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PROFESSOR: So last
time we started

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00:00:22,590 --> 00:00:25,620
talking about superfluid helium.

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00:00:25,620 --> 00:00:32,070
And we said that the
phase diagram of helium-4,

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00:00:32,070 --> 00:00:36,710
the isotope that is a boson
has the following interesting

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00:00:36,710 --> 00:00:37,700
properties.

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00:00:37,700 --> 00:00:40,530
First of all, helium
stays a liquid

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00:00:40,530 --> 00:00:43,000
all the way down
to 0 temperature

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00:00:43,000 --> 00:00:45,990
because of the combinations
of its light mass and heat

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00:00:45,990 --> 00:00:47,960
interactions.

17
00:00:47,960 --> 00:00:53,970
And secondly, that [INAUDIBLE]
to cool down helium

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00:00:53,970 --> 00:00:57,580
through this process
of evaporated cooling,

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00:00:57,580 --> 00:00:59,550
one immediately observes
something interesting

20
00:00:59,550 --> 00:01:03,200
happening at temperatures
below 2 degrees

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00:01:03,200 --> 00:01:07,020
Kelvin, where it becomes
this superfluid that

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00:01:07,020 --> 00:01:10,000
has a number of
interesting properties.

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00:01:10,000 --> 00:01:13,430
And in particular as
pertaining to viscosity,

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00:01:13,430 --> 00:01:15,080
we made two observations.

25
00:01:15,080 --> 00:01:18,400
First of all, you can
make these capillaries--

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00:01:18,400 --> 00:01:20,880
and I'll show you been
movie in more detail later

27
00:01:20,880 --> 00:01:24,710
on where it flows through
capillaries as if there

28
00:01:24,710 --> 00:01:28,440
is no resistance
and there is nothing

29
00:01:28,440 --> 00:01:31,010
that sticks to the walls
of the capillaries.

30
00:01:31,010 --> 00:01:33,770
It flows without
viscosity, whereas there

31
00:01:33,770 --> 00:01:37,190
was this experiment of
Andronikashvili, in which you

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00:01:37,190 --> 00:01:40,080
had something that
was oscillating

33
00:01:40,080 --> 00:01:44,760
and you were calculating
how much of the helium

34
00:01:44,760 --> 00:01:47,710
was stuck to the plates
of the container.

35
00:01:47,710 --> 00:01:50,920
And the result was
something like this,

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00:01:50,920 --> 00:01:58,080
that is there was a decrease
in the amount of fluid

37
00:01:58,080 --> 00:01:59,460
that is stuck to the plates.

38
00:01:59,460 --> 00:02:02,705
But it doesn't go
immediately down to 0.

39
00:02:02,705 --> 00:02:04,780
It has a kind of
form such as this

40
00:02:04,780 --> 00:02:10,560
that I will draw more
clearly now and explain.

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00:02:10,560 --> 00:02:16,250
So what we did last time was
to note that people observed

42
00:02:16,250 --> 00:02:17,860
that there were
some similarities

43
00:02:17,860 --> 00:02:20,010
between this
superfluid transition

44
00:02:20,010 --> 00:02:22,640
and Bose-Einstein condensation.

45
00:02:22,640 --> 00:02:26,360
But what I would
like to highlight

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00:02:26,360 --> 00:02:28,200
in the beginning
of this lecture is

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00:02:28,200 --> 00:02:32,450
that there are also very
important differences.

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00:02:32,450 --> 00:02:40,310
So let's think about
these distinctions

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00:02:40,310 --> 00:02:48,126
between Bose-Einstein condensate
add the superfluid helium.

50
00:02:58,690 --> 00:03:05,120
One set of things we would
like to take from the picture

51
00:03:05,120 --> 00:03:12,660
that I have over there, which
is diffraction of the fluid that

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00:03:12,660 --> 00:03:15,370
is stuck to the plates
and in some sense

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00:03:15,370 --> 00:03:19,030
behaves like a normal fluid.

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00:03:19,030 --> 00:03:24,050
Now let me make the analogy
to Bose-Einstein condensation.

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00:03:24,050 --> 00:03:27,400
You know that in the
Bose-Einstein condensation

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00:03:27,400 --> 00:03:32,260
there was also this phenomenon
that there was a separation

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00:03:32,260 --> 00:03:36,230
into two parts of
the total density.

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00:03:36,230 --> 00:03:39,650
And be regarded as a
function of temperature

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00:03:39,650 --> 00:03:45,750
some part of the density as
belonging to the normal state.

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00:03:45,750 --> 00:03:50,840
So when you are above
Tc of n, everything

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00:03:50,840 --> 00:03:53,860
is essentially normal.

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00:03:53,860 --> 00:04:02,600
And then what happens is
that when you hit Tc of n

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00:04:02,600 --> 00:04:07,530
you can no longer put
all of the particles

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00:04:07,530 --> 00:04:11,320
that you have in
the excited states.

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00:04:11,320 --> 00:04:13,870
So the fraction that goes
in the excited states

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00:04:13,870 --> 00:04:20,590
goes down and eventually
goes to 0 at 0 temperature.

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00:04:23,460 --> 00:04:31,770
And essentially, this would
be the reverse of the curve

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00:04:31,770 --> 00:04:34,180
that we have in that
figure over there.

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00:04:37,320 --> 00:04:39,610
Basically, there's
a portion that

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00:04:39,610 --> 00:04:42,940
would be the normal part that
would be looking like this.

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00:04:46,040 --> 00:04:52,430
Now, the way that we
obtained this result

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was that basically there
was a fraction that

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00:04:54,860 --> 00:04:57,100
was in the normal state.

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00:04:57,100 --> 00:05:00,040
The part that was
excited was described

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00:05:00,040 --> 00:05:05,500
by this simple formula that was
g over lambda cubed [INAUDIBLE]

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00:05:05,500 --> 00:05:07,410
of 3/2.

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00:05:07,410 --> 00:05:10,960
So it went to 0 as T to the 3/2.

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00:05:10,960 --> 00:05:12,990
So basically, the
proportionality

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00:05:12,990 --> 00:05:16,040
here is T to the 3/2.

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00:05:16,040 --> 00:05:17,950
And then basically,
the curve would

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00:05:17,950 --> 00:05:22,270
come down here and
go to 0 linearly.

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00:05:22,270 --> 00:05:26,260
Now what is shown
in the experiment

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00:05:26,260 --> 00:05:30,000
is that the curve
actually goes to 0

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00:05:30,000 --> 00:05:33,600
in a much more sharp fashion.

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00:05:33,600 --> 00:05:40,250
And actually, when people try
to fit a curve through this,

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00:05:40,250 --> 00:05:44,010
the curve looks something like
Tc minus T to the 2/3 over.

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00:05:48,410 --> 00:05:54,140
But also it goes to 0 much
more rapidly than the curve

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00:05:54,140 --> 00:05:57,100
that we have for
Bose-Einstein condensation.

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00:05:57,100 --> 00:05:59,900
Indeed, it goes through
0 proportionately to T

90
00:05:59,900 --> 00:06:02,650
to the fourth.

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00:06:02,650 --> 00:06:06,100
And so that's something that we
need to understand and explain.

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00:06:12,200 --> 00:06:18,070
Now all of the properties of
the Bose-Einstein condensate

93
00:06:18,070 --> 00:06:21,840
was very easy to
describe once we realized

94
00:06:21,840 --> 00:06:26,010
that all of the things that
correspond to excitation,

95
00:06:26,010 --> 00:06:28,500
such as the energy,
heat capacity,

96
00:06:28,500 --> 00:06:32,350
pressure, come from
this fraction that

97
00:06:32,350 --> 00:06:35,210
is in the excited state.

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00:06:35,210 --> 00:06:39,010
And we can calculate,
say, the contribution

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00:06:39,010 --> 00:06:42,060
to energy heat
capacity, et cetera.

100
00:06:42,060 --> 00:06:45,730
And in particular, if we look
at the behavior of the heat

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00:06:45,730 --> 00:06:50,400
capacity as a function
of temperature,

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00:06:50,400 --> 00:06:57,240
for this Bose-Einstein
condensate, the behavior

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00:06:57,240 --> 00:07:01,090
that we had was that again
simply at low temperatures

104
00:07:01,090 --> 00:07:08,000
it was going proportionately
to T to the 3/2,

105
00:07:08,000 --> 00:07:12,160
because this was the number
of excitations that you had.

106
00:07:12,160 --> 00:07:15,090
So these two T to the
3/2 are very much related

107
00:07:15,090 --> 00:07:16,960
to each other.

108
00:07:16,960 --> 00:07:22,300
And then this curve would
basically go along its way

109
00:07:22,300 --> 00:07:26,450
until it hit Tc of
n at some point.

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00:07:26,450 --> 00:07:28,930
And we separately
calculated the behavior

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00:07:28,930 --> 00:07:30,640
coming from high temperatures.

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00:07:30,640 --> 00:07:33,140
And the behavior from
high temperatures

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00:07:33,140 --> 00:07:35,480
would start with
the classical result

114
00:07:35,480 --> 00:07:39,500
that the heat capacity
is 3/2 per particle

115
00:07:39,500 --> 00:07:42,800
due to the kinetic energy that
you can put in these things.

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00:07:42,800 --> 00:07:46,327
And then it would rise, and it
would then join this curve over

117
00:07:46,327 --> 00:07:49,189
here.

118
00:07:49,189 --> 00:07:54,760
Now when you look at the
actual heat capacity,

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00:07:54,760 --> 00:07:57,380
indeed the shape up
the heat capacity

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00:07:57,380 --> 00:08:01,370
is the thing that gives this
transition the name of a lambda

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00:08:01,370 --> 00:08:02,180
transition.

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00:08:02,180 --> 00:08:03,805
It kind of looks like a lambda.

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00:08:09,090 --> 00:08:13,390
And at Tc there are
divergences approaching

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00:08:13,390 --> 00:08:17,600
from the two sides that
behave like the log.

125
00:08:24,450 --> 00:08:27,680
And again more
importantly, what we

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00:08:27,680 --> 00:08:30,770
find is that at
0 temperature, it

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00:08:30,770 --> 00:08:34,730
doesn't go through 0, the
heat capacity as T to the 3/2,

128
00:08:34,730 --> 00:08:38,909
but rather as T to
the third power.

129
00:08:38,909 --> 00:08:46,060
So the red curve
corresponds to superfluid,

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00:08:46,060 --> 00:08:51,710
the green curve corresponds
to Bose-Einstein condensate.

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00:08:51,710 --> 00:08:56,550
And so they're clearly
different from each other.

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00:08:56,550 --> 00:09:00,980
So that's what we would
like you to understand.

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00:09:00,980 --> 00:09:07,010
Well the thing that is easiest
to understand and figure out

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00:09:07,010 --> 00:09:10,780
is the difference between
these heat capacities.

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00:09:10,780 --> 00:09:13,510
And the reason for that
is that we had already

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00:09:13,510 --> 00:09:17,400
seen a form that was of
the heat capacity that

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00:09:17,400 --> 00:09:18,600
behaved like T cubed.

138
00:09:18,600 --> 00:09:23,050
That was when we were looking
at phonons in a solid.

139
00:09:23,050 --> 00:09:25,880
So let's remind
you why was it that

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00:09:25,880 --> 00:09:28,740
for the Bose-Einstein
condensate we

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00:09:28,740 --> 00:09:34,010
were getting this T
to the 3/2 behavior?

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00:09:34,010 --> 00:09:38,520
The reason for that was that
the various excitations,

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00:09:38,520 --> 00:09:46,612
I could plot as a function
of k, or p, which is h bar k.

144
00:09:46,612 --> 00:09:49,450
They're very much
related to each other.

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00:09:49,450 --> 00:09:53,890
And for the
Bose-Einstein condensate,

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00:09:53,890 --> 00:09:58,310
the form was simply
a parabola, which

147
00:09:58,310 --> 00:10:03,144
is this p squared over
2 mass of the helium.

148
00:10:03,144 --> 00:10:06,330
Let's say, assuming that
what we are dealing with

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00:10:06,330 --> 00:10:12,620
is non-interacting particles
with mass of helium.

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00:10:12,620 --> 00:10:17,390
And this parabolic
curve essentially

151
00:10:17,390 --> 00:10:28,050
told us that various quantities
behave as T to the 3/2.

152
00:10:28,050 --> 00:10:32,920
Roughly, the idea is that
at some temperature that

153
00:10:32,920 --> 00:10:37,520
has energy of the order
of kT, you figure out

154
00:10:37,520 --> 00:10:41,800
how far you have excited things.

155
00:10:41,800 --> 00:10:45,000
And since this
form is a parabola,

156
00:10:45,000 --> 00:10:48,700
the typical p is going to
scale like T to the 1/2.

157
00:10:51,220 --> 00:10:55,810
You have a volume in three
dimensions in p space.

158
00:10:55,810 --> 00:10:59,110
If the radius goes
like T to the 1/2,

159
00:10:59,110 --> 00:11:02,030
the volume goes
like T to the 3/2.

160
00:11:02,030 --> 00:11:04,880
That's why you have all kinds
of excitations such as this.

161
00:11:08,150 --> 00:11:13,090
And the reason for the
Bose-Einstein condensation

162
00:11:13,090 --> 00:11:19,790
was that you would start to fill
out all of these excitations.

163
00:11:19,790 --> 00:11:25,430
And when you were adding all
of the mean occupation numbers,

164
00:11:25,430 --> 00:11:27,783
the answer was not
coming up all the way

165
00:11:27,783 --> 00:11:29,800
to the total number
of particles.

166
00:11:29,800 --> 00:11:32,400
So then you have to put
an excess at p close

167
00:11:32,400 --> 00:11:35,880
to 0, which corresponds to the
ground state of this system.

168
00:11:38,680 --> 00:11:41,090
Now of course when
you look at helium,

169
00:11:41,090 --> 00:11:44,310
helium molecules, helium
atoms have the interactions

170
00:11:44,310 --> 00:11:46,440
between them that we discussed.

171
00:11:46,440 --> 00:11:50,130
In particular, you can't really
put them on top of each other.

172
00:11:50,130 --> 00:11:53,150
There is a hard exclusion
when you bring things

173
00:11:53,150 --> 00:11:55,460
close to each other.

174
00:11:55,460 --> 00:11:58,470
So the ground
state of the system

175
00:11:58,470 --> 00:12:00,760
must look very different
from the ground state

176
00:12:00,760 --> 00:12:04,180
of the Bose-Einstein condensate
in which the particles freely

177
00:12:04,180 --> 00:12:08,000
occupy the entire box.

178
00:12:08,000 --> 00:12:12,030
So there is a very
difficult story

179
00:12:12,030 --> 00:12:15,020
associated with
figuring out what

180
00:12:15,020 --> 00:12:18,930
the ground state of this
combination of interacting

181
00:12:18,930 --> 00:12:22,880
particles that make
up liquid helium is.

182
00:12:22,880 --> 00:12:23,930
What is the behavior?

183
00:12:23,930 --> 00:12:26,260
What's the many-body wave
function at 0 temperature?

184
00:12:29,470 --> 00:12:34,630
Now as we see here, in order
to understand the heat capacity

185
00:12:34,630 --> 00:12:37,130
we really don't
need to know what

186
00:12:37,130 --> 00:12:40,060
is happening at
the ground state.

187
00:12:40,060 --> 00:12:43,350
What we need to know in
order to find heat capacity

188
00:12:43,350 --> 00:12:47,170
is how to put more energy in the
system above the ground state.

189
00:12:47,170 --> 00:12:51,090
So we need to know
something about excitations.

190
00:12:51,090 --> 00:12:54,930
And so that's the
perspective that Landau took.

191
00:12:54,930 --> 00:12:58,900
Landau said, well, this is
the spectrum of excitations

192
00:12:58,900 --> 00:13:03,470
if you had [? plain ?] particles
without any interactions.

193
00:13:03,470 --> 00:13:06,120
Let's imagine what
happens if we gradually

194
00:13:06,120 --> 00:13:08,220
tune in the interactions,
the particles

195
00:13:08,220 --> 00:13:10,960
start to repel each
other, et cetera.

196
00:13:10,960 --> 00:13:13,120
This non-interacting
ground state

197
00:13:13,120 --> 00:13:16,480
that we had in which the
particles were uniformly

198
00:13:16,480 --> 00:13:18,590
distributed across
the system will

199
00:13:18,590 --> 00:13:22,910
evolve into some complicated
ground state I don't know.

200
00:13:22,910 --> 00:13:25,210
And then, presumably
there would be a spectrum

201
00:13:25,210 --> 00:13:30,260
of excitations around
that ground state.

202
00:13:30,260 --> 00:13:34,760
Now the excitations around
the non-interacting ground

203
00:13:34,760 --> 00:13:39,250
state we can label by
this momentum peak.

204
00:13:39,250 --> 00:13:40,990
And it kind of
makes sense that we

205
00:13:40,990 --> 00:13:45,080
should be able to have a
singular label for excitations

206
00:13:45,080 --> 00:13:48,850
around the ground state of
these interacting particles.

207
00:13:48,850 --> 00:13:50,550
And this is where
you sort of needed

208
00:13:50,550 --> 00:13:53,330
a little bit of Landau's
type of insight.

209
00:13:53,330 --> 00:13:55,590
He said, well,
presumably what you

210
00:13:55,590 --> 00:13:59,770
do when you have
excitations of momentum p

211
00:13:59,770 --> 00:14:03,940
is to distort the wave
function in a manner that

212
00:14:03,940 --> 00:14:10,330
is consistent with having these
kind of excitations of momentum

213
00:14:10,330 --> 00:14:11,580
p.

214
00:14:11,580 --> 00:14:14,680
And he said, well
we typically know

215
00:14:14,680 --> 00:14:17,990
that if you have
a fluid or a solid

216
00:14:17,990 --> 00:14:21,850
and we want to impart some
momentum above the ground

217
00:14:21,850 --> 00:14:24,870
state, if will go in
the form of phonons.

218
00:14:24,870 --> 00:14:28,870
These are distortions in
which the density will

219
00:14:28,870 --> 00:14:35,060
vary in some sinusoidal or
cosine way across the system.

220
00:14:35,060 --> 00:14:37,920
So he said that maybe
what is happening

221
00:14:37,920 --> 00:14:40,370
is that for these
excitations you

222
00:14:40,370 --> 00:14:43,730
have to take what ever this
interacting ground state is--

223
00:14:43,730 --> 00:14:47,590
which we don't know and
can't write that down--

224
00:14:47,590 --> 00:14:49,900
but hope that
excitations around it

225
00:14:49,900 --> 00:14:52,950
correspond to these
distortions in density.

226
00:14:52,950 --> 00:14:56,410
And that by analogy with
what happens for fluids,

227
00:14:56,410 --> 00:15:01,350
that the spectrum of excitations
will then become a linear.

228
00:15:01,350 --> 00:15:04,240
You would have something
like a sound wave

229
00:15:04,240 --> 00:15:07,060
that you would have in
a liquid or a solid.

230
00:15:07,060 --> 00:15:14,290
So if you do this, if you
have a linear spectrum

231
00:15:14,290 --> 00:15:16,895
then we can see what happen.

232
00:15:16,895 --> 00:15:21,540
For a particular energy
of the order of kT,

233
00:15:21,540 --> 00:15:24,670
we will go here,
occupied momenta that

234
00:15:24,670 --> 00:15:27,160
would be of the order
of kT over H bar.

235
00:15:30,920 --> 00:15:35,560
The number of excitations would
be something like this cubed,

236
00:15:35,560 --> 00:15:37,300
and so you would
imagine that you

237
00:15:37,300 --> 00:15:41,000
would get a heat capacity that
is proportional to this times

238
00:15:41,000 --> 00:15:42,870
kB.

239
00:15:42,870 --> 00:15:45,570
And if you do things
correctly, like really

240
00:15:45,570 --> 00:15:47,840
for the case of
phonons or photons,

241
00:15:47,840 --> 00:15:52,010
you can even figure out what
the numerical prefactor is here.

242
00:15:52,010 --> 00:15:56,240
And there's a velocity
here because this curve

243
00:15:56,240 --> 00:15:59,280
goes like H bar vP.

244
00:16:05,980 --> 00:16:09,010
So then you can
compare what you have

245
00:16:09,010 --> 00:16:15,330
over here with the coefficient
of the T cubed over here,

246
00:16:15,330 --> 00:16:19,140
and you could even figure
out what this velocity is.

247
00:16:19,140 --> 00:16:23,030
And it turns out to be of the
order of 240 meters per second.

248
00:16:27,640 --> 00:16:31,450
A typical sound wave that
you would have in a fluid.

249
00:16:31,450 --> 00:16:35,390
So that's kind of
a reasonable thing.

250
00:16:35,390 --> 00:16:38,700
Now of course, when you go
to higher and higher momenta,

251
00:16:38,700 --> 00:16:41,420
it corresponds to essentially
shorter and shorter

252
00:16:41,420 --> 00:16:42,520
wavelengths.

253
00:16:42,520 --> 00:16:44,880
You expect that when
you get wavelengths

254
00:16:44,880 --> 00:16:49,860
that is of the order of
the interatomic spacing,

255
00:16:49,860 --> 00:16:52,670
then the interactions become
less and less important.

256
00:16:52,670 --> 00:16:55,080
You have particles
rattling in a cage

257
00:16:55,080 --> 00:16:57,230
that is set up by
everybody else.

258
00:16:57,230 --> 00:17:00,320
And then you should regain
this kind of spectrum

259
00:17:00,320 --> 00:17:03,120
at high values of momentum.

260
00:17:03,120 --> 00:17:06,920
And so what Landau
did was he basically

261
00:17:06,920 --> 00:17:10,270
joined these things
together and posed

262
00:17:10,270 --> 00:17:14,760
that there is a spectrum
such as this that

263
00:17:14,760 --> 00:17:17,310
has what is called
a phonon part, which

264
00:17:17,310 --> 00:17:24,359
is this linear part where
energy goes like H bar,

265
00:17:24,359 --> 00:17:30,290
like the velocity
times the momentum.

266
00:17:30,290 --> 00:17:35,010
And it has a part that in
the vicinity of this point,

267
00:17:35,010 --> 00:17:37,930
you can expand parabolically.

268
00:17:37,930 --> 00:17:38,960
And it's called rotons.

269
00:17:41,900 --> 00:17:46,690
There is a gap
delta, and then H bar

270
00:17:46,690 --> 00:17:52,390
squared over 2, some effective
mass, k minus k0 squared.

271
00:17:54,970 --> 00:17:58,110
This k0 turns out to
be roughly of the order

272
00:17:58,110 --> 00:18:00,930
of the inverse
[INAUDIBLE], two Angstrom

273
00:18:00,930 --> 00:18:04,470
inverse, between particles.

274
00:18:04,470 --> 00:18:09,430
This mu is of the order
of mass of [INAUDIBLE].

275
00:18:14,580 --> 00:18:21,330
So about 10 years
or so after Landau,

276
00:18:21,330 --> 00:18:25,540
people are able to get to this
whole spectrum of excitation

277
00:18:25,540 --> 00:18:29,330
through neutron scattering
and other scattering

278
00:18:29,330 --> 00:18:30,700
types of experiments.

279
00:18:30,700 --> 00:18:33,160
And so this picture
was confirmed.

280
00:18:39,710 --> 00:18:41,344
So, yes?

281
00:18:41,344 --> 00:18:43,320
AUDIENCE: What is a
roton? [INAUDIBLE]

282
00:18:49,874 --> 00:18:51,540
PROFESSOR: Over here
what you are seeing

283
00:18:51,540 --> 00:18:56,500
is essentially particles
rattling in the cage.

284
00:18:56,500 --> 00:18:59,840
It is believed that
what is happening here

285
00:18:59,840 --> 00:19:02,700
are collections of
three or four atoms that

286
00:19:02,700 --> 00:19:06,300
are kind of rotating
in a bigger cage.

287
00:19:06,300 --> 00:19:08,750
So something, the
picture that people draw

288
00:19:08,750 --> 00:19:12,192
is three or four
particles rotating around.

289
00:19:12,192 --> 00:19:13,178
Yes?

290
00:19:13,178 --> 00:19:15,643
AUDIENCE: Is there
some [INAUDIBLE] curve

291
00:19:15,643 --> 00:19:17,615
where energy's
decreasing [INAUDIBLE]?

292
00:19:21,559 --> 00:19:26,060
Does the transition between
photon and roton [INAUDIBLE]?

293
00:19:26,060 --> 00:19:29,630
PROFESSOR: There is no
thermodynamic or other rule

294
00:19:29,630 --> 00:19:32,746
that says that the energy should
be one of [INAUDIBLE] momentum

295
00:19:32,746 --> 00:19:36,040
that I know.

296
00:19:36,040 --> 00:19:37,290
Yes?

297
00:19:37,290 --> 00:19:41,877
AUDIENCE: Is there an
expression for that k0

298
00:19:41,877 --> 00:19:47,416
in terms of temperature and
other properties of the system?

299
00:19:47,416 --> 00:19:49,880
PROFESSOR: This
curve of excitations

300
00:19:49,880 --> 00:19:53,020
is supposed to be property
of the ground state.

301
00:19:53,020 --> 00:19:56,780
That is, you take this
system in its ground state,

302
00:19:56,780 --> 00:19:58,970
and then you create
an excitation that

303
00:19:58,970 --> 00:20:01,660
has some particular
momentum and calculate

304
00:20:01,660 --> 00:20:04,930
what the energy of that is.

305
00:20:04,930 --> 00:20:08,300
Actually, this whole
curve is phenomenological,

306
00:20:08,300 --> 00:20:12,650
because in order to get
the excitations you better

307
00:20:12,650 --> 00:20:16,430
have an expression for
what the ground state is.

308
00:20:16,430 --> 00:20:19,910
And so writing a
kind of wave function

309
00:20:19,910 --> 00:20:23,590
that describes the ground state
of this interacting system

310
00:20:23,590 --> 00:20:26,270
is a very difficult task.

311
00:20:26,270 --> 00:20:30,800
I think Feynman has some
variational type of wave

312
00:20:30,800 --> 00:20:33,490
function that we can start
and work with, and then

313
00:20:33,490 --> 00:20:37,760
calculate things approximately
in terms of that.

314
00:20:37,760 --> 00:20:38,350
Yes?

315
00:20:38,350 --> 00:20:40,058
AUDIENCE: What inspired
Landau to propose

316
00:20:40,058 --> 00:20:42,510
that there was a [INAUDIBLE]?

317
00:20:42,510 --> 00:20:46,460
PROFESSOR: Actually,
it was not so much

318
00:20:46,460 --> 00:20:49,300
I think looking at
this curve, but which

319
00:20:49,300 --> 00:20:51,090
I think if you
want to match that

320
00:20:51,090 --> 00:20:53,890
and that, you have to
have something like this.

321
00:20:53,890 --> 00:20:58,610
But this was really that the
whole experimental version

322
00:20:58,610 --> 00:21:01,440
of the heat capacity,
it didn't seem

323
00:21:01,440 --> 00:21:04,620
like this expression
was sufficient.

324
00:21:04,620 --> 00:21:08,800
And then there was some amount
of excitation and energy

325
00:21:08,800 --> 00:21:12,710
at the temperatures that were
experimentally accessible

326
00:21:12,710 --> 00:21:17,100
that heated at the presence
of the rotons in the spectrum.

327
00:21:17,100 --> 00:21:17,860
Yes?

328
00:21:17,860 --> 00:21:21,270
AUDIENCE: So
continuing [INAUDIBLE],

329
00:21:21,270 --> 00:21:26,230
if you raise the thermal energy
kBT, [INAUDIBLE] level where

330
00:21:26,230 --> 00:21:28,256
you have multiple
roots of this curve.

331
00:21:28,256 --> 00:21:29,030
PROFESSOR: Yes.

332
00:21:29,030 --> 00:21:33,010
AUDIENCE: So you will be
able to excite some states

333
00:21:33,010 --> 00:21:38,780
and have some kind of gap, like
a gap of momenta which are not.

334
00:21:38,780 --> 00:21:42,240
PROFESSOR: OK, so at any
finite temperatures-- and I'll

335
00:21:42,240 --> 00:21:45,380
do the calculation for
you shortly-- there

336
00:21:45,380 --> 00:21:50,430
is a finite probability for
exciting all of these states.

337
00:21:50,430 --> 00:21:52,280
What you are saying
is that when there

338
00:21:52,280 --> 00:21:56,070
is more occupation at this
momentum compared to that,

339
00:21:56,070 --> 00:21:59,420
but much less compared to this.

340
00:21:59,420 --> 00:22:06,600
So that again does not
violate any condition.

341
00:22:06,600 --> 00:22:13,570
So it is like, again, trying to
shake this system of particles.

342
00:22:13,570 --> 00:22:16,170
Let's imagine that
you have grains,

343
00:22:16,170 --> 00:22:17,780
and you are trying
to shake them.

344
00:22:17,780 --> 00:22:20,900
And it may be that at
some shaking frequencies,

345
00:22:20,900 --> 00:22:22,820
then there are things
that are taking place

346
00:22:22,820 --> 00:22:25,702
at short distances in
addition to some waves

347
00:22:25,702 --> 00:22:26,660
that you're generating.

348
00:22:32,361 --> 00:22:32,860
Yeah?

349
00:22:32,860 --> 00:22:34,901
AUDIENCE: I have a question
about the methodology

350
00:22:34,901 --> 00:22:36,880
of getting this
spectrum, because if we

351
00:22:36,880 --> 00:22:40,290
have a experimental result
of the [INAUDIBLE] capacity,

352
00:22:40,290 --> 00:22:42,700
then if we assume
there's a spectrum,

353
00:22:42,700 --> 00:22:44,840
there has to be
this one, because it

354
00:22:44,840 --> 00:22:47,250
gives a one to one
correspondence.

355
00:22:47,250 --> 00:22:54,840
So we can get the spectrum
directly from the c.

356
00:22:54,840 --> 00:22:55,808
So--

357
00:22:55,808 --> 00:23:00,460
PROFESSOR: I'm not sure,
because in reality this

358
00:23:00,460 --> 00:23:04,680
is going to be spectrum in
three-dimensional space.

359
00:23:04,680 --> 00:23:07,870
And there is certainly
an expression

360
00:23:07,870 --> 00:23:12,950
that relates the heat capacity
to the excitation spectrum.

361
00:23:12,950 --> 00:23:15,340
What I'm not sure is
whether mathematically

362
00:23:15,340 --> 00:23:18,840
that expression
uniquely invertible.

363
00:23:18,840 --> 00:23:22,840
It is given an epsilon-- c,
you have a unique epsilon of p.

364
00:23:22,840 --> 00:23:27,090
Certainly, given an epsilon
of p, you have a unique c.

365
00:23:27,090 --> 00:23:27,932
Yes?

366
00:23:27,932 --> 00:23:30,307
AUDIENCE: But if the excitation
spectrum only depends on,

367
00:23:30,307 --> 00:23:33,344
let's say, k squared, not on
the three-dimensional components

368
00:23:33,344 --> 00:23:36,296
of k, then maybe
it's much easier

369
00:23:36,296 --> 00:23:39,531
to draw a one to
one correspondence?

370
00:23:39,531 --> 00:23:41,030
PROFESSOR: I don't
know, because you

371
00:23:41,030 --> 00:23:43,230
have a function of
temperature and you

372
00:23:43,230 --> 00:23:47,130
want to convert it to
a function of momentum

373
00:23:47,130 --> 00:23:49,490
that after some
integrations will give you

374
00:23:49,490 --> 00:23:51,950
that function of temperature.

375
00:23:51,950 --> 00:23:56,530
I don't know the difficulty
of mathematically doing that.

376
00:23:56,530 --> 00:24:01,300
I know that I can't off my head
think of an inversion formula.

377
00:24:01,300 --> 00:24:03,426
It's not like the function
that you're inversing.

378
00:24:13,640 --> 00:24:20,940
So the Landau spectrum
can explain this part.

379
00:24:20,940 --> 00:24:24,800
It turns out that the Landau
spectrum cannot explain this

380
00:24:24,800 --> 00:24:26,470
logarithmic divergence.

381
00:24:26,470 --> 00:24:27,334
Yes?

382
00:24:27,334 --> 00:24:29,250
AUDIENCE: Sorry, one
more question about this.

383
00:24:32,040 --> 00:24:34,538
The allowed values of
k, do they get modified,

384
00:24:34,538 --> 00:24:38,410
or are they thought
to be the same?

385
00:24:38,410 --> 00:24:39,010
PROFESSOR: No.

386
00:24:39,010 --> 00:24:43,990
So basically at some point,
I have to change perspective

387
00:24:43,990 --> 00:24:49,520
from a sum over k to an integral
over k or an integral over p.

388
00:24:49,520 --> 00:24:52,120
The density of
states in momentum

389
00:24:52,120 --> 00:24:57,390
is something that is
kind of invariant.

390
00:24:57,390 --> 00:25:00,920
It is a very slight
function of shape.

391
00:25:00,920 --> 00:25:03,540
So the periodic
boundary conditions

392
00:25:03,540 --> 00:25:06,930
and the open boundary
conditions, et cetera,

393
00:25:06,930 --> 00:25:10,882
give you something slightly
different over here.

394
00:25:10,882 --> 00:25:14,190
But by the time you
go to the continuum,

395
00:25:14,190 --> 00:25:17,520
it's a property
of dimension only.

396
00:25:17,520 --> 00:25:20,673
It doesn't really depend
on the underlying shape.

397
00:25:20,673 --> 00:25:22,048
AUDIENCE: So we
still change sums

398
00:25:22,048 --> 00:25:23,650
for integrals with
the same rules?

399
00:25:23,650 --> 00:25:24,275
PROFESSOR: Yes.

400
00:25:26,550 --> 00:25:30,780
It's a sort of general
density of state property.

401
00:25:30,780 --> 00:25:34,060
So there's some nice
formula that tells you

402
00:25:34,060 --> 00:25:38,110
what the density of state
is for an arbitrary shape,

403
00:25:38,110 --> 00:25:41,740
and the leading term is
always proportional to volume

404
00:25:41,740 --> 00:25:44,260
or area [INAUDIBLE]
the density of state

405
00:25:44,260 --> 00:25:46,180
that you have been calculating.

406
00:25:46,180 --> 00:25:47,910
And then there are
some leading terms

407
00:25:47,910 --> 00:25:52,140
that are proportional to
if it is volume to area,

408
00:25:52,140 --> 00:25:55,260
or number of edges, et cetera.

409
00:25:55,260 --> 00:25:58,626
But those are kind of subleading
the thermodynamic sense.

410
00:26:04,500 --> 00:26:09,050
So I guess Feynman
did a lot of work

411
00:26:09,050 --> 00:26:15,100
on formalizing these
ideas of Landau getting

412
00:26:15,100 --> 00:26:19,600
some idea of what the ground
state does, is, and excitations

413
00:26:19,600 --> 00:26:22,090
that you can have
about the spectrum.

414
00:26:22,090 --> 00:26:24,890
And so he was very
happy at being

415
00:26:24,890 --> 00:26:28,990
able to explain this,
including the nature of rotons,

416
00:26:28,990 --> 00:26:29,920
et cetera.

417
00:26:29,920 --> 00:26:31,890
And he was worried
that somehow he

418
00:26:31,890 --> 00:26:35,060
couldn't get this
logarithmic divergence.

419
00:26:35,060 --> 00:26:37,080
And that bothered
him a little bit,

420
00:26:37,080 --> 00:26:39,990
but Onsager told him
that that's really

421
00:26:39,990 --> 00:26:42,280
a much more fundamental
property that

422
00:26:42,280 --> 00:26:45,960
depends on critical phenomena,
and for resolving that issue,

423
00:26:45,960 --> 00:26:49,700
you have to come to 8.334.

424
00:26:49,700 --> 00:26:52,460
So we will not discuss
that, nor will we

425
00:26:52,460 --> 00:26:57,165
discuss why this is Tc minus
T to the 2/3 power and not

426
00:26:57,165 --> 00:26:58,820
a linear dependence.

427
00:26:58,820 --> 00:27:01,910
It again is one of these
critical properties.

428
00:27:01,910 --> 00:27:06,180
But we should be able to
explain this T to the fourth.

429
00:27:06,180 --> 00:27:07,900
And clearly, this
T to the fourth

430
00:27:07,900 --> 00:27:13,880
is not as simple as saying this
exponent changed from 3/2 to 3,

431
00:27:13,880 --> 00:27:16,060
this 3/2 should
also change the 3.

432
00:27:16,060 --> 00:27:21,420
No, it went to T to the fourth,
so what's going on over here?

433
00:27:21,420 --> 00:27:24,580
So last time at the
end of the lecture

434
00:27:24,580 --> 00:27:35,100
I wrote a statement that
the BEC is not superfluid.

435
00:27:40,310 --> 00:27:44,110
And what that really
means is that it

436
00:27:44,110 --> 00:27:48,646
has too many excitations,
low energy excitations.

437
00:27:56,410 --> 00:27:59,630
So imagine the
following, that maybe we

438
00:27:59,630 --> 00:28:03,650
have a container-- I don't
know, maybe we have a tube--

439
00:28:03,650 --> 00:28:13,480
and we have our superfluid going
through this with velocity v

440
00:28:13,480 --> 00:28:15,390
sub s.

441
00:28:15,390 --> 00:28:21,660
We want it to maintain that
velocity without experiencing

442
00:28:21,660 --> 00:28:23,870
friction, which it
seems to do in going

443
00:28:23,870 --> 00:28:25,050
through these capillaries.

444
00:28:25,050 --> 00:28:29,580
You don't have to push it, it
seems to be going by itself.

445
00:28:29,580 --> 00:28:36,480
And so the question is,
can any of these pictures

446
00:28:36,480 --> 00:28:42,230
that we drew for excitations
be consistent with this?

447
00:28:42,230 --> 00:28:45,690
Now, why am I talking about
excitations and consistency

448
00:28:45,690 --> 00:28:47,810
with superfluid?

449
00:28:47,810 --> 00:28:51,440
Because what can
happen in principle

450
00:28:51,440 --> 00:28:57,190
is that within your system,
you can spontaneously

451
00:28:57,190 --> 00:29:00,120
generate some excitation.

452
00:29:00,120 --> 00:29:04,120
This excitation will
have some momentum p

453
00:29:04,120 --> 00:29:08,410
and some energy epsilon of p.

454
00:29:08,410 --> 00:29:13,500
And if you spontaneously can
create these excitations that

455
00:29:13,500 --> 00:29:19,310
would take away energy from this
kinetic energy of the flowing

456
00:29:19,310 --> 00:29:23,950
superfluid, gradually the
superfluid will slow down.

457
00:29:23,950 --> 00:29:27,420
Its energy will be
dissipated and the superfluid

458
00:29:27,420 --> 00:29:29,997
itself will heat up
because you generated

459
00:29:29,997 --> 00:29:31,080
these excitations with it.

460
00:29:34,000 --> 00:29:38,690
So let's see what happens.

461
00:29:38,690 --> 00:29:42,750
If I were to create
such an excitation,

462
00:29:42,750 --> 00:29:51,320
actually I have to worry
about momentum conservation

463
00:29:51,320 --> 00:29:56,120
because I created something
that carried momentum p.

464
00:29:56,120 --> 00:30:04,690
Now initially, let's say
that this whole entity, all

465
00:30:04,690 --> 00:30:08,330
of the fluid that are
superflowing with velocity Vs

466
00:30:08,330 --> 00:30:13,055
have mass M. So the initial
momentum would be MVs.

467
00:30:16,280 --> 00:30:21,280
Now, I created some
excitation that

468
00:30:21,280 --> 00:30:24,970
is carrying away
some momentum p.

469
00:30:24,970 --> 00:30:28,740
So the only thing that
can ensure this happens

470
00:30:28,740 --> 00:30:32,510
is that I have to
slightly change

471
00:30:32,510 --> 00:30:34,285
the velocity of the fluid.

472
00:30:37,240 --> 00:30:40,425
Now this change in
velocity is infinitesimal.

473
00:30:40,425 --> 00:30:48,470
It is Vs minus p
divided by M. M is huge,

474
00:30:48,470 --> 00:30:53,292
so why bother
thinking about this?

475
00:30:53,292 --> 00:30:57,040
Well, let's see what
the change in energy is.

476
00:31:02,250 --> 00:31:07,160
Delta E. Let's say, well,
you created this excitation

477
00:31:07,160 --> 00:31:09,280
so you have energy epsilon of p.

478
00:31:12,310 --> 00:31:15,295
But I say, in
addition to that there

479
00:31:15,295 --> 00:31:20,500
is a change in the kinetic
energy of the superfluid.

480
00:31:20,500 --> 00:31:23,830
I'm now moving at
Vs prime squared,

481
00:31:23,830 --> 00:31:28,130
whereas initially when this
excitation was not present,

482
00:31:28,130 --> 00:31:32,650
I was moving at Vs.

483
00:31:32,650 --> 00:31:36,300
And so what do you have here?

484
00:31:36,300 --> 00:31:44,566
We have epsilon of p, M
over 2 Vs minus p over M,

485
00:31:44,566 --> 00:31:49,920
this infinitesimal change
in velocity squared minus M

486
00:31:49,920 --> 00:31:53,760
over 2 Vs squared.

487
00:31:53,760 --> 00:31:56,830
We can see that the leading
order of the kinetic energy

488
00:31:56,830 --> 00:32:01,780
goes away, but that there
is a cross term here

489
00:32:01,780 --> 00:32:05,660
in which the M contribution--
the contribution of the mass--

490
00:32:05,660 --> 00:32:07,460
goes away.

491
00:32:07,460 --> 00:32:15,140
And so the change in energy is
actually something like this.

492
00:32:18,020 --> 00:32:25,330
So if I had a system that
when stationary, the energy

493
00:32:25,330 --> 00:32:30,830
to create an excitation of
momentum p was epsilon of p,

494
00:32:30,830 --> 00:32:35,960
when I put that in a frame that
is moving with some velocity

495
00:32:35,960 --> 00:32:40,250
Vs, you have the
ability to borrow

496
00:32:40,250 --> 00:32:45,040
some of that kinetic energy
and the excitation energy

497
00:32:45,040 --> 00:32:47,045
goes down by this amount.

498
00:32:58,610 --> 00:33:07,330
So what happens if I take
the Bose-Einstein type

499
00:33:07,330 --> 00:33:10,810
of excitation spectrum
that is p squared over 2M

500
00:33:10,810 --> 00:33:15,860
and then subtract
a v dot p from it?

501
00:33:15,860 --> 00:33:20,520
Essentially there is a
linear subtraction going on,

502
00:33:20,520 --> 00:33:24,870
and I would get a
curve such as this.

503
00:33:24,870 --> 00:33:28,630
So I probably exaggerated
this by a lot.

504
00:33:28,630 --> 00:33:31,150
I shouldn't have
subtracted so much.

505
00:33:31,150 --> 00:33:37,590
Let me actually not
subject so much,

506
00:33:37,590 --> 00:33:40,260
because we don't want to go
all the way in that range.

507
00:33:45,860 --> 00:33:50,590
But you can see that there
is a range of momenta where

508
00:33:50,590 --> 00:33:56,710
you would spontaneously gain
energy by creating excitations.

509
00:33:56,710 --> 00:34:00,630
If the spectrum was
initially p squared over 2M,

510
00:34:00,630 --> 00:34:05,430
basically you just have too
many low energy excitations.

511
00:34:05,430 --> 00:34:07,960
As soon as you
start moving it, you

512
00:34:07,960 --> 00:34:11,280
will spontaneously
excite these things.

513
00:34:11,280 --> 00:34:14,080
Even if you were initially
at 0 temperature,

514
00:34:14,080 --> 00:34:18,340
these phonon excitations
would be created spontaneously

515
00:34:18,340 --> 00:34:19,159
in your system.

516
00:34:19,159 --> 00:34:20,840
They would move
all over the place.

517
00:34:20,840 --> 00:34:22,610
They would heat up your system.

518
00:34:22,610 --> 00:34:25,800
There is no way
that you can pass

519
00:34:25,800 --> 00:34:28,889
the Bose-Einstein
condensate-- actually,

520
00:34:28,889 --> 00:34:32,350
there's no way that you can even
move it without losing energy.

521
00:34:35,300 --> 00:34:38,900
But you can see that
this red curve does not

522
00:34:38,900 --> 00:34:40,830
have that difficulty.

523
00:34:40,830 --> 00:34:44,580
If I were to shift this curve
by an amount that is linear,

524
00:34:44,580 --> 00:34:45,520
what do I get?

525
00:34:45,520 --> 00:34:47,202
I will get something like this.

526
00:34:53,880 --> 00:35:00,030
So the Landau spectrum
is perfectly fine

527
00:35:00,030 --> 00:35:04,980
as far as excitations
is concerned.

528
00:35:04,980 --> 00:35:11,010
At zero temperature, even if the
whole fluid is moving then it

529
00:35:11,010 --> 00:35:14,440
cannot spontaneously
create excitations,

530
00:35:14,440 --> 00:35:17,490
because you would increase
energy of the system.

531
00:35:20,230 --> 00:35:22,530
So there's this difference.

532
00:35:22,530 --> 00:35:25,430
Ultimately, you would
say that the first time

533
00:35:25,430 --> 00:35:32,830
you would get excitation is
if you move it fast enough

534
00:35:32,830 --> 00:35:36,690
so that some portion of
this curve goes to 0.

535
00:35:36,690 --> 00:35:39,890
And indeed, if you were
to try to stir or move

536
00:35:39,890 --> 00:35:43,140
a superfluid fast
enough, there's

537
00:35:43,140 --> 00:35:45,240
a velocity at which
it breaks down,

538
00:35:45,240 --> 00:35:47,860
it stops being a superfluid.

539
00:35:47,860 --> 00:35:52,040
But it turns out that that
velocity is much, much smaller

540
00:35:52,040 --> 00:35:56,430
than you would predict based on
this roton spectrum going down.

541
00:35:56,430 --> 00:35:59,530
There are some other
many body excitations

542
00:35:59,530 --> 00:36:04,830
that come before and cause
the superfluid to lose energy

543
00:36:04,830 --> 00:36:06,650
and break down.

544
00:36:06,650 --> 00:36:12,050
But the genetic idea as
to why a linear spectrum

545
00:36:12,050 --> 00:36:16,140
for k close to 0 is
consist with superfluidity

546
00:36:16,140 --> 00:36:18,942
but the quadratic one
is not remains correct.

547
00:36:25,310 --> 00:36:31,910
Now suppose I am
in this situation.

548
00:36:31,910 --> 00:36:34,410
I have a moving super
fluid, such as the one

549
00:36:34,410 --> 00:36:36,950
that I have described over here.

550
00:36:36,950 --> 00:36:40,730
The spectrum is going to
be somewhat like this,

551
00:36:40,730 --> 00:36:43,450
but I'm not at 0 temperature.

552
00:36:43,450 --> 00:36:46,365
I want to try to describe
this T to the fourth behavior,

553
00:36:46,365 --> 00:36:49,880
so I want to be at some
finite temperature.

554
00:36:49,880 --> 00:36:52,480
So if I'm at some
finite temperature,

555
00:36:52,480 --> 00:36:57,070
there is some probability to
excite these different states,

556
00:36:57,070 --> 00:37:03,380
and the number that would
correspond to some momentum p

557
00:37:03,380 --> 00:37:07,170
would be given by this
general formula you have,

558
00:37:07,170 --> 00:37:14,490
1 over Z inverse e to the
beta epsilon of p minus 1.

559
00:37:14,490 --> 00:37:22,230
Furthermore, if I think that
I am in the regime where

560
00:37:22,230 --> 00:37:27,430
the number of excitations
is not important because

561
00:37:27,430 --> 00:37:32,840
of the same reason that I had
for Bose-Einstein condensate,

562
00:37:32,840 --> 00:37:36,740
I would have this formula
except that I would use epsilon

563
00:37:36,740 --> 00:37:41,540
of p that is appropriate
to this system.

564
00:37:41,540 --> 00:37:44,800
Actually, what is it
appropriate to this system

565
00:37:44,800 --> 00:37:53,160
is that my epsilon of
p was velocity times p.

566
00:37:53,160 --> 00:37:58,056
But then I started to move
with this superfluid velocity.

567
00:37:58,056 --> 00:37:59,850
Actually, maybe I'll
call this c so that I

568
00:37:59,850 --> 00:38:04,580
a distinction between c, which
is the linear spectrum here,

569
00:38:04,580 --> 00:38:06,430
and the superfluid
velocity dot p.

570
00:38:17,580 --> 00:38:21,565
No, this is actually
a vectorial product.

571
00:38:25,390 --> 00:38:28,570
And because of that,
I only drew one part

572
00:38:28,570 --> 00:38:31,920
of this curve that corresponds
to positive momentum.

573
00:38:31,920 --> 00:38:33,590
If I had gone to
negative momentum,

574
00:38:33,590 --> 00:38:37,460
actually, this curve
would have continued

575
00:38:37,460 --> 00:38:41,120
and whereas one branch
the energy is reduced,

576
00:38:41,120 --> 00:38:45,600
if I go to minus p,
the energy goes up.

577
00:38:45,600 --> 00:38:50,740
So whereas if the
superfluid was not moving,

578
00:38:50,740 --> 00:38:54,190
I can generate as many
excitations with momentum p

579
00:38:54,190 --> 00:38:56,510
as momentum minus p.

580
00:38:56,510 --> 00:39:02,450
Once the superfluid is
moving, there is a difference.

581
00:39:02,450 --> 00:39:04,100
One of them has a v dot p.

582
00:39:04,100 --> 00:39:06,660
The other has minus v dot p.

583
00:39:06,660 --> 00:39:11,020
So because of that,
there is a net momentum

584
00:39:11,020 --> 00:39:14,180
that is carried by
these excitations.

585
00:39:14,180 --> 00:39:16,660
This net momentum is
obtained by summing

586
00:39:16,660 --> 00:39:19,600
over all of these
things, multiplying

587
00:39:19,600 --> 00:39:20,850
with appropriate momentum.

588
00:39:20,850 --> 00:39:31,230
So I have beta is CP
minus v dot p minus 1.

589
00:39:31,230 --> 00:39:33,920
This is the momentum
of the excitation.

590
00:39:33,920 --> 00:39:39,080
This is the net momentum of
the system for one excitation.

591
00:39:39,080 --> 00:39:43,060
But then I have to sum
over all possible P's.

592
00:39:43,060 --> 00:39:45,100
Sum over P's, as
we've discussed,

593
00:39:45,100 --> 00:39:48,550
I can replace with an integral.

594
00:39:48,550 --> 00:39:54,960
And sum over k be replaced with
V. Integral over K, K and P

595
00:39:54,960 --> 00:39:58,540
are simply related
by a factor of h bar.

596
00:39:58,540 --> 00:40:02,930
So whereas before for k I
had 2 pi cubed for v cubed,

597
00:40:02,930 --> 00:40:04,135
I have 2 pi h bar cubed.

598
00:40:07,510 --> 00:40:11,360
So this is what I
have to calculate.

599
00:40:11,360 --> 00:40:13,160
Now what happens for small v?

600
00:40:18,760 --> 00:40:23,360
I can make an expansion
in vs. The 0 order term

601
00:40:23,360 --> 00:40:25,090
in the expansion
is what we would

602
00:40:25,090 --> 00:40:27,490
have for non-moving fluid.

603
00:40:27,490 --> 00:40:29,550
Momenta in the two
directions are the same,

604
00:40:29,550 --> 00:40:32,190
so that contribution goes away.

605
00:40:32,190 --> 00:40:35,740
The first contribution
that I'm going to get

606
00:40:35,740 --> 00:40:47,760
is going to come from expanding
this to lowest order in P.

607
00:40:47,760 --> 00:40:52,390
So there is a P that
is sitting out front.

608
00:40:52,390 --> 00:40:54,240
When I make the
expansion, I will

609
00:40:54,240 --> 00:40:59,000
get a vs dot P
times the derivative

610
00:40:59,000 --> 00:41:02,410
of the exponential
function gives me

611
00:41:02,410 --> 00:41:08,760
a factor of beta
e to the beta CP.

612
00:41:08,760 --> 00:41:15,410
And down here I will have e to
the beta CP minus 1 squared.

613
00:41:22,970 --> 00:41:25,890
Now, in the problem set you
have to actually evaluate

614
00:41:25,890 --> 00:41:27,260
this integral.

615
00:41:27,260 --> 00:41:28,660
It's not that difficult.

616
00:41:28,660 --> 00:41:31,020
It's related to Zeta functions.

617
00:41:31,020 --> 00:41:33,990
But what I'm really
only interested in what

618
00:41:33,990 --> 00:41:37,170
is the temperature dependence?

619
00:41:37,170 --> 00:41:43,660
You can see that I can rescale
this combination, call it x.

620
00:41:46,250 --> 00:41:50,540
Essentially what it says is that
whenever I see a factor of p,

621
00:41:50,540 --> 00:41:55,590
replace it by Kt over Cx.

622
00:41:55,590 --> 00:41:57,562
And how many P's do I have?

623
00:41:57,562 --> 00:42:01,715
I have three, four, five.

624
00:42:04,610 --> 00:42:10,740
So I have five factors of P. So
I will have five factors of Kt.

625
00:42:10,740 --> 00:42:13,910
One of them gets
killed by the beta,

626
00:42:13,910 --> 00:42:16,450
so this whole thing
is proportional to T

627
00:42:16,450 --> 00:42:17,380
to the fourth power.

628
00:42:23,170 --> 00:42:25,480
So what have we found?

629
00:42:25,480 --> 00:42:31,820
We have found that as
this fluid is moving

630
00:42:31,820 --> 00:42:35,240
at some finite
temperature T, it will

631
00:42:35,240 --> 00:42:38,220
generate these excitations.

632
00:42:38,220 --> 00:42:41,200
And these excitations
are preferably

633
00:42:41,200 --> 00:42:44,520
along the direction
of the momentum.

634
00:42:44,520 --> 00:42:49,610
And they correspond to
an additional momentum

635
00:42:49,610 --> 00:42:53,460
of the fluid that is
proportional to the volume.

636
00:42:57,210 --> 00:43:02,400
It's proportional to temperature
to the fourth and something.

637
00:43:02,400 --> 00:43:05,990
And of course, proportional
to the velocity.

638
00:43:09,200 --> 00:43:14,830
Now, we are used to thinking of
the proportionality of momentum

639
00:43:14,830 --> 00:43:18,050
and velocity to be
some kind of a mass.

640
00:43:21,280 --> 00:43:24,040
If I divide that
mass by the volume,

641
00:43:24,040 --> 00:43:28,710
I have a density of
these excitations.

642
00:43:28,710 --> 00:43:31,480
And what we have established
is that the density

643
00:43:31,480 --> 00:43:35,000
of those excitations
is proportional to t

644
00:43:35,000 --> 00:43:36,430
to the fourth.

645
00:43:36,430 --> 00:43:40,190
And what is happening in this
Andronikashvili experiment

646
00:43:40,190 --> 00:43:45,170
is that as these plates are
moving, by this mechanism

647
00:43:45,170 --> 00:43:47,230
the superfluid
that is in contact

648
00:43:47,230 --> 00:43:50,740
with them will
create excitations.

649
00:43:50,740 --> 00:43:53,720
And the momentum of
those excitations

650
00:43:53,720 --> 00:43:57,130
would correspond to some
kind of a density that

651
00:43:57,130 --> 00:44:00,540
vanishes as T to
the fourth, again

652
00:44:00,540 --> 00:44:06,370
in agreement with
what we've seen here.

653
00:44:06,370 --> 00:44:07,790
OK?

654
00:44:07,790 --> 00:44:08,420
Yes?

655
00:44:08,420 --> 00:44:10,825
AUDIENCE: So in an
integral expression

656
00:44:10,825 --> 00:44:13,086
you have vs as part
of a dot product.

657
00:44:13,086 --> 00:44:13,711
PROFESSOR: Yes.

658
00:44:13,711 --> 00:44:16,597
AUDIENCE: And then in the
next line [INAUDIBLE].

659
00:44:16,597 --> 00:44:18,490
So it's in that direction.

660
00:44:18,490 --> 00:44:23,530
PROFESSOR: So let's give these
indices p in direction alpha.

661
00:44:23,530 --> 00:44:26,520
This is p in direction alpha.

662
00:44:26,520 --> 00:44:29,250
This is p in direction alpha.

663
00:44:29,250 --> 00:44:32,330
This is v in direction
beta, p in direction

664
00:44:32,330 --> 00:44:33,590
beta, sum over beta.

665
00:44:36,220 --> 00:44:40,880
Now, I have to do an
angular integration that

666
00:44:40,880 --> 00:44:43,440
is spherically symmetric.

667
00:44:43,440 --> 00:44:48,170
And then somewhere inside
there it has a p alpha p beta.

668
00:44:48,170 --> 00:44:54,120
That angular
integration will give me

669
00:44:54,120 --> 00:44:59,080
a p squared over three delta
alpha beta, which then converts

670
00:44:59,080 --> 00:45:02,770
this v beta to a
b alpha, which is

671
00:45:02,770 --> 00:45:04,284
in the direction
of the momentum.

672
00:45:11,280 --> 00:45:11,780
Yes?

673
00:45:11,780 --> 00:45:13,155
AUDIENCE: Will
you say once again

674
00:45:13,155 --> 00:45:15,130
what happened to the
integral dimension?

675
00:45:15,130 --> 00:45:15,760
PROFESSOR: OK.

676
00:45:15,760 --> 00:45:24,020
So when we are in the
Bose-Einstein condensate,

677
00:45:24,020 --> 00:45:28,050
as far as the
excitations of concerned

678
00:45:28,050 --> 00:45:30,980
we have zero chemical potential.

679
00:45:30,980 --> 00:45:33,090
Whatever number of
particle that we

680
00:45:33,090 --> 00:45:36,560
have in excess of what
can be accommodated

681
00:45:36,560 --> 00:45:40,920
through the excitations we put
together in the ground state.

682
00:45:40,920 --> 00:45:43,980
So if you like the ground
state, the kp equals to 0

683
00:45:43,980 --> 00:45:47,430
or k equals to 0,
is a reservoir.

684
00:45:47,430 --> 00:45:49,410
You can add as many
particles there

685
00:45:49,410 --> 00:45:52,230
or bring as many particles
out of it as you like.

686
00:45:52,230 --> 00:45:55,000
So effectively, you have
no conservation number

687
00:45:55,000 --> 00:45:56,540
and no need for a [? z. ?]

688
00:46:03,350 --> 00:46:05,880
Of course, that we
only know for the case

689
00:46:05,880 --> 00:46:08,420
of the true
Bose-Einstein condensate.

690
00:46:08,420 --> 00:46:10,470
We are kind of
jumping and giving

691
00:46:10,470 --> 00:46:13,880
that concept relevance for
the interacting superfluid.

692
00:46:18,960 --> 00:46:21,407
OK?

693
00:46:21,407 --> 00:46:22,240
Any other questions?

694
00:46:26,230 --> 00:46:29,660
So this is actually
the last item

695
00:46:29,660 --> 00:46:34,280
I wanted to cover for
going on the board.

696
00:46:34,280 --> 00:46:37,170
The rest of the hour,
we have this movie

697
00:46:37,170 --> 00:46:39,846
that I had promised you.

698
00:46:39,846 --> 00:46:42,660
I will let that movie run.

699
00:46:42,660 --> 00:46:47,240
I also have all the connection
of problem sets, and exams,

700
00:46:47,240 --> 00:46:49,760
and test that you
have not picked up.

701
00:46:49,760 --> 00:46:53,450
So while the movie runs, you
are welcome to sit and enjoy it.

702
00:46:53,450 --> 00:46:54,940
It's very nice.

703
00:46:54,940 --> 00:46:58,510
Or you can go and take your
stuff and go your own way

704
00:46:58,510 --> 00:47:02,020
or do whatever you like.

705
00:47:02,020 --> 00:47:05,025
So let's go back.

706
00:47:29,244 --> 00:47:32,730
[VIDEO PLAYBACK]

707
00:48:29,314 --> 00:48:30,855
PROFESSOR: There
will be more action.

708
00:49:58,968 --> 00:50:00,960
-We just made a transfer
from liquid helium

709
00:50:00,960 --> 00:50:06,438
out of the storage tank into
our own experimental equipment.

710
00:50:06,438 --> 00:50:08,928
It is a remarkable [INAUDIBLE].

711
00:50:08,928 --> 00:50:11,418
It has two different and
easily distinguishable

712
00:50:11,418 --> 00:50:15,402
liquid phases-- a
warmer and a colder one.

713
00:50:15,402 --> 00:50:19,137
The warmer phase is called
liquid helium I and the colder

714
00:50:19,137 --> 00:50:21,876
phase liquid helium II.

715
00:50:21,876 --> 00:50:25,860
The two stages are separated
by a transition temperature,

716
00:50:25,860 --> 00:50:28,570
known as the lambda point.

717
00:50:28,570 --> 00:50:31,725
When liquid helium is pulled
down through the lambda point,

718
00:50:31,725 --> 00:50:36,610
a transition from helium I to
helium II is clearly visible.

719
00:50:36,610 --> 00:50:39,830
We will show it to you
later in this film.

720
00:50:39,830 --> 00:50:44,720
The two liquids behave nothing
like any other known liquid,

721
00:50:44,720 --> 00:50:48,620
although it could be said that
helium I, the warmer phase,

722
00:50:48,620 --> 00:50:52,140
approximates the behavior
of common liquids.

723
00:50:52,140 --> 00:50:55,620
But it is helium II,
the colder phase,

724
00:50:55,620 --> 00:50:57,690
which is truly different.

725
00:50:57,690 --> 00:51:01,676
Because of this, it is
called a superfluid.

726
00:51:01,676 --> 00:51:04,730
The temperatures involved when
working with liquid helium

727
00:51:04,730 --> 00:51:06,730
are quite low.

728
00:51:06,730 --> 00:51:10,154
Helium boils at
4.2 degrees Kelvin

729
00:51:10,154 --> 00:51:12,850
under conditions of
atmospheric pressure.

730
00:51:12,850 --> 00:51:17,400
And the lambda point lies
at roughly 2.2 degrees.

731
00:51:17,400 --> 00:51:21,810
Note that this
corresponds to minus 269

732
00:51:21,810 --> 00:51:26,514
and minus 271
degrees centigrade.

733
00:51:26,514 --> 00:51:29,390
The properties of liquid
helium that I have just

734
00:51:29,390 --> 00:51:31,738
been telling you about
are characteristic

735
00:51:31,738 --> 00:51:36,202
of the heavy isotope
if helium, helium-4.

736
00:51:36,202 --> 00:51:40,178
The element occurs in the
form of two stable isotopes.

737
00:51:40,178 --> 00:51:43,330
[INAUDIBLE] The second
and lighter one,

738
00:51:43,330 --> 00:51:46,620
helium-3, is very rare.

739
00:51:46,620 --> 00:51:51,738
Its abundance is only
about 1 part of 10 million.

740
00:51:51,738 --> 00:51:55,126
Pure liquid helium-3
is the subject

741
00:51:55,126 --> 00:51:57,950
of intensive study
at the present time,

742
00:51:57,950 --> 00:52:02,690
but so far no second
superfluid liquid phase

743
00:52:02,690 --> 00:52:04,868
has been found to
exist for helium-3.

744
00:52:08,330 --> 00:52:10,670
The low temperature at
which we'll be working

745
00:52:10,670 --> 00:52:12,980
calls for well-insulated
containers.

746
00:52:12,980 --> 00:52:15,045
The dewar meets
our requirements.

747
00:52:15,045 --> 00:52:17,620
The word "dewar" is
a scientific name

748
00:52:17,620 --> 00:52:19,758
given to a double-walled
vessel with the space

749
00:52:19,758 --> 00:52:22,500
between the walls evacuated.

750
00:52:22,500 --> 00:52:24,000
When these dewars
are made of glass,

751
00:52:24,000 --> 00:52:26,800
the surface of this
inner space is usually

752
00:52:26,800 --> 00:52:30,045
filtered to cut down heat
transfer by radiation.

753
00:52:30,045 --> 00:52:33,920
However, our dewars will
have to be transparent

754
00:52:33,920 --> 00:52:37,280
so that we can look at
what's going on inside.

755
00:52:37,280 --> 00:52:40,420
Now, liquid helium is commonly
stored in double dewars.

756
00:52:40,420 --> 00:52:44,035
The design is quite
simple, just put one

757
00:52:44,035 --> 00:52:49,676
inside the other like this.

758
00:52:49,676 --> 00:52:52,360
In the inner dewar, we
put the liquid helium,

759
00:52:52,360 --> 00:52:54,820
and in the space between
the inner and outer dewar,

760
00:52:54,820 --> 00:52:57,240
we maintain a supply
of liquid air.

761
00:53:00,300 --> 00:53:02,955
Here is a double dewar
exactly like the one

762
00:53:02,955 --> 00:53:07,530
we will be using in our
demonstration experiment.

763
00:53:07,530 --> 00:53:10,700
The inner dewar is filled
with liquid helium.

764
00:53:10,700 --> 00:53:12,910
The outer dewar
contains liquid air.

765
00:53:16,530 --> 00:53:19,450
The normal boiling
temperature of liquid air

766
00:53:19,450 --> 00:53:23,740
is about 80 degrees
Kelvin, 75 or more degrees

767
00:53:23,740 --> 00:53:26,190
hotter than liquid helium.

768
00:53:26,190 --> 00:53:28,620
The purpose of the
liquid air is twofold.

769
00:53:31,710 --> 00:53:34,450
First, we put the liquid air
in the outer dewar well ahead

770
00:53:34,450 --> 00:53:37,810
of putting liquid helium
in the inner dewar.

771
00:53:37,810 --> 00:53:42,003
In this way, the inner
dewar is pre-cooled.

772
00:53:42,003 --> 00:53:44,861
Secondly, we maintain
a supply of liquid air

773
00:53:44,861 --> 00:53:47,240
in the outer dewar
because it provides

774
00:53:47,240 --> 00:53:49,910
an additional
[INAUDIBLE] of insulation

775
00:53:49,910 --> 00:53:53,010
now that the liquid helium
is in the inner dewar.

776
00:53:53,010 --> 00:53:55,480
The [INAUDIBLE] liquid
air attests to the fact

777
00:53:55,480 --> 00:53:57,510
that it is absorbing
some of the heat which

778
00:53:57,510 --> 00:53:58,510
enters the double dewar.

779
00:54:01,410 --> 00:54:03,850
Even with the boiling
of the liquid air,

780
00:54:03,850 --> 00:54:06,690
the liquid helium
is clearly visible.

781
00:54:06,690 --> 00:54:09,140
Later, we will use
liquid air cooled

782
00:54:09,140 --> 00:54:12,228
below its boiling temperature
to reduce or eliminate

783
00:54:12,228 --> 00:54:13,811
the air bubbles for
better visibility.

784
00:54:17,870 --> 00:54:20,050
Now the liquid
air is cooled down

785
00:54:20,050 --> 00:54:22,540
and we have eliminated boiling.

786
00:54:22,540 --> 00:54:25,340
The smaller bubbles of
the boiling liquid helium

787
00:54:25,340 --> 00:54:27,424
are clearly visible.

788
00:54:27,424 --> 00:54:32,210
The cover over the inner dewar
has a port, at present open.

789
00:54:32,210 --> 00:54:35,750
The liquid helium is at
atmospheric pressure,

790
00:54:35,750 --> 00:54:39,200
so its temperature is
4.2 degrees Kelvin.

791
00:54:43,200 --> 00:54:45,546
In other words, what
we have in here now

792
00:54:45,546 --> 00:54:49,800
is liquid helium one, the
warmer of the two phases.

793
00:54:49,800 --> 00:54:53,740
Before we cool it down to take
a look at the superfluid phase,

794
00:54:53,740 --> 00:54:57,710
I want to dwell greatly on the
properties of helium I. I've

795
00:54:57,710 --> 00:55:00,665
told you before
that even helium I

796
00:55:00,665 --> 00:55:04,361
is different from
the normal liquids.

797
00:55:04,361 --> 00:55:06,610
The distance between neighboring
atoms and this liquid

798
00:55:06,610 --> 00:55:08,186
is quite large.

799
00:55:08,186 --> 00:55:10,150
The atoms are not
as closely packed

800
00:55:10,150 --> 00:55:11,700
as in the classical liquids.

801
00:55:11,700 --> 00:55:14,890
The reason for this
is quantum mechanics.

802
00:55:14,890 --> 00:55:17,700
The zero point energy is
relatively more important

803
00:55:17,700 --> 00:55:19,270
here than in any other liquid.

804
00:55:21,817 --> 00:55:23,275
As a consequence,
liquid helium has

805
00:55:23,275 --> 00:55:29,460
a very low mass density, only
about 13% the density of water,

806
00:55:29,460 --> 00:55:31,430
and a very low optical density.

807
00:55:31,430 --> 00:55:35,455
The index of refraction
is quite close to 1.

808
00:55:35,455 --> 00:55:38,510
This makes its surface hard
to see with the naked eye

809
00:55:38,510 --> 00:55:42,160
under ordinary
lighting conditions.

810
00:55:42,160 --> 00:55:44,434
You are no doubt
familiar with the fact

811
00:55:44,434 --> 00:55:49,150
that the helium atom has
closed shell atomic structure.

812
00:55:49,150 --> 00:55:52,500
This explains why helium is
a chemically inert element.

813
00:55:52,500 --> 00:55:55,820
It also accounts for the fact
that the force of attraction

814
00:55:55,820 --> 00:55:58,905
between neighboring helium
atoms, the so-called van der

815
00:55:58,905 --> 00:56:02,760
Waals force, is small.

816
00:56:02,760 --> 00:56:05,580
It takes little energy to
pull two helium atoms apart,

817
00:56:05,580 --> 00:56:10,170
as for example in evaporation.

818
00:56:10,170 --> 00:56:12,550
This gives liquid helium
a better small latent

819
00:56:12,550 --> 00:56:14,660
heat of vaporization.

820
00:56:14,660 --> 00:56:18,242
Only five calories are
needed to evaporate one gram.

821
00:56:18,242 --> 00:56:21,900
Compare this with water,
where evaporation requires

822
00:56:21,900 --> 00:56:24,105
between 500 and 600
calories per gram.

823
00:56:26,780 --> 00:56:30,745
The low van der Waals force
combined with a large zero

824
00:56:30,745 --> 00:56:34,200
point energy also accounts for
the fact that liquid helium

825
00:56:34,200 --> 00:56:38,150
does not freeze, cannot
be solidified at ordinary

826
00:56:38,150 --> 00:56:41,490
pressure, no matter
how far we cool it.

827
00:56:41,490 --> 00:56:44,700
However, liquid helium has been
solidified at high pressure.

828
00:56:47,450 --> 00:56:51,465
The liquid helium in the
dewar is at 4,2 degrees.

829
00:56:51,465 --> 00:56:54,720
We now want to cool it
down to the lambda point

830
00:56:54,720 --> 00:56:58,396
and show you the transition
to the [INAUDIBLE].

831
00:56:58,396 --> 00:57:02,132
Our method will be
cooling by evaporation

832
00:57:02,132 --> 00:57:04,562
using a vacuum pump.

833
00:57:04,562 --> 00:57:09,960
Now, the lambda point lies at
2.2 degrees, only 2 degrees

834
00:57:09,960 --> 00:57:14,500
colder than the [INAUDIBLE]
temperature of the liquid.

835
00:57:14,500 --> 00:57:17,020
What's more, not
very much heat has

836
00:57:17,020 --> 00:57:19,105
been removed from
the liquid helium

837
00:57:19,105 --> 00:57:22,455
now in the dewar to bring
it to the lambda point.

838
00:57:22,455 --> 00:57:27,305
It amounts to only
about 250 calories.

839
00:57:27,305 --> 00:57:31,013
Nevertheless, don't get the
idea that this cooling process

840
00:57:31,013 --> 00:57:32,456
is easy.

841
00:57:32,456 --> 00:57:35,830
On the contrary,
it's quite difficult.

842
00:57:35,830 --> 00:57:39,240
More than 1/3 of the liquid
helium now in the dewar

843
00:57:39,240 --> 00:57:41,790
has to be knocked
away in vapor form

844
00:57:41,790 --> 00:57:46,830
before we can get what remains
behind to the lambda point.

845
00:57:46,830 --> 00:57:49,763
That requires an
awful lot of pumping

846
00:57:49,763 --> 00:57:54,020
and explains why we use this
large and powerful vacuum

847
00:57:54,020 --> 00:57:55,510
pump over here.

848
00:58:00,812 --> 00:58:03,080
Even with this pump,
the cooling process

849
00:58:03,080 --> 00:58:04,960
takes a considerable
amount of time.

850
00:59:09,272 --> 00:59:11,268
Let me explain why
it is so difficult

851
00:59:11,268 --> 00:59:14,262
to cool liquid helium
to the lambda point.

852
00:59:14,262 --> 00:59:17,006
I have already mentioned
that liquid helium has

853
00:59:17,006 --> 00:59:20,250
a remarkably small
[INAUDIBLE] vaporization,

854
00:59:20,250 --> 00:59:23,244
only five calories per gram.

855
00:59:23,244 --> 00:59:26,737
At the same time, liquid
helium at 4.2 degrees

856
00:59:26,737 --> 00:59:31,750
has a high specific heat,
almost calorie per gram.

857
00:59:31,750 --> 00:59:34,355
Therefore, 1 gram of
the vapor pumped away

858
00:59:34,355 --> 00:59:36,830
carries with it
an amount of heat

859
00:59:36,830 --> 00:59:42,230
which can cool only 5 or 6 grams
of liquid helium by 1 degree.

860
00:59:42,230 --> 00:59:44,460
That's not very much cooling.

861
00:59:44,460 --> 00:59:46,930
It is less by a
factor of almost 100

862
00:59:46,930 --> 00:59:51,110
than when we cool
water by evaporation.

863
00:59:51,110 --> 00:59:53,710
The situation gets even
worse as cooling progresses

864
00:59:53,710 --> 00:59:57,588
below 4.2 degrees because the
specific heat of liquid helium

865
00:59:57,588 --> 01:00:00,570
rises astonishingly.

866
01:00:00,570 --> 01:00:05,280
As we approach 2,17
degrees, the lambda point.

867
01:00:05,280 --> 01:00:07,440
The heat of vaporization,
on the other hand,

868
01:00:07,440 --> 01:00:10,190
remains roughly the same.

869
01:00:10,190 --> 01:00:12,460
So a given amount
of vapor carried

870
01:00:12,460 --> 01:00:14,960
off produces less
and less cooling

871
01:00:14,960 --> 01:00:20,240
as we approach 2.17 degrees,

872
01:00:20,240 --> 01:00:23,130
Our thermometer here
is a low pressure gauge

873
01:00:23,130 --> 01:00:26,520
connected to the space
above the liquid helium.

874
01:00:26,520 --> 01:00:29,272
The needle registers
the pressure there.

875
01:00:29,272 --> 01:00:33,240
It is the saturated vapor
pressure of liquid helium.

876
01:00:33,240 --> 01:00:37,430
The gauge is calibrated for
the corresponding temperature.

877
01:00:37,430 --> 01:00:41,412
We call it a vapor
pressure thermometer.

878
01:00:41,412 --> 01:00:43,620
As we approach 2.17
degrees, boiling

879
01:00:43,620 --> 01:00:44,885
becomes increasingly violent.

880
01:00:48,662 --> 01:00:51,142
Suddenly it stops.

881
01:00:51,142 --> 01:00:55,120
This was the transition.

882
01:00:55,120 --> 01:00:57,732
The liquid you now
see is helium II.

883
01:00:57,732 --> 01:01:02,500
Even though evaporation does
continue, there is no boiling.

884
01:01:02,500 --> 01:01:06,309
The normal liquids, such
as the water in this basin,

885
01:01:06,309 --> 01:01:10,660
boil because of their relatively
low heat conductivity.

886
01:01:10,660 --> 01:01:13,199
Before heat,
[INAUDIBLE] at one point

887
01:01:13,199 --> 01:01:15,075
can be carried away
to a cooler place

888
01:01:15,075 --> 01:01:19,300
in the liquid bubbles
of the vapor form.

889
01:01:19,300 --> 01:01:23,390
Helium I behaves like a
normal liquid in this respect.

890
01:01:23,390 --> 01:01:25,470
The absence of
boiling in helium II

891
01:01:25,470 --> 01:01:27,790
reveals that this
phase acts as if it

892
01:01:27,790 --> 01:01:31,546
had a large heat conductivity.

893
01:01:31,546 --> 01:01:34,844
As a matter of fact, as
the liquid helium passed

894
01:01:34,844 --> 01:01:36,302
through the lambda
point transition

895
01:01:36,302 --> 01:01:38,926
you just saw, its
heat conductivity

896
01:01:38,926 --> 01:01:41,290
increased by the fantastic
factor of one million.

897
01:01:43,880 --> 01:01:47,340
The heat conductivity of
helium II is many times

898
01:01:47,340 --> 01:01:50,270
greater than in the
metals silver and copper,

899
01:01:50,270 --> 01:01:52,870
which are among the best
solid heat conductors.

900
01:01:52,870 --> 01:01:55,265
And yet here we
deal with a liquid.

901
01:01:55,265 --> 01:01:57,220
For this alone,
helium II deserves

902
01:01:57,220 --> 01:01:59,500
the name of superfluid.

903
01:01:59,500 --> 01:02:02,610
Actually, the way in
which helium II transports

904
01:02:02,610 --> 01:02:05,304
such large quantities
of heat so rapidly

905
01:02:05,304 --> 01:02:08,750
is totally different from
the classical concept

906
01:02:08,750 --> 01:02:11,040
for heat conduction.

907
01:02:11,040 --> 01:02:12,550
I'll come back to
the subject later

908
01:02:12,550 --> 01:02:15,440
in connection with an
experiment demonstrating

909
01:02:15,440 --> 01:02:20,260
the phenomenon of second
sound in helium II.

910
01:02:20,260 --> 01:02:23,710
Remember that this great
change in heat conductivity

911
01:02:23,710 --> 01:02:26,880
occurred at a single
and fixed transition

912
01:02:26,880 --> 01:02:29,246
temperature, the lambda point.

913
01:02:29,246 --> 01:02:32,630
We do indeed deal with
a change in phase,

914
01:02:32,630 --> 01:02:38,440
only here it is a change from
one liquid to another liquid.

915
01:02:38,440 --> 01:02:42,050
As we told you before, the
specific heat of liquid helium

916
01:02:42,050 --> 01:02:44,590
is very large as a lambda point.

917
01:02:44,590 --> 01:02:48,900
In fact, it behaves abnormally
even below the lambda point

918
01:02:48,900 --> 01:02:53,280
and falls again very rapidly
with the temperature.

919
01:02:53,280 --> 01:02:55,880
This discontinuity
in specific heat

920
01:02:55,880 --> 01:02:57,570
is another reflection
of the fact

921
01:02:57,570 --> 01:02:59,440
that we are dealing
with a change

922
01:02:59,440 --> 01:03:01,720
in the phase of the substance.

923
01:03:01,720 --> 01:03:05,415
By the way, the curve resembles
the Greek letter lambda.

924
01:03:05,415 --> 01:03:07,760
The transition
temperature got its name

925
01:03:07,760 --> 01:03:08,970
from the shape of this curve.

926
01:03:12,881 --> 01:03:14,370
[INAUDIBLE]

927
01:03:14,370 --> 01:03:19,200
The next one has to do with
the viscosity of liquid helium.

928
01:03:19,200 --> 01:03:21,285
When a normal liquid
flows through a tube,

929
01:03:21,285 --> 01:03:23,270
it will resist the flow.

930
01:03:23,270 --> 01:03:26,460
In this experiment, we
shall cause some glycerin

931
01:03:26,460 --> 01:03:29,800
to flow to a tube
under its own weight.

932
01:03:29,800 --> 01:03:32,043
The top layer is
colored glycerin.

933
01:03:36,960 --> 01:03:41,140
The liquid layer closest to
the tube wall adheres to it.

934
01:03:41,140 --> 01:03:45,040
The layer next in from the one
touching the wall flows by it

935
01:03:45,040 --> 01:03:49,530
and is retarded as it flows due
to the interatomic, the van der

936
01:03:49,530 --> 01:03:51,850
Waals force of attraction.

937
01:03:51,850 --> 01:03:54,930
The second layer in
turn drags on the third,

938
01:03:54,930 --> 01:03:59,260
and so on inward from the
wall, producing fluid friction,

939
01:03:59,260 --> 01:03:59,820
or viscosity.

940
01:04:07,770 --> 01:04:09,910
The narrower the
tube, the slower

941
01:04:09,910 --> 01:04:11,972
the liquid rate
of flow through it

942
01:04:11,972 --> 01:04:14,750
under a given head of pressure.

943
01:04:14,750 --> 01:04:17,680
Here I have a beaker with
an unglazed ceramic bottom

944
01:04:17,680 --> 01:04:21,600
of ultra-fine [INAUDIBLE].

945
01:04:21,600 --> 01:04:24,420
Many capillary channels run
through this ceramic disk.

946
01:04:24,420 --> 01:04:28,360
The diameter is quite small,
about one micron which is

947
01:04:28,360 --> 01:04:32,390
1/10,000 of a centimeter.

948
01:04:32,390 --> 01:04:34,410
There is liquid
helium in the beaker.

949
01:04:34,410 --> 01:04:39,611
It is 4.2 degrees Kelvin,
helium I, the normal phase.

950
01:04:39,611 --> 01:04:41,916
The capillaries in the
disk are fine enough

951
01:04:41,916 --> 01:04:44,845
to prevent the liquid now in
the beaker from flowing through

952
01:04:44,845 --> 01:04:46,930
under its on weight.

953
01:04:46,930 --> 01:04:49,750
Clearly, helium I is viscous.

954
01:04:49,750 --> 01:04:52,730
To be sure, its
viscosity is very small.

955
01:04:52,730 --> 01:04:55,210
That's why we had to choose
extremely fine capillaries

956
01:04:55,210 --> 01:04:56,584
to demonstrate it.

957
01:05:00,560 --> 01:05:04,110
Here you see the lambda
point transition.

958
01:05:08,770 --> 01:05:11,700
The helium II all poured out.

959
01:05:11,700 --> 01:05:15,070
The rate of pouring would
not be noticeably slower

960
01:05:15,070 --> 01:05:18,770
if the [INAUDIBLE]
were made yet finer.

961
01:05:18,770 --> 01:05:20,938
We call this kind
of flow a superflow.

962
01:05:24,842 --> 01:05:30,220
The temperature is
now at 1.6 degrees.

963
01:05:30,220 --> 01:05:33,304
The superflow is even faster.

964
01:05:33,304 --> 01:05:36,200
The viscosity of helium
II in this experiment

965
01:05:36,200 --> 01:05:38,532
is so small that it has
not been possible to find

966
01:05:38,532 --> 01:05:40,620
a value for it.

967
01:05:40,620 --> 01:05:43,990
It is less than the experimental
uncertainty incurred

968
01:05:43,990 --> 01:05:46,760
in attempts to measure it.

969
01:05:46,760 --> 01:05:50,064
We now believe that
helium II, the superfluid,

970
01:05:50,064 --> 01:05:54,306
has zero viscosity, although
we should be more precise here.

971
01:05:54,306 --> 01:06:00,675
We believe its viscosity is zero
when observing capillary flow.

972
01:06:00,675 --> 01:06:03,280
Bear this statement
in mind, for we

973
01:06:03,280 --> 01:06:05,720
will come up with a
contradiction to it

974
01:06:05,720 --> 01:06:07,916
in the next experiment,
where we will

975
01:06:07,916 --> 01:06:11,576
look for viscosity by
a different method.

976
01:06:11,576 --> 01:06:14,016
There is a copper cylinder
in the liquid helium,

977
01:06:14,016 --> 01:06:18,896
so mounted as we can turn
it about a vertical axis.

978
01:06:18,896 --> 01:06:22,090
In order to turn it smoothly
and with as little vibration as

979
01:06:22,090 --> 01:06:25,110
possible, we laid the
cylinder into the [INAUDIBLE]

980
01:06:25,110 --> 01:06:27,542
of a simple induction
motor energized

981
01:06:27,542 --> 01:06:29,024
from outside the dewar.

982
01:06:29,024 --> 01:06:31,494
The four horizontal
coils you see

983
01:06:31,494 --> 01:06:34,458
provide the torque which
turns the cylinder.

984
01:06:34,458 --> 01:06:37,230
The liquid helium is
electrically non-conducting.

985
01:06:37,230 --> 01:06:40,410
The coil exerts no
torque on it directly.

986
01:06:40,410 --> 01:06:44,350
Yet as we turn on our
motor, the liquid layer

987
01:06:44,350 --> 01:06:47,048
bounding the cylinder is
dragged along behind it.

988
01:06:47,048 --> 01:06:50,280
The boundary layer in turn
drags on the next layer,

989
01:06:50,280 --> 01:06:51,310
and so on outward.

990
01:06:58,955 --> 01:07:01,640
Finally a circulation
showing up in the helium

991
01:07:01,640 --> 01:07:05,432
due to its own viscosity and
the wooden panels we [INAUDIBLE]

992
01:07:05,432 --> 01:07:06,358
is turned along.

993
01:07:08,987 --> 01:07:10,820
What we have just seen
occurred in helium I,

994
01:07:10,820 --> 01:07:15,290
the normal phase at
4.2 degrees Kelvin.

995
01:07:15,290 --> 01:07:17,952
That is to say,
this demonstration

996
01:07:17,952 --> 01:07:22,470
is consistent with our results
for helium I by capillary flow.

997
01:07:22,470 --> 01:07:26,254
Helium I is viscous.

998
01:07:26,254 --> 01:07:28,470
Here you see the liquid
cooled down and passing

999
01:07:28,470 --> 01:07:30,400
into the superfluid
phase, helium II.

1000
01:07:42,960 --> 01:07:44,020
Let's turn on the motor.

1001
01:07:55,428 --> 01:07:58,404
The paddle wheel starts again.

1002
01:07:58,404 --> 01:08:01,248
What does this mean?

1003
01:08:01,248 --> 01:08:04,980
First of all, let me
emphasize that, like helium I,

1004
01:08:04,980 --> 01:08:08,560
helium II is also non-conducting
in the electrical sense.

1005
01:08:08,560 --> 01:08:10,710
In other words, the
circulation in the experiment

1006
01:08:10,710 --> 01:08:15,910
can only have been caused
through viscous drag.

1007
01:08:15,910 --> 01:08:19,250
So we conclude from the
rotating cylinder observations

1008
01:08:19,250 --> 01:08:22,499
that helium II is viscous
and from the method

1009
01:08:22,499 --> 01:08:26,370
of capillary flow that
it has zero viscosity.

1010
01:08:26,370 --> 01:08:29,649
Our experimentation has
come up with a paradox.

1011
01:08:29,649 --> 01:08:32,415
No normal classical
liquid is known

1012
01:08:32,415 --> 01:08:35,860
to behave so inconsistently, in
capillary flow on the one hand

1013
01:08:35,860 --> 01:08:39,158
and in bulk flow on the other.

1014
01:08:39,158 --> 01:08:42,180
This state of affairs
forces us to think

1015
01:08:42,180 --> 01:08:46,560
of helium II, the
superfluid, not as a single,

1016
01:08:46,560 --> 01:08:48,996
but as a dual liquid.

1017
01:08:48,996 --> 01:08:52,390
It appeared as if helium
II had two separate and yet

1018
01:08:52,390 --> 01:08:54,817
interpenetrating
component liquids.

1019
01:08:54,817 --> 01:08:56,813
We shall call one
component normal.

1020
01:08:56,813 --> 01:09:00,306
It is this component
which we call

1021
01:09:00,306 --> 01:09:02,240
responsible for the
appearance of viscosity

1022
01:09:02,240 --> 01:09:05,565
below the lambda point in the
rotating cylinder experiment.

1023
01:09:05,565 --> 01:09:09,016
The normal component,
as the name suggests,

1024
01:09:09,016 --> 01:09:14,439
behaves like a normal liquid,
and therefore as viscosity.

1025
01:09:14,439 --> 01:09:17,910
It is the one which the cylinder
drags along as its turned.

1026
01:09:17,910 --> 01:09:20,910
But the normal components
cannot flow through the narrow

1027
01:09:20,910 --> 01:09:25,410
channels of the ceramic disc
because of its viscosity.

1028
01:09:25,410 --> 01:09:28,410
The second component
has zero viscosity,

1029
01:09:28,410 --> 01:09:30,910
and it's called the
superfluid component.

1030
01:09:30,910 --> 01:09:33,910
We think that it does
not participate at all

1031
01:09:33,910 --> 01:09:36,810
in the rotating cylinder
experiment below the lambda

1032
01:09:36,810 --> 01:09:37,639
point.

1033
01:09:37,639 --> 01:09:40,034
It stays at rest.

1034
01:09:40,034 --> 01:09:43,546
On the other hand, it can flow
through channels of one micron

1035
01:09:43,546 --> 01:09:45,864
diameter with the greatest
of ease and countering

1036
01:09:45,864 --> 01:09:50,834
no resistance whatever
because it has no viscosity.

1037
01:09:50,834 --> 01:09:53,819
As we'll see later, this
flow is not repeated even

1038
01:09:53,819 --> 01:09:56,070
when the capillary
diameters are made

1039
01:09:56,070 --> 01:09:58,720
far smaller than one micron.

1040
01:09:58,720 --> 01:10:01,822
This [INAUDIBLE] construction
is called the two fluid

1041
01:10:01,822 --> 01:10:04,237
model for liquid helium II.

1042
01:10:04,237 --> 01:10:05,927
Whether it is correct
or not depends

1043
01:10:05,927 --> 01:10:08,190
on further tests
comparing the theory

1044
01:10:08,190 --> 01:10:13,750
based on this model with
experimental results.

1045
01:10:13,750 --> 01:10:17,120
We now go on to another
phenomenon, the fountain

1046
01:10:17,120 --> 01:10:17,620
effect.

1047
01:10:17,620 --> 01:10:20,470
What you see here
is a tube which

1048
01:10:20,470 --> 01:10:23,125
narrows down and then
opens into a bulb.

1049
01:10:23,125 --> 01:10:26,240
A small piece of cotton is
stuffed into the [INAUDIBLE]

1050
01:10:26,240 --> 01:10:27,930
section between the
tube and the bulb.

1051
01:10:27,930 --> 01:10:31,311
And the bulb has been tightly
packed with one of the finest

1052
01:10:31,311 --> 01:10:35,660
powders available, [INAUDIBLE].

1053
01:10:35,660 --> 01:10:39,388
And second wad of cotton
keeps the powder in the bulb.

1054
01:10:39,388 --> 01:10:44,278
This powder presents extremely
fine capillary channels.

1055
01:10:44,278 --> 01:10:47,020
Their average diameter is a
small fraction of 1 micron.

1056
01:10:51,000 --> 01:10:53,590
This device has been
placed in the dewar.

1057
01:10:53,590 --> 01:10:57,590
The liquid helium is
below the lambda point.

1058
01:10:57,590 --> 01:11:01,110
We submerge the bulb, and then
we'll send a beam of light

1059
01:11:01,110 --> 01:11:05,610
from this lamp to a
point near the top.

1060
01:11:05,610 --> 01:11:09,375
You will see the light beam
when the lamp is turned on.

1061
01:11:09,375 --> 01:11:13,080
It focuses some heat in the
form of infrared radiation

1062
01:11:13,080 --> 01:11:15,000
on the point in question.

1063
01:11:15,000 --> 01:11:17,880
The temperature will rise
above the temperature

1064
01:11:17,880 --> 01:11:20,760
of the rest of the apparatus.

1065
01:11:20,760 --> 01:11:21,720
Let us turn it on.

1066
01:11:29,600 --> 01:11:33,102
Liquid helium flows through the
hole in the bottom of the bulb,

1067
01:11:33,102 --> 01:11:35,940
through the fine
powder, and rises

1068
01:11:35,940 --> 01:11:38,310
above the level of
liquid helium outside.

1069
01:11:38,310 --> 01:11:40,225
The height to which
it will go depends

1070
01:11:40,225 --> 01:11:42,205
on the temperature
increase produced

1071
01:11:42,205 --> 01:11:45,700
bu the lamp focused on the bulb.

1072
01:11:45,700 --> 01:11:48,422
We can very well ask, where
does the mechanical energy

1073
01:11:48,422 --> 01:11:51,866
come from that does the
work necessary to pump

1074
01:11:51,866 --> 01:11:55,310
the liquid above
the ambient level?

1075
01:11:55,310 --> 01:11:57,662
Before we attempt to
discuss this question,

1076
01:11:57,662 --> 01:12:00,690
there are two other facts
that should be noted.

1077
01:12:00,690 --> 01:12:03,780
The first is by now obvious.

1078
01:12:03,780 --> 01:12:07,780
The upward flow through the bulb
must clearly be a superfluid.

1079
01:12:07,780 --> 01:12:13,280
Only the superfluid component
of helium II could get through.

1080
01:12:13,280 --> 01:12:15,780
The second fact is
more significant.

1081
01:12:15,780 --> 01:12:19,700
Let me explain it this way, the
superfluid flows spontaneously

1082
01:12:19,700 --> 01:12:23,440
from a to b,from a
cooler to a warmer place.

1083
01:12:23,440 --> 01:12:25,970
Point a is in the
cold liquid, but b

1084
01:12:25,970 --> 01:12:28,139
is being heated
with infrared rays.

1085
01:12:28,139 --> 01:12:32,576
The second law of thermodynamics
positively says that heat

1086
01:12:32,576 --> 01:12:36,273
cannot of itself flow from
a point of lower to a point

1087
01:12:36,273 --> 01:12:38,492
of higher temperature.

1088
01:12:38,492 --> 01:12:40,628
What does this mean
to us here, knowing

1089
01:12:40,628 --> 01:12:42,436
as we do that the
superfluid is flowing

1090
01:12:42,436 --> 01:12:45,790
from a colder to a warmer spot?

1091
01:12:45,790 --> 01:12:51,272
Simply this, it carries no
heat, no thermal energy.

1092
01:12:51,272 --> 01:12:55,060
Any internal energy
[INAUDIBLE] is no longer

1093
01:12:55,060 --> 01:12:56,964
thermally available.

1094
01:12:56,964 --> 01:13:01,724
To say it precisely,
it has zero entropy.

1095
01:13:01,724 --> 01:13:03,730
We have discovered another
remarkable property

1096
01:13:03,730 --> 01:13:05,248
of helium II.

1097
01:13:05,248 --> 01:13:09,072
Its superfluid component
not only is friction free,

1098
01:13:09,072 --> 01:13:11,940
it also contains no heat.

1099
01:13:11,940 --> 01:13:13,852
The heat energy
contained in helium II

1100
01:13:13,852 --> 01:13:19,595
as a whole resides, all of
it, in the normal component.

1101
01:13:19,595 --> 01:13:23,060
We may, of course, add heat
to the superfluid component,

1102
01:13:23,060 --> 01:13:27,030
as we are doing when it passes
the spot heated by the lamp.

1103
01:13:27,030 --> 01:13:28,910
But in doing so, we
are converting it

1104
01:13:28,910 --> 01:13:32,908
into the normal component.

1105
01:13:32,908 --> 01:13:36,360
Let me return briefly to
a question posed earlier.

1106
01:13:36,360 --> 01:13:38,808
Mechanical work is done
in pumping the liquid

1107
01:13:38,808 --> 01:13:40,804
above equilibrium level.

1108
01:13:40,804 --> 01:13:42,800
Where does it come from?

1109
01:13:42,800 --> 01:13:45,295
I cannot answer this
question here in full,

1110
01:13:45,295 --> 01:13:49,287
but it suffice to tell you that
we are dealing here with a heat

1111
01:13:49,287 --> 01:13:49,800
engine.

1112
01:13:49,800 --> 01:13:51,508
The mechanical energy
comes from the heat

1113
01:13:51,508 --> 01:13:54,304
added at the light spot.

1114
01:13:54,304 --> 01:13:57,400
An amusing demonstration
of the same phenomenon

1115
01:13:57,400 --> 01:14:00,522
again uses a bulb
packed with rouge,

1116
01:14:00,522 --> 01:14:03,994
but this one opens
into a capillary.

1117
01:14:03,994 --> 01:14:07,962
Light is beamed on a spot
just below the capillary,

1118
01:14:07,962 --> 01:14:09,610
and it produces a
helium fountain.

1119
01:14:14,273 --> 01:14:17,195
The phenomenon in this and
the previous experiment

1120
01:14:17,195 --> 01:14:19,143
has become known as
the thermomechanical,

1121
01:14:19,143 --> 01:14:23,526
or the fountain, effect.

1122
01:14:23,526 --> 01:14:27,160
Below the lambda point,
the superfluid component

1123
01:14:27,160 --> 01:14:29,980
of liquid helium creeps
up along the walls

1124
01:14:29,980 --> 01:14:32,758
of its container in an
extremely thin film.

1125
01:14:32,758 --> 01:14:34,740
It is known as a Rollin film.

1126
01:14:37,674 --> 01:14:41,586
This creeping film is
a variety of superflow.

1127
01:14:41,586 --> 01:14:45,752
It is difficult to make the film
itself directly visible to you.

1128
01:14:45,752 --> 01:14:49,215
To show it indirectly,
we've put some liquid helium

1129
01:14:49,215 --> 01:14:51,370
into a glass vessel.

1130
01:14:51,370 --> 01:14:54,360
It is below the lambda point.

1131
01:14:54,360 --> 01:14:57,140
There is no part porous
bottom in this vessel.

1132
01:14:57,140 --> 01:14:59,920
The film rises along
the inside wall

1133
01:14:59,920 --> 01:15:01,310
and comes down
along the outside,

1134
01:15:01,310 --> 01:15:03,880
collecting in drops
at the bottom.

1135
01:15:03,880 --> 01:15:06,265
The thickness of
this creeping film

1136
01:15:06,265 --> 01:15:09,604
is only a small fraction of
1 micron and of the order

1137
01:15:09,604 --> 01:15:12,480
of 200 to 300 angstrom.

1138
01:15:12,480 --> 01:15:16,512
Its speed, while small just
below the lambda point,

1139
01:15:16,512 --> 01:15:18,902
may reach a value as
high as 35 centimeters

1140
01:15:18,902 --> 01:15:22,815
per second at
lower temperatures.

1141
01:15:22,815 --> 01:15:27,120
Our next experiment deals with
the phenomenon of second sound.

1142
01:15:27,120 --> 01:15:29,915
We are all familiar with wave
motion in elastic materials,

1143
01:15:29,915 --> 01:15:34,120
be they solids,
liquids, or gases.

1144
01:15:34,120 --> 01:15:37,630
Elastic energy of deformation,
carried away from its source

1145
01:15:37,630 --> 01:15:40,690
in the form of waves with
a characteristic speed,

1146
01:15:40,690 --> 01:15:43,842
the speed of sound.

1147
01:15:43,842 --> 01:15:45,600
Liquid helium is an
elastic substance

1148
01:15:45,600 --> 01:15:48,320
both above and below
the lambda point.

1149
01:15:48,320 --> 01:15:51,300
Both helium one and two
support sound waves.

1150
01:15:51,300 --> 01:15:54,640
Now helium II, the
superfluid phase,

1151
01:15:54,640 --> 01:15:57,650
also conducts heat
in the form of waves.

1152
01:15:57,650 --> 01:16:01,580
This remarkable property is
shared by no other substance.

1153
01:16:01,580 --> 01:16:06,593
For better or for worse, it
has been called second sound.

1154
01:16:06,593 --> 01:16:09,540
Normal heat conduction
is a diffusion process.

1155
01:16:09,540 --> 01:16:12,400
The rate of flow of heat is
proportional to the temperature

1156
01:16:12,400 --> 01:16:13,310
differences.

1157
01:16:13,310 --> 01:16:18,104
But in helium II it
is a wave process.

1158
01:16:18,104 --> 01:16:23,170
Heat flows through helium II
with a characteristic speed,

1159
01:16:23,170 --> 01:16:26,210
the speed of second sound.

1160
01:16:26,210 --> 01:16:30,360
We shall send small heat pulses
into helium II from a heater.

1161
01:16:30,360 --> 01:16:32,682
They will spread away
from the heater uniformly,

1162
01:16:32,682 --> 01:16:34,140
carrying the heat
energy with them.

1163
01:16:39,712 --> 01:16:41,337
The speed of second
sound is small just

1164
01:16:41,337 --> 01:16:43,960
below the lambda point.

1165
01:16:43,960 --> 01:16:46,420
In the neighborhood
of 1.6 degrees Kelvin,

1166
01:16:46,420 --> 01:16:49,210
it reaches a value of
roughly 20 meters per second,

1167
01:16:49,210 --> 01:16:54,010
and it is in this range that
we will run our demonstration.

1168
01:16:54,010 --> 01:16:57,142
The experimental
procedure is as follows.

1169
01:16:57,142 --> 01:17:01,081
There are two disks
in the liquid helium.

1170
01:17:01,081 --> 01:17:04,490
They are carbon resistors
with the carbon applied

1171
01:17:04,490 --> 01:17:06,925
in thin layers on one
side of each disk.

1172
01:17:06,925 --> 01:17:09,360
In this way, good
thermal contact

1173
01:17:09,360 --> 01:17:13,256
is established between the
resistor and the liquid helium.

1174
01:17:13,256 --> 01:17:15,920
The following resistor
will be used as a heater.

1175
01:17:15,920 --> 01:17:18,580
Electric currents will
be sent through it

1176
01:17:18,580 --> 01:17:22,435
in pulses from this pulse
generator by means of the cable

1177
01:17:22,435 --> 01:17:24,295
you see here.

1178
01:17:24,295 --> 01:17:26,620
The [INAUDIBLE] of
the generator is also

1179
01:17:26,620 --> 01:17:31,490
connected via a second cable
to a dual-trace oscilloscope,

1180
01:17:31,490 --> 01:17:34,760
where it will be recorded
on the bottom [? trip. ?]

1181
01:17:34,760 --> 01:17:37,951
In other words, it will record
the heat pulse as it enters

1182
01:17:37,951 --> 01:17:41,220
the liquid helium.

1183
01:17:41,220 --> 01:17:43,790
The pulses have been turned on.

1184
01:17:43,790 --> 01:17:45,770
They themselves trigger
the horizontal sweep

1185
01:17:45,770 --> 01:17:50,214
of [INAUDIBLE], which
records [INAUDIBLE].

1186
01:17:50,214 --> 01:17:53,060
It is calibrated
at 1 millisecond

1187
01:17:53,060 --> 01:17:54,785
per unit on the scale.

1188
01:17:54,785 --> 01:17:58,330
The pulses are 1
millisecond long.

1189
01:17:58,330 --> 01:18:00,185
The pulses leave the
heater at the bottom

1190
01:18:00,185 --> 01:18:02,420
in the form of
second sound and move

1191
01:18:02,420 --> 01:18:05,790
up to where they strike the
carbon resistor at the top.

1192
01:18:05,790 --> 01:18:10,375
Being heat pulses, they
greatly raise is temperature.

1193
01:18:10,375 --> 01:18:13,550
The carbon resistor is
quite sensitive to changes

1194
01:18:13,550 --> 01:18:14,175
in temperature.

1195
01:18:14,175 --> 01:18:17,590
It acts as a thermometer.

1196
01:18:17,590 --> 01:18:19,480
So the heat pulse
of second sound

1197
01:18:19,480 --> 01:18:21,582
creates a pulse-like
change in the resistance

1198
01:18:21,582 --> 01:18:24,300
of the [INAUDIBLE] up here.

1199
01:18:24,300 --> 01:18:26,410
It isn't hard to
convert this resistance

1200
01:18:26,410 --> 01:18:28,200
pulse into a [INAUDIBLE].

1201
01:18:28,200 --> 01:18:31,260
What we will do is to
maintain a small DC

1202
01:18:31,260 --> 01:18:33,211
current in the top resistor.

1203
01:18:33,211 --> 01:18:36,650
It is supplied from a
battery in this metal box.

1204
01:18:36,650 --> 01:18:38,757
The box shields the
circuits in order

1205
01:18:38,757 --> 01:18:41,619
to reduce electronic noise.

1206
01:18:41,619 --> 01:18:44,004
The voltage pulse is small.

1207
01:18:44,004 --> 01:18:50,080
In this second box
we have an amplifier.

1208
01:18:50,080 --> 01:18:53,527
The amplified output is fed
into the oscilloscope, where

1209
01:18:53,527 --> 01:18:55,660
it will appear on
the upper trace.

1210
01:18:55,660 --> 01:18:57,556
The horizontal
timescale on this trace

1211
01:18:57,556 --> 01:19:00,860
is exactly the same as
for the bottom trace.

1212
01:19:00,860 --> 01:19:03,710
However, the upper trace
records both exchanges

1213
01:19:03,710 --> 01:19:06,520
as they occur in
the top resistor,

1214
01:19:06,520 --> 01:19:09,852
a detector of second sound.

1215
01:19:09,852 --> 01:19:36,172
The temperature of the liquid
is about 1.65 degrees Kelvin.

1216
01:19:36,172 --> 01:19:42,110
The battery has been turned
on, and now the amplifier.

1217
01:19:42,110 --> 01:19:45,515
Among noise and other
distortions in the upper trace,

1218
01:19:45,515 --> 01:19:49,760
a clear-cut voltage pulse
appears about 4 and 1/2 units

1219
01:19:49,760 --> 01:19:53,100
to the right, 4 and
1/2 milliseconds later

1220
01:19:53,100 --> 01:19:57,687
than the pulse
entering the heater.

1221
01:19:57,687 --> 01:20:02,724
This pulse in the upper trace is
also about 1 millisecond long.

1222
01:20:02,724 --> 01:20:07,188
It is the second sound as it
arrives at the upper resistor.

1223
01:20:07,188 --> 01:20:10,164
The upper trace also shows
a strong voltage pulse

1224
01:20:10,164 --> 01:20:13,636
at the left, simultaneous
to the heater pulse.

1225
01:20:13,636 --> 01:20:16,612
That's due to pick-up
by electromagnetic waves

1226
01:20:16,612 --> 01:20:20,084
with the heater acting as
transmitter and the detector

1227
01:20:20,084 --> 01:20:21,076
as receiver.

1228
01:20:28,020 --> 01:20:30,996
We're moving the detector
toward the heater.

1229
01:20:30,996 --> 01:20:33,476
The pulse moves
with it to the left.

1230
01:20:33,476 --> 01:20:37,865
Notice the echos of second sound
which appear on the upper trace

1231
01:20:37,865 --> 01:20:41,360
while the detector
is near the heater.

1232
01:20:41,360 --> 01:20:44,270
They're caused by multiple
reflections between the two

1233
01:20:44,270 --> 01:20:45,725
resistors.

1234
01:20:45,725 --> 01:20:50,090
A total of three echos
is clearly visible.

1235
01:20:50,090 --> 01:20:51,060
[END VIDEO PLAYBACK]

1236
01:20:51,060 --> 01:20:55,400
PROFESSOR: OK, you can watch
the rest of it at home.