Classical behavior of a magnetic moment in a magnetic field

If a magnetic moment μ is place in a uniform magnetic field B it experiences a torque τ = μ × B.  Let the magnetic moment be proportional to the intrinsic angular momentum S,  μ = γS, and let B point into the z-direction  B = B₀k.  Then τ = dS/dt = ω₀S × k, where ω₀ = γB₀.

Therefore dS_z/dt = 0,  dS_x/dt = ω₀S_y,  dS_y/dt = -ω₀S_x.  Solving these coupled differential equations we obtain

S_x = Acos(ω₀t + φ),  S_y = -Asin(ω₀t + φ). 

A and φ are arbitrary constants determined by the initial conditions.  The component of S perpendicular to B rotates clockwise (cw) about the z-axis with angular frequency ω₀.

If you place yourself into a coordinate system rotating cw about the z-axis with angular frequency ω₀, you observe a stationary magnetic moment.  In your accelerating frame a fictitious magnetic field -B₀k, cancels out the real magnetic field B = B₀k.

Assume that at t = 0 a time-varying field B₁(t) is added to B₀.  Assume that for t > 0 we have B = (B₁cosω₀t, -B₁sinω₀t, B₀).  The z-component of B is constant, while the component of B perpendicular to the z-axis rotates cw about the z-axis with frequency ω₀.  If you place yourself into the rotating coordinate system, you observe a constant magnetic field B₁, pointing along the x’ axis of that system.  In this coordinate system S rotates about the x’ axis with angular frequency ω₁ = γB₁, as long as B₁(t) is turned on.  When B₁(t) is turned on, S again becomes stationary.  Starting out with S pointing along  the z-axis, we can turn on B₁(t) for a time t = T₁/4, with T₁= 2π/ω₁, and rotate S into the xy-plane, or turn on B₁(t) for a time t = T₁/2 and flip S.

Quantum Mechanics
In quantum mechanics we can only calculate the mean or average value of the Cartesian components of S.  We find that <S_x>, <S_y>, and <S_z> behave like the classical components of S.  So while we cannot make definite predictions about the orientation of an individual magnetic moment μ, we can make predictions about the orientation of the magnetization M, which is calculated by averaging over many magnetic moments μ.

Since <S_x>, <S_y>, and <S_z> rotate like the classical components of S, the components of the individual spins, even though their values cannot all be known, must rotate through the same angles as well.  This allows us to manipulate spin components in a well defined way, without actually knowing their value.

NMR

A continuous wave experiment can be performed by sweeping the frequency of the rotating field B₁ through the resonance frequency ω₀ and by monitoring the power output of the power supply that supplies the current that produces B₁.  Oscillations in the power output indicate that the frequency is sweeping through the resonance frequency.  If γ is known, ω₀ = γB₀ yields B₀.  In this way the strength of an unknown magnetic field can be measured.  If B₀ is known, ω₀ = γB₀ yields γ.  Instead of sweeping the frequency ω of the rotating field, a continuous wave experiment can also be performed by sweeping B₀ while holding ω constant.

Experiments can also be performed with pulsed magnetic fields.  Assume that at time t₁ = π/(2ω₁) we turn off B₁.  Then for t > t₁ the magnetization lies in the x-y plane and rotates cw about the z-axis.  As the magnetization rotate about the z-axis, it will induce a current in a pick-up coil located with its axis along the x-axis.  Plotting this current versus time yields a sinusoidal wave form.  This current will decay as a function of time due to dephasing.  This is called free-induction decay.  The spin-spin relaxation time constant T₂ is the time characterizing the decay of the transverse magnetization.  M_xy = M_xy0e^-t/T2.  Loss of transverse magnetization is due to molecular interactions and field inhomogeneity due to local magnetic fields.

At equilibrium the net magnetization is M = (0,0,M_z).  As we have seen, it is possible to make M_z = 0.  The time constant characterizing the rate with which M_z returns back to its equilibrium value is called the spin-lattice relaxation constant T₁.  M_z = M₀(1 - e^-t/T1).  Spin lattice relaxation is the result of the nuclei transferring energy to the surrounding molecules as thermal energy.

In pure water T₁ ~ T₂ ~ 2 - 3s.  In biological materials T₂ < T₁.