H.-J. BRRNHARDT VEB Carl Zeiss Jena

The Reversion of Smoky Quartz Coloration in Crystals with Induced Growth Striations

Induced growth striations in quartz crystals grown from Al-containing material have heen investigated after X-raying them. Earlier results (RERNHARDT, ALTER) showed t h a t the stripes may occur as strong colored ones on a light background or as light stripes on high coloration background. This behaviour depends on the growth region regarded, and t h e applied X-ray dose. The position of the stripes m a y become displaced i n dependence of the X - r a y dose. The present paper developes a model which m a y describe this effect only itsing the diffusion law, the srnoky quartz centre model according O’RRIEN and the A13+/ Na+-distribution in the crystal. Oiir results mus t he taken into account when interpreting the growth striations.

Indusierte Wachstumsstreifen an aus Al-haltigem Material gezuchteten Quarmkristallen wrirtlen nach Rontgenverflirbung untersucht. Dahei zeigten fruhere Ergebnisse (BERN- HARDT, ALTER), da13 in den Wachstumsstreifen eine Erhohung oder Erniedrigung der Itaiichqnarzfarbe auftreten kann. Dan jeweilige Verhalten zeigte sinh von der hetrachteten Wachstumszone u n d der Hestrahlringsdosis unahhiingig. I m s-Rereich wurde sogar eine (1osisal)hiingige Umkehr des Schwlirziingsbildes heohachtet. Die Position der Streifen ver- schiebt sich tfabei. IXe vorliegeude Arheit modelliert tliesen Effekt zwanglos und ohne zusatzliche Annahrnen airs dern Rauchqnarzzentrenmodell von O’BRIEN, drr Al-Vertei- lung und dern Uiffusionsgesetz. JXe Ergelmisse sind h i der Auswertirng indnzierter Wachs- tirinsstreifen zu I)eriicksich tigen.

1. Introduction

The model of smoky quartz centre creation according with O’BRIEN has been accepted in the literature until now. Some aspects have been modernized every now and then (see SCHIRMER et al.). The present paper contains investigations which do not deal with the smoky quartz centre structure but the dynamics of creation and annihilation during the process of X-raying.

We stated in the previous paper that the pattern of local smoky quartz centre distribution changes in dependence of the applied X-ray dose. Crystals containing induced growth striations have been used for these experiments (BERNHARDT, ALTER). A modulation of the temperature during the growth of the crystals has been applied to influence the growing behaviour of the quartz crystals.

The distribution of smoky quartz centres responsed by forming stripes along the growth front. The X-ray induced coloration of the striations has been measured photometrically to get a distribution pattern of the smoky quartz centre concen- tration. Such photometer curves have been presented with respect to all growth regions of the crystals. The temperature modulation had the shape of a sawtooth and had been arranged in groups of three or six “teeth” and modulation free regions between them. The frequency and amplitude of the modulation have been varied according Figure 1. We observed the following surprising phenomena of coloration :

H.-J. BERNHARDT 372

Fig. 1. Modulation scheme of thc induced growth striations in quartz

1.1. Conversion of the optical density in the R region

Figure 2 represents parts of the s region which have been X-rayed with increasing dose. High exposition creates a light boundary at the steep flank of the “teeth”. These light boundaries convert into “negative” saw teeth on deep dark background, when X-raying with saturation dose.

50-

I 1

Fig. 2. Dose dependence of the coloration pattern in the 6-region of a quartz crystal with indu- ced growth striations applied X-ray doses in the sub-areas: in lO/r: A: 1,08, B: 2,16, C: 3 2 3 , D: 4,07, E: 4,56, F: 4,91

Reversion of Smoky Quartz Coloration in Crystals with Growth Striations

1.2. Missing conversion in the -x and z regions

373

Increasing X-ray dose results in increasing optical density. The shapes of all struc- tures remain nearly unchanged. The stripes are dark ones on light background in the -x region and light ones on weakly coloured background in the z region.

1.3. Partial conversion in the +x region

The +x-region is characterized by high inhoniogeneity of the growing front. We observed as well regions in which the growing front is parallel to s as such with paral- lelity to -x. The shape of the stripes growing parallel to -x remains unchanged while the stripes being parallel to s behave like the s-region (see l.l), see li’igure 6 in our previous paper (BERNTTARDT et al.).

1.4. Variation of the blackening of different stripes in a group of homo- geneous modulation

The maxiniuin optical density of each stripe of a group is observed to vary in a typical way. The group with the lowest stripe distance and largest modulation amplitude shows an increase of optical density in the first stripes and a decrease in the last stripes, when counting the stripes in growing direction, see Figure 3.

The teniperature modulation has been equal, however.

seed edge

a?mm cpsfai H

Fig. 3. Fhotometcr curve of induced smoky quartz stripe ein the -9-region: different coloration density of different stripes of one group with equal treatment

374 H.-J. BERNKARDT

1.5. The local resolution of the narrow stripes in the - x region is higher than had been expected according a time constant of about 1 d-l

The decrease of the steep flanks of the stripes makes us assume a time constant of the reaction of the crystal of the modulation of the order of magnetude of 1 d-l. The narrow stripes have a distance of one day with respect to the growth. There ougth to be expected a strong smoothing of these stripes. This is a contradiction to the observation (BERNHARDT et al.).

The afore mentioned effects 1.1 to 1.5 are of unusual character. Unterstanding them is important for the interpretation of induced smoky quartz striations and of the dynamics of the smoky quartz centre creation. We found the key for understanding in them i. the symmetry of the saw teeth shape, ii. the concentration ratio of AP+- and interstitial ions and iii. the orientation of the growth front with respect to the channels in the quartz

structure.

2. Investigations of the coloration effects

2.l.Basic ideas The process of smoky quartz centre creation is illustrated by Figure 4 (above). In the case of high background concentration of A13+ and interstitial ions the emigrating Nao of neighbouring smoky quartz centres may retrap holes. This leads to reconi- hination processes. This elementary act includes as well the creation as the subsequent decay of smoky quartz centres, see Figure 4 (lower part). In the case of saw-tooth shaped distribution of the foreign ions the direction of predominant diffusion of the migrating Na" is given by the concentration gradient. The diffusion in the direction of the steep side of the stripe is predominating but even in the less steep flank a migration of Na" particles is induced by the X-rays. The reconibination probability is estimated by the general concentration of potential smoky quartz centres and the strength of the ionic current. The latter one is dependent on the anisotropic mobility in the quartz lattice.

e+

Fig. 4. Smoky quartz centre formation during X-raying according to O'BRIEN (above); equili- brium betwecn formation and recombination of smoky quartz centres in tho case of local neighbourhood of two centres (lower part)

Reversion of Smoky Quartz Coloration in Crystals with Growth Striations 375

The following chapter will describe this model mathematically. The important parameters influencing the recombination probability shall be discussed subsequently.

2.2. The mathematical model 2.2.1. Diffusion in a tooth-shaped profile

Let us solve the diffusion equation with a tooth-shaped profile in the initial state. The distribution of the Na+ ions before X-raying shall be assumed as to be tooth- shaped.

The diffusion law 3.S d2.S

at dz2 -.=K.-

with - concentration of alkali ions

z - local parameter K -. diffusion constant

has the following particular solutions: cosoc'x , = 2, . e - Ka"t

= pe - Kd21 sin ci'z . The general solution has been found as a linear conibination of solutions (3)

sgen = xi p i e - f i a i z t Sinai z (4) The development of a saw-tooth-shaped distribution as a Fourier series has the aim to use them as initial condition

m. l P ( z ) = n - 2 2 x T s i n i z . 1 %

Comparing the relations (4) and ( 5 ) for t = 0, calculating the coefficients and normali- zing sgen to unity we obtain

1 m e--i*K.t

Sgen = 0.5 - - xi sin iz. (6) Z l 8

Figure 5 represents the behaviour of an initial tooth-shaped distribution according the diffusion law.

Fig. 5. Time dependent smoothing of an initial saw-tooth-like impurity distribution due to diffusion: s - concentration, z - position coordinate, - time, k - diffusivity. full line: t = 0; dashed line: t = 0 . 1 . K ; fat points: t = 0 . 5 . K ; small points: t = 1 . 0 . K ; equal distribution: t -+ 00

3 76

2.2.2. Diffusion and recombination

H.-J. BERNHARDT

We assume the following properties of the recombination probability : i. it depends on the strength of the current fornied by the diffusion of X-ray created

Na" particles, ii. on the locally found smoky quartz centre concentration and iii. on the duration of the diffusion. The diffusion through a cross-section a t the position z is given by

- K -- dt lL:i (7)

with K - factor containing the diffusion constant.

The decrease of radiation induced centres due to reconibination as a follow of the diffusion is proportional to the expression (7) and the centre concentration found a t the position z :

with c - F ( z ) + u F ( z ) - tooth-shaped distrihiition I / - bacliground. Equation (8) gives us the following solution after having siihstitrited F ( z ) by (6):

sin iz x 1 33 e - i 2 1 i 7 C( t ) = (1 - e-") u + n - 2 zip-

1 i

The pre-factor (1 - e-ef) is an artificial term which have been introduced into the solution. It describes the creation of smoky quartz centres in dependence of time and has assumed to be expongntial. So the irradiation process has been divided into two steps: i. the creation of all potential smoky quartz centres according the irradiation time

(or dose) and excluding reconibination and ii. diffusion and recombination for the same time.

The numerical calculation of a Fourier developed solution includes some numerical problems which are connected with the limited accuracy of the development. These inaccuracies become multiplied by the exponentiation and differentiation procedures included in (9). These numerical effects should be minimized by the following cal- culation procedures : i. numerical smoothing of the function (9) in the interval z = + O . l and using three

points in each interval. ii. replacing of the differential quotient by the steepness of the secant in a migrating

interval Remaining overtone frequencies do not injure essentially the general solution.

The numerical deviations increase a t large t .

Reversioti of Stnoky Quartz Coloratioti i t i Crystals with Growth Striatioils

2.2.3. Special case: distribution of the optical density in a group of stripes (compare 1.4)

377

The envelope of a group of six stripes which had been falsificated by a time constant increases from the first to the last stripe according an exponential law. The centre of gravity of this group is accordingly out of the middle. This fact influences the pre- ferential direction of the diffusion. The diffusion is preferentially directed from the first to the last stripe. To calculate this problem analogously to (l)-(ci) the relation must be influenced by a time constant :

F'(z ) = epnt [ J c x F ( z ) eqt dt + C] P"(z ) - saw-tooth structure siiioothed by u tinie cotistatit t - titiie.

F' is the new initial condition in (1) now. We found another way to by-pass this ciiniber~onie calculating procedure and

nevertheless to investigate the behavionr of a group like (10): i. Approximation of the envelope by a Fourier series. The envelope is assunicd to be of the type E = KO (1 - e P r z ) . ii. Addition of a cos-function to the envelope to siniulate the stripes. This constructed functicn has tlie look of a group of stripes which had been snioothed by a time constant. It is more suitable to use it as initial equation in (1). The deviation of the modulation from the real saw-tooth-like shape may be neglected here. The further calculation is analogous to (1) to (9).

A. Results

The effects discussed in 1.1 to 1.5 have been simulated by relations of the type (9). These five cascs differ from each other by the following paranieters used in the cal- culation : 11' - coloratioii I)aukgroritid of the tooth-like structure I< - (lill'iisioti coiistutit of tlie iiiterstitids - fuotor of proportionality with respect to the recoii~binstioti.

3.1. Saw tooth without background

Calculating with (9) and us;ng the parameters K = 1, ,3 = 9, U = 0 resiilts in thc sininlation of a saw-tooth-like structure with zero background. Figure (i illustrates this ease for three different irradiation stages. The coloration increases wit,hout any change of profile. Assuming additionally a vanishing diffusion constant K the relat,ion (9) will be reduced to t.he following expression:

This rel'resents a saw-tooth-like structure of an aliiplitrrde increasing with t h o irra- diation dose. This case describes the smoky quartz stripes in the -x and +s regions. The growing fronts in this crystal regions are parallel to z and the structural channels of the quartz structure. The diffusion of interstitials in quartz is known to take place along these channels, however. The diffusion coefficient in the perpendicular direction is low a t room temperature indeed. The missing of any coloration inversion may be understood by these facts.

3 78 H . J . BERNHARDT

5

4

3

S

2

7

0 L

Fig. 6. Dose depcndeimc of the smoky quartz stripes in the case of diininialiing coloratioii back- ground, calculated with the abovementioncd parameters; S - optical density, z - local coordiwatc

3.2. Saw-tooth with background

This cast: has been simulated using the parameters K = 1, U = 5 (that denotes a 5-times stronger colorated background compared with the saturation value of the stripe coloring), B = 9. The results are given in Figure 7 and are to be compared with

Z Fig. 7. Coloration conversion in the easc of a saw-tooth-shaped structurc with remarkable coloration background. Numerical smoothing with help of a gliding iiiterval of three points, calculated using the aboveinentiorled valucs. left-handed : genenil view of two teeth; right- handed: a stcep flank in detail, shift of the light/dark edge denoted by arrows; increasing X- ray dose from t , to t , in arbitr. units

Reversion of Sinoky Quartz Coloration in Crystals with Growth Striations 349

the observations in the s-region of our quartz samples (see 1.1). Figure 7 illustrates an initial increasing optical density without changes of the shape but a light contour a t the steep flank to the tooth-shaped structure after high exposure of the crystal. I n the same time the position of the maximum density is shifted. Figure 7 shows two teeth in the general view (left-handed) and the behaviour of the region a t the steep flank in detail (right-handed).

3.3. Coloring behaviour of a group

Fig. 8 a represents the idealized envelope of a group smoothed by a time constant, Figure 8 b the Fourier-like developement of the order 6. Eigure 8 c represents the simulation of a group of stripes smoothed by a time constant according the descrip- tion given in 2.2.3. The curves d, e and f show the calculated change of shape due to diffusion and recombination assunling large diffusion times and for different diffusion constants. The model describes a depletion of the 6. stripe (in growth direction) in satisfactory agreement with the experiment.

L

Fig. 8 . CIiaiigc of t h o slrape of a gioiip of six stripes due to sriioothiiig according a tiriio const,ant, diffusion aiid rcconibiiiatio~~ (cal culatcd). a : envelope; b: Fourier-like devclopnicnt of “a” (6. order); c : “b” ~ l u s ( - 0 , l sin 6 2 ) ; d: “c” aftcr X-raying and diffusion using t + co, K = 1 ; e : like “d” but K = 2 ; f: like “d” but K = 10

3.4. The resolution of individual stripes of a group

ICstiniating roughly the time constant of the system autoclave/crystal resulted in :in approxiniate value of about 1d-l. A remarkable smoothing of the daily growth stripes is expected. The experiment proves a good resolution of the individual stripes, however. This observation niay be understood like the case 3.2 (stripes on Iwkground). The smoothing due to the time constant may be understood as an

380 H.-J. BERNHARDT

increase of the background compared with the remaining modulation of the group. The diffusion of the Na” particles favours the direction from the coloration maxima to the niiniiria and elevates the reconibination probability a t the position of the minima. This results in a better resolution of the stripes than had been expected before.

3.5. Yecularities of the growth regions z, s, x and -x

A substantial difference between the several growth regions is the each resulting component of the concentration gradient of the stripes in the direction of the struc- tural channels of the quartz crystal ( z direction). This resulting component varies froni zero to unity comparing the -x and z regions, respectively. This fact, resulting in a variation of the effective diffusion constant points out the possibility to under- stand the partially observed conversion of the coloration in partial areas of the +x region with changed direction of the growing frong (1.3). The cause is a non- diminishing A- in these parts of the crystal.

The “negative” stripes which have been observed already in the initial irradiation stage of the z region might be understood by the high diffusivity in the z direction. This fact must be investigated inore intensively, however. The general coloration of this region is too small to investigate a dose dependence of the stripes in our crystals.

4. Conclusions

Several unusual coloration effects in quartz crystals with inhomogeneous distribution of smoky quartz centres may be understood with help of one niodel. Further exaniina- tion of this niodel with help of niicrophotonietric methods in the ir region are neces- sary. Investigations of the electrolytical coloring of samples regarded in the present paper are in work. The present results are not only interesting with respect to crystals with induced striations but also such with unwished inhomogeneous At-distribution such as natural ones.

References

BERNIIARDT, H.-J.; ALTER, U.: Cryst. Res. Techiiol.,.lS, 453 (1984) O’BRIEN, M. C. M.: Pror. Roy. SOC., A 231, 404 (1955) SCHIRMER, 0. F.: Sol. St. Cointtiuns., 18, 1349 (1976)

Author’s address : Dr. H.-J. BXRNHARUT VEB Carl-Zeiss-Jena DDR-6900 Jona Carl-Zeiss Str. 1

(Roceived April 16, 1984)