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MICHAEL SHORT: So since
I know series decay is

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a difficult topic to jump into,
I wanted to quickly re-go over

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the derivation today and
then specifically go over

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00:00:30,900 --> 00:00:34,110
the case of nuclear activation
analysis, which reminds me,

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did you guys bring in your
skin flakes and food pieces?

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00:00:37,680 --> 00:00:38,222
We have time.

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So if you didn't remember,
start thinking about what you

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00:00:40,555 --> 00:00:41,933
want to bring in, what you got.

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00:00:41,933 --> 00:00:42,760
AUDIENCE: Aluminum foil.

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00:00:42,760 --> 00:00:43,890
MICHAEL SHORT: OK, so
you've got aluminum foil.

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You want to see what in it
is not aluminum-- excellent.

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00:00:46,410 --> 00:00:48,102
Well, what else
did folks bring in?

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00:00:48,102 --> 00:00:49,102
AUDIENCE: [INAUDIBLE]

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00:00:49,102 --> 00:00:50,560
MICHAEL SHORT: OK,
rubber stopper--

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00:00:50,560 --> 00:00:52,140
sound perfect.

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00:00:52,140 --> 00:00:54,040
Anyone else bring something in?

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00:00:54,040 --> 00:00:55,920
AUDIENCE: [INAUDIBLE]

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00:00:55,920 --> 00:00:59,970
MICHAEL SHORT: OK, so tell you
what, when you bring stuff in,

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00:00:59,970 --> 00:01:01,742
bring it in a little
plastic baggie.

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00:01:01,742 --> 00:01:03,450
I can supply those if
you don't have them

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00:01:03,450 --> 00:01:05,580
with your name on them just so
we know whose samples are what

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00:01:05,580 --> 00:01:07,290
because that's going to be
the basis for another one

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00:01:07,290 --> 00:01:08,530
of your homeworks
where are you going

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00:01:08,530 --> 00:01:10,350
to use the stuff that
we're learning today

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00:01:10,350 --> 00:01:14,870
to determine which impurities
and how much are in whatever

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00:01:14,870 --> 00:01:16,290
thing that you looked at.

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00:01:16,290 --> 00:01:18,180
And, of course, you're not
going to get all the impurities

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00:01:18,180 --> 00:01:19,555
because in order
to do that, we'd

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00:01:19,555 --> 00:01:22,170
have to do a long nuclear
activation analysis,

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00:01:22,170 --> 00:01:24,873
irradiate for days, and
count for a longer time.

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00:01:24,873 --> 00:01:26,790
So you'll just be
responsible for the isotopes

39
00:01:26,790 --> 00:01:29,390
on the shortlist, which we've
posted on the learning module

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00:01:29,390 --> 00:01:31,410
site.

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00:01:31,410 --> 00:01:33,840
So again, bring
in your whatever,

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00:01:33,840 --> 00:01:36,480
as long as it's not hair
because, apparently, that's

43
00:01:36,480 --> 00:01:40,350
a pain to deal with or salty
because the sodium activates

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00:01:40,350 --> 00:01:43,800
like crazy or fissionable, which
you shouldn't have, anyway.

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00:01:43,800 --> 00:01:46,890
I hope none of you have
fissionable material at home.

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00:01:46,890 --> 00:01:52,230
So let's get back
into series decay.

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00:01:52,230 --> 00:01:56,430
We very quickly went over
the definition of activity

48
00:01:56,430 --> 00:01:59,460
which is just the decay constant
times the amount of stuff

49
00:01:59,460 --> 00:02:01,200
that there is, the
decay constants,

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00:02:01,200 --> 00:02:04,160
and units of 1 over second.

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00:02:04,160 --> 00:02:06,600
The amount of
stuff-- let's call it

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00:02:06,600 --> 00:02:12,780
a number density-- could be like
an atoms per centimeter cubed,

53
00:02:12,780 --> 00:02:13,970
for example.

54
00:02:13,970 --> 00:02:17,670
So the activity would
give you the amount of,

55
00:02:17,670 --> 00:02:22,845
let's say decays, per
centimeter cubed per second.

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00:02:22,845 --> 00:02:25,470
If you wanted to do this for an
absolute amount of a substance,

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00:02:25,470 --> 00:02:28,020
like you knew how much of
the substance there was,

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00:02:28,020 --> 00:02:30,180
you just ditched the volume.

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00:02:30,180 --> 00:02:33,360
And you end up with the
activity in decays per second.

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00:02:33,360 --> 00:02:37,260
That unit is better known
as becquerels or BQ,

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00:02:37,260 --> 00:02:39,180
named after Henri
Becquerel, though I don't

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know if I'm saying that right.

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00:02:40,080 --> 00:02:41,700
But my wife's
probably going to yell

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00:02:41,700 --> 00:02:43,860
at me when she sees this video.

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00:02:43,860 --> 00:02:45,090
But so becquerel is simple.

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00:02:45,090 --> 00:02:47,550
It's simply 1 decay
per second, and there's

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00:02:47,550 --> 00:02:51,180
another unit called the curie,
which is just a whole lot more

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00:02:51,180 --> 00:02:52,080
decays per second.

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00:02:52,080 --> 00:02:54,690
It's a more manageable
unit of the case

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00:02:54,690 --> 00:02:57,630
because becquerel-- the activity
of many things in becquerels

71
00:02:57,630 --> 00:03:00,540
tends to be in the millions
or billions or trillions

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00:03:00,540 --> 00:03:03,290
or much, much more for something
that's really radioactive.

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00:03:03,290 --> 00:03:05,040
And it gets annoying
writing all the zeros

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00:03:05,040 --> 00:03:07,130
or all the scientific notation.

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00:03:07,130 --> 00:03:10,470
And so last time we looked
at a simple situation--

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let's say you have
some isotope N1 which

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00:03:13,800 --> 00:03:16,620
decays with the k
constant lambda 1

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00:03:16,620 --> 00:03:23,970
to isotope N2, which decays with
the k constant lambda 2 and N3.

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00:03:23,970 --> 00:03:26,130
And we decided to
set up our equations

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00:03:26,130 --> 00:03:29,040
in the form of change.

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00:03:29,040 --> 00:03:32,220
Everyone is just a change
equals a production

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00:03:32,220 --> 00:03:38,130
minus a destruction
for all cases.

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00:03:38,130 --> 00:03:40,240
So let's forget the
activation part.

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00:03:40,240 --> 00:03:42,240
For now, we're just going
to assume that we have

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00:03:42,240 --> 00:03:45,060
some amount of isotope, N1.

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00:03:45,060 --> 00:03:50,790
We'll say we have
N1 0 at t equals 0.

87
00:03:50,790 --> 00:03:53,910
And it decays to
N2 and decays N3.

88
00:03:53,910 --> 00:03:55,830
So what are the
differential equations

89
00:03:55,830 --> 00:04:00,520
describing the rate of change
of each of these isotopes?

90
00:04:08,270 --> 00:04:10,825
So how about N1?

91
00:04:10,825 --> 00:04:12,200
Is there any method
of production

92
00:04:12,200 --> 00:04:15,220
of isotope N1 in this scenario?

93
00:04:15,220 --> 00:04:17,860
No, we just started
off with some N1,

94
00:04:17,860 --> 00:04:22,048
but we do have destruction
of N1 via radioactive decay.

95
00:04:22,048 --> 00:04:23,590
And so the amount
of changes is going

96
00:04:23,590 --> 00:04:26,230
to be equal to
negative the activity.

97
00:04:26,230 --> 00:04:29,800
So for every decay of
N1, we lose an N1 atom.

98
00:04:29,800 --> 00:04:33,850
So we just put
minus lambda 1 N1.

99
00:04:33,850 --> 00:04:37,810
For every N1 atom that
decays, it produces an N2.

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00:04:37,810 --> 00:04:44,020
So N2 has an equal but
different sign production term

101
00:04:44,020 --> 00:04:46,030
and has a similar
looking destruction term.

102
00:04:49,470 --> 00:04:54,000
Meanwhile, since
N2 becomes N3, then

103
00:04:54,000 --> 00:04:57,103
we just have this
simple term right there,

104
00:04:57,103 --> 00:04:58,770
and these are the
differential equations

105
00:04:58,770 --> 00:05:01,390
which we want to solve.

106
00:05:01,390 --> 00:05:04,750
We knew from last time that
the solution to this equation

107
00:05:04,750 --> 00:05:07,665
is pretty simple.

108
00:05:07,665 --> 00:05:09,540
I'm not going to re-go
through the derivation

109
00:05:09,540 --> 00:05:13,290
there since I think that's
kind of an easy one.

110
00:05:13,290 --> 00:05:16,355
And N3, we know
is pretty simple.

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00:05:16,355 --> 00:05:17,730
We used the
conservation equation

112
00:05:17,730 --> 00:05:21,660
to say that the total amount
of all atoms in the system

113
00:05:21,660 --> 00:05:25,830
has to be equal to N10 or N10.

114
00:05:25,830 --> 00:05:31,170
So we know we have N10
equals and N1 plus N2

115
00:05:31,170 --> 00:05:33,930
plus N3 for all time.

116
00:05:33,930 --> 00:05:37,170
So we don't really have to
solve for N3 because we can just

117
00:05:37,170 --> 00:05:39,020
deal with it later.

118
00:05:39,020 --> 00:05:40,610
The last thing that
we need to derive

119
00:05:40,610 --> 00:05:42,960
is what is the solution to N2.

120
00:05:42,960 --> 00:05:44,570
And I want to correct
a mistake that I

121
00:05:44,570 --> 00:05:47,360
made because I'm going to
chalk that up to exhaustion

122
00:05:47,360 --> 00:05:49,310
assuming that integrating
factor was zero.

123
00:05:49,310 --> 00:05:53,240
It's not zero, so I want to
show you why it's not now.

124
00:05:53,240 --> 00:05:55,220
So how do we go
about solving this?

125
00:05:55,220 --> 00:05:56,270
What method did we use?

126
00:05:59,360 --> 00:06:01,130
We chose the integration
factor method

127
00:06:01,130 --> 00:06:03,080
because it's a nice clean one.

128
00:06:03,080 --> 00:06:06,500
So we rewrite this equation
in terms of let's just

129
00:06:06,500 --> 00:06:07,550
say N2 prime--

130
00:06:11,460 --> 00:06:18,570
I'm sorry-- plus lambda
2 N2 minus lambda 1 N1.

131
00:06:18,570 --> 00:06:22,080
And we don't necessarily
want to have an N1 in there

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00:06:22,080 --> 00:06:24,330
because we want to
have one variable only.

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00:06:24,330 --> 00:06:28,080
So instead of N1, we can
substitute this whole thing

134
00:06:28,080 --> 00:06:29,830
in there.

135
00:06:29,830 --> 00:06:36,250
So N10 e to the minus
lambda 1t equals zero.

136
00:06:36,250 --> 00:06:39,670
And let's just draw a
little thing around here

137
00:06:39,670 --> 00:06:41,710
to help visually separate.

138
00:06:41,710 --> 00:06:45,460
We know how to solve this
type of differential equation

139
00:06:45,460 --> 00:06:48,340
because we can define
some integrating factor mu

140
00:06:48,340 --> 00:06:52,690
equals e to the minus whatever
is in front of the N2.

141
00:06:52,690 --> 00:06:54,460
That's not too hard.

142
00:06:54,460 --> 00:06:57,100
I'm sorry, just e
to the integral, not

143
00:06:57,100 --> 00:07:01,870
minus, of lambda 2 dt.

144
00:07:01,870 --> 00:07:05,610
We're just equal to
just e lambda 2t.

145
00:07:05,610 --> 00:07:08,290
And we multiply every
term in this equation

146
00:07:08,290 --> 00:07:12,760
by mu because we're going to
make sure that the stuff here--

147
00:07:12,760 --> 00:07:16,840
after we multiply
by mu and mu and nu

148
00:07:16,840 --> 00:07:19,900
and mu for completeness--

149
00:07:19,900 --> 00:07:22,060
that stuff in here should
be something that looks

150
00:07:22,060 --> 00:07:23,470
like the end of a product rule.

151
00:07:27,273 --> 00:07:28,690
So if we multiply
that through, we

152
00:07:28,690 --> 00:07:35,350
get N2 prime e to
the lambda 2t plus,

153
00:07:35,350 --> 00:07:39,670
let's see, mu times
lambda 2 e to the lambda

154
00:07:39,670 --> 00:07:49,360
2t times n2 minus e to
the lambda 2t lambda 1 n10

155
00:07:49,360 --> 00:07:54,640
e to the minus lambda
1t equals zero.

156
00:07:54,640 --> 00:07:57,670
And indeed, we've
got right here what

157
00:07:57,670 --> 00:08:00,430
looks like the end result
of the product rule

158
00:08:00,430 --> 00:08:03,610
where we have something,
let's say, one function times

159
00:08:03,610 --> 00:08:06,130
the derivative of another
plus the derivative

160
00:08:06,130 --> 00:08:09,310
of that function times the
original other function.

161
00:08:12,390 --> 00:08:17,720
So to compact that up, we
can call that, let's say, N2

162
00:08:17,720 --> 00:08:20,780
e to the lambda t--

163
00:08:20,780 --> 00:08:25,980
sorry, lambda 2t prime minus--

164
00:08:25,980 --> 00:08:30,920
and I'm going to combine these
two exponents right here.

165
00:08:30,920 --> 00:08:39,950
So we'll have minus
lambda 1 and 10e to the--

166
00:08:39,950 --> 00:08:49,400
let's see, is it lambda 2
minus lambda 1t equals zero.

167
00:08:49,400 --> 00:08:51,920
Just going to take this
term to the other side

168
00:08:51,920 --> 00:08:57,625
of the equals sign, so I'll just
do that, integrate both sides.

169
00:09:01,200 --> 00:09:09,550
And we get N2 e to
the lambda 2t equals--

170
00:09:09,550 --> 00:09:17,920
let's see, that'll be lambda 1
N10 over lambda 2 minus lambda

171
00:09:17,920 --> 00:09:20,375
1 times all that stuff.

172
00:09:28,050 --> 00:09:31,010
I'm going to divide each
side of the equation by--

173
00:09:31,010 --> 00:09:34,740
I'll use a different color
for that intermediate step--

174
00:09:34,740 --> 00:09:35,850
e to the lambda 2t.

175
00:09:40,760 --> 00:09:43,510
And that cancels these out.

176
00:09:43,510 --> 00:09:46,570
That cancels these out.

177
00:09:46,570 --> 00:09:49,520
And I forgot that integrating
factor again, didn't I?

178
00:09:49,520 --> 00:09:54,700
Yeah, so there's going to
be a plus C somewhere here.

179
00:09:54,700 --> 00:10:00,360
And we're just going to
absorb this e to the lambda 2t

180
00:10:00,360 --> 00:10:02,840
into this integrating
constant because it's

181
00:10:02,840 --> 00:10:03,840
an integrating constant.

182
00:10:03,840 --> 00:10:05,280
We haven't defined it yet.

183
00:10:05,280 --> 00:10:08,380
Did someone have a
question I thought I saw?

184
00:10:08,380 --> 00:10:14,650
OK, and so now this is
where I went wrong last time

185
00:10:14,650 --> 00:10:17,410
because I think I was exhausted
and commuted in from Columbus.

186
00:10:17,410 --> 00:10:19,630
I just assumed right
away that C equals zero,

187
00:10:19,630 --> 00:10:21,190
but it's not the case.

188
00:10:21,190 --> 00:10:27,010
So if we plug-in the
condition at t equals 0--

189
00:10:27,010 --> 00:10:29,723
and two should equal 0--

190
00:10:29,723 --> 00:10:30,640
let's see what we get.

191
00:10:35,050 --> 00:10:38,400
That would become a zero.

192
00:10:38,400 --> 00:10:42,270
That t would be a zero,
which means that we just

193
00:10:42,270 --> 00:10:48,570
end up with the
equation lambda 1 N10

194
00:10:48,570 --> 00:10:53,790
over lambda 2 minus
lambda 1 plus c equals

195
00:10:53,790 --> 00:10:57,430
zero so obviously the
integration constant

196
00:10:57,430 --> 00:11:01,050
is not like we thought it was.

197
00:11:06,590 --> 00:11:14,590
So then C equals
negative that stuff.

198
00:11:19,590 --> 00:11:20,980
That make more sense.

199
00:11:20,980 --> 00:11:23,490
So you guys see why the
integrating constants not zero.

200
00:11:23,490 --> 00:11:24,348
So in the end--

201
00:11:24,348 --> 00:11:26,640
I'm going t5o skip ahead a
little of the math because I

202
00:11:26,640 --> 00:11:28,560
want to get into nuclear
activation analysis--

203
00:11:28,560 --> 00:11:33,360
we end up with N2
should equal, let's see,

204
00:11:33,360 --> 00:11:40,760
lambda 1 N10 over lambda
1 minus lambda 1 times--

205
00:11:40,760 --> 00:11:43,450
did I write that twice?

206
00:11:43,450 --> 00:11:44,180
I think I did--

207
00:11:48,560 --> 00:11:52,410
times e to the--

208
00:11:52,410 --> 00:11:55,860
let's see, e to the
minus lambda 1t minus e

209
00:11:55,860 --> 00:11:59,780
to the minus lambda 2t.

210
00:11:59,780 --> 00:12:03,910
And so since we know
N1, we've found N10.

211
00:12:03,910 --> 00:12:06,640
We know N3 from this
conservation equation.

212
00:12:06,640 --> 00:12:08,260
We've now fully
determined what is

213
00:12:08,260 --> 00:12:10,780
the concentration
of every isotope

214
00:12:10,780 --> 00:12:13,000
in this system for all time.

215
00:12:13,000 --> 00:12:16,150
And because the solution to
this is not that intuitive--

216
00:12:16,150 --> 00:12:18,900
like I can't picture what the
function looks like in my head.

217
00:12:18,900 --> 00:12:20,440
I don't know about you guys.

218
00:12:20,440 --> 00:12:21,700
Anyone?

219
00:12:21,700 --> 00:12:22,200
No?

220
00:12:22,200 --> 00:12:24,770
OK, I can't either.

221
00:12:24,770 --> 00:12:28,340
I coded them up in this handy
graphing calculator where

222
00:12:28,340 --> 00:12:32,420
you can play around with the eye
of the concentration N10, which

223
00:12:32,420 --> 00:12:34,610
is just a multiplier
for everything,

224
00:12:34,610 --> 00:12:39,083
and the relative half lives
lambda 1 and lambda 2.

225
00:12:39,083 --> 00:12:40,500
And I'll share
this with you guys,

226
00:12:40,500 --> 00:12:43,810
so you can actually see
generally how this works.

227
00:12:43,810 --> 00:12:47,580
So let's start looking
at a couple of cases--

228
00:12:47,580 --> 00:12:50,560
move this a little over
so we can see the axes.

229
00:12:50,560 --> 00:12:53,100
Let's say don't worry about
anything before T equals zero.

230
00:12:53,100 --> 00:12:55,360
That's kind of an invalid
part of the solution.

231
00:12:55,360 --> 00:12:59,320
So I'll just shrink
us over there.

232
00:12:59,320 --> 00:13:01,530
And so I've coded up all
three of these equations.

233
00:13:01,530 --> 00:13:04,410
There is the solution to
N1 highlighted right there.

234
00:13:04,410 --> 00:13:07,420
That's as you'd expect
simple exponential decay.

235
00:13:07,420 --> 00:13:09,660
All N1 knows is
that it's decaying

236
00:13:09,660 --> 00:13:13,890
according to its own half life
or exponential decay equation.

237
00:13:13,890 --> 00:13:18,970
And two, here in the blue,
which expands, of course,

238
00:13:18,970 --> 00:13:21,370
looks a little more complicated.

239
00:13:21,370 --> 00:13:26,280
So what we notice here is
that N2 is tied directly

240
00:13:26,280 --> 00:13:30,090
to the slope of N1.

241
00:13:30,090 --> 00:13:32,130
That should follow
pretty intuitively

242
00:13:32,130 --> 00:13:34,650
from the differential
equations because if you

243
00:13:34,650 --> 00:13:38,520
look at the slope of N2,
well, it depends directly

244
00:13:38,520 --> 00:13:40,410
on the value of N1.

245
00:13:40,410 --> 00:13:43,380
For very, very
short times, this is

246
00:13:43,380 --> 00:13:46,110
the sort of limiting behavior in
the and the graphical guidance.

247
00:13:46,110 --> 00:13:47,943
I want to give you two
solve questions like,

248
00:13:47,943 --> 00:13:51,690
what's on the exam or how to do
a nuclear activation analysis.

249
00:13:51,690 --> 00:13:54,570
Is everyone comfortable with me
hiding this board right here?

250
00:13:54,570 --> 00:13:55,070
OK.

251
00:13:58,110 --> 00:14:03,180
So let's say at time
is approximately zero,

252
00:14:03,180 --> 00:14:09,570
we know that there's going to be
N1 is going to equal about N10.

253
00:14:09,570 --> 00:14:11,580
What's the value of N2
going to be very, very

254
00:14:11,580 --> 00:14:13,574
close to 2 equals 0?

255
00:14:13,574 --> 00:14:14,460
AUDIENCE: Zero.

256
00:14:14,460 --> 00:14:15,780
MICHAEL SHORT: Zero.

257
00:14:15,780 --> 00:14:19,800
So N2 is going to equal 0.

258
00:14:19,800 --> 00:14:22,080
But what's the slope
of N2 going to be?

259
00:14:28,387 --> 00:14:29,970
This is how we can
get started solving

260
00:14:29,970 --> 00:14:31,920
these graphically
without even knowing

261
00:14:31,920 --> 00:14:32,930
what the real forms are.

262
00:14:35,810 --> 00:14:38,810
So we've already said
that at a very short time,

263
00:14:38,810 --> 00:14:40,640
N2 is approximately 0.

264
00:14:40,640 --> 00:14:43,670
So if that's zero,
then that whole term

265
00:14:43,670 --> 00:14:46,520
is zero, which means
that the slope of N2

266
00:14:46,520 --> 00:14:52,970
is approximately lambda 1
N1, just the activity of N1.

267
00:14:52,970 --> 00:14:55,137
And hopefully, that
follows intuitively

268
00:14:55,137 --> 00:14:56,720
because it says for
really short times

269
00:14:56,720 --> 00:15:04,820
before you get any
buildup, the slope of N1

270
00:15:04,820 --> 00:15:07,520
determines the value of N2.

271
00:15:07,520 --> 00:15:10,460
So if we were to
start graphing these--

272
00:15:10,460 --> 00:15:13,490
let's just start looking at some
limiting behavior-- that's t,

273
00:15:13,490 --> 00:15:17,575
and we're going to need
some colors for this.

274
00:15:17,575 --> 00:15:19,200
Let's stick with the
ones on the board.

275
00:15:19,200 --> 00:15:22,720
Oh, hey, awesome.

276
00:15:22,720 --> 00:15:30,180
Make N1 red, N2
blue, and N3 green.

277
00:15:30,180 --> 00:15:33,240
So let's start drawing
some limiting behavior.

278
00:15:33,240 --> 00:15:37,430
So we know that N1
starts here at N10.

279
00:15:37,430 --> 00:15:41,280
And we know it's going to
start decaying exponentially.

280
00:15:41,280 --> 00:15:46,200
So the slope here
is just going to be

281
00:15:46,200 --> 00:15:51,740
minus lambda 1 and N1,
which is going to be

282
00:15:51,740 --> 00:15:56,840
the negative slope of N2--

283
00:15:56,840 --> 00:15:59,840
looks pretty
similar, doesn't it?

284
00:15:59,840 --> 00:16:02,930
So we know N2 for
very short times

285
00:16:02,930 --> 00:16:07,720
is going to start growing at the
same rate that N1 is shrinking.

286
00:16:14,442 --> 00:16:16,150
So we already know
what sort of direction

287
00:16:16,150 --> 00:16:18,428
these curves are
starting to go in.

288
00:16:18,428 --> 00:16:18,970
How about N3?

289
00:16:21,740 --> 00:16:24,215
What's the value of N3
for very short times?

290
00:16:28,230 --> 00:16:29,240
Anyone call it out.

291
00:16:36,520 --> 00:16:38,820
Well, we've got kind of
a solution right here.

292
00:16:38,820 --> 00:16:43,150
If we know that N2 is about
zero for very short times,

293
00:16:43,150 --> 00:16:46,020
what would the value
of n' three have to be?

294
00:16:46,020 --> 00:16:48,510
Also zero.

295
00:16:48,510 --> 00:16:50,220
And what about the slope of N3?

296
00:16:55,320 --> 00:16:57,300
Also, about zero.

297
00:16:57,300 --> 00:17:02,510
If there's no N2 built up, then
there's nothing to create N3.

298
00:17:02,510 --> 00:17:04,310
So we know that our
end three curve is

299
00:17:04,310 --> 00:17:07,910
going to start out pretty flat.

300
00:17:12,390 --> 00:17:15,490
Now how do we find some
other limiting behavior?

301
00:17:15,490 --> 00:17:18,270
Let's now take the case--

302
00:17:18,270 --> 00:17:20,665
let's see, I want to rewrite
that a little closer here,

303
00:17:20,665 --> 00:17:21,540
so we have some room.

304
00:17:26,680 --> 00:17:30,110
So that's at t
equals about zero.

305
00:17:30,110 --> 00:17:37,340
And at t equals infinity,
what sort of limiting behavior

306
00:17:37,340 --> 00:17:40,190
do you think we'll have?

307
00:17:40,190 --> 00:17:44,880
What's the value of N1 going
to be at infinite time?

308
00:17:44,880 --> 00:17:48,350
Zero-- it will have
all decayed away,

309
00:17:48,350 --> 00:17:50,840
will equal zero at
t equals infinity.

310
00:17:50,840 --> 00:17:52,050
How about N2?

311
00:17:52,050 --> 00:17:52,910
AUDIENCE: Zero.

312
00:17:52,910 --> 00:17:54,920
MICHAEL SHORT: Zero.

313
00:17:54,920 --> 00:17:56,205
How about N3?

314
00:17:56,205 --> 00:17:57,540
AUDIENCE: [INAUDIBLE]

315
00:17:57,540 --> 00:18:00,230
MICHAEL SHORT: N10-- correct.

316
00:18:00,230 --> 00:18:03,000
Because of that conservation
equation right here.

317
00:18:03,000 --> 00:18:08,450
So we know for limiting cases,
N1 one is going to be 0.

318
00:18:08,450 --> 00:18:11,180
N2 is going to be 0.

319
00:18:11,180 --> 00:18:14,900
And N3, it's going to be N0.

320
00:18:14,900 --> 00:18:17,900
So we've now filled in all
the four corners of the graph

321
00:18:17,900 --> 00:18:22,025
just intuitively without solving
the differential equations.

322
00:18:22,025 --> 00:18:23,900
Now let's start to fill
in some middle parts.

323
00:18:26,520 --> 00:18:28,860
What other sorts
of things can we

324
00:18:28,860 --> 00:18:33,780
determine, like, for example,
where H2 has a maximum?

325
00:18:33,780 --> 00:18:36,720
That shouldn't be too hard.

326
00:18:36,720 --> 00:18:39,300
So let's make another
separation here.

327
00:18:39,300 --> 00:18:48,687
So what if we want to find out
when does the N2 dt equal 0?

328
00:18:48,687 --> 00:18:49,520
What do we do there?

329
00:18:58,490 --> 00:19:01,430
Anyone have an idea?

330
00:19:01,430 --> 00:19:03,820
Using the equations
we have up here.

331
00:19:03,820 --> 00:19:06,497
AUDIENCE: [INAUDIBLE]

332
00:19:06,497 --> 00:19:08,080
MICHAEL SHORT: Yeah,
well, we can just

333
00:19:08,080 --> 00:19:09,370
take this equation right here.

334
00:19:09,370 --> 00:19:11,680
We can figure that
out in terms of N1.

335
00:19:11,680 --> 00:19:14,200
So if D and 2D
equals zero, then we

336
00:19:14,200 --> 00:19:20,560
know that lambda 1 N1 is
going to equal lambda 2 N2.

337
00:19:20,560 --> 00:19:23,470
What this says intuitively is
that the rate of production

338
00:19:23,470 --> 00:19:28,510
of N2 by decaying on 1 equals
the rate of destruction of N2

339
00:19:28,510 --> 00:19:30,190
by its own decay.

340
00:19:30,190 --> 00:19:37,650
So at some point, the N2 is
going to have to level off.

341
00:19:37,650 --> 00:19:40,920
When that point is depends
on the relative differences

342
00:19:40,920 --> 00:19:42,960
between those half lives.

343
00:19:42,960 --> 00:19:46,290
So we already know if we were
to just kind of fill in smoothly

344
00:19:46,290 --> 00:19:48,750
what's going to
happen, N2 is probably

345
00:19:48,750 --> 00:19:53,910
going to follow something
roughly looking like this.

346
00:19:53,910 --> 00:19:55,620
We already know
the solution to N1.

347
00:19:55,620 --> 00:19:58,050
I think we can figure
that out graphically.

348
00:19:58,050 --> 00:19:59,535
It's simply exponential decay.

349
00:20:04,340 --> 00:20:07,100
The only trick now is
how does N3 shape up?

350
00:20:10,140 --> 00:20:11,320
What do you guys think?

351
00:20:11,320 --> 00:20:13,590
How would we go
about graphically

352
00:20:13,590 --> 00:20:17,730
plotting these solutions
without solving them?

353
00:20:17,730 --> 00:20:20,820
I don't think yet I've
given you the full form.

354
00:20:20,820 --> 00:20:24,190
It's kind of ugly, and I
doubt that if you looked at it

355
00:20:24,190 --> 00:20:26,880
you'd be able to tell me
exactly what it would do.

356
00:20:26,880 --> 00:20:29,220
So this is just the
mathematical expression

357
00:20:29,220 --> 00:20:31,590
of N10 minus N1 minus N2.

358
00:20:34,800 --> 00:20:37,790
So how do we figure out
all the stuff about N3?

359
00:20:44,650 --> 00:20:45,150
Yeah.

360
00:20:45,150 --> 00:20:47,340
AUDIENCE: You could
just draw a curve

361
00:20:47,340 --> 00:20:50,760
so that you get all
three curves and always

362
00:20:50,760 --> 00:20:52,017
add it to the same number.

363
00:20:52,017 --> 00:20:52,850
MICHAEL SHORT: Yeah.

364
00:20:52,850 --> 00:20:54,797
AUDIENCE: [INAUDIBLE]

365
00:20:54,797 --> 00:20:56,880
MICHAEL SHORT: Absolutely,
that's totally correct.

366
00:20:56,880 --> 00:21:00,630
Yeah, if you just take
N10 minus N1 and N2,

367
00:21:00,630 --> 00:21:02,520
that gives you the value of N3.

368
00:21:02,520 --> 00:21:05,160
That's completely correct.

369
00:21:05,160 --> 00:21:07,410
So you could do that sort
of one point at a time

370
00:21:07,410 --> 00:21:11,028
and say, well, maybe around
there, maybe around there.

371
00:21:11,028 --> 00:21:12,570
It might take a
little while, though.

372
00:21:12,570 --> 00:21:14,570
So I want to think what's
another intuitive way?

373
00:21:17,390 --> 00:21:20,070
What would the value or
the slope of N3 track--

374
00:21:20,070 --> 00:21:22,700
what other variable
in the system?

375
00:21:22,700 --> 00:21:26,995
Or in other words, how
are they directly related?

376
00:21:26,995 --> 00:21:27,998
Yeah.

377
00:21:27,998 --> 00:21:31,407
AUDIENCE: The slope
of n2 are they equal?

378
00:21:31,407 --> 00:21:32,240
MICHAEL SHORT: Yeah.

379
00:21:32,240 --> 00:21:34,970
The slope of n3 depends
directly on the value

380
00:21:34,970 --> 00:21:37,040
of n2 and nothing else.

381
00:21:37,040 --> 00:21:43,640
So initially, you can see
the value of n2 is almost 0,

382
00:21:43,640 --> 00:21:46,790
so the slope of n3 is almost 0.

383
00:21:46,790 --> 00:21:52,610
As the value of n2 picks up,
so should the slope of n3

384
00:21:52,610 --> 00:21:55,510
until we reach here.

385
00:21:55,510 --> 00:21:57,112
What happens at that point?

386
00:21:57,112 --> 00:21:59,633
AUDIENCE: N2 decreases.

387
00:21:59,633 --> 00:22:00,425
MICHAEL SHORT: Yep.

388
00:22:00,425 --> 00:22:03,740
The rate of production of n2
decreases because the val--

389
00:22:03,740 --> 00:22:06,410
I'm sorry-- yeah, the
rate of production of n3

390
00:22:06,410 --> 00:22:09,200
decreases because the
rate of production of n3

391
00:22:09,200 --> 00:22:12,650
is just dependent directly
on the value of n2.

392
00:22:12,650 --> 00:22:15,895
So the maximum slope
of n3 has to be

393
00:22:15,895 --> 00:22:17,270
right there at
which point it has

394
00:22:17,270 --> 00:22:23,850
to start leveling off and
eventually reaching 0.

395
00:22:23,850 --> 00:22:26,670
You're going to see this kind
of problem on the homework.

396
00:22:26,670 --> 00:22:29,580
You're going to see this
kind of problem on the exam.

397
00:22:29,580 --> 00:22:30,860
I guarantee you.

398
00:22:30,860 --> 00:22:32,990
But it's not going to
have this exact form.

399
00:22:32,990 --> 00:22:34,650
But what I'll want
you to be able to do

400
00:22:34,650 --> 00:22:36,810
is follow this example.

401
00:22:36,810 --> 00:22:40,140
Let's say I pose you a small
set of these first order

402
00:22:40,140 --> 00:22:41,760
differential equations.

403
00:22:41,760 --> 00:22:45,120
Can you use any method that you
want-- intuitive, graphical,

404
00:22:45,120 --> 00:22:46,260
mathematical--

405
00:22:46,260 --> 00:22:50,250
to predict what the values
and slopes of these isotopes

406
00:22:50,250 --> 00:22:52,410
are going to be as
a function of time?

407
00:22:52,410 --> 00:22:54,600
So in order to get nuclear
activation analysis right

408
00:22:54,600 --> 00:22:56,580
you need to be able to do this.

409
00:22:56,580 --> 00:23:02,730
In nuclear activation
analysis it's just one twist.

410
00:23:02,730 --> 00:23:04,605
I'm going to move this
over to add the twist.

411
00:23:08,350 --> 00:23:13,390
You're also producing isotope
n1 with a reaction rate.

412
00:23:17,200 --> 00:23:22,120
By some either isotope and not
what you put in the reactor.

413
00:23:22,120 --> 00:23:25,630
So if you want to know what
your impurities and naught were

414
00:23:25,630 --> 00:23:28,900
to undergo what's called
nuclear activation analysis,

415
00:23:28,900 --> 00:23:31,360
then you can figure out,
depending on which one you

416
00:23:31,360 --> 00:23:34,700
count, what they could be.

417
00:23:34,700 --> 00:23:35,740
So this right here.

418
00:23:35,740 --> 00:23:39,823
Let's look at the units of
this versus the units of this.

419
00:23:39,823 --> 00:23:41,240
First of all, if
we're adding them

420
00:23:41,240 --> 00:23:44,220
together they'd better be
in the same units, right?

421
00:23:44,220 --> 00:23:48,060
So we already talked about the
units of this decay equation.

422
00:23:48,060 --> 00:23:51,113
It's like number of
decays per second.

423
00:23:51,113 --> 00:23:52,530
So this reaction
right here better

424
00:23:52,530 --> 00:23:55,770
give us a number of atoms
produced per second,

425
00:23:55,770 --> 00:23:58,380
or we're kind of
messed up in the units.

426
00:23:58,380 --> 00:24:01,740
So anyone remember
what is the units of--

427
00:24:01,740 --> 00:24:04,680
I'll make a little
extra piece right here--

428
00:24:04,680 --> 00:24:08,640
what's the units of microscopic
cross sections or barns?

429
00:24:08,640 --> 00:24:10,190
What is that in some
sort of SI unit?

430
00:24:10,190 --> 00:24:13,190
AUDIENCE: Centimeter squared.

431
00:24:13,190 --> 00:24:14,070
MICHAEL SHORT: Yep.

432
00:24:14,070 --> 00:24:16,870
It's like a centimeter squared.

433
00:24:16,870 --> 00:24:18,030
And what about flux?

434
00:24:23,030 --> 00:24:25,940
This one you may not
know, but it definitely

435
00:24:25,940 --> 00:24:27,470
depends on the
number of neutrons

436
00:24:27,470 --> 00:24:30,022
or the number of
particles that are there.

437
00:24:30,022 --> 00:24:32,943
AUDIENCE: Is it
barns [INAUDIBLE]

438
00:24:32,943 --> 00:24:33,860
MICHAEL SHORT: Almost.

439
00:24:33,860 --> 00:24:35,930
So the flux describes
how many particles

440
00:24:35,930 --> 00:24:38,940
pass through a surface
in a given time.

441
00:24:38,940 --> 00:24:41,180
So we have how many
particles per unit

442
00:24:41,180 --> 00:24:46,010
surface, per unit time.

443
00:24:46,010 --> 00:24:49,070
Ends up being neutrons per
centimeter squared per second.

444
00:24:49,070 --> 00:24:51,770
Just like the flux of
photons through a space

445
00:24:51,770 --> 00:24:53,950
or the flux of any
particle through anywhere,

446
00:24:53,950 --> 00:24:56,000
it describes how
many particles go

447
00:24:56,000 --> 00:24:58,400
through a space
in a certain time.

448
00:24:58,400 --> 00:25:02,400
And then there's the number
of particles that are there.

449
00:25:02,400 --> 00:25:06,740
If we're going with
atoms, it's just atoms.

450
00:25:06,740 --> 00:25:08,660
These are all
multiplied together.

451
00:25:08,660 --> 00:25:11,270
The centimeter squared cancel.

452
00:25:11,270 --> 00:25:19,360
And we end up with some sort
of a atoms per second produced.

453
00:25:19,360 --> 00:25:21,930
We can put in a little hidden
unit in the cross section.

454
00:25:21,930 --> 00:25:27,180
If there is a reaction going
on where in goes a neutron

455
00:25:27,180 --> 00:25:31,610
and out goes an
atom or something,

456
00:25:31,610 --> 00:25:33,680
that should cancel
all things out.

457
00:25:33,680 --> 00:25:35,340
Let's not get into that now.

458
00:25:35,340 --> 00:25:37,340
The whole point is we
have the same sort of unit

459
00:25:37,340 --> 00:25:40,190
going on here, which is some
number of atoms produced

460
00:25:40,190 --> 00:25:41,610
per second.

461
00:25:41,610 --> 00:25:45,553
Same thing as number of
atoms decayed per second.

462
00:25:45,553 --> 00:25:46,970
So it's the
production-destruction

463
00:25:46,970 --> 00:25:49,180
equivalent of each other.

464
00:25:49,180 --> 00:25:53,400
So, in that way, we can have a
reaction rate that we impose,

465
00:25:53,400 --> 00:25:55,020
something artificial,
by sticking

466
00:25:55,020 --> 00:25:58,320
something in the reactor and
controlling its power level.

467
00:25:58,320 --> 00:26:00,630
And then follow
the decay process

468
00:26:00,630 --> 00:26:03,920
which is a natural
radioactivity event.

469
00:26:03,920 --> 00:26:06,890
And this is one of
the simplest governing

470
00:26:06,890 --> 00:26:09,740
equations for nuclear
activation analysis.

471
00:26:09,740 --> 00:26:12,290
Now, one, I might give
this to you on an exam

472
00:26:12,290 --> 00:26:14,870
and say OK now draw the
curves for nuclear activation

473
00:26:14,870 --> 00:26:16,030
analysis.

474
00:26:16,030 --> 00:26:18,530
And maybe calculate what's the
impurity level if you measure

475
00:26:18,530 --> 00:26:20,110
this many counts of something.

476
00:26:20,110 --> 00:26:22,350
Then you just work
backward through the math.

477
00:26:22,350 --> 00:26:24,350
But I want to get you
guys thinking conceptually

478
00:26:24,350 --> 00:26:25,880
right now.

479
00:26:25,880 --> 00:26:29,300
What are the real equations for
nuclear activation analysis?

480
00:26:34,880 --> 00:26:38,600
Let's just do these in
terms of n1, n2, n3.

481
00:26:45,586 --> 00:26:48,490
That's dt, d, and 3dt.

482
00:26:51,525 --> 00:26:52,900
We'll start with
the stuff that's

483
00:26:52,900 --> 00:26:56,140
up there minus lambda 1 n1.

484
00:26:56,140 --> 00:27:00,740
Plus some cross section times
the flux times some other atom

485
00:27:00,740 --> 00:27:01,240
n0.

486
00:27:03,910 --> 00:27:05,410
What other things
are we missing?

487
00:27:05,410 --> 00:27:08,110
Are there any other methods
of production or destruction

488
00:27:08,110 --> 00:27:10,270
of isotope n1 that
we need to consider?

489
00:27:17,460 --> 00:27:20,300
Well, we've got isotope
n1 in a reactor.

490
00:27:20,300 --> 00:27:25,430
It can decay, or it can absorb
one of the neutrons nearby.

491
00:27:25,430 --> 00:27:27,830
So how do we write that
term-- that destruction term?

492
00:27:31,950 --> 00:27:32,450
Yep.

493
00:27:32,450 --> 00:27:37,116
AUDIENCE: Flux times the
absorption cross section

494
00:27:37,116 --> 00:27:38,847
times n0.

495
00:27:38,847 --> 00:27:39,680
MICHAEL SHORT: Yeah.

496
00:27:39,680 --> 00:27:42,440
So let's say that's the
absorption of atom 1 times

497
00:27:42,440 --> 00:27:43,722
n what did you say?

498
00:27:43,722 --> 00:27:44,430
AUDIENCE: Naught.

499
00:27:44,430 --> 00:27:48,200
MICHAEL SHORT: And would
it be n0 or would it be n1?

500
00:27:48,200 --> 00:27:50,508
If you want to know how
quick is n1 being destroyed--

501
00:27:50,508 --> 00:27:51,050
AUDIENCE: OK.

502
00:27:51,050 --> 00:27:53,280
MICHAEL SHORT: --By
absorbing neutrons.

503
00:27:53,280 --> 00:27:57,630
So then let's call
this absorption of n0.

504
00:27:57,630 --> 00:27:59,830
And is it a plus or a minus?

505
00:27:59,830 --> 00:28:01,205
If it's a destruction rate.

506
00:28:01,205 --> 00:28:02,080
AUDIENCE: [INAUDIBLE]

507
00:28:02,080 --> 00:28:03,247
MICHAEL SHORT: It's a minus.

508
00:28:03,247 --> 00:28:03,870
Yep.

509
00:28:03,870 --> 00:28:06,270
So what's really
going on here is

510
00:28:06,270 --> 00:28:08,430
you've got some precursor
isotope, whatever

511
00:28:08,430 --> 00:28:12,810
impurity you want to
measure n0, producing n1.

512
00:28:12,810 --> 00:28:16,560
And you're looking at n1's decay
signature, like its activity,

513
00:28:16,560 --> 00:28:17,910
to determine how much was there.

514
00:28:17,910 --> 00:28:20,340
But you also have to
account for the fact

515
00:28:20,340 --> 00:28:23,550
that isotope n1 can be
burned in the reactor.

516
00:28:23,550 --> 00:28:25,020
So this is like producing.

517
00:28:27,750 --> 00:28:29,850
This is decaying.

518
00:28:29,850 --> 00:28:34,630
And this is we'll
call it being burned.

519
00:28:34,630 --> 00:28:37,510
This isn't burned in the
sense of creating of fuel--

520
00:28:37,510 --> 00:28:39,660
creating energy
by burning fuel--

521
00:28:39,660 --> 00:28:42,310
but we will refer
to this sort of in--

522
00:28:42,310 --> 00:28:44,710
colloquially too burning--
because we're then

523
00:28:44,710 --> 00:28:48,610
absorbing neutrons by
n1 and removing that

524
00:28:48,610 --> 00:28:51,250
from the available
decay signature.

525
00:28:51,250 --> 00:28:52,030
How about n2?

526
00:28:52,030 --> 00:28:54,640
How do we modify our
equation to account correctly

527
00:28:54,640 --> 00:28:57,610
for the production
and destruction of n2?

528
00:29:01,555 --> 00:29:03,180
And by the way this
is not in the book,

529
00:29:03,180 --> 00:29:06,434
so I don't expect you to know
it off the top of your head.

530
00:29:06,434 --> 00:29:08,588
AUDIENCE: It's the same
type of thing, flux.

531
00:29:08,588 --> 00:29:10,630
MICHAEL SHORT: Yep So
let's first take every term

532
00:29:10,630 --> 00:29:12,190
that we have up there.

533
00:29:12,190 --> 00:29:17,630
We have lambda 1 n1
minus lambda 2 n2.

534
00:29:17,630 --> 00:29:19,370
And what else do we
have to account for?

535
00:29:23,710 --> 00:29:24,340
Yep.

536
00:29:24,340 --> 00:29:25,780
AUDIENCE: N2 also being burnt.

537
00:29:25,780 --> 00:29:26,290
MICHAEL SHORT: That's right.

538
00:29:26,290 --> 00:29:27,940
So n2 is also being
burned so we'll

539
00:29:27,940 --> 00:29:35,500
have a minus, a flux times the
absorption cross section for n2

540
00:29:35,500 --> 00:29:36,760
times the amount of n2.

541
00:29:39,520 --> 00:29:40,380
How about n3?

542
00:29:40,380 --> 00:29:43,880
We'll start with
what we had there.

543
00:29:43,880 --> 00:29:46,760
Lambda 2 n2.

544
00:29:46,760 --> 00:29:49,100
And, just like before,
we've got to account

545
00:29:49,100 --> 00:29:51,810
for the burning of n3.

546
00:29:51,810 --> 00:29:55,810
So then we'll have minus
flux times the absorption

547
00:29:55,810 --> 00:29:59,530
cross section of 3 times n3.

548
00:29:59,530 --> 00:30:02,830
And these equations hold
true only for the time

549
00:30:02,830 --> 00:30:06,030
that your material
is in the reactor.

550
00:30:06,030 --> 00:30:10,228
What happens when you take the
material out of the reactor?

551
00:30:10,228 --> 00:30:12,180
AUDIENCE: You go right
back to zero readings.

552
00:30:12,180 --> 00:30:12,630
MICHAEL SHORT: You do.

553
00:30:12,630 --> 00:30:13,130
Yep.

554
00:30:13,130 --> 00:30:19,075
When you come out of the reactor
all of the fluxes go to zero.

555
00:30:19,075 --> 00:30:20,200
And that's the end of that.

556
00:30:22,820 --> 00:30:23,405
Yeah.

557
00:30:23,405 --> 00:30:26,030
AUDIENCE: Why don't you account
for the production of n2 and n3

558
00:30:26,030 --> 00:30:30,330
in the burn rate?

559
00:30:30,330 --> 00:30:33,037
MICHAEL SHORT: Ah, so did
I necessarily specify--

560
00:30:33,037 --> 00:30:35,370
the question was why don't
we account for the production

561
00:30:35,370 --> 00:30:37,590
of n2 and n3 by the burning.

562
00:30:37,590 --> 00:30:38,835
Right?

563
00:30:38,835 --> 00:30:39,460
AUDIENCE: Yeah.

564
00:30:39,460 --> 00:30:42,610
MICHAEL SHORT: Did we specify
that absorbing a neutron

565
00:30:42,610 --> 00:30:44,498
is the way to make n2?

566
00:30:44,498 --> 00:30:45,040
AUDIENCE: No.

567
00:30:45,040 --> 00:30:46,930
MICHAEL SHORT:
Oftentimes it's not.

568
00:30:46,930 --> 00:30:51,730
So if you burn n1 by
absorbing a neutron,

569
00:30:51,730 --> 00:30:53,780
then you will make
another isotope

570
00:30:53,780 --> 00:30:56,890
that has the same proton
number and one more neutron.

571
00:30:56,890 --> 00:30:59,680
And it may decay by some other
crazy way, or it may be stable.

572
00:30:59,680 --> 00:31:00,790
Who knows.

573
00:31:00,790 --> 00:31:07,090
But by decay-- this could
be by beta, positron, alpha,

574
00:31:07,090 --> 00:31:08,362
spontaneous fission--

575
00:31:08,362 --> 00:31:09,820
not gamma because
then you wouldn't

576
00:31:09,820 --> 00:31:12,170
have a different isotope.

577
00:31:12,170 --> 00:31:13,970
But oftentimes you won't have--

578
00:31:13,970 --> 00:31:17,180
the burning process won't
produce the same isotopes

579
00:31:17,180 --> 00:31:18,650
as the decay.

580
00:31:18,650 --> 00:31:21,320
So the situation we looked at
on Friday when we said let's

581
00:31:21,320 --> 00:31:22,590
escalate things.

582
00:31:22,590 --> 00:31:24,410
That was a purely
hypothetical situation

583
00:31:24,410 --> 00:31:27,220
where isotope n2 could
be burned to make n0.

584
00:31:27,220 --> 00:31:30,470
I'm not saying it can't
happen, but it's not likely.

585
00:31:30,470 --> 00:31:31,850
But still we can model it.

586
00:31:31,850 --> 00:31:33,595
We can model anything.

587
00:31:33,595 --> 00:31:35,220
That just wasn't a
realistic situation.

588
00:31:35,220 --> 00:31:36,540
This is.

589
00:31:36,540 --> 00:31:38,790
This is what you guys are
going to have to look at

590
00:31:38,790 --> 00:31:40,470
to understand how
much impurities there

591
00:31:40,470 --> 00:31:42,180
are in each of your materials.

592
00:31:46,300 --> 00:31:50,630
So this I would say is
the complete description

593
00:31:50,630 --> 00:31:53,535
of nuclear activation
analysis in the reactor.

594
00:31:53,535 --> 00:31:55,160
At which point you
then have to account

595
00:31:55,160 --> 00:31:58,510
for what happens when
you turn the reactor off.

596
00:32:01,882 --> 00:32:02,590
So what actually?

597
00:32:02,590 --> 00:32:04,840
What physically happens when
you turn the reactor off?

598
00:32:10,020 --> 00:32:10,520
Yep.

599
00:32:10,520 --> 00:32:11,093
Oh Yeah.

600
00:32:11,093 --> 00:32:12,010
You've answered a lot.

601
00:32:12,010 --> 00:32:12,840
So Chris, yeah.

602
00:32:12,840 --> 00:32:15,946
CHRIS: Well, you try to--
you put your control rods all

603
00:32:15,946 --> 00:32:17,810
the way in and try to
stop as many neutrons

604
00:32:17,810 --> 00:32:20,300
as you can to stop
the chain reaction.

605
00:32:20,300 --> 00:32:21,460
MICHAEL SHORT: Yep.

606
00:32:21,460 --> 00:32:23,120
So normally to shut
down the reactor

607
00:32:23,120 --> 00:32:25,690
you'd put the control rods
in and shut down the reactor.

608
00:32:25,690 --> 00:32:28,163
Or the easier thing is
just pull the rabbit out.

609
00:32:28,163 --> 00:32:29,830
Remember those little
polyethylene tubes

610
00:32:29,830 --> 00:32:31,000
I showed you?

611
00:32:31,000 --> 00:32:34,120
This way we can keep the reactor
on and remove your samples

612
00:32:34,120 --> 00:32:35,240
without changing anything.

613
00:32:35,240 --> 00:32:37,630
So it makes the reactor folks--

614
00:32:37,630 --> 00:32:39,520
angry would be an
understatement--

615
00:32:39,520 --> 00:32:43,660
to constantly change the
power level of the reactor.

616
00:32:43,660 --> 00:32:47,120
Reactors, especially power
reactors and research reactors,

617
00:32:47,120 --> 00:32:49,270
they're kind of
like Mack trucks.

618
00:32:49,270 --> 00:32:51,598
If they're moving they
want to stay moving,

619
00:32:51,598 --> 00:32:53,890
and if they're not moving
they don't want to be moving.

620
00:32:53,890 --> 00:32:57,040
And it takes an awful lot
of effort to change that.

621
00:32:57,040 --> 00:32:59,710
It also happens to
screw up experiments.

622
00:32:59,710 --> 00:33:03,550
If you are irradiating something
like I was a couple months ago

623
00:33:03,550 --> 00:33:06,340
for 30 days, you want
to have a constant flux

624
00:33:06,340 --> 00:33:08,020
so that your
calculations are easy.

625
00:33:08,020 --> 00:33:10,810
You don't want 15 students
to come in and turn the knobs

626
00:33:10,810 --> 00:33:12,250
all up and down,
and then you have

627
00:33:12,250 --> 00:33:14,920
to account for
that in your data.

628
00:33:14,920 --> 00:33:16,090
Which has happened.

629
00:33:16,090 --> 00:33:18,640
So you guys are going to
be manipulating the reactor

630
00:33:18,640 --> 00:33:22,060
power when the experiments are
out, and it's at low power.

631
00:33:22,060 --> 00:33:24,700
So you're won't be infuriating
anyone else on campus

632
00:33:24,700 --> 00:33:26,790
like we did last year.

633
00:33:26,790 --> 00:33:29,290
So if you didn't account for
that, but they still let us in.

634
00:33:29,290 --> 00:33:30,190
So they're bad.

635
00:33:30,190 --> 00:33:30,870
Whatever.

636
00:33:30,870 --> 00:33:31,570
[LAUGHING]

637
00:33:31,570 --> 00:33:32,350
Yeah.

638
00:33:32,350 --> 00:33:35,050
So after you either
shut down the reactor

639
00:33:35,050 --> 00:33:37,750
or pull the rabbit out
of the reactor then

640
00:33:37,750 --> 00:33:40,390
the production and destruction
by neutrons is over

641
00:33:40,390 --> 00:33:42,460
but the decay keeps going.

642
00:33:42,460 --> 00:33:44,890
Which means if you wait
too long, like for some

643
00:33:44,890 --> 00:33:48,170
of those short isotopes, if
you wait more than a day or so,

644
00:33:48,170 --> 00:33:50,080
you'll have so
little activity left

645
00:33:50,080 --> 00:33:52,160
that you won't be
able to measure it.

646
00:33:52,160 --> 00:33:54,880
So what we're going to be
doing is sticking your samples

647
00:33:54,880 --> 00:33:57,730
into the reactor for
maybe an hour or so,

648
00:33:57,730 --> 00:33:59,920
pulling them out, and
immediately running them over

649
00:33:59,920 --> 00:34:03,498
to the detector, so that we get
the most signal per unit time.

650
00:34:03,498 --> 00:34:06,040
Because the things are going to
be the hottest when they come

651
00:34:06,040 --> 00:34:09,219
right out of the reactor,
and every second you lose

652
00:34:09,219 --> 00:34:11,020
from there you lose signal.

653
00:34:11,020 --> 00:34:13,090
Which means you have
to account for longer

654
00:34:13,090 --> 00:34:14,980
to get the same
amount of information

655
00:34:14,980 --> 00:34:16,397
with the same certainty.

656
00:34:16,397 --> 00:34:17,980
This is a nice segue
way to what we'll

657
00:34:17,980 --> 00:34:21,130
be talking about Thursday
which is statistics, certainty,

658
00:34:21,130 --> 00:34:22,270
and precision.

659
00:34:22,270 --> 00:34:24,639
How long do you have
to count something

660
00:34:24,639 --> 00:34:27,760
to be confident
within some interval

661
00:34:27,760 --> 00:34:30,000
that you've got the
correct activity?

662
00:34:30,000 --> 00:34:34,730
For background counts-- who here
is made a NSC Geiger counter?

663
00:34:34,730 --> 00:34:36,080
Hopefully almost all of you.

664
00:34:36,080 --> 00:34:37,580
Maybe you guys
remember how long you

665
00:34:37,580 --> 00:34:40,980
had to count to be 95% sure
that your background rate was

666
00:34:40,980 --> 00:34:42,870
accurate.

667
00:34:42,870 --> 00:34:46,510
It ends up being about 67
minutes or over an hour,

668
00:34:46,510 --> 00:34:49,040
and the reason is because
the count rate is very low.

669
00:34:49,040 --> 00:34:51,460
So I'll do a little
flash forward to Thursday

670
00:34:51,460 --> 00:34:53,380
since we're talking about it.

671
00:34:53,380 --> 00:34:56,080
When you count something
with a very low count rate,

672
00:34:56,080 --> 00:34:58,750
you have to account for
longer to be as confident

673
00:34:58,750 --> 00:35:00,470
that your number is correct.

674
00:35:00,470 --> 00:35:03,460
So let's say you want
to be 95% confident

675
00:35:03,460 --> 00:35:07,120
or within plus or minus 2
standard deviations or 2 sigma.

676
00:35:07,120 --> 00:35:09,340
You have to count for
longer and longer.

677
00:35:09,340 --> 00:35:10,990
For something that's
really radioactive

678
00:35:10,990 --> 00:35:14,590
you can be sure, or 95%
sure, that the count

679
00:35:14,590 --> 00:35:18,010
rate you measured is accurate
for a shorter counting time.

680
00:35:18,010 --> 00:35:20,480
So everything in this class
seems to come up in trade-offs.

681
00:35:20,480 --> 00:35:20,980
Right?

682
00:35:20,980 --> 00:35:23,530
You trade off stability
for a half-life.

683
00:35:23,530 --> 00:35:26,200
You trade off decay
constant for half-life.

684
00:35:26,200 --> 00:35:29,220
You trade off binding
energy for excess mass.

685
00:35:29,220 --> 00:35:32,880
You trade off counting
time and precision.

686
00:35:32,880 --> 00:35:34,830
You trade off exposure
and dose. which

687
00:35:34,830 --> 00:35:36,550
you're going to get into later.

688
00:35:36,550 --> 00:35:38,710
We'll see if anyone
wants to use a cell phone

689
00:35:38,710 --> 00:35:40,690
or eat irradiated
food afterwards.

690
00:35:40,690 --> 00:35:44,470
And I do all the time, so that
should tell you the answer.

691
00:35:44,470 --> 00:35:47,172
So in the last
seven minutes or so,

692
00:35:47,172 --> 00:35:48,880
I want to walk you
through playing around

693
00:35:48,880 --> 00:35:51,670
with what happens when you
change the values of lambda

694
00:35:51,670 --> 00:35:53,710
1 and lambda 2?

695
00:35:53,710 --> 00:35:57,010
So what do they look like
when the half-lives are

696
00:35:57,010 --> 00:36:01,160
roughly equal and when one is
much larger than the other one?

697
00:36:01,160 --> 00:36:03,010
So let's set them
to be about equal.

698
00:36:03,010 --> 00:36:04,270
These are just unitless.

699
00:36:04,270 --> 00:36:06,670
So let's set them equal to one.

700
00:36:06,670 --> 00:36:10,750
I think the system explodes
when we set them exactly equal

701
00:36:10,750 --> 00:36:14,700
because that term right there.

702
00:36:14,700 --> 00:36:18,250
So let's say that's 1.001.

703
00:36:18,250 --> 00:36:20,500
It's about as close
as we can get,

704
00:36:20,500 --> 00:36:23,610
and let's confirm that we get
the same sort of behavior.

705
00:36:23,610 --> 00:36:26,910
So isotope n1 just
follows exponential decay.

706
00:36:26,910 --> 00:36:28,890
There's nothing
that changes that.

707
00:36:28,890 --> 00:36:34,710
Isotope 2, its slope tracks
the value of isotope 1

708
00:36:34,710 --> 00:36:37,920
for a little while until
you build up enough n2

709
00:36:37,920 --> 00:36:39,690
that it starts to decay.

710
00:36:39,690 --> 00:36:42,270
You can find when
that point is when

711
00:36:42,270 --> 00:36:46,470
lambda 1 n1 equals lambda 2 n2.

712
00:36:46,470 --> 00:36:48,930
There's one little step that
we didn't fill in if you

713
00:36:48,930 --> 00:36:51,148
want to find the value of n2.

714
00:36:51,148 --> 00:36:53,190
So then you can just
rearrange this a little bit,

715
00:36:53,190 --> 00:36:56,100
and you'll say n2 would
have to equal lambda

716
00:36:56,100 --> 00:37:00,940
1 over lambda 2 times n1.

717
00:37:00,940 --> 00:37:06,960
Which is n1 0 e to
the minus lambda 1 t.

718
00:37:06,960 --> 00:37:11,400
So if you want to find that
point right there in time,

719
00:37:11,400 --> 00:37:14,730
you can solve this.

720
00:37:14,730 --> 00:37:17,130
Then let's look at n3.

721
00:37:17,130 --> 00:37:22,440
So n3 when n2 is almost
0, n3 slope is almost 0.

722
00:37:22,440 --> 00:37:24,555
It's a little hard
to see because--

723
00:37:24,555 --> 00:37:27,180
I'll tell you what-- let's make
all the half lives longer which

724
00:37:27,180 --> 00:37:29,640
kind of expands the graph.

725
00:37:29,640 --> 00:37:30,390
Wrong way.

726
00:37:33,890 --> 00:37:35,492
Let's make it--

727
00:37:35,492 --> 00:37:42,650
Ah, we'll just move
that decimal point 0.1.

728
00:37:42,650 --> 00:37:43,357
There we go.

729
00:37:43,357 --> 00:37:44,690
That's like expanding the graph.

730
00:37:44,690 --> 00:37:45,190
Right?

731
00:37:45,190 --> 00:37:50,120
So when n2 is almost 0, the
slope of n3 is almost 0.

732
00:37:50,120 --> 00:37:55,100
And when n2 reaches a maximum,
so does the slope of n3.

733
00:37:55,100 --> 00:38:02,090
Just like we predicted using
our graphical method right here.

734
00:38:02,090 --> 00:38:03,440
And then over longer times--

735
00:38:03,440 --> 00:38:05,870
let's put the half lives
back to the way they were.

736
00:38:09,210 --> 00:38:13,778
Over long times
n3 trends to n1 0.

737
00:38:13,778 --> 00:38:15,320
Don't let that little
piece fool you.

738
00:38:15,320 --> 00:38:19,120
Again t equals less than 0
is not a valid time for this,

739
00:38:19,120 --> 00:38:20,780
so we're not
accounting for that.

740
00:38:20,780 --> 00:38:25,790
And n3 tracks right to
here to the value of n1 0,

741
00:38:25,790 --> 00:38:28,250
and n2 and n3 turn to 0.

742
00:38:28,250 --> 00:38:31,700
So for this case where you
have the half-lives roughly

743
00:38:31,700 --> 00:38:35,770
equal to each other, you can
expect a pretty big bump in n2.

744
00:38:35,770 --> 00:38:40,750
What's going to happen when
lambda 1 is extraordinarily

745
00:38:40,750 --> 00:38:45,026
big meaning the half-life of
n1 is extraordinarily short.

746
00:38:45,026 --> 00:38:46,484
What do you guys
think will happen?

747
00:38:51,790 --> 00:38:55,350
Not mathematically
but physically.

748
00:38:55,350 --> 00:38:59,624
If n1 just kind of goes ba-boom
and instantly decays away.

749
00:38:59,624 --> 00:39:02,155
AUDIENCE: There would
be a lot of n2--

750
00:39:02,155 --> 00:39:02,655
right then.

751
00:39:02,655 --> 00:39:03,447
MICHAEL SHORT: Yep.

752
00:39:03,447 --> 00:39:04,322
AUDIENCE: [INAUDIBLE]

753
00:39:04,322 --> 00:39:05,822
MICHAEL SHORT: Your
n1 is just going

754
00:39:05,822 --> 00:39:07,560
to turn into n2
right away, and n2

755
00:39:07,560 --> 00:39:10,800
is going to take its
sweet time decaying to n3.

756
00:39:10,800 --> 00:39:13,140
So let's see what
that looks like.

757
00:39:13,140 --> 00:39:20,190
If lambda 2 is much
bigger than lambda 1,

758
00:39:20,190 --> 00:39:23,860
let's make the maxima
a little different.

759
00:39:23,860 --> 00:39:27,580
Change our slider value a bit.

760
00:39:27,580 --> 00:39:34,430
So if l2 is big and
l1 is small, well,

761
00:39:34,430 --> 00:39:37,980
let me change the
actual axes to make

762
00:39:37,980 --> 00:39:40,630
this a little easier to see.

763
00:39:40,630 --> 00:39:42,780
There we go.

764
00:39:42,780 --> 00:39:45,020
You can see that much,
much more quickly

765
00:39:45,020 --> 00:39:48,470
than we have it in this graph
right here l1 just decays away

766
00:39:48,470 --> 00:39:49,670
right away.

767
00:39:49,670 --> 00:39:51,640
L-- I'm sorry-- n1
decays right away.

768
00:39:51,640 --> 00:39:55,640
n2 builds up to a much
higher relative value

769
00:39:55,640 --> 00:39:58,940
because it's produced
faster than it's destroyed

770
00:39:58,940 --> 00:40:00,620
for short amounts of time.

771
00:40:00,620 --> 00:40:03,710
So you can end up with
a great spike in n2

772
00:40:03,710 --> 00:40:08,500
which slowly decays away to n3.

773
00:40:08,500 --> 00:40:10,180
How about the opposite effect?

774
00:40:10,180 --> 00:40:14,410
What if l1 lambda 1 is
really, really small

775
00:40:14,410 --> 00:40:17,080
indicating a very
long half-life,

776
00:40:17,080 --> 00:40:19,450
and lambda 2 is
really, really large

777
00:40:19,450 --> 00:40:22,260
indicating a small half-life.

778
00:40:22,260 --> 00:40:22,760
Yeah.

779
00:40:22,760 --> 00:40:25,180
AUDIENCE: It would
basically go from n1 to n3.

780
00:40:25,180 --> 00:40:27,833
As soon as it goes to n2,
it's going to decay to n3.

781
00:40:27,833 --> 00:40:29,000
MICHAEL SHORT: That's right.

782
00:40:29,000 --> 00:40:31,010
In this case you've
got n2 as soon

783
00:40:31,010 --> 00:40:33,493
as its created self-destructs.

784
00:40:33,493 --> 00:40:34,910
So let's see what
that looks like.

785
00:40:34,910 --> 00:40:41,010
So we can just slide l2 to
be big, slide l1 to be small,

786
00:40:41,010 --> 00:40:46,270
and you can actually
graphically see n2 just shrink

787
00:40:46,270 --> 00:40:48,030
towards the x-axis.

788
00:40:48,030 --> 00:40:50,980
And it's almost like you
only have two equations.

789
00:40:50,980 --> 00:40:52,720
It's like you just
have n1 and n3

790
00:40:52,720 --> 00:40:55,150
and n2 basically doesn't exist.

791
00:40:55,150 --> 00:40:57,850
Where the slope of--

792
00:40:57,850 --> 00:41:01,990
where the slope of n3, except
at extremely short times,

793
00:41:01,990 --> 00:41:04,598
just tracks the value of n1.

794
00:41:04,598 --> 00:41:06,140
And I know in the
book they're called

795
00:41:06,140 --> 00:41:08,120
secular or transient equilbria.

796
00:41:08,120 --> 00:41:11,045
I'm not going to require that
you memorize those terms.

797
00:41:11,045 --> 00:41:12,920
It's more important to
me that I can give you

798
00:41:12,920 --> 00:41:14,600
a real physical situation.

799
00:41:14,600 --> 00:41:17,470
Say here's these three
isotopes, for four isotopes,

800
00:41:17,470 --> 00:41:20,690
or six doesn't matter because we
can solve these pretty quickly.

801
00:41:20,690 --> 00:41:22,190
Tell me what's going
to happen based

802
00:41:22,190 --> 00:41:25,070
on the relative
half-lives as long as they

803
00:41:25,070 --> 00:41:26,960
decay in a nice linear chain.

804
00:41:26,960 --> 00:41:28,850
I'm not going to give
you something where

805
00:41:28,850 --> 00:41:31,760
n1 can beget n1 or n4 or n6.

806
00:41:31,760 --> 00:41:35,360
Because at that point you
can construct the equations,

807
00:41:35,360 --> 00:41:39,450
but I don't expect you to be
able to graphically solve them.

808
00:41:39,450 --> 00:41:42,450
And I may also
throw you curveballs

809
00:41:42,450 --> 00:41:44,910
like nuclear activation
analysis to see well

810
00:41:44,910 --> 00:41:48,640
what happens when you turn
on or turn off a reactor.

811
00:41:48,640 --> 00:41:50,300
I've got an example of that too.

812
00:41:55,000 --> 00:41:59,680
We're right at this point
here, I guess, t equals 50.

813
00:41:59,680 --> 00:42:00,190
Yeah.

814
00:42:00,190 --> 00:42:03,340
I've set it up such that
you turn off the reactor

815
00:42:03,340 --> 00:42:07,120
and n3 is stable right
there, but n1 and n2

816
00:42:07,120 --> 00:42:09,020
continue to decay.

817
00:42:09,020 --> 00:42:10,920
So it's not hard to cad these--

818
00:42:10,920 --> 00:42:12,582
to code these
sorts of things up.

819
00:42:12,582 --> 00:42:14,290
I'll share the links
with these equations

820
00:42:14,290 --> 00:42:16,570
so you guys can play
with them yourselves.

821
00:42:16,570 --> 00:42:19,960
Add to them yourselves, and try
just getting an intuitive feel

822
00:42:19,960 --> 00:42:23,052
for how series
radioactive decay happens.

823
00:42:23,052 --> 00:42:25,510
So I want to know now that we
spent a couple of days on it,

824
00:42:25,510 --> 00:42:28,420
would you guys be
comfortable setting up

825
00:42:28,420 --> 00:42:30,310
sets of differential
equations like this?

826
00:42:30,310 --> 00:42:33,090
Say yes, no, maybe?

827
00:42:33,090 --> 00:42:34,970
I see a lot of up and
down shaking heads.

828
00:42:34,970 --> 00:42:36,813
That's a promising sign.

829
00:42:36,813 --> 00:42:38,980
If not I'm willing to spend
a little more time on it

830
00:42:38,980 --> 00:42:40,980
on Thursday if folks would
like a bit of review.

831
00:42:42,918 --> 00:42:44,460
And if you're afraid
to tell me, just

832
00:42:44,460 --> 00:42:46,110
send me an email
anonymous or not.

833
00:42:46,110 --> 00:42:46,373
Yeah?

834
00:42:46,373 --> 00:42:48,040
AUDIENCE: Do you think
maybe Thursday we

835
00:42:48,040 --> 00:42:50,820
could do a like a real example?

836
00:42:50,820 --> 00:42:51,720
MICHAEL SHORT: Yeah.

837
00:42:51,720 --> 00:42:53,697
AUDIENCE: Of a like a series.

838
00:42:53,697 --> 00:42:54,780
MICHAEL SHORT: I think so.

839
00:42:54,780 --> 00:42:57,035
Yeah, with real example
with numbers and everything.

840
00:42:57,035 --> 00:42:57,660
AUDIENCE: Yeah.

841
00:42:57,660 --> 00:42:58,493
MICHAEL SHORT: Sure.

842
00:42:58,493 --> 00:42:59,640
OK.

843
00:42:59,640 --> 00:43:02,250
Well, we can make one of
those up for Thursday.

844
00:43:02,250 --> 00:43:03,930
Cool.

845
00:43:03,930 --> 00:43:06,060
And what about the
graphical solution

846
00:43:06,060 --> 00:43:10,740
method since I don't know
whether they teach that

847
00:43:10,740 --> 00:43:13,230
in the GIRs, but what I
do want to be able to do

848
00:43:13,230 --> 00:43:14,670
is look at the limiting cases.

849
00:43:14,670 --> 00:43:17,430
In other words, fill in the
four corners of the graph.

850
00:43:17,430 --> 00:43:20,790
At t equals 0, what are
things actually doing?

851
00:43:20,790 --> 00:43:23,440
n1 is just decaying
at its half-life.

852
00:43:23,440 --> 00:43:25,270
There is no n2 yet.

853
00:43:25,270 --> 00:43:27,440
So these slopes are
equal and opposite.

854
00:43:27,440 --> 00:43:31,260
And there's no n3 yet, so
that there is no slope of n3.

855
00:43:31,260 --> 00:43:35,158
So I would like you guys
to try reproducing this.

856
00:43:35,158 --> 00:43:37,700
And I will-- again I'll provide
pictures of these blackboards

857
00:43:37,700 --> 00:43:39,120
so you guys can
see, but it would

858
00:43:39,120 --> 00:43:41,850
be very helpful for you guys to
try to reproduce these graphs

859
00:43:41,850 --> 00:43:43,200
as we saw them.

860
00:43:43,200 --> 00:43:46,160
Then you can check them here
on the graphical calculator,

861
00:43:46,160 --> 00:43:49,250
and then play around with
the amount of n0, or--

862
00:43:49,250 --> 00:43:50,820
I think I just broke it.

863
00:43:50,820 --> 00:43:52,930
Let's just call that one.

864
00:43:52,930 --> 00:43:54,250
There we go.

865
00:43:54,250 --> 00:43:59,830
And play around with sliders
or values of n1, n2, or n3.

866
00:43:59,830 --> 00:44:02,530
That's an interesting solution.

867
00:44:02,530 --> 00:44:04,440
So since it's about
four or five of,

868
00:44:04,440 --> 00:44:07,310
I want to open it up to any
questions you guys may have.

869
00:44:07,310 --> 00:44:07,826
Yeah.

870
00:44:07,826 --> 00:44:10,350
AUDIENCE: I have a question
from the [INAUDIBLE] First

871
00:44:10,350 --> 00:44:11,015
time do this.

872
00:44:11,015 --> 00:44:14,360
Did you have-- do you know
what integrated video because

873
00:44:14,360 --> 00:44:16,620
like there are endless
possibilities if you

874
00:44:16,620 --> 00:44:18,840
just [INAUDIBLE] add up
to the right mass number?

875
00:44:18,840 --> 00:44:20,443
MICHAEL SHORT: Oh yeah.

876
00:44:20,443 --> 00:44:22,110
The questions for a
spontaneous fission.

877
00:44:22,110 --> 00:44:24,700
What fission products
do you choose?

878
00:44:24,700 --> 00:44:26,790
I'll say they're all good.

879
00:44:26,790 --> 00:44:31,270
As long as you pick something
with roughly equalish masses,

880
00:44:31,270 --> 00:44:33,900
so you don't pick like it fizzes
into hydrogen and something

881
00:44:33,900 --> 00:44:35,790
quite smaller, which
should be better known

882
00:44:35,790 --> 00:44:37,350
as just proton emission.

883
00:44:37,350 --> 00:44:40,090
You're going to get
roughly the same answer.

884
00:44:40,090 --> 00:44:40,810
Yep.

885
00:44:40,810 --> 00:44:43,500
AUDIENCE: Would that be
something we just like set up.

886
00:44:43,500 --> 00:44:44,950
Like the top number is 80.

887
00:44:44,950 --> 00:44:50,750
We just look at whatever 40,
42, 38 arms pick a number.

888
00:44:50,750 --> 00:44:53,340
MICHAEL SHORT: Roll a D 80.

889
00:44:53,340 --> 00:44:56,920
You'll get basically the same
result. Roll an 80 sided dice.

890
00:44:56,920 --> 00:45:00,540
Hopefully at MIT you could
find one, or write a program

891
00:45:00,540 --> 00:45:05,220
to make random number between
about 10 and 80 or 10 and 70.

892
00:45:05,220 --> 00:45:09,930
Let's go to the actual problem
set to see what you guys mean.

893
00:45:09,930 --> 00:45:13,570
I want to make sure I'm
answering the correct question.

894
00:45:13,570 --> 00:45:17,541
Problem statement Yep.

895
00:45:17,541 --> 00:45:19,896
Yep.

896
00:45:19,896 --> 00:45:21,900
So allowable this
is for this one

897
00:45:21,900 --> 00:45:23,280
allowable nuclear reactions.

898
00:45:23,280 --> 00:45:24,030
Yep.

899
00:45:24,030 --> 00:45:26,370
So for a spontaneous
fission just pick one

900
00:45:26,370 --> 00:45:28,470
you think would be likely.

901
00:45:28,470 --> 00:45:30,450
You can also look up
what sort of isotopes

902
00:45:30,450 --> 00:45:32,393
are created when elements fizz.

903
00:45:32,393 --> 00:45:33,810
It's not straight
down the middle,

904
00:45:33,810 --> 00:45:36,300
so like uranium won't
often split into two

905
00:45:36,300 --> 00:45:38,340
equally sized fission products.

906
00:45:38,340 --> 00:45:40,410
They'll have roughly
different masses,

907
00:45:40,410 --> 00:45:42,143
but which ones you pick?

908
00:45:42,143 --> 00:45:44,310
You're still going to get
the same general solution.

909
00:45:44,310 --> 00:45:45,310
Yep.

910
00:45:45,310 --> 00:45:47,390
AUDIENCE: I'm so confused
on how to find one.

911
00:45:47,390 --> 00:45:50,954
Like a situation where it is
unlikely possible because is

912
00:45:50,954 --> 00:45:55,360
it spontaneous fission or is it
generally possible for heavier

913
00:45:55,360 --> 00:45:57,798
elements like
transuranic elements?

914
00:45:57,798 --> 00:46:00,340
MICHAEL SHORT: So the question
is when is spontaneous fission

915
00:46:00,340 --> 00:46:00,860
possible?

916
00:46:00,860 --> 00:46:03,280
Is it only for heavy elements?

917
00:46:03,280 --> 00:46:05,380
There is a difference
between energetically

918
00:46:05,380 --> 00:46:08,500
possible and observed.

919
00:46:08,500 --> 00:46:11,140
That's part of the
trick to this problem.

920
00:46:11,140 --> 00:46:14,350
If you do out the Q equation
to find out for add fission

921
00:46:14,350 --> 00:46:20,593
products that you picked, you
may be surprised at the result.

922
00:46:20,593 --> 00:46:21,510
However, you're right.

923
00:46:21,510 --> 00:46:24,390
You don't tend to see
spontaneous fission happen

924
00:46:24,390 --> 00:46:27,670
until you get to really
heavy things like uranium.

925
00:46:27,670 --> 00:46:31,050
So there's more to will
something spontaneously fizz

926
00:46:31,050 --> 00:46:35,450
than does the Q value
allow it to happen.

927
00:46:35,450 --> 00:46:36,950
So I don't want to
give away anymore

928
00:46:36,950 --> 00:46:40,140
but I will say if you're
surprised at your result,

929
00:46:40,140 --> 00:46:42,710
you might be right.

930
00:46:42,710 --> 00:46:43,430
Yep.

931
00:46:43,430 --> 00:46:47,210
AUDIENCE: On this question
for electron capture.

932
00:46:47,210 --> 00:46:48,710
In the equation
you gave us, you're

933
00:46:48,710 --> 00:46:50,670
calculating Q in an
electronic capture.

934
00:46:50,670 --> 00:46:56,210
Its the massive parents
minus the dollars minus the I

935
00:46:56,210 --> 00:47:00,056
think binding energy of the
electrons is what you wanted.

936
00:47:00,056 --> 00:47:02,960
When I try find out what the
binding energy of an electron

937
00:47:02,960 --> 00:47:05,005
is, it says it depends
on the shell that its in.

938
00:47:05,005 --> 00:47:06,260
MICHAEL SHORT: Yep.

939
00:47:06,260 --> 00:47:08,970
AUDIENCE: So how
do we know which

940
00:47:08,970 --> 00:47:11,285
electron the nucleus is after?

941
00:47:11,285 --> 00:47:14,400
Do we assume its from
the innermost shell?

942
00:47:14,400 --> 00:47:15,350
MICHAEL SHORT: Yep.

943
00:47:15,350 --> 00:47:17,900
So the question was if you're
doing electron capture where

944
00:47:17,900 --> 00:47:20,090
you have some parent
nucleus and you've

945
00:47:20,090 --> 00:47:23,990
got a lot of electron shells.

946
00:47:23,990 --> 00:47:26,450
The binding energy of every
electron is different.

947
00:47:26,450 --> 00:47:28,440
Which one goes in?

948
00:47:28,440 --> 00:47:30,960
One, you find the data
on the nest tables.

949
00:47:30,960 --> 00:47:34,980
Two, chances are it
will be the closest one.

950
00:47:34,980 --> 00:47:37,650
So a roughly 80% of
the time these things

951
00:47:37,650 --> 00:47:41,310
happen from the K shell with
very decreasing probabilities

952
00:47:41,310 --> 00:47:42,390
from the outer shells.

953
00:47:42,390 --> 00:47:46,010
So you can pick either
the k or the L shell,

954
00:47:46,010 --> 00:47:48,030
like both things may happen.

955
00:47:48,030 --> 00:47:51,030
But I would say for simplicity's
sake assume it's an inner most

956
00:47:51,030 --> 00:47:52,255
shell electron.

957
00:47:52,255 --> 00:47:54,630
And you can look up the binding
energy on the NIST tables

958
00:47:54,630 --> 00:47:55,797
on the learning module site.

959
00:47:58,890 --> 00:47:59,920
So any other questions?

960
00:47:59,920 --> 00:48:01,120
Yeah, Luke.

961
00:48:01,120 --> 00:48:05,840
LUKE: On graphing the
spectrum, the satellite

962
00:48:05,840 --> 00:48:10,000
intensity vs. the
energy for 4 2.

963
00:48:10,000 --> 00:48:11,890
MICHAEL SHORT: For 4 2.

964
00:48:11,890 --> 00:48:12,520
Ah yes.

965
00:48:12,520 --> 00:48:14,530
So graphing the--
this would be like

966
00:48:14,530 --> 00:48:16,360
if you had an electron detector.

967
00:48:16,360 --> 00:48:18,290
Is that what I asked for?

968
00:48:18,290 --> 00:48:20,420
4 2, write the full
nuclear reactions

969
00:48:20,420 --> 00:48:23,660
and draw the energy spectrum you
expect from each released form

970
00:48:23,660 --> 00:48:27,140
of radiation including
secondary ejections of particles

971
00:48:27,140 --> 00:48:28,410
or photons.

972
00:48:28,410 --> 00:48:33,522
So by a spectrum, I mean,
yep, energy versus intensity.

973
00:48:33,522 --> 00:48:34,490
LUKE: OK.

974
00:48:34,490 --> 00:48:35,948
MICHAEL SHORT: So
there you'll have

975
00:48:35,948 --> 00:48:38,660
to account for the spectrum
like the various range

976
00:48:38,660 --> 00:48:40,490
of the betas that
can be released,

977
00:48:40,490 --> 00:48:45,560
any ejected electrons, any
Auger electrons, any photons

978
00:48:45,560 --> 00:48:48,380
from X-ray emission from
electrons falling down

979
00:48:48,380 --> 00:48:49,130
and energy levels.

980
00:48:49,130 --> 00:48:50,422
Yeah, Alex, you had a question?

981
00:48:50,422 --> 00:48:51,080
ALEX: Yeah.

982
00:48:51,080 --> 00:48:53,083
What are the Auger electrons?

983
00:48:53,083 --> 00:48:54,500
MICHAEL SHORT: The
Auger electrons

984
00:48:54,500 --> 00:48:57,680
is that funny case where
in our mental model

985
00:48:57,680 --> 00:49:01,020
a gamma ray hits an
inner shell electron,

986
00:49:01,020 --> 00:49:05,120
and it's usually an inner shell
electron, shooting it out.

987
00:49:05,120 --> 00:49:08,780
Then another electron
will fall down

988
00:49:08,780 --> 00:49:12,660
to fill that hole
emitting an X-ray.

989
00:49:12,660 --> 00:49:14,370
And the Auger process
can be thought

990
00:49:14,370 --> 00:49:18,060
of that second X-ray hits
an electron on the way out

991
00:49:18,060 --> 00:49:20,340
and fires out the electron.

992
00:49:20,340 --> 00:49:23,610
So this here would be
the Auger electron.

993
00:49:23,610 --> 00:49:26,370
They tend to be
particularly low energy.

994
00:49:26,370 --> 00:49:27,285
Yeah, Luke.

995
00:49:27,285 --> 00:49:31,602
LUKE: If you have that cascade
of electrons during an electron

996
00:49:31,602 --> 00:49:35,450
capture, are they still
Auger [INAUDIBLE] radiation?

997
00:49:35,450 --> 00:49:36,510
MICHAEL SHORT: Yep.

998
00:49:36,510 --> 00:49:41,070
As long as you have a higher
level shell coming down

999
00:49:41,070 --> 00:49:43,230
to a lower shell and the
ejection of an outer shell

1000
00:49:43,230 --> 00:49:43,730
electron.

1001
00:49:43,730 --> 00:49:46,440
That's than Auger
electron emission process.

1002
00:49:46,440 --> 00:49:48,690
Regardless of whether
it started with gamma

1003
00:49:48,690 --> 00:49:51,154
or started with
electron capture.

1004
00:49:51,154 --> 00:49:51,654
Yep.

1005
00:49:51,654 --> 00:49:53,490
AUDIENCE: How do we
know if that happens?

1006
00:49:53,490 --> 00:49:55,530
MICHAEL SHORT: You can
actually sense or detect

1007
00:49:55,530 --> 00:49:57,180
the energy of those
Auger electrons

1008
00:49:57,180 --> 00:50:01,100
with a very sensitive
Auger electron detector.

1009
00:50:01,100 --> 00:50:03,020
These are in the sort
of hundreds or thousands

1010
00:50:03,020 --> 00:50:04,855
of EV range.

1011
00:50:04,855 --> 00:50:07,022
AUDIENCE: But for like the
context of this question,

1012
00:50:07,022 --> 00:50:09,380
how do we know--

1013
00:50:09,380 --> 00:50:10,984
where would we go to find them?

1014
00:50:10,984 --> 00:50:13,067
MICHAEL SHORT: Oh, to get
the Auger electron data?

1015
00:50:13,067 --> 00:50:13,810
AUDIENCE: Yeah.

1016
00:50:13,810 --> 00:50:15,477
MICHAEL SHORT: For
that you can actually

1017
00:50:15,477 --> 00:50:19,990
look up the binding energies
of an outer shell electron,

1018
00:50:19,990 --> 00:50:21,880
and you can do
that energy balance

1019
00:50:21,880 --> 00:50:25,000
where it would be E2 minus E1.

1020
00:50:25,000 --> 00:50:26,890
Whatever the energy
of that X-ray

1021
00:50:26,890 --> 00:50:34,060
is minus the binding energy
of the emitted electron.

1022
00:50:34,060 --> 00:50:35,860
And because there's
infinite possibilities.

1023
00:50:35,860 --> 00:50:39,280
I mean you could eject any
electron, just pick one.

1024
00:50:39,280 --> 00:50:41,500
And say here's an
Auger electron,

1025
00:50:41,500 --> 00:50:43,540
or draw a couple
of lines in places.

1026
00:50:43,540 --> 00:50:45,935
I don't have I don't want
you to get every single line.

1027
00:50:45,935 --> 00:50:47,560
If we asked you to
do this for uranium,

1028
00:50:47,560 --> 00:50:50,613
there's like you
know 92 electrons

1029
00:50:50,613 --> 00:50:52,030
and a lot of
different transitions

1030
00:50:52,030 --> 00:50:53,810
that's not what we're going for.

1031
00:50:53,810 --> 00:50:55,518
I want to make sure
you know the physics.

1032
00:50:55,518 --> 00:50:57,340
Not that you can draw
92 lines accurately

1033
00:50:57,340 --> 00:50:59,622
with a fine-toothed pencil.

1034
00:50:59,622 --> 00:51:01,240
Do you need a question two?

1035
00:51:01,240 --> 00:51:01,865
AUDIENCE: Yeah.

1036
00:51:01,865 --> 00:51:04,600
So for 2 1 if we write two
possible nuclear reactions

1037
00:51:04,600 --> 00:51:07,830
for 239 on [INAUDIBLE]
the right was only

1038
00:51:07,830 --> 00:51:09,580
off the case of any
decision and that it's

1039
00:51:09,580 --> 00:51:12,390
a state which decay processes
and repeating processes may

1040
00:51:12,390 --> 00:51:15,490
be possible for each
general type of reaction.

1041
00:51:15,490 --> 00:51:16,620
What exactly does that--

1042
00:51:16,620 --> 00:51:19,350
I find the answer for
number two is alpha.

1043
00:51:19,350 --> 00:51:22,284
MICHAEL SHORT: Maybe
that's the answer.

1044
00:51:22,284 --> 00:51:24,500
AUDIENCE: Oh, OK.

1045
00:51:24,500 --> 00:51:28,590
MICHAEL SHORT: And does anything
compete with alpha decay?

1046
00:51:28,590 --> 00:51:32,082
Does anything compete
with spontaneous fission?

1047
00:51:32,082 --> 00:51:33,880
OK.

1048
00:51:33,880 --> 00:51:34,656
Cool.

1049
00:51:34,656 --> 00:51:36,024
[INTERPOSING VOICES]

1050
00:51:36,024 --> 00:51:38,310
AUDIENCE: Is that an
indication of [INAUDIBLE]

1051
00:51:38,310 --> 00:51:39,393
MICHAEL SHORT: Beta decay.

1052
00:51:39,393 --> 00:51:40,985
Well, is there--
well, for that you

1053
00:51:40,985 --> 00:51:42,360
can look up the
table of nuclides

1054
00:51:42,360 --> 00:51:45,680
which I've got up here.

1055
00:51:45,680 --> 00:51:49,720
So let's take a look
at plutonium 239,

1056
00:51:49,720 --> 00:51:54,560
and it precedes by alpha
or spontaneous fission.

1057
00:51:54,560 --> 00:51:56,510
So every year I
switch up the isotope

1058
00:51:56,510 --> 00:51:59,220
and make sure that there's at
least a couple of decay modes,

1059
00:51:59,220 --> 00:52:02,220
and therefore the answers are
going to change every year.

1060
00:52:02,220 --> 00:52:05,355
But the general
question doesn't,

1061
00:52:05,355 --> 00:52:07,480
So this year I happened to
pick an interesting one.

1062
00:52:07,480 --> 00:52:10,180
Yeah, it's kind of
a mind game, right?

1063
00:52:10,180 --> 00:52:11,290
What are you missing?

1064
00:52:11,290 --> 00:52:12,040
AUDIENCE: Nothing.

1065
00:52:12,040 --> 00:52:13,140
MICHAEL SHORT: Nothing.

1066
00:52:13,140 --> 00:52:13,786
Yeah.

1067
00:52:13,786 --> 00:52:14,950
[LAUGHING]

1068
00:52:15,450 --> 00:52:18,620
Yeah, go with your
physical intuition.

1069
00:52:18,620 --> 00:52:21,416
Any other questions, and
maybe time for one more.

1070
00:52:21,416 --> 00:52:31,370
AUDIENCE: For 3 2, I could
only find one nuclear reaction.

1071
00:52:31,370 --> 00:52:35,831
The action played after
the nuclear reaction.

1072
00:52:35,831 --> 00:52:37,020
MICHAEL SHORT: Ah Yeah,

1073
00:52:37,020 --> 00:52:39,020
AUDIENCE: That's very
curious you're really

1074
00:52:39,020 --> 00:52:40,237
[INAUDIBLE] decay.

1075
00:52:40,237 --> 00:52:41,070
MICHAEL SHORT: Yeah.

1076
00:52:41,070 --> 00:52:42,890
AUDIENCE: And, so I
was wondering where

1077
00:52:42,890 --> 00:52:46,170
the molybdenum [INAUDIBLE]

1078
00:52:46,170 --> 00:52:47,670
MICHAEL SHORT: Yeah,
so the question

1079
00:52:47,670 --> 00:52:51,720
is I specifically wrote
which nuclear reactions could

1080
00:52:51,720 --> 00:52:53,790
make 99 molybdenum,
despite there

1081
00:52:53,790 --> 00:52:56,740
being only one natural one.

1082
00:52:56,740 --> 00:53:00,320
So what could you
induce artificially?

1083
00:53:00,320 --> 00:53:03,330
And if you can do that
profitably, I'll guarantee you

1084
00:53:03,330 --> 00:53:05,130
there's a startup in it for you.

1085
00:53:05,130 --> 00:53:06,870
So what are all the
different particles

1086
00:53:06,870 --> 00:53:10,260
that something could absorb
to create molybdenum 99,

1087
00:53:10,260 --> 00:53:13,530
and which of those are
allowable nuclear reactions?

1088
00:53:13,530 --> 00:53:15,750
And if they're not
allowable, how much energy

1089
00:53:15,750 --> 00:53:18,510
do you have to put
in an accelerator

1090
00:53:18,510 --> 00:53:20,550
to make that reaction happen?

1091
00:53:20,550 --> 00:53:23,370
And is the price of
electricity in the accelerator

1092
00:53:23,370 --> 00:53:26,540
worth the Molly 99
that you create?

1093
00:53:26,540 --> 00:53:28,410
There are actually
quite a few startups

1094
00:53:28,410 --> 00:53:29,827
working on this
problem right now.

1095
00:53:29,827 --> 00:53:33,173
So the answer to this
question is be creative.

1096
00:53:33,173 --> 00:53:35,340
Think about all the different
particles you know of,

1097
00:53:35,340 --> 00:53:38,010
and how they could
create Molly 99.

1098
00:53:38,010 --> 00:53:41,270
And figure out are any of
those processes allowed,

1099
00:53:41,270 --> 00:53:43,430
and if they're not
allowed, how energetic

1100
00:53:43,430 --> 00:53:45,805
do you have to make the incoming
particles to allow them?

1101
00:53:47,980 --> 00:53:49,850
Ah, good question.

1102
00:53:49,850 --> 00:53:52,370
There's some creativity
hiding in these problems.

1103
00:53:52,370 --> 00:53:53,360
So it's 10:02.

1104
00:53:53,360 --> 00:53:56,090
I want to cut it off here,
and we'll start off Thursday

1105
00:53:56,090 --> 00:54:01,550
with a numerical
example of this stuff.

1106
00:54:01,550 --> 00:54:04,240
Nuclear activation analysis.