An experimental assessment of X-ray diffraction theory in high purity aluminium crystals

Materials Science and Engineering, 55 (1982) 219 - 229

219

An Experimental Assessment of X-ray Diffraction Theory in High Purity Aluminium Crystals J. J. URCOLA, J. FAUSTMANN* and M. FUENTES Centro de Investigaciones Tdcnicas de Guip~zcoa, Escuela Superior de Ingenieros, San Sebastidn (Spain) (Received November 9, 1981)

SUMMARY

An experimental test of Zachariasen's X-ray diffraction formula was carried out for high purity aluminium crystals with a wide range o f mosaic block sizes and mosaic block misorientations. These crystals were produced by cold rolling to different strains three samples (crystals 1, 2 and 3) cut from a large single crystal grown by the Bridgman method. The (100), (200)and (220)X-ray integrated intensities were measured for each crystal, and the block sizes and block misorientations were obtained using transmission electron microscopy techniques (the mean linear intercept method for the block size and Kikuchi line techniques for the block misorientation). With the exception o f crystal 2 in the undeformed condition, for a given state the three crystals showed a common diffraction efficiency, the value o f which increased with increasing deformation. The different diffraction efficiency o f crystal 2 with respect to crystals 1 and 3 in the undeformed condition is rationalized in terms o f a segregation o f impurities at the bottom of the large single crystal. Comparison between the integrated intensities determined experimentally and the intensities calculated by substituting the measured block size and block misorientation parameters into Zachariasen's formula showed good agreement between the two sets of values. In view of this, it was concluded that Zachariasen's formula is closely consistent with the diffraction efficiency o f the crystals under study; the behaviour o f the crystals is characteristic o f Zachariasen's type H crystals, i.e. the extinction is primarily controlled by

*Present address: Junta de Energia Nuclear, Madrid-3, Spain.

0025-5416/82/0000-0000]$02.75

the mosaic block size rather than by the mosaic block misorientation.

1. INTRODUCTION

Rigorous theoretical treatment of X-ray diffraction in Bragg geometry is limited to the special cases of the "ideally perfect" crystal (dynamic theory) and ~he "ideally mosaic" crystal (kinematic theory). General formulations for crystals whose diffraction properties lie between the t w o ideal cases referred to above (i.e. aggregates of perfect crystal domains separated by discontinuous surfaces that interrupt the lattice periodicity and with small relative misorientations between contiguous blocks) aim at a general expression for the extinction factor that must contain the two limiting cases previously mentioned, i.e. the extinction factors yielded by the kinematic and dynamic theories. Although different expressions for the extinction factor have been given in the literature [ 1 - 3 ] it is generally accepted that Zachariasen's formula [3], despite its mathematical rigidity [4, 5], is more amenable to experimental verification than are other theories [6]. Therefore the present assessment of X-ray diffraction theory is only concerned with Zachariasen's formula. Given the inherent complexity of the derivation of the extinction factor, a number of approximations have been introduced during its deduction that, although believed to be theoretically justified, cast d o u b t on the validity of the final formula. Although it is generally agreed that the ultimate test of the theory must be the agreement between predictions and experiment [3, 6], only a very small number of workers have attempted to relate the crystal structural parameters © Elsevier Sequoia/Printed in The Netherlands

220

obtained by using both theory and X-ray measurements with those determined from metallographic observations for the same crystal [4, 7 - 9]. Furthermore, the types of aggregate so far tested have been very few and the technique used to examine their microstructures too limited to allow for (i) an unambiguous resolution of the mosaic boundaries and (ii) a reliable determination of their misorientations. In view of these shortcomings the present investigation was undertaken in order to assess the validity of Zachariasen's formula by using data that are more general and more accurate (these data consist of the block sizes, block misorientations and integrated intensities) than those previously reported. These data were obtained during a wider investigation of high purity aluminium crystals aimed at elucidating the effect of the dislocation substructure on the quantitative measurement of crystallographic textures and were re-examined in more detail with respect to the measurement of crystallographic textures. The results of this analysis will be submitted for publication shortly.

2. BACKGROUND The integrated intensity of the diffracted beam and the extinction factor, E and y respectively, obey the equation Q

E=C--y

(1)

2p

where the constant C = IoSo/¢O consists of the incident beam intensity I0 at the crystal surface, its cross-sectional area So and the angular velocity co with which the crystal is rotated, p represents the absorption coefficient for the radiation wavelength adopted and Q is defined by

N2)k3 [ e 2 ~2 2 1 + c0s200 Vhkz - sin 200 ['mcc2] fh~l 2

(2)

where 00 and Fhkt denote respectively the Bragg angle and the structure factor for the crystallographic planes (hkl) and N, m, e, and c have their usual meanings. The extinction factor y is given by the expression 1 I + K 4 _ ~-1/2 s =

+ 2---=.Xo,

+ K"

(a)

with K = cos 200 and Xo =

Qo a t +

(~-~)2}1/2

(4)

where t and T represent the mosaic block size and the crystal effective size respectively and a and g are given by -

sin 20o

(5)

X g-

1 21r 1/2~?

(6)

which are functions of the block size t and the standard deviation ~? respectively of the orientation distribution: 7rl/2 ~?= - - ~ (7) 2 with ~ equal to the mean misorientation [ 10].

3. EXPERIMENTAL DETAILS A 99.999% high purity aluminium single crystal was grown under vacuum by the Bridgman technique. From this cylindrical single crystal (30 mm in diameter and 80 mm in length), three discs 6 mm thick (crystals 1, 2 and 3) with face normals [123] were cut and studied using X-ray and transmission electron microscopy (TEM) techniques (i) in the undeformed condition, (ii) after they had been reduced in thickness by rolling about 21% and (iii) after they had been reduced in thickness by rolling about 45% . Before the X-ray and TEM studies were carried out, these crystals were prepared as follows. (i) The top face of each disc was polished (first mechanically and then electrolytically), annealed for 4 h at 500 °C, electropolished again in order to remove the superficial layer deformed in the cutting operation and oxidized during the annealing treatment. The polished surfaces were then exposed to Cu Ka nickel-filtered X-ray radiation and the resulting (111), (200) and {220) pole diffracted intensities were recorded in chart form. Sheets about 0.5 mm thick were finally spark machined from these surfaces and TEM specimens were prepared by conventional techniques: after the disc-shaped samples had been spark machined, a preliminary electrolytic polishing in a 10%HCIO4-90%CH 3CH2OH

221 solution followed b y a profiling polish and a final polish to perforation in the same solution were carried o u t in a Polaron electropolishing unit. (ii) The three discs were subsequently rolled to a b o u t a 21% reduction in thickness and their faces were electrolytically polished again until the superficial layer had been removed. After the (111), (200) and (220) integrated intensities had been measured, TEM specimens were produced following the same procedure outlined earlier. (iii) A new reduction in thickness up to approximately 45% was performed and the same sequence of operations (the removal of the superficial layer, the recording of X-ray diffracted intensities and thin foil preparation) was repeated again. 3.1. X-ray measurements A modified version of the X-ray scanning technique c o m m o n l y used was applied in the present investigation [ 1 1 ] . The main features of this modified technique which allows a more precise analysis of the intensity distribution of the diffraction m a x i m u m than the conventional m e t h o d are n o w described. Let the surface of Fig. 1 represent schematically the intensity distribution a b o u t the diffraction m a x i m u m of a given crystallo-

.jJ --4aC

j Fig. 1. Schematic intensity distribution about a diffraction maximum.

Fig. 2. Schematic stereographic projection of the itinerary followed in the scanning of the intensity distribution about a diffraction maximum. graphic set of planes. In order to define this distribution the following procedure was adopted. Once the t w o rotations ~ and required to bring the normal of the (hkl) planes into coincidence with the direction that bisects the incident and diffracted beams had been determined and the crystal, which had been previously attached to a Norelco texture goniometer, had then been rotated (Fig. 2), a series of measurements of the diffracted intensities E(oL)~=constant as a function of ~ for successive values of ~, of equal spacing A~ = 0.5, were performed {Figs. 1 and 2). An example of these measurements for the (111) plane is shown in Fig. 3 where each peak and the associated number represent respectively the intensity distribution E(~)~=constant (i.e. the intensity distribution resulting from the intersection of the surface E(0~) with the plane ~ = constant) and the total number of pulses recorded from a scan of ~ at constant ~. From these fl = constant contours, the magnitude of E ( ~ ) was obtained first by integrating numerically over the whole range of ~: E =

fE{ ) d~

(8)

222

Fig. 3. Scanning of the (111) intensity distribution for crystal 2 which was reduced in thickness about 21%.

Then this value was corrected b y a factor

substructural parameters, 95% confidence limits were obtained.

sin 0hkt sin ~hkl which takes into account the different durations of exposure of the X-ray detector with varying c~. For each crystal and each reflection an average of six intensity measurements was performed for every given state: four in the centre of the disc and two a b o u t 10 mm away from the centre. The mean intensity values and 95% confidence intervals were subsequently obtained.

3.2. Electron microscopy measurements TEM specimens were examined in a Philips EM 300 electron microscope provided with a tilt-rotation goniometer at 100 kV. The mean linear intercept cell or subgrain size was determined for the three crystals in each state over a minimum o f 20 low magnification (5000X) pictures obtained from at least three different foils. Since contiguous cell (or subgrain) misorientations were n o t resolved b y conventional diffraction patterns, "split" Kikuchi line techniques [12] were extensively used to measure the mean block misorientation (an average value was obtained from approximately 15 - 20 individual measurements performed using the three foils from the previous experiment. For both

4. RESULTS

4.1. Substructural parameters Crystals in the as-grown and annealed condition exhibited poorly developed block boundaries with the interior of the blocks almost completely free of dislocations (Fig. 4(a)). The block size in crystals 1 and 3 (about 5 #m) was distinctly larger than in crystal 2 (about 3 pro), as can be seen in Table 1 where the actual sizes and their 95% confidence limits are shown. As could have been foreseen from Fig. 3, in which the contrasts under given diffraction conditions were similar on both sides of the boundaries, the block misorientation deduced from the corresponding Kikuchi line patterns was extremely small {Table 1), around 20'. The first reduction in thickness induced a substructure c o m p o s e d of well-developed cells (Fig. 4(b)) whose size was essentially the same for the three crystals, a b o u t 3.5 #m (Table 1). Therefore, although the cell size in crystals 1 and 3 decreased with this deformation, in crystal 2 it remained nearly constant. Dislocations were also confined to the walls where they adopted an arrangement with a high degree of regularity (Fig. 4(b)). As

TABLE 1 Mean block sizes and mean block misorientations Crystal

1 2 3

45% reduced state

As-grown and annealed state

21% reduced state

t (b~m)

/~ (rain)

t (pm)

/~ (min)

t (pm)

/~ (min)

5.525±0.34 2.86±0.51 4.96±0.58

18.7±4.7 17.4±5.7 24.1±7.8

3.80±0.34 3.10±0.29 3.29±0.40

40.4±5.6 30.0±7.2 44.3±5.4

2.51±0.52 2.28±0.22 1.99±0.16

69.5±15.6 56.0±12.0 68.1±17.5

223

(a)

I 2~33

(a)

(b)

,2um,

(b)

(c)

,2~rq

(c)

Fig. 4. Typical microstructural features of crystal 1 : (a) in the as-grown and annealed condition; (b) reduced in thickness about 21%; (c) reduced in thickness about 45%.

Fig. 5. Increase in the crystal 3 block misorientation with reduction in thickness as revealed by the enhanced separation of split Kikuchi lines: (a) in the as-grown and annealed condition; (b) reduced in thickness about 21%; (c) reduced in thickness about 45%.

revealed b y the different diffraction contrasts on both sides o ' f a given cell wall, the misorientations were larger than in the as-grown

and annealed conditions; the present values are approximately twice the previous ones (Table 1).

224 With increasing deformation, Le. with a further reduction in thickness up to 45%, a true subgrain microstructure was formed (Fig. 4(c)). As can be seen in Fig. 4, the initial cell walls, which are made up of threedimensional arrangements of dislocations, thinned out with increasing deformation until well-defined bidimensional dislocation boundaries {enclosing free dislocation volumes) were produced. Concomitantly with these changes, the misorientation increased to a value about three times the corresponding values in the as-grown and annealed state (Table 1). This steady variation in the misorientation is qualitatively illustrated in the Kikuchi line patterns (Fig. 5); Fig. 5 shows the increased separation of any pair of either "excess" or "defect" lines with deformation and even more strikingly the separate evolution of those pairs of lines associated with

6 5. 4

_z 3

2 1

I

I

(a)

60 z F-

5O

4o

S 20

S

./7

///

10 I 01

012

I 03

4.2. Integrated intensities Figures 7(a), 7(b) and 7(c) exhibit the diffraction efficiency of (111), {200) and (220) reflections for crystals 1, 2 and 3 respectively. The main feature that is prominent from an examination of these diagrams is the different diffraction behaviour of crystals 1 and 3 with respect to crystal 2. In both crystals 1 and 3 the {220) integrated intensity remained nearly constant with deformation while the {111) and (200) intensities increased by an amount that, for the crystals reduced in thickness by 45%, varies between 20% and 40% of the initial value, depending on the pole and crystal considered; these increases are somewhat higher in crystal I than in crystal 3. In contrast with this behaviour the (111), (200) and {220) pole integrated intensities of crystal 2 did not show a significant variation with strain. It can also be seen that the initial integrated intensities, i.e. the intensities relative to the as-grown and annealed crystals, are distinctly higher in crystal 2 than in crystals 1 and 3.

5. DISCUSSION

/

0

crystallographic planes approximately parallel to the boundary plane, i.e. the actual misorientation values summarized in Table 1, and consequently the tilt components of the boundaries, were obtained from these latter pairs of lines. For conciseness, the mean block size and the mean block misorientation dependences on the degree of cold working are plotted in Figs. 6(a) and 6(b) respectively. It is worth noting that for the three crystals the mean size and mean misorientation decreased and increased respectively with deformation.

I 04

I 0,5

I 06

E

(b) Fig. 6. Variations in (a) the mean block size and (b) the mean block misorientation with strain: o, crystal 1;D, crystal 2; A, crystal 3.

From eqns. (3) and (4) it can be seen that the contribution of the first term of eqn. (4) {which represents the effect of primary extinction) depends on the mosaic block size and not on the mosaic block misorientation distribution. However, in view of the large value of the effective crystal size in comparison with the mosaic block size (T >> t) the contribution of the second term of eqn. (4) (which represents the effect of secondary extinction) can be taken as dependent on the

225

20000

20 000

A

5

........~{111}

E

15000

15.000

25 <

25

I

>.

c_

[111)

{200) r-

E

10.000

1200) 10.000

~n C r-

5 000 -

(220)

4

I 21

SO00

/"-"-''~

~

12201

I

I

I

45

21

G5

Reduction

Reduction in thickness C°/o)

in thickness(0/o)

(c)

(a)

20000

u~

t

{ 1111)

c 15.000

.o

<

~ ~

~

12001

1O0O0

(220)

5 000

I

I

21

4S

Reduction in t h i c k n e s s (°/o)

(b)

Fig. 7. Variation in the diffraction efficiency of (111), (200) and (220) reflections with reduction in thickness: (a) crystal 1; (b) crystal 2; (c) crystal 3.

226

mosaic block misorientation distribution b u t independent of the mosaic block size. To test h o w closely eqns. (3) and (4) represent the actual diffracted intensity, the agreement between the experimental integrated intensity values and those predicted by the primary and secondary extinction models will be examined. The increase in integrated intensity with progressive deformation predicted by the calculated secondary extinction factors and the increase in the actual measurements will first be compared. Table 2 summarizes the secondary extinction factors obtained by first substituting the angular deviation values from Table 1 in the second term of eqn. (4) and then replacing this term in eqn. (3). The lattice parameter [ 1 3 ] , the wavelength (ref. 14, p. 65) and the atomic scattering factors [15] used to calculate the required Qhkl values are also recorded in Table 2. It can be seen that the integrated intensities yielded by these secondary extinction factors do n o t agree with the measured integrated intensities. While the secondary extinction factors lead to intensities that increase up to a b o u t 3% from the as-grown and annealed state to the approximately 45% deformed state, the increase in the actual measurements is up to an order of magnitude higher, i.e. an increase of 30% for the (111) and (200) reflections in crystals 1 and 3. If the 45% deformed crystals are taken as a reference state, the integrated intensity increase would

require, on the basis of the secondary extinction model, a misorientation in the as-grown and annealed state of a b o u t 2', a value that is some ten times smaller than the measured misorientation (Table 1). Furthermore, the secondary extinction model also falls to explain the " a n o m a l o u s " diffraction efficiency of crystal 2 in the undeformed condition since its block misorientation does n o t differ appreciably from the misorientations of crystals 1 and 3 (Table 1). It is therefore concluded that, in the three crystals under study, secondary extinction does not play an important role. From eqn. (1) it can be seen that the ratio Ely is independent of both the crystal considered and its substructure. Therefore for a given reflection the ratio of the measured integrated intensity to the corresponding primary extinction factor should be constant. Tables 3 and 4 summarize respectively the calculated primary extinction factors (obtained by first substituting the mosaic block sizes from Table 1 into the first term of eqn. (4) and then substituting these terms into eqn. (3)) and the Ely ratios. It can be seen that for a given reflection the discrepancy between the E/y ratios and their mean value does n o t exceed a b o u t 10% of the mean value. To test further the agreement of the data summarized in Tables 3 and 4 with the primary extinction model, the values of the t w o sides (E/C and (Qhkl/2p)Y) of the rewritten equation, eqn. (1),

TABLE 2 Secondary extinction factors

Crystal

Reflection

Secondary extinction factor for the following states As grown and annealed

21% reduced

45% reduced

(111) (200) (220)

0.957 0.967 0.981

0.979 0.984 0.991

0.988 0.991 0,995

(111) (200) (220)

0.954 0.964 0.980

0.972 0.979 0.988

0.985 0.988 0.933

(111) (200) (220)

0.966 0.974 0.985

0.981 0.985 0.992

0.987 0.990 0.995

a(20 °C) = 4.0496 x 10 - 8 cm; ~CuKa= 1.5418 X 10 - 8 cm; f ( l l l ) ( 2 0 °C) = 8.45, f(200)(20 °C) = 7.93, f(220)(20 °C) = 6.52.

227 TABLE 3 Calculated primary extinction factors Crystal

Calculated primary extinction factor for the following states

Reflection

1

(111) (200) (220)

2

(111) (200)

(220) 3

(111) (200) (220)

As grown and annealed

21% reduced

45% reduced

0.586 0.609{0.633 0.615 0.637{0.661 0.679 0.700{0.723

]0.716 0.745 ]0.775 10.741 0.769 ~0.797 0.795

0.814 0.860 {0.905 10.833 0.876 ~0.917 0.873

0.783 0.829{0.875 0.805 0.848/0.889 /0.849 0.884/0.917

0.781 0.807 {0.834 0.803 0.827 {0.852 0.848 0.868 {0.888

10.861 0.881 }0.899 /0.877 0.895/0.912 /0.908 0.922 (0.935

0.608 0.650 ~/0.696 0.636 0.678 {0.722 10.699 0"738 [0.778

0.754 0"790{0.826 /0.778 0.811 [0.845 0.827 0.855 {0.882

0.892 0"905 {0.918 /0.905 0.917 [,0.928

0"819{0.843

0"907{0.939

n a~a 10.929 . . . . . /0.947

TABLE 4 Ratio of the integrated intensity E to the primary extinction factor y Reflection

(111)

(200)

(220)

E

C

Qhkl

-y 2p

Crystal

E/y for the following states As grown and annealed

21% reduced

45% reduced

Mean value

1 2 3

21051 19578 20631

21208 20136 20190

19744 18584 19050

20000

1 2 3

13140 12924 14233

15345 15080 14352

13664 13162 12334

13800

1 2 3

6714 6210 6436

6166 6277 6433

5654 5607 5463

6100

(1')

(where C equals the mean value of the quotients (E/y)/(Qhkd2p) with p = 131 cm -1 (ref. 14, p. 194)) are plotted in Fig. 8 as functions of deformation. These diagrams show first the close agreement between the t w o sets of points yielded by both sides of eqn. (1') and secondly the gradual approach of these points to the respective a s y m p t o t e Qhkz/2P with increasing strain. Thus, as the

features of the strain-induced substructure approach those envisaged in the "ideal mosaic", the extinction factor tends to unity. Also it is n o t e w o r t h y that the diffraction behaviour of crystal 2 (so far referred to as anomalous) is consistent with the primary extinction model since its smaller initial block size compared with that of crystals 1 and 3 should give rise to a higher integrated intensity. It is suggested that the smaller block size of crystal 2 which was machined from the b o t t o m of the single crystal is due

228 10 -4

'°:'t

am_

2p

2~

c_

g

E

0220 2p

, -----f

I

I

21

L,5

0

I

I

21

&5

Reduction in thickness [°/=)

Reduction in thickness (°/o)

(a)

(c)

10-4

2~

=~ - - S ~ -

(} 0200 2p

01 ~J

"0 C~

J

-F

2

0220 2p

. ~ _ - -

0

l 21

Reduction in thickness (°/o)

(b)

I.5

Fig. 8. Plot of the two sides (E/C (o) and (Qhkt/21~)Y (e)) of eqn. (1') vs. reduction in thickness: (a) crystal 1 ; (b) crystal 2; (c) crystal 3.

229 t o its c o m p a r a t i v e l y high i m p u r i t y c o n t e n t ; these impurities hinder, t o a greater e x t e n t t h a n in crystals 1 and 3, t h e processes t h a t c o n t r o l the final b l o c k size o f t h e as-grown and a n n e a l e d crystals. This suggestion is in line w i t h t h e i n f o r m a t i o n gained f r o m crystal g r o w t h , which indicates t h a t impurities t e n d t o segregate at the b o t t o m o f single crystals produced by the Bridgman technique.

6. CONCLUSIONS C o m p a r i s o n b e t w e e n the i n t e g r a t e d intensities e x p e r i m e n t a l l y d e t e r m i n e d and the intensities c a l c u l a t e d b y substituting t h e m e a s u r e d mosaic b l o c k size and mosaic b l o c k m i s o r i e n t a t i o n p a r a m e t e r s into Zachariasen's f o r m u l a indicates t h a t in t h e t h r e e high p u r i t y a l u m i n i u m crystals u n d e r s t u d y (i) s e c o n d a r y e x t i n c t i o n d o e s n o t play an i m p o r t a n t role and (ii) the p r i m a r y e x t i n c t i o n m o d e l provides a c o n s i s t e n t r a t i o n a l e f o r the d a t a obtained. Since t h e e x t i n c t i o n is c o n t r o l l e d b y the mosaic b l o c k size r a t h e r t h a n b y t h e mosaic b l o c k m i s o r i e n t a t i o n , it is c o n c l u d e d t h a t t h e d i f f r a c t i o n b e h a v i o u r o f the crystals u n d e r s t u d y is characteristic o f Zachariasen's t y p e II crystals,

REFERENCES 1 R. W. James, The Optical Principles o f the Diffraction o f X-rays: the Crystalline State,

Vol. II, Bell, London, 1962, pp. 272, 282. 2 P. B. Hirsch and G. N. Ramachandran, Acta Crystallogr., 3 (1950) 187. 3 W. H. Zachariasen, Acta Crystallogr., 23 (1967) 558. 4 D. Michell, A. P. Smith and T. H. Sabine, Acta Crystallogr., Sect. B, 25 (1968) 2453. 5 F. Zigan, Neues Jahrb. Mineral., Monatsh., 8 (1970) 374. 6 D. B. Brown and M. Fatemi, J. Appl. Phys., 45 (4) (1974) 1544. 7 L. J. Azaroff, in G. N. Ramachandran (ed.), Crystallography

8 9 10 11 12 13

and

Crystal

Reflections,

Academic Press, New York, 1963, p. 109. E. I. Raikhel, I. V. Smundikov and V. M. Trembach, Soy. Phys. -- Solid State, 10 (1968) 1330. D. B. Brown, M. Fatemi and L. S. Birks, J. Appl. Phys., 45 (4) (1974) 1555. C. G. Dunn and E. F. Koch, Acta Metall., 5 (1957) 548. J. J. Urcola, Doctoral Thesis, Escuela Superior de Ingenieros, Universidad de Navarra, San Sebastian, 1976. A. J. Hartley and S. R. Keown, Micron, 3 (1972) 374. N. B. Pearson, A Handbook o f Lattice Spacings and Structures o f Metals and Alloys, Vol. 2, Pergamon, Oxford, 1967, p. 80.

14 International Tables for X-ray Crystallography, Vol. III, Kynoch, Birmingham, 1968, p. 65. 15 J. J. De Marco, Philos. Mag., 15 (1967) 483.