Nuclear Magnetic Resonance and the Measurement of Relaxation Times of Acetone with Gadolinium Xia Lee and Albert Tsai June 15 2006 1 1 Introduction Nuclear magnetic resonance NMR is a spectroscopic technique frequently employed by chemists to elicit the structures of chemicals especially organic molecules in a liquid or solid form In the field of medicine where a 2 dimensional version of NMR termed magnetic resonance imaging MRI is employed this technique has also had significant impacts on the development of non invasive diagnosis of diseases First introduced in the 1940 s by Felix Bloch and Edward Purcell to measure the magnetic moment of nuclei in liquids and solids NMR is based on protons in a nucleus having an intrinsic spin angular momentum thus a magnetic moment When a constant magnetic field is applied a nucleus will resonate like a mechanical oscillator when driven in to an excited energy state by an EM wave in the radio frequency for NMR at the correct frequency which is determined by the strength of the magnetic field and the magnetic moment of the nucleus Traditionally NMR was done by sweeping the magnetic field strength while applying a continuous EM wave and measuring where the signal from the nuclei emitted peaked continuous wave NMR or CW NMR However modern NMR is usually done by applying a pulse of RF waves pulsed NMR which contains a broad spectrum of frequency components and then measuring the resulting radio signal emitted from the nuclei after the pulse This signal is termed the free induction decay FID as it decays similar a damped oscillator However modern NMR is usually done by applying a pulse of RF wave pulsed NMR which contains a broad spectrum of frequency components and measuring the radio signal termed the free induction decay FID as the signal decays like a damped oscillator emitted from the nuclei after the pulse Using a Fourier transform of the FID signal the resonance frequency of the nuclei can be determined Moreover pulsed NMR can provide information concerning the physical properties of the nuclei measured One of the properties is the relaxation times or the time required for the nuclei in the sample to return to their ground state after being saturated by a strong EM pulse of the sample In this experiment we attempt to measure the relaxation times of acetone using proton NMR with various concentrations of gadolinium added as well as glycerol which will serve as a standard to compare our results to measurements from other experiments In this experiment we attempt to measure using proton NMR the relaxation times of acetone with various concentrations of gadolinium added as well as glycerol which will serve as a standard by which to compare our results to measurements from other experiments 2 Theory of Experiment For a formal and comprehensive treatment on the theory of NMR and relaxation times please refer to a textbook on the subject of NMR spectroscopy 1 This section will instead focus primarily on the theory directly relevant to our experiment 2 2 1 Measuring T1 To measure T1 the spin lattice relaxation time a 90o pulse is first applied to the sample so that the z component of the total magnetization vector is rotated onto the xy plane which induces a FID Then another 90o pulse is applied after a certain time to again rotate the magnetization that decayed back to the z axis into the xy plane where the precession of the magnetic moment becomes measurable The 90o pulse can be repeated in series to obtain a clearer signal through averaging Mathematically this process can be described by a simplified version of the Bloch equations in the z component 2 dMz M0 Mz dt T1 This first order differential equation has a solution of Mz M0 1 e T1 Linearizing this equation gives Mz ln 1 M0 T1 Mz is directly related to the signal strength of the FID and M0 can be measured from the o FID by applying a 90 pulse after allowing the sample to settle for several seconds Therefore Mz by plotting ln 1 M0 versus T1 can be derived from the slope of the linear fitting function 2 2 Measuring T2 In theory T2 can be measured by examining the envelope of the FID directly However inhomogeneity in the magnetic field applied causes the individual magnetic moments of the nuclei to precess at slightly different frequencies and de phase where they will begin to partially cancel each other as the signal decays This will reduce the component of the total magnetization vector in the xy plane faster than by T2 alone This effective decay time is termed T2 inhomogeneous and is always less than the pure T2 To solve this problem a 180o pulse is applied after a time after the initial 90o pulse This reverses the phase of the individual magnetic moments of the nuclei so that the higher frequency moments now lag the lower frequency ones and vice versa After a time following the 180o pulse the magnetic moments will be in phase again and generate a signal a spin echo whose strength depends solely on the pure T2 and the time that has elapsed since the 90o pulse 2 Mathematically this process can also be described using a simplified version of the Bloch equations in the transverse direction 2 dMxy Mxy dt T2 3 This equation has a solution of Mxy Mxy0 e T2 Linearizing this equation gives Mxy ln Mxy ln Mxy0 ln Mxy0 T2 By plotting ln Mxy 2 versus 2 T2 can be obtained from the slope of the linear fitting function 2 3 Effects of Gadolinium on Relaxation Times The longitudinal relaxation process T1 termed the spin lattice relaxation is determined by the time required for an excited proton in a molecule to transfer its spin angular momentum to other materials in its environment Gadolinium having 7 unpaired electrons in its highest f orbital is a paramagnetic atom that provides a route to quickly allow the protons to transfer their spin angular momentum Thus adding a gadolinium compound into a solution is expected to decrease the T1 of the solution On the other hand the transverse relaxation process T2 termed the spin spin relaxation is determined by the time needed for excited protons in a molecule to transfer their spin angular momentum to each other Since this entirely an intra molecular process addition of gadolinium into the solution should have little or no effect on T2 3 3 1 Results Experimental Setup The sample is placed in a glass test tube and secured in a sample holder between two Helmholtz coils where the magnetic field generated by the coils is relatively homogeneous The Helmholtz coils