Measurement Of T₂ Relaxation Times Near Phase Transition For Cyclohexane

Rory Alt

Methods of Experimental Physics Spring 2013

University of Minnesota

Abstract

Investigation of change in the spin-spin relaxation time, T₂, was conducted by pulsed nuclear magnetic resonance (NMR) spectroscopy techniques for cyclohexane, C₆H₁₂, as the sample went through phase transition from a solid state to a liquid state. It has been established that temperature and sample phase state effects NMR measurements, though there is not complete understanding regarding the behavior of nuclei during a phase transition. The individual solid and liquid T₂ relaxation times were measured simultaneously as a function of time while the sample melted in order to attempt to measure their fraction. An asymptotic approach to an all liquid sample was observed and the solid and liquid T₂ times were successfully separated from a single signal.

Introduction

With vast application to the medical and chemistry fields, nuclear magnetic resonance spectroscopy is a versatile technique. Knowledge of the technology is essential for development and utilization of new applications [1]. NMR occurs when certain atomic nuclei are immersed in a static magnetic field, providing a polarization, and subjected to a second, oscillatory magnetic field from which can be determined relaxation times T₁ and T₂ [2]. The oscillatory field is used to rotate the net magnetization from alignment with the static field, where its return to equilibrium is characterized by the two relaxation times.

This study investigated the resonance of the hydrogen nuclei in cyclohexane molecules. Cyclohexane was chosen due to its drastic difference in T₂ times between solid and liquid states, with solid being on the order of milliseconds and liquid being on the order of seconds [3]. This made it easier to differentiate the two when measured simultaneously. This type of measurement is analogous to the differentiation of tissue types in MRI since different substances in different states will have distinguishable characteristic times. Successful separation of the solid and liquid signals was accomplished and the rate at which the frozen sample approached 100% liquid was found to be asymptotical.

Theory

The Hydrogen nucleus, having a net spin from its single proton, was the component in cyclohexane that was studied. In order for the magnetic resonance of a substance to be observable, its magnetic constituents must have both a magnetic moment,

, and an angular momentum, . Thinking of hydrogen’s proton nucleus effectively as a small spinning bar magnet with an effective north and south pole, its magnetic moment and angular momentum can be related by [2]:

(1)

where γ is the gyromagnetic ratio, the ratio of a particle’s magnetic dipole moment to its angular momentum. When surrounded by a static magnetic field in the z-direction, B₀, the magnetic energy is given by [2]:

(2)

where I_z is the net spin of the nucleus in the z direction.

Since the proton has a spin of one half, quantum mechanics allows only two values of quantum spin, ±1/2. This allows for only two magnetic energy states for a proton in such a magnetic field; the low-energy state, where it is aligned parallel to the external field, and the high energy-state, where it is aligned anti-parallel to the external field. These two states have an energy separation, which is relatable to an angular frequency by [2]:

(3)

where ω₀ gives the fundamental resonance condition. The gyromagnetic ratio of the proton is 2.675x10⁴ rad/s-gauss, which can be applied to equation (3) to give the resonant frequency of the proton in the constant magnetic field known as the Larmor frequency [2]:

(4)

Figure 1: Left: Spin modeled as a rotating magnetic moment vector. Right: net spin magnetization of multiparticle system.

In the cyclohexane sample, there is initially no net nuclear magnetization due to local equilibrium alignments. Once the group of spins is placed in a magnetic field in the z-direction, however, a nuclear magnetization in this direction is eventually established. A vectorial schematic of the system is shown in Figure 1. Each spin aligns in one of the two possible orientations with the number of spins per unit volume in the lower energy level, N⁺, slightly outnumbering that of the upper level, N⁻. The population ratio when in thermal equilibrium can then be determined from Boltzmann statistics [2]:

(5)

where k is Boltzmann’s constant and T is the temperature. The longitudinal magnetization and thermal equilibrium magnetization per unit volume are given respectively, at equilibrium, by [2]:

(6)

where N=N⁺+N^-. This net magnetization can be changed when the nuclear spin system is exposed to energy of a frequency equal to the energy difference between the spin states, on the order of 10^-7eV in this experiment. Putting this energy into the system can saturate the spin system, making Mz=0. The system, due to the constant magnetic field B0 in the z-direction, will then proceed to return to the equilibrium state. Assuming this rate of approach is proportional to the separation from equilibrium, it can be described by the differential equation [2]:

(7)

where the time constant, T₁, which describes how M_z returns to its equilibrium value, is called the spin-lattice relaxation time [2]. This equation is illustrated in Figure 2. The rate at which the magnetization approaches its equilibrium value depends on the type of nucleus, the magnetic-field strength, and the chemical environment [3].

Figure 2: Exponential growth of z^--component of magnetic field as a function of time. Spin-lattice relaxation time is shown along with the relaxed magnitude of the magnetic field [3]

In thermal equilibrium, there is no transverse component of the net magnetization of the sample in the x-y plane. One can be induced, however, by application of a time dependent rotating magnetic field. Modeling the system as appropriately small current loops in the magnetic field, the torque,

, on each loop is

, which causes the angular momentum of these loops to change as. For our sample of protons, this gives the generalization as the classical motion of the net magnetization for the entire sample [2]:

(8)

with being the total magnetic field. If a rotating, circularly polarized magnetic field of frequency ω in the x-y plane is applied along with the constant magnetic field, , the total field is given by [2]:

(9)

In order to analyze this magnetization, it is helpful to consider a noninertial rotating coordinate system rotating at the same angular frequency as the magnetic field, with its axis in the z’=z direction. Here, the field appears stationary and aligned with the x’-axis. This gives rise to an effective magnetic field along the z’-direction with magnitude

must always be zero no matter its orientation. This is only true if an effective field of is added. Applying the rotating coordinate system to the total magnetic field from equation (9) gives the total magnetic field in the rotating frame [2]:

(10)

This can then be substituted into equation (8), thus showing that the magnetization will precess about this effective magnetic field in the rotating frame. If the rotating magnetic field were at the frequency ω=ω₀= γB₀, then

. The magnetization, M_z, then begins to precess about this magnetic field at a rate . This argument is illustrated in Figure 3. Removing the B₁ field at the instant the magnetization reaches the x-y plane then defines a 90° pulse that caused a transient realignment where there is a net magnetization of M_xy in the x-y plane. Intuitively, if the magnetic field were applied for twice the time, it would produce a 180° pulse with magnetization –M_z. In the rest frame, then, the magnetization not only precesses about B₁, but also rotates about the z-axis during the pulse. Once the transmitter pulse is over, the spectrometer is able to detect the net magnetization precessing about the constant magnetic field,

in the x-y plane [2].

Figure 3: Left: Effective magnetic field in rotating frame. Right: Magnetization rotation due to pulsed magnetic field. [3]

If a 90° pulse is applied to a sample in thermal equilibrium, the net equilibrium magnetization will be rotated into the x-plane where it precesses about the constant magnetic field. This x-y magnetization will decay according to the differential equations [2]:

(11)

where T₂ is the spin-spin relaxation time. This is illustrated in Figure 4.

Figure 4: Exponential decay of x and y magnetization after excitation as a function of time. Spin -- spin relaxation time is labeled. [3]

Time constant T₂ gives information about the distribution of local fields at the nuclear sites. After a 90° pulse, the majority of the protons are in phase but soon get out of phase, reducing the net x-y magnetization to zero, due to the protons precessing about

at a distribution of frequencies dictated by local field effects at the nuclear sites. Simply plotting the decay of M_xy after a 90° pulse gives a signal known as the free induction decay (FID). Provided that the magnet’s field is perfectly uniform over the entire sample volume, the time constant associated with this decay would be precisely T₂. In reality, the magnet’s non-uniformity is responsible for the FID constant for samples whose T₂ times are greater that the apparatus’s resolution, in this case accurate for T₂<.3ms, dictated by the uniformity of the field. To overcome this limitation, multi-pulse sequencing was developed. Suppose a first 90° pulse is produced with a second, 180° pulse turned on a time τ later. The 180° pulse allows the x-y magnetization to rephase to the value it would have had with a perfect magnet by allowing the flipped precession to rephrase completely to a maximum value and decay again, creating an “echo” signal a time € τ after the second pulse. Stochastic fluctuation in the local fields at the nuclear sites are, however, not rephasable by the 180° pulse and lend to a loss of transverse magnetization. A series of 90°- €τ - 180° pulse experiments, varying € τ , and plotting the echo height as a function of time between FID and the echo will give the correct T2 value, with the transverse magnetization measurable by the maximum echo height as [2]:

(12)

Experimental Setup and Data Collection

The main apparatus includes a TeachSpin Pulsed NMR setup consisting of two permanent magnets that produce a highly uniform magnetic field of approximately 3500G, an RF pulse generator, a mixer, and an amplifier, as depicted in Figure 5 [2].

Figure 5: Simplified block diagram of experimental apparatus representing most important junctions of each modular component of the spectrometer. [2]

With this apparatus, samples are typically placed in a vial, which fits in a sample probe between the permanent magnets. An illustration of the sample probe is in Figure 6 and illustrates the Helmholtz transmitter coil, perpendicular to the constant magnetic field, and the receiver pickup solenoid coil around the vial, which produces an electromagnetic field, EMF, due to precessing magnetization, which is amplified [2]. The pulse generator produces is capable of producing two tunable RF pulses, denoted A and B, of variable duration between 1 and 30ms, that correspond to the 90° and 180° pulses when tuned to the proper duration. The time between these two pulses, known as the delay time, is adjustable between 1 millisecond and 10 seconds, and the time between a sequence of A and B pulses, known as the repetition time, is adjustable from 10ms to 10 seconds. This signal is amplified and sent through the RF coils, creating a time dependent rotating homogeneous 12G magnetic field [4]. The x-y component of magnetization from the protons then induces an EMF signal in the pickup coils that is sent to the receiver. From the receiver, a split signal is sent with one part sent to the mixer where it is multiplied with the original RF signal, which is then used to find the difference between the originally sent frequency and the received frequency from the pickup coils [2]. Resonance is inferred when this difference is zero. Equation (4) gives that, for this particular apparatus, a frequency around 15MHz must be generated and implemented via the Helmholtz coils in order to induce resonance. The second part of the split signal is sent to the detector where it is amplified [2]. Both the mixer and detector signals are sent to the oscilloscope from which computer interfacing were utilized for data acquisition and storage by use of a LabView program.

Figure 6: Artistic sketch of sample probe. [2]

It was first necessary to determine a rough value for T₁ in order to determine the minimum repetition time, quoted as around 3T₁, so that the sample had time to return to equilibrium [2]. Since [3] suggested a T₁ on the order of seconds, the repetition time was initially set to the maximum, 10s value to accommodate. This spin-lattice time was determined by first producing a 180° A-pulse and a 90° B-pulse tuned to the resonant frequency. The 90° pulse was verified as such at the maximum observed response signal [2,4]. The minimum induced EMF corresponded to a successful 180° pulse. A plot of these results is included in Figure 7. Since the data followed the expected curve as compared to Figure 2, equation (7) was solved for

, where M₀ is the maximum value, so that a plot of this with the measured values would produce a linear curve whose slope was equal to the negative inverse of T₁. This resulted in a value of ~1.49s, meaning that the maximum repetition time was necessary since the next lowest setting was one second.

Figure 7: Left: Raw amplitude curve for liquid cyclohexane. Right: T1 linearized fit for liquid cyclohexane. [original]

To determine T2 from equation (11), an A-pulse sequence of 90° followed by 90, the maximum, B pulses of 180° was used to create echo signals, as observable in Figure 8 [2,4]. Using a peak finding program in LabView that allowed for distinguishing between minimum expected amplitude and peak width, these echoes were extracted from the A and B-pulses in the data. These echo pulses follow equation (11) as their amplitudes decrease with time. The linearization of this gave:

(13)

As can be seen in equation (13), T₂ was determinable by the slope of the linear curve as a function of time.

Once the methods of determining T₂ were shown to be successful for liquid cyclohexane, a frozen sample was measured while melting. No more than an ice bath was necessary to freeze the cyclohexane since its melting point is 5.5ºc.

Figure 8: Spin echo decay as a function of time for determination of the x^--y magnetization. [4]

Results

An example of the combined peak signal vs. time can be seen in Figure 9.

Figure 9: Combined components of solid and liquid characteristic curves. Distinguishable by the drastic difference in slope. [Original]

The isolation of these two curves was achieved by first realizing that since the solid and liquid T₂ times were expected to be different by a factor of about 1 part in 500, peaks in the combined signal that were at the tail would have negligible contributions from the solid signal since the decay rate would be much quicker. With the sample melting completely in ~100s, each run had about 8 to 9 useful sets of data for every ten second interval of the repetition time. The sample data in Figure 9 was the first in the set acquired and was initiated ~5s after removal from the ice bath. Using equation (11) to estimate the order of magnitude signal drop of the solid and liquid signals, peaks after .042s were used to represent the liquid portion with a theoretical amplitude difference of ~5 parts in 1000. The linearization of these liquid peaks and the solid peaks after subtraction can be seen in Figure 10. A linear least-squares fitting program was utilized to establish goodness of fit. Uncertainties in the liquid signal was determined by estimating the uncertainty of the peak data, based on the noise seen in the signal and the resolution of points over each peak, and propagated through equation (13). For the isolated solid signal, the uncertainties from the peak resolution and the subtracted liquid signal were propagated through equation (13) for the linearization. The reduced chi² for the isolated solid signal was found to be 1.098, close to the expected value of 1. The resulting T₂ for solid cyclohexane was found to be (3.46±0.0581)ms, on the order of the estimate from [3].

Figure 10: Left: Initial determination of liquid T2 for separation from solid signal. Right: separated and linearized T2 signal for solid [Original]

From the T₂ value obtained for the isolated solid portion of the signal, a more accurate value for the liquid portion was obtained by subtracting the solid fit from the raw data for which all the data points were able to be utilized. These results are given in Figure 11. Combining all uncertainties up to this point and adding in quadrature, the uncertainty for the linearization of the isolated liquid came out to be lengthy and difficult to read from a concise equation. Thus, it can be followed for each step in the appended spreadsheet. As seen in Figure 11, the fit proved sufficient with a reduced chi² of 0.835, suggesting a slight overestimate. The final value for this data gave T₂=(334±20.1)ms, which is accordingly ~2 orders of magnitude larger than that of solid cyclohexane but also short of the order of seconds that was expected. This was likely due to the more influential effects of temperature near the phase transition.

Figure 11: Left: Linearized liquid signal after subtracting fit curve for solid signal. Right:Chi plot for each point. [Original]

Having established a method for getting accurate fits to differentiate the solid portion of the sample from the liquid, this method was expanded to each of the 8 sets in the run. From equations (6), (12), and (13) it can be shown that the intercept of the linear slope is proportional the total number of hydrogen molecules in either state, N. A plot of the actual intercept-time relation is given in Figure 12, relating the values to that of liquid cyclohexane at equilibrium. While a way to definitively determine the actual fraction of liquid vs. solid was unable to be quantitatively established, the dependence did appear to approach an asymptote as hypothesized.

Figure 12: Left: Fraction of liquid to solid as a function of time. Right : Linearized liquid cyclohexane at room temperature giving T2=(1.647±0.029)s. [Original]

Conclusion

Though a definitive method to determine the percent of each phase during a phase transition based on the difference in T₂ signals of the solid vs. liquid phase was inconclusive, there was great success in distinguishing between the T₂ signals of solid and liquid cyclohexane, giving values on the order of milliseconds and seconds, as respectively expected. Systematic error in this experiment could have come from imperfect A and B-pulse durations, background noise in the electronics of the apparatus, imperfect alignment of sample in the magnetic field, drifting from resonance over time as measurements were being taken, impurities in the sample, or fluctuating lab temperature. With further work with the apparatus, it is plausible that one might be able to measure thermal properties such as the specific heat of fusion, based on what was found here.

References

[1] Hornak, Joseph P., Ph.D. "Basics of NMR." Basics of NMR. N.p., 1997. Web. 11 Mar. 2013. <http://www.cis.rit.edu/htbooks/nmr/bnmr.htm>.

[2] Wolff-Reichert, Barbara. "Instructional Pulsed NMR Apparatus Instruction Manual." N.p., 2003. Web. 11 Mar. 2013. <http://www.phys.utk.edu/labs/modphys/TeachSpin%20NMR%20Manual.pdf>.

[3] O'Reilly, D. E., E. M. Peterson, and D. L. Hogenboom. "Self-Diffusion Coefficients and Rotational Correlation Times in Polar Liquids. V. Cyclohexane, Cyclohexanone, and Cyclohexanol." Journal of Chemical Physics. AIP, n.d. Web.

[4] Madhavan, Arun S., and Abe Hanson. "Pulsed NMR Analysis of the Changes in Spin-Lattice and Spin-Spin Relaxation Times for a Five-Minute Curing Epoxy." University of Minnesota Methods of Experimental Physics (2005): n. pag. Web. 11 Mar. 2013. <http://mxp.physics.umn.edu/s05/Projects/S05NMR