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MICHAEL SHORT: Want
to give a quick review

9
00:00:23,150 --> 00:00:26,030
so that we can launch into some
more technical stuff today.

10
00:00:26,030 --> 00:00:29,840
We started talking about this
reaction for boron neutron

11
00:00:29,840 --> 00:00:33,430
capture therapy was the
focus of today's lecture.

12
00:00:33,430 --> 00:00:35,180
And I want to get this
one up on the board

13
00:00:35,180 --> 00:00:39,170
since I'm going to move
to different slides later.

14
00:00:39,170 --> 00:00:42,650
We had boron 10
captures a neutron

15
00:00:42,650 --> 00:00:48,980
and becomes lithium 7, helium 4.

16
00:00:48,980 --> 00:00:50,360
There's a gamma ray.

17
00:00:50,360 --> 00:00:54,560
And there's going to be
some q value, or energy

18
00:00:54,560 --> 00:00:57,470
either released or consumed,
in this case, released

19
00:00:57,470 --> 00:01:01,010
in the form of the kinetic
energy of the recoil products

20
00:01:01,010 --> 00:01:03,410
as well as the gamma ray energy.

21
00:01:03,410 --> 00:01:07,710
So let's also give a quick
review of the Q equation

22
00:01:07,710 --> 00:01:10,010
since I think we
covered that last week.

23
00:01:10,010 --> 00:01:12,770
If you remember, if you have a
general system of, let's say,

24
00:01:12,770 --> 00:01:19,100
a small initial nucleus i firing
into a larger nucleus capital

25
00:01:19,100 --> 00:01:24,650
I, after the reaction, off
comes some small final nucleus

26
00:01:24,650 --> 00:01:27,480
and some large final nucleus.

27
00:01:27,480 --> 00:01:29,210
If I were to draw
these arrows to scale,

28
00:01:29,210 --> 00:01:33,220
the little one would
probably be moving faster.

29
00:01:33,220 --> 00:01:37,370
Then we can figure out how
much mass and kinetic energy

30
00:01:37,370 --> 00:01:40,310
each of these nuclei are by
just conserving everything.

31
00:01:40,310 --> 00:01:43,400
So we can write, let's
say, the mass of nucleus i,

32
00:01:43,400 --> 00:01:46,820
c squared plus the
kinetic energy of i

33
00:01:46,820 --> 00:01:50,000
plus the mass of
big I c squared.

34
00:01:50,000 --> 00:01:55,850
Plus the kinetic energy of big I
has to equal the mass of little

35
00:01:55,850 --> 00:02:00,920
f c squared plus the
kinetic energy of little f

36
00:02:00,920 --> 00:02:07,160
plus the mass of big F c
squared plus its kinetic energy.

37
00:02:07,160 --> 00:02:10,250
And what this tells us is
if the total amount of mass,

38
00:02:10,250 --> 00:02:12,500
or the total kinetic
energy change,

39
00:02:12,500 --> 00:02:15,900
they've got to exchange
energy kind of equally.

40
00:02:15,900 --> 00:02:20,780
So we can write the difference
in mass or energy by just,

41
00:02:20,780 --> 00:02:22,760
let's say, taking
all of the mass terms

42
00:02:22,760 --> 00:02:24,620
and putting them on one side.

43
00:02:24,620 --> 00:02:26,670
So we can say that--

44
00:02:26,670 --> 00:02:29,360
let's just say everything here
is multiplied by c squared.

45
00:02:29,360 --> 00:02:37,730
We have M i plus M I minus
M little f minus M big F

46
00:02:37,730 --> 00:02:44,210
has got to equal the sum of
the final kinetic energies

47
00:02:44,210 --> 00:02:53,390
minus the initial ones, which
we also call this Q value.

48
00:02:53,390 --> 00:02:56,300
And so by getting the
difference in the masses

49
00:02:56,300 --> 00:02:58,400
or the kinetic
energies at the end,

50
00:02:58,400 --> 00:03:00,080
you can figure out
whether this reaction

51
00:03:00,080 --> 00:03:02,360
is exothermic or endothermic.

52
00:03:02,360 --> 00:03:06,010
If you remember, we said
if Q is greater than 0,

53
00:03:06,010 --> 00:03:07,370
it's exothermic.

54
00:03:07,370 --> 00:03:11,660
If Q is less than
zero, it's endothermic.

55
00:03:11,660 --> 00:03:14,390
In a nuclear reaction
like a chemical reaction,

56
00:03:14,390 --> 00:03:17,000
you've got to input extra
energy into the system

57
00:03:17,000 --> 00:03:19,700
beyond just the rest
masses of the particles

58
00:03:19,700 --> 00:03:22,880
to make an endothermic
reaction happen.

59
00:03:22,880 --> 00:03:26,340
For example, since we found
out that this BNCT reaction is

60
00:03:26,340 --> 00:03:29,510
endothermic, you can
make it go the other way,

61
00:03:29,510 --> 00:03:32,570
but you have to impart
kinetic energy to one

62
00:03:32,570 --> 00:03:38,000
or both of those nuclei to
overcome the Q value that--

63
00:03:38,000 --> 00:03:42,170
let's see-- yeah, to overcome
the Q value that you'd get.

64
00:03:42,170 --> 00:03:44,720
And I want to put up a
couple of other terms just

65
00:03:44,720 --> 00:03:46,470
to quickly review.

66
00:03:46,470 --> 00:03:50,120
We started looking at
the table of nuclides.

67
00:03:50,120 --> 00:03:52,430
We learned how to read it
specifically to find things

68
00:03:52,430 --> 00:03:54,950
like excess mass
and binding energy,

69
00:03:54,950 --> 00:03:58,080
the definitions for which I want
to leave up here on the board.

70
00:03:58,080 --> 00:04:03,230
So we had the excess mass,
which, again, doesn't

71
00:04:03,230 --> 00:04:04,910
have a physical significance.

72
00:04:04,910 --> 00:04:08,210
It's the difference between
the actual mass of a nucleotide

73
00:04:08,210 --> 00:04:10,790
and the integer
approximation of its mass

74
00:04:10,790 --> 00:04:13,640
just from the number of
protons and neutrons,

75
00:04:13,640 --> 00:04:15,440
and the binding energy.

76
00:04:15,440 --> 00:04:18,230
Since some of you asked
for a nucleus uniquely

77
00:04:18,230 --> 00:04:23,060
defined by its total mass number
a and number of protons z.

78
00:04:23,060 --> 00:04:25,340
So let's say this is
a functional quantity

79
00:04:25,340 --> 00:04:27,590
because you can have any
nucleus with a certain number

80
00:04:27,590 --> 00:04:31,220
of protons and a certain
number of protons plus neutrons

81
00:04:31,220 --> 00:04:33,740
would be the sum--

82
00:04:33,740 --> 00:04:43,970
let's see-- the sum of
its individual nucleons

83
00:04:43,970 --> 00:04:48,590
minus its actual
mass, which also

84
00:04:48,590 --> 00:04:51,650
is a function of z
and a, or its proton

85
00:04:51,650 --> 00:04:54,210
number and its total
atomic mass number.

86
00:04:54,210 --> 00:04:58,710
So let's leave these
things up on the board

87
00:04:58,710 --> 00:05:00,400
while we move into
some new stuff.

88
00:05:05,750 --> 00:05:08,750
So back to where
we were on Tuesday.

89
00:05:08,750 --> 00:05:10,640
We wanted to figure
out, well, how do we

90
00:05:10,640 --> 00:05:12,140
calculate this Q value?

91
00:05:12,140 --> 00:05:14,430
You can do it any of three ways.

92
00:05:14,430 --> 00:05:16,880
You can use the masses,
like we have over here.

93
00:05:16,880 --> 00:05:21,080
So the masses in amu, or atomic
mass units, times c squared

94
00:05:21,080 --> 00:05:23,360
gives you the Q value in MeV.

95
00:05:23,360 --> 00:05:25,490
Or the difference
in kinetic energies

96
00:05:25,490 --> 00:05:27,890
gives you the Q value in MeV,
although you don't usually

97
00:05:27,890 --> 00:05:30,350
just know these off
the bat, especially

98
00:05:30,350 --> 00:05:31,970
for the final products.

99
00:05:31,970 --> 00:05:34,040
Or you can use the
binding energies.

100
00:05:34,040 --> 00:05:36,710
Because the binding
energy is directly related

101
00:05:36,710 --> 00:05:40,580
to the mass additively, you can
substitute in binding energies

102
00:05:40,580 --> 00:05:43,040
here and you'll
get the same thing.

103
00:05:43,040 --> 00:05:45,620
And so if we have the
table of nuclides,

104
00:05:45,620 --> 00:05:48,860
you can either use
the atomic mass

105
00:05:48,860 --> 00:05:50,570
of any nucleus or
its binding energy

106
00:05:50,570 --> 00:05:52,130
and just look it up directly.

107
00:05:52,130 --> 00:05:53,780
The thing with the
fewest steps is just

108
00:05:53,780 --> 00:05:55,988
to use the binding energies
because those are already

109
00:05:55,988 --> 00:05:57,590
given in MeV or keV.

110
00:05:57,590 --> 00:06:00,010
And then it's just an
addition subtraction problem.

111
00:06:00,010 --> 00:06:02,000
If you use the
masses, don't forget

112
00:06:02,000 --> 00:06:03,540
to multiply by c squared.

113
00:06:03,540 --> 00:06:07,160
And remember our
conversion formula,

114
00:06:07,160 --> 00:06:17,940
which should not be rounded,
which is this right here.

115
00:06:17,940 --> 00:06:22,330
Again, 931.49, not 931,
not even 931.5, or else

116
00:06:22,330 --> 00:06:24,545
you're off by 10
kiloelectron volts.

117
00:06:24,545 --> 00:06:26,170
And so then the
question is, all right,

118
00:06:26,170 --> 00:06:29,320
let's try to calculate the
Q value from this reaction.

119
00:06:29,320 --> 00:06:32,860
I have the individual energies
and kinetic energies up here.

120
00:06:32,860 --> 00:06:34,990
But just so we have
a worked out example,

121
00:06:34,990 --> 00:06:36,950
let's actually do this.

122
00:06:36,950 --> 00:06:40,210
So the Q value of
this reaction should

123
00:06:40,210 --> 00:06:48,550
be the binding energy of lithium
7 plus the binding energy of--

124
00:06:48,550 --> 00:06:55,060
what do we have-- helium 4 minus
the binding energy of boron 10

125
00:06:55,060 --> 00:06:59,620
minus the binding
energy of a neutron.

126
00:06:59,620 --> 00:07:00,920
Now first of all, the easy one.

127
00:07:00,920 --> 00:07:03,670
What's the binding
energy of a low neutron?

128
00:07:03,670 --> 00:07:04,860
Let's call it out.

129
00:07:04,860 --> 00:07:07,260
0, yeah.

130
00:07:07,260 --> 00:07:10,630
A lone nucleon is not bound
to anything, so that's easy.

131
00:07:10,630 --> 00:07:11,472
That's 0.

132
00:07:11,472 --> 00:07:13,180
And we can just use
the table of nuclides

133
00:07:13,180 --> 00:07:14,590
to look up the other three.

134
00:07:14,590 --> 00:07:18,650
Luckily I've got
it live over here.

135
00:07:18,650 --> 00:07:20,980
So let's just punch these in.

136
00:07:20,980 --> 00:07:24,450
We'll have boron 10.

137
00:07:24,450 --> 00:07:36,880
And our binding energy
right here is 64.75 MeV.

138
00:07:36,880 --> 00:07:38,385
Let's look up helium 4.

139
00:07:42,230 --> 00:07:42,730
Good.

140
00:07:42,730 --> 00:07:45,230
That's showing up on the screen.

141
00:07:45,230 --> 00:07:47,540
I think I'll make it bigger
so it's easier for everyone

142
00:07:47,540 --> 00:07:49,010
to see.

143
00:07:49,010 --> 00:07:57,680
The binding energy
is 28.296 MeV.

144
00:07:57,680 --> 00:07:59,310
And notice everything's
already in MeV,

145
00:07:59,310 --> 00:08:02,250
so this is nice and
easy to deal with.

146
00:08:02,250 --> 00:08:04,740
And lithium 7.

147
00:08:07,440 --> 00:08:22,340
39.245 plus minus minus 0 MeV.

148
00:08:22,340 --> 00:08:23,060
Let's see.

149
00:08:23,060 --> 00:08:29,790
Actually, I think my sines for
exo and endo are backwards.

150
00:08:29,790 --> 00:08:33,960
Let me fix that right
now because the idea is

151
00:08:33,960 --> 00:08:37,140
if you release energy--

152
00:08:37,140 --> 00:08:38,442
there we go.

153
00:08:38,442 --> 00:08:39,150
Sorry about that.

154
00:08:44,110 --> 00:08:45,210
No, wait a minute.

155
00:08:45,210 --> 00:08:48,240
Let's calculate this
out, figure what we get.

156
00:08:48,240 --> 00:08:50,280
And again, I can't do
six digits in my head,

157
00:08:50,280 --> 00:08:53,430
but I will do it as
fast as I can here.

158
00:08:53,430 --> 00:08:59,432
245 plus 28.296 minus 0.75.

159
00:08:59,432 --> 00:09:01,550
Ah, indeed.

160
00:09:01,550 --> 00:09:04,340
2.79 MeV.

161
00:09:04,340 --> 00:09:05,860
I had it right the first time.

162
00:09:05,860 --> 00:09:08,300
So good.

163
00:09:08,300 --> 00:09:09,550
Shouldn't second guess myself.

164
00:09:15,990 --> 00:09:16,490
Cool.

165
00:09:20,810 --> 00:09:25,120
So now we know the total
Q value of this reaction.

166
00:09:25,120 --> 00:09:27,325
And I'll bring the
reaction back up here.

167
00:09:27,325 --> 00:09:29,450
We also know, which you
can find from measurements,

168
00:09:29,450 --> 00:09:38,630
that the gamma ray comes off
with an energy of 0.48 MeV,

169
00:09:38,630 --> 00:09:40,310
leaving--

170
00:09:40,310 --> 00:09:42,290
let's see how much--

171
00:09:42,290 --> 00:09:48,770
leaving 2.13 MeV for the
sum of the kinetic energies

172
00:09:48,770 --> 00:09:51,410
of the lithium and
the helium nucleus.

173
00:09:51,410 --> 00:09:58,460
So let's say Tli 7
plus t for helium.

174
00:09:58,460 --> 00:09:59,960
Now the question
is how did I get

175
00:09:59,960 --> 00:10:03,890
to those numbers for the split
between the kinetic energies

176
00:10:03,890 --> 00:10:06,045
of those two?

177
00:10:06,045 --> 00:10:06,920
Anyone have any idea?

178
00:10:10,420 --> 00:10:11,582
Yeah.

179
00:10:11,582 --> 00:10:12,947
AUDIENCE: Related to their mass?

180
00:10:12,947 --> 00:10:13,780
MICHAEL SHORT: Yeah.

181
00:10:13,780 --> 00:10:16,690
So it's definitely related
to their relative masses.

182
00:10:16,690 --> 00:10:18,910
More specifically, we
have this conservation

183
00:10:18,910 --> 00:10:21,340
of energy equation,
but we've still

184
00:10:21,340 --> 00:10:23,320
got two variables
and one equation.

185
00:10:23,320 --> 00:10:24,670
We need a second equation.

186
00:10:24,670 --> 00:10:26,170
That certainly does
relate the mass.

187
00:10:26,170 --> 00:10:26,670
Yep.

188
00:10:26,670 --> 00:10:28,003
AUDIENCE: Conservation momentum.

189
00:10:28,003 --> 00:10:30,080
MICHAEL SHORT: You can
relate their momentum.

190
00:10:30,080 --> 00:10:34,870
So we can say that if this
initial kinetic energy of boron

191
00:10:34,870 --> 00:10:37,870
and of the neutron was
approximately 0, then

192
00:10:37,870 --> 00:10:40,150
it's like these two
nuclei, lithium and helium,

193
00:10:40,150 --> 00:10:42,520
were kind of standing
still and all of a sudden

194
00:10:42,520 --> 00:10:44,710
moved off in opposite
directions, which

195
00:10:44,710 --> 00:10:48,260
means they've got to have
equal and opposite momenta.

196
00:10:48,260 --> 00:10:52,150
So let's say the absolute value
of the momentum of lithium

197
00:10:52,150 --> 00:10:53,890
has got to equal
the absolute value

198
00:10:53,890 --> 00:10:57,580
of the momentum of helium.

199
00:10:57,580 --> 00:10:59,350
This is our second
equation, where

200
00:10:59,350 --> 00:11:01,390
we'll use the first one
to figure out what are

201
00:11:01,390 --> 00:11:03,550
the relative kinetic energies.

202
00:11:03,550 --> 00:11:06,010
And so we're going to use a
quick trick to say if momentum

203
00:11:06,010 --> 00:11:10,060
equals mass times
velocity, we can also

204
00:11:10,060 --> 00:11:14,470
say it equals the
kinetic energy, which

205
00:11:14,470 --> 00:11:18,490
is 1/2 mv squared, and then
multiply by things in order

206
00:11:18,490 --> 00:11:19,990
to make it mv.

207
00:11:19,990 --> 00:11:22,690
So we can multiply by 2.

208
00:11:22,690 --> 00:11:26,590
Let's make a little
bit of space here.

209
00:11:26,590 --> 00:11:32,020
If we take the kinetic energy,
multiply by 2, multiply by m,

210
00:11:32,020 --> 00:11:35,800
and take the square
root, we have t.

211
00:11:35,800 --> 00:11:37,210
Let's see, that
would give us 2 m

212
00:11:37,210 --> 00:11:41,170
squared v squared inside
the square root gives us mv.

213
00:11:41,170 --> 00:11:42,730
So now we can take
these expressions

214
00:11:42,730 --> 00:11:49,390
and we can say that the root
2 mass lithium, t lithium,

215
00:11:49,390 --> 00:11:56,000
equals root 2 mass
helium t helium.

216
00:11:56,000 --> 00:11:59,600
And they both have
a square root of 2.

217
00:11:59,600 --> 00:12:03,050
We can square both
sides of both equations.

218
00:12:03,050 --> 00:12:08,240
And we end up with mass
of lithium t lithium

219
00:12:08,240 --> 00:12:14,060
equals mass of helium t helium.

220
00:12:14,060 --> 00:12:17,360
Now we take this equation,
rearrange it a little bit.

221
00:12:17,360 --> 00:12:21,830
Let's just call that Q to
keep things in variable space.

222
00:12:21,830 --> 00:12:31,270
And we can say that t helium
equals Q minus t lithium.

223
00:12:31,270 --> 00:12:36,960
We can take this t
helium, stick it in here.

224
00:12:36,960 --> 00:12:40,920
We end up with m
lithium t lithium

225
00:12:40,920 --> 00:12:49,000
equals the mass of helium
times Q minus t lithium.

226
00:12:49,000 --> 00:12:50,700
There's a missing h right there.

227
00:12:50,700 --> 00:12:51,960
There we go.

228
00:12:51,960 --> 00:12:53,040
And then from here--

229
00:12:53,040 --> 00:12:53,540
oh good.

230
00:12:53,540 --> 00:12:55,950
We've got some blank
space right here.

231
00:12:55,950 --> 00:12:56,450
Let's see.

232
00:12:56,450 --> 00:12:58,200
So we'd have-- I'll
do out all the steps--

233
00:13:00,930 --> 00:13:08,600
equals mass of helium times
Q minus mass of helium times

234
00:13:08,600 --> 00:13:11,160
t lithium.

235
00:13:11,160 --> 00:13:16,860
And we're solving for t lithium,
so we can put this term over

236
00:13:16,860 --> 00:13:18,640
on the other side.

237
00:13:18,640 --> 00:13:23,610
So let's say t lithium
times the mass of lithium

238
00:13:23,610 --> 00:13:31,610
plus the mass of helium equals
Q times the mass of helium.

239
00:13:31,610 --> 00:13:33,710
Then we can just
divide both sides

240
00:13:33,710 --> 00:13:45,510
by the sum of those masses

241
00:13:45,510 --> 00:13:47,640
Those two terms
cancel, and we're

242
00:13:47,640 --> 00:13:53,120
left with the expression for
the kinetic energy of lithium,

243
00:13:53,120 --> 00:13:56,060
which is the mass of helium
over the sum of the masses

244
00:13:56,060 --> 00:13:58,750
times the total Q value.

245
00:13:58,750 --> 00:14:00,500
Looks like the two of
those might actually

246
00:14:00,500 --> 00:14:02,390
be backwards, huh?

247
00:14:02,390 --> 00:14:06,340
Because the lithium
one should be smaller.

248
00:14:06,340 --> 00:14:10,070
I'll correct that for the
notes when I put them up online

249
00:14:10,070 --> 00:14:14,180
because this ratio should
be smaller than 0.5 Q.

250
00:14:14,180 --> 00:14:16,880
But at any rate, this is
how you actually get them.

251
00:14:16,880 --> 00:14:18,590
And I want to
point out something

252
00:14:18,590 --> 00:14:21,860
a little flash forward
to some of the decay

253
00:14:21,860 --> 00:14:24,800
that you're going to be looking
at in terms of nuclear decays.

254
00:14:24,800 --> 00:14:27,440
Let's talk for a second
about alpha decay.

255
00:14:27,440 --> 00:14:30,890
In alpha decay, you have a
low nucleus sitting around.

256
00:14:30,890 --> 00:14:33,890
And then all of a sudden it
emits a helium nucleus as well

257
00:14:33,890 --> 00:14:35,540
as a recoil nucleus.

258
00:14:35,540 --> 00:14:38,240
And one of the questions
that came up a lot last year

259
00:14:38,240 --> 00:14:41,420
is why isn't the Q
value of an alpha decay

260
00:14:41,420 --> 00:14:46,070
reaction-- let's just write one
up on the board, possibly one

261
00:14:46,070 --> 00:14:50,780
that you'll be dealing
with very soon, hands on.

262
00:14:50,780 --> 00:14:58,390
Uranium 235 can spontaneously
go to a helium nucleus.

263
00:14:58,390 --> 00:15:06,490
That's 92 plus
looks like 90 to 31.

264
00:15:06,490 --> 00:15:08,200
What comes before uranium?

265
00:15:08,200 --> 00:15:11,050
I think that is thorium,
although I wouldn't-- yeah,

266
00:15:11,050 --> 00:15:12,423
I think that's thorium.

267
00:15:12,423 --> 00:15:13,840
And the idea here
is that there is

268
00:15:13,840 --> 00:15:16,780
some Q value associated with
the kinetic energy of both

269
00:15:16,780 --> 00:15:18,260
of these.

270
00:15:18,260 --> 00:15:21,380
But it's not the
alpha kinetic energy

271
00:15:21,380 --> 00:15:25,190
because the thorium nucleus
would have to take away

272
00:15:25,190 --> 00:15:26,630
some of that kinetic energy.

273
00:15:26,630 --> 00:15:29,520
And I want to show you
this on the diagrams.

274
00:15:29,520 --> 00:15:34,040
So let's look at U 235.

275
00:15:34,040 --> 00:15:38,880
And we can see that it has an
alpha decay to thorium 231.

276
00:15:38,880 --> 00:15:39,380
Awesome.

277
00:15:39,380 --> 00:15:41,040
Got the symbol right.

278
00:15:41,040 --> 00:15:46,170
And it has a decay
energy of 4.679 MeV.

279
00:15:46,170 --> 00:15:47,610
Let's take a look
at the diagram,

280
00:15:47,610 --> 00:15:50,910
which actually lists all of the
possible alpha decay energies,

281
00:15:50,910 --> 00:15:54,220
of which there are many.

282
00:15:54,220 --> 00:15:56,584
Had to zoom out a
little bit there.

283
00:15:56,584 --> 00:15:57,970
Yeah?

284
00:15:57,970 --> 00:15:59,590
Anyone have a question?

285
00:15:59,590 --> 00:16:01,010
OK.

286
00:16:01,010 --> 00:16:06,680
So notice that the difference
in energy levels is 4.676 MeV.

287
00:16:06,680 --> 00:16:08,730
And if we look at the
highest energy alpha ray,

288
00:16:08,730 --> 00:16:11,422
it's less than that.

289
00:16:11,422 --> 00:16:13,380
And that's because just
like we showed up here,

290
00:16:13,380 --> 00:16:15,180
the thorium nucleus
has to take away

291
00:16:15,180 --> 00:16:17,970
some of that kinetic energy
to conserve both energy

292
00:16:17,970 --> 00:16:19,630
and momentum.

293
00:16:19,630 --> 00:16:22,430
So this is a question that came
up quite a few times last year,

294
00:16:22,430 --> 00:16:25,180
and I want to make sure you guys
don't get tripped up like this.

295
00:16:25,180 --> 00:16:26,610
So again, I think I've
said it every single day,

296
00:16:26,610 --> 00:16:28,027
and I'll say it
again because it's

297
00:16:28,027 --> 00:16:30,360
the next day, is make sure
to conserve mass energy

298
00:16:30,360 --> 00:16:30,930
and momentum.

299
00:16:30,930 --> 00:16:32,910
That's the whole
theme of this class.

300
00:16:36,960 --> 00:16:37,972
Yes.

301
00:16:37,972 --> 00:16:39,430
AUDIENCE: How do
you get that value

302
00:16:39,430 --> 00:16:42,710
for [INAUDIBLE]
mega electron volts?

303
00:16:42,710 --> 00:16:44,750
MICHAEL SHORT: Should
have been 2.79 minus-- oh,

304
00:16:44,750 --> 00:16:48,180
did I do a little
mental math mistake?

305
00:16:48,180 --> 00:16:49,070
That should be-- oh.

306
00:16:49,070 --> 00:16:49,590
Yeah, no.

307
00:16:49,590 --> 00:16:51,010
A little dyslexia thing.

308
00:16:51,010 --> 00:16:53,520
3, 1.

309
00:16:53,520 --> 00:16:54,350
Yeah.

310
00:16:54,350 --> 00:16:55,080
There we go.

311
00:16:55,080 --> 00:16:55,580
Thank you.

312
00:16:58,460 --> 00:17:01,240
Cool.

313
00:17:01,240 --> 00:17:02,930
OK.

314
00:17:02,930 --> 00:17:05,290
Is everyone clear on how
to calculate Q values

315
00:17:05,290 --> 00:17:08,859
from nuclear reactions using
either kinetic energies, which

316
00:17:08,859 --> 00:17:12,040
you won't typically know, or
masses or binding energies,

317
00:17:12,040 --> 00:17:15,109
which you can look up directly
from the table of nuclides?

318
00:17:15,109 --> 00:17:15,730
Yeah.

319
00:17:15,730 --> 00:17:18,849
AUDIENCE: So what did you do
down there in the bottom right

320
00:17:18,849 --> 00:17:20,145
corner of the chalkboard?

321
00:17:20,145 --> 00:17:21,270
MICHAEL SHORT: Of this one?

322
00:17:21,270 --> 00:17:22,047
AUDIENCE: Yeah.

323
00:17:22,047 --> 00:17:22,839
MICHAEL SHORT: Yep.

324
00:17:22,839 --> 00:17:26,950
So I took this expression
right here, which

325
00:17:26,950 --> 00:17:28,720
is to say the Q
value has got to be

326
00:17:28,720 --> 00:17:30,940
the sum of the kinetic
energies of the lithium

327
00:17:30,940 --> 00:17:35,760
and helium nuclei, rearranged
it thusly to isolate--

328
00:17:35,760 --> 00:17:37,490
ah, there we go--

329
00:17:37,490 --> 00:17:41,240
yep, to isolate the
helium kinetic energy,

330
00:17:41,240 --> 00:17:45,230
and then substituted
that expression in here

331
00:17:45,230 --> 00:17:46,760
to get this one right there.

332
00:17:46,760 --> 00:17:47,697
AUDIENCE: OK.

333
00:17:47,697 --> 00:17:48,530
MICHAEL SHORT: Yeah.

334
00:17:48,530 --> 00:17:49,883
So this way, we have--

335
00:17:49,883 --> 00:17:51,800
in this case, we had,
let's say, two variables

336
00:17:51,800 --> 00:17:52,592
and three unknowns.

337
00:17:52,592 --> 00:17:54,948
But because we have this
equation relating them,

338
00:17:54,948 --> 00:17:56,990
we're left down with two
variables, two unknowns,

339
00:17:56,990 --> 00:17:59,250
and we can actually
solve this thing.

340
00:17:59,250 --> 00:18:00,771
Yep.

341
00:18:00,771 --> 00:18:04,260
Yeah, good question.

342
00:18:04,260 --> 00:18:05,420
Yes.

343
00:18:05,420 --> 00:18:08,110
AUDIENCE: The energy of the
gamma, is that just a known?

344
00:18:08,110 --> 00:18:09,097
Like the .48 MeV?

345
00:18:09,097 --> 00:18:10,680
MICHAEL SHORT: That's
something either

346
00:18:10,680 --> 00:18:13,500
I tell you or you would
measure, let's say.

347
00:18:13,500 --> 00:18:15,640
That's just for completeness
to say all right,

348
00:18:15,640 --> 00:18:17,680
this reaction actually
gives off a gamma,

349
00:18:17,680 --> 00:18:21,570
and I want to give the right
value for the kinetic energies.

350
00:18:21,570 --> 00:18:24,230
And we'll get into what
gamma transitions are allowed

351
00:18:24,230 --> 00:18:26,960
and then how you measure them
in the next couple of weeks,

352
00:18:26,960 --> 00:18:28,430
actually.

353
00:18:28,430 --> 00:18:29,810
Yes.

354
00:18:29,810 --> 00:18:33,110
AUDIENCE: So do we refer to Q as
the Q you calculated up there,

355
00:18:33,110 --> 00:18:36,110
or that 2.31?

356
00:18:36,110 --> 00:18:38,732
MICHAEL SHORT: They're
the same one, actually.

357
00:18:38,732 --> 00:18:39,440
AUDIENCE: The 2.7

358
00:18:39,440 --> 00:18:42,190
MICHAEL SHORT: Oh, I see.

359
00:18:42,190 --> 00:18:44,270
That's a good point.

360
00:18:44,270 --> 00:18:46,210
So this wouldn't
really be Q, would it?

361
00:18:46,210 --> 00:18:49,300
But it is the sum of
the kinetic energies.

362
00:18:49,300 --> 00:18:54,380
This is like Q minus
the gamma ray energy.

363
00:18:54,380 --> 00:18:55,520
Let me stick that in there.

364
00:19:00,000 --> 00:19:01,980
L, i.

365
00:19:01,980 --> 00:19:02,580
Yep.

366
00:19:02,580 --> 00:19:03,080
Yeah.

367
00:19:03,080 --> 00:19:03,580
Good point.

368
00:19:08,420 --> 00:19:09,500
OK.

369
00:19:09,500 --> 00:19:12,250
Any other questions
before I move on?

370
00:19:12,250 --> 00:19:15,040
We're going to get into
a universal formula

371
00:19:15,040 --> 00:19:18,940
to predict in a so-so way what
the binding energy of any given

372
00:19:18,940 --> 00:19:21,490
nucleus will be and start
looking at stability trends

373
00:19:21,490 --> 00:19:24,885
so you can predict, just
from the number of protons

374
00:19:24,885 --> 00:19:27,490
and the number of neutrons,
how stable a nucleus will

375
00:19:27,490 --> 00:19:31,840
be with a few exceptions,
which we will go over.

376
00:19:31,840 --> 00:19:34,270
And this is what's referred
to as the semi-empirical mass

377
00:19:34,270 --> 00:19:35,510
formula.

378
00:19:35,510 --> 00:19:38,590
So I'm going to
erase some stuff.

379
00:19:38,590 --> 00:19:43,510
Has everyone got the notes on
this bottom board right here?

380
00:19:43,510 --> 00:19:44,170
OK.

381
00:19:44,170 --> 00:19:47,350
Let me know when you're
ready, and let's see.

382
00:19:49,790 --> 00:19:50,290
Yeah.

383
00:19:50,290 --> 00:19:52,870
I want to make sure I
move at your guys' pace.

384
00:20:06,610 --> 00:20:13,400
Let's say going to have a graph
of binding energy per nucleon

385
00:20:13,400 --> 00:20:17,027
on versus nucleons.

386
00:20:17,027 --> 00:20:18,610
Well, anyway, I'll
leave that up there

387
00:20:18,610 --> 00:20:20,650
and I'll do the work on
this board right here.

388
00:20:20,650 --> 00:20:23,890
So let's say we wanted to
figure out a weighted graph

389
00:20:23,890 --> 00:20:27,340
or to predict the binding
energy per nucleon.

390
00:20:27,340 --> 00:20:29,170
So I have this
binding energy term

391
00:20:29,170 --> 00:20:32,380
over a, where a is the
total number of nucleons,

392
00:20:32,380 --> 00:20:35,050
as a function of the
number of nucleons--

393
00:20:35,050 --> 00:20:36,100
in a generalized way.

394
00:20:36,100 --> 00:20:40,420
Not accounting for magic numbers
or anything else that we'll

395
00:20:40,420 --> 00:20:41,530
get into pretty soon.

396
00:20:41,530 --> 00:20:43,720
And I don't like the
term magic numbers,

397
00:20:43,720 --> 00:20:46,390
but that is the parlance
that's used in this field

398
00:20:46,390 --> 00:20:48,170
so I'm going to stick with it.

399
00:20:48,170 --> 00:20:51,520
Let's try and think about
if you imagine the nucleus

400
00:20:51,520 --> 00:20:53,290
as a kind of drop
of liquid-- and one

401
00:20:53,290 --> 00:20:55,660
of the other words for the
semi-empirical mass formula

402
00:20:55,660 --> 00:20:59,650
is called the liquid drop
formula or the liquid drop

403
00:20:59,650 --> 00:21:02,420
model.

404
00:21:02,420 --> 00:21:05,030
It assumes that the
nucleus takes the shape

405
00:21:05,030 --> 00:21:06,980
roughly of a liquid
drop, and you can kind of

406
00:21:06,980 --> 00:21:11,147
treat some of the energy
terms accordingly.

407
00:21:11,147 --> 00:21:13,480
This is why I have all the
different colors of chalk out

408
00:21:13,480 --> 00:21:14,660
for this.

409
00:21:14,660 --> 00:21:18,210
Makes it a little
visually easier to see.

410
00:21:18,210 --> 00:21:20,660
So let's start writing
a general expression

411
00:21:20,660 --> 00:21:24,268
for the binding energy
as a function of a and z,

412
00:21:24,268 --> 00:21:26,060
and start thinking
about what sort of terms

413
00:21:26,060 --> 00:21:28,940
would add to or
decrease the stability

414
00:21:28,940 --> 00:21:32,540
of a given liquid drop nucleus,
where all the nucleons are just

415
00:21:32,540 --> 00:21:35,660
kind of there in some sort of
floating, crazy, coulombic,

416
00:21:35,660 --> 00:21:38,000
strong nuclear force soup.

417
00:21:38,000 --> 00:21:40,430
First of all, as
you add nucleons

418
00:21:40,430 --> 00:21:42,620
to a given nucleus,
what tends to happen

419
00:21:42,620 --> 00:21:44,600
to the binding
energy, in general?

420
00:21:47,280 --> 00:21:50,030
Without knowing anything else.

421
00:21:50,030 --> 00:21:53,390
Assemble more nucleons, you
convert more mass to energy.

422
00:21:53,390 --> 00:21:56,870
And you end up increasing
the binding energy.

423
00:21:56,870 --> 00:21:59,240
So let's call this
the volume term.

424
00:21:59,240 --> 00:22:02,310
As you increase the volume
of this liquid drop,

425
00:22:02,310 --> 00:22:05,700
its total binding energies
start to increase.

426
00:22:05,700 --> 00:22:07,550
So let's say there's
some term that's

427
00:22:07,550 --> 00:22:13,190
going to be proportional to A,
the number of nucleons that's

428
00:22:13,190 --> 00:22:15,027
in this liquid drop.

429
00:22:15,027 --> 00:22:15,860
And we'll draw this.

430
00:22:15,860 --> 00:22:17,970
Let's see if I can
do the trick right.

431
00:22:17,970 --> 00:22:18,920
Yes.

432
00:22:18,920 --> 00:22:21,700
I love doing that.

433
00:22:21,700 --> 00:22:23,990
If we were to graph
binding energy per nucleon

434
00:22:23,990 --> 00:22:27,620
as a function of number of
nucleotides for this term,

435
00:22:27,620 --> 00:22:30,230
it would just be a flat line
because it's related to A.

436
00:22:30,230 --> 00:22:35,820
And there's going to be
some constant, which we're

437
00:22:35,820 --> 00:22:39,335
going to call the volume
constant, that says, well,

438
00:22:39,335 --> 00:22:40,710
there's going to
be some relation

439
00:22:40,710 --> 00:22:42,690
between the actual amount
of stability gained

440
00:22:42,690 --> 00:22:44,100
and the number of nucleons.

441
00:22:44,100 --> 00:22:45,840
We don't know what
it is yet, but what

442
00:22:45,840 --> 00:22:48,090
we're really concerned
with is the functional

443
00:22:48,090 --> 00:22:50,040
form of this thing.

444
00:22:50,040 --> 00:22:52,720
It's proportional to A.

445
00:22:52,720 --> 00:22:56,410
Next up, what also
happens to a liquid drop

446
00:22:56,410 --> 00:22:57,917
as you increase its volume?

447
00:22:57,917 --> 00:22:59,500
What other parameters
do you increase?

448
00:23:02,755 --> 00:23:03,880
AUDIENCE: The surface area?

449
00:23:03,880 --> 00:23:04,838
MICHAEL SHORT: Exactly.

450
00:23:04,838 --> 00:23:05,740
The surface area.

451
00:23:05,740 --> 00:23:08,440
The idea here is that
if this liquid drop

452
00:23:08,440 --> 00:23:10,810
is made of all sorts
of different nucleons--

453
00:23:13,660 --> 00:23:15,920
and let's pretend that they're
like atoms in a crystal

454
00:23:15,920 --> 00:23:19,460
and they're all
binding to each other--

455
00:23:19,460 --> 00:23:24,860
the ones on the outside aren't
bound to as many nucleons

456
00:23:24,860 --> 00:23:26,990
as the ones on the inside.

457
00:23:26,990 --> 00:23:29,870
And so the more nuclei
there are near the surface

458
00:23:29,870 --> 00:23:33,380
as opposed to inside the
liquid droplets, say,

459
00:23:33,380 --> 00:23:36,020
inside some little radius
where all it sees around

460
00:23:36,020 --> 00:23:39,470
it are other nucleons, then
they're not quite as bound.

461
00:23:39,470 --> 00:23:42,020
And how does the surface
area of a liquid drop

462
00:23:42,020 --> 00:23:44,833
scale with its volume?

463
00:23:44,833 --> 00:23:46,375
To what function or
to what exponent?

464
00:23:51,520 --> 00:23:52,780
2/3.

465
00:23:52,780 --> 00:23:56,800
I mean, let's take a quick
look at the volume of a sphere

466
00:23:56,800 --> 00:24:00,050
is 4/3 pi r cubed.

467
00:24:00,050 --> 00:24:04,070
And the surface area of a
sphere is 4 pi r squared.

468
00:24:04,070 --> 00:24:06,160
So if you want to get
some expression for how

469
00:24:06,160 --> 00:24:11,080
does area scale as
volume, I said cube,

470
00:24:11,080 --> 00:24:12,720
then I wrote squared.

471
00:24:12,720 --> 00:24:15,460
It's going to end up
looking like something times

472
00:24:15,460 --> 00:24:18,220
r to the 2-- let's see.

473
00:24:18,220 --> 00:24:18,790
Oh, yeah.

474
00:24:18,790 --> 00:24:20,998
I'm sorry, that's not the
expression I want to write.

475
00:24:20,998 --> 00:24:24,680
But the idea here is it's going
to scale with r to the 2/3.

476
00:24:24,680 --> 00:24:28,300
So let's pick a
different color and say

477
00:24:28,300 --> 00:24:33,940
we're going to have some surface
term times number of nucleons

478
00:24:33,940 --> 00:24:35,830
to the 2/3.

479
00:24:35,830 --> 00:24:39,670
And if we then adjust
this formula to also take

480
00:24:39,670 --> 00:24:42,220
into account this
surface area term--

481
00:24:42,220 --> 00:24:44,930
which is to say for
very small nuclei,

482
00:24:44,930 --> 00:24:48,070
there's a lot of nucleotides
near the surface,

483
00:24:48,070 --> 00:24:50,040
and as the nucleus
gets bigger and bigger,

484
00:24:50,040 --> 00:24:52,540
more and more of them are in
the juicy center and don't know

485
00:24:52,540 --> 00:24:54,130
they're near the surface--

486
00:24:54,130 --> 00:24:59,710
we'd have some modification
that looks like that.

487
00:24:59,710 --> 00:25:01,960
Now I'm going to erase the
stuff over here because I'm

488
00:25:01,960 --> 00:25:02,793
running out of room.

489
00:25:05,900 --> 00:25:10,460
Now these nucleons aren't just
untagged, anonymous nucleons.

490
00:25:10,460 --> 00:25:12,440
They're either
protons or neutrons.

491
00:25:15,618 --> 00:25:17,910
And what happens when you
try and cram a lot of protons

492
00:25:17,910 --> 00:25:19,413
into one space?

493
00:25:19,413 --> 00:25:21,080
AUDIENCE: They want
to repel each other.

494
00:25:21,080 --> 00:25:21,872
MICHAEL SHORT: Yep.

495
00:25:21,872 --> 00:25:24,390
They want to repel each
other by coulombic forces.

496
00:25:24,390 --> 00:25:26,220
And so every proton--

497
00:25:26,220 --> 00:25:29,800
let's pick a different color
for the coulombic forces--

498
00:25:29,800 --> 00:25:31,550
and that should be
a minus if I want

499
00:25:31,550 --> 00:25:34,950
to stick with all the notation.

500
00:25:34,950 --> 00:25:37,650
There's going to be some other
term to account for the fact

501
00:25:37,650 --> 00:25:40,560
that the nuclei,
specifically the protons,

502
00:25:40,560 --> 00:25:42,668
are trying to repel each other.

503
00:25:42,668 --> 00:25:45,210
So in this case, it's going to
be proportional to, let's say,

504
00:25:45,210 --> 00:25:48,450
the number of
protons that we have.

505
00:25:48,450 --> 00:25:51,300
And every proton should
feel a repulsive force

506
00:25:51,300 --> 00:25:52,325
from every other proton.

507
00:25:55,800 --> 00:25:59,260
So let's say it's times
z times z minus 1,

508
00:25:59,260 --> 00:26:02,220
so that every proton feels the
force of every other proton

509
00:26:02,220 --> 00:26:04,290
except for itself.

510
00:26:04,290 --> 00:26:08,310
And that's going to be
mediated by the total number

511
00:26:08,310 --> 00:26:09,040
of nucleons.

512
00:26:09,040 --> 00:26:11,160
So if there are more
neutrons in the way,

513
00:26:11,160 --> 00:26:13,280
it won't be quite as bad.

514
00:26:13,280 --> 00:26:14,850
And there's going
to be some other--

515
00:26:14,850 --> 00:26:18,430
we'll call it a C for
the coulombic term--

516
00:26:18,430 --> 00:26:21,040
and that will say that as
you make a bigger and bigger

517
00:26:21,040 --> 00:26:23,200
nucleus, you start to get
more and more coulombic

518
00:26:23,200 --> 00:26:26,060
repulsion trying
to rip it apart.

519
00:26:26,060 --> 00:26:29,350
So if we were to then
modify the purple curve--

520
00:26:29,350 --> 00:26:30,100
oops.

521
00:26:30,100 --> 00:26:35,700
Trying to get it to go the same
as the nucleus gets bigger.

522
00:26:35,700 --> 00:26:37,590
I want to make sure it's
really to scale-ish.

523
00:26:42,310 --> 00:26:44,080
As the nuclei get
bigger and bigger,

524
00:26:44,080 --> 00:26:46,485
it's going to be a
little less stable.

525
00:26:46,485 --> 00:26:47,860
And already we're
starting to get

526
00:26:47,860 --> 00:26:51,822
a curve that is getting close to
looking like the binding energy

527
00:26:51,822 --> 00:26:52,780
curve from the reading.

528
00:26:52,780 --> 00:26:55,570
But there are a couple
more terms to reckon with.

529
00:26:55,570 --> 00:26:58,623
So let's pick a fourth color.

530
00:26:58,623 --> 00:27:00,040
What other sort
of trends that you

531
00:27:00,040 --> 00:27:02,623
notice in the reading about the
stability of different nuclei?

532
00:27:06,210 --> 00:27:09,060
Let's say you were to take a
common nucleus like carbon-12.

533
00:27:09,060 --> 00:27:11,850
It's got six protons
and six neutrons,

534
00:27:11,850 --> 00:27:13,830
and it's exceptionally stable.

535
00:27:13,830 --> 00:27:15,600
What about carbon 6?

536
00:27:15,600 --> 00:27:18,930
A nucleus of just 6 protons?

537
00:27:18,930 --> 00:27:19,620
Doesn't exist.

538
00:27:19,620 --> 00:27:21,600
Exceptionally unstable.

539
00:27:21,600 --> 00:27:23,560
What about carbon 24?

540
00:27:23,560 --> 00:27:25,500
18 neutrons, six protons.

541
00:27:29,020 --> 00:27:31,120
Sound stable or not?

542
00:27:31,120 --> 00:27:32,050
Not at all.

543
00:27:32,050 --> 00:27:35,140
So there's some sort of
asymmetry term going on.

544
00:27:35,140 --> 00:27:37,030
When the number of
protons and neutrons

545
00:27:37,030 --> 00:27:40,480
is roughly in balance,
especially for light nuclei,

546
00:27:40,480 --> 00:27:42,580
the nucleus tends
to be more stable.

547
00:27:42,580 --> 00:27:44,410
So we can write
some sort of term--

548
00:27:44,410 --> 00:27:48,340
let's call it an
asymmetry term--

549
00:27:48,340 --> 00:27:51,970
that relates to the
number of neutrons

550
00:27:51,970 --> 00:27:54,840
minus the number of protons.

551
00:27:54,840 --> 00:27:58,780
And in this case, for reasons
I'm not going to get into,

552
00:27:58,780 --> 00:28:01,050
but are derived in a
reference in your reading,

553
00:28:01,050 --> 00:28:02,050
there's a squared on it.

554
00:28:02,050 --> 00:28:04,930
But suffice to say if the
number of neutrons and number

555
00:28:04,930 --> 00:28:07,630
of protons are equal,
then the nucleus is

556
00:28:07,630 --> 00:28:09,490
predicted to be pretty stable.

557
00:28:09,490 --> 00:28:11,560
And this works out quite
well for light nuclei.

558
00:28:11,560 --> 00:28:16,090
It starts to break down a
little bit for heavier nuclei.

559
00:28:16,090 --> 00:28:21,070
And then divide by the number
of nuclei that there are.

560
00:28:21,070 --> 00:28:25,230
I also see a missing
1/3 because let's say

561
00:28:25,230 --> 00:28:28,230
this nucleus has a volume
that scales with roughly

562
00:28:28,230 --> 00:28:29,880
the number of nucleons.

563
00:28:29,880 --> 00:28:31,920
Then the distance of
that coulombic force

564
00:28:31,920 --> 00:28:34,890
is going to be like A
to the 1/3 or the radius

565
00:28:34,890 --> 00:28:36,810
of this nuclear drop.

566
00:28:36,810 --> 00:28:40,100
So let's take the
asymmetry term.

567
00:28:40,100 --> 00:28:44,630
That's going to give us a
further modification slightly

568
00:28:44,630 --> 00:28:46,690
downward.

569
00:28:46,690 --> 00:28:49,030
And finally, there's what's
called the pairing term.

570
00:28:49,030 --> 00:28:50,800
What's the last
color I haven't used?

571
00:28:55,860 --> 00:28:57,210
This pairing term delta.

572
00:28:57,210 --> 00:28:58,710
And this is not a
smooth function.

573
00:28:58,710 --> 00:29:01,140
It's a piecewise
function that depends on

574
00:29:01,140 --> 00:29:04,260
whether you have an
odd or an even number

575
00:29:04,260 --> 00:29:07,513
of each type of nucleon,
protons or neutrons.

576
00:29:07,513 --> 00:29:08,930
And so what this
means, it's going

577
00:29:08,930 --> 00:29:11,370
to add a little
bit of jaggedness

578
00:29:11,370 --> 00:29:16,630
to the beginning of the curve
and equal out in the end

579
00:29:16,630 --> 00:29:18,730
because this delta
term can be something

580
00:29:18,730 --> 00:29:23,320
like plus, let's
call it an A pairing.

581
00:29:23,320 --> 00:29:30,220
And it scales with the square
root of A, or minus, or 0,

582
00:29:30,220 --> 00:29:32,860
depending on if the
nuclei are odd odd,

583
00:29:32,860 --> 00:29:35,680
like odd number of protons,
odd number of neutrons,

584
00:29:35,680 --> 00:29:39,460
even even, or odd even.

585
00:29:39,460 --> 00:29:42,160
Now I know that the derivation
is a little hand wavy.

586
00:29:42,160 --> 00:29:44,380
That's why we call
it semi-empirical.

587
00:29:44,380 --> 00:29:46,450
We're taking each of
these additive terms

588
00:29:46,450 --> 00:29:50,920
and saying it kind of comes
from a fairly OK, a little poor

589
00:29:50,920 --> 00:29:52,630
approximation of the nucleus.

590
00:29:52,630 --> 00:29:55,570
But what we end up
with is a formula

591
00:29:55,570 --> 00:30:00,580
whose constants are fit, whose
terms, the actual variables,

592
00:30:00,580 --> 00:30:04,360
are derived somewhat
from physical intuition.

593
00:30:04,360 --> 00:30:07,772
These constants were then fit
later by some other folks,

594
00:30:07,772 --> 00:30:09,730
and the references for
this are in the reading.

595
00:30:09,730 --> 00:30:13,630
They're all in MeV, and this
gives you a binding energy

596
00:30:13,630 --> 00:30:17,210
in MeV for a given nucleus.

597
00:30:17,210 --> 00:30:20,820
Now it works some places and it
doesn't work in other places.

598
00:30:20,820 --> 00:30:21,320
Yeah.

599
00:30:21,320 --> 00:30:24,054
AUDIENCE: So the lower
case a [INAUDIBLE]

600
00:30:24,054 --> 00:30:26,030
no matter what the
nucleus looks like?

601
00:30:26,030 --> 00:30:27,260
MICHAEL SHORT: That's right.

602
00:30:27,260 --> 00:30:31,340
So this is the universal,
semi-empirical, usually works

603
00:30:31,340 --> 00:30:35,480
formula for the binding
energy of a nucleus.

604
00:30:35,480 --> 00:30:38,390
So the constants don't change
because the variables here

605
00:30:38,390 --> 00:30:41,840
are z, [INAUDIBLE] a, and n.

606
00:30:41,840 --> 00:30:43,520
And don't forget--
because you'll

607
00:30:43,520 --> 00:30:45,260
need to remember this
on the homework--

608
00:30:45,260 --> 00:30:48,830
that A equals z plus n.

609
00:30:48,830 --> 00:30:52,790
So for example, if
you want to express

610
00:30:52,790 --> 00:30:55,700
what is the most
stable nucleus, you

611
00:30:55,700 --> 00:30:58,040
could take the derivative
of this formula with respect

612
00:30:58,040 --> 00:30:59,450
to A or z.

613
00:30:59,450 --> 00:31:03,520
And don't forget that you can
substitute this expression

614
00:31:03,520 --> 00:31:06,020
into there.

615
00:31:06,020 --> 00:31:09,190
That's giving you guys
a hint for the homework.

616
00:31:09,190 --> 00:31:10,940
And let's look at what
this actually looks

617
00:31:10,940 --> 00:31:14,480
like as far as theory
compared to experiment.

618
00:31:14,480 --> 00:31:17,240
So the red points are
theoretical predictions.

619
00:31:17,240 --> 00:31:19,940
The black points are
experimental predictions.

620
00:31:19,940 --> 00:31:22,770
And all of the different
nuclei are shown here.

621
00:31:22,770 --> 00:31:25,460
First of all, the curve looks
quite a bit like the one

622
00:31:25,460 --> 00:31:28,100
that we just hand wavy made
on the board right here.

623
00:31:28,100 --> 00:31:31,010
And second of all, there
aren't too many exceptions.

624
00:31:31,010 --> 00:31:32,810
It's hard to see what
the exceptions are,

625
00:31:32,810 --> 00:31:35,300
so it's a little
easier to draw them

626
00:31:35,300 --> 00:31:36,590
in terms of relative error.

627
00:31:36,590 --> 00:31:38,780
So you can see where
does this formula work

628
00:31:38,780 --> 00:31:40,910
and where does this formula not?

629
00:31:40,910 --> 00:31:43,430
So if you notice,
for the small nuclei,

630
00:31:43,430 --> 00:31:45,320
approximating it
as a liquid drop

631
00:31:45,320 --> 00:31:47,690
is not a very good
approximation because you

632
00:31:47,690 --> 00:31:50,570
can't treat this
as a homogenized,

633
00:31:50,570 --> 00:31:51,800
smeared liquid drop.

634
00:31:51,800 --> 00:31:53,630
It's much more--
well, there's either

635
00:31:53,630 --> 00:31:57,320
two or three nucleons, and
very few protons and neutrons

636
00:31:57,320 --> 00:31:58,340
in each.

637
00:31:58,340 --> 00:32:00,650
But then as you get to
larger and larger nuclei,

638
00:32:00,650 --> 00:32:05,435
it starts to hit very
close with a few exceptions

639
00:32:05,435 --> 00:32:08,980
that I want to
point out right now.

640
00:32:08,980 --> 00:32:10,870
If you zoom in on that
part, you can actually

641
00:32:10,870 --> 00:32:14,350
see that at certain neutron
numbers, or certain proton

642
00:32:14,350 --> 00:32:17,560
numbers, there is an
exceptionally high stability

643
00:32:17,560 --> 00:32:18,700
of a lot of those nuclei.

644
00:32:18,700 --> 00:32:21,010
And that's as you
start to approach

645
00:32:21,010 --> 00:32:22,510
these what's called
magic numbers,

646
00:32:22,510 --> 00:32:26,110
or numbers of nucleons
which, say, fill all energy

647
00:32:26,110 --> 00:32:27,880
levels at a certain level.

648
00:32:27,880 --> 00:32:30,850
And again, it's not for
every nucleus as a function

649
00:32:30,850 --> 00:32:31,700
of neutron number.

650
00:32:31,700 --> 00:32:34,370
But even drawing an
envelope around this curve,

651
00:32:34,370 --> 00:32:37,060
you can see that the
nuclei around 82,

652
00:32:37,060 --> 00:32:41,110
around 50, around 28, are
a whole lot more stable

653
00:32:41,110 --> 00:32:42,190
than the ones in between.

654
00:32:42,190 --> 00:32:45,700
And this pattern kind of
repeats with larger and larger

655
00:32:45,700 --> 00:32:47,200
periodicity.

656
00:32:47,200 --> 00:32:48,790
And it kind of looks
like right here,

657
00:32:48,790 --> 00:32:51,400
at the edge of our
knowledge of nuclei,

658
00:32:51,400 --> 00:32:53,935
we haven't quite gotten
to the next peak yet.

659
00:32:53,935 --> 00:32:56,560
This is something we're going to
talk about Friday on the quest

660
00:32:56,560 --> 00:32:59,110
for super heavy
elements, or SHE's,

661
00:32:59,110 --> 00:33:00,640
as you'll see in the reading.

662
00:33:00,640 --> 00:33:01,190
Yeah.

663
00:33:01,190 --> 00:33:05,680
AUDIENCE: So the most stable
nuclei peaks were closest to 0.

664
00:33:05,680 --> 00:33:08,260
MICHAEL SHORT: Closest to
0 is the closest agreement

665
00:33:08,260 --> 00:33:11,290
between experiment and theory.

666
00:33:11,290 --> 00:33:13,470
So the ones that are
exceptionally stable,

667
00:33:13,470 --> 00:33:17,040
which are not predicted by
this very simple formula,

668
00:33:17,040 --> 00:33:19,680
are up here at the peaks
of these magic numbers.

669
00:33:19,680 --> 00:33:22,410
And actually, I want you to
take a look right here at some

670
00:33:22,410 --> 00:33:24,190
of these very small nuclei.

671
00:33:24,190 --> 00:33:27,390
Like helium 4 is probably
way up here somewhere, all

672
00:33:27,390 --> 00:33:28,680
the way over on the right.

673
00:33:28,680 --> 00:33:30,990
It's an exceptionally
stable nucleus

674
00:33:30,990 --> 00:33:33,390
that is not very well
approximated by liquid drop

675
00:33:33,390 --> 00:33:35,460
model because it's
got four nucleons.

676
00:33:35,460 --> 00:33:36,750
They're all on the surface.

677
00:33:36,750 --> 00:33:41,100
There's nothing on the inside
of a helium nucleus, let's say.

678
00:33:41,100 --> 00:33:43,210
And then if you look at
stability trends in terms

679
00:33:43,210 --> 00:33:46,990
of are the nuclei more stable
if they have odd numbers

680
00:33:46,990 --> 00:33:50,050
or even numbers, you can
graph the two separately

681
00:33:50,050 --> 00:33:52,690
and look at the number
of stable nuclei

682
00:33:52,690 --> 00:33:56,020
that have an odd total mass
number or an even total mass

683
00:33:56,020 --> 00:33:56,890
number.

684
00:33:56,890 --> 00:33:58,880
And there's a few
things to note here.

685
00:33:58,880 --> 00:34:01,480
One of them is the even
numbers tend to have

686
00:34:01,480 --> 00:34:03,310
a lot more stable nuclei.

687
00:34:03,310 --> 00:34:06,050
This is something I mentioned
on the second day of class.

688
00:34:06,050 --> 00:34:10,030
If you look at the [INAUDIBLE]
table of nuclides--

689
00:34:10,030 --> 00:34:12,370
let's go to their home page--

690
00:34:12,370 --> 00:34:17,210
and just look at it
sort of in a color way.

691
00:34:17,210 --> 00:34:19,550
The blue colors
are stable nuclei,

692
00:34:19,550 --> 00:34:21,770
and you notice that
every other row of pixels

693
00:34:21,770 --> 00:34:25,219
here has a whole lot
more stable ones.

694
00:34:25,219 --> 00:34:28,190
And that's the same thing
that we're seeing right here,

695
00:34:28,190 --> 00:34:32,480
is that there's a lot more even
nuclei that are more stable.

696
00:34:32,480 --> 00:34:36,540
If I jump back to our
semi-empirical maths formula,

697
00:34:36,540 --> 00:34:40,162
notice that this binding energy
goes up for even, even nuclei.

698
00:34:40,162 --> 00:34:41,870
So when there's an
even number of protons

699
00:34:41,870 --> 00:34:45,380
and an even number of neutrons,
the semi-empirical mass formula

700
00:34:45,380 --> 00:34:48,710
does predict an
increase in stability,

701
00:34:48,710 --> 00:34:53,909
which you can actually see
on the table of nuclides,

702
00:34:53,909 --> 00:34:56,630
and on this sort
of stability trend.

703
00:34:56,630 --> 00:34:58,020
And so let's look
a little closer

704
00:34:58,020 --> 00:35:03,150
and see how many nuclei for each
proton number or each neutron

705
00:35:03,150 --> 00:35:04,860
number are actually stable.

706
00:35:04,860 --> 00:35:07,622
And we graphed the odd and
the even ones separately.

707
00:35:07,622 --> 00:35:09,330
And what's important
to note here is one,

708
00:35:09,330 --> 00:35:11,550
the odd is way
lower than the even.

709
00:35:11,550 --> 00:35:16,590
There's usually either 2 1 or
0 stable nuclei at that number.

710
00:35:16,590 --> 00:35:19,800
And what other sort of features
do you guys notice about this?

711
00:35:24,390 --> 00:35:26,603
It's not smooth, first of all.

712
00:35:26,603 --> 00:35:27,520
Where are those peaks?

713
00:35:27,520 --> 00:35:29,603
Where do you tend to find
that most stable nuclei?

714
00:35:33,210 --> 00:35:33,710
4?

715
00:35:33,710 --> 00:35:37,367
Where do you tend
to find the least?

716
00:35:37,367 --> 00:35:38,700
What about these two right here?

717
00:35:38,700 --> 00:35:41,530
No stable nuclei at
these proton numbers.

718
00:35:41,530 --> 00:35:44,340
And remember, proton number
uniquely defines an element.

719
00:35:44,340 --> 00:35:47,643
Anyone know what
these two might be?

720
00:35:47,643 --> 00:35:48,560
What sort of elements?

721
00:35:48,560 --> 00:35:50,268
Look to the back of
the room if you want.

722
00:35:50,268 --> 00:35:52,940
There's a periodic
table in the back wall.

723
00:35:52,940 --> 00:35:55,280
And you can see, except
for the super heavy things

724
00:35:55,280 --> 00:35:56,840
down at the bottom,
there's a couple

725
00:35:56,840 --> 00:36:00,150
of elements that have
no stable isotopes.

726
00:36:00,150 --> 00:36:03,920
These are technetium
and promethium, which

727
00:36:03,920 --> 00:36:05,750
are relatively light isotopes--

728
00:36:05,750 --> 00:36:08,570
I say relatively light compared
to things like uranium--

729
00:36:08,570 --> 00:36:10,490
with no stable isotopes.

730
00:36:10,490 --> 00:36:13,460
They're also fairly far away
from these so-called magic

731
00:36:13,460 --> 00:36:17,480
numbers or other regions,
where you tend to have a spike

732
00:36:17,480 --> 00:36:20,360
in the number of
stable nuclei due to--

733
00:36:20,360 --> 00:36:22,700
well, things that
you'll learn in 22.02,

734
00:36:22,700 --> 00:36:26,540
in terms of nuclear shell
occupancy and stability.

735
00:36:26,540 --> 00:36:30,700
But you see the same thing when
you graph the neutron number.

736
00:36:30,700 --> 00:36:32,290
You can see a couple
of sudden spikes

737
00:36:32,290 --> 00:36:38,693
right here at 20,
28, 50, 82, and 126.

738
00:36:38,693 --> 00:36:41,110
When everything gets really
stable, all of a sudden you've

739
00:36:41,110 --> 00:36:45,010
got one last gasp
of a stable isotope

740
00:36:45,010 --> 00:36:48,600
before you go off
into nowhere land.

741
00:36:48,600 --> 00:36:51,540
So let's start looking at
relative stabilities of nuclei,

742
00:36:51,540 --> 00:36:54,510
let's say, for a given mass
number or a given proton

743
00:36:54,510 --> 00:36:56,340
number.

744
00:36:56,340 --> 00:36:57,840
Anyone mind if I
cover this board

745
00:36:57,840 --> 00:36:59,210
because you can't roll it up.

746
00:36:59,210 --> 00:37:01,581
You all got the notes from here?

747
00:37:01,581 --> 00:37:04,510
Cool.

748
00:37:04,510 --> 00:37:07,840
That one has less to erase.

749
00:37:07,840 --> 00:37:10,590
And I want to keep these
formulas up for our reference.

750
00:37:22,210 --> 00:37:24,075
Let's say I pose this problem.

751
00:37:24,075 --> 00:37:26,700
I want to find out, make sure I
solve the right one-- actually,

752
00:37:26,700 --> 00:37:28,780
I'm going to check
my notes real quick--

753
00:37:28,780 --> 00:37:33,250
that what is the-- for a given
A, or for a given mass number--

754
00:37:38,295 --> 00:37:40,045
what is the most stable
number of protons?

755
00:37:47,980 --> 00:37:50,890
for A, given A.

756
00:37:50,890 --> 00:37:53,350
This is the question that
I like to answer here.

757
00:37:53,350 --> 00:37:54,940
How would you
approach this question

758
00:37:54,940 --> 00:37:57,700
using the semi-empirical
mass formula

759
00:37:57,700 --> 00:37:59,830
that-- well, you can't see
here, so I will bring it

760
00:37:59,830 --> 00:38:02,384
up back on the screen.

761
00:38:02,384 --> 00:38:02,884
Here.

762
00:38:05,540 --> 00:38:06,707
How would you find this out?

763
00:38:16,950 --> 00:38:20,170
Well, let's say for
a given mass number

764
00:38:20,170 --> 00:38:24,000
A, for a given poor
approximation of the total mass

765
00:38:24,000 --> 00:38:28,110
of a nucleus, the more binding
energy it has, the more stable

766
00:38:28,110 --> 00:38:29,220
it is.

767
00:38:29,220 --> 00:38:37,500
Therefore, if you want to
find the minimum of a mass

768
00:38:37,500 --> 00:38:43,540
for A and Z, given
a fixed A, that

769
00:38:43,540 --> 00:38:45,310
will give you the
most stable nucleus

770
00:38:45,310 --> 00:38:48,760
because it will tell
you which value of Z

771
00:38:48,760 --> 00:38:50,530
gives you the
smallest m, or rather,

772
00:38:50,530 --> 00:38:52,810
the most tightly
bound nucleus that

773
00:38:52,810 --> 00:38:55,387
has the most binding energy.

774
00:38:55,387 --> 00:38:56,720
So let's start writing this out.

775
00:38:56,720 --> 00:38:59,690
First of all, we can use
one of the two equations

776
00:38:59,690 --> 00:39:02,660
we already have up here, a
relation between the mass

777
00:39:02,660 --> 00:39:04,200
and the binding energy.

778
00:39:04,200 --> 00:39:07,040
The second one, well,
we have it right here.

779
00:39:07,040 --> 00:39:11,510
So let's substitute in our
binding energy equation

780
00:39:11,510 --> 00:39:15,590
and express it in terms of mass.

781
00:39:15,590 --> 00:39:25,970
So let's say our mass of a
nucleus A and Z is equal to Z

782
00:39:25,970 --> 00:39:34,360
times the mass of hydrogen
plus A minus Z times the mass

783
00:39:34,360 --> 00:39:39,222
of a neutron minus
the binding energy

784
00:39:39,222 --> 00:39:41,180
because in this case,
what I've done right here

785
00:39:41,180 --> 00:39:44,510
is I've added mass to each side
of the equation, subtracted

786
00:39:44,510 --> 00:39:46,690
binding energy from each
side of the equation,

787
00:39:46,690 --> 00:39:48,770
and we can just take
negative that expression

788
00:39:48,770 --> 00:39:50,610
and write it all out together.

789
00:39:50,610 --> 00:40:04,510
So we have minus AVA plus A
surface A to the 2/3 plus AC z

790
00:40:04,510 --> 00:40:09,670
times z minus 1
over A to the 1/3,

791
00:40:09,670 --> 00:40:22,690
and plus AA symmetry and minus z
squared over A and minus delta.

792
00:40:22,690 --> 00:40:26,220
So we've got one expression
for the total mass.

793
00:40:26,220 --> 00:40:28,200
We've fixed the value
of A because we're

794
00:40:28,200 --> 00:40:29,580
going to take some fixed--

795
00:40:29,580 --> 00:40:31,620
we're going to choose
some fixed value of A.

796
00:40:31,620 --> 00:40:33,945
Let's say A equals 93.

797
00:40:33,945 --> 00:40:36,570
And that's the example that I've
kind of worked out in my head.

798
00:40:36,570 --> 00:40:40,680
It so happens that niobium has a
stable isotope that a mass of A

799
00:40:40,680 --> 00:40:41,745
equals 93.

800
00:40:41,745 --> 00:40:43,620
And we just found out
in some of our research

801
00:40:43,620 --> 00:40:45,700
that niobium doesn't stick
to chromium very well.

802
00:40:45,700 --> 00:40:47,200
That's why I've got
it on the brain.

803
00:40:47,200 --> 00:40:49,710
So this is what I was
thinking about this morning.

804
00:40:49,710 --> 00:40:52,770
So for a fixed A
equals 93, we want

805
00:40:52,770 --> 00:40:56,280
to find what is the most
stable A. How do we do that?

806
00:41:00,690 --> 00:41:02,160
Anyone have an idea?

807
00:41:07,550 --> 00:41:08,050
Yeah.

808
00:41:08,050 --> 00:41:09,050
AUDIENCE: Differentiate?

809
00:41:09,050 --> 00:41:10,280
MICHAEL SHORT: Differentiate.

810
00:41:10,280 --> 00:41:11,580
Sure.

811
00:41:11,580 --> 00:41:16,418
Let's take, we'll just say
the derivative of-- oh, no,

812
00:41:16,418 --> 00:41:17,960
it is a partial
derivative because we

813
00:41:17,960 --> 00:41:19,080
have two variables here.

814
00:41:22,010 --> 00:41:25,670
Take the derivative
as a function of z,

815
00:41:25,670 --> 00:41:27,140
set it equal to 0.

816
00:41:27,140 --> 00:41:29,960
This will give us the
z number that gives us

817
00:41:29,960 --> 00:41:32,690
the minimum mass for a
fixed A. So let's actually

818
00:41:32,690 --> 00:41:34,850
do this right now.

819
00:41:34,850 --> 00:41:36,140
So let's see.

820
00:41:36,140 --> 00:41:38,870
This gives us mh.

821
00:41:38,870 --> 00:41:46,880
And this term expanded
out is AMN minus ZMN.

822
00:41:46,880 --> 00:41:49,240
So we have a minus MN.

823
00:41:49,240 --> 00:41:51,410
I'm going to make one quick
correction right here.

824
00:41:51,410 --> 00:41:54,140
I want to make sure
everything's in the same units.

825
00:41:54,140 --> 00:41:58,570
All of these semi-empirical
mass terms are in MeV.

826
00:41:58,570 --> 00:42:01,960
These right here are in
AMU, atomic mass units.

827
00:42:01,960 --> 00:42:07,344
What do we have to add in
order to get these all in MeV?

828
00:42:07,344 --> 00:42:11,405
AUDIENCE: [INAUDIBLE]
the conversion factor.

829
00:42:11,405 --> 00:42:13,030
MICHAEL SHORT:
Conversion factor, yeah.

830
00:42:13,030 --> 00:42:16,640
Or in this case, we'll
just stick a C squared.

831
00:42:16,640 --> 00:42:18,390
I'll make a little
bit of room so I can

832
00:42:18,390 --> 00:42:21,510
stick the C squared in there.

833
00:42:21,510 --> 00:42:22,660
C squared.

834
00:42:22,660 --> 00:42:24,120
Now everything's in MeV.

835
00:42:24,120 --> 00:42:26,140
We're all in the same units.

836
00:42:26,140 --> 00:42:31,140
So let's say we
have mh minus mn.

837
00:42:31,140 --> 00:42:33,240
These are in C squared.

838
00:42:33,240 --> 00:42:48,750
And minus VC plus 2/3 AS times
A to the negative 1/3 plus--

839
00:42:48,750 --> 00:42:54,240
I'm going to expand this out
to call it z squared minus z.

840
00:42:54,240 --> 00:43:00,270
I'm also going to stick
in n equals A minus Z

841
00:43:00,270 --> 00:43:06,420
so that this expands out
to A minus 2Z squared.

842
00:43:06,420 --> 00:43:07,980
Is everyone with me here?

843
00:43:07,980 --> 00:43:08,630
Yeah.

844
00:43:08,630 --> 00:43:11,673
AUDIENCE: Shouldn't the A
terms, like the AVA and the AS,

845
00:43:11,673 --> 00:43:12,400
stay at 2/3?

846
00:43:12,400 --> 00:43:13,798
MICHAEL SHORT: Oh.

847
00:43:13,798 --> 00:43:14,340
You're right.

848
00:43:14,340 --> 00:43:18,000
I'm deriving with respect
to the wrong variable.

849
00:43:18,000 --> 00:43:19,270
Thank you.

850
00:43:19,270 --> 00:43:19,770
Yep.

851
00:43:19,770 --> 00:43:21,840
So we want to do this
as a function of z.

852
00:43:21,840 --> 00:43:25,260
So that term disappears.

853
00:43:25,260 --> 00:43:27,630
That term disappears.

854
00:43:27,630 --> 00:43:28,752
Thank you.

855
00:43:28,752 --> 00:43:30,210
Let's work on these
ones right now.

856
00:43:30,210 --> 00:43:34,590
So we have AC time z
squared over A to the 1/3.

857
00:43:34,590 --> 00:43:43,260
So that will give us A plus
AC over A to the 1/3 times 2z.

858
00:43:46,830 --> 00:43:50,090
And then we have--

859
00:43:50,090 --> 00:43:54,960
let's see- minus AC
over A to the 1/3.

860
00:43:57,900 --> 00:44:00,270
That's it, actually.

861
00:44:00,270 --> 00:44:03,600
Oh, times plus 1, OK?

862
00:44:03,600 --> 00:44:05,943
And then we have the A minus 2z.

863
00:44:05,943 --> 00:44:08,610
So let's expand this out just so
we can see it all on the board.

864
00:44:08,610 --> 00:44:15,750
So we have AA over A
times A minus 4z squared

865
00:44:15,750 --> 00:44:24,810
plus 4 minus 4 AZ
plus 4z squared.

866
00:44:24,810 --> 00:44:28,170
So let's take the derivative
with respect to z of that.

867
00:44:28,170 --> 00:44:30,900
That term goes away.

868
00:44:30,900 --> 00:44:33,840
That becomes 4A.

869
00:44:33,840 --> 00:44:39,720
So we have plus AA
over A times 4A.

870
00:44:39,720 --> 00:44:44,830
And there's a minus sign there.

871
00:44:44,830 --> 00:44:46,580
And that becomes 8z.

872
00:44:49,470 --> 00:44:56,070
So we have plus AA
symmetry over A times 8z.

873
00:44:56,070 --> 00:44:58,470
And the delta term goes
away because there's

874
00:44:58,470 --> 00:45:00,090
no z dependence.

875
00:45:00,090 --> 00:45:03,690
And what we end up here is the
solution for what is the most

876
00:45:03,690 --> 00:45:06,120
stable z as a function of A?

877
00:45:06,120 --> 00:45:08,040
This is a linear equation.

878
00:45:08,040 --> 00:45:09,840
There's only one
solution for it.

879
00:45:09,840 --> 00:45:14,640
If we actually want to graph
this m as a function of A

880
00:45:14,640 --> 00:45:23,700
and Z, we end up with what's
called a mass parabola, which

881
00:45:23,700 --> 00:45:26,700
is to say you can graph the
binding energy per nucleon,

882
00:45:26,700 --> 00:45:31,500
or the mass, or pretty much
similar things, of a nucleus,

883
00:45:31,500 --> 00:45:35,640
of all nuclei with A
given as a function of z.

884
00:45:35,640 --> 00:45:38,670
Think I can do this on the
remaining space right here.

885
00:45:38,670 --> 00:45:44,940
So let's say 4A
equals 93 if this is Z

886
00:45:44,940 --> 00:45:49,170
and this is m as a function
of A and Z. Let's actually

887
00:45:49,170 --> 00:45:51,100
look at a concrete example.

888
00:45:51,100 --> 00:45:54,450
So let's go live to
the chart of nuclides

889
00:45:54,450 --> 00:45:57,735
and start looking at things
with a mass number of 93.

890
00:46:00,380 --> 00:46:02,070
Looks like I clicked
a little too high.

891
00:46:05,000 --> 00:46:06,780
There it is.

892
00:46:06,780 --> 00:46:07,610
Let's see.

893
00:46:07,610 --> 00:46:11,620
Moly 93 I was
looking at, and that

894
00:46:11,620 --> 00:46:16,790
becomes niobium 93, which is the
stable isotope I was thinking.

895
00:46:16,790 --> 00:46:20,050
So let's put niobium right here.

896
00:46:20,050 --> 00:46:22,000
I haven't given an
actual scale to this

897
00:46:22,000 --> 00:46:24,800
because I just want to show
you in sort of relative terms.

898
00:46:24,800 --> 00:46:27,710
So let's say niobium
is the stable one.

899
00:46:27,710 --> 00:46:30,370
So it's going to have the lowest
actual mass, even though it's

900
00:46:30,370 --> 00:46:32,650
got an A number of 93.

901
00:46:32,650 --> 00:46:35,650
If you look here on
the chart of nuclides,

902
00:46:35,650 --> 00:46:38,080
you can see it could
have come from a couple

903
00:46:38,080 --> 00:46:41,110
of different places,
either from zirconium 93

904
00:46:41,110 --> 00:46:43,120
or from molybdenum 93.

905
00:46:43,120 --> 00:46:44,950
And now is a good time
to start introducing

906
00:46:44,950 --> 00:46:49,190
these different modes of decay
so you can figure out, well,

907
00:46:49,190 --> 00:46:52,810
how would a nucleus decay to
get to the most stable place?

908
00:46:52,810 --> 00:46:58,240
Let's say it came from
zirconium 93 and--

909
00:46:58,240 --> 00:47:02,580
let's see, miobium has
a proton number of 41.

910
00:47:02,580 --> 00:47:05,670
So if we go to
zirconium, it beta

911
00:47:05,670 --> 00:47:11,310
decays to niobium 93 with
an energy of .091 MeV.

912
00:47:11,310 --> 00:47:13,560
Very, very close.

913
00:47:13,560 --> 00:47:16,710
So we'll draw it
slightly higher.

914
00:47:16,710 --> 00:47:19,560
That's about 91 keV.

915
00:47:19,560 --> 00:47:24,300
Zirconium 93 could have come
from the beta decay of yttrium

916
00:47:24,300 --> 00:47:27,250
93 you can see right here.

917
00:47:27,250 --> 00:47:30,840
So let's go up the mass
parabola and keep exploring.

918
00:47:30,840 --> 00:47:37,590
And now we see that yttrium can
decay by beta decay with 3 MeV.

919
00:47:37,590 --> 00:47:43,860
So if we put yttrium on this
graph, it would be way higher.

920
00:47:43,860 --> 00:47:46,960
Yttrium itself could
have come from--

921
00:47:46,960 --> 00:47:50,410
well, let's see, strontium
93 with the decay

922
00:47:50,410 --> 00:47:52,720
energy of 4.1 MeV.

923
00:47:52,720 --> 00:47:55,930
So let's put strontium here.

924
00:47:55,930 --> 00:48:00,190
And I think 4 MEV would
be, like, off the chart.

925
00:48:00,190 --> 00:48:00,880
But whatever.

926
00:48:00,880 --> 00:48:02,170
That's the way we drew it.

927
00:48:02,170 --> 00:48:05,140
Already we've got the
makings of a parabola.

928
00:48:05,140 --> 00:48:10,010
And each one of these
can decay by beta decay,

929
00:48:10,010 --> 00:48:11,950
or does decay by
beta decay, in order

930
00:48:11,950 --> 00:48:15,640
to get to the most
stable nucleus.

931
00:48:15,640 --> 00:48:17,350
So let's write the
nuclear reaction

932
00:48:17,350 --> 00:48:20,470
for beta decay of one of these,
let's say, from zirconium

933
00:48:20,470 --> 00:48:21,970
to niobium.

934
00:48:21,970 --> 00:48:27,610
So we'd have 9340
zirconium spontaneously

935
00:48:27,610 --> 00:48:36,730
goes to 9341 niobium
plus a beta and

936
00:48:36,730 --> 00:48:39,332
plus an electron anti-neutrino.

937
00:48:39,332 --> 00:48:41,290
That's the part I don't
expect you to know yet.

938
00:48:41,290 --> 00:48:44,590
But that's the whole
energy conservation thing.

939
00:48:44,590 --> 00:48:46,330
A little bit of a flash forward.

940
00:48:46,330 --> 00:48:49,012
The beta decay energy
is not necessarily

941
00:48:49,012 --> 00:48:50,470
the energy of the
electron that you

942
00:48:50,470 --> 00:48:52,137
will measure because
some of that energy

943
00:48:52,137 --> 00:48:54,700
is taken away by
the anti-neutrino.

944
00:48:54,700 --> 00:48:58,640
But we'll get into how those
relate probably next week.

945
00:48:58,640 --> 00:49:01,090
So let's now look on the
other side of the parabola

946
00:49:01,090 --> 00:49:03,790
and confirm that the
semi-empirical mass

947
00:49:03,790 --> 00:49:05,680
formula, which
predicts something

948
00:49:05,680 --> 00:49:08,920
parabolic here and
here with respect to z,

949
00:49:08,920 --> 00:49:11,750
actually checks out.

950
00:49:11,750 --> 00:49:16,640
So let's back up to
niobium 93 and notice

951
00:49:16,640 --> 00:49:20,370
that it could also have
come from electron capture

952
00:49:20,370 --> 00:49:22,650
from molybdenum 93.

953
00:49:22,650 --> 00:49:26,160
So let's put
molybdenum right here.

954
00:49:26,160 --> 00:49:33,420
And it decays with
an energy of 0.4 MeV

955
00:49:33,420 --> 00:49:38,650
into niobium, which let's
say it's around here.

956
00:49:38,650 --> 00:49:40,270
Let's keep going
through the chain.

957
00:49:40,270 --> 00:49:43,950
Anyone have any questions
so far while we keep going?

958
00:49:43,950 --> 00:49:44,930
Cool.

959
00:49:44,930 --> 00:49:47,160
Let's trace it
back up the chain.

960
00:49:47,160 --> 00:49:50,060
Technetium 93 can
beget molybdenum 93

961
00:49:50,060 --> 00:49:55,917
by electron capture with a
much higher energy, 3.201.

962
00:49:55,917 --> 00:49:58,250
I'm going to extend our graph
because we need the space.

963
00:50:01,220 --> 00:50:05,280
Technetium, another 3 MeV.

964
00:50:05,280 --> 00:50:08,630
And let's go one more back.

965
00:50:08,630 --> 00:50:11,950
Technetium can be made by
electron capture from rubidium

966
00:50:11,950 --> 00:50:16,293
93 with an even
higher energy, which

967
00:50:16,293 --> 00:50:18,460
means a higher difference
in mass between these two.

968
00:50:18,460 --> 00:50:24,820
So so far-- let's see, what's
6 MeV for rubidium here?

969
00:50:24,820 --> 00:50:27,670
It would be like there, I guess.

970
00:50:30,280 --> 00:50:33,492
There's our mass parabola,
right from the data.

971
00:50:33,492 --> 00:50:35,950
So I like doing this better
than just showing you a diagram

972
00:50:35,950 --> 00:50:38,200
because you can actually
try it for yourself.

973
00:50:38,200 --> 00:50:41,470
Pick a fixed A, change
A, and construct

974
00:50:41,470 --> 00:50:43,540
the mass parabolas yourself.

975
00:50:43,540 --> 00:50:47,800
Now the question is how could
these decay into niobium 93,

976
00:50:47,800 --> 00:50:49,730
which is the stable isotope?

977
00:50:49,730 --> 00:50:51,340
I have negative
one minutes left,

978
00:50:51,340 --> 00:50:55,150
so I'm very quickly going
to tell you for large energy

979
00:50:55,150 --> 00:50:58,760
changes, it can either be
positron decay or electron

980
00:50:58,760 --> 00:50:59,260
capture.

981
00:50:59,260 --> 00:51:03,770
And we'll go over what these
modes of decay are next week.

982
00:51:03,770 --> 00:51:08,440
This can be, again, positron
or electron capture.

983
00:51:08,440 --> 00:51:11,800
And for small amounts
of decay energy,

984
00:51:11,800 --> 00:51:17,740
it can only be electron capture
because in order for positron

985
00:51:17,740 --> 00:51:22,060
decay to happen, you have to
be able to create the positron.

986
00:51:22,060 --> 00:51:24,460
And the positron plus
the extra electron

987
00:51:24,460 --> 00:51:32,040
ejected to balance charge
has got to be 1.022 MeV,

988
00:51:32,040 --> 00:51:35,490
or same thing as what's
known as two times the rest

989
00:51:35,490 --> 00:51:38,700
mass of the electron.

990
00:51:38,700 --> 00:51:40,500
I'm going to stop
there, and we'll

991
00:51:40,500 --> 00:51:43,920
pick up with lots of examples
and questions tomorrow.

992
00:51:43,920 --> 00:51:46,500
The last thing-- well, I'll
go over the next problem

993
00:51:46,500 --> 00:51:47,167
set tomorrow.

994
00:51:47,167 --> 00:51:48,750
I want to make sure
everyone's seen it

995
00:51:48,750 --> 00:51:50,460
seven days before it's due.

996
00:51:50,460 --> 00:51:53,540
And the best way to do that
is to show it on the board.