1
00:00:07,820 --> 00:00:09,920
PROFESSOR: OK, why don't we
you get started again?

2
00:00:09,920 --> 00:00:13,040
I can understand why there is
an air of excitement in the

3
00:00:13,040 --> 00:00:14,910
room since tomorrow's
a holiday.

4
00:00:14,910 --> 00:00:20,230
But we still got two hours
before the day is over.

5
00:00:20,230 --> 00:00:23,695
Any questions on what we have
done up to this point?

6
00:00:26,750 --> 00:00:28,920
No, that's good.

7
00:00:28,920 --> 00:00:33,470
Let me erase some of this art
work then and ask why one

8
00:00:33,470 --> 00:00:40,170
would indulge in this rather
bizarre exercise of taking the

9
00:00:40,170 --> 00:00:43,520
elements of something that
represents a physical property

10
00:00:43,520 --> 00:00:48,450
and constructing a surface
out of them.

11
00:00:48,450 --> 00:00:53,700
Well, the name suggests that
maybe these surfaces, be they

12
00:00:53,700 --> 00:00:56,170
hyperboloids of one or two
sheets or imaginary

13
00:00:56,170 --> 00:01:00,280
ellipsoids, have something to
say about the property that

14
00:01:00,280 --> 00:01:03,270
the tensor that formed the
coefficient of these

15
00:01:03,270 --> 00:01:07,440
functions is doing.

16
00:01:07,440 --> 00:01:12,480
So let me take the case
of an ellipsoid.

17
00:01:12,480 --> 00:01:14,700
That would be the case
where all the

18
00:01:14,700 --> 00:01:17,230
coefficients are positive.

19
00:01:17,230 --> 00:01:22,310
And let's let this be
x1 and this x2.

20
00:01:22,310 --> 00:01:26,860
And let me ask a
first question.

21
00:01:26,860 --> 00:01:31,920
If this is to be a well deserved
function, does the

22
00:01:31,920 --> 00:01:36,010
surface transform in exactly
the same way

23
00:01:36,010 --> 00:01:37,650
as the tensor elements?

24
00:01:37,650 --> 00:01:42,990
In other words, if we take the
tensor in one coordinate

25
00:01:42,990 --> 00:01:45,430
system and then change the
coordinate system to a

26
00:01:45,430 --> 00:01:48,740
different coordinate system, are
the coefficients in front

27
00:01:48,740 --> 00:01:53,810
of x1, x2, and x3 still the
elements of the tensor in that

28
00:01:53,810 --> 00:01:55,050
new coordinate system?

29
00:01:55,050 --> 00:01:59,000
So let me show you that this
is, in fact, the case.

30
00:01:59,000 --> 00:02:04,760
Suppose our original relation,
sticking to conductivity, is

31
00:02:04,760 --> 00:02:09,419
that x sigma ij xi
xj equals 1.

32
00:02:09,419 --> 00:02:12,340
And then for some reason or
another, we change axes.

33
00:02:12,340 --> 00:02:17,010
So the new equation will be some
different coefficients.

34
00:02:17,010 --> 00:02:19,560
Because as we change axes,
they're going to wink on and

35
00:02:19,560 --> 00:02:23,440
off, still the same surface, but
now referred to different

36
00:02:23,440 --> 00:02:28,990
axes xi prime and xj prime
still equal to 1.

37
00:02:28,990 --> 00:02:36,190
And let me now use the reverse
transformation to put xi prime

38
00:02:36,190 --> 00:02:40,430
in the terms of xi.

39
00:02:40,430 --> 00:02:50,160
If we do that, we can say that
xi prime is cli x sub l prime.

40
00:02:50,160 --> 00:02:51,960
That's the reverse
transformation.

41
00:02:51,960 --> 00:02:57,210
So notice the inverted order
of the direction cosine

42
00:02:57,210 --> 00:02:58,520
subscripts.

43
00:02:58,520 --> 00:03:06,210
And then xj prime is going
to be equal to cmj

44
00:03:06,210 --> 00:03:08,825
times x sub ep prime.

45
00:03:08,825 --> 00:03:12,250
To find this, let's just
rearrange these terms in a

46
00:03:12,250 --> 00:03:13,420
trivial fashion.

47
00:03:13,420 --> 00:03:31,640
Then I will have cx sigma ij
cil cmj times xl prime xm

48
00:03:31,640 --> 00:03:34,600
prime equals 1.

49
00:03:34,600 --> 00:03:35,900
And what is this?

50
00:03:35,900 --> 00:03:45,370
This is a summation of direction
cosines where the

51
00:03:45,370 --> 00:03:50,210
first subscript goes with the
subscript on sigma prime.

52
00:03:50,210 --> 00:03:51,550
What did I do wrong?

53
00:03:51,550 --> 00:03:52,820
This is mj.

54
00:03:58,060 --> 00:04:01,120
Why is that not coming first?

55
00:04:01,120 --> 00:04:02,370
AUDIENCE: [INAUDIBLE]

56
00:04:04,550 --> 00:04:06,620
PROFESSOR: Well, what I would
like to get this in the form

57
00:04:06,620 --> 00:04:10,830
of is a summation of sigma
ij prime over two dummy

58
00:04:10,830 --> 00:04:14,246
indices l and m.

59
00:04:14,246 --> 00:04:15,668
What did I do wrong?

60
00:04:20,890 --> 00:04:23,450
Ah, yes, right, right, right,
right, this is li.

61
00:04:23,450 --> 00:04:24,730
Thank you somebody said it.

62
00:04:24,730 --> 00:04:26,080
And my nose was right in it.

63
00:04:26,080 --> 00:04:27,160
And I didn't notice it.

64
00:04:27,160 --> 00:04:30,830
OK, this is a summation of all
the original elements sigma ij

65
00:04:30,830 --> 00:04:35,780
prime times dummy
indices l and m.

66
00:04:35,780 --> 00:04:41,520
And this actually
is sigma ijn.

67
00:04:41,520 --> 00:04:43,730
So it's the same set
of coefficients.

68
00:04:43,730 --> 00:04:47,100
And that is a technicality,
which probably you wouldn't

69
00:04:47,100 --> 00:04:48,470
want to worry about.

70
00:04:48,470 --> 00:04:53,100
So anyway, the elements of the
tensor transform in the same

71
00:04:53,100 --> 00:04:56,310
way that the equations for
the surfaces transform.

72
00:04:56,310 --> 00:04:59,210
So if you change coordinate
system, the coefficients that

73
00:04:59,210 --> 00:05:03,670
you should use should be the
elements that are in the

74
00:05:03,670 --> 00:05:06,410
transform tensor.

75
00:05:06,410 --> 00:05:07,890
OK, now I'm going to
ask a question.

76
00:05:07,890 --> 00:05:13,280
Suppose I define some direction
by a set of

77
00:05:13,280 --> 00:05:21,990
direction cosines li and I ask
the value of the radius of the

78
00:05:21,990 --> 00:05:26,450
surface in that direction.

79
00:05:26,450 --> 00:05:29,810
So this is some radius vector.

80
00:05:29,810 --> 00:05:39,940
The radius vector will have
components, R1, R2, R3 or Ri.

81
00:05:39,940 --> 00:05:46,350
And each of those components
will be equal to the magnitude

82
00:05:46,350 --> 00:05:51,330
of R times the direction
cosines li.

83
00:05:51,330 --> 00:05:52,116
Yes, sir?

84
00:05:52,116 --> 00:05:55,924
AUDIENCE: Could you briefly rego
over what you just stated

85
00:05:55,924 --> 00:05:57,828
before you erased
it [INAUDIBLE]?

86
00:05:57,828 --> 00:05:58,780
PROFESSOR: Oh, OK.

87
00:05:58,780 --> 00:06:01,370
I'm sorry.

88
00:06:01,370 --> 00:06:06,070
I said aij xi xj equals
1 is the original

89
00:06:06,070 --> 00:06:08,020
equation for the quadric.

90
00:06:08,020 --> 00:06:14,480
If we change the coordinate
system, we're going to have

91
00:06:14,480 --> 00:06:19,170
some new elements aij prime
times xi prime xj prime.

92
00:06:19,170 --> 00:06:20,840
Because we've got a new
coordinate system.

93
00:06:20,840 --> 00:06:23,900
And therefore, the coefficients
in front of the

94
00:06:23,900 --> 00:06:26,330
equation for the quadric
have to change.

95
00:06:26,330 --> 00:06:33,350
And now what I did was to
express xi prime and xj prime

96
00:06:33,350 --> 00:06:39,810
in terms of xi and xj using the
reverse transformation.

97
00:06:39,810 --> 00:06:49,360
And xi prime in terms of the
original x's will be c sub l

98
00:06:49,360 --> 00:06:52,930
sub i x sub l prime.

99
00:06:52,930 --> 00:07:00,320
Usually, we write it xl
equals cil x sub l.

100
00:07:00,320 --> 00:07:02,920
This is the reverse
transformation where the order

101
00:07:02,920 --> 00:07:04,710
of the subscripts is reversed.

102
00:07:04,710 --> 00:07:07,850
So this will give
me x of i prime.

103
00:07:07,850 --> 00:07:16,340
The expression for x sub j prime
will be given by cmj x

104
00:07:16,340 --> 00:07:19,830
sub j where m is
a dummy index.

105
00:07:19,830 --> 00:07:21,600
And that's equal to 1.

106
00:07:21,600 --> 00:07:33,920
And if I just rearrange this,
then I will have cli cmj times

107
00:07:33,920 --> 00:07:43,800
aij prime equals
1 times xl xj.

108
00:07:43,800 --> 00:07:55,860
And this is going
to be alx lxm.

109
00:07:55,860 --> 00:08:04,370
And this will be alm times xl
xm equals 1, which is the

110
00:08:04,370 --> 00:08:05,890
equation for the quadric in the

111
00:08:05,890 --> 00:08:08,470
original coordinate system.

112
00:08:08,470 --> 00:08:10,750
I think that was better when I
was standing in front of it so

113
00:08:10,750 --> 00:08:12,000
you couldn't see it.

114
00:08:14,150 --> 00:08:20,280
It's just that the surface
transforms formally to the

115
00:08:20,280 --> 00:08:25,020
surface that we would create
if we used the new tensor

116
00:08:25,020 --> 00:08:28,130
elements for the same change of
coordinates system, which

117
00:08:28,130 --> 00:08:30,630
you would probably will be
willing to take my word for.

118
00:08:35,550 --> 00:08:37,520
AUDIENCE: [INAUDIBLE]

119
00:08:37,520 --> 00:08:39,730
PROFESSOR: This is a direction
cosine for the--

120
00:08:39,730 --> 00:08:40,630
AUDIENCE: [INAUDIBLE]
right below.

121
00:08:40,630 --> 00:08:41,286
PROFESSOR: Right below?

122
00:08:41,286 --> 00:08:42,536
AUDIENCE: [INAUDIBLE]

123
00:08:46,880 --> 00:08:52,440
PROFESSOR: cli cmj aij prime
xl xm, and then I collected

124
00:08:52,440 --> 00:08:56,520
the cli cmj times alm.

125
00:08:56,520 --> 00:08:59,710
And that is the transfer.

126
00:08:59,710 --> 00:09:01,146
Yes?

127
00:09:01,146 --> 00:09:05,130
AUDIENCE: In the previous
[INAUDIBLE], you defined the

128
00:09:05,130 --> 00:09:14,094
derivitave side as being
xi prime equals cil xl.

129
00:09:14,094 --> 00:09:15,344
[INAUDIBLE]?

130
00:09:18,078 --> 00:09:26,325
PROFESSOR: You can say that x
prime is cil times x sub l.

131
00:09:26,325 --> 00:09:30,480
And if we want to use the same
direction cosine scheme but do

132
00:09:30,480 --> 00:09:38,395
it in reverse, we would say that
x sub l is equal to c--

133
00:09:51,280 --> 00:10:01,255
yeah, OK, x sub i is cli
times x sub i, right,

134
00:10:01,255 --> 00:10:02,872
no, times x sub l--

135
00:10:02,872 --> 00:10:04,423
is that right-- x sub l prime.

136
00:10:08,106 --> 00:10:13,810
AUDIENCE: On the second line
to the third line, you say

137
00:10:13,810 --> 00:10:17,034
that xi prime equals
cli [INAUDIBLE].

138
00:10:21,994 --> 00:10:32,446
It's either xi equals xcli xl
prime or xi prime equals cil.

139
00:10:37,330 --> 00:10:41,160
PROFESSOR: OK, you want to say
this. xi prime and cil becomes

140
00:10:41,160 --> 00:10:43,050
xl, yes, yeah.

141
00:10:43,050 --> 00:10:45,360
And then mj times x--

142
00:10:52,720 --> 00:10:55,650
that's x sub m.

143
00:10:55,650 --> 00:11:07,310
OK, and then this says that cil
cjm times aij prime should

144
00:11:07,310 --> 00:11:11,240
be alm, right, OK, OK.

145
00:11:11,240 --> 00:11:12,615
So it's OK then.

146
00:11:15,285 --> 00:11:17,980
OK, that was supposed to be a
small point that everybody

147
00:11:17,980 --> 00:11:18,730
would accept.

148
00:11:18,730 --> 00:11:20,730
But now it's done correctly.

149
00:11:20,730 --> 00:11:23,970
OK, let's get back to
this, which is more

150
00:11:23,970 --> 00:11:25,200
hair raising and exciting.

151
00:11:25,200 --> 00:11:27,870
What is the radius of the
quadric in a given direction

152
00:11:27,870 --> 00:11:32,690
that we specify by three
direction cosines l1, l2, l3?

153
00:11:32,690 --> 00:11:35,540
So there's some radius from the
center of the quadric out

154
00:11:35,540 --> 00:11:39,090
to the surface in that
particular direction specified

155
00:11:39,090 --> 00:11:41,050
by three direction cosines.

156
00:11:41,050 --> 00:11:44,580
And we know what those three
components are, sub i, are

157
00:11:44,580 --> 00:11:48,200
going to be in terms of the
direction cosines, magnitude

158
00:11:48,200 --> 00:11:50,920
of R times li.

159
00:11:50,920 --> 00:12:02,650
And those points at the terminus
of the radius vector

160
00:12:02,650 --> 00:12:05,630
go to one point on the surface
of the quadric.

161
00:12:05,630 --> 00:12:10,190
So these values of R are
coordinates that satisfy the

162
00:12:10,190 --> 00:12:11,420
surface of the quadric.

163
00:12:11,420 --> 00:12:19,280
So we can say that just
substitute Ri for the

164
00:12:19,280 --> 00:12:23,260
different values xi in the
equation for the quadric.

165
00:12:23,260 --> 00:12:33,270
And this says that sigma ij
magnitude of R times li, that

166
00:12:33,270 --> 00:12:38,220
would correspond to x sub
i times magnitude of

167
00:12:38,220 --> 00:12:41,740
R times l sub j.

168
00:12:41,740 --> 00:12:44,990
That's the magnitude of xj.

169
00:12:44,990 --> 00:12:47,640
That should be equal to 1.

170
00:12:47,640 --> 00:12:51,150
And if I rearranged this
slightly, this says that the

171
00:12:51,150 --> 00:12:58,690
magnitude of R squared is going
to be equal to 1 over

172
00:12:58,690 --> 00:13:03,040
sigma ij times li lj.

173
00:13:05,980 --> 00:13:11,260
So the radius of the quadric
gives me not the value of the

174
00:13:11,260 --> 00:13:12,790
property in that direction.

175
00:13:12,790 --> 00:13:16,770
This is the value of the
property that will occur in

176
00:13:16,770 --> 00:13:20,610
the direction specified by
the direction cosines lj.

177
00:13:20,610 --> 00:13:25,410
The radius of the quadric is
going to be equal to 1 over

178
00:13:25,410 --> 00:13:28,410
the square root of
the value of the

179
00:13:28,410 --> 00:13:30,675
property in that direction.

180
00:13:36,100 --> 00:13:40,410
So this is why it's called the
representation quadric.

181
00:13:40,410 --> 00:13:45,000
If you construct the surface
from the tensor elements, you

182
00:13:45,000 --> 00:13:49,060
will have defined a quadratic
form, which has the property

183
00:13:49,060 --> 00:13:52,850
that, as you go in different
directions look at the

184
00:13:52,850 --> 00:13:56,755
distance out to the surface of
the quadric in that direction,

185
00:13:56,755 --> 00:14:00,990
that that distance squared is
going to be equal to 1 over

186
00:14:00,990 --> 00:14:05,530
the property in that direction
or, alternatively, the radius

187
00:14:05,530 --> 00:14:07,520
is going to be 1 over the square
root of the value of

188
00:14:07,520 --> 00:14:09,570
the property.

189
00:14:09,570 --> 00:14:12,740
There is an enormous
implication here.

190
00:14:12,740 --> 00:14:18,390
This says that the value of the
property, if the quadratic

191
00:14:18,390 --> 00:14:21,920
form is an ellipsoid, the value
of the property as we go

192
00:14:21,920 --> 00:14:25,170
around in different directions
in the crystal is going to be

193
00:14:25,170 --> 00:14:28,760
a smooth, uniformly
varying function.

194
00:14:28,760 --> 00:14:30,820
They're going to be no
lobes sticking out.

195
00:14:30,820 --> 00:14:34,250
They're going to be no
dimples, no lumps.

196
00:14:34,250 --> 00:14:37,540
It's going to be an almost
monotonous property, not very

197
00:14:37,540 --> 00:14:37,980
interesting.

198
00:14:37,980 --> 00:14:40,590
It's going to change
in a uniform way.

199
00:14:40,590 --> 00:14:44,120
And in fact, we could put the
two surfaces side by side.

200
00:14:44,120 --> 00:14:49,710
If this is the value of the
quadric as a function of

201
00:14:49,710 --> 00:14:53,990
direction, if we make a polar
plot of the property as a

202
00:14:53,990 --> 00:14:57,170
function of direction, it's
going to look sort of like the

203
00:14:57,170 --> 00:14:58,200
reciprocal of this.

204
00:14:58,200 --> 00:15:01,630
The minimum value of the
property will be in the

205
00:15:01,630 --> 00:15:06,980
direction of the maximum value
of the radius of the quadric.

206
00:15:06,980 --> 00:15:11,640
And the maximum value of the
property is going to go in the

207
00:15:11,640 --> 00:15:13,260
direction of the
minimum value.

208
00:15:13,260 --> 00:15:16,830
So the value of the property as
a function of direction is

209
00:15:16,830 --> 00:15:19,900
going to be a quasi-ellipsoidal
sort of

210
00:15:19,900 --> 00:15:22,440
variation but not really
an ellipsoid.

211
00:15:22,440 --> 00:15:25,260
It's going to go as this
inverse square of

212
00:15:25,260 --> 00:15:26,760
the radii in ellipsoid.

213
00:15:26,760 --> 00:15:29,140
But the thing is it's going to
be something that varies

214
00:15:29,140 --> 00:15:33,690
uniformly between extreme values
of the maximum and

215
00:15:33,690 --> 00:15:36,160
minimum value of the property.

216
00:15:36,160 --> 00:15:39,260
We are, I assure you, for higher
ranked tensor property

217
00:15:39,260 --> 00:15:45,830
going to look at some absolutely
wild surfaces with

218
00:15:45,830 --> 00:15:50,460
surfaces that do have lobes and
extreme values and very,

219
00:15:50,460 --> 00:15:53,360
very irregular variation of
properties with directions but

220
00:15:53,360 --> 00:15:55,195
not for second ranked
tensor properties.

221
00:15:57,810 --> 00:16:02,080
There will be a few variations
on this theme, which are also

222
00:16:02,080 --> 00:16:03,405
interesting to touch upon.

223
00:16:06,750 --> 00:16:11,310
In principle, we can get other
quadratic forms from the

224
00:16:11,310 --> 00:16:16,190
tensor if some of the elements
of the tensor are negative.

225
00:16:16,190 --> 00:16:19,450
And in particular, what
would we see--

226
00:16:19,450 --> 00:16:21,590
and I'll draw this
relative to the

227
00:16:21,590 --> 00:16:23,470
principal axes of the surface--

228
00:16:23,470 --> 00:16:28,315
what would we see for an
hyperboloid of one sheet?

229
00:16:33,770 --> 00:16:42,670
This is the sort of surface
that might result if one

230
00:16:42,670 --> 00:16:47,130
principal value of the tensor
had a negative value.

231
00:16:47,130 --> 00:16:50,220
Well, this is the quadric.

232
00:16:50,220 --> 00:16:54,390
And this is a radius in one
particular direction.

233
00:16:54,390 --> 00:16:57,240
What would this say about
the property?

234
00:16:57,240 --> 00:17:00,850
Well, let me, to make it clear,
look at one of these

235
00:17:00,850 --> 00:17:05,990
sections through the quadric
that our hyperbola.

236
00:17:05,990 --> 00:17:10,150
In directions like this, we
have a radius that is a

237
00:17:10,150 --> 00:17:13,690
minimum out to the surface.

238
00:17:13,690 --> 00:17:16,119
In other directions,
the radius gets

239
00:17:16,119 --> 00:17:17,773
progressively larger.

240
00:17:21,220 --> 00:17:28,540
And then if we plot the
reciprocal of the square root

241
00:17:28,540 --> 00:17:31,800
of that, that says that, in
this direction, we get the

242
00:17:31,800 --> 00:17:34,090
maximum value of the property.

243
00:17:34,090 --> 00:17:38,430
And as the direction approaches
the asymptote, the

244
00:17:38,430 --> 00:17:41,050
reciprocal will go down to 0.

245
00:17:41,050 --> 00:17:43,310
But how did I get
two y-axes here?

246
00:17:46,880 --> 00:17:49,490
OK, so the maximum will
be in the direction

247
00:17:49,490 --> 00:17:50,790
of the minimum radius.

248
00:17:50,790 --> 00:17:55,396
And then it will go down
to 0 for the asymptote.

249
00:17:55,396 --> 00:17:58,790
And that's going to be
symmetrical on either side of

250
00:17:58,790 --> 00:18:00,160
this principle axis.

251
00:18:03,080 --> 00:18:09,710
So what then are the radii in
directions outside of the

252
00:18:09,710 --> 00:18:12,570
asymptotes of this
hyperboloid?

253
00:18:12,570 --> 00:18:16,640
Within this range, the
radius is imaginary.

254
00:18:21,000 --> 00:18:25,320
But if you square it, you
get a negative number.

255
00:18:25,320 --> 00:18:27,490
So within these two lobes,
the value of

256
00:18:27,490 --> 00:18:29,940
the property is positive.

257
00:18:29,940 --> 00:18:33,730
As you go away from the
asymptotes, you'll get another

258
00:18:33,730 --> 00:18:37,800
lobe like this and another lobe
like this where the value

259
00:18:37,800 --> 00:18:39,050
of the property is negative.

260
00:18:42,170 --> 00:18:43,940
How in the world could
you get anything

261
00:18:43,940 --> 00:18:44,830
that looked like that?

262
00:18:44,830 --> 00:18:48,140
Well, a good example of a
property that has this

263
00:18:48,140 --> 00:18:53,054
behavior is thermal expansion.

264
00:18:53,054 --> 00:18:54,695
And let me give you
two examples.

265
00:18:58,030 --> 00:19:02,780
The structure of selenium
and tellurium

266
00:19:02,780 --> 00:19:05,750
is a hexagonal structure.

267
00:19:05,750 --> 00:19:11,910
And there are chain-like
molecules that are pairs of

268
00:19:11,910 --> 00:19:21,360
bonds that rise up around
a threefold screw axis.

269
00:19:21,360 --> 00:19:26,510
So this is two coordinated atoms
in the structure just

270
00:19:26,510 --> 00:19:31,510
spirals up in this triangular
spiral around the threefold

271
00:19:31,510 --> 00:19:32,760
screw axis.

272
00:19:35,470 --> 00:19:41,060
So this is a material in which
the bonding is very, very

273
00:19:41,060 --> 00:19:42,750
anisotropic.

274
00:19:42,750 --> 00:19:46,940
The bonding within these
covalently bonded spirals,

275
00:19:46,940 --> 00:19:50,610
which are like springs that you
might put on your screen

276
00:19:50,610 --> 00:19:54,180
door in the summertime, the
bonding is very strong.

277
00:19:54,180 --> 00:19:57,280
Between these individual
molecular chains, the bonding

278
00:19:57,280 --> 00:19:58,570
is very weak.

279
00:19:58,570 --> 00:20:01,180
So what happens when you
heat this stuff up?

280
00:20:01,180 --> 00:20:03,790
It expands like the dickens
in the direction

281
00:20:03,790 --> 00:20:05,270
of these weak bonds.

282
00:20:08,320 --> 00:20:13,630
But the spirals are, in part,
held in this extended form by

283
00:20:13,630 --> 00:20:16,380
repulsive interactions
in like chains.

284
00:20:16,380 --> 00:20:21,670
So when these spirals move apart
as a result of large

285
00:20:21,670 --> 00:20:24,730
thermal expansion in the plane
normal to the chain, the

286
00:20:24,730 --> 00:20:27,360
chains relax a little bit.

287
00:20:27,360 --> 00:20:31,460
So you have a large positive
thermal expansion here.

288
00:20:31,460 --> 00:20:35,960
But in one direction along the
normal to the hexagonal in the

289
00:20:35,960 --> 00:20:38,980
structure, the thermal expansion
is negative.

290
00:20:38,980 --> 00:20:42,940
And that gives you a variation
of property with direction

291
00:20:42,940 --> 00:20:44,350
that looks exactly like this.

292
00:20:48,150 --> 00:20:50,070
Negative value of the property,
that means the

293
00:20:50,070 --> 00:20:52,100
structure, contracts
in that direction,

294
00:20:52,100 --> 00:20:53,980
positive values this way.

295
00:20:53,980 --> 00:20:55,230
The structure expands.

296
00:20:55,230 --> 00:20:57,030
So there's a good
example of this.

297
00:21:06,640 --> 00:21:09,580
There's another example of a
material, which, again, has a

298
00:21:09,580 --> 00:21:12,250
negative thermal expansion
in one direction.

299
00:21:12,250 --> 00:21:18,030
And this is calcium carbonate,
which is a complicated

300
00:21:18,030 --> 00:21:19,910
hexagonal structure.

301
00:21:19,910 --> 00:21:23,480
But it looks very, very much
like the structure of rock

302
00:21:23,480 --> 00:21:27,490
salt in a distorted form.

303
00:21:27,490 --> 00:21:30,094
These are the calcium atoms.

304
00:21:30,094 --> 00:21:36,120
And calcite is CaCL3 And
the calciums are in a

305
00:21:36,120 --> 00:21:38,050
face-centered cubic
arrangement just

306
00:21:38,050 --> 00:21:40,320
like in rock salt.

307
00:21:40,320 --> 00:21:46,710
The carbonate groups are very
tightly bonded little

308
00:21:46,710 --> 00:21:49,190
triangles with the carbon
in the middle.

309
00:21:49,190 --> 00:21:54,730
And these things are arranged
on the edges of the cell.

310
00:21:54,730 --> 00:21:57,950
And this is going to
be very schematic.

311
00:21:57,950 --> 00:22:07,090
These triangles are
normal to the body

312
00:22:07,090 --> 00:22:08,190
diagonal of the cells.

313
00:22:08,190 --> 00:22:11,100
So you have one family of
triangles that are all

314
00:22:11,100 --> 00:22:12,560
parallel to one another.

315
00:22:12,560 --> 00:22:16,430
On the next edge, the triangles
are anti-parallel to

316
00:22:16,430 --> 00:22:17,590
this first orientation.

317
00:22:17,590 --> 00:22:18,770
So think of rock salt.

318
00:22:18,770 --> 00:22:20,810
Tip it up on its
body diagonal.

319
00:22:20,810 --> 00:22:24,270
Every place you have a chlorine
anion, place a

320
00:22:24,270 --> 00:22:27,570
triangle in an orientation that
is perpendicular to the

321
00:22:27,570 --> 00:22:32,830
body's diagonal of the
rock salt structure.

322
00:22:32,830 --> 00:22:36,900
Again, these tightly bonded
little triangles don't do much

323
00:22:36,900 --> 00:22:38,580
as you increase temperature.

324
00:22:38,580 --> 00:22:41,910
But the bonding between the
calcium and the triangles is

325
00:22:41,910 --> 00:22:43,000
rather weak.

326
00:22:43,000 --> 00:22:48,280
So again, you find that the
structure expands in one

327
00:22:48,280 --> 00:22:51,810
direction so strongly that
it contracts to the other

328
00:22:51,810 --> 00:22:54,235
direction so calcium
carbonate.

329
00:22:54,235 --> 00:22:57,670
The calcite form also has the
distinction of having a

330
00:22:57,670 --> 00:23:00,420
negative thermal expansion
coefficient.

331
00:23:00,420 --> 00:23:03,920
We'll talk quite a bit about
expansion coefficients as one

332
00:23:03,920 --> 00:23:07,050
example of a tensor property.

333
00:23:07,050 --> 00:23:09,600
Interestingly, and we'll say
more about this and I'll give

334
00:23:09,600 --> 00:23:15,220
you some references when we come
to this point, can you

335
00:23:15,220 --> 00:23:19,410
get a material that has a
negative thermal expansion

336
00:23:19,410 --> 00:23:21,146
coefficient in all directions?

337
00:23:24,250 --> 00:23:28,620
No, that seems as though it
would violate in a flagrant

338
00:23:28,620 --> 00:23:32,220
way some vast law of
thermodynamics that things

339
00:23:32,220 --> 00:23:36,410
have to increase their volume
when you increase temperature.

340
00:23:36,410 --> 00:23:41,810
There were, however, a few odd
ball materials that over a

341
00:23:41,810 --> 00:23:47,310
very, very limited temperature
range would contract but only

342
00:23:47,310 --> 00:23:50,290
by a tiny amount in
all directions.

343
00:23:50,290 --> 00:23:53,020
And then one of the very
interesting discoveries of

344
00:23:53,020 --> 00:23:58,910
recent years done primarily by
a crystal chemist named Art

345
00:23:58,910 --> 00:24:02,970
Slate, who's out at the
University of Oregon, he

346
00:24:02,970 --> 00:24:06,780
discovered a family of materials
that have large

347
00:24:06,780 --> 00:24:10,150
negative expansion coefficients
in all directions

348
00:24:10,150 --> 00:24:11,580
over a considerable range of

349
00:24:11,580 --> 00:24:13,910
temperatures, like 100 degrees.

350
00:24:13,910 --> 00:24:15,870
So these are materials, when
you heat them up, they

351
00:24:15,870 --> 00:24:19,710
contract amazing
as that seems.

352
00:24:19,710 --> 00:24:22,320
And there's a structural
reason for it.

353
00:24:22,320 --> 00:24:25,920
These are tetrahedral frameworks
in which one corner

354
00:24:25,920 --> 00:24:28,860
of a tetrahedron is dangling.

355
00:24:28,860 --> 00:24:33,710
And as you heat it up, this
tetrahedral corner can get

356
00:24:33,710 --> 00:24:39,110
closer to other tetrahedra,
which takes energy to do.

357
00:24:39,110 --> 00:24:43,570
But in so doing, you actually
are changing the net volume

358
00:24:43,570 --> 00:24:47,090
occupied by the solid
so very, very

359
00:24:47,090 --> 00:24:49,170
anomalous set of compounds.

360
00:24:49,170 --> 00:24:50,390
More of these have been found.

361
00:24:50,390 --> 00:24:53,830
There are probably a dozen
materials now that have

362
00:24:53,830 --> 00:24:55,750
negative thermal expansion
coefficients in all

363
00:24:55,750 --> 00:24:56,250
directions.

364
00:24:56,250 --> 00:25:02,060
So is the imaginary ellipsoid
a viable representation

365
00:25:02,060 --> 00:25:06,240
quadric for real materials?

366
00:25:06,240 --> 00:25:09,105
Yes, in the case of thermal
expansion coefficients.

367
00:25:13,450 --> 00:25:18,150
OK, so the representation
quadric then is a dandy device

368
00:25:18,150 --> 00:25:21,940
for seeing with one function
how a property

369
00:25:21,940 --> 00:25:23,530
will vary with direction.

370
00:25:23,530 --> 00:25:31,040
For an ellipsodial quadric, the
variation of the property

371
00:25:31,040 --> 00:25:35,230
itself with direction is not
really ellipsodial but

372
00:25:35,230 --> 00:25:38,120
quasi-ellipsodial in that
there's a small principal

373
00:25:38,120 --> 00:25:40,270
axis, a large principal axis.

374
00:25:40,270 --> 00:25:44,350
And the large value of the
property goes with the short

375
00:25:44,350 --> 00:25:45,600
axis of the quadric.

376
00:25:47,980 --> 00:25:54,860
So we have a nice device for
representing the value of the

377
00:25:54,860 --> 00:25:57,935
property as a function
of direction.

378
00:25:57,935 --> 00:26:00,560
It looks as though we've lost
all information about the

379
00:26:00,560 --> 00:26:03,850
direction of the resulting
vector, the generalized

380
00:26:03,850 --> 00:26:06,370
displacement.

381
00:26:06,370 --> 00:26:12,180
It looks as though that's
not in here at all.

382
00:26:12,180 --> 00:26:17,650
Well, there was an ad years
ago for a bottle spaghetti

383
00:26:17,650 --> 00:26:23,160
sauce, which offends many cooks
who are very fiercely

384
00:26:23,160 --> 00:26:25,030
proud of their own
spaghetti sauce.

385
00:26:25,030 --> 00:26:29,280
So here's a wife who's using
the canned stuff.

386
00:26:29,280 --> 00:26:34,170
And her husband comes home and
looks at it very, very

387
00:26:34,170 --> 00:26:37,925
skeptically and says,
where's the basil?

388
00:26:37,925 --> 00:26:40,140
And the housewife says,
it's in there.

389
00:26:40,140 --> 00:26:41,950
It's in there.

390
00:26:41,950 --> 00:26:44,730
And he sniffs again and says,
where is the basil?

391
00:26:44,730 --> 00:26:46,820
And she says finally,
it's in there.

392
00:26:46,820 --> 00:26:47,330
It's in there.

393
00:26:47,330 --> 00:26:49,180
Well, this is a similar
situation.

394
00:26:49,180 --> 00:26:52,390
Where is the direction of
the generalized force?

395
00:26:52,390 --> 00:26:54,820
And I say, it's in there.

396
00:26:54,820 --> 00:26:55,990
It's in there.

397
00:26:55,990 --> 00:26:58,890
Not obvious, but
it's in there.

398
00:26:58,890 --> 00:27:02,310
So let me now show you
where it is lurking.

399
00:27:05,350 --> 00:27:07,295
I think we have time to
carry this through.

400
00:27:12,160 --> 00:27:17,260
Let me describe how it's
embodied in the quadric and

401
00:27:17,260 --> 00:27:19,990
then prove to you that this
is indeed the case.

402
00:27:19,990 --> 00:27:26,930
And I'll use a general quadric
in the form of an ellipsoid.

403
00:27:30,780 --> 00:27:33,140
That looks more like an
egg than an ellipsoid.

404
00:27:33,140 --> 00:27:34,735
It has a pointed end.

405
00:27:42,660 --> 00:27:44,670
OK, this is something
that's called the

406
00:27:44,670 --> 00:27:45,985
radius normal property.

407
00:27:56,170 --> 00:28:01,370
And what it says, in words, is
that, if you pick a particular

408
00:28:01,370 --> 00:28:08,240
direction relative to the
quadric, we know that its

409
00:28:08,240 --> 00:28:10,990
length is going to be inversely
proportional to the

410
00:28:10,990 --> 00:28:13,350
square root of the value
of the property.

411
00:28:13,350 --> 00:28:17,420
If you want to know what the
direction of the resulting

412
00:28:17,420 --> 00:28:21,840
vector is, for example, in our
relation for conductivity now

413
00:28:21,840 --> 00:28:28,230
getting very warm, it says the
current flow is given by a

414
00:28:28,230 --> 00:28:30,790
linear combination of
every component of

415
00:28:30,790 --> 00:28:32,250
the electric field.

416
00:28:32,250 --> 00:28:35,020
The radius normal properties is
that, if you want to know

417
00:28:35,020 --> 00:28:40,060
where J is for this particular
direction, look at the point

418
00:28:40,060 --> 00:28:44,040
where the radius vector
intersects the surface of the

419
00:28:44,040 --> 00:28:49,190
quadric and, at that point,
construct a perpendicular to

420
00:28:49,190 --> 00:28:52,700
the surface, which, in general,
will not be parallel

421
00:28:52,700 --> 00:28:57,950
to R. And this will be the
direction of J. It won't give

422
00:28:57,950 --> 00:28:58,770
you the magnitude.

423
00:28:58,770 --> 00:29:00,210
But it'll give you the
direction of it.

424
00:29:18,300 --> 00:29:27,380
OK, let me now prove to you
that the quadric does have

425
00:29:27,380 --> 00:29:28,630
this property.

426
00:29:32,020 --> 00:29:41,360
In our conductivity relation,
the direction of J is going to

427
00:29:41,360 --> 00:29:43,960
be given by the tensor
relation.

428
00:29:43,960 --> 00:29:51,200
Namely that J sub i is equal
to sigma ij times E sub j,

429
00:29:51,200 --> 00:29:57,950
which can be written as sigma ij
times the direction cosines

430
00:29:57,950 --> 00:30:04,590
of E sub j times the
magnitude of E.

431
00:30:04,590 --> 00:30:09,310
I'm going to want to compare
this expression, with which

432
00:30:09,310 --> 00:30:13,910
you're very familiar term by
term, with the normal to the

433
00:30:13,910 --> 00:30:17,350
surface that we can
compute from its

434
00:30:17,350 --> 00:30:20,740
derivative at that point.

435
00:30:20,740 --> 00:30:26,980
So this will say that J1 is
equal to sigma 1, 1 l1 times

436
00:30:26,980 --> 00:30:34,390
the magnitude of E plus sigma
1, 2 times l2 times the

437
00:30:34,390 --> 00:30:39,795
magnitude of E plus sigma 1, 3
times l3 times the magnitude

438
00:30:39,795 --> 00:30:42,910
of E. And I don't need
to write much more.

439
00:30:42,910 --> 00:30:50,640
J2 will be sigma 2, 1 times l1
magnitude of E plus sigma 2, 2

440
00:30:50,640 --> 00:30:53,772
l2 times the magnitude
of E and so on.

441
00:30:58,200 --> 00:31:03,020
What is going to be the normal
to the surface as a function

442
00:31:03,020 --> 00:31:04,270
of direction?

443
00:31:07,110 --> 00:31:14,740
OK, to do this I'm going to
say that the normal to the

444
00:31:14,740 --> 00:31:22,760
surface is going to be, if I
have some function of xyz, the

445
00:31:22,760 --> 00:31:25,330
normal to that function
is the gradient.

446
00:31:30,250 --> 00:31:34,570
Let me say that that's
G of xyz.

447
00:31:34,570 --> 00:31:41,360
And we can say then that the x1
component of the normal is

448
00:31:41,360 --> 00:31:45,290
not going to be equal to but
proportional to the gradient

449
00:31:45,290 --> 00:31:48,790
of the equation for the quadric
with respect to x1,

450
00:31:48,790 --> 00:31:51,650
dx2, and dx3.

451
00:31:51,650 --> 00:31:55,080
So it's going to be proportional
to the

452
00:31:55,080 --> 00:31:58,340
differential of the function
that gives us the surface with

453
00:31:58,340 --> 00:32:03,810
respect to x1 times i plus the
differential of the function

454
00:32:03,810 --> 00:32:15,370
with respect to x2 times
j plus dF dx3 times k.

455
00:32:15,370 --> 00:32:18,730
So this is the normal
entirely.

456
00:32:18,730 --> 00:32:24,030
So if we split this into
components, N1 is going to be

457
00:32:24,030 --> 00:32:27,660
equal to dF dx1.

458
00:32:27,660 --> 00:32:30,800
And if I differentiate the
equation for the quadric,

459
00:32:30,800 --> 00:32:36,630
that's going to be 2 sigma
1, 1 times x1.

460
00:32:36,630 --> 00:32:44,950
Remember the equation for the
quadrant is sigma 1, 1 times

461
00:32:44,950 --> 00:32:49,080
x1 squared plus sigma
1, 2 times x1 x2.

462
00:32:49,080 --> 00:32:50,610
So if I differentiate--
let me write it down.

463
00:33:01,390 --> 00:33:04,910
If I take those terms and
differentiate with respect to

464
00:33:04,910 --> 00:33:09,010
x1, I'm going to get 2
sigma 1, 1 times x1.

465
00:33:09,010 --> 00:33:13,910
Here I'll get sigma
1, 2 times x2.

466
00:33:13,910 --> 00:33:18,540
But then down in this line where
I have a sigma 2, 1 x2

467
00:33:18,540 --> 00:33:21,960
x1, if I differentiate with
respect to x1, I'll have

468
00:33:21,960 --> 00:33:25,200
another term sigma 2, 1.

469
00:33:25,200 --> 00:33:29,540
And if I differentiate with
respect to x3, I'll have sigma

470
00:33:29,540 --> 00:33:34,850
1, 3 plus sigma 3, 1 times x3.

471
00:33:34,850 --> 00:33:39,060
And a similar thing for the x2
component of the normal,

472
00:33:39,060 --> 00:33:41,240
that's going to be proportional
to the gradient

473
00:33:41,240 --> 00:33:43,150
with respect to x2.

474
00:33:43,150 --> 00:33:47,650
And that's going to be equal to
sigma 1, 2 plus sigma 2, 1

475
00:33:47,650 --> 00:33:58,920
times x1 plus 2 sigma 2, 2 times
x2 plus sigma 2, 3 plus

476
00:33:58,920 --> 00:34:04,610
sigma 3, 2 times x3 and
similarly for N3.

477
00:34:07,980 --> 00:34:11,840
OK, I've made my case if I
can demonstrate that each

478
00:34:11,840 --> 00:34:18,310
component of J, J sub i, is
proportional to each component

479
00:34:18,310 --> 00:34:21,190
of the gradient to the function

480
00:34:21,190 --> 00:34:23,989
that defines the quadric.

481
00:34:23,989 --> 00:34:30,700
If you look at them, they're
close but no cigar.

482
00:34:30,700 --> 00:34:32,230
The first term is OK.

483
00:34:32,230 --> 00:34:36,219
I've got a 2 out in front of
sigma 1, 1 for the x1 term.

484
00:34:41,130 --> 00:34:49,010
Here, though, I have the sum
of two off-diagonal terms.

485
00:34:49,010 --> 00:34:55,590
And for this expression,
I have just sigma 1, 2.

486
00:34:55,590 --> 00:35:00,700
And down here I've just the
single term sigma 2, 1.

487
00:35:00,700 --> 00:35:07,230
So the conclusion we're forced
to draw is that these two

488
00:35:07,230 --> 00:35:10,460
expressions would have
components of a vector that

489
00:35:10,460 --> 00:35:15,200
are parallel to one another
if sigma 1, 2 was

490
00:35:15,200 --> 00:35:18,490
equal to sigma 2, 1.

491
00:35:18,490 --> 00:35:21,660
Because if these were equal,
I could write just

492
00:35:21,660 --> 00:35:23,940
twice sigma 1, 2.

493
00:35:23,940 --> 00:35:28,200
And down here I could write
twice sigma 2, 1 and do that

494
00:35:28,200 --> 00:35:30,380
all the way through these
two expressions.

495
00:35:30,380 --> 00:35:33,060
And then this factor of 2 would
just be part of the

496
00:35:33,060 --> 00:35:35,900
proportionality constant.

497
00:35:35,900 --> 00:35:46,210
So the radius normal property
will be true or valid only for

498
00:35:46,210 --> 00:35:47,460
symmetric tensors.

499
00:35:54,660 --> 00:35:58,400
That is it's going to be true
only if sigma ij is

500
00:35:58,400 --> 00:36:01,570
equal to sigma ji.

501
00:36:01,570 --> 00:36:03,900
We mentioned when we first
started talking about second

502
00:36:03,900 --> 00:36:09,980
ranked tensors that a tensor
does not have to be symmetric.

503
00:36:09,980 --> 00:36:15,070
And showing that the tensor is
symmetric has nothing to do

504
00:36:15,070 --> 00:36:16,250
with symmetry.

505
00:36:16,250 --> 00:36:19,550
Because it's symmetric in an
algebraic sense not in the

506
00:36:19,550 --> 00:36:23,770
literal sense of geometrical
symmetry.

507
00:36:23,770 --> 00:36:28,770
And there are some properties,
the thermal electricity tensor

508
00:36:28,770 --> 00:36:32,450
is one notable example, where
the tensor is a second ranked

509
00:36:32,450 --> 00:36:37,360
tensor but it decidedly
is not symmetric.

510
00:36:37,360 --> 00:36:53,445
So this is OK for most
properties but not all.

511
00:37:08,850 --> 00:37:13,820
OK, so here are properties of
the representation surface

512
00:37:13,820 --> 00:37:21,380
that we can construct from the
representation quadric.

513
00:37:21,380 --> 00:37:29,230
And what I would like to do
following this and after the

514
00:37:29,230 --> 00:37:34,340
inevitable unpleasantness of
a quiz is to look at some

515
00:37:34,340 --> 00:37:37,500
specific properties that
illustrate second ranked

516
00:37:37,500 --> 00:37:39,460
tensor properties.

517
00:37:39,460 --> 00:37:45,500
And in particular, I would like
to look at some other

518
00:37:45,500 --> 00:37:49,040
sorts of second ranked tensor
properties that represent

519
00:37:49,040 --> 00:37:53,120
generalized forces, namely
stress and strain.

520
00:37:53,120 --> 00:37:56,760
You've seen these before in
other contexts perhaps not

521
00:37:56,760 --> 00:38:00,570
actually defined rigorously
in terms of tensors.

522
00:38:00,570 --> 00:38:04,580
Because, obviously, stress and
strain can be regarded as

523
00:38:04,580 --> 00:38:05,740
generalized forces.

524
00:38:05,740 --> 00:38:08,810
A stress is something that
you can apply to a solid.

525
00:38:08,810 --> 00:38:11,840
And that will cause various
things to happen.

526
00:38:11,840 --> 00:38:16,280
In addition to mechanical
behavior, different properties

527
00:38:16,280 --> 00:38:17,910
and effects can result.

528
00:38:17,910 --> 00:38:21,850
So that is going to involve a
generalized force, which is a

529
00:38:21,850 --> 00:38:23,410
second ranked tensor.

530
00:38:23,410 --> 00:38:26,050
And a thing that might happen
could be a vector.

531
00:38:26,050 --> 00:38:28,420
It could be another second
ranked tensor.

532
00:38:28,420 --> 00:38:32,040
And this is going to introduce
us to the nasty world of

533
00:38:32,040 --> 00:38:34,640
higher ranked tensor properties
and their

534
00:38:34,640 --> 00:38:36,830
representation surfaces.

535
00:38:36,830 --> 00:38:41,180
So this is a nice place to quit
and pause for a quiz.

536
00:38:41,180 --> 00:38:46,390
When we resume, we'll look at
stress and strain in terms of

537
00:38:46,390 --> 00:38:47,640
our tensor algebra.