This manual is best viewed at the URL address " http://solidstate.physics.sunysb.edu/teach/phy515/nmr/ ". Most equations need the SYMBOL typeface. If a printed version is made the relevant links (the description of the spectrometer, the short history of NMR and the sample spectra) should also be printed. .
Before coming to the lab, read this manual and look up at least one or two books/articles from the list of references. During the first day of the lab work, read the equipment manual for the magnet, the spectrometer and the oscilloscope; explore the computer software. You can find questions at various places in this manual (they appear green on the computer screen or on a color printout); be ready to answer them in you report or in the discussions with the instructor.
Completion of all studies described in this manual are not required to obtain a good grade. Discuss your plans with the TA or the professor responsible for the measurement at an early stage of the project. Keep a good record of your work. In your report explain your observations, and compare them to published results (if you find any). The better you do the parts you agreed to do, the better will be your grade. Also, be ready to answer the questions in this manual. They are typeset in green in the color printout.
Nuclear magnetic resonance (NMR) is widely used in physics and chemistry to characterize materials (see the Appendix for a short history). NMR is a microscopic method, in the sense that it probes the nuclei and their immediate surroundings. It is possible, for example, that there are various local magnetic fields at various sites within a certain solid. A magnetic measurement by a magnetometer measures an average field. NMR, on the other hand, is capable to measure the local fields at atomic nuclei. Another example could be a molecule containing an atom, for example carbon, at various different different configurations. The NMR signal is different for each one of these configurations. The microscopic nature of the NMR measurement makes it extremely useful, and sometimes unique. Of course, in order to have a signal of detectable magnitude, we need many of the same configuration to occur in the sample. The nuclei investigated in this experiment, hydrogen (proton) and fluorine are very abundant, and they have particularly strong signals.
In a continuous wave (cw) NMR measurement the energy absorption of the nuclear spins is studied as a function of frequency. The more advanced pulsed NMR (pNMR) instrument offers a wide range of opportunities for the study of spin dynamics. You will use a pNMR spectrometer made by "TeachSpin" for educational purposes. The principles of operation of this device are identical to a top-of-the-line NMR instrument.
See the Appendix or the equipment manuals of the spectrometer and oscilloscope for details.
IMPORTANT
The power supply and the magnet need water cooling. The power supply should be switched on and off only after turning the current setting potentiometer to ZERO.
Do not disconnect the sample box from the transmitter. This (blue) cable is the only one that is the "screw on" type.
If you need further help for instrument settings, check the Appendix for typical spectra obtained on this spectrometer.
If the equations look funny, read the introduction.
For nuclei with non-zero magnetic moment m, a static magnetic field B0 leads to energy levels spaced at intervals DE = g(h/2p) B0 , where g = m /J is the gyromagnetic ratio and J is the angular momentum quantum number of the nuclei. A spin resonance occurs if a radiofrequency (rf) field is applied so that the energy quantum equals the energy level spacing. The energy absorption of the nuclear spins exhibits a resonance at the frequency w = gB0. The various interactions lead to a finite resonace width Dw .
Subjected to a static magnetic field, a portion of the nuclear spins lines up parallel to the field. The application of an appropriate radiofrequency pulse (the p/2 pulse) rotates the spins perpendicular to the field. Still under the influence of the static field, the spins start to precess at the frequency determined by the magnetic moment and the field. This precession creates a weak rf signal that is picked up by the receiver, and it is mixed with the original reference frequency. Over time the coherent precession becomes incoherent, and the signal decays. This is the free induction decay or FID.
If the conditions are right, another strong rf pulse (the p pulse) sets up the spins so that the coherence is gradually recovered, and then lost again. The rf signal picked up in this process is called the spin echo. The time delay between the spin echo and the second pulse is exactly the same as the time between the first and second pulse. More elaborate pulse sequences can create a train of repeated echoes.
Spins subjected to a combination of external static magnetic field and an rf oscillating field of magnitude B1 and frequency w, have a rather complex response. The situation is further complicated by the interactions the spins are subjected to. Fortunately, we do not need to consider every detail of the spin response to understand our experiments. A convenient phenomenological theory, due to A. Bloch, incorporates most of the basic ideas we need to know. Bloch treated the spins as classical objects, and represented the effect of the interactions by relaxation times, as discussed later. Advanced textbooks (like Abragam's) may be consulted to understand why does classical mechanics work so well for spins that are fundamentally quantum mechanical objects. Notice that the Bloch equtaions do not describe the behavior of any individual spin, but they work reasonably well for a collection of many spins.
In the absence of oscillating field the equilibrium value of the z component of the magnetization is related to the population difference of the up spins and down spins (n+ and n-, respectively) by Mz = M0 = g (h/2p) (n+ - n-)/2 . As discussed by Melissinos (p. 352) and many other books, for reasonable fields and temperatures the thermal energy is much larger than the magnetic energy (make and estimate for the proton, with the values of B0=3.5kG and room temperature!), the thermal equilibrium occupation of the two spin states is nearly equal. The small fractional difference between n+ and n- is responsible for our signal.
Assume the spins are kicked out of the equilibrium. How will they return? Bloch assumed a simple relaxational dynamics, described by the rate equations
| dMz/dt = (M0 - Mz)/T1 | |
| dMx/dt = - Mx/T2 | Eq. (1) |
| dMy/dt = - My/T2 |
T1 and T2 are called the longitudinal and the transverse relaxation times, respectively. The inverse of arelaxation time, 1/T , is called a relaxation rate.
There is a clear difference between the z axis direction (the direction of the static external field) and the other two directions is the system. Due to the conservation of energy, the interaction between the spins cannot change Mz, but no similar restriction applies to Mx and My . The relaxation time T1 is larger than or equal to T2; sometimes T1 can reach minutes or hours. The relaxation of the z component of the magnetization involves energy transfer from the spin system to the surrounding thermal bath (atoms, electrons, spins, etc.); therefore the longitudinal relaxation time is often called the spin-lattice relaxation time. This term is used even for liquids where there is no crystal "lattice".
Along the z direction the magnetization will reach M0. Along the other two directions the equilibrium magnetization will be zero, as the magnitude of the magnetization follows
| M(t) = M(0) exp {- t / T2 }. | Eq.(2) |
To describe the time development of the average magnetization of the spin system in magnetic field, the Bloch equations include the term dM/dt = g M x B. A frame of reference, rotating with angular frequency w is introduced (for details, see the recommended literature), and
| dMz / dt = - gMyB1+ (M0 - Mz) / T1 | |
| dMx / dt = gMyb0 - Mx / T2 | Eq.(3) |
| dMy / dt = g(MzB1-Mxb0) - My / T2 |
Here the M's are components of the magnetization and b0 = B0 - w/g. It is assumed the the external field is along the z axis and the rf field is represented by B1 pointing in the x direction of the rotating frame of reference.
The Bloch equations work for cw and pulsed NMR as well. In the presence of a weak continuous rf field the spin resonance causes an absorption of electromagnetic energy around the resonance frequency w = g B0. The solution of the Bloch equations tells us that the resonance is a Lorentz curve, with a full width at half maximum of Dw = 2 / T2.
More important to us is the understanding of the effects of rf pulses on the spin system. If w = g B0 , the effective field is zero, b0 = 0. For a time period much shorter than T1 or T2 the equations become dMz / dt = - gMyB1, dMx / dt = 0 and dMy / dt = gMzB1 , describing the rotation of the spin along the x axis. If the rf is applied for the right time (depending on the rf amplitude B1), the angle of rotation can be exactly 90o, and the spins end up perpendicular to the external field. This is the p/2 pulse. The p pulse is twice las long, and it creates spins that are turned to the direction exactly opposite to their original direction. The time separation of the p/2 and p pulses is traditionally denoted by t. In the TeachSpin instrument the p/2 pulse called "A pulse" and the p pulses is the "B pulse".
The signal in the receiver coil is proportional to the magnetization component perpendicular to the z axis.The amplitude of the FID signal follows M(t) as described by |Eq. (2), and the integrated intensity of the FID signal, plotted as a function of the frequency, follows the same Lorentzian resonance curve that is obtained in cw NMR. The decay time of the FID amplitude is inversely proportional to the width of the resonance and T2 Dw/2 = 1.
For a collection of non-interacting spins in a perfectly homogeneous magnetic field, T1 and T2 are infinitely long, and the FID signal would never decay. (Is it really possible to have a system like that?) In a real magnet the field is always inhomogeneous, and that leads to a slightly different precession rate for each spin. Right after the p/2 pulse, all magnetic moments are perfectly aligned perpendicular to the external field, and the net moment is large. Over time, since some spins precess faster, and others slower, the spin vectors do not stay aligned. The net rotating moment, being the vector sum of the individual moments, begins to decay. A good analogy to the direction of the spins is the position of the cars on the racetrack: they start as one bunch, and they are close to each other for a while, but they spread all over the track during the race.
In the framework of the Bloch theory the "spread out time" is an important source of the transverse relaxation. Calculate: for a given inhomogeneity DB, how much time does it take for the slower spins to lag behind so that they are just opposite to the faster spins? The result of this calculation is a good estimate for the transverse relaxation time T2 due to inhomogeneous broadening.
Imagine to have a strange car race: 20 minutes after the start, when the cars are spread out evenly over the track, a referee gives a signal, and every car turns back and starts to race in the opposite direction, at exactly the same speed as before. As long as the cars do not obstruct each other, and there are no mechanical failures, the seemingly disorganized collection of speeding racecars will exhibit a remarkable show: in another 20 minutes they cross the start line again IN ONE CLOSE GROUP.
The p pulse in the spin echo experiment acts like the referee in the car race. (The analogy is somewhat flawed, since the cars stay in place and race backwards, but the spins switch orientation and keep precessing in the same direction). The echo corresponds to the racecars bunching together at the starting line. As long as there is no interaction between the spins, and the precession frequency of each spin is independent of the time, the echo intensity does not depend on t and equals to the original FID signal intensity.
What if in the "echo race" the cars obstruct each other's motion? What if some of them break down, or cannot keep their speed? All of these factors will eventually lead to the same result: not all cars will come back together at start line. This corresponds to a decrease of the echo intensity in our experiment. By measuring the echo amplitude we eliminate the unwanted effects of magnetic field inhomogneities. In terms of the Bloch equations, the transverse relaxation time can be (approximately) written as as the sum of relaxation rates due to field inhomogeneity and all other effects:
| 1/T2*= 1/T2 '+ 1/T2, | Eq.(4) |
where 1/T2 ' may be due to the imperfections of our magnet, and 1/T2 originates from the intrinsic spin relaxation. While T2* is obtained from the free induction decay, the relaxation time T2 is measured by looking at the decay of the echo amplitude as a function of t :
| Mecho(t) = M(0) exp {- t / T2 }. | Eq.(5) |
The relaxation rate 1/T2 can be further split into several components depending on the processes causing the decay of the echo amplitude. There are several sources of spin relaxation we can study in our apparatus: spin diffusion in an inhomogeneous static field, spin-spin interactions, and fluctuating internal magnetic fields.
The spins reach thermal equilibrium over a time scale of T1. When t is comparable or longer than T1, the spins will stop precessing before they have a chance to produce an echo (in our car race analogy, the most competitors give up before the race ends). Therefore the echo intensity will decay at least as fast as 1/T1; in other words T2 may not be longer than T1. In systems where the spin lattice relaxation is strong, all other contributions can be neglected, and the deacay rate of the echo amplitude is a measure of T1.
Assume now that T1 is long relative to T2. How can we measure T1? Consider this idea: each time we do a spin echo sequence the nuclei are kicked out of equilibrium. If the next sequence comes too soon (that is within a time interval comparable to T1), the signal intensity will decay, since Mz can not recover its thermal equilibrium value. In fact, for samples of really slow spin-lattice relaxation, the signal may entirely disappear if the repetition rate is too fast. This phenomenon is used to measure T1, by recording the intensity of the signal as a function of the repetition rate.
Up to this point we focused our attention to the nuclear spins moving under the influence of the external field. But each nucleus creates a magnetic field of its own, and other nuclei will be influenced by this field. Estimate: what is the average distance a between the H nuclei (protons) in water? What is the "typical" magnetic field dB due to the magnetic dipole moment of a proton at distance a? (see Melissinos Eq. 3.16) The real theory of spin interactions involves quantum mechanics, but an estimate of spin-spin relaxation time can be obtained by treating this field as a random magnetic field inhomogeneity, leading to the loss of phase coherence between the spins. We obtain 1/T2 = gdB . This estimate is valid for a solid, where the spins do not move, and the perturbation field can be treated as quasi static. Spin-spin interactions conserve the energy of the system, and therefore they do not contribute to the spin lattice relaxation T1.
Magnetism due to electrons is much stronger than nuclear magnetism. (Why?) Fortunately for NMR experimentalists, in a solid or liquid most of the quantum states are occupied by electrons with two opposite spins, therefore the magnetic field is compensated. If a magnetic phase transition occurs in the system, the electron spins develop a static ordering and the NMR line suffers enormous shift or broadening (see, e.g Abragam p. 210). In our experiments, we can add a controlled amount of magnetic impurities (Fe ions) to water. Estimate the average distance d between Fe ions in a 1 mole solution, and what is the magnitude of the average magnetic field of dipole moment mB (Bohr magneton) at a distance of d? This random field can cause very strong broadening of the signal. The interaction of the NMR spins and the magnetic impurities is also an important channel for the spin-lattice relaxation.
In a liquid the atoms move around freely, and the nuclear spin experiences a fluctuating magnetic field. The average field is equal to the externally applied field; the fluctuations are due to the random dipole fields of the constituents of the liquid (in our experiments these fields may include the magnetic filed of the Fe ions, other protons, etc.) If the fluctuations are fast then each nucleus will feel the same average field, resulting in a sharp resonance line and long relaxation times.
An interesting question to ask is: how fast should the field switch from one (random) value to the next one in order to have a motional narrowing? What is the critical time scale separating the broad and the narrow signal? A natural guess would be that the relevant time is the period of the oscillation frequency, 1/w . Detailed calculations show that in order to see motional narrowing it is enough to have much slower fluctuations, on the time scale of 1/Dw, where Dw is the width of the resonance line before the narrowing occurs.
The issue of motional narrowing can be addressed without facing the complicated mathematics of the spin dynamics. Imagine we have a wave generator, providing a variable frequency sine wave output. We analyze the output by a spectrum analyzer or by driving a speaker and hearing the sound. Let us assume that we switch between 1000Hz and 1010Hz in random time intervals of about 1 second. There is no doubt that two peaks appear in the spectrum and we will hear two distinct tones. If the switching happens very fast, like in every 0.001sec, one expects to have one peak in the spectrum, and hear only one tone. The question is: how will the cross-over happen between the two limits? Think about this problem. If you decide to do calculations, make sure that the wave is continuous, i.e. there are no sudden jumps in the wave when the frequency is switched.
The echo method works to eliminate the effects of magnetic field inhomogeneity only if the fastest spin during the forward "race" is also the fastest when the race is run backward. If the atoms move and the field felt by the spin changes, the precession frequency also changes, and the echo is imperfect; the amplitude is reduced. In liquids random diffusion describes the motion quite well. According to microscopic calculations (see, for example, Slichter, p. 42) the echo amplitude decays as
| M(t) = M(0) exp {-(2/3)Dg2t3(dB/dz)2}, | Eq.(6) |
where D is the constant of diffusion, and the magnetic field varies in the z direction with a gradient dB/dz. Notice that, in contrast to Eq (5), in this case the decay of the echo amlitude is not a simple exponential.
When a magnetic field is applied on a sample, it will influence the electronic states as well. The change in electronic states will be felt by the nuclei in two ways: the electrons will create local magnetic fields, and they will also interact with the nuclei via the Fermi contact interaction. These processes are well described by a local magnetic field that is proportional to the external field, and leads to a shift of the resonance frequency relative to the frequency of the "bare" nucleus. The amount of shift (the chemical shift) depends on the electronic configuration, and it is used extensively to characterize the atomic or molecular environment of the nucleus.
Do not attempt to do all of the projects listed below, but discuss your plans with the instructor before starting the work.
This is a long list, and you do not have to read all. But you should certainly take time to read some, so that you understand the basic ideas.
Books
Papers
Handbook
CRC Handbook of Chemistry and Physics. Reference for the NMR frequencies.
Created by Laszlo Mihaly in 1997; last updated 8/26/2004. Portions of this lab notes are based on the Equipment Manual provided by "TeachSpin", the manufacturer of the pulsed NMR spectrometer.