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Appendix 12: Magnetic Precession in Static and Oscillating Magnetic Fields:
The Rotating Coordinate System
Mara Prentiss
Ronald Walsworth
September 2002
Before learning the full quantum mechanical treatment of a spin 1/2 particle in a
magnetic field, it is useful to consider the more
r familiar and intuitive problem of the evolution of
a classical particle with magnetic moment M . The quantum mechanical problem will turn out to
be analogous.
The unperturbed system treated here is a single object or particle with magnetic moment
r
†
M in the presence of a strong static magnetic field in the z direction. The perturbation will be a
weak rotating magnetic field in the x-y plane (which is largely equivalent to an oscillating field
in the x direction).
†
Unperturbed System
The energy of the unperturbed system depends on the angle q between the magnetic
moment and the magnetic field.
r
B = B0 zˆ
r r
r
U = - M ⋅ B = - M B0 cos q
For a negatively charged particle
†
†
r
r
M = -gL
r
where g is called the gyromagnetic ratio, and L is the angular momentum of the particle (which
for an electron is also the spin).
†
The torque on the magnetic moment is given by
†
r
r dL r r
t=
= M¥B
dt
Hence
r
r
†dM
r r
r r
dL
= -g
= -gM ¥ B = g 2 L ¥ B .
dt
dt
Note that the magnitude of the magnetic moment does not change since
†
Appendix 12 - 1
r2
dL
r r
d L⋅ L
(
) = 2Lr ⋅ dLr = 2Lr ⋅
r
r
r
r
r
( M ¥ B) = 2L ⋅ (-gL ¥ B)
= 0
dt
dt
r
r
r
Similarly the angle between M and B (i.e., the projection of M along the z axis) does not
change, since
†
r r
r
r dL r r † r
r r r
† d B†⋅ L
= 0+ B⋅
= B ⋅ M ¥ B = -gB ⋅ L ¥ B = 0
dt
dt
dt
=
(
)
(
)
(
)
In the quantum mechanical problem of a particle with spin there will be only two possible
orientations of the spin, up or down. Both spin up and spin down will be eigenstates, just as the
z component†of the spin is a constant in the classical case.
The equations of motion for the magnetic moment become
dM x
= -w 0 M y
dt
dM y
= w0 Mx
dt
dM z
=0
dt
where w 0 = gB0 is known as the Larmor frequency.
†
With the initial condition
r
M (0) = M x (0) xˆ + M z (0) zˆ
†
i.e., q ≠ 0 , The solutions are
†
M x (t) = M x (0)cosw 0 t
M y (t) = M x (0)sinw 0 t
†
That is, the magnetic moment precesses around the z-axis at the Larmor frequency.
†
Rotating Frame
Now consider the problem in the reference frame rotating at the Larmor frequency w 0 . In
r
this reference frame, M is a constant, which makes the time evolution trivial. The time
evolution in the lab frame can then be obtained by transforming from the rotating frame back to
the lab frame.
†
†
Appendix 12 - 2
Consider in general the transformation to a reference frame rotating at a frequency w.
The rotation matrix for the transformation between the two frames is
Ê coswt sin wt 0ˆ
Á
˜
R = Á-sin wt coswt 0˜
Á
˜
0
1¯
Ë 0
The magnetic moment in the new reference frame is given by
†
M x ¢ (t) = M x cos wt + M y sin wt
M y ¢ (t) = -M x sinwt + M y cos wt
M z¢ (t) = M z
The equations or motion in the rotating frame are
†
dM x ¢
= (w - w 0 ) M y ¢
dt
dM y ¢
= -(w - w 0 ) M x ¢
dt
dM z¢
=0
dt
These equations can be recombined into a vector equation
†
r
r
r
dM ¢
= -gM ¢ ¥ Beff
dt
where
†
r
Ê
wˆ
D
Beff = Á B0 - ˜zˆ = - zˆ
g¯
g
Ë
and D = w - w 0 .
†
r
†
Note that at resonance, one has w = w 0 and thus Beff = 0 , so there is no precession of the
magnetic moment in the rotating frame. For any other frequency there is a precession because the
rotating frame is out of synch with the Larmor frequency. The equations above can be
transformed back into the lab frame:
†
†
Appendix 12 - 3
r
r
M Æ M¢
r
r
dM
dM ¢ r r
Æ
+ w ¥ M¢
dt
dt
r
r wr
Beff Æ B g
Perturbation by a Rotating Magnetic Field
†
Next, consider an applied
B field, which precesses in the x-y plane at frequency w:
r
B pert = B1 (cos wt xˆ + sin wt yˆ )
[Note: in the laboratory it is often impractical to create a rotating magnetic field, so one
commonly employs a field that oscillates along a particular direction, e.g., the x axis. This
†
oscillating field is equivalent
to the sum of two counter-rotating fields: one field rotates with the
same helicity as the magnetic moment's rotation, while the other field rotates with opposite
helicity and can usually be ignored because it is effectively very far off resonance (at w @ 2w 0 ).]
r
With the addition of B pert the equations of motion become
†
r
r r
r
dM
= -gM ¥ B = -gM ¥ [ B0 zˆ + B1 (cos wt xˆ + sin wt yˆ )]
dt
†
In the frame rotating at w this becomes
r
˘
r
r
r ÈÊ
dM ¢
wˆ
= -gM ¢ ¥ Beff = -gM ¢ ¥ ÍÁ B0 - ˜ zˆ + B1 xˆ ˙
dt
g¯
ÎË
˚
†
In component form one has
†
dM x¢
= DM y
dt
dM y¢
= -DM x - gB1 M z
dt
dM z¢
= gB1 M y
dt
On resonance in the rotating frame, i.e., at w = w 0 , the magnetic field lies ONLY along
the x direction, so the magnetic moment
simply precesses around x. This means that Mx is
r
†
constant and the
r component of M perpendicular to x rotates in the y-z plane. Thus the
component of M which is along the z-axis
at time t=0 will be along the -z axis at t = p gB1 . In
r†
other words, all of the projection of M along the positive z-axis will be transferred to a
†
†
†
Appendix 12 - 4
†
projection along the negative z-axis. Classically and geometrically, this is a consequence of the
axis of rotation being perpendicular to the y-z plane. Quantum mechanically, the applied
rotating field allows a coupling between the spin up and spin down states. In either picture, this
inversion effect is known as resonant Rabi oscillation.
Mathematically, the solutions are
M x¢ (t) = M x¢ (0)
M y¢ (t) = - M z¢ (0)sin gB1t + M y¢ (0)cos gB1t
M z¢ (t) = - M y¢ (0)sin gB1t + M z¢ (0)cos gB1t
Thus the magnetic moment rotates around the x-axis at the Rabi frequency W = gB1 .
r
†
If B pert is tuned off resonance, such that w @ w 0 , then the effective magnetic field will
r
have some z component as well as an x component. Thus the component
of M in the positive z
†
direction will NEVER rotate completely into the negative z direction (and visa versa). As the
detuning of w from w0 gets larger and larger,
the effective field approaches the z direction and
†
†
the rotations between the positive and negative components of the magnetic moment approaches
†
zero.
r
r
Mathematically, in the rotating frame M ¢ precesses about B at a frequency
r 2
Weff = g Beff = D2 + (gB1 ) 2 = D2 + W2
†
†
This is the off-resonant Rabi oscillation frequency.
†
The effect of the field is thus to set up a correlation between the spin up and spin down
states since something, which is in spin up at one time, will rotate into spin down at another time.
In the stationary frame, the moment precesses around the moving magnetic field vector.
Mathematically, in the rotating frame, the time evolution of the components becomes
M x¢ (t) = M x¢ (0)
(gB1 )
2
+ D2 cosWeff t
D
DW
+ M y¢ (0)
sinWeff t + M z¢ (0) 2 (cosWeff t - 1)
2
Weff
Weff
Weff
M y¢ (t) = - M x¢ (0)
†
D
W
sinWeff t + M y¢ (0)cosWeff t - M z¢ (0) 2 sinWeff t
Weff
Weff
2
È Ê
˘
ˆ
DW
W
W
Í
M z¢ (t) = M x¢ (0) 2 (cosWeff t - 1) + M y¢ (0) 2 sinWeff t + M z¢ (0) 1+ ÁÁ
˜ (cosWeff t - 1)˙
†
Í Ë Weff ˜¯
˙
Weff
Weff
Î
˚
†
Appendix 12 - 5
Appendix 12 - 6