Miller indices and Family of the Planes

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1 SOLID4 Miller Indices ltest Fmily of Plnes nd Miller indices; Miller indices nd Fmily of the Plnes The geometricl fetures of the crystls represented by lttice points re clled Rtionl. Thus lttice point (or site in lttice) with respect to nother lttice point is clled Rtionl Point. A row of lttice point is clled Rtion Line nd the the plne defined by (rtionl) lttice point is clled Rtionl Plne. All other fetures re clled Irrtionl Fetures. It is necessry to hve n pproprite nottions to describe the rtionl fetures. A nottion used for describing these rtionl fetures ( point, line nd plne) is clled Miller Indices. 1 out of 10:Solid stte Chemistry

2 Indices for point or site indices A position of point or site in the lttice is lwys described with respect to rbitrrily Z chosen one of the lttice point s n origin nd representing it in terms of crtisin coordintes. These coordintes re expressed in the following p [mnp] P forms; x=m, y=nb, nd z=pc. Where,b, nd c re lttice constnt nd m,n, nd p re integers. The cite O m n X indices of the point P is [[mnp]]. For the negtive indices the br is written over the index. For the Y site with the coordintes x = -, y =1b nd z=-3c, the site indices re written s [[ 13 ]] Problem: Determine the indices of the sites depicted in the two dimensionl net. Indices of Direction: To describe direction in crystl lttice, stright line pssing through origin nd prllel to the line of direction under the investigtion is drwn. Then the Determine the indices of sites. b O (x,y,z Brek Point coordintes of point through which the chosen line emerge out or brek open the unit cell is determined. The

3 coordintes of the brek open point in the given exmple is 1,0,0 in terms of,b, nd c respectively. It is indicted by [100]. The sme line will lso pss through (x y z), (3x 3y 3z), (4x 4y 4z).nd will hve common coordintes [x y z]. The x,y nd z re rrnged to be set of smllest possible integer, by dividing nd multiplying through common fctor. b c e.g. The coordintes of the brek open point is (1,1/,0). Multiplying by to ll the numbers, the set of smllest possible integers is (,1,0) nd the miller indices re [10], [10], [40], [630] will hvesme direction nd ll will desribed by the set of smllest possible intergers i.e. [10]. The coordintes of the brek open points is (-1, 0.66, 1) or (-1,/3,1). Multiply this by 3 leds to coordintes (-3,,3) thus the Miller indices re [ 33 ] c b O Miller indices for lttice plne W hen we shine X-ry rdition on crystl, it is observed tht the X-rys sctters selectively t prticulr direction. These nisotropic scttering of X-rys from the crystl, force us to believe presence of tomic plnes (though hypotheticl) in the crystls which 3

4 re oriented in the prticulr direction, nd ct s mirror to reflect the x-ry t prticulr direction. Such reflecting plnes re termed s lttice plnes. Thus Lttice plnes re just imginry equidistnt surfces on which most of the lttice points re lying. We cn strt with two dimensionl lttice i.e. NET. In this cse, The plne is just substituted by line. One cn drw infinite such lines chrcterized by the perpendiculr distnce between the two plnes for given set of plne clled interplnner spcing designted by d. Some times, it lso clled d-spcing. We cn give some rtionl nmes to these fmily of pln es clled Miller indices. We will strt with simple two dimensionl exmple which will extrpolte to three dimension. Consider the two dimensionl rectngulr lttice Here we hve chosen set of plne pssing through the lttic points. In order the determine the Miller indices, First chose the rbitr ry origin nd the unit cell such tht one plnes from the set is lying on the origin Then determine the intercept of the very next plne of the set 4

on x nd y xes in terms of the unit lengths nd b respectively.

Thus, the miller indices of the plne is (1 1) e.g.

1) (1) Intercept 1 nd 1/b Therfore miller indices re In the third exmple, the intercept next to the plne pssing

5 on x nd y xes in terms of the unit lengths nd b respectively. The intercepts re 1 nd 1b divide this number by respective unit lengths i.e. nd b 1 1 b =1 nd =1 b The lst step is to tke reciprocl of the number. Thus, the miller indices of the plne is (1 1) e.g. The coordinte re nd 1b deviding by nd b the intecepts re nd 1 receptively, tking the reciprocl of the number (0 1) (1) Intercept 1 nd 1/b Therfore miller indices re In the third exmple, the intercept next to the plne pssing through origin is 1 nd 1 there fore the indices re (11) If we consider the nother plne, the indices would hve been (1 1) These two set of plnes re equivlent nd designted by symbol {1 1}. 5

6 Miller indices in three Dimensionl Lttice Consider the following exmple.. Here point O is the chosen origin of the unit cell nd set of plnes pssing through the unit cell. In order to determine the miller indices of chosen set of plne, the first step is to determine member of set which psses through origin. After identifying such plne, determine the intercept of n immedite neighbor to ll three chosen xis in terms of,b nd c. In given exmple plne cuts the t /. It cuts to b xis t 1 nd c xis t c/3. The intercepts re therefore, 1/, 1, 1/3. c Third step is to tke reciprocl of ll three o b numbers. Those would be,1,3. These re conventionlly depicted s (13) Threrefore, the miller indices of chosen set of plne is (13). This set of pln hs chrcteristics d vlue which is depicted s d 13. The symbol, { } is used to indicte the set of plnes which re equivlent. Eg. The set (100), (010) nd (001) re equivlent t represented s {100}. 6

7 Exmples: Determine The miller indices of the following lttice: The Miller indices of Hexgonl Lttice. or Miller-Brvis indices As discussed bove, for ll crystl systems, three no-coplnner xes re sufficient to describe the miller indices of the plne. However, the hexgonl unit cell re n exceptions to it. Four indices re often used for tht purpose, nmely (hkil). Insted of three xes, four c xes re used nmely, 1,, 3 nd c. For exmple, the plne which is shown in the figure cuts 1 xis t (-1), 3 7 1

8 xis t 1/, 3 xis t (-1) nd c xis t infinite. Therefore the miller indices re (1 1 0) 8

9 PROBLEMS 1. In crystl, plne cuts intercepts of 3b nd 6c long the three crystllogrphic xes, Determine the miller indices of the plne. Intercept 3b 6c Division by lttice constnts /= 3b/b=3 6c/c=6 Reciprocl ½ 1/3 1/6 After clering frctions (multiply by 6) 3 1 Miller indices (31). Determine the Miller indices of plne which is prllel to the x-xis nd cuts intercepts of nd ½, respectively long y nd z xes. Intercept b 1/c Division by lttice constnts /= b/b= 1/c/c=1/ Reciprocl 0 ½ After clering frctions (multiply by ) Miller indices (014) 3 An orthorhombic Crystl whose primitive trnsltions re = 1.1A, b=1.84a nd C=1.97A respectively. If plne of miller indices (3 1 ) cuts n intercept of 1.1A long x xis find the length of intercept t y nd z xes. h:k:l =:3: 1 Rtios of the intercepts would be 1.1 : 1.84 : Now its cuts t x xis t 1.1A, then obviously we hve to multiply ll the numbers by nd we get 1.1:1.:-3.94 Answer: it will intercept t 1.A nd 3.94A t y nd z xes respectively. 9

10 4.The distnce between the consecutive (111) plne in cubic crystl is A determine the lttice prmeter. Solution: For the cubic crystls, we hve d = h k l = A 3 5. In the tergonl crystl, the lttice prmeter =b=.4a nd c=1.74a, determine the inteplnner spcing between consecutive (101) plnes Solution: For the tetrgonl cell, 1 d h k l c = 10

IV. CONDENSED MATTER PHYSICS

IV. CONDENSED MATTER PHYSICS UNIT I CRYSTAL PHYSICS Lecture - II Dr. T. J. Shinde Deprtment of Physics Smt. K. R. P. Kny Mhvidyly, Islmpur Simple Crystl Structures Simple cubic (SC) Fce centered cubic