03 Drude Theory of Electrons in Metals Sommerfeld Free Electron Theory of Electrons in Metals.srt

1
00:00:00,000 --> 00:00:03,689
在我们开始之前，我想对如何
before we get going I want to do a
little bit of philosophizing about how

2
00:00:03,689 --> 00:00:07,740
我们开始的一周开始就是在物理学中学习东西
it is we learn things in in physics at
the beginning of the week we started

3
00:00:07,740 --> 00:00:11,910
用固体振动的玻尔兹曼模型，然后我们决定
with the Boltzmann model of vibrations
in solids and then we decided that

4
00:00:11,910 --> 00:00:15,570
不太正确，因此我们必须使用固体的爱因斯坦模型对其进行改进
wasn't quite right so we had to improve
it with the Einstein model of the solid

5
00:00:15,570 --> 00:00:18,840
那是相当不错的，但是没有正确的低温行为
and that was pretty good but it didn't
get the low-temperature behavior right

6
00:00:18,840 --> 00:00:22,199
所以我们不得不用德拜固体理论对其进行改进，甚至
so we had to improve it with the Debye
theory of solids and that was even

7
00:00:22,199 --> 00:00:25,019
更好，但今年晚些时候我们也将不得不对其进行改进
better but we're going to have to
improve it later on in the year as well

8
00:00:25,019 --> 00:00:28,949
一个完全正确的问题是我们为什么不从中学到正确的东西
and a perfectly valid question is why
didn't we learn the right thing at the

9
00:00:28,949 --> 00:00:32,969
开始，而不必经历所有这些错误的模型，并且
beginning and not have to go through all
of these models which are all wrong and

10
00:00:32,969 --> 00:00:37,530
这不仅仅是因为我喜欢告诉您了解有关所有
it's not just because I like telling you
know history stories about how all of

11
00:00:37,530 --> 00:00:41,160
这是开发出来的，但我们这样做的原因是因为这始终是我们的方式
this was developed but the reason we do
this is because this is always how we

12
00:00:41,160 --> 00:00:43,920
在物理学中学习事物，即使它是
learn things in physics you always learn
the simple model even though it's

13
00:00:43,920 --> 00:00:47,789
首先是错误的，因为这样容易思考，例如
incorrect first because it's a lot
easier to think about so for example you

14
00:00:47,789 --> 00:00:51,149
首先要学习古典力学，这是不对的，因为你知道
learn in classical mechanics first and
it's not really right because you know

15
00:00:51,149 --> 00:00:54,059
有相对论和量子力学，然后将它们叠加
there's special relativity and there's
quantum mechanics and you layer those on

16
00:00:54,059 --> 00:00:58,469
回到顶部，但是您的直觉总是回到简单的模型上
top later but your intuition always
falls back to the simple model it's a

17
00:00:58,469 --> 00:01:01,559
更容易思考，这就是我们这样做的原因，而这正是
lot easier to think about and that's why
we do this and this is exactly what

18
00:01:01,559 --> 00:01:06,380
今天我们要谈论上次开始的金属时要做的
we're going to be doing today when we
talk about metals we started last time

19
00:01:06,380 --> 00:01:13,020
金属和我们用于许多事物的简单原油模型是德鲁伊
metals and the simple crude model we use
for many many things is the druid

20
00:01:13,020 --> 00:01:15,869
电子和金属的图片
apicture
of electrons and metals which is

21
00:01:15,869 --> 00:01:24,530
基本上只是动力学理论。 
basically just kinetic theory kinetic
theory for electrons for electrons and

22
00:01:24,530 --> 00:01:32,610
在最后一次结束时，我们导出了dru来传递方程D动量DT 
at the end of last time we derived the
dru to transport equation D momentum DT

23
00:01:32,610 --> 00:01:39,840
等于在某些现象学散射上电子减去P上的力
equals the force on the electron minus P
over some phenomenological scattering

24
00:01:39,840 --> 00:01:48,570
时间力是洛伦兹力的时间tau 
time tau where the force is Lorentz
force the general force that the

25
00:01:48,570 --> 00:01:52,770
电子的感觉，所以基本上只是牛顿方程，增加了阻力
electron feels so it's basically just
Newton's equation with an added drag

26
00:01:52,770 --> 00:01:57,960
 P超过tau的力P代表散射
force P over tau which sort of
represents scattering sort of slows the

27
00:01:57,960 --> 00:02:01,799
现在，对于许多实验，电子都会以某种方式下降，我们将对此感兴趣
electron down in some way now for many
experiments we're going to be interested

28
00:02:01,799 --> 00:02:05,939
在我们实际上正在进行某种稳态实验，您可能会应用
in we are actually doing some sort of
steady state experiment you might apply

29
00:02:05,939 --> 00:02:10,080
稳定的电场，您可能会施加稳定的电流，并且您对此感兴趣
a steady electric field you might apply
a steady current and you're interested

30
00:02:10,080 --> 00:02:15,590
处于稳定状态的结果，因此大大简化了寻找工作
in a steady state result so it's
simplifies things a lot to look for

31
00:02:15,590 --> 00:02:25,189
稳态，所以DP DT等于零，所以我们尝试解决，使GP DT等于
steady states so DP DT equals zero so
let's try to solve that so GP DT equals

32
00:02:25,189 --> 00:02:28,939
零，我们将在右侧施加力，因此我要使用的力
zero we'll put in the force on the right
hand side here so the force I'm going to

33
00:02:28,939 --> 00:02:36,680
用洛伦兹力加V叉B代替，然后我们需要P超过tau 
replace that by the Lorentz force plus V
cross B and then we need P over tau over

34
00:02:36,680 --> 00:02:42,139
在这里，但是我要用tau的MV代替P，因为我们有一个速度
here but I'm going to replace P by M V
over tau just because we have a velocity

35
00:02:42,139 --> 00:02:46,970
哦，这里有相同的速度，我们现在要在其中放置另一个速度
here oh there's the same velocity here
we'll put another velocity there now

36
00:02:46,970 --> 00:02:51,260
速度是一个非常好的数量，但实际上不是你所需要的
velocity is a perfectly good quantity
but it's not actually what you are

37
00:02:51,260 --> 00:02:54,620
有可能在实验中进行测量，而您更有可能进行测量
likely to measure in the experiment
which you're much more likely to measure

38
00:02:54,620 --> 00:03:02,030
实验是电流密度电流密度与
an experiment is the current density
current density very closely related to

39
00:03:02,030 --> 00:03:10,750
速度J是电子数或电子密度
velocity J is the number of electron or
the density of electrons density of

40
00:03:10,750 --> 00:03:16,400
电子乘以电子上的电荷至负e乘以电子
electrons times the charge on the
electron goes to minus e times his

41
00:03:16,400 --> 00:03:21,109
速度好，所以电流密度就是你拥有多少电子，但是
velocity okay so the current density is
just how many electrons you have but the

42
00:03:21,109 --> 00:03:26,269
给他们充电，移动速度有多快，所以每次我有速度时
charge they have and how fast they're
moving so every time I have velocity in

43
00:03:26,269 --> 00:03:30,859
该方程式我将插入J并除以N和负E 
that equation I'm going to plug in J
instead and divide by an N and a minus E

44
00:03:30,859 --> 00:03:36,979
第一步，我还要将e移到另一边
and one step I'm also going to move e
over to the other side and get this

45
00:03:36,979 --> 00:03:53,840
方程e在整个J交叉B上等于1，然后在N平方的tau时间上等于M 
equation e equals one over n e J cross B
and then plus M over N squared tau times

46
00:03:53,840 --> 00:04:05,239
 J对一次太多的步骤感到非常高兴，是的，所以我只是我
J is that good happy with that too many
steps at once yeah so I just I just

47
00:04:05,239 --> 00:04:07,819
插入j4v，然后将D移动到另一侧
plugged in j4v
and then move D to the other side and

48
00:04:07,819 --> 00:04:13,069
通过任何驱动器将其划分为我的密钥，好吧，所以这有两个术语
divide it through by any drive to my key
okay good so there's two terms in this

49
00:04:13,069 --> 00:04:15,739
等式，我们称它们为两个不同的东西
equation and we'll call them two
different things

50
00:04:15,739 --> 00:04:23,530
让我们将此称为e并行，我们将其称为e暂停
let's call this e parallel and we'll
call this e halt

51
00:04:23,530 --> 00:04:29,620
我用这个画一个图，所以我们有一块金属
let me draw a diagram with this so we
have is we have a block of metal like

52
00:04:29,620 --> 00:04:35,710
这将使电流像这样的电流流过金属
this we're going to run a current
through the metal like this current

53
00:04:35,710 --> 00:04:40,360
像我们可能会施加磁场
density in content C out like that we
might apply a magnetic field

54
00:04:40,360 --> 00:04:46,300
也许像这样的B垂直于金属，然后我们将有一个平行
maybe perpendicular to the metal like
this B and then we will have a parallel

55
00:04:46,300 --> 00:04:50,680
在这个方向上的电场在那个方向上然后是霍尔垂直
in this direction electric field in that
direction and then a Hall perpendicular

56
00:04:50,680 --> 00:05:00,580
电流和垂直于磁场以及一个大厅，它是
to the current and perpendicular to the
magnetic field as well a hall and it's

57
00:05:00,580 --> 00:05:04,960
之所以称为霍尔电场是因为它是由埃德温·霍尔（Edwin Hall）发现的， 
called the hall electric field because
it was discovered by Edwin Hall who is

58
00:05:04,960 --> 00:05:09,370
正是在1879年进行了这类实验，他发现当他
doing exactly these kind of experiments
in 1879 and he discovered that when he

59
00:05:09,370 --> 00:05:13,270
在磁场中使电流流过金属，最终导致
ran a current through a metal in a
magnetic field he ended up with an

60
00:05:13,270 --> 00:05:15,880
垂直于它们两个的电场，您之前可能已经遇到过
electric field perpendicular both of
them you may have run into this before

61
00:05:15,880 --> 00:05:20,110
这是洛伦兹力量的一个非常明显的结果，发生的是你在
it's a pretty clear result of the
Lorentz force what's happening is you're

62
00:05:20,110 --> 00:05:23,980
使电子穿过材料，并且它们试图弯曲，因为
running electrons through the material
and they're trying to curve because of

63
00:05:23,980 --> 00:05:29,410
磁场并建立一个电场，每个人都满意
the magnetic field and that builds up an
electric field good everyone happy with

64
00:05:29,410 --> 00:05:36,310
差不多，所以让我们尝试了解一下这个方程
that more or less okay so let's try to
you know think about this equation a

65
00:05:36,310 --> 00:05:42,780
让我们更简单地考虑一个磁场为零的简单情况
little more closely let's take a simple
case case one magnetic field equals zero

66
00:05:42,780 --> 00:05:48,580
所以在那种情况下，我们只有电场让我绕过另一个
so in that case we just have electric
field let me turn it around the other

67
00:05:48,580 --> 00:05:50,860
将J移到他们旁边的方式
way
move the J to their side so J the

68
00:05:50,860 --> 00:05:58,390
然后，电流在m倍于电场的平方n上被平方
current is then n e squared Tao over m
times the electric field this quantity

69
00:05:58,390 --> 00:06:07,450
这是电导率Sigma电导率这是电导率，因为
here is a conductivity Sigma
conductivity it's a conductivity because

70
00:06:07,450 --> 00:06:12,640
它把电场与电流联系起来
it relates the electric field to the
current now the expression for the

71
00:06:12,640 --> 00:06:16,750
我们在m上推导出e平方的Tao的电导率是Deuter 
conductivity that we derived an e
squared Tao over m is the Deuter

72
00:06:16,750 --> 00:06:20,350
电导率是我们将在Drude理论中计算出的电导率
conductivity is the conductivity we
would calculate in the Drude theory we

73
00:06:20,350 --> 00:06:24,729
只是计算了一下，我们对什么是什么应该有一个很容易的直觉
just calculated it and we should have
pretty easy intuition for what's what's

74
00:06:24,729 --> 00:06:31,300
继续讲，电导率具有最高密度的因素，因为
going on here the conductivity has a
factor of density up top because the

75
00:06:31,300 --> 00:06:34,240
电子越多，电流就越多
more electrons you have the more current
you're going to get and the more

76
00:06:34,240 --> 00:06:37,840
您将要获得的电导率是tau的系数
conductivity you're going to get it has
a factor of tau up top

77
00:06:37,840 --> 00:06:41,950
因为越长的道越大道意味着更长的散射时间和更长的时间
because longer Tao bigger Tao means
longer scattering time and the longer

78
00:06:41,950 --> 00:06:45,040
散布时间或更少的事情变得更好
scattering time you have or the less
things you have to run into the better

79
00:06:45,040 --> 00:06:48,700
现在你的电导率会变成事实，我正在楼下，看起来有点
your conductivity is going to be now the
fact I'm mass downstairs looks a little

80
00:06:48,700 --> 00:06:53,470
稍微复杂一点，但实际上如果质量等于F等于MA 
bit more complicated but actually it
just comes from F equals MA if your mass

81
00:06:53,470 --> 00:06:58,030
对于固定力来说很小，您的加速度必须更大，所以您
is small for a fixed force your
acceleration has to be larger so you

82
00:06:58,030 --> 00:07:02,710
施加一定的固定力，如果电子质量较小，则电子运动得更快
apply out some fixed force the electrons
move faster if their mass were smaller

83
00:07:02,710 --> 00:07:08,650
这就是为什么现在我们可能要问楼下传出大量东西的原因，或者保罗提请德
that's why the mass comes out downstairs
now we might ask or Paul drew de might

84
00:07:08,650 --> 00:07:11,020
问这是一个好答案吗？ 
ask is this a good answer is this a bad
answer

85
00:07:11,020 --> 00:07:17,320
现在我们实际上不知道，因为它是一个未知数
and right now we actually don't know
because how is some unknown number it's

86
00:07:17,320 --> 00:07:21,550
他不知道如何计算的一些现象，我们只能把
some phenomena he doesn't know how to
calculate it we can just we have to put

87
00:07:21,550 --> 00:07:25,120
从现象学上讲，所以实际上它几乎适合任何东西
it in phenomenologically so really it
would fit pretty much anything at this

88
00:07:25,120 --> 00:07:27,760
点，所以你可能会想到它，你知道排序
point
so you might think of it you know sort

89
00:07:27,760 --> 00:07:31,660
这反过来说明电导率的测量实际上是
of turn this on its head that
measurement of conductivity is actually

90
00:07:31,660 --> 00:07:37,110
参数tau的度量正常，这通常是如何查看的
a measurement of the parameter tau okay
and that's frequently how it's viewed

91
00:07:37,110 --> 00:07:43,780
好吧，让我们做一些稍微复杂一些的事
all right let's do a little something a
little bit more complicated case to case

92
00:07:43,780 --> 00:07:48,789
 B等于零的两个，我们可能会以多种方式思考
two which is B not equal to zero and
there are various ways we might think

93
00:07:48,789 --> 00:07:53,350
关于B不等于零的一种可能
about B not equal to zero one
possibility of doing this experiment is

94
00:07:53,350 --> 00:08:00,070
我们可能想像一下，假设我们知道，所以将其称为假设我们知道一对
we might imagine suppose we know so it's
called its to a suppose we know a couple

95
00:08:00,070 --> 00:08:03,880
假设我们知道电子的密度，当然我们知道
things suppose we know the density of
our electrons of course we know the

96
00:08:03,880 --> 00:08:12,250
电子上的电荷假设我们知道电流并且我们测量我们测量一个
charge on the electron suppose we know
the current and we measure we measure a

97
00:08:12,250 --> 00:08:19,660
然后根据犹大理论停下来，我们知道然后我们得到一个磁场B 
halt then according to Judah Theory we
then know then we get a magnetic field B

98
00:08:19,660 --> 00:08:25,479
实际上，这是一种非常常用的测量磁场的方法
and in fact this is a very frequently
used method for measuring magnetic

99
00:08:25,479 --> 00:08:34,679
在现场它被称为霍尔传感器，并且您知道它仍然经常在
fields it's known as a hall sensor and
you know it's still used frequently in

100
00:08:34,679 --> 00:08:39,218
现在，让我们稍微详细一点

101
00:08:39,219 --> 00:08:46,870
通常在这里，如果您在磁场中有电流，您将会得到一些东西
here generally if you have a current in
a magnetic field you will get something

102
00:08:46,870 --> 00:08:54,550
的形式，我们定义此数量rhv cross j注意转订单
of the form we define this quantity
rhv cross j notice that turn the order

103
00:08:54,550 --> 00:08:59,110
 J和B在那边的-在这里这只是常用的定义
of J and B over there - over here this
is just a definition of a commonly used

104
00:08:59,110 --> 00:09:03,060
称为霍尔系数的数量
quantity known as the Hall coefficient

105
00:09:05,850 --> 00:09:13,990
所以嗯，你知道你是否知道J是什么，B就是II 
so um you know if you know what J is you
know what B is measure II get the Hall

106
00:09:13,990 --> 00:09:25,120
例如，系数如此正确，因此吸引了犹大理论比较
coefficient for example so right so in
drew the theory Judah Theory comparing

107
00:09:25,120 --> 00:09:31,480
这个方程式，所以这是犹大理论中的B霍尔对不起G霍尔
to that equation so this is B Hall sorry
G Hall in Judah Theory comparing to that

108
00:09:31,480 --> 00:09:41,110
那里的等式，霍尔系数RH在密度乘以1时得到-e减去
equation over there the hall coefficient
RH is 1 over density times - e the minus

109
00:09:41,110 --> 00:09:45,040
信号来自这样一个事实，即我将这两个顺序颠倒了
sign comes from the fact that I flipped
the order of these two compared to over

110
00:09:45,040 --> 00:09:49,450
那里的霍尔电场
there
so the Hall electric field that you

111
00:09:49,450 --> 00:09:53,890
将测量与霍尔系数成正比，如果你是
would measure would be proportional to
the Hall coefficient here and if you're

112
00:09:53,890 --> 00:09:58,630
试图建立一个霍尔传感器以找到准确的测量方法
trying to build a hall sensor in order
to find accurate ways of measuring

113
00:09:58,630 --> 00:10:03,910
使用电子设备或使用电压测量结果产生的磁场
magnetic fields using electronics or
using voltage measurements you really

114
00:10:03,910 --> 00:10:08,290
希望霍尔电压尽可能大，以便
want the hall voltage to be large as
large as possible so that you have a

115
00:10:08,290 --> 00:10:13,090
要测量大电场，以便这样做，您通常会选择一个
large electric field to measure so to do
that what you do is you usually choose a

116
00:10:13,090 --> 00:10:22,950
密度较小的材料，所以使用时要用较小的密度来变大
material with a small density so use use
use small density to get big to get big

117
00:10:22,950 --> 00:10:31,090
大e和典型的霍尔传感器由半导体和其他材料制成
big e and typical Hall sensors are built
with semiconductors and other materials

118
00:10:31,090 --> 00:10:36,430
电子密度很小，我们稍后将介绍半导体
that have a small electron density and
we'll come to semiconductors later on in

119
00:10:36,430 --> 00:10:41,740
这个词现在让我们开始实验吧，大概是
the term now let's turn this experiment
now on its head probably the way that

120
00:10:41,740 --> 00:10:49,270
保罗·德鲁（Paul Drude）对他的案子有一个想法，他可能不知道有什么问题， 
Paul Drude a thought about it in his
case he probably knew no the dent up and

121
00:10:49,270 --> 00:10:52,870
因此，在他的情况下，他可能知道磁场，电子电荷和
so in his case he probably knew the
magnetic field the electron charge and

122
00:10:52,870 --> 00:11:02,260
当前测得的霍尔数和得到的结果
the current measured measure Hall and
what you get

123
00:11:02,260 --> 00:11:07,970
你得到的是样品中电子的密度，因为他当然
what you get is the density of electrons
in your sample because of course he

124
00:11:07,970 --> 00:11:12,590
不太了解他材料中的电子密度，所以让我们尝试一下
didn't know the density of electrons in
his materials well so let's try this

125
00:11:12,590 --> 00:11:16,760
假设我们针对多种不同的材料执行此操作，因此我们将采用
suppose we do this for a bunch of
different materials so we'll take a

126
00:11:16,760 --> 00:11:23,200
一堆金属，例如锂钠，钾，铜，铜
bunch of metals like lithium sodium
potassium copper are sort of typical

127
00:11:23,200 --> 00:11:29,240
好的金属，保罗·犹大会注意到的第一件事是
good metals and the first thing that
Paul Judah would have noticed is that

128
00:11:29,240 --> 00:11:35,360
对于所有这些金属，相对湿度均小于零，这是他所预测的
for all of these metals RH is less than
zero which is what he predicted by his

129
00:11:35,360 --> 00:11:40,010
公式在这里，所以很多东西都很好，然后他可以实际输入
formula over here so that much is good
and then he can actually put in you know

130
00:11:40,010 --> 00:11:45,830
测量e霍尔的幅度并尝试提取电子的密度。 
measure the magnitude of e Hall and try
to extract the density of electrons and

131
00:11:45,830 --> 00:11:51,770
他得到的是每个原子约0.8个电子，这里约1.2 
what he would have gotten was about 0.8
electrons per atom here about 1.2

132
00:11:51,770 --> 00:11:56,000
每个原子电子约1.1个电子/原子1.5个电子/个
electrons per atom here about 1.1
electrons per atom 1.5 electrons per

133
00:11:56,000 --> 00:12:00,560
原子似乎是个合理的数字，您知道想知道几个电子
atom seems like reasonable numbers a
couple of electrons you know wondering

134
00:12:00,560 --> 00:12:05,060
每个原子或多或少有一个电子，这是一个好答案吗？ 
one electron per atom more or less is
this a good answer is this a bad answer

135
00:12:05,060 --> 00:12:09,200
好吧，如果你考虑一秒钟，你会记得你的周期表
well if you think about it for a second
you'll remember your periodic table

136
00:12:09,200 --> 00:12:13,940
铜是元素周期表中的第29个元素，因此它具有29个质子
copper is the twenty-ninth element on
the periodic table so it has 29 protons

137
00:12:13,940 --> 00:12:18,890
和29个电子，我们测得每个原子1.5个电子
and 29 electrons and we measured 1.5
electrons per atom

138
00:12:18,890 --> 00:12:24,590
其他27.5到哪里去了，所以这可能有点令人费解
where'd the other 27.5 go so this might
have been a little bit puzzling to to

139
00:12:24,590 --> 00:12:29,000
犹大，但实际上我们现在对原子结构有所了解
Judah but actually now that we know a
little bit about the atomic structure we

140
00:12:29,000 --> 00:12:31,670
了解原子的壳结构，我们对化学有一点了解
know about the shell structure of atoms
we know a little bit about chemical

141
00:12:31,670 --> 00:12:34,250
键合，我们甚至会在稍后的内容中谈论化学键合
bonding we'll even talk a little bit
about chemical bonding later on in the

142
00:12:34,250 --> 00:12:39,650
一年，我们知道实际上一个原子中的许多电子实际上在
year and we know that in fact many of
the electrons in an atom are actually in

143
00:12:39,650 --> 00:12:44,630
这些核心轨道电子非常靠近原子核的轨道
orbitals very close to the to the
nucleus these core orbital electrons are

144
00:12:44,630 --> 00:12:53,570
基本上卡住了，所以核心轨道上的littlez电子或明显的电子
basically stuck so core orbitals
littlez electrons or overt electrons

145
00:12:53,570 --> 00:13:06,230
不要移动唯一能移动外壳电子的电子，也就是
don't move the only ones that do move
outer shell electrons also known as

146
00:13:06,230 --> 00:13:09,220
价电子
valence electrons

147
00:13:10,070 --> 00:13:16,020
移动，因此在固体中运行的电子是
move so the electrons that are running
around in the in the solid are the

148
00:13:16,020 --> 00:13:22,590
所谓的价电子，您知道化学作用，实际上我们知道
so-called valence electrons and you know
do from chemistry we actually know how

149
00:13:22,590 --> 00:13:27,270
这些材料具有许多价电子，实际上所有这四个
many valence electrons these materials
have and in fact all these four

150
00:13:27,270 --> 00:13:33,210
材料的化合价为1，意味着在最外层壳中只有一个电子
materials have valence one meaning only
one electron in the outermost shell so

151
00:13:33,210 --> 00:13:36,660
如果我们假设只有最外面的电子移动最外面的电子
if we assume only the outermost
electrons move the ones in the outermost

152
00:13:36,660 --> 00:13:40,770
壳，我们应该预测每个原子一个电子
shell
we should predict one electron per atom

153
00:13:40,770 --> 00:13:44,760
实际上，您知道与测得的结果相当一致
and that's actually you know in fairly
good agreement with what is measured

154
00:13:44,760 --> 00:13:50,400
从实验上来说，这对于juda理论非常有用
experimentally so that's pretty good for
for juda theory we get roughly one

155
00:13:50,400 --> 00:13:57,570
每个原子移动的电子，所以我们可能会为成功感到鼓舞
electron per atom moving around so we
might you know emboldened by our success

156
00:13:57,570 --> 00:14:02,780
我们可能会尝试其他一些材料，所以让我们尝试一些价
we might try some other materials so
let's try some with valence to valence

157
00:14:02,780 --> 00:14:07,920
等于2我们有像铍这样的材料我们有镁
equals two we have materials like
beryllium we have magnesium

158
00:14:07,920 --> 00:14:13,500
他们两个都有价，突然之间我们遇到了一个大问题
both of them have valence two and all of
a sudden we have a big problem the big

159
00:14:13,500 --> 00:14:19,830
问题是它的相对湿度霍尔系数现在大于零，这是
problem is it RH the Hall coefficient is
now greater than zero and this is

160
00:14:19,830 --> 00:14:24,060
从这张犹大计算机的图片中完全令人困惑的是，应该
completely puzzling from this picture of
Judah computer says that it should be

161
00:14:24,060 --> 00:14:28,020
只是密度乘以负电荷的电子电荷，所以
just the density times the charge of the
electron which is negative and so

162
00:14:28,020 --> 00:14:32,190
在犹大理论中，我们不可能得到霍尔系数
there's no way in Judah Theory we're
going to get a Hall coefficient which is

163
00:14:32,190 --> 00:14:36,570
积极，这一定使保罗·德鲁德（Paul Drude）非常困惑，并在当年晚些时候
positive and this must have puzzled Paul
Drude a terribly and later on the year

164
00:14:36,570 --> 00:14:40,560
当我们研究材料的能带结构时会明白为什么
when we study band structure of
materials will understand why this is

165
00:14:40,560 --> 00:14:44,940
但看起来好像发生了两件事之一，可能是
but it kind of looks like one of two
things is going on possibility one is

166
00:14:44,940 --> 00:14:48,780
电子密度由于某种原因变为负数
that the density of electrons has gone
negative for some reason if that makes

167
00:14:48,780 --> 00:14:52,410
任何有意义的可能性二是充电
any sense
possibility two is that the charge

168
00:14:52,410 --> 00:14:56,310
携带电荷的东西不是电子
carrier the thing that's moving around
carrying the charge is not the electron

169
00:14:56,310 --> 00:14:59,910
但是其他带有正电荷的东西似乎都没有
but something else with a positive
charge neither of those seem to make

170
00:14:59,910 --> 00:15:04,760
现在很有道理，但希望以后会更有意义
much sense right now but they'll make a
little bit more sense later on hopefully

171
00:15:04,760 --> 00:15:10,050
但是你知道犹大是一个非常勇敢的人，他决定我们要去
but you know Judah was a very brave
person and he decided that we're going

172
00:15:10,050 --> 00:15:14,460
忽略这两种材料，这些材料似乎不适合它们的价
to ignore these two materials that don't
seem to fit the materials with valence

173
00:15:14,460 --> 00:15:19,560
两个霍尔系数为正，我们将继续尝试
two that have Hall coefficient positive
and we're going to go ahead and try to

174
00:15:19,560 --> 00:15:23,690
计算动力学理论或犹大理论的其他数量
calculate some other quantity
that kinetic theory or Judah theory

175
00:15:23,690 --> 00:15:28,430
应该可以预测，您去年和一年以前学习了动力学理论
should be able to predict now you
studied kinetic theory last year and one

176
00:15:28,430 --> 00:15:32,570
您能在动力学理论中计算的事物是热的
of the things that you were able to
calculate in kinetic theory was thermal

177
00:15:32,570 --> 00:15:40,520
电导率我不会进行整个计算，因为它是
conductivity I'm not going to go through
the whole calculation because it's

178
00:15:40,520 --> 00:15:43,760
您去年可能学得很好的东西可能不值得
something that you probably learned very
well last year it's probably not worth

179
00:15:43,760 --> 00:15:47,540
再次经历，但我会写下答案，你可能应该
going through again but I'll write down
the answer that you probably well should

180
00:15:47,540 --> 00:15:51,200
可能看起来很熟悉单原子气体的导热系数
probably look familiar the thermal
conductivity for a monatomic gas is

181
00:15:51,200 --> 00:15:58,810
颗粒密度每颗粒热容量的三分之一，这是N之上的CV 
one-third the density of particles heat
capacity per particle this is CV over N

182
00:15:58,810 --> 00:16:05,600
然后有一个速度和一个散射长度，即速度
and then there's a velocity and a
scattering length which is velocity

183
00:16:05,600 --> 00:16:11,200
乘以散射时间，在动力学理论中，我们可以将速度设为
times the scattering time and in kinetic
theory we can take the velocity to be

184
00:16:11,200 --> 00:16:18,320
 pi乘以pi的倍数为8 KBT的平方根
square root of eight KBT over pi times
the mass does this all look familiar is

185
00:16:18,320 --> 00:16:23,900
去年大概有点熟悉，希望还可以，所以我们要做的是
vaguely familiar from last year
hopefully okay so what we do is we

186
00:16:23,900 --> 00:16:27,770
实际上只是将其插入V平方，因为它发生了
actually just plug this in it's going to
be V squared because because it occurs

187
00:16:27,770 --> 00:16:32,870
在两个地方将插入动力学理论CD在N上的表达式
in two places will plug in the
expression from kinetic theory CD over N

188
00:16:32,870 --> 00:16:42,920
是经典气体物理学的三分之二kb，而我们得到的是热
is three-halves kb from classical gas
physics and what we get is the thermal

189
00:16:42,920 --> 00:16:49,070
电导率在pi上是4，所以让我们看一下PI就是从这里下来的
conductivity is 4 over pi so the let's
see so the PI comes from down here

190
00:16:49,070 --> 00:16:53,150
平方之后，我猜我们那里有一个2，它取消了8以使
having gotten squared I guess we got a 2
over there which cancel the 8 to make a

191
00:16:53,150 --> 00:17:03,290
 4那么我们在n上有n tau KB平方T，所以这是对
4 then we have n tau K B squared T over
m so that's the prediction for the

192
00:17:03,290 --> 00:17:08,780
在犹大理论中的热导率现在我们再次有这个问题是
thermal conductivity in in judah theory
now again we have this question is this

193
00:17:08,780 --> 00:17:12,650
一个好答案还是一个不好的答案，我们真的不知道，因为我们有
a good answer or is this a bad answer
and we don't really know because we have

194
00:17:12,650 --> 00:17:18,319
这个tau这个等式中的未知数量tau但是还不错
this tau this unknown quantity tau in
this equation but it's not so bad

195
00:17:18,319 --> 00:17:24,439
因为我们也有Sigma是t大于m的平方的平方，我们可以看一下比率

196
00:17:24,440 --> 00:17:29,330
这两个数量中的1个，然后去掉tau，所以我们做的就是
of these two quantities and get rid of
tau so let's do that we'll take what is

197
00:17:29,330 --> 00:17:36,590
称为L洛伦兹burr洛伦兹编号这不是
known as L the Lorentz
burr lorentz number number this is not

198
00:17:36,590 --> 00:17:40,700
实际上是洛伦兹转型和洛伦兹力量的同一个人
actually the same guy who is Lorentz
transformations and Lorentz force

199
00:17:40,700 --> 00:17:45,890
这是洛伦兹（Lorenz），他名字中没有T，每个人似乎都有洛伦兹（Lorentz）号
this is Lorenz without the T in his name
Lorentz number everyone seems to have

200
00:17:45,890 --> 00:17:50,380
 1800年代在德国同名啊
the same name in Germany in the 1800s ah

201
00:17:50,410 --> 00:17:56,930
我们可以将导热率与T乘以Sigma的比率
we so we can take the ratio of the
thermal conductivity to T times Sigma

202
00:17:56,930 --> 00:18:05,360
在这里，Taos取消了，我们只剩下四个PI和KB / e平方
here the Taos cancel and we're left with
just four over PI and KB / e squared

203
00:18:05,360 --> 00:18:12,320
就是这样，当您输入数字时，此结果是10的1倍
that's it and this result while you put
in numbers you discover it's 1 times 10

204
00:18:12,320 --> 00:18:18,080
到每开尔文平方减去-8瓦欧姆，这个结果有点
to the minus 8 watt ohms per Kelvin
squared and this result is kind of

205
00:18:18,080 --> 00:18:22,400
有趣，因为这就是我们所说的通用性，它并不依赖于
interesting because it's it's what we
call universal it doesn't depend on

206
00:18:22,400 --> 00:18:25,850
温度与电子的质量无关
temperature it doesn't depend on the
mass of the electron it doesn't depend

207
00:18:25,850 --> 00:18:28,700
电子的散射时间与电子的密度无关
on the scattering time of the electrons
doesn't pend on the density of electrons

208
00:18:28,700 --> 00:18:32,870
除了这些基本数量之外，它不依赖任何其他因素
it doesn't depend on anything except
these fundamental quantities Boltzmann's

209
00:18:32,870 --> 00:18:38,420
常数和电子的电荷，这是一个有趣的预测
constant and the charge of the electron
it's kind of an interesting prediction

210
00:18:38,420 --> 00:18:47,420
实际上，洛伦兹数固定为L的事实对于
and in fact the fact that the Lorentz
number is fixed L the same the same for

211
00:18:47,420 --> 00:18:58,880
所有材料的所有材料在所有场合都是已知的
all materials for all materials was
known in all t t um this is known as the

212
00:18:58,880 --> 00:19:07,880
维生素Franz法维生素Franz，自1800年代中期以来就广为人知， 
vitamin Franz law vitamin Franz and it
was known since the mid 1800s and no one

213
00:19:07,880 --> 00:19:11,270
有一个最雾的想法，为什么它是真的，他们只是测量了维生素和弗朗兹
had the foggiest idea why it was true
they just measured it vitamin and Franz

214
00:19:11,270 --> 00:19:15,890
是德国人，他们喜欢测量热导率电
were Germans who liked to measure things
like thermal conductivity electrical

215
00:19:15,890 --> 00:19:20,510
电导率在1800年代中期，他们注意到，如果采用该比率
conductivity in the mid-1800s and they
noticed that if you take this ratio it

216
00:19:20,510 --> 00:19:25,250
我猜如果我们想成为所有材料的结果几乎相同
comes out pretty much the same for all
materials I guess if we want to be

217
00:19:25,250 --> 00:19:31,160
实际上，对于锂来说，它的精确度是2.2瓦乘以10 
precise about it in fact for lithium it
comes out about 2.2 watt times 10 to the

218
00:19:31,160 --> 00:19:39,350
铜的每开尔文平方减去8瓦欧姆，铁约为2.0 
minus 8 watt ohms per Kelvin squared for
copper it's about 2.0 for iron

219
00:19:39,350 --> 00:19:44,390
大约是2.6，所以非常接近犹大的
it's about 2.6 so it's pretty close to
judah's

220
00:19:44,390 --> 00:19:48,650
预测好吧，它偏离了两倍，但至少它是正确的
prediction okay it's off by a factor of
two but at least it's in the right

221
00:19:48,650 --> 00:19:53,660
球场这是一个非常粗糙的理论，在犹大之前没有人有任何想法
ballpark it's a very crude theory and
before Judah no one had the had any idea

222
00:19:53,660 --> 00:19:59,960
为什么要为所有材料完全固定这个比例，所以这是
why this ratio should be fixed for all
materials all and at all so this was

223
00:19:59,960 --> 00:20:03,770
确实向前迈出了一大步，我们应该拥有直觉
really a great step forward and the
intuition that we should have in our

224
00:20:03,770 --> 00:20:08,900
头实际上不是那么复杂的直觉应该是
heads is actually not that complicated
the intuition should be that whether

225
00:20:08,900 --> 00:20:11,390
我们正在考虑传热或他们在考虑什么
we're thinking about heat transport or
whatever they're thinking about

226
00:20:11,390 --> 00:20:15,920
电传输我们实际上只是在考虑电子在运动
electrical transport we're really just
thinking about electrons moving if we're

227
00:20:15,920 --> 00:20:18,890
考虑到常规电导率，我们有点在数
thinking about regular conductivity
we're sort of counting how many

228
00:20:18,890 --> 00:20:22,669
电子移动，每个电子上都有一定量的电荷，而
electrons move and each electron has a
certain amount of charge on it whereas

229
00:20:22,669 --> 00:20:26,750
如果我们在考虑导热系数，我们仍在计算
if we're thinking about thermal
conductivity we're still counting how

230
00:20:26,750 --> 00:20:30,650
许多电子在运动，但是每个电子在运动一定量的
many electrons are moving but each
electron is moving a certain amount of

231
00:20:30,650 --> 00:20:34,970
热量与温度成正比，因此热量比
heat which is proportional to
temperature so that ratio of thermal

232
00:20:34,970 --> 00:20:40,250
电导率除以温度，然后除以常规电导率
conductivity divided by temperature then
divided by regular conductivity should

233
00:20:40,250 --> 00:20:43,160
大致恒定，因为在所有情况下，您都只是在数
come out roughly constant because in
both in all cases you're just counting

234
00:20:43,160 --> 00:20:48,290
可以移动的电子数，所以这是一个非常好的结果
the number of electrons that move okay
so this was a really good result from

235
00:20:48,290 --> 00:20:52,490
来自犹大的会议，解释了维南法和弗朗茨法，但其中有一个
meeting from from Judah explaining the
vinum and Franz law but there's a really

236
00:20:52,490 --> 00:21:04,370
大难题，而难题在于，我们过去使用了相当容易的陈述
big puzzle and the puzzle is that we
used we used rather glibly the statement

237
00:21:04,370 --> 00:21:10,190
 N上的CV为三分之二kb，对于
the CV over N is three-halves kb which
is a perfectly good result for a

238
00:21:10,190 --> 00:21:17,380
单原子气体，但对于金属中的电子而言并非如此
monatomic gas but it's just not true for
electrons in the metal not true for

239
00:21:17,380 --> 00:21:22,760
金属中的电子为什么不能很好地测量，我们发现它不是
electrons in metal why not
well we measured it and we saw it's not

240
00:21:22,760 --> 00:21:27,980
真实记住金属中的热容量是立方T，我们
true remember in metals the heat
capacity is alpha T cubed and we

241
00:21:27,980 --> 00:21:34,040
将此alpha识别为振动或Debye Theory加gamma 
identified this alpha as being
vibrations or Debye Theory plus gamma

242
00:21:34,040 --> 00:21:40,100
乘以T，实际上，如果您将此伽马乘以T，对于金属来说是特殊的，所以这是
times T and in fact if you this gamma
times T is special for metals so this is

243
00:21:40,100 --> 00:21:44,000
电子在金属中以及任何周围流动的热容量
the heat capacity of the electrons
running around in the metal and for any

244
00:21:44,000 --> 00:21:49,640
合理的温度伽玛乘以T远小于三分之二Kb 
reasonable temperature gamma times T is
much much less than three halves Kb as

245
00:21:49,640 --> 00:21:59,080
只要T不是巨大的，就意味着您知道的巨大
long as T is not ginormous
enormous that means huge like you know

246
00:21:59,080 --> 00:22:03,940
只要T不为10,000开尔文10,000开尔文伽马T远小于
10,000 Kelvin as long as T is not 10,000
Kelvin gamma T is much less than

247
00:22:03,940 --> 00:22:08,320
即使我们使用三分之二的Kb 
three-halves Kb
so somehow or other even though we use

248
00:22:08,320 --> 00:22:11,800
这东西显然是不正确的，我们用电子给了热量
this thing that was clearly incorrect we
use we gave the electrons the heat

249
00:22:11,800 --> 00:22:17,620
每个容量可减少三分之二KB，而实际上如果测量热量
capacity of three-halves KB each whereas
in fact if you measure the heat capacity

250
00:22:17,620 --> 00:22:21,430
金属的热容量不只是唯一的热容量
of the metal that heat capacity is just
not there the only heat capacity the

251
00:22:21,430 --> 00:22:24,340
电子具有的这个伽马乘以T远小于
electrons have is this gamma times T
which is much much less than

252
00:22:24,340 --> 00:22:29,230
三分之二Kb，那么我们如何正确地处理这件事呢
three-halves Kb so how did we get this
right what what what's going on here

253
00:22:29,230 --> 00:22:34,540
好了，这里的问题这里的问题实际上变得更加明显
well the problem here the problem here
actually becomes much much more obvious

254
00:22:34,540 --> 00:22:39,660
如果我们看其他数量，我们来看看热电特性
if we look at some other quantities
let's look at thermoelectric properties

255
00:22:39,660 --> 00:22:52,210
热电性能电性能是一种
thermo electric properties electric
properties properties this is a kind of

256
00:22:52,210 --> 00:22:56,140
我们要看的实验这里是一块金属，我们要运行电流
experiment we want to look at here's a
block of metal we want to run a current

257
00:22:56,140 --> 00:23:00,730
所以这是电流源，我们将电流流过金属
so here's a current source we run the
current through the metal so the current

258
00:23:00,730 --> 00:23:06,370
通过这种方式将电子拖过金属，然后因为电流是
drags electrons through the metal this
way and then because the current is

259
00:23:06,370 --> 00:23:08,830
穿过金属，这意味着电子正在穿过金属
moving through the metal it means the
electrons are moving through the metal

260
00:23:08,830 --> 00:23:14,440
因此热量就是我让电子以这种方式移动
and therefore heat is I just have the
electrons moving this way a current

261
00:23:14,440 --> 00:23:18,610
向相反的方向移动，但是电子以这种方式移动，因此
moves in the opposite direction but the
electron is moving this way so then the

262
00:23:18,610 --> 00:23:23,380
加热时，热电流被拖向同一方向，因为每个电子
heat the heat current gets dragged in
the same direction because each electron

263
00:23:23,380 --> 00:23:30,580
用它拖动一定量的热量，所以可以定义J q JQ现在
is dragging a certain amount of heat
with it so one can define J q JQ is now

264
00:23:30,580 --> 00:23:42,730
热流密度热流密度J q等于某个系数
the heat current density heat current
density J q is equal to some coefficient

265
00:23:42,730 --> 00:23:49,270
 pi乘以常规电流密度J和pi称为珀耳帖
pi times the regular current density J
and pi is known as the Peltier

266
00:23:49,270 --> 00:23:59,510
珀尔帖系数，这称为
coefficient Peltier coefficient and this
is known as a Peltier effect after

267
00:23:59,510 --> 00:24:05,210
先生。佩莱捷（Pelletier）在1834年发现了这种效应， 
mr. Pelletier who discovered this effect
in 1834 another one of these people from

268
00:24:05,210 --> 00:24:12,770
不管怎样，无论如何，我们都喜欢测量热电传输
the 1800s who like to measure thermal
and electrical transport anyway we can

269
00:24:12,770 --> 00:24:17,570
实际上，除了珀尔帖效应是
actually I should comment here as a bit
of an aside that Peltier effect is

270
00:24:17,570 --> 00:24:22,130
实际上，了解您的技术效果非常有用
actually a very technologically useful
effect to know about you can actually

271
00:24:22,130 --> 00:24:25,970
用它建造一个非常好的没有移动部件的冰箱，这个想法是
build a very nice refrigerator with it
which has no moving parts the idea is

272
00:24:25,970 --> 00:24:30,890
如果您想将热量从冰箱评价器的一侧转移到
that if you want to move heat from one
side the inside of your fridge rater to

273
00:24:30,890 --> 00:24:33,140
冰箱的外部将热量转移出去
the outside of the refrigerator to move
heat out

274
00:24:33,140 --> 00:24:36,710
你只需要运行电流，它就会带走热量，使这一面变冷
you just run current and it drags heat
with it and leaving this side cold

275
00:24:36,710 --> 00:24:41,390
事实上，您可以购买这样构造的冰箱
making that side hot in fact you can buy
refrigerators that are built like this

276
00:24:41,390 --> 00:24:44,420
他们没有水泵之类的东西他们不大声他们很安静
they don't have pumps and things like
that they're not loud they're very quiet

277
00:24:44,420 --> 00:24:49,190
实际上，它们经常被用作葡萄酒冰箱，我不知道为什么，但是
frequently they're actually used as wine
refrigerators I don't know why but but

278
00:24:49,190 --> 00:24:54,290
这些天，您可以在公开市场上购买Peltier冰箱，但是
you can buy a Peltier refrigerator you
know on the open market these days but

279
00:24:54,290 --> 00:24:58,190
在Peltier冰箱中必须记住的一件事是
one thing that wuntch has to keep in
mind in a Peltier refrigerator is that

280
00:24:58,190 --> 00:25:01,280
你不能使自己不能仅仅继续运行越来越多的电流， 
you can't make you can't just keep
running more and more current and

281
00:25:01,280 --> 00:25:04,550
越来越冷，因为在某个时候会有一个我
getting a colder and colder because at
some point there's going to be an I

282
00:25:04,550 --> 00:25:10,040
平方R的散热量，因此您将消耗功率，如果我平方R 
squared R heat dissipation so you'll be
dissipating power and if I squared R

283
00:25:10,040 --> 00:25:14,150
变得太大，那么您就开始加热东西，所以对于小电流来说
gets too big then you start heating
things up so for small currents this

284
00:25:14,150 --> 00:25:18,020
一侧会变冷一侧会变热但对于大电流而言
side will get cold this side will get
hot but for large currents everything

285
00:25:18,020 --> 00:25:23,450
会变热，因此建造一台好的帕尔贴冰箱的艺术形式是
gets hot so the the art form in building
a good Peltier refrigerator is that you

286
00:25:23,450 --> 00:25:27,350
必须找到一种具有大珀尔帖系数但很小的材料
have to find a material with a large
Peltier coefficient but a small

287
00:25:27,350 --> 00:25:32,560
电阻率，找到这些材料的人很多
resistivity and there's many many people
whose job it is to find these materials

288
00:25:32,560 --> 00:25:36,550
好吧，无论如何，我们将尝试计算此帕尔帖系数um 
alright anyway we're going to try to
calculate this Peltier coefficient um

289
00:25:36,550 --> 00:25:43,490
我们如何做得好，热电流密度JQ 
how do we do that well the heat current
density JQ and we have the electrical

290
00:25:43,490 --> 00:25:46,460
电流信号，我们知道电流密度是多少
current ensign we know what the
electrical current density is it's the

291
00:25:46,460 --> 00:25:50,060
速度电子正在移动电子和电荷的密度
velocity electrons are moving the
density of the electrons and the charge

292
00:25:50,060 --> 00:25:54,200
在电子上，热含量非常相似， 
on the electrons the heat content see
very similar it's sort of a lot while

293
00:25:54,200 --> 00:25:57,290
如果您老实说，这是三分之一，这是通常的情况
there's a factor of one-third if you're
honest about it that's a this usual

294
00:25:57,290 --> 00:25:59,540
几何因素不必太担心
geometric factor don't worry about it
too much

295
00:25:59,540 --> 00:26:02,960
电子的速度正在移动电子的密度和热量
the velocity electrons are moving the
density of the electrons and the heat

296
00:26:02,960 --> 00:26:07,410
每个电子携带的是CVT 
that each electron is carrying which is
CVT

297
00:26:07,410 --> 00:26:12,170
我们将这两件事的比值求得珀尔帖系数pi为
we take the ratio of these two things
we'll get a Peltier coefficient pi is

298
00:26:12,170 --> 00:26:20,010
如果插入通常的CV动力学理论结果，则CVT超过e的三倍
CVT over three times minus e if we plug
in the usual kinetic theory result of CV

299
00:26:20,010 --> 00:26:26,390
是三分之二的KB，我要警告您，这不是一个好主意，我们最终会得到
is three-halves KB which I'm re warning
you is not a good idea we end up getting

300
00:26:26,390 --> 00:26:34,380
 KB的两倍乘以e乘以T，人们经常做的是
KB over two times minus e times T and
frequently what people do is they

301
00:26:34,380 --> 00:26:40,410
实际考虑的比率称为s，即PI与T的比称为
actually consider the ratio known as s
which is PI over T s is known as the

302
00:26:40,410 --> 00:26:46,530
塞贝克系数塞贝克是1800年代的另一个人， 
Seebeck coefficient Seebeck was another
person from the 1800s who liked to

303
00:26:46,530 --> 00:26:49,920
我相信他来自
measure electrical and thermal
properties he I believe he was from

304
00:26:49,920 --> 00:26:56,250
爱沙尼亚实际上任何人无论如何都恰好是爱沙尼亚人，所以如果你把那个
Estonia actually anyone who happens to
be Estonian anyway so if you take that

305
00:26:56,250 --> 00:27:02,010
我们只得到2倍减去e的通用常数KB 
ratio we just get the universal constant
KB over 2 times minus e which has a

306
00:27:02,010 --> 00:27:09,660
负0.4的数值乘以10至负10到负4伏
numerical value of minus 0.4 times 10 to
the minus 10 to the minus 4 volts per

307
00:27:09,660 --> 00:27:14,640
开尔文（Kelvin），问题就在这里，如果您实际测量塞贝克（Seebeck） 
Kelvin and the problem is here that if
you actually measure the Seebeck

308
00:27:14,640 --> 00:27:26,660
您实际上发现的任何金属的系数s小100倍
coefficient for any metal you discover
actually actually s is 100 times smaller

309
00:27:28,580 --> 00:27:35,730
所以这在动力学理论动力学中是一个明显得多的明显错误
so this is an obvious much more obvious
error in in kinetic theory kinetic

310
00:27:35,730 --> 00:27:40,590
理论是错误的，这个问题的根源是我们正在使用
theory is going way wrong and the source
of this problem is that we were using

311
00:27:40,590 --> 00:27:45,360
电子的热容量是错误的，问题就来了
this heat capacity for the electrons
that is just wrong and the problem comes

312
00:27:45,360 --> 00:27:54,540
从为什么为什么这样的问题开始，因为实际上C比N大得多
from why why the problem by the problem
because in fact C over N is much much

313
00:27:54,540 --> 00:28:00,870
如果少于三分之二kb的话，如果
less than three-halves kb c vo ran much
much less than three-halves kb well if

314
00:28:00,870 --> 00:28:07,230
如果我们使用的热量如此之大，那是真的，为什么
this is true if we're using this heat
capacity that's so wrong why is it that

315
00:28:07,230 --> 00:28:12,450
当我们计算热量时，我们向上计算的方法还可以
we did okay when we calculated way up
there when we calculated the thermal

316
00:28:12,450 --> 00:28:16,980
导热率我们得到了导热率与常规导热率之比
conductivity we got the ratio of the
thermal conductivity to the regular

317
00:28:16,980 --> 00:28:20,400
电导率我们得到的洛伦兹数几乎完全正确
conductivity we got the lorentz number
almost exactly right where is the

318
00:28:20,400 --> 00:28:23,190
塞贝克的效率是完全错误的，所以
Seebeck
efficient is completely wrong so

319
00:28:23,190 --> 00:28:26,550
某种程度上的矛盾或其他V需求弗朗兹定律即将到来
something's inconsistent somehow or
other the V demand Franz law is coming

320
00:28:26,550 --> 00:28:29,790
正确，塞贝克系数完全错误， 
out right the Seebeck coefficient is
coming out completely wrong and the

321
00:28:29,790 --> 00:28:34,980
发生这种情况的原因我们获得了Vietminh Franz法律正确的原因
reason that this happens the reason we
got the Vietminh Franz law right reason

322
00:28:34,980 --> 00:28:46,860
 CAPA之所以正确的原因是因为我们犯了两个取消错误，两个取消
why why CAPA is right is because we made
two canceling errors two cancelling

323
00:28:46,860 --> 00:28:55,410
错误，这是我提到的第一个vit C over n CV over N 
errors there's the first one I've
mentioned already vit C over n CV over N

324
00:28:55,410 --> 00:29:00,660
比kb的三分之二少得多，我们使用kb，但还有另一半
is much much less than three-halves kb
and we use kb but there's another one we

325
00:29:00,660 --> 00:29:07,110
也使用V的V平方期望值，我们插入了动力学理论结果8 
also use V squared expectation of V we
plugged in the kinetic theory result 8

326
00:29:07,110 --> 00:29:14,100
 KB T超过PI m，实际上V平方实际上要大得多
KB T over PI m and in actuality V
squared is actually much much greater

327
00:29:14,100 --> 00:29:21,600
在导热系数方面比PI M的KB还要大，它们几乎
than a KB over PI M in these errors in
the thermal conductivity they almost

328
00:29:21,600 --> 00:29:25,110
完全抵消，所以我们得到的导热系数几乎完全正确
exactly cancel so we get the thermal
conductivity almost exactly right

329
00:29:25,110 --> 00:29:30,240
而这个V平方的量不会输入塞贝克系数，所以我们
whereas this quantity V squared doesn't
enter in the Seebeck coefficient so we

330
00:29:30,240 --> 00:29:35,280
完全弄错了，因为我们使用了这个，但现在还不行
get it completely wrong because we use
this but not this okay now the first

331
00:29:35,280 --> 00:29:40,470
一年中有机会赢得极好的奖品一种高质量的巧克力
chance of the year to win fabulous
prizes one very high quality chocolate

332
00:29:40,470 --> 00:29:43,740
棒实际上我最喜欢的巧克力棒已经卖完了，所以这有点少了
bar actually my favorite chocolate bar
was sold out so this is a slightly less

333
00:29:43,740 --> 00:29:49,140
而不是高质量的巧克力棒，所以如果有人可以告诉我我们为什么做
than a very high quality chocolate bar
so if anyone who can tell me why we made

334
00:29:49,140 --> 00:29:55,500
这两个错误我们该怎么办我们会遗漏吗
both of these mistakes what do we do
wrong would we leave out come on come on

335
00:29:55,500 --> 00:30:00,950
来吧来吧来吧来吧我们该怎么忘记某人某人是
come on come on come on come on what do
we forget someone someone yes

336
00:30:04,810 --> 00:30:08,420
你还好，所以我不会成为那个巧克力棒，但这很好
you're okay so I'm not going to be the
chocolate bar for that but it's a good

337
00:30:08,420 --> 00:30:13,790
通常使用V对X的期望的想法是UV平方
idea generally generally using
expectation of V versus X is UV squared

338
00:30:13,790 --> 00:30:16,670
我们会犯一个错误，但是通常只是两种错误的一个因素，所以这是一个
we'll make an error buts usually only a
factor of two kind of error so it's a

339
00:30:16,670 --> 00:30:20,690
小错误，因此有各种各样的争论，关于哪个更适合用于此
small error so there's various debates
as to which one is better to use in this

340
00:30:20,690 --> 00:30:22,850
的情况，但我同意这是一个潜在的问题
case but I agree that that's a potential
problem

341
00:30:22,850 --> 00:30:32,270
其他人是是是是费米子是，所以我会扔给
anyone else yes yes yes that's it
fermions yes so I would throw this to

342
00:30:32,270 --> 00:30:35,080
你，但它可能会错过，所以你可以出来，然后再拿出来，所以
you but it would probably miss so you
can come it out and get it afterwards so

343
00:30:35,080 --> 00:30:39,380
我们遗漏的是保利排斥原理电子是
the thing we left out was Pauli
exclusion principle electrons are

344
00:30:39,380 --> 00:30:42,320
费米子不能使所有电子处于相同的状态，它们不是
fermions you can't put all the electrons
in the same state they're not a

345
00:30:42,320 --> 00:30:47,450
无论如何，经典气体都有费米气体，所以我们将不得不
classical gas in any sense there are a
Fermi gas and so we're going to have to

346
00:30:47,450 --> 00:30:50,480
妥善对待，也许能够读下来，因为它真的
treat that properly maybe able to read
that write that down because it's really

347
00:30:50,480 --> 00:30:59,620
重要的费米统计费米统计，所以我们必须处理
important Fermi statistics Fermi
statistics so we have to deal with that

348
00:30:59,620 --> 00:31:03,650
这就是我们下一步要做的，但是在此之前，您可以做一点
that's what we're going to do next but
before doing that you can do a little

349
00:31:03,650 --> 00:31:09,470
犹大理论摘要犹大摘要um 
bit of a summary of juda theory
juda summary um

350
00:31:09,470 --> 00:31:15,590
所以提请总结很多事情，很多事情，很多运输
so drew to summary many things right
many things right many transport

351
00:31:15,590 --> 00:31:20,630
正确的物业，我认为您要学习家庭作业， 
properties you get right and I think for
homework you'll study a couple of other

352
00:31:20,630 --> 00:31:24,530
在犹大理论中您正确的事情仍然被大量使用
things that you get right in judah
theory it's still used very heavily drew

353
00:31:24,530 --> 00:31:27,470
该理论被广泛用于理解金属和半导体
the theory is used very heavily for
understanding metals and semiconductors

354
00:31:27,470 --> 00:31:35,030
特别适用于半导体，但存在一些问题
particularly good for semiconductors but
it has problems and some of the problems

355
00:31:35,030 --> 00:31:47,150
好吧，我们的H可能有错误的符号，错误的符号，您可以让CV超过
are well our H can have wrong sign have
wrong sign sign um you can have CV over

356
00:31:47,150 --> 00:31:51,620
 N实际上远小于我们使用的三分之二kb 
N is actually much much less than
three-halves kb where we use

357
00:31:51,620 --> 00:31:56,470
在计算中减少了三分之二kb，这样您就可以知道Seebeck错误
three-halves kb in the calculations and
you can get you know Seebeck wrong by

358
00:31:56,470 --> 00:32:02,570
错误一百一百零零等等，实际上您可以得到
wrong by a hundred long by a hundred and
so forth in fact you can get the sign of

359
00:32:02,570 --> 00:32:06,590
 Seebeck系数也是错误的，但是由于我们被Mac和
the Seebeck coefficient wrong as well
but being that we're off by a mac and

360
00:32:06,590 --> 00:32:10,880
幅度高达一百倍，似乎有点多余
magnitude by a factor of a hundred it
seems a little bit superfluous to worry

361
00:32:10,880 --> 00:32:14,750
无论如何，关于标志的一切都突然好起来了，这些是其中的一些
about the sign all of a sudden okay
anyway so these are some of the

362
00:32:14,750 --> 00:32:18,520
犹大理论的总结，但尽管如此， 
summaries of juda theory but nonetheless
this was

363
00:32:18,520 --> 00:32:23,950
人们如何在犹大之前的1900年代首次了解奖牌
how people understood medals for the
first time in the 1900s before Judah

364
00:32:23,950 --> 00:32:27,730
人们根本不了解金属，但对理论仍然很满意
people didn't understand metals at all
and still do to theories extremely good

365
00:32:27,730 --> 00:32:32,620
粗略了解金属的方法，但是大约25步并没有真正进展
way to roughly understand metals but
nothing really progressed for about 25

366
00:32:32,620 --> 00:32:36,910
十年，犹大理论就存在了，人们不明白为什么这些
years and Judah theory was all there was
and people didn't understand why these

367
00:32:36,910 --> 00:32:41,980
事情并没有解决，但是在1925年左右，先生
things didn't come out right
but around 1925 there are a lot of sir

368
00:32:41,980 --> 00:32:46,210
物理上同时取得的巨大进步是1925年的Pauli排除法
simultaneous enormous advances in
physics 1925 was the Pauli exclusion

369
00:32:46,210 --> 00:32:52,090
原理1926年是薛定inger方程，1926年是费米·狄拉克
principle 1926 was the Schrodinger
equation also 1926 was was Fermi Dirac

370
00:32:52,090 --> 00:33:00,190
统计资料，1927年有人来了一个名叫Arnold Sommerfeld的家伙
statistics and in 1927 someone came
along guy named Arnold Sommerfeld 1927

371
00:33:00,190 --> 00:33:08,950
索默菲尔德索默菲尔德获得诺贝尔奖提名81次从未失败
Sommerfeld Sommerfeld was nominated for
nobel prize 81 times never got it poor

372
00:33:08,950 --> 00:33:14,830
家伙，但他的想法是他要用费米统计来治疗金属
guy but his idea was he was going to
treat metals with fermi statistics treat

373
00:33:14,830 --> 00:33:26,500
用费米统计与费米统计换句话说就是
metals metals with fermi statistics with
fermi statistics in other words we're

374
00:33:26,500 --> 00:33:31,420
将解释电子实际上是费米子这样的事实
going to account for the fact that the
electrons are actually fermions and so

375
00:33:31,420 --> 00:33:35,800
现在，我们需要回顾一下关于费米的知识
now we have to do a little bit of a
review of what we know about fermi

376
00:33:35,800 --> 00:33:39,640
统计数据，我们要记住的最重要的是费米
statistics and the most important thing
we have to remember is the Fermi

377
00:33:39,640 --> 00:33:48,580
 βε的职业函数n F减去mu比βε的e大1 
occupation function n F of beta epsilon
minus mu is 1 over e to the beta epsilon

378
00:33:48,580 --> 00:33:56,470
减亩加1处视图是化学势γ势和
minus mu plus 1 view here is the
chemical potential gamma potential and

379
00:33:56,470 --> 00:34:06,970
这个NF事物NF是概率桶处于本征态的概率是一个鸡蛋。 
this NF thing NF is the probability the
prob bin an eigenstate it's an egg an

380
00:34:06,970 --> 00:34:17,130
鸡蛋能量在能量能量epsilon被占领
egg at energy at energy energy epsilon
is occupied

381
00:34:18,829 --> 00:34:26,809
好是是问题否是好的好吧好吧，让我们来画这个东西
good yes yes question No yeah good all
right good so so let's plot this thing

382
00:34:26,809 --> 00:34:34,049
所以我们在这个轴上有能量，在这个轴上还有NF 
so we have energy on this axis we have
NF on this axis and somewhere over here

383
00:34:34,049 --> 00:34:39,659
我们在零温度下的化学势μ费米函数
we have the chemical potential mu at
zero temperature the Fermi function goes

384
00:34:39,659 --> 00:34:44,069
从一到零作为应该在那儿放平的阶跃函数
from one to zero as a step function
that's supposed to be flat up there so

385
00:34:44,069 --> 00:34:48,989
这等于T等于零，这是从1下降到零的完美步骤
this is at T equals zero it's a perfect
step from one down to zero at the

386
00:34:48,989 --> 00:34:54,418
 T处的化学势不等于零，看起来更平滑
chemical potential at T not equal to
zero it's somewhat smoother looks like

387
00:34:54,418 --> 00:35:00,000
因此，这是T大于零，并且宽度从一开始下降
this so this is T greater than zero and
the width over which it drops from one

388
00:35:00,000 --> 00:35:06,150
设为零，这里的宽度大约是KBT好的，这看起来有点模糊吗
to zero that this width here is roughly
KBT okay does this look vaguely familiar

389
00:35:06,150 --> 00:35:10,650
从去年开始，我希望是的，所以现在对
from last year
I hope yeah okay so now it's useful to

390
00:35:10,650 --> 00:35:16,650
有几个定义，我们将使用定义化学
have a couple of definitions that we're
going to use definition the chemical

391
00:35:16,650 --> 00:35:25,740
 t处的电位mu等于零称为费米能量EF 
potential mu at t equals zero is known
as is called the Fermi energy EF a Fermi

392
00:35:25,740 --> 00:35:33,809
能量费米能量，您会发现很多东西都是以先生的名字命名的
energy Fermi energy you'll discover that
a lot of things are named after mister

393
00:35:33,809 --> 00:35:41,279
 Fermi好吧，现在我应该警告您， 
Fermi okay now I should warn you that
there's some disagreement in the

394
00:35:41,279 --> 00:35:44,609
书中有关您所谓的费米能量的文献
literature in the books as to what
you're supposed to call the Fermi energy

395
00:35:44,609 --> 00:35:48,690
有人认为化学势和费米能是
some people think that the chemical
potential and the Fermi energy are

396
00:35:48,690 --> 00:35:53,430
实际上，同义词是大多数人都相信的东西，我认为
actually synonymous mean the same thing
most people I believe think that the

397
00:35:53,430 --> 00:35:57,869
 0温度下的化学势是费米能， 
chemical potential at 0 temperature is
the Fermi energy and that's the

398
00:35:57,869 --> 00:36:01,140
我们将要使用的定义实际上是有意义的
definition we're going to work with it's
actually it makes sense to do that

399
00:36:01,140 --> 00:36:04,260
因为为什么只用两个词来表示完全相同的事情
because why would you just have two
terms that mean exactly the same thing

400
00:36:04,260 --> 00:36:07,500
如果您指的是化学势，就称其为化学势
just call it the chemical potential if
you mean chemical potential Kelvin

401
00:36:07,500 --> 00:36:13,049
零温度下的电势我们称费米能为电子
potential at zero temperature we call
the Fermi energy for electrons which are

402
00:36:13,049 --> 00:36:17,490
你知道自由电子波我们通常有H bar平方K平方
you know free electron waves we
generally have H bar squared K squared

403
00:36:17,490 --> 00:36:24,660
如果您的波向量为K，则超过2m的能量就是能量
over 2m is the energy is an energy if
you have a wave with wave vector K its

404
00:36:24,660 --> 00:36:30,300
能量将为H bar平方K平方超过2m，因此如果我们将其固定为
energy will be H bar squared K squared
over 2m so if we fix this thing to be

405
00:36:30,300 --> 00:36:38,040
费米能量然后我们定义KF来满足这个方程，所以定义了KF 
the Fermi energy then we define KF to
satisfy this equation so KF is defined

406
00:36:38,040 --> 00:36:50,640
通过这个方程式，可以确定KF为费米动量费米动量
by this equation this defines KF is the
Fermi momentum fermi momentum so if we

407
00:36:50,640 --> 00:36:55,080
知道费米能量我们只是从这个方程计算费米
know the Fermi energy we just calculate
from this equation what the Fermi

408
00:36:55,080 --> 00:37:00,870
每个人都对此感到高兴，所以如果你想知道我们是否
momentum is okay everyone happy with
that all right so if you wonder if we

409
00:37:00,870 --> 00:37:04,110
有一些物理系统，我们知道那个物理中有多少个电子
have some physical system we know how
many electrons are in that physical

410
00:37:04,110 --> 00:37:08,610
我们写n的系统电子总数是所有电子的总和
system we write n the total number of
electrons is the sum over all

411
00:37:08,610 --> 00:37:16,140
本征态数据ε的费米占有因子的本征态卵
eigenstates eggs of the Fermi occupation
factor of data epsilon of the eigenstate

412
00:37:16,140 --> 00:37:21,030
减去亩，为什么我要这样写所以这是所有可能的总和
minus mu so why do I write it like this
so this is the sum over all possible

413
00:37:21,030 --> 00:37:25,620
状态在系统中被占据的概率，以及
states in the system the probability
that that state will be occupied and if

414
00:37:25,620 --> 00:37:29,430
对所有本征态求和，就得到了粒子的总数
you sum up over all eigenstates you get
the total number of particles in the

415
00:37:29,430 --> 00:37:35,820
系统不错，是的，如果您知道
system good yeah you can also turn this
on the on its head if you know the

416
00:37:35,820 --> 00:37:39,240
系统中粒子的数量，您可以使用此公式计算
number of particles in the system you
could use this equation to figure out

417
00:37:39,240 --> 00:37:44,610
如果您必须像现在那样做，那么化学潜力是什么
what the chemical potential is if you
have to do that now as we did in the

418
00:37:44,610 --> 00:37:51,060
德拜理论，我们不得不经常对所有可能的平面波求和，所以
Debye theory we had to frequently sum
over all possible plane waves so one

419
00:37:51,060 --> 00:37:54,720
这时候我们应该熟悉的事情我向你保证，我们会用这个
thing we should be familiar with at this
time I promised you we would use this a

420
00:37:54,720 --> 00:38:00,090
很多是所有可能的平面波的总和被D 3代替
lot is the sum over all possible plane
waves gets replaced by an integral a D 3

421
00:38:00,090 --> 00:38:06,840
 K超过2 PI的立方，因此每个本征态都是我们盒子中的特定平面波，因此
K over 2 PI cubed so every eigenstate is
a particular plane wave in our box so

422
00:38:06,840 --> 00:38:10,950
总和成为2 pi乘以体积因子的整数d 3 
the sum becomes an integral d 3 cubed
over 2 pi times a factor of the volume

423
00:38:10,950 --> 00:38:14,850
唯一不同的是，我将前面的系数设为2 
the only thing that's different is I'm
going to put a factor of 2 out front and

424
00:38:14,850 --> 00:38:21,360
对于自旋，因子2是电子具有两个可能的自旋
the factor of 2 is for spins that
electrons have two possible spins in one

425
00:38:21,360 --> 00:38:25,170
特定的平面波可以旋转或向下旋转，因此有两种可能
particular plane wave it can be either
spin up or spin down so two possible

426
00:38:25,170 --> 00:38:33,540
状态，然后我们将费米因子beta epsilon K减去Mew ok 
States and then we're integrating the
Fermi factor beta epsilon K minus Mew ok

427
00:38:33,540 --> 00:38:41,040
到目前为止，还不错，所以在低温，低T下
so far so good
alright so at low temperature at low T

428
00:38:41,040 --> 00:38:49,290
洛蒂这东西是一个阶梯功能这是阶梯
loti this thing here is a step function
this is step which the step function

429
00:38:49,290 --> 00:38:54,180
会告诉我们，我们应该整合直到将第一相加直到
would tell us we should integrate up
until integrate the number one up until

430
00:38:54,180 --> 00:38:58,740
能量就是费米能量，换句话说，所有电子都被充满
the energy is the Fermi energy in other
words all the electrons are filled all

431
00:38:58,740 --> 00:39:01,830
这些州被充满在费米能量之下，而所有东西都被塞满
the states are filled below the Fermi
energy and everything above the Fermi

432
00:39:01,830 --> 00:39:11,960
能量为空，所以我们可以重写n为2v，让我们拉出2个PI立方体
energy is empty so we can rewrite n is
2v let's pull out the 2 PI cubed

433
00:39:11,960 --> 00:39:21,510
积分d3k小于K小于KF，因此说K的绝对值为
integral d3k up to K less than KF so
saying that the absolute value of K is

434
00:39:21,510 --> 00:39:26,490
费米动量小于KF等于说能量必须是
less than KF the fermi momentum is
equivalent to saying the energy must be

435
00:39:26,490 --> 00:39:33,540
小于费米能量好，是的，现在这告诉我们的是
less than the Fermi energy good yeah ok
now this what this is telling us is that

436
00:39:33,540 --> 00:39:49,070
此处的填充状态填充状态形成半径KF的球，并且
the filled States here filled States
States form a ball ball of radius KF and

437
00:39:49,070 --> 00:39:54,420
零温度下的充满状态通常又被称为费米海
filled States at zero temperature are
usually known as the Fermi sea again

438
00:39:54,420 --> 00:40:02,180
以费米和球表面命名的东西被称为
something named after Fermi and the
surface of the ball surface is known as

439
00:40:02,180 --> 00:40:11,160
费米表面，所以这种恐惧和球体周围的所有地方都是
the Fermi surface so this fear and
everywhere around the sphere are the

440
00:40:11,160 --> 00:40:16,260
这个球的表面所有能量都是EF，所以如果您有一个半径球体
surface of this ball the energies are
all EF so if you have a sphere of radius

441
00:40:16,260 --> 00:40:20,880
 KF那个球体表面上的所有东西都具有相同的能量，并且
KF everything on the surface of that
sphere has the same energy and the

442
00:40:20,880 --> 00:40:28,050
能量就是费米能量，好，所以我们要做的最后一件事就是
energy is the Fermi energy good alright
so last thing we have to do is we have

443
00:40:28,050 --> 00:40:34,860
实际计算球与球的结合量，所以我们有2 V 
to actually calculate the volume of that
sphere bond with a ball so we have 2 V

444
00:40:34,860 --> 00:40:38,820
超过2个PI的立方，该积分的值只会给我们
over 2 PI cubed and the value of that
integral is just going to give us the

445
00:40:38,820 --> 00:40:46,970
体积为4/3 pi KF立方，然后进行一些重新排列
volume which is 4/3 pi K F cubed and
then with a little bit of rearrangement

446
00:40:46,970 --> 00:40:51,450
我们可以将V移交给该副律师
we can move the V over to this side
counsel

447
00:40:51,450 --> 00:41:00,200
因子将在V上得到n，这是密度和电子密度e密度
factors will get n over V which is the
density and electron density e density

448
00:41:00,200 --> 00:41:16,860
这是KF超过3 pi平方的立方，然后求解此KF将得到KF等于3 
which is KF cubed over 3 pi squared then
solve this for KF will get KF equals 3

449
00:41:16,860 --> 00:41:21,660
 PI的平方乘以密度的1/3，所以如果我们知道电子的密度
PI squared times the density to the 1/3
so if we know the density of electrons

450
00:41:21,660 --> 00:41:27,930
我们知道KF，然后我们可以将KF重新插入方程式中
we know KF and then we can plug KF back
in to our equation which I'm just about

451
00:41:27,930 --> 00:41:36,230
向下滚动以获取EF，因此EF为H bar平方KF平方2 m 
to scroll off the top to get EF so EF is
H bar squared K F squared over 2 m

452
00:41:36,230 --> 00:41:48,090
等于H bar平方超过2m乘以3 pi平方n等于2/3，这是
equals H bar squared over 2m times 3 pi
squared n to the 2/3 and this is a

453
00:41:48,090 --> 00:41:56,520
相当重要的方程式，我们稍后将使用它，它告诉我们什么是
rather important equation we're going to
use it later on so it tells us what is

454
00:41:56,520 --> 00:42:01,260
费米能量的密度，所以你知道费米能量有多大
the Fermi energy in terms of the density
so you know how big is the Fermi energy

455
00:42:01,260 --> 00:42:04,890
好吧，我们首先要弄清楚密度是多少
well let's first we have to figure out
what the density is

456
00:42:04,890 --> 00:42:11,190
应该使用什么密度好让我们尝试每个原子大约1个电子的密度
what density should use well let's try
try density of about 1 electron per atom

457
00:42:11,190 --> 00:42:16,710
至少对于福特Ruda理论而言，这似乎非常有效
which seemed to work pretty well for
Ford Ruda theory at least for some of

458
00:42:16,710 --> 00:42:25,430
这些金属，如果我们这样说，对于铜II X铜，我们得到的EF为
these metals if we do that say for
copper II X copper we get EF is

459
00:42:25,430 --> 00:42:31,740
大约7电子伏特，是个好数量还是个好数量
approximately 7 electron volts and is
that a big number or small number well

460
00:42:31,740 --> 00:42:35,400
将其转换为费米温度非常有用
it's useful to convert it to a
temperature which is known as the Fermi

461
00:42:35,400 --> 00:42:38,940
温度，所以我将再次定义费米温度
temperature so I'll define Fermi
temperature again something named after

462
00:42:38,940 --> 00:42:52,230
费米温度TF等于EF超过KB，对于铜或铜TF约为
Fermi temp which is T F equals EF over
KB and for copper or copper TF is about

463
00:42:52,230 --> 00:42:55,310
 80,000开尔文
80,000 Kelvin

464
00:42:55,760 --> 00:43:01,410
巨大的巨大巨大的巨大的费米能量和费米温度
huge huge huge huge so typical Fermi
energies and Fermi temperatures in

465
00:43:01,410 --> 00:43:04,890
金属绝对是巨大的，其原因是
metals are absolutely enormous and the
reason for the

466
00:43:04,890 --> 00:43:08,490
这是因为如果每个原子有一个电子，那么会有很多电子
this is because there's a lot a lot of
electrons if one electron per atom

467
00:43:08,490 --> 00:43:12,330
当您开始将电子放入系统时，会有很多原子
there's a lot of atoms when you start
putting electrons into your system you

468
00:43:12,330 --> 00:43:16,560
填充小的低能态首先是具有最小K的态，然后
fill up the small the low energy states
first the ones with smallest K but then

469
00:43:16,560 --> 00:43:18,960
那些被填满了，你必须开始越来越多地填满
those are filled and you have to start
filling up higher and higher and higher

470
00:43:18,960 --> 00:43:23,100
能量状态，当你把最后一个电子放进去的时候
energy states and by the time you put
that last electron in you've gotten up

471
00:43:23,100 --> 00:43:26,640
消耗巨大的能量，因为这些电子的密度确实
to an enormous energy because the
density of these electrons is really

472
00:43:26,640 --> 00:43:30,090
很大，所以让我们实际绘制费米
really big
so let's actually plot the Fermi

473
00:43:30,090 --> 00:43:38,010
再次起作用，所以这里是F这里是EF这里的化学势
function again so here's an F here's one
here's EF the chemical potential

474
00:43:38,010 --> 00:43:44,850
从KB中取出约80,000开尔文，费米功能将
approximately 80,000 Kelvin from take
out a KB and the Fermi function will

475
00:43:44,850 --> 00:43:53,520
看起来像这样，它的范围很小，是KBT茶室
look like this it drops in a very very
small range this is KBT tearoom it's

476
00:43:53,520 --> 00:43:58,200
比我画的还要夸张，比我画的还要窄
even more exaggerated than I've drawn
it's even narrower little drop than I've

477
00:43:58,200 --> 00:44:03,000
然后我画了一下，因为这里的距离是8万
then I've drawn because this distance
here as a temperature is 80 thousand

478
00:44:03,000 --> 00:44:09,270
开尔文与我们的室温300开尔文之间的距离
Kelvin versus our room temperature which
is 300 Kelvin so this distance over

479
00:44:09,270 --> 00:44:15,630
费米函数从一下降到零的范围非常狭窄
which the the Fermi function drops from
from one to zero is a very very narrow

480
00:44:15,630 --> 00:44:22,050
在费米表面的顶部银条，实际上这张图片给了我们
sliver at the top of the Fermi surface
and in fact this picture here gives us

481
00:44:22,050 --> 00:44:26,790
已经暗示了为什么金属的热容量如此之低
already a hint as to why the heat
capacity of the metal is so low so much

482
00:44:26,790 --> 00:44:30,300
比我们想像的要低，以具有热容量，您必须
lower than we would have guessed in
order to have heat capacity you have to

483
00:44:30,300 --> 00:44:34,560
能够吸收一些能量，所以你想象一个电子处于本征态
be able to absorb some energy so you
imagine an electron in some eigenstate

484
00:44:34,560 --> 00:44:39,090
它吸收一些能量，然后一点点跳到附近的本征态
it absorbs some energy and it jumps up
to a nearby eigenstate with a little

485
00:44:39,090 --> 00:44:45,720
所有这些电子都在这里，它们吸收不了任何能量
more energy well all of these electrons
down here they can't absorb any energy

486
00:44:45,720 --> 00:44:49,050
因为它们附近的所有本征态都已被填充，所以无处可去
because all the eigenstates near them
are already filled there's nowhere for

487
00:44:49,050 --> 00:44:52,830
他们走了，他们完全冻结了，他们根本无法吸收任何能量
them to go they're completely frozen
they can't absorb any energy at all the

488
00:44:52,830 --> 00:44:56,460
只有能吸收能量的东西才是费米表面附近的东西
only things that can absorb energy are
the things close to the Fermi surface

489
00:44:56,460 --> 00:45:00,780
他们可能会跳到费米表面上方并吸收能量，因为
where they could jump above the Fermi
surface and absorb energy because

490
00:45:00,780 --> 00:45:04,110
如果没有空的状态可以进入一个空状态
there's an empty state there for them to
go into if there's no empty state to go

491
00:45:04,110 --> 00:45:09,660
他们根本无法吸收任何能量，因此给了我们一个提示
into there's no way they can absorb any
energy at all so that gives us a hint as

492
00:45:09,660 --> 00:45:16,110
为什么热容量这么低最后一件事是我们可以看看
to why the the heat capacity is so low
one final thing is we can look at the

493
00:45:16,110 --> 00:45:21,930
典型速度典型速度称为
typical velocity
typical velocity which is known as the

494
00:45:21,930 --> 00:45:33,500
费米速度费米速度显然可以，VF可以是h-bar KF 
Fermi velocity Fermi velocity apparently
right okay VF which would be h-bar K F

495
00:45:33,500 --> 00:45:41,460
除以质量好的KF，如果把它放进去，就会得到很大的数目
divided by the mass okay K F and if you
put that in you get a huge number

496
00:45:41,460 --> 00:45:47,130
巨大约等于光速的1％ 
huge equals huge approximately 1% of the
speed of light

497
00:45:47,130 --> 00:45:51,720
这可能会让您感到惊讶，因为您曾经接触过的每种金属
which might surprise you this every
metal that you've ever run into contact

498
00:45:51,720 --> 00:45:56,970
像铜，铅，银，钨，钨等具有电子的东西
with things like copper lead silver
tungsten whatever it is it has electrons

499
00:45:56,970 --> 00:46:01,830
在它周围以一定速度奔跑时将具有光速的1％甚至更大
in it running around at speeds will have
1% the speed of light or even greater

500
00:46:01,830 --> 00:46:06,300
现在这听起来让您感到惊讶，而实际上相对论开始变得
now that might sound surprising to you
and in fact relativity starts to become

501
00:46:06,300 --> 00:46:11,400
重要的是，如果您以这些速度小心地做事，但实际上
important if you're doing things
carefully at those speeds but in fact

502
00:46:11,400 --> 00:46:15,570
一想到有多少电子就不足为奇了
it's not surprising once you think about
how many electrons there are there are

503
00:46:15,570 --> 00:46:18,840
每个原子一个或一个原子几个电子
tons and tons and tons of electrons one
per every atom or several per every atom

504
00:46:18,840 --> 00:46:22,260
在某些情况下，所有低能状态都被您完全充满了
in some cases and so all the low energy
states are completely filled you just

505
00:46:22,260 --> 00:46:25,110
必须不断建立越来越高的能量状态， 
have to keep building up to higher and
higher and higher energy states and the

506
00:46:25,110 --> 00:46:29,100
费米表面上的费米表面顶部的电子
electrons on the top of the Fermi
surface on the Fermi on the Fermi

507
00:46:29,100 --> 00:46:33,480
费米球外边缘的表面异常高
surface in the outer edge of the fermi
ball are an extraordinarily high have

508
00:46:33,480 --> 00:46:37,500
非凡的高能量，高动能，因此从外部
extraordinary high energy high kinetic
energy therefore externally a high

509
00:46:37,500 --> 00:46:41,560
速度，我想我们停在那里，我想我周一见
velocity and I guess we stop there and I
guess I see you Monday

510
00:46:41,560 --> 00:46:45,150
 [掌声] 
[Applause]