The goal of this experiment is to measure the approach to thermal equilibrium of a system of nuclear spins. The spins (protons) are placed in a large, static magnetic field that induces a nuclear magnetization. The spins (or magnetization) are driven out of equilibrium by a short radio-frequency (rf) pulse. Microscopically, the proton spins are coupled to the heat reservoir in which they sit by: (1) interactions with unlike spins in the sample such as paramagnetic ions, (2) resonant interactions with like spins precessing at the same frequency (other protons), and (3) interactions with other degrees of freedom such as lattice vibrations in a solid or collisions in a liquid. These interactions cause the magnetization to relax back toward its equilibrium value.
The approach to equilibrium is detected through nuclear magnetic resonance (NMR) using a pulsed NMR spectrometer from TeachSpin Enterprise. The spectrometer includes magnets that provide a uniform, static field of about 0.28 tesla, an rf coil to apply an alternating magnetic field in a direction perpendicular to the static field, and a sensing coil to detect the presence of a net magnetization precessing in a transverse direction.
Consider an aqueous solution. The water molecules contain proton nuclei
that have two spin orientations with respect to a static field B0,
‘spin-up’ or ‘spin-down’. The energies for the two orientations are 
,
in which m is the projection of the proton’s
magnetic moment along the direction of B0.
When allowed to come into thermal equilibrium, the number of ‘spin-up’
protons, N+, (with moments along the field)
will be greater than the number of spin-down protons, N-.
The relative probability for finding spins in either of the two orientations
in equilibrium is given by the Boltzmann factors 
.
The excess of ‘spin-up’ protons amounts to a nuclear magnetization along
the static field.
Due to the torque exerted by the static field on the canted moments,
they precess around the field with a frequency given by 
,
in which h is Planck’s constant and Iz=
½ is the projection of the spin along the B-direction. The
precession frequency for protons is 
,
or f0~ 15 MHz for B0~0.28
tesla.. The phenomenological Block equations describe the motion
of the longitudinal and transverse components of the magnetization under
the influences of the torque exerted by the static field and microscopic
relaxation processes. The Block equations are (see, e.g., Kittel or Slichter):
, (1)
in which Mz is the component of the magnetization along B0, M0 is the equilibrium magnetization in the z-direction, and g is the proton’s magnetogyric ratio. Thus, Mz relaxes back to M0 with a phenomenological relaxation time T1. T1 is called the longitudinal, or spin-lattice, relaxation time. For the transverse component x, one has
, (2)
and similarly for y, in which T2 is the transverse, or spin-spin, relaxation time..
A resonance condition occurs when there is an rf-field applied transverse to the static field at the precession frequency. It will rotate the ensemble of magnetic moments away from the static-field direction. The spectrometer is designed to generate rf-pulses of well-defined field-amplitude and duration that rotate the ensemble through desired angles. Especially useful for analysis of nuclear relaxation are sequences of pulses that rotate the magnetization through angles of p and p/2. Please read how particular pulse sequences can be used to measure relaxation times by reading the introductory TeachSpin notes or in Slichter.
The sample volume in the vial is about 1 cm3. Estimate the magnetization induced when a vial filled with pure water is placed in a field of 0.35 tesla at room temperature (293 K).
9.2 Getting started
Read and follow the "Getting Started" section in the TeachSpin manual ito become familiar with the operation of the spectrometer and digital oscilloscope. A few cautionary statements: (a) The magnetic fields near the permanent magnets are strong, so keep your analog watch away. (b) Sample vials are stored in the two plastic boxes. The vial tops just ‘pop’ off. Please be sure to place an o-ring around the top of a vial before inserting it in the magnet.
9.3 Experiments
Choose from one of the following two experiments or from alternatives listed in the TeachSpin manual.
A. Relaxation times for water doped with paramagnetic ions.
Paramagnetic ions have large electronic magnetic moments that strongly affect relaxation times of protons. Solutions of such ions can be made by dissolving salts of transition metal or rare-earth ions such as Fe(No3)3, NiSO4, CuSO4. Such salts often have bright colors. Relaxation times T1 and T2 can be measured to determine how various factors affect the relaxation.
A.1 Dependence of relaxation times on concentration of one kind of ion. Make solutions of one salt with a wide range of concentrations.
A.2 Dependence of relaxation times on the type of dissolved paramagnetic ion. Compare relaxation times at a single concentrationwith literature values of the moments of the ions.
B. Relaxation times in glycerine-water mixtures.
Measure relaxation times in solutions of different strength and determine what dependence there may be on the glycerine concentration or viscosity.