Radioactive Decay


 
 Radioactive
Decay
 


Grace
Elwell
 Thomas
Sullivan
 November
11,
2008
 
 
 Abstract:
 
 This
experiment
sought
to
measure
the
decay
rate
and
determine
the
 lifetimes
of
the
unstable
silver
isotopes
 108 Ag 
and 110 Ag 
found
in
a
quarter
after
it
 had
been
irradiated
in
a
neutron
howitzer,
which
served
as
a
neutron
source.

A
 Geiger‐Müller
counter
was
used
to
measure
radiation,
which
was
sorted
by
a
multi
 channel
analyzer
according
to
time.

This
allowed
for
the
calculation
of
rate
of
decay
 € € and
thus
the
lifetimes
of
the
isotopes
with
were
found
to
be
(149.25
±
122.52)s
and
 (30.03
±
16.68)s
respectively
which
were
comparable
to
expected
values
of
209s
 and
35.5s
respectively,
thus
reinforcing
the
validity
of
previous
experiments.
 
 
 
 
 
 
 
 
 
 
 


Introduction:
 


Some
of
the
biggest
discoveries
in
radioactive
decay
occurred
just
over
one


hundred
years
ago.
Around
the
year
1900
Ernest
Rutherford
and
Fredrick
Soddy
 proved
that
radioactive
decay
involved
the
transmutation
of
one
element
into
 another.

 In
this
lab
we
look
at
two
silver
isotopes,
 107 Ag 
and
 109 Ag ,
found
in
a
quarter
 minted
before
1964
at
ratios
of
about
53%
to
47%
respectively.

After
being


€ € 108 irradiated
with
a
neutron
source
these
isotopes
become
 Ag 
and
 110 Ag ,
which
will
 transmute
into
cadmium.

We
sought
to
measure
the
decay
rate
of
these
unstable


€ € 108 isotopes
and
determine
their
lifetime.

We
found
that
the
lifetime
for
 Ag 
was
 (149.25
±
122.52)s
and
for
 110 Ag 

it
was
(30.03
±
16.68)s,
which
were
comparable
to


€ the
expected
lifetime
values
of
209s
and
35.5s
respectively.
 €




Experimental
Methods:
 


Our
source
of
neutron
radiation
was
a
“neutron
howitzer”
which
used


9 plutonium
( 242 94 Pu )
and
beryllium
( 4 Be )
to
create
neutrons
through
a
three‐step


process:
 


€

€


 


242 94 9 4

€ €

€

4 Pu→238 92 U+ 2 α 


(1)


Be+ 42 α →136 C * 


(2)


13 6

C*→126 C+ 01n 


(3)


where
 136 C * 
is
unstable
because
it
is
excited
due
to
kinetic
energy
brought
into
the
 nucleus
by
the
α
particle.
 136 C * 
gives
off
a
neutron
to
release
kinetic
energy
from
the


€

nucleus
and
becomes
stable
as
 126 C .
 


€ We
irradiated
quarters
by
exposing
them
to
this
process
and
then
measured


€ the
rate
of
decay
in
the
quarters
using
a
Geiger‐Müller
counter
(GMC).

A
GMC
 produces
a
voltage
pulse
when
an
electron
strikes
a
wire,
but
due
to
a
high
 operating
voltage
it
is
sensitive
to
radiation
but
not
energy.
 


The
signal
from
the
counter
is
sent
to
a
multichannel
analyzer
where
the


signal
is
sorted
into
bins
according
to
their
arrival
time.

This
allows
us
to
see
the
 change
in
radiation
over
time.
 


We
irradiated
the
quarter
for
10
minutes.
During
the
irradiation
period
we


monitored
the
number
of
background
counts
from
the
GMC
for
several
minutes
to
 determine
the
average
background
rate.

Then
we
quickly
removed
the
quarter
from
 the
neutron
howitzer
and
placed
it
in
the
GMC
where
we
monitored
the
decay
rate
 for
5
minutes.
 
 Results:
 


First
we
calculated
the
average
background
and
subtracted
it
from
the
rates


we
got
for
radioactive
decay
of
the
silver
in
the
quarter.
So
let
“data”
refer
to
the
 data
after
accounting
for
background.

We
knew
that
the
expression
for
the
rate
of
 radioactive
decay
is
 


R[t] = R0 exp[−t / τ ]


€

(4)


where
t
is
the
half‐life
and
τ
is
the
lifetime.

To
make
a
linear
plot
we
graphed
the
log
 of
the
data
we
collected
(ln[R])
against
time.
Since
 110 Ag 
decays
at
a
much
faster
rate
 than
 108 Ag 
we
could
see
two
trends
on
the
graph;
an
initial
steep
rate
of
decay
and
a


€ second
slower
rate
later
in
time
after
essentially
all
of
the
 110 Ag 
had
decayed
(see
 Figure
1).


€

Data
(Minus
Background)
 6
 5
 4
 ln[R]


€

3
 2
 1
 0
 0


50


100


150


200


250


300


Time
(seconds)



 Figure
1.
Natural
log
of
the
rate
of
decay
v.
time.
 110 Ag 
is
decayed
by
100seconds,
so
data
 after
100
seconds
is
only
 108 Ag 
decay.


€


 


€ Based
on
the
data
and
the
graph
we
estimated
that
the
change
in
slope,
and


thus
the
end
of
effective
 110 Ag 
radioactive
decay,
was
at
100seconds.
Data
taken
 after
100
seconds
was
treated
as
only
 108 Ag 
decay.

So
we
made
a
separate
plot
of


€ €

data
collected
after
100
seconds,
ln[R]
against
time,
and
this
time
we
found
a
single
 slope
of
‐0.0067/s
(see
Figure
2).
 





Ag108


y
=
‐0.0067x
+
3.7976


4.5
 4
 3.5


ln[R]


3
 2.5
 2
 1.5
 1
 0.5
 0
 0


50


100


150


200


250


300


Time
(sec)



 Figure
2.
Decay
of
 108 Ag 
as
ln[R]
v.
time.

Data
does
not
start
until
100
seconds
because
 prior
to
100
seconds
the
decay
rate
is
for
both
 108 Ag 
and
 110 Ag .
The
slope
of
‐0.0067/s
is


€ the
opposite
inverse
of
the
lifetime
τ 108.
 
 


€

€

Having
found
the
rate
for
 108 Ag 
we
could
then
find
the
rate
for
 110 Ag .

Looking


at
the
data
from
prior
to
100
seconds,
which
includes
both
 108 Ag 
and
 110 Ag 
decay,
we


€ € used
the
rate
for
 108 Ag 
and
extended
it
backwards
to
t=0
to
estimate
which
data
 € € from
the
first
100
seconds
was
due
to
 108 Ag 
decay.


This
estimation
at
each
time
was
 € subtracted
from
each
total
data
point
to
isolate
 110 Ag 
decay.

The
isolated
 110 Ag 
data
 € €

€

was
graphed
in
the
same
way
as
 108 Ag 
after
100
seconds,
the
slope
was
‐0.0333/s
 (see
Figure
3).
Data
between
60
and
100
seconds
was
left
out
due
to
extremely
large


€ 110 uncertainties.
This
seemed
reasonable
since
the
time
at
which
 Ag 
decay
became
 inconsequential
was
estimated
and
between
60
and
100
seconds
the
number
of


€

counts
is
low
that
they
are
highly
inaccurate.
 


y
=
‐0.0333x
+
4.6915


Ag110
 6
 5


ln[R]


4
 3
 2
 1
 0
 0


10


20


30


40


50


60


70


Time
(seconds)



 Figure
3.
Decay
of
 110 Ag 
as
ln[R]
v.
time.

This
data
was
found
by
subtracting
estimated
 values
for
 108 Ag 
based
on
the
rate
found
previously.

The
slope
of
‐0.0333/s
represents
the


€

rate
for
 110 Ag .
 
€

€


When
we
take
the
slope
back
out
of
log
from
we
see
that





R[t] = R0 exp[−t / τ ] = exp[−0.0067t + 3.7976] 


€

(5)


whence
 





τ108 =

−1 = 149.25s
 −0.0067 /s

(6)


τ110 =

−1 = 30.03s 
 −0.0333/s

(7)


and
in
the
same
way


€










With
errors,
due
to
rate
uncertainties,
the
life
times
are
τ108
=
(149.25
±


€ 122.5)s
and
τ110
=
(30.03
±
16.7)s.
However
the
theoretical
data
we
were
given
was
 in
half‐lives.

The
half‐life
is
the
time
it
takes
for
one
half
of
substance
to
decay.

To
 convert
the
given
half‐lives
into
lifetimes
we
used
 











exp[t / τ ] = .5 


(8)


where
t
is
the
half
life
and
τ
is
the
lifetime.

The
theoretical
half‐lives
were
t108
=
 € 145s
and
t110
=
24.6s.
So
the
theoretical
lifetimes
were
209s
and
35.5s
respectively.



 Discussion:
 


While
the
errors
on
the
lifetimes
are
pretty
large,
they
do
include
the


expected
values
in
the
range
of
uncertainty.

Additionally,
the
data
still
clearly
shows
 the
difference
in
the
lifetimes
of
 108 Ag 
and 110 Ag .

We
could
improve
the
accuracy
of
 the
data
with
more
refined
and
timely
lab
methods,
such
as
a
quicker
transfer
from


€ € the
neutron
howitzer
to
the
Geiger‐Müller
counter,
or
possibly
comparing
the
 effects
of
a
different
neutron
source.
 
 108

The
cross‐section
for
absorption
of
 110 Ag 
is
significantly
higher
than
that
of


Ag .

This
property
of
their
respective
nuclei
is
seen
in
the
y‐intercepts
of
their
 €

€

isolated
graphs
since
there
was
approximately
the
same
amount
of
both
types
of
 silver
in
the
quarter.
The
difference
would
be
even
more
apparent
if
there
were
a
 way
to
start
counting
more
immediately
when
the
quarter
was
removed
from
the
 neutron
howitzer
since
obviously
more
of
the
 110 Ag 
decayed
during
the
few
seconds
 between
removal
and
counting,
thus
closing
the
gap
between
the
initial
rates
of
the


€ two
isotopes.

Further
experiments
could
be
done
to
explain
why
these
nuclei,
which
 seem
otherwise
very
similar
display
such
different
affinities
toward
absorption.
 
 
 Acknowledgements:
 


I
would
like
to
thank
my
lab
partner
Mr.
Thomas
Sullivan
for
his


collaboration
and
the
internet
ACE
Hardware
man
for
assuring
me
that
Geiger‐ Müller
counters
are
not
that
easy
to
come
by
in
this
world.

Additionally,
I
would
like
 to
thank
the
Carleton
Physics
Department,
the
Carleton
Mathematics
Department,
 and
the
Wellness
Center
collectively
for
supplying
me
with
all
the
candy
I
need
to
 make
it
through
a
long
day
of
lab
write‐ups
and
problem
sets.