<s>
In	O
physics	O
,	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
is	O
the	O
average	O
distance	O
over	O
which	O
a	O
moving	O
particle	O
(	O
such	O
as	O
an	O
atom	B-Language
,	O
a	O
molecule	O
,	O
or	O
a	O
photon	B-Application
)	O
travels	O
before	O
substantially	O
changing	O
its	O
direction	O
or	O
energy	O
(	O
or	O
,	O
in	O
a	O
specific	O
context	O
,	O
other	O
properties	O
)	O
,	O
typically	O
as	O
a	O
result	O
of	O
one	O
or	O
more	O
successive	O
collisions	O
with	O
other	O
particles	O
.	O
</s>
<s>
The	O
magnitude	O
of	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
depends	O
on	O
the	O
characteristics	O
of	O
the	O
system	O
.	O
</s>
<s>
Assuming	O
that	O
all	O
the	O
target	O
particles	O
are	O
at	O
rest	O
but	O
only	O
the	O
beam	O
particle	O
is	O
moving	O
,	O
that	O
gives	O
an	O
expression	O
for	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
:	O
</s>
<s>
where	O
is	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
,	O
is	O
the	O
number	O
of	O
target	O
particles	O
per	O
unit	O
volume	O
,	O
and	O
is	O
the	O
effective	O
cross-sectional	O
area	O
for	O
collision	O
.	O
</s>
<s>
where	O
is	O
the	O
area	O
(	O
or	O
,	O
more	O
formally	O
,	O
the	O
"	O
scattering	O
cross-section	O
"	O
)	O
of	O
one	O
atom	B-Language
.	O
</s>
<s>
whose	O
solution	O
is	O
known	O
as	O
Beer	O
–	O
Lambert	O
law	O
and	O
has	O
the	O
form	O
,	O
where	O
is	O
the	O
distance	O
traveled	O
by	O
the	O
beam	O
through	O
the	O
target	O
,	O
and	O
is	O
the	O
beam	O
intensity	O
before	O
it	O
entered	O
the	O
target	O
;	O
is	O
called	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
because	O
it	O
equals	O
the	O
mean	O
distance	O
traveled	O
by	O
a	O
beam	O
particle	O
before	O
being	O
stopped	O
.	O
</s>
<s>
In	O
the	O
kinetic	O
theory	O
of	O
gases	O
,	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
a	O
particle	O
,	O
such	O
as	O
a	O
molecule	O
,	O
is	O
the	O
average	O
distance	O
the	O
particle	O
travels	O
between	O
collisions	O
with	O
other	O
moving	O
particles	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
kinetic	B-Data_Structure
diameter	I-Data_Structure
of	O
a	O
molecule	O
is	O
defined	O
in	O
terms	O
of	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
different	O
definitions	O
of	O
the	O
molecular	O
diameter	O
,	O
as	O
well	O
as	O
different	O
assumptions	O
about	O
the	O
value	O
of	O
atmospheric	O
pressure	O
(	O
100	O
vs	O
101.3	O
kPa	O
)	O
and	O
room	O
temperature	O
(	O
293.17	O
K	O
vs	O
296.15	O
K	O
or	O
even	O
300	O
K	O
)	O
can	O
lead	O
to	O
slightly	O
different	O
values	O
of	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
.	O
</s>
<s>
In	O
gamma-ray	O
radiography	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
a	O
pencil	O
beam	O
of	O
mono-energetic	O
photons	B-Application
is	O
the	O
average	O
distance	O
a	O
photon	B-Application
travels	O
between	O
collisions	O
with	O
atoms	O
of	O
the	O
target	O
material	O
.	O
</s>
<s>
It	O
depends	O
on	O
the	O
material	O
and	O
the	O
energy	O
of	O
the	O
photons	B-Application
:	O
</s>
<s>
In	O
X-ray	B-Library
radiography	O
the	O
calculation	O
of	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
is	O
more	O
complicated	O
,	O
because	O
photons	B-Application
are	O
not	O
mono-energetic	O
,	O
but	O
have	O
some	O
distribution	O
of	O
energies	O
called	O
a	O
spectrum	O
.	O
</s>
<s>
As	O
photons	B-Application
move	O
through	O
the	O
target	O
material	O
,	O
they	O
are	O
attenuated	O
with	O
probabilities	O
depending	O
on	O
their	O
energy	O
,	O
as	O
a	O
result	O
their	O
distribution	O
changes	O
in	O
process	O
called	O
spectrum	O
hardening	O
.	O
</s>
<s>
Because	O
of	O
spectrum	O
hardening	O
,	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
the	O
X-ray	B-Library
spectrum	O
changes	O
with	O
distance	O
.	O
</s>
<s>
Sometimes	O
one	O
measures	O
the	O
thickness	O
of	O
a	O
material	O
in	O
the	O
number	O
of	O
mean	B-Algorithm
free	I-Algorithm
paths	I-Algorithm
.	O
</s>
<s>
Material	O
with	O
the	O
thickness	O
of	O
one	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
will	O
attenuate	O
to	O
37%	O
(	O
1/e	O
)	O
of	O
photons	B-Application
.	O
</s>
<s>
This	O
concept	O
is	O
closely	O
related	O
to	O
half-value	O
layer	O
(	O
HVL	O
)	O
:	O
a	O
material	O
with	O
a	O
thickness	O
of	O
one	O
HVL	O
will	O
attenuate	O
50%	O
of	O
photons	B-Application
.	O
</s>
<s>
A	O
standard	O
x-ray	B-Library
image	O
is	O
a	O
transmission	O
image	O
,	O
an	O
image	O
with	O
negative	O
logarithm	O
of	O
its	O
intensities	O
is	O
sometimes	O
called	O
a	O
number	O
of	O
mean	B-Algorithm
free	I-Algorithm
paths	I-Algorithm
image	O
.	O
</s>
<s>
In	O
macroscopic	O
charge	O
transport	O
,	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
a	O
charge	O
carrier	O
in	O
a	O
metal	O
is	O
proportional	O
to	O
the	O
electrical	O
mobility	O
,	O
a	O
value	O
directly	O
related	O
to	O
electrical	O
conductivity	O
,	O
that	O
is	O
:	O
</s>
<s>
In	O
thin	O
films	O
,	O
however	O
,	O
the	O
film	O
thickness	O
can	O
be	O
smaller	O
than	O
the	O
predicted	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
,	O
making	O
surface	O
scattering	O
much	O
more	O
noticeable	O
,	O
effectively	O
increasing	O
the	O
resistivity	O
.	O
</s>
<s>
Electron	O
mobility	O
through	O
a	O
medium	O
with	O
dimensions	O
smaller	O
than	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
electrons	O
occurs	O
through	O
ballistic	O
conduction	O
or	O
ballistic	O
transport	O
.	O
</s>
<s>
If	O
one	O
takes	O
a	O
suspension	O
of	O
non-light-absorbing	O
particles	O
of	O
diameter	O
d	O
with	O
a	O
volume	O
fraction	O
Φ	O
,	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
the	O
photons	B-Application
is	O
:	O
</s>
<s>
In	O
an	O
otherwise	O
empty	O
cavity	O
,	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
of	O
a	O
single	O
particle	O
bouncing	O
off	O
the	O
walls	O
is	O
:	O
</s>
<s>
In	O
particle	O
physics	O
the	O
concept	O
of	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
is	O
not	O
commonly	O
used	O
,	O
being	O
replaced	O
by	O
the	O
similar	O
concept	O
of	O
attenuation	O
length	O
.	O
</s>
<s>
In	O
particular	O
,	O
for	O
high-energy	O
photons	B-Application
,	O
which	O
mostly	O
interact	O
by	O
electron	O
–	O
positron	O
pair	O
production	O
,	O
the	O
radiation	O
length	O
is	O
used	O
much	O
like	O
the	O
mean	B-Algorithm
free	I-Algorithm
path	I-Algorithm
in	O
radiography	O
.	O
</s>
