i m u m n used in the calculat ion, to the smoo th ing pow- er ( smoo th ing tends to increase T d wi th respec t to Te) and to the cu tof f va lue used for the dis t r ibut ion, w h e r e a s the e x t r a p o l a t e d m e a n is i n s e n s i t i v e to changes in these parameters .

By ana lyz ing more than 1 ref lect ion order, the m e a n and the d is t r ibut ion can be cor rec ted for s t ra in b road- ening, w h i c h does not s ignif icant ly af fec t the 001 and 002 reflect ions, but wh ich may affect h ighe r orders. Loga r i thms o f Four ier coeff ic ients for the 001, 002, 003 and 005 ref lect ions (Equa t ion [27]) are plot ted against the square o f the ref lect ion order (F) for a r ange of n ' s (Figure 9A). The in tercepts of these l ines on the Y-axis yield the s t ra in-correc ted Four ie r coef- ficients, H(n), which then can be used to de t e rmine the m e a n th ickness and th ickness dis t r ibut ion, as in Fig- ures 7A and 7B. The slope__s of th__ese l ines yie ld the m e a n square o f the s train (cr, 2 = ntrl 2) for each n. I f the

rat ios (rffn are constant , then strain (el, in A uni ts) has a Gauss ian d is t r ibut ion (Figure 9C) wi th a s tandard dev ia t ion (~2),/2. Cor rec t ion o f the Four ie r coeff ic ients for s train b roaden ing yie lds good m e a n th ickness val- ues (as will be discussed) , bu t may distort th ickness dis t r ibut ions. However , i f smoo th ing is appl ied to the s t ra in-correc ted Four ie r coeff ic ients ( smoo th ing p o w e r = 3 in the M u d M a s t e r p rogram) , then a d is t r ibut ion is r ecove red that is s imi lar to that found f rom the 001 ref lect ion (Figure 7C).

Vol. 46, No. 1, 1998 XRD measurement of illite crystallite thickness distributions 47

_=

o.o ~ , ~. " " n-ol

- . --n=l t ~ n-2

-0.5 n=3

0 n=6

-2.5 I I I I

1 4 9 ~ 2 25

A

0.25

.t

0.20

0.15

0.10

0.05

O.OG

Region chosen

All strain B

5 10 15 20

\ Thickness (nm)

0.03

0.02

o c 0

.0

" 0.01

0.00 -0 .6 -0 .4 -0 .2 0.0 0.2 0.4 0.6

Strain ( E 1 ,A)

Figure 9. Strain determinations for the Zempleni sample made according to BWA theory. See Figures 5 and 7 for more infor- mation concerning this sam__ple. A = plot used to determine the mean square of the strain (~ ) as a function of n; B = the plateau in the root mean square strain data indicates that the strain (el) has a Gaussian distribution; C = Gaussian strain (el, in A) dis- tribution (~/(~0; see Equation [24]) for this sample.

Table 2. Parameters of crystallite thickness distributions measured by the BWA method and by the integral peak width method (Drits et al. 1997) from an XRD pattern of a K- exchanged and dehydrated sample of almost nonswelling il- lite (SG1). A smoothing power of 1 was used for the BWA distributions.

Reflec- ~ Td ~, Id(QOI) tion (nm) (nm) (nm) a /3 : (A)

001 21.9 22.1 18.7 3.006 0.181 10.026 002 24.3 24.4 19.2 3.082 0.229 10.022 003 18.4 21.6 19.0 2.945 0.284 10.008 004 21.7 22.7 17.4 3.037 0.179 10.004 005 19.0 19.9 15.6 2.861 0.278 10.010

Key: See Table 1 for explanation of symbols.

The B W A techn ique was fur ther tes ted us ing sample SG1, an illi te tha t in its natural state con ta ins a lmos t no swel l ing inter layers . Resul ts (Table 2) indica te tha t all o f the first 5 ref lect ions can be used to de t e rmine Te (mean de t e rmined by extrapola t ion) , Td ( m e a n de- t e rmined f rom the dis t r ibut ion) , c~ and [32. The m e a n s indica te tha t s t rain b roaden ing is min imal , because such b r o a d e n i n g would tend to regular ly decrease wi th increas ing ref lect ion order. Near ly ident ical re- sults were found w h e n in tens i ty data were cor rec ted for the p resence of Kot 2 radiat ion. A n o t h e r pure i l l i te (Kaube) gave s imilar resul ts for s t ra in analysis , sug- ges t ing that s t ra in is m i n i m a l wi th in fundamen ta l i l l i te par t ic les (Nadeau et al. 1984), a l though it is p resent in dehydra t ed K-smec t i t e inter layers , as wil l be d i scussed below.

Resul ts ob ta ined for 00 l ref lect ions f rom all o f the dehydra t ed samples are p resen ted in Table 3. Area- we igh ted m e a n th icknesses ob ta ined f rom an in tegra l peak wid th s tudy (Ti) o f the same X R D pa t te rns (Dri ts et ai. 1997) are close to those ob ta ined by the B W A m e t h o d (Tables 2 and 3 and Figure 10). Our data set does not al low us to evalua te the accuracy of B W A m e a s u r e m e n t s of 132 for na tura l samples .

M e a n th icknesses for fundamen ta l i l l i te par t ic les (N) can be ca lcula ted f rom m e a n th icknesses for the de- hydra t ed mixed- laye r crysta l l i tes (73 and expandab i l i ty (%S) (af ter Drits et al. 1997):

100T N = - [34]

( T - 1)(%S) + 100

Values of N ca lcula ted us ing m e a n th icknesses ob- ta ined for 001 ref lect ions f rom the B W A c o m p a r e wel l wi th those de t e rmined f rom th icknesses ob ta ined by the in tegral peak wid th m e t h o d (Figure 11).

All K-dehydra t ed samples were ana lyzed for s t ra in ( layer th ickness f luctuat ions) us ing var ious combina - t ions o f the 001, 002, 003 and 005 reflect ions. Cal- cu la t ions tha t used all 4 peaks p roduced the mos t re- l iable results. S t ra in correc ted values for Te ob ta ined by correc t ing Te of the 005 ref lect ion for s t ra in (To), ca lcula ted us ing the 0 0 1 , 0 0 2 , 003 and 005 ref lect ions,

48 Drits, Eberl and Srodofi Clays and Clay Minerals

Table 3. Parameters of crystallite thickness distributions measured by the B W A (using a smoothing power o f 1) and by the integral peak width methods from 001 reflections of XRD patterns of K-exchanged and dehydrated samples. Units are in n m unless otherwise indicated.

S a m p l e N f i x N % S 0 0 1 0 0 1 0 0 2 0 0 3 0 0 5 0 0 5 (~r~) ~r2

Kaube 45.0 22.0 0 18.1 22.0 20.3 16.4 16.0 21.6 0.061 SG1 11.1 15.4 2 18.7 21.9 24.3 18.4 19.0 23.3 0.049 RM30 10.0 11.8 2 13.4 15.1 14.5 13.2 10.7 15.3 0.099 M l l 5.3 7.6 5 9.5 11.6 10.5 8.3 7.9 10.6 0.114 L F I 0 6.8 8.3 6 13.3 15.7 14.6 12.2 9.9 15.7 0.113 M8 4.1 5.5 7 7.2 8.4 7.8 5.9 5.6 8.2 0.138 RM8 6.3 5.2 10 8.3 9.7 8.0 6.9 6.5 8.6 0.124 RM35A 5.6 4.8 12 8.2 10.0 7.6 6.2 6.1 8.7 0.104 Zempleni 4.5 4.3 15 8.9 10.5 10.1 7.3 6.0 9.9 0.145 T9 2.72 3.1 20 5.6 6.6 5.6 4.3 3.8 6.0 0.177 M10 2.23 2.4 31 5.4 6.1 5.6 4.3 3.7 5.7 0.204 Ch5 1.96 2.2 37 6.2 7.5 6.7 5.0 4.3 7.1 0.178 MB 1.82 2.2 38 7.1 7.3 6.2 4.6 3.7 6.7 0.228 M D 1.15 1.1 88 5.9 5.7 4.9 3.7 3.1 5.5 0.268 2M9 1.09 1.1 88 6.0 7.0 5.6 3.4 3.4 6.7 - -

Key: Nfix = thickness of fundamental illite particles calculated by the fi_xed cation method (Srodori et al. 1992); N = thickness of fundamen_tal illite particles calculated by Equation [34], us ing T c for the 001 reflection and %S (percentage o f smecti te interlayers); T c = strain corrected thickness extrapolated from the 001, 002, 003 and 005 refections; and (trlz) 1/2 = root mean square of the strain calculated from a combinat ion of the 001, 002, 003 and 005 XRD reflections. Remaining symbols are explained in Table 1. The samples have been described in Srodo6 et al. 1986, Eberl et al. 1987, Srodofi et al. 1992 and Srodofi and Elsass 1994.

c o m p a r e c l o s e l y w i t h t h o s e n o t c o r r e c t e d fo r s t r a in de -

t e r m i n e d f r o m the 001 a n d 0 0 2 r e f l e c t i o n s (Tab le 3),

i n d i c a t i n g tha t t h e e f f e c t o f s t r a i n o n Tr w a s r e m o v e d

c o m p l e t e l y fo r t h e 005 r e f l e c t i o n a n d tha t t h e 001 a n d

0 0 2 r e f l e c t i o n s c a n be u s e d to d e t e r m i n e T w i t h o u t

c o r r e c t i o n fo r s t ra in . H o w e v e r , w i t h o u t t he s t r a in cor-

r ec t ion , T m e a s u r e d f r o m 0 0 5 p e a k c a n b e u n d e r e s t i - m a t e d b y u p to 5 0 % fo r t h e m o s t s m e c t i t i c s a m p l e s .

A s w a s m e n t i o n e d p r e v i o u s l y , r e m o v a l o f s t r a in m a y

d i s to r t C S D d i s t r i b u t i o n s . T h e r e f o r e , w h e r e a s s t r a in -

c o r r e c t e d m e a s u r e m e n t s c a n b e u s e d to d e t e r m i n e a n d (tr2) 1/2 ( the roo t m e a n s q u a r e o f t he s t ra in ) , r e l i ab l e

C S D t h i c k n e s s d i s t r i b u t i o n s c a n o n l y b e o b t a i n e d f r o m

001 or 0 0 2 r e f l e c t i o n s tha t h a v e n o t b e e n c o r r e c t e d fo r

strain. P l o t s o f (cr2)1/2 v e r s u s t h i c k n e s s ( for e x a m p l e , F i g u r e

9 B ) i n d i c a t e t ha t (~r2) 1/2 is i n d e p e n d e n t o f t h i c k n e s s ,

t h e r e b y c o n f i r m i n g t he G a u s s i a n d i s t r i b u t i o n o f s t r a i n

( E q u a t i o n [27]) for K - d e h y d r a t e d I - S . V a l u e s o f (cr2)1/2

30

20 e-

W m

e-

.o

2 5

Figure 10. Relation between mean thicknesses measured by the B W A extrapolation method (Te) and the integral peak breadth method (T~; Drits et al. 1997) for 5 reflection orders. Data f rom Table 3.

c~' y = 0.38 + 0.84x R"2 = 0.996

20

S

~ s

o . . . . i . . . . i . . . . . . . . . . . . . .

i = 0 5 10 15 20 25 1 0 20 30

TI t h i c k n e s s ( n m ) Thickness from BWA (nm)

Figure 11. Relation between fundamental illite particle thicknesses calculated integral peak width thic_knesses (T~) and expandability, and from B W A thickness (T0) and ex- pandability us ing Equation [34].

Vol. 46, No. 1, 1998 XRD measurement of illite crystallite thickness distributions 49

0.30 . . . . . . . . . . . . . . . , - �9 -

~ * 0.25 _= m

0.20 .=

U" �9 �9

0.15 �9 �9 | - E

o.lo O n. y ~ 0.098 + .00196x R^2 = 0.81

0.05 I�9 0.00

0 20 40 60 80 100

% Smectlte

Figure 12. Relation between (~)% measured from dehy- drated I-S, and percent smectite layers, measured from gly- col-solvated samples prior to dehydration.

r ange f rom 0.049 to 0 .268 .~ (Table 3) and correlate pos i t ive ly wi th the pe rcen tage of smect i te layers (Fig- ure 12).

S U M M A R Y A N D C O N C L U S I O N S

The M u d M a s t e r p r og r am (Eberl et al. 1996) was deve loped to s tudy crysta l l i te th ickness d is t r ibut ions and s t ra in in minera ls , par t icular ly in c lay minera ls , by the B W A method. There fore the p r og r am conta ins sev- eral features that dif fer f rom commerc ia l ly ava i lab le p rograms des igned to s tudy metals . These features in- clude: the analys is of raw data, ra ther than m o d e l e d X R D peaks , so tha t actual C SD dis t r ibut ions can be ex t rac ted f rom peak shape; the r emova l of the LpC~ con t r ibu t ion to the X R D intensi t ies , a con t r ibu t ion that can distort peak shapes for clays; and the f l ipping of in te r fe rence func t ion peaks tha t are s i tuated too close- ly to zero va lues for the L p G 2 funct ion.

The B W A m e t h o d has b e e n appl ied successful ly to s tudy crysta l l i te th ickness d is t r ibut ions for i l l i tes and I - S up to a m e a n th ickness of about 20 nm, us ing bo th ca lcula ted and natural samples . Crys ta i l i te size distri- bu t ions for I -S , wh ich can be s tudied by X- ray ing K- saturated, dehydra t ed samples , genera l ly can be ap- p rox ima ted by lognorma l funct ions . I ndependen t ver- i f icat ion of the accuracy of the X R D m eas u r em en t s was ob ta ined by c o m p a r i n g the m e a n th ickness of the fundamen ta l par t ic les ca lcula ted f r o m Te and expand- abil i ty (Equa t ion [34]), wi th the m e a n th ickness cal- cu la ted f rom T~ and expandab i l i ty (Equat ion [34], Ta- ble 3 and Figure 11).

K-sa tura t ion and dehydra t ion of I - S leads to s train b roaden ing for h igher -o rder reflections. The effects of this b roaden ing on m e a s u r e d T can be avo ided by an- a lyzing the 001 or 002 reflect ions, or by correc t ing the Four ie r coeff ic ients of the h igher -order ref lect ions for

strain. Values for the root m e a n square of the s t ra in increase wi th inc reas ing expandabi l i ty and fo l low a symmet r i c (Gauss ian) dis t r ibut ion. We specula te tha t s t ra in is induced by 2 p h e n o m e n a : 1) K ions loca ted in dehydra t ed smect i te in ter layer space in pos i t ions outs ide o f the d i t r igonal holes in the te t rahedra l sheets, the reby m a k i n g the d - spac ing larger than that of illite; and 2) a def ic iency of K ions in o ther areas of the same interlayer, w h i c h makes the spac ing smal ler than that of illite.

A C K N O W L E D G M E N T S

This research was made possible thanks to financial support from the USGS to V. Drits and J. Srodofi. Special thanks to R. Krushensky and E Hearn of the Office of International Geology. We thank J. Neil, R. Niiesch and R. Pollastro for reviewing the initial manuscript. The use of trade names is for identification purposes only and does not constitute en- dorsement by the U.S. Geological Survey.

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