Spin precession and the Rotating Frame
N. Chandrakumar
The spin magnetic moment (µ) is expressed in terms of the spin
angular momentum (hI) as:
r
r
µ = γ hI
Here, the constant of proportionality is the magnetogyric ratio (or
gyromagnetic ratio).
The motion of a magnetic moment (µ) in a magnetic field (B) is given
by the classical expression that equates the torque to the rate of change of
angular momentum (hI):
h
r
dI
dt
⇒
r r
= µ×B
r
dµ
dt
ˆi
r r
= γ µ × B ≡ γ µx
Bx
(
)
ˆj
kˆ
µy
µz
By
Bz
This vector equation may be decomposed into its three components,
resulting in a set of coupled first order differential equations. With a constant
Zeeman magnetic field B0 being oriented along the z-direction, ie, Bx = 0, By
= 0, Bz = B0, we have:
µ& x = γ ( µ y B0 )
µ& y = −γ ( µ x B0 )
µ& z = 0
The last equation may be integrated at once, giving: µz = constant.
The first two equations may be uncoupled by taking the second derivative
and then integrating, satisfying the conditions on the first derivatives. This
results in the solutions:
µ&&x = γµ& y B0 = γ B0 ( −γµ x B0 ) = −γ 2 B02 µ x
⇒ µ x ( t ) = µ x ( 0 ) cos ( γ B0t ) + µ y ( 0 ) sin ( γ B0t )
µ&&y = −γµ& x B0 = −γ B0 ( γµ y B0 ) = −γ 2 B02 µ y
⇒ µ y ( t ) = − µ x ( 0 ) sin ( γ B0t ) + µ y ( 0 ) cos ( γ B0t )
The time evolution of the magnetic moment vector may thus be
visualized as a precession on the surface of a cone, maintaining constant zcomponent, while the x- and y- components rotate at the angular velocity ω0
= − γ B0, the so-called Larmor frequency. Note that the initial values of the
x- and y- components are arbitrary. In the random phase approximation
(RPA), it is assumed that the magnetic moment vectors of individual spins in
an ensemble are initially oriented randomly on the surface of the precession
cone. This corresponds to the fact that the expectation value of the x- and ycomponents of spin and magnetic moment are predicted to be 0, given spin
quantization in the direction of the magnetic field (z).
The actual precession frequency of a test nuclear spin magnetic
moment (in solution state) is in fact modified from the Larmor frequency by
chemical shifts and spin-spin couplings, which arise respectively from the
influence of electrons in the neighborhood of the test nucleus and from the
influence of other nuclear spins in the neighborhood of the test nucleus.
These contributions to the precession frequency are typically several orders
of magnitude smaller than the Larmor precession frequency.
Accurate measurement of the precession frequency shifts – and
visualization of the magnetization trajectory – under the influence of
chemical shifts and spin-spin couplings is facilitated by ‘removing’ the large
Larmor frequency! This is accomplished experimentally by the special
process of signal ‘demodulation’ known as phase sensitive detection and
corresponds to subtracting the Larmor frequency from the actual signal
frequency.
The process may be visualized as observation of the spin
magnetization in a co-ordinate frame that is rotating about z at the same
frequency and in the same sense as the spin precession itself. Such a frame
may be labeled as the x'y'z' frame, in contrast to the stationary xyz frame
defined in the laboratory (the ‘laboratory frame’). Here z' coincides with z
and the x'y' plane coincides with the xy plane. In such a rotating frame, the
spins appear stationary, ie, there appears to be no magnetic field (except for
the effect of resonance offsets or shifts and couplings)! This is rather like
observing a moving object in a beam of light that is strobed: when the strobe
frequency matches the motional velocity, the moving object appears
stationary in the strobed light beam. Another contemporary example is a
geo-stationary satellite, which is stationary relative to the earth’s motion.
[Note: In practice, most descriptions in the rotating frame drop the primes
on the axis labels for convenience.]
How does a radiofrequency magnetic field appear in the rotating
frame? The rf field is produced by passing an alternating current through a
coil that forms part of a tank circuit tuned to the appropriate resonance
frequency. The rf magnetic field that results from the rf current may be
represented as a linearly polarized field given by i2B1cos(ωt+ϕ), the axis of
the coil in the fixed (stationary) laboratory co-ordinate frame being x, the
peak-to-peak amplitude of the field being 4B1, its frequency and phase being
respectively ω and ϕ. i represents the unit vector in the x-direction. Such a
linearly polarized field may be decomposed in terms of two circularly
polarized fields that rotate in the opposite sense (ie, counter-rotating fields),
one counter-clockwise and the other clockwise:
(
) (
)
i 2 B1 cos ωt = B1 ⎡ ˆi cos ωt + ˆj sin ωt + ˆi cos ωt − ˆj sin ωt ⎤
⎣
⎦
Clearly, the rotating rf component that has the same sense of motion as the
rotating frame [ie, matches the sense of Larmor precession] and matches its
frequency appears stationary in the rotating frame. In this frame, the other rf
component appears to be moving away at twice the rate.
What is the influence of the two counter-rotating resonant rf fields on
the spin magnetic moments? The orientation of a spin magnetic moment
may be influenced by the resonant field only if it exerts a non-zero torque on
the spin. The torque exerted during a period of the rf wave would be
constant if there were no relative motion between the spin magnetic moment
and the rf magnetic field during this time – else it would average to zero.
[Recall that the torque is a function of the angle between the magnetic
moment and the magnetic field, and is oriented along the right handed
normal to the plane formed by them.]
Clearly therefore, we may expect that the rf field that has the ‘correct’
sense of rotation influences the spin orientation, while the counter-rotating
component that has the ‘wrong’ sense would not, to first order, influence it.
Accordingly, standard treatments of spin resonance ignore the counterrotating rf component, to first order. [However, note that the counter-rotating
component does lead to a resonance frequency shift, termed the BlochSiegert shift: the shift being measurable especially with strong rf fields.]
Phase sensitive detection
The signal voltage is multiplied by a reference voltage in a mixer
driven in the linear regime. The resulting output, which contains two
frequencies – viz., the sum and difference of the signal and reference
frequencies – is passed through a low pass filter that rejects the sum
frequency and passes the difference frequency.
S = s cos (ωs t + ϕ s )
R = r cos (ωr t + ϕ r )
SR = sr cos (ωs t + ϕ s ) cos (ωr t + ϕr ) ≡
→
sr
2
cos ( (ωs − ωr ) t + ϕ s − ϕ r )
sr
2
⎡cos ( (ωs + ωr ) t + ϕ s + ϕ r ) ⎤
⎢
⎥
⎢⎣ + cos ( (ωs − ωr ) t + ϕ s − ϕ r ) ⎥⎦
The low pass filtered output of the phase sensitive detector is a function of
the phase difference between the signal and reference voltages – and when
on resonance (ie, ωs = ωr) is maximum if their phase difference vanishes.