Biomedical Engineering Reference
In-Depth Information
4.10 Analysis of Diffusive Offsets Due to Static B Fields
While viscous theory is consistent with random walk theory
developedabovefortheEfield,differencesrelatingtothestochastic
nature of the random walk emerge when B field results are
compared for the two models. This is due to the random directions
oftheBfieldoffsetscomparedwiththeunidirectionalEfieldoffsets.
Thereisalsoafurtherdifferenceintheorbitalandrotationaleffects
between models.
WhereaBfield,
B
0
,isdirectedalongthe
z
axis,andtheionmoves
inthedirectionofthe
x
axisfollowingcollision,theLorentzequation
can besolvedto yield the helicalmotion illustrated in Fig. 4.10a.
=
v
x
0
x
ω
c
sin(
ω
c
t
)
(4.15a)
y
=
v
x
0
ω
c
(1
−
cos(
ω
c
t
))
(4.15b)
where
ω
c
=
qB
0
/
m
is the cyclotron frequency. For small
t
(
t
<<
1),
we obtain an approximation suitable for the steps of the random
walk.
=
v
x
0
t
x
(4.16a)
v
x
0
B
z
m
1
2
q
t
2
=
y
(4.16b)
ThisdemonstratesthatforaBfieldexposure,theeffect(theforce
anditsconsequences)isalwaysorthogonaltothevelocityandsono
work ordiffusiveeffect on equilibriumconditionscan occur.
Similar to the preliminary test used for the static E field random
walk, a single-step numerical test was performed using a static B
field of 60
μ
Tdirectedalongthe
z
direction and an initial velocity
of 200 m s
−
1
in the
x
direction. Here, the displacement in the
y
direction was numerically determined to be 4.94
×
10
−
24
m,
comparingwellwiththeanalyticresult4.80
×
10
−
24
m.Thissimple
single-step test is of course only a single time step of the overall
random walk. What was required was to see how a sequence of
collisionsaffects the cyclotron motion.
Unliketheviscousmodel,therandomwalkmodelpredictsthatin
the transverse plane, due to the piece-wise helical motion between
