1
00:00:00,480 --> 00:00:04,830
Welcome back. It's now the 14th lecture of the condensed matter course.

2
00:00:05,640 --> 00:00:07,830
More or less, we've finished talking about scattering.

3
00:00:08,430 --> 00:00:14,430
There's a couple of little bits and pieces of scattering that I'm going to try to pick up later, maybe at the end of some other lectures.

4
00:00:14,700 --> 00:00:22,859
Instead, I'd like to move on and use what we've learned when we study scattering in order to go back and study other types of waves in crystals,

5
00:00:22,860 --> 00:00:28,290
the type of waves we were interested in originally where electron waves and vibrational waves are phonons.

6
00:00:28,560 --> 00:00:32,430
So let me remind you a few things we had learned about electron waves in one dimension,

7
00:00:32,580 --> 00:00:36,780
and then we're going to try to generalise up to higher dimensions from what we've learned.

8
00:00:37,770 --> 00:00:48,870
So let's recall the type finding model type, binding model for electrons, binding model in one day, in one day.

9
00:00:48,870 --> 00:00:55,440
And we're going to consider the case of two orbitals with two orbitals per unit cell orbits per unit cell.

10
00:00:57,780 --> 00:01:06,210
And that can be two atoms, one orbital in each atom, or it could be two orbitals on the same atom, one atom per unit cell.

11
00:01:06,690 --> 00:01:10,530
And we can, we can draw out the dispersion in reduced zone scheme.

12
00:01:10,530 --> 00:01:21,780
Here's K, here's energy, here's pi over a, here's minus pi over a that's the barren zone, the big green zone.

13
00:01:22,170 --> 00:01:31,560
We can write, draw the dispersion kind of like this and like this two modes at each wave vector K in the bronze zone.

14
00:01:31,950 --> 00:01:36,840
Or we can unfold that picture in the extended zone scheme.

15
00:01:38,740 --> 00:01:53,290
Looking like this. Here's K, here's pi over a year minus pi over a year to pi over A and here's minus two pi over a.

16
00:01:55,270 --> 00:02:02,950
So we have the same picture for the first mode and then for the second mode, we're going to shift things over, shift the thing on the left,

17
00:02:02,950 --> 00:02:06,609
over to the right by two pi over eight to make it look like this and shift the

18
00:02:06,610 --> 00:02:12,130
piece on the right over by two pi over a to the left to make it look like this.

19
00:02:13,120 --> 00:02:16,269
So we have what we called the first boron zone here.

20
00:02:16,270 --> 00:02:27,310
First we're on zone busy here, and then these pieces here where the second bronze zone, second bronze zone.

21
00:02:30,660 --> 00:02:33,540
And so it's useful. There's chemicals there right in the middle.

22
00:02:33,840 --> 00:02:38,640
It's useful to try to generalise this concept to three dimensions or higher dimensions.

23
00:02:39,060 --> 00:02:48,480
So let's let's do that. So generally, the Britain zone for one zone, since we're thinking about electrons in three dimensions,

24
00:02:48,900 --> 00:02:55,500
is going to be always defined as a unit cell in reciprocal space.

25
00:03:00,620 --> 00:03:05,750
And we're going to have to figure out what we mean by first bronze owned, second bronze on and so forth.

26
00:03:06,170 --> 00:03:13,820
So it's general and bronze on general, general and bronze zone.

27
00:03:14,690 --> 00:03:24,140
We define as follows. So we'll start with the first bronze on points in space.

28
00:03:27,950 --> 00:03:31,220
That are closer to. That are. Closer to.

29
00:03:32,120 --> 00:03:41,250
That are. Closer to. K equals zero, k equals zero.

30
00:03:42,040 --> 00:04:03,950
Then do any other any other reciprocal lattice point receptor lat point are the first boron zone are the first boron zone first B's.

31
00:04:05,020 --> 00:04:14,860
So all these points in the first boron zone are closer to the point of k equals zero, then to the other reciprocal lattice points.

32
00:04:14,860 --> 00:04:17,680
Here's the reciprocal last point here. At two minus two pi over eight,

33
00:04:17,710 --> 00:04:23,890
here's another reciprocal lattice point here at two pi over a and everything between minus pi over and pi over a

34
00:04:24,070 --> 00:04:29,410
constitutes the first boron zone because it's closer to K equals zero than it is to any other reciprocal lattice point.

35
00:04:29,800 --> 00:04:36,250
This definition might look a little familiar to you because in fact it's the same definition as the big nearside cell.

36
00:04:36,580 --> 00:04:40,660
So this equals the big inner site cell sites.

37
00:04:41,350 --> 00:04:46,330
Cell of the cake was zero point of k equals zero.

38
00:04:46,840 --> 00:04:49,960
In reciprocal lattice. In reciprocal lattice.

39
00:04:52,840 --> 00:04:59,469
Does that sound familiar? I hope all points that are closer to a given reciprocal lattice point than to any all points that are

40
00:04:59,470 --> 00:05:04,000
closer to a given lattice point than to any other lattice point constitute its bigger site cell.

41
00:05:04,240 --> 00:05:07,569
So all points that are closer to K equals zero than to any other reciprocal lattice

42
00:05:07,570 --> 00:05:12,850
point constitute the vendor site cell of the cake or zero point in reciprocal space.

43
00:05:13,000 --> 00:05:17,260
Good. Yes. Okay, good. So how about the second foreign zone?

44
00:05:18,400 --> 00:05:26,379
Points where k equals zero is the second close.

45
00:05:26,380 --> 00:05:32,760
This is the second closest. Recep lat point.

46
00:05:35,220 --> 00:05:38,760
What point are the second brands on?

47
00:05:38,880 --> 00:05:42,700
Are the. Second.

48
00:05:44,980 --> 00:05:50,050
Second to B.C. So, for example, if we pick pick a point here.

49
00:05:50,800 --> 00:05:56,020
X call point X, the first closest reciprocal lattice point to this point X is here.

50
00:05:56,350 --> 00:06:00,400
The second closest reciprocal last point to the point x is equal zero.

51
00:06:00,610 --> 00:06:04,180
Therefore, this point x is in the second burn zone. Good.

52
00:06:04,870 --> 00:06:09,790
Okay, that's the definition. So a couple of notes about these definitions.

53
00:06:10,030 --> 00:06:14,140
First of all, one bronze zone boundary.

54
00:06:17,360 --> 00:06:23,390
Two opposite to opposite opposite busy boundary.

55
00:06:27,000 --> 00:06:31,230
Is a reciprocal lattice factor is a reciprocal factor.

56
00:06:31,860 --> 00:06:35,040
Sep lad back.

57
00:06:35,310 --> 00:06:41,670
Let's see. So here's a bronze. On boundary is the boundary between the first bronze on the second bronze zone and minus pi over a.

58
00:06:41,730 --> 00:06:45,070
Here's another one. It's the opposite one from minus pile raised plus pile ray.

59
00:06:45,510 --> 00:06:51,750
Distance between them is two pi over a, which is indeed a reciprocal lattice vector.

60
00:06:51,930 --> 00:06:57,090
Similarly, here's a boundary between the second Barone zone and out here actually would be the third bronze zone.

61
00:06:57,510 --> 00:07:05,819
And so it's a bronze zone boundary, the distance from it to the opposite Barone zone boundary over here at plus

62
00:07:05,820 --> 00:07:10,830
two pi over a is four pi over A and that is also a reciprocal lattice factor.

63
00:07:10,830 --> 00:07:21,060
That is generally a true statement. Okay. There's a similar so one, a sort of caveat, a corollary of this is that these boundaries.

64
00:07:24,870 --> 00:07:28,440
Or maybe, maybe it's not a corollary, but it's almost equivalent statement.

65
00:07:28,740 --> 00:07:32,100
These boundaries are points.

66
00:07:33,600 --> 00:07:47,030
Where? Absolute value of K equals absolute value of k plus g for some g for some reciprocal lattice vector g.

67
00:07:48,680 --> 00:08:00,320
Okay. So for example, you need, you need to have K and K plus G and see what's K and k k posted having the same magnitude for,

68
00:08:00,560 --> 00:08:02,660
for G being some reciprocal lattice vector.

69
00:08:02,660 --> 00:08:09,800
So if you take minus pi over a, you add a reciprocal lattice vector to pi over eight to it you get plus pi over a those have the same magnitude.

70
00:08:09,980 --> 00:08:15,860
So you're on the bronze on boundary. Similarly, you take minus two pi over a at a reciprocal lattice vector to it.

71
00:08:15,860 --> 00:08:19,460
For pi over eight you get two plus two pi over a that has the same magnitude.

72
00:08:19,700 --> 00:08:26,120
So you're on a bronze on boundary. The second statement, which is important about bronze zones,

73
00:08:26,450 --> 00:08:41,570
is each DC bronze zone has the same the same area or volume in three dimensions, same area and and represents.

74
00:08:45,600 --> 00:08:50,280
Represents each k point. Each crystal momentum.

75
00:08:50,970 --> 00:08:56,790
Each crystal wave vector, I guess. Wave vector k once.

76
00:08:59,140 --> 00:09:02,950
That was our general idea of constructing the bronze zone in the first place,

77
00:09:03,190 --> 00:09:08,200
that every physically different wave is represented exactly once in the bronze zone.

78
00:09:08,380 --> 00:09:11,140
The second bronze zone is equivalent to the first bronze zone.

79
00:09:11,140 --> 00:09:20,020
We've just removed the pieces around into different places, but each wave vector K is represented exactly once inside each bronze zone.

80
00:09:20,680 --> 00:09:24,459
Incidentally, is it obvious from the second bronze zone that you would define?

81
00:09:24,460 --> 00:09:32,890
The third bronze zone analogously points where the archaic zero is the third closest reciprocal at this point would be the third bronze zone.

82
00:09:32,890 --> 00:09:35,710
So you have the first, second, third. You can have as many bronze zones as you want.

83
00:09:36,730 --> 00:09:45,940
The final the final statement, which is useful, I think this is a homework assignment on the third homework set that the number of K states,

84
00:09:45,940 --> 00:09:56,470
number of K states, we even prove this in lecture in each busy B.C. equals the number of units cells.

85
00:09:57,280 --> 00:10:01,690
Unit cells. In system. In system.

86
00:10:03,860 --> 00:10:09,140
Does that sound familiar? We prove this in one dimension. I think for homework, you're supposed to prove it in three dimensions.

87
00:10:09,410 --> 00:10:17,390
It is also generally a true statement. So actually, let me do an example of this in two dimensions to make this more clear.

88
00:10:18,110 --> 00:10:21,200
So this we can start with a square lattice and real space.

89
00:10:21,410 --> 00:10:27,260
The reciprocal lattice of a square lattice is also a square lattice. So I've drawn a square last is supposed to be the reciprocal lattice.

90
00:10:27,260 --> 00:10:32,209
So it's a square lattice. Again, I've coloured some of the points differently just to make them easy to talk about.

91
00:10:32,210 --> 00:10:37,970
The point of k equals zero. I've coloured black in the centre and so first we're going to try to construct the

92
00:10:37,970 --> 00:10:43,280
first boron zone that is the inner site cell of this black point in the centre.

93
00:10:43,280 --> 00:10:44,599
So how do you construct the bigger site?

94
00:10:44,600 --> 00:10:51,770
So you put down perpendicular by sectors to the nearest points, perpendicular bisected to the red points here we'll make this point.

95
00:10:51,770 --> 00:10:56,210
That's a pretty good bisect here. Then this this plane, this perpendicular by sector,

96
00:10:56,450 --> 00:11:03,950
these two also giving all the perpendicular bisect is to the red points and then you colour in the area that is the first boron zone.

97
00:11:04,760 --> 00:11:12,229
Okay. Now the distance across the first boron zone is two pi over a which is a reciprocal lattice vector the way it's supposed to be.

98
00:11:12,230 --> 00:11:18,230
That distance there from perpendicular, bisected or perpendicular by sector is a reciprocal lattice vector.

99
00:11:18,590 --> 00:11:21,829
Now let's look for the second brian zone.

100
00:11:21,830 --> 00:11:26,990
How do we do that? It's a very similar construction. You just start putting down more perpendicular by sectors.

101
00:11:27,260 --> 00:11:30,829
So here the second closest sets of points are these blue points here.

102
00:11:30,830 --> 00:11:35,959
So let's put down perpendicular bisected to those like this, like this, like this, like this.

103
00:11:35,960 --> 00:11:44,750
So these are the perpendicular bisected to the blue points out here. We'll colour in the walled off area blue and this is now the second branch zone.

104
00:11:44,960 --> 00:11:47,030
Now it's this check that affects the definition.

105
00:11:47,210 --> 00:11:54,800
If you take a point like here in the second Barone zone, the first closest reciprocal lattice point is this red point here.

106
00:11:55,130 --> 00:11:58,610
The second closest reciprocal lattice point is k equals zero.

107
00:11:58,880 --> 00:12:04,430
Therefore, this point here somewhere in this blue region is in the second Brian zone.

108
00:12:04,730 --> 00:12:10,610
Good. Okay. Now it may not be obvious that the distance across the bronze zone from brown

109
00:12:10,610 --> 00:12:13,670
zone boundaries to bronze on boundary here is a reciprocal lattice vector,

110
00:12:13,940 --> 00:12:18,200
but I'll show you that it is this is a reciprocal lattice vector from zero to the blue point.

111
00:12:18,410 --> 00:12:27,170
You just shift it like that. It's indeed the distance across from beyond zone boundary to bronze on boundary so we can keep going on looking for the

112
00:12:27,200 --> 00:12:33,470
third baron zone and to construct the third bronze zone we have to take put perpendicular by sectors to these green points.

113
00:12:33,680 --> 00:12:40,880
So let's do that, put down those green lines which are perpendicular, bisected to the green points and you fill in the area now green.

114
00:12:41,120 --> 00:12:43,279
And let's check that it satisfies the definition.

115
00:12:43,280 --> 00:12:49,490
If you pick a point here in the green area, the first closest reciprocal lattice point is this red point here.

116
00:12:49,700 --> 00:12:55,700
The second closest is the blue point out here. The third closest is the cable zero point in the centre.

117
00:12:55,700 --> 00:12:57,140
So this is in the third Brian zone.

118
00:12:57,560 --> 00:13:07,310
Now also the distance from Brown Zone boundary to brown zone boundary across the system is four pi over a, which is also a reciprocal lattice vector.

119
00:13:07,670 --> 00:13:12,950
Now, another thing that you've been promised from these definitions is that each Barone zone has the same area,

120
00:13:13,240 --> 00:13:18,650
looks like the kind of might and that they should each represent each k point once.

121
00:13:18,890 --> 00:13:22,640
So the bronze zones represent every possible point. Exactly.

122
00:13:22,880 --> 00:13:28,910
Wants to see this. What we're going to do is we're going to take the pieces of this puzzle and with some very crude animation,

123
00:13:28,910 --> 00:13:33,710
we're going to move them by reciprocal lattice vectors and show that they're actually the same shape.

124
00:13:34,010 --> 00:13:35,240
So let's see how that works.

125
00:13:35,600 --> 00:13:41,959
So I'm going to take this red section, the first boron zone, and I'm going to get that exact same shape of the first bronze zone,

126
00:13:41,960 --> 00:13:47,510
same area, same shape, exactly by moving these pieces of the second bronze zone over.

127
00:13:47,750 --> 00:13:53,060
So this piece, I'm going to move by a reciprocal lattice vector like that, two pi over a over.

128
00:13:53,300 --> 00:13:57,680
Then I'll take this piece, I'll move it to pi over a over this way and I'll take the top piece,

129
00:13:57,680 --> 00:14:01,370
move it to pi over eight down and the other one to pi over a up.

130
00:14:01,610 --> 00:14:05,300
And it fills exactly the first boron zone, so. Well, okay.

131
00:14:05,390 --> 00:14:09,170
My animation skills are not not perfect, which you see, the other one is even worse.

132
00:14:10,220 --> 00:14:15,440
But. But you get the picture that it's supposed to fill the first bronze on exactly one side we by shifting things

133
00:14:15,650 --> 00:14:20,790
by two pi over a we actually didn't change the waves at all because if you shift by two pi over eight,

134
00:14:20,810 --> 00:14:25,310
you're getting back exactly the same crystal momentum, the same physical wave by doing this.

135
00:14:26,180 --> 00:14:30,979
So the second bronze zone is effectively exactly equivalent to the first bronze zone.

136
00:14:30,980 --> 00:14:37,280
Now let's do the same thing for the third bronze zone. We'll shift this over by two pi over a this over by t by over a okay.

137
00:14:37,280 --> 00:14:44,030
Now it's really going to get bad, but but you get the idea that it's supposed to is supposed to fit perfectly.

138
00:14:44,330 --> 00:14:48,050
And had I been better with my animation skills, it would have fit perfectly.

139
00:14:48,380 --> 00:14:52,550
Okay. So that's the general general idea of O'Brien's answers.

140
00:14:52,550 --> 00:14:54,890
A couple facts about Brian Jones in 3D.

141
00:14:55,370 --> 00:15:02,600
So in 3D, the lattices that we're interested in right down a list of the lattices we're interested in are mainly in.

142
00:15:03,420 --> 00:15:07,350
And we have simple cubic, simple cubic.

143
00:15:07,770 --> 00:15:15,059
We have BTC and we have FCC. So the corresponding reciprocal lattice is very simple.

144
00:15:15,060 --> 00:15:22,290
That is well, a simple cubic lattice has has a reciprocal lattice which is also simple cubic.

145
00:15:24,060 --> 00:15:28,740
The VC lattice. This is something I mentioned and I think it's a revision homework problem.

146
00:15:29,310 --> 00:15:36,300
The reciprocal lattice of a BC lattice is FCC and the reciprocal lattice of an FCC lattice is C.

147
00:15:36,630 --> 00:15:44,700
So the shape of the first Barone zone, first being B.C. shape is well for a simple cubic lattice.

148
00:15:44,700 --> 00:15:51,180
The Fichtner site cell is a cube. So the first Barone zone for the simple cubic lattice is a cube.

149
00:15:51,600 --> 00:15:55,230
For the BC lattice, the reciprocal lattice is FCC.

150
00:15:55,470 --> 00:16:02,070
So we get something that's the shape of the bigger site cell for FCC, which was a funny shape.

151
00:16:02,070 --> 00:16:08,820
It was a truncated octahedron. I'll show your picture in a second. And for FCC lattice, the reciprocal lattice goes back.

152
00:16:09,240 --> 00:16:15,960
And so you get the bigger site cell for BBC here.

153
00:16:16,890 --> 00:16:20,280
Actually, I said this. This one is the truncated octahedron.

154
00:16:21,030 --> 00:16:24,780
Now this one is a truncated octahedron. And this is the let me just show it on the slide.

155
00:16:25,260 --> 00:16:28,319
Okay. So this is the first Bruns on an FCC lattice.

156
00:16:28,320 --> 00:16:34,440
This one's a chunk. This one is a truncated octahedron. This one is the truncated the dodecahedron.

157
00:16:34,650 --> 00:16:43,320
This one is 12 sides. This one has 14 sides. All right. Whatever. These are the shapes of the of the of the bronze zone.

158
00:16:43,320 --> 00:16:48,209
So in this case, the FCC lattice has a reciprocal lattice, which is boxy.

159
00:16:48,210 --> 00:16:49,500
And this is the Vicknair site.

160
00:16:49,500 --> 00:16:57,450
So the BC lattice, and this is the bigger site cell of the SCC lattice, which is also the first boron zone of the BC lattice.

161
00:16:57,450 --> 00:17:02,370
Now you'll notice that these pictures are drawn in K, space, K, X, Y and Z.

162
00:17:02,580 --> 00:17:07,290
And you'll also notice that there are some letters dropped on to these pictures.

163
00:17:07,590 --> 00:17:14,730
These letters are there because people typically, when they describe points on in the bronze zone, they usually say,

164
00:17:15,000 --> 00:17:21,960
we're talking about the Gamma Point or the X point or the L point, and that's just shorthand nomenclature that you don't need to know.

165
00:17:22,200 --> 00:17:26,999
But it's worth knowing that they have these when they say X point, they mean some particular point in the bronze zone.

166
00:17:27,000 --> 00:17:30,150
So the point in the middle of a square face is called the X Point.

167
00:17:30,390 --> 00:17:35,400
This point, this point at this point, the point in the middle of one of the hexagonal faces is called the L point.

168
00:17:35,640 --> 00:17:39,300
The point in the centre is always called gamma. You don't need to know these things.

169
00:17:40,200 --> 00:17:47,669
All right. So now that we we've we have these pictures of the bronze zone for, you know,

170
00:17:47,670 --> 00:17:51,600
for our various lattices, we can start thinking about the excitation spectra,

171
00:17:52,110 --> 00:17:58,290
either the electronic expectation spectra or the phone on excitation spectra for various different crystals.

172
00:17:59,160 --> 00:18:03,209
And so everywhere in the bronze zone there will be different like in the reduced zone scheme,

173
00:18:03,210 --> 00:18:09,420
you'll have different excitation modes at each possible value of K in the in the bronze zone.

174
00:18:09,720 --> 00:18:16,380
So let's, let's do an example being this Valentine's Day happy Valentine's Day everyone is we're going to do diamond

175
00:18:16,740 --> 00:18:24,060
for s all right so well we do diamond every year so just happens to be a Valentine's Day this year.

176
00:18:25,260 --> 00:18:34,050
So Diamond, you'll recall, is an FCC lattice with two atom basis, one carbon and 000 and one carbon at one quarter, one quarter, one quarter.

177
00:18:34,260 --> 00:18:37,740
This is the shape of the bronze zone as we had in the previous slide.

178
00:18:37,860 --> 00:18:45,630
Now the first question and opportunities, wind, chocolate, how many phonon modes should there be at each k vector in the first boron zone?

179
00:18:47,310 --> 00:18:54,550
How many phone unloads? No, no, no, no.

180
00:18:56,340 --> 00:19:00,060
Oh, my gosh. So there's there's two atoms in the.

181
00:19:00,180 --> 00:19:03,930
I'm going to eat this one myself. All right. I'm going to gain weight this year.

182
00:19:04,890 --> 00:19:10,500
Now, there's two atoms in the primitive unit, so each atom can move in three possible directions.

183
00:19:10,680 --> 00:19:15,800
So that means you have six modes. Yeah. So. Let's see if we have a picture of it.

184
00:19:15,810 --> 00:19:19,010
Yeah. So here's here's a picture of the diamond phonon spectrum.

185
00:19:19,880 --> 00:19:26,060
So the way it's drawn is that you take cuts through the barren zone along certain directions that you're interested in.

186
00:19:26,270 --> 00:19:33,980
So, for example, this line. So here's the frequency vertical and then the K vector is horizontal, but this given to you in a cut.

187
00:19:34,190 --> 00:19:41,240
So this is cutting from Gamma to X along the 100 direction gamma two X along 100 direction along this dotted line here.

188
00:19:41,600 --> 00:19:48,680
And then from here to here is Gamma two l along the one, one, one direction from here to here, along that dotted line.

189
00:19:49,070 --> 00:19:53,090
This one is a little more complicated. It's X to gamma along the 110 direction.

190
00:19:53,300 --> 00:20:00,200
So that actually goes from this X through this K into the gamma in the centre of the next brown zone over.

191
00:20:00,440 --> 00:20:07,429
So that was a little complicated, but. All right. Now you'll notice that if you pick a point in the centre here, you will count exactly six modes.

192
00:20:07,430 --> 00:20:11,690
One, two, three, four, five, six. If you count over here, you'll discover there's only four modes.

193
00:20:12,440 --> 00:20:18,799
Why is that? Well, the reason is because two of them are degenerate. And you'll see these two modes coming in to this point here.

194
00:20:18,800 --> 00:20:21,890
And then they have the same energy all the way down from here to here.

195
00:20:22,070 --> 00:20:27,590
And then two modes here are degenerate up here, and they have the same energy or same frequency all the way over here.

196
00:20:27,710 --> 00:20:33,590
And the reason for the degeneracy is because you're looking along a particularly symmetric direction where

197
00:20:33,590 --> 00:20:38,180
vibrations transverse in this direction and vibrations are transverse in that direction look identical.

198
00:20:38,420 --> 00:20:42,290
Whereas along this direction, it's a very symmetric direction.

199
00:20:42,290 --> 00:20:45,859
So all directions of vibration actually have different energies.

200
00:20:45,860 --> 00:20:50,089
You can see a couple of interesting things in this. In this figure, the gamma point is K equals zero.

201
00:20:50,090 --> 00:20:55,640
You can see the acoustic modes coming down to 22k equals zero linearly.

202
00:20:55,850 --> 00:20:59,360
And these are the optical modes up here at at high frequency.

203
00:21:00,080 --> 00:21:04,280
And one of these modes is longitudinal. Two of them are transverse.

204
00:21:04,580 --> 00:21:09,050
Now we can ask the same questions about an electronic excitation spectrum.

205
00:21:09,410 --> 00:21:14,059
And you when you calculate an electronic excitation spectrum in this type binding approximation,

206
00:21:14,060 --> 00:21:20,080
you have to decide how many orbitals you're going to consider on each on each atom.

207
00:21:20,090 --> 00:21:24,320
So for example, in that picture up there, we considered two orbitals per unit cell.

208
00:21:24,680 --> 00:21:32,659
So given that a typical type of calculation for carbon, you don't want to keep all the orbitals out to infinity.

209
00:21:32,660 --> 00:21:35,690
So generally you just keep the orbitals you're interested in.

210
00:21:36,050 --> 00:21:44,750
What is usually interested in is the orbitals where there's some sort of interesting action going on that the, the one, the one orbitals.

211
00:21:44,750 --> 00:21:48,110
There was a core orbitals that just completely filled. Nothing interesting going on there.

212
00:21:48,380 --> 00:21:51,800
The orbitals that are three three has three p and so forth.

213
00:21:51,800 --> 00:21:57,740
Those are just completely empty, totally uninteresting. The ones that are interesting are the ones they're sort of partially filled, partially empty.

214
00:21:57,740 --> 00:22:04,280
Those are two S and the 32p orbitals. So given that we're considering the three S and and the 32p orbitals,

215
00:22:04,730 --> 00:22:09,800
how many excitation modes eigenvalues should we find at each k point in the bronze zone?

216
00:22:11,580 --> 00:22:16,560
One more chance. We're considering four orbitals per atom.

217
00:22:18,730 --> 00:22:23,690
What was the answer? Eight. Did you say? Yeah. Okay, so almost.

218
00:22:23,730 --> 00:22:27,690
I got that. All right. Eight. Yeah. So there's two orbitals in your account.

219
00:22:27,690 --> 00:22:32,219
What you need to do is you need to count the total number of orbitals per primitive unit

220
00:22:32,220 --> 00:22:38,550
cell and that will give you the total number of excitation modes per per k vector.

221
00:22:38,760 --> 00:22:43,500
Now this diagram, this is an excitation spectrum, an electronic excitation spectrum of diamond.

222
00:22:43,770 --> 00:22:46,860
These things are frequently known as spaghetti diagrams for obvious reasons,

223
00:22:46,860 --> 00:22:50,490
because it looks like some sort of complicated ball of spaghetti that you can't make any sense of.

224
00:22:50,820 --> 00:22:55,860
But but it's actually it's fairly simple. Well, not that not that complicated.

225
00:22:56,130 --> 00:23:03,030
What's again, you're looking at cuts through the barren zone. So this is a line between the L and Gamma Point along this lot dotted line.

226
00:23:03,270 --> 00:23:08,220
This is a line from Gamma to X along this dotted line, then X to K, K to gamma.

227
00:23:08,610 --> 00:23:10,740
So it's basically just taking cuts to the branch zone.

228
00:23:10,980 --> 00:23:19,139
And at each K point, there are eight modes one, two, three, four, five, six, seven, eight and some in some directions.

229
00:23:19,140 --> 00:23:22,830
It looks like there's only six. Let's see, for example, this direction.

230
00:23:22,830 --> 00:23:26,820
One, two, three, four, five, six. Because two of them have become degenerate here.

231
00:23:27,120 --> 00:23:31,800
See, these two came together. Oh, no, hang on. These two came together here to make only one.

232
00:23:31,950 --> 00:23:35,490
That's because are actually two modes with the same energy. So you have to be a little bit careful about that.

233
00:23:38,250 --> 00:23:48,330
Four Now carbon has a valence of four, which means each carbon donates four electrons and that means with two carbons per primitive unit cell,

234
00:23:48,480 --> 00:23:53,700
we have donated eight electrons total. So you fill the bottom four bands with both spins.

235
00:23:53,940 --> 00:23:57,690
So the bottom four bands are filled and there's a gap in the upper four bands.

236
00:23:57,900 --> 00:24:00,510
And in fact, as we'll discuss in a couple of lectures later on,

237
00:24:00,660 --> 00:24:06,840
that means what we have is we have a really good insulator filled bands, then a big gap and then a bunch of empty bands.

238
00:24:06,960 --> 00:24:13,380
It's also what makes Diamond nice and transparent, so it makes Diamond beautiful, appropriately enough, for Valentine's Day.

239
00:24:13,380 --> 00:24:16,680
All right. So let's move on from here.

240
00:24:16,680 --> 00:24:18,720
I think that's enough of diamond for today.

241
00:24:20,130 --> 00:24:26,460
So one of the things that's fairly important about this, this story is that with our tight binding spectra,

242
00:24:26,580 --> 00:24:31,170
we always have the principle that if you take K to some K plus G,

243
00:24:31,470 --> 00:24:37,740
you get back exactly the same wave and you have the same principle for vibrations as well.

244
00:24:38,010 --> 00:24:45,840
Now, you might ask whether this are the principle that if when you shift K by a reciprocal lattice vector,

245
00:24:46,020 --> 00:24:51,989
whether that is actually special to the tight binding approximation we use or whether this is actually a general statement,

246
00:24:51,990 --> 00:25:01,020
let's remind ourselves why it is that we found this for an exit, for r for a tight binding model, basically the tight binding wave function.

247
00:25:01,110 --> 00:25:11,919
This is a slightly different language than I wrote it before. But you can you can write it as either the I KKR times electrons sitting on

248
00:25:11,920 --> 00:25:18,720
lattice point R is this we used to call this thing phi and we gave this a no.

249
00:25:18,750 --> 00:25:23,309
I think it's more or less I guess the difference is now we're doing it in three dimensions rather than one dimension.

250
00:25:23,310 --> 00:25:30,270
That's why it looks a little different. But this is basically the type binding wave function that we wrote down before and

251
00:25:30,270 --> 00:25:35,970
this thing is obviously invariant undertaking K to K plus g because either the I k,

252
00:25:36,300 --> 00:25:44,340
either the Igot R when you shift k by g g that R equals one because g is in the reciprocal lattice and R is in the direct lattice.

253
00:25:44,340 --> 00:25:47,820
So if cake is shifted by g, it doesn't change this factor here and you get back the same wave.

254
00:25:48,210 --> 00:25:54,480
So but you might ask whether this is just a property of the tight binding model or if this is actually something that's general.

255
00:25:54,480 --> 00:26:02,459
And this is actually a really important question and it's a question that came up very shortly after the discovery of the shortening equation scoring.

256
00:26:02,460 --> 00:26:08,280
I discovered his equation in 1926, in 1928, that people were already worried about this issue.

257
00:26:08,550 --> 00:26:14,880
And it was resolved very famously by Bloch, Felix Bloch, and what's known as Bloch's Theorem.

258
00:26:18,390 --> 00:26:23,340
Block's theorem. All right. Okay. Which is frequently.

259
00:26:24,490 --> 00:26:28,510
Referred to as the underpinning of all of material science.

260
00:26:28,720 --> 00:26:31,120
It's an underpinning of all of semiconductor physics.

261
00:26:31,330 --> 00:26:38,050
It's an extremely important principle which may not seem so important until we talk about some of its implications.

262
00:26:38,350 --> 00:26:42,670
But let me state the theorem first, then we'll prove the theorem. Then we'll talk about some of the implications.

263
00:26:43,120 --> 00:26:47,800
So the statement is an electron in a periodic potential.

264
00:26:48,160 --> 00:26:51,729
In periodic potential potential.

265
00:26:51,730 --> 00:26:53,950
Periodic vrr, I guess.

266
00:26:53,980 --> 00:27:04,000
Periodic means V of our equals vrr plus v of capital r with capital being a lot of spectre has eigen states of the form as eigen states.

267
00:27:07,190 --> 00:27:14,070
Of the form. Okay.

268
00:27:14,230 --> 00:27:24,370
There's a lot of symbols here for a second size sub K of R equals e to the i k

269
00:27:24,370 --> 00:27:37,540
dot are times you superscript alpha some k of our where where you is periodic.

270
00:27:41,070 --> 00:27:51,150
So it's the same if you shift it from you, you've r equals your R plus r and ks.

271
00:27:51,300 --> 00:28:03,460
And KS. Can be chosen in the first bronze zone, can be chosen in first bronze zone.

272
00:28:07,030 --> 00:28:10,780
In other words, K is crystal momentum. K is crystal wave.

273
00:28:10,780 --> 00:28:24,530
Vector. Crystal wave vector. This form is known as this form of SCI.

274
00:28:27,800 --> 00:28:37,280
OSSI is known as a modified plain wave is known as as a modified plain wave.

275
00:28:38,180 --> 00:28:43,430
A modified plain wave. Modified plain wave.

276
00:28:47,390 --> 00:28:55,700
That that should be fairly obvious. It's a plane wave, the iCar, and it's modified by being multiplied by a periodic function.

277
00:28:55,700 --> 00:28:58,700
You. That's the modification of the plane wave.

278
00:29:00,620 --> 00:29:04,820
The index. Alpha. Alpha is the band index.

279
00:29:08,930 --> 00:29:11,120
And the band index is meant to describe.

280
00:29:11,770 --> 00:29:18,200
You know, you get a different wave function if you're talking about this point here or this point here at the same k vector,

281
00:29:18,200 --> 00:29:21,320
at the same K vector, you can have more than one eigen state.

282
00:29:21,740 --> 00:29:43,160
So the energy's eigen energies. R e superscript alpha of k so e one and e two depending on which which band we're talking about,

283
00:29:43,160 --> 00:29:47,660
which which which energy level, the lowest or the next lowest or the next next lowest and so forth.

284
00:29:48,200 --> 00:29:52,610
Okay. So this is a pretty big bunch of statements we've made here,

285
00:29:52,910 --> 00:29:58,940
but it's fairly important because it tells you about the general structure of iron states in a periodic potential.

286
00:29:58,940 --> 00:30:04,190
So let's first prove this theorem and then we'll talk about why this theorem is so important

287
00:30:04,190 --> 00:30:09,200
and why it underlies so much of material science and semiconductor physics and so forth.

288
00:30:09,590 --> 00:30:17,150
So first, okay, so this is sort of a quasi proof. It's actually fairly rigorous, but we'll we'll not fill in the details.

289
00:30:17,600 --> 00:30:19,470
But the quasi proof goes like like this.

290
00:30:19,490 --> 00:30:27,020
First we'll consider a Hamiltonian, which is h, not just the free electron Hamiltonian plus or potential V, which is periodic.

291
00:30:27,020 --> 00:30:34,190
So V is periodic and H not is the usual p squared over two m.

292
00:30:37,170 --> 00:30:41,280
Now, if we throw away the periodic potential, we would know the iron states.

293
00:30:41,310 --> 00:30:42,810
Those are just plain wave eigen states.

294
00:30:42,810 --> 00:30:55,440
Plane waves are e not k k where enought you know of k is just the squared k squared over two m we usually have.

295
00:30:57,420 --> 00:31:01,800
So so far so good. We know we have the plane waves, then we have to add the periodic potential.

296
00:31:02,220 --> 00:31:12,960
Now, what can the periodic potential do? The periodic potential can scatter you from some K to some K prime via a matrix element of this form.

297
00:31:13,320 --> 00:31:20,040
Now we know a lot about matrix elements of this form. From what we learned in scattering theory, this thing here,

298
00:31:20,040 --> 00:31:24,779
this matrix element from a plane wave K to a plane wave k prime via a periodic potential

299
00:31:24,780 --> 00:31:32,670
v is the for you transform for a transform a vrr evaluated at k minus k prime.

300
00:31:34,590 --> 00:31:38,940
And that is the following it is zero.

301
00:31:39,960 --> 00:31:54,090
If K minus K prime came out as k prime is not a reciprocal lattice vector is not equal to g and we'll call it v sub g, f k minus k prime equals g.

302
00:31:54,490 --> 00:31:57,900
I mean it's called k prime minus cable's g that.

303
00:31:59,210 --> 00:32:00,860
And this is just Allawi's condition.

304
00:32:02,200 --> 00:32:09,690
You can scatter from a wave vector K to a wave vector k prime only if the difference between and K prime is a reciprocal lattice vector.

305
00:32:10,200 --> 00:32:14,130
We learned this when we studied scattering of X-rays and neutrons and so forth.

306
00:32:14,610 --> 00:32:19,440
So this gets us almost to the proof. So we can almost conclude the result here.

307
00:32:19,650 --> 00:32:27,150
Why? Well, the point here is that the Hamiltonian with the scattering potential is blocked diagonal.

308
00:32:27,490 --> 00:32:31,680
No, no pun intended. BLOCK, diagonal.

309
00:32:31,680 --> 00:32:39,870
BLOCK of the K diagonal. What I mean by this is that you can scatter from K to a k prime if k k prime are

310
00:32:39,870 --> 00:32:44,160
separated by G and then from k prime you can scatter to another k double prime.

311
00:32:44,340 --> 00:32:49,170
If those are separated by a g and you can scatter to another k prime and so forth and so on and so forth.

312
00:32:49,470 --> 00:32:51,630
But you can never change the crystal momentum.

313
00:32:51,840 --> 00:32:59,790
All of the ks that you can get to by scattering by the via the periodic potential, all of them differ from each other by reciprocal lattice vectors.

314
00:33:00,030 --> 00:33:04,290
So the Hamiltonian breaks up and if you go into these sectors, if you like.

315
00:33:04,920 --> 00:33:10,020
And so the wave function must be of the form, cy k must be of the form,

316
00:33:11,130 --> 00:33:22,800
some over g of some coefficients of column AMG plus k times either the i k plus g dotted with are.

317
00:33:23,280 --> 00:33:29,849
Now why should it have to be of this form? Well, you start with some of the iCar and then you mix into that.

318
00:33:29,850 --> 00:33:38,370
The Hamiltonian mixes into that different plane waves whose wave vector K differs by a reciprocal reliance vector g.

319
00:33:38,490 --> 00:33:42,090
And you can mix in as many of those as you want and they all get some coefficients a.

320
00:33:42,420 --> 00:33:49,500
But at the end of the day, the wave function must be a sum only of things that have the same crystal momentum k plus some reciprocal lattice vector g.

321
00:33:49,680 --> 00:33:53,820
There's no way that I potentially can take you can change your crystal momentum.

322
00:33:53,830 --> 00:33:57,120
That's the important realisation. Good. Yeah.

323
00:33:57,540 --> 00:34:04,259
Happy? Good. All right. So from here we're pretty much done because you can just factor out the, you know,

324
00:34:04,260 --> 00:34:13,230
the I k dot are and then you have some over g of a of g plus k into the I,

325
00:34:15,120 --> 00:34:31,650
g r and this piece here is the is the periodic function u u of R which and you can tell this thing must be periodic because if you take e to the I,

326
00:34:32,730 --> 00:34:40,620
g g dotted with R plus r, move it by a direct space lattice vector.

327
00:34:40,860 --> 00:34:43,920
This thing is the same as the i k that are.

328
00:34:44,580 --> 00:34:50,330
So this thing is periodic. As as claimed.

329
00:34:50,840 --> 00:35:00,250
Okay. Furthermore, we can also check what happens if you take and you shift it by a reciprocal lattice vector g.

330
00:35:00,270 --> 00:35:05,510
So let's let's actually take this form of K up here and let's shift K by G.

331
00:35:06,050 --> 00:35:15,860
So I have K plus g, I actually I've already used the letter G, so let's call this a G prime shifted by K to Kate plus g prime.

332
00:35:16,370 --> 00:35:29,570
So we'll write it as some over g a sub g plus k plus g prime e to the i r k plus g, prime plus g.

333
00:35:30,650 --> 00:35:35,060
But the sum is over a dummy index so we can define a dummy index.

334
00:35:35,360 --> 00:35:42,259
It's called g twiddle equals g g twiddle of vector equals g plus g prime vector.

335
00:35:42,260 --> 00:35:57,649
And then if you substitute that in, that becomes sum over g twiddle vector a of g twiddle of vector plus k into the i k r k plus g

336
00:35:57,650 --> 00:36:04,010
twiddle vector dot r which is exactly the same expression that we started with up up there.

337
00:36:04,160 --> 00:36:10,190
So if you shift K by reciprocal lattice vector G, you get back exactly the same wave function,

338
00:36:10,190 --> 00:36:12,410
which was what we were trying to answer in the first place.

339
00:36:12,740 --> 00:36:18,799
Is it special that when you shift K by reciprocal last vector you get back the exactly the same wave function?

340
00:36:18,800 --> 00:36:23,780
Is that special to the tight binding model? It's not. It's a property of the shortening equation in the periodic potential.

341
00:36:24,080 --> 00:36:33,560
Okay. All right. So the implications of this theorem, which is viewed as so important, is one implications.

342
00:36:38,030 --> 00:36:46,310
One are an excitation are all excitations can be described all excitations.

343
00:36:48,870 --> 00:37:03,359
Can be described be described in one bronze zone in one B.C., because whenever you go outside of the bronze zone,

344
00:37:03,360 --> 00:37:09,030
you can just zip by a reciprocal lattice vector to get back into the first back into that bronze zone.

345
00:37:11,190 --> 00:37:18,660
So that's an important statement. The second important statement, maybe more important is that a periodic potential,

346
00:37:19,200 --> 00:37:31,860
a periodic potential does not change, does not scatter crystal momentum.

347
00:37:36,450 --> 00:37:43,020
And this is something that I mentioned before when we discussed the type finding model that the crystal momentum is conserved.

348
00:37:44,490 --> 00:37:50,400
The crystal momentum is always conserved in the fact that we have this very strong periodic potential doesn't ruin that.

349
00:37:50,850 --> 00:37:57,929
The reason this is so important, remember, we had this puzzle when we studied the Sommerfeld model that you have this mean free path of electrons,

350
00:37:57,930 --> 00:38:01,500
which is enormously long, hundreds of atoms or thousands of atoms long.

351
00:38:01,770 --> 00:38:07,679
And we couldn't understand it because there's a big nucleus that the atom can hit every few angstroms.

352
00:38:07,680 --> 00:38:14,030
It could scatter off of tons and tons of nuclei. The point here is that the you don't have pure plane waves.

353
00:38:14,040 --> 00:38:15,689
What you have is modified plane waves.

354
00:38:15,690 --> 00:38:21,240
You have these plane waves times a periodic function, but once you modify the plane wave by multiplying it by this periodic function,

355
00:38:21,480 --> 00:38:25,860
that is perfectly good plane wave to go all the way across the system without scattering at all.

356
00:38:26,250 --> 00:38:31,230
So if you take a, you know, an electron, you put in a little wave packet of this modified plane wave,

357
00:38:31,350 --> 00:38:36,360
it would travel clear across the system as long as it doesn't hit any impurities or or nasty things like that.

358
00:38:36,480 --> 00:38:40,080
But it could go all the way across the system without scattering at all.

359
00:38:40,440 --> 00:38:47,520
So this sort of resolves our our initial issue of why it is that electrons can travel so far without without scattering.

360
00:38:48,360 --> 00:38:51,780
Okay. And where did this this result come from?

361
00:38:51,780 --> 00:38:56,280
It's important to realise that this important result really came from Louise condition.

362
00:38:56,490 --> 00:39:01,320
It came from the fact that a periodic potential only scatters you by reciprocal wave vectors.

363
00:39:02,070 --> 00:39:05,820
All right, so the one thing that this doesn't do for us.

364
00:39:07,030 --> 00:39:12,429
Is it does not manage to. It does not manage to help us actually solve the shorthand equations.

365
00:39:12,430 --> 00:39:21,879
So solving solving Schrödinger equation is still hard and we've already discussed one way to do it.

366
00:39:21,880 --> 00:39:31,230
One way to do it is tight binding. And by type binding, what we're really doing is we start with.

367
00:39:31,740 --> 00:39:41,500
Start with atomic orbitals. With atomic orbitals. And allow week hopping, allow week hopping.

368
00:39:46,090 --> 00:39:51,650
But that might sound a little non generic because maybe the hopping is strong from one place to another.

369
00:39:52,390 --> 00:39:55,090
So with useful too,

370
00:39:55,600 --> 00:40:01,330
it's actually very useful to think about it from a very complementary viewpoint where instead of starting with a type binding model,

371
00:40:01,510 --> 00:40:20,090
we start with plane waves, start with with plane waves, plane waves and add week periodic potential, the air.

372
00:40:20,530 --> 00:40:23,980
So it's entirely the opposite way of looking at things.

373
00:40:24,310 --> 00:40:34,420
This approach, approach to two is known as the nearly free electron model, nearly free electron model,

374
00:40:38,860 --> 00:40:46,750
and is a very useful model for understanding a lot of the physics of semiconductors and band structure in solids.

375
00:40:47,020 --> 00:40:55,149
So let's let's go about that. Let's start it. We probably won't finish it today, but the so how do we do this?

376
00:40:55,150 --> 00:41:00,850
We start with our H0 on k are plane waves without the periodic potential.

377
00:41:01,210 --> 00:41:09,910
So is zero k on k where is zero is the usual h bar squared k squared over two m?

378
00:41:11,140 --> 00:41:17,080
It's a good place to start. And then let's assume this is weak, assume weak.

379
00:41:17,080 --> 00:41:21,250
V All right.

380
00:41:21,250 --> 00:41:27,520
So what do you do when you have a weak perturbation to a Hamiltonian? Well, you use perturbation theory, which you presumably learned last year.

381
00:41:28,090 --> 00:41:32,709
So you remember what happens to the energies in perturbation theory.

382
00:41:32,710 --> 00:41:45,730
So let's start with first order of perturbation theory. First order, order, purge the energy this probably look familiar the energy of the wave.

383
00:41:45,730 --> 00:41:54,700
The Eigen State K is the bare energy without the perturbation plus the first order of perturbation theory term which is k,

384
00:41:55,150 --> 00:42:02,140
v, k and this matrix element here when I scrolled it off the top of the board.

385
00:42:02,740 --> 00:42:12,550
No, maybe I didn't, maybe I did. I think I did. We defined it as the sub zero because k minus k is zero and this v sub zero is the same.

386
00:42:13,450 --> 00:42:17,830
Same for all K for all k.

387
00:42:18,880 --> 00:42:24,370
So this just gives an overall constant energy shift of the the eigen state.

388
00:42:24,380 --> 00:42:28,240
So the the I can say t to get shifted up in energy or they get shifted down in energy.

389
00:42:28,240 --> 00:42:35,950
This is sort of the the zero thorium mode of this potentials where a constant potential is added on top of any fluctuating potentials,

390
00:42:35,950 --> 00:42:42,999
things that change in space. So the zero three mode of the of the potential just shifts up and down the overall energy of K.

391
00:42:43,000 --> 00:42:46,030
Does this expression look familiar first or probation theory. You just take the exponent.

392
00:42:46,030 --> 00:42:52,209
Yeah. Okay, good. So this is not interesting. And as a matter of fact, people frequently drop V zero because it's so uninteresting.

393
00:42:52,210 --> 00:42:57,550
They just get rid of it. I'll try to keep it, but maybe I'll forget it by mistake at some point.

394
00:42:57,850 --> 00:43:01,240
So second order is more interesting.

395
00:43:01,570 --> 00:43:08,469
So it's second order you have ic equals e k not that's the they are ps plus v zero.

396
00:43:08,470 --> 00:43:14,500
That's the first order piece. And then you have the second order piece which is sum of k prime not equal to k.

397
00:43:14,920 --> 00:43:24,810
Then we have k prime v k squared over e sub k not minus e k prime not.

398
00:43:24,820 --> 00:43:28,959
Is that the right order? Yeah. Does that look familiar from second order of probation theory?

399
00:43:28,960 --> 00:43:33,010
Good. All right, now, what do we know about this second order term?

400
00:43:33,010 --> 00:43:41,320
Well, again, we have lousy condition for the upstairs matrix element that in order for that matrix element to be non-zero,

401
00:43:41,590 --> 00:43:47,020
you must have K prime minus K must equal a reciprocal lattice vector g.

402
00:43:48,130 --> 00:43:54,550
So this term here can be rewritten as some over g of reciprocal.

403
00:43:54,550 --> 00:44:10,120
Last factor is g of v sub g squared where g is k prime minus k divided by e sub k not minus e sub k plus g not.

404
00:44:11,630 --> 00:44:14,960
Good luck. Happy with that? Somewhat.

405
00:44:15,830 --> 00:44:22,510
Okay. So this is our expression for the second order shift in energy of the plane waves that we started with.

406
00:44:22,940 --> 00:44:25,940
But we have to be a little bit careful about expressions that look like this.

407
00:44:25,940 --> 00:44:34,040
And the reason we have to be careful is because you might have a situation where the two terms in the denominator are either equal to each other,

408
00:44:34,040 --> 00:44:39,140
in which case you get zero downstairs or very close to each other, in which case you get something very small downstairs.

409
00:44:39,290 --> 00:44:43,640
And in either case, the secondary term would blow up, in which case you have a divergence.

410
00:44:44,030 --> 00:44:48,080
And that means the second order perturbation theory is not correct.

411
00:44:49,220 --> 00:44:52,670
So you must be careful.

412
00:44:53,510 --> 00:44:59,360
Be careful when he sub k.

413
00:45:00,050 --> 00:45:05,900
That's just right. Is SMK not approximately equal to E sub k plus g?

414
00:45:06,530 --> 00:45:12,439
Not now. When does that happen? Well, e sub k is actually just proportional to k squared.

415
00:45:12,440 --> 00:45:17,150
So that will happen when absolute k equals absolute k plus g.

416
00:45:17,660 --> 00:45:23,090
Well where's that. You'll remember somewhere here.

417
00:45:25,130 --> 00:45:29,360
There it is. It's the condition for being on a bronze on boundary.

418
00:45:29,600 --> 00:45:36,290
When you're on bronze on boundary e sub K and it's a K plus g are actually the same.

419
00:45:36,380 --> 00:45:50,390
Let's actually draw a picture of that. So what we have is we have a this is our bare spectrum K and E, the bare spectrum is just a parabola.

420
00:45:52,120 --> 00:46:04,450
So this is e sub k is h bar squared not k squared over two m and here is pi over a here is minus pi over a.

421
00:46:05,080 --> 00:46:13,990
And the energy of these two, these are separated by a reciprocal lattice vector to pi over a and their energy is identical.

422
00:46:14,440 --> 00:46:17,980
So this would give us a divergence in second order perturbation theory.

423
00:46:18,280 --> 00:46:25,180
Similarly, if we go to minus two pi over a and two pi over a,

424
00:46:27,910 --> 00:46:33,969
these are separated by a reciprocal lattice vector for pi over a and their energies are identical.

425
00:46:33,970 --> 00:46:37,480
So they will give you a divergence in second order perturbation theory.

426
00:46:37,690 --> 00:46:44,920
So the place where second order perturbation theory fails is exactly when you're on a bronze zone boundary.

427
00:46:46,150 --> 00:46:54,340
So what we're going to need to do is we are going to need to are probably going to have to do this next time.

428
00:46:54,880 --> 00:47:04,450
But what we're going to do to fix this problem, fix the problem by using using degenerate perturbation theory,

429
00:47:05,530 --> 00:47:15,880
degenerate perturbation theory, effort theory, which hopefully you've learned something about already.

430
00:47:17,320 --> 00:47:20,889
So we're not going to have time to launch into that today.

431
00:47:20,890 --> 00:47:25,510
So I'm going to actually take the opportunity to insert one piece of information or maybe two

432
00:47:25,510 --> 00:47:29,710
pieces of information about scattering that I promised you we would talk about some time,

433
00:47:30,070 --> 00:47:34,060
and that is scattering off of. So maybe I'll put that over here.

434
00:47:34,360 --> 00:47:41,310
Scattering off of x rays and neutron scattering x ray and neutron scattering.

435
00:47:41,320 --> 00:47:44,690
This is different subject just inserted in this last 5 minutes.

436
00:47:44,710 --> 00:47:49,450
Neutron scattering from.

437
00:47:50,380 --> 00:47:54,910
From liquids. Liquids and amorphous solids.

438
00:47:55,780 --> 00:47:58,790
Amorphous. Solids.

439
00:47:59,810 --> 00:48:04,610
So in liquids, in amorphous solids, we do not have periodic structures.

440
00:48:04,820 --> 00:48:12,110
We do not have periodic arrangement of atoms. But nonetheless, you get a lot of information out of the neutron X-ray scattering.

441
00:48:12,110 --> 00:48:20,929
If you think way back to when we did Fermi's Golden Rule, that the scattering amplitude is still the 48 transform of the periodic of the potential.

442
00:48:20,930 --> 00:48:24,230
But it's not a periodic potential, it's Fourier transform of the non periodic potential.

443
00:48:24,560 --> 00:48:27,920
But does that mean there's no information in it? No, there's still information in it.

444
00:48:28,130 --> 00:48:31,690
It's just that you don't get sharp peaks anymore. What you get is here.

445
00:48:31,820 --> 00:48:36,110
There it is. So this is X-ray scattering on liquid aluminium.

446
00:48:36,140 --> 00:48:39,290
It's heated up to some high temperature. So it's so it's a liquid.

447
00:48:40,220 --> 00:48:44,210
There's a function of a scattering angle or the reciprocal wave vector.

448
00:48:44,540 --> 00:48:48,350
That's the way vector along this axis.

449
00:48:48,590 --> 00:48:56,540
The amplitude of scattering it has peaks. And the peaks represent roughly the distance between between atoms.

450
00:48:56,550 --> 00:49:03,710
It's one over the distance between atoms. The same way the peaks represent the the periodicity of the unit cell when it becomes crystalline.

451
00:49:03,860 --> 00:49:10,430
Now what happens is as you cooled down a liquid and it gets more and more crystalline, that becomes more and more local order.

452
00:49:10,550 --> 00:49:12,770
And these peaks get sharper and sharper and sharper.

453
00:49:12,980 --> 00:49:21,090
And finally, when the thing actually crystallises, they become a delta function peaks the way we expected them to for for a crystal.

454
00:49:21,110 --> 00:49:25,490
But if it's a liquid, it's you can sort of think of a liquid as locally being like a solid.

455
00:49:25,880 --> 00:49:30,060
Each atom has a couple of neighbours, but if you sort of look farther away, it's not ordered any more.

456
00:49:30,080 --> 00:49:36,790
It sort of loses its order. But if you sort of look just in some small region, it probably has, you know, it's organised over a small region.

457
00:49:36,800 --> 00:49:41,360
So you get these sort of broad peaks in in the scattering spectrum.

458
00:49:42,510 --> 00:49:50,670
Okay. I think maybe I won't try to cover any further issues today and we'll start again on Wednesday, I think.

459
00:49:50,690 --> 00:49:51,380
Have a good weekend.