1 00:00:10,570 --> 00:00:11,690 STUDENT: Hello. 2 00:00:11,690 --> 00:00:15,220 Today we are going to talk about real and reciprocal space 3 00:00:15,220 --> 00:00:17,230 in two dimensions and three dimensions. 4 00:00:19,880 --> 00:00:24,210 Crystallography is a major topic within material science. 5 00:00:24,210 --> 00:00:27,620 A crystal is a highly ordered solid material 6 00:00:27,620 --> 00:00:32,450 made up of a lattice and a periodic arrangement of atoms. 7 00:00:32,450 --> 00:00:35,780 Atoms are located on every lattice site. 8 00:00:35,780 --> 00:00:38,030 Due to the periodic nature of the structure, 9 00:00:38,030 --> 00:00:42,600 all crystals display what's called translational symmetry. 10 00:00:42,600 --> 00:00:45,000 Crystals are defined by their lattice vectors, 11 00:00:45,000 --> 00:00:46,980 which are different in two dimensions and three 12 00:00:46,980 --> 00:00:48,810 dimensions. 13 00:00:48,810 --> 00:00:51,270 In two dimensions, lattice vectors 14 00:00:51,270 --> 00:00:54,420 are the shortest and next to shortest translations 15 00:00:54,420 --> 00:00:56,390 from a given point. 16 00:00:56,390 --> 00:01:00,570 These lattice vectors are called a and b. 17 00:01:00,570 --> 00:01:04,870 Four types of general 2D lattices exist-- 18 00:01:04,870 --> 00:01:10,630 oblique, rectangular, square, and hexagonal. 19 00:01:10,630 --> 00:01:13,760 In an oblique lattice, a and b are different. 20 00:01:13,760 --> 00:01:19,350 And the angle between them is greater than 90 degrees. 21 00:01:19,350 --> 00:01:21,740 In a rectangular lattice, a and b are different. 22 00:01:21,740 --> 00:01:24,920 And the angle between them is 90 degrees. 23 00:01:24,920 --> 00:01:27,860 In a square lattice, a and b are the same. 24 00:01:27,860 --> 00:01:30,320 And the angle between them is 90 degrees. 25 00:01:30,320 --> 00:01:34,610 And in a hexagonal lattice, the a and b are the same. 26 00:01:34,610 --> 00:01:39,340 And the angle between them is 120 degrees. 27 00:01:39,340 --> 00:01:42,550 On the other hand, lattices in three dimensions 28 00:01:42,550 --> 00:01:47,500 are defined by the three lattice vectors a, b, 29 00:01:47,500 --> 00:01:53,710 and c, which have angles of alpha, theta, 30 00:01:53,710 --> 00:01:56,620 and gamma between them. 31 00:01:56,620 --> 00:01:59,440 There are seven types of simple crystal systems, 32 00:01:59,440 --> 00:02:01,690 which you can see here. 33 00:02:01,690 --> 00:02:04,840 These structures range from simple cubic, 34 00:02:04,840 --> 00:02:06,760 where all the lattice vectors are equal 35 00:02:06,760 --> 00:02:09,610 and all the angles are 90 degrees 36 00:02:09,610 --> 00:02:12,280 to triclinic, where all the lattice vectors 37 00:02:12,280 --> 00:02:15,470 and angles are different. 38 00:02:15,470 --> 00:02:19,430 One thing to note about this hexagonal crystal system 39 00:02:19,430 --> 00:02:21,620 is it's actually made up of three unit 40 00:02:21,620 --> 00:02:25,820 cells, which you can see here. 41 00:02:25,820 --> 00:02:31,000 In 3D, there are also non-simple crystal systems. 42 00:02:31,000 --> 00:02:33,310 With the non-simple crystal systems added, 43 00:02:33,310 --> 00:02:37,000 there are 14 total possible crystal structures. 44 00:02:37,000 --> 00:02:42,370 For example, body centered cubic and face centered cubic 45 00:02:42,370 --> 00:02:44,455 are non-simple cubic crystals. 46 00:02:48,550 --> 00:02:51,090 Crystals exist in real space. 47 00:02:51,090 --> 00:02:53,670 This is the physical world around us. 48 00:02:53,670 --> 00:02:56,640 Unfortunately, not all concepts in materials science 49 00:02:56,640 --> 00:03:00,000 can be properly represented in real space. 50 00:03:00,000 --> 00:03:03,630 Reciprocal space is a non-physical space. 51 00:03:03,630 --> 00:03:05,610 In material science, it is mainly 52 00:03:05,610 --> 00:03:09,310 used to portray diffraction phenomena. 53 00:03:09,310 --> 00:03:11,710 For those of you who know, reciprocal space 54 00:03:11,710 --> 00:03:14,240 is the four-year transform of real space. 55 00:03:14,240 --> 00:03:16,720 However, this information is not important to understand 56 00:03:16,720 --> 00:03:19,310 the rest of the video. 57 00:03:19,310 --> 00:03:21,890 Reciprocal lattice vectors relate to sets 58 00:03:21,890 --> 00:03:24,950 of planes in real space. 59 00:03:24,950 --> 00:03:27,290 Reciprocal space has some key properties 60 00:03:27,290 --> 00:03:29,330 that related to real space. 61 00:03:29,330 --> 00:03:31,400 These properties include the units 62 00:03:31,400 --> 00:03:35,750 of reciprocal space or an inverse length. 63 00:03:35,750 --> 00:03:38,870 The volume of a reciprocal unit cell 64 00:03:38,870 --> 00:03:42,620 is inverse of the real space volume. 65 00:03:42,620 --> 00:03:45,350 And the plane spacing is inverted. 66 00:03:45,350 --> 00:03:47,390 See the attached notebook for more information 67 00:03:47,390 --> 00:03:52,400 about reciprocal space and how to calculate reciprocal lattice 68 00:03:52,400 --> 00:03:54,770 vectors. 69 00:03:54,770 --> 00:03:58,530 Consequently, a reciprocal space is a very important concept. 70 00:03:58,530 --> 00:04:00,350 For example, it is a key property 71 00:04:00,350 --> 00:04:02,810 in understanding X-ray diffraction. 72 00:04:02,810 --> 00:04:05,690 In X-ray diffraction, a beam of incident X-rays 73 00:04:05,690 --> 00:04:07,580 are diffracted in a material. 74 00:04:07,580 --> 00:04:10,170 And the diffraction angle is measured. 75 00:04:10,170 --> 00:04:14,450 This produces an electron diffraction pattern seen here. 76 00:04:14,450 --> 00:04:19,160 Here the electron diffraction pattern is of silicon carbide. 77 00:04:19,160 --> 00:04:21,890 Points in the pattern originate from a set 78 00:04:21,890 --> 00:04:23,780 of planes in the crystal. 79 00:04:23,780 --> 00:04:27,970 And each point represents a reciprocal lattice vector. 80 00:04:27,970 --> 00:04:30,930 Therefore, electron diffraction patterns 81 00:04:30,930 --> 00:04:32,475 exist in reciprocal space. 82 00:04:36,370 --> 00:04:38,710 Now we are going to compare unit cells 83 00:04:38,710 --> 00:04:42,220 in real and reciprocal space to better visualize the change 84 00:04:42,220 --> 00:04:44,770 from real to reciprocal space. 85 00:04:44,770 --> 00:04:49,630 As I said earlier, there are four types of 2D lattices. 86 00:04:49,630 --> 00:04:54,640 In this image, you can see the real space lattice. 87 00:04:54,640 --> 00:04:58,350 And in this image, you can see the reciprocal space lattice. 88 00:04:58,350 --> 00:05:02,950 In real space, there are lattice vectors a and b. 89 00:05:02,950 --> 00:05:06,730 And in reciprocal space, there are lattice vectors a star 90 00:05:06,730 --> 00:05:08,920 and b star, which are perpendicular 91 00:05:08,920 --> 00:05:11,840 to their real counterpart. 92 00:05:11,840 --> 00:05:15,580 As you can see here, a change in real space 93 00:05:15,580 --> 00:05:21,030 produces an inverse result in reciprocal space. 94 00:05:21,030 --> 00:05:27,260 As the real unit cell shrinks or expands, 95 00:05:27,260 --> 00:05:32,300 the reciprocal unit cell does the opposite. 96 00:05:32,300 --> 00:05:34,940 The same holds true as the angles change. 97 00:05:41,370 --> 00:05:46,160 In three dimensions, it is a little more complicated. 98 00:05:46,160 --> 00:05:48,950 As I said before, there are seven types 99 00:05:48,950 --> 00:05:50,105 of simple crystal systems. 100 00:05:52,770 --> 00:05:55,500 This image is the real space unit cell. 101 00:05:55,500 --> 00:05:58,740 And this image is the reciprocal space unit cell. 102 00:05:58,740 --> 00:06:01,350 Similar to two dimensions, the real lattice 103 00:06:01,350 --> 00:06:03,930 vectors are a, b, and c. 104 00:06:03,930 --> 00:06:05,430 And the reciprocal lattice vectors 105 00:06:05,430 --> 00:06:09,480 are a star, b star, and c star. 106 00:06:09,480 --> 00:06:11,920 The points at the vertices represents 107 00:06:11,920 --> 00:06:14,560 atoms in the simple unit cell. 108 00:06:19,570 --> 00:06:25,550 As you can see, as I change the size of the real unit cell, 109 00:06:25,550 --> 00:06:31,520 the volume between the two cells move in opposite directions. 110 00:06:31,520 --> 00:06:37,370 As the lattice vectors change, the change 111 00:06:37,370 --> 00:06:41,730 in shape and volume of the unit cells are clearly different. 112 00:06:41,730 --> 00:06:45,540 For the simplest example in cubic, 113 00:06:45,540 --> 00:06:50,360 if we shrink the unit cell in real space, 114 00:06:50,360 --> 00:06:55,910 it expands in reciprocal space. 115 00:06:55,910 --> 00:06:58,640 You can see this kind of behavior 116 00:06:58,640 --> 00:06:59,990 in any of the unit cells. 117 00:07:05,850 --> 00:07:08,100 Hopefully this information helps you 118 00:07:08,100 --> 00:07:10,470 to visualize reciprocal space. 119 00:07:10,470 --> 00:07:12,290 Thank you.