Generation and Detection of NMR Signals

Generation and Detection of NMR Signals Hanudatta S. Atreya NMR Research Centre Indian Institute of Science

NMR Spectroscopy Spin (I)=1/2h B 0 Energy 0 = B 0 Classical picture (B 0 ) Quantum Mechanical picture Larmor Precession 0 = B 0

Equilibrium Bulk Magnetization z z Larmor Precession 0 = B 0 y x For large number of nuclei y x At equilibrium, the vector sum of magnetic moment in the x-y plane is zero and a small net moment in the +ve z direction is obtained. This net magnetic moment is termed as magnetization z y x 3

Sensitivity of NMR According to Boltzman law, the population ratio or difference is governed as: N / N α = e - E = e - hb/kt E = hb To increase the population difference, we have to increase energy gap between the α and the state (i.e., increase E) Increased Sensitivity (increased E)

Sensitivity of NMR E= hb 0 /kt can be increased by: (1) Going to higher (2) Increasing the magnetic field (B 0 ) (3) Lowering the temperature (T) Sensitivity can also be increased by increasing the total number of nuclei (i.e., increasing the concentration of the sample)

Isotope Natural % Abundance Sensitivity of NMR Spin (I) Magnetic Moment (μ) Gyromagne tic Ratio (γ) * 1 H 99.9844 1/2 2.7927 26.753 2 H 0.0156 1 0.8574 4,107 13 C 1.108 1/2 0.7022 6,728 15 N 0.037 5/2-1.8930-3,628 19 F 100.0 1/2 2.6273 25,179 29 Si 4.700 1/2-0.5555-5,319 31 P 100.0 1/2 1.1305 10,840 * γ has units of 10 7 rad T -1 sec -1

Observing the NMR signal B 0 Quantum mechanical picture Energy 0 = B 0 z RF irradiation z Classical picture Larmor Precession 0 = B 0 y x

Observing the NMR signal The RF irradiation can be visualized as an circulating wave y 0 = B 0 0 = B 0 y t B 1 y y=acos( t) y=b 1 cos( t) (B 1 =amplitude or strength of applied RF We look only at the magnetic field component of the EM wave) y=b 1 cos( )=B 1 cos( t) z Observed along y-axis If the RF irradiation is applied along x or y axis, the precessional frequency of the spin will be in resonance with applied frequency ( 0 = B 0 ) y 0 = B 0 x 0 = B 0

Observing the NMR signal z In the frame of reference of the circulating wave, the spin will appear static. This is called rotating frame of reference B 1 =0 y x In this frame of reference, the nucleus sees the applied magnetic field (B 1 ) as static external magnetic field and starts Larmor precession around this field. z y x The angular frequency of precession is now given as: 1 = B 1 equivalent to the precession of the spin around B 0 B 1 1 = B 1

Observing the NMR signal z z This picture can be extended to bulk magnetization y x Bulk magnetization z z y x RF irradiation along y-axis with frequency: y 1 = B 1 x B 1 Bulk magnetization Bulk magnetization precesses around B 1 (in x-z plane)

Observing the NMR signal How is the duration (length) of the pulse decided? Usually a 90 0 or 180 0 pulse is applied. z z 90 0 Pulse y After time: t=( /2)/ 1 y 1 = B 1 x RF pulse is switched off x B 1 B 1 180 0 Pulse z z B 1 y 1 = B 1 x After time: t=( )/ 1 RF pulse is switched off B 1 y x

Observing the NMR signal After a RF pulse is switched off, the nuclei relax to equilibrium conditions z z B 0 y y x x B 1 During relaxation two things happen: The signal decays to zero in the X-Y plane due to T 2 relaxation The signal builds up along the Z-axis due to T 1 relaxation These two processes happen simultaneously and hence the magnetization can be thought of as spiraling towards the z-axis.

Intensity Observing the NMR signal During T 2 decay, the magnetization starts in-phase and starts de-phasing z B 0 z B 0 y x T 2 relaxation y x Signal intensity maximum along x-axis Spins start de-phasing. Signal intensity starts decreasing x-axis The individual spins precess with B 0 Signal as observed along x-axis time Free Induction Decay (FID) Frequency of precession

Intensity Observing the NMR signal The oscillation of the signal in the X-Y plane and the de-phasing of the signal due to T 2 relaxation occur simultaneously z B 0 y x Thus, the FID can be decomposed into two components: time * time time Oscillation=cos( *t) Decay=exp(-t/T 2 ) FID=cos( *t)e -t/t 2

Intensity Observing the NMR signal time Free Induction Decay (FID) Frequency of precession in the rotating frame of reference (After subtraction from the reference signal) The FID is digitized and subject to Fourier transform This gives the NMR spectrum eff, rot, i Frequency

Intensity Intensity Intensity Fourier transform Fourier transformation is a mathematical technique to represent a signal which varies with time in a frequency scale. Performing a FT helps us to know the frequency and the relative intensities of the different signals present Consider a signal consisting of two cosine waves with frequencies 1 and 2 with different intensities time Signal (FID) time + 1 2 time time Fourier transform 0 1 2 frequency

Intensity Intensity Intensity Observing the NMR signal If there are n nuclei in the molecule, each of the nucleus will give an FID having its characteristic oscillation and decay. The resultant FID is a simple addition of all the individual FIDs. A CH 3 - CH 2 - OH B C time time time cos( CH3 t)*e -t/t 2CH3 cos( CH2 t)*e -t/t 2CH2 cos( OH t)*e -t/t 2OH Observed FID: A + B + C = cos( CH3 t)*e -t/t 2CH3 + cos( CH2 t)*e -t/t 2CH2 + cos( OH t)*e -t/t 2OH

Intensity Intensity Observing the NMR signal Intensity Intensity A CH 3 - CH 2 - OH B C time time time cos( CH3 t)*e -t/t 2CH3 cos( CH2 t)*e -t/t 2CH2 cos( OH t)*e -t/t 2OH Observed FID: A + B + C = cos( CH3 t)*e -t/t 2CH3 + cos( CH2 t)*e -t/t 2CH2 + cos( OH t)*e -t/t 2OH Fourier transform gives NMR spectrum (after quadrature detection) CH3 CH2 OH

NMR experiment Protein sample in magnet (~ 1mM) Radio frequency Pulse (microseconds) Time domain signal FID Process the data (Fourier Transform) time Frequency domain signal 1D NMR Chemical shift (ppm) 19

Basic NMR Setup B 0 Dewar to hold superconducting coils generating B 0 Sample NMR Console Coil to supply and receive RF energy Computer 20

The Probe Probe 21

MAGNET 22

NMR Probe Room temperature (RT) Probe The RF coil and other Parts are at room temperature Cryogenic Probes The RF coil and other Parts are cryo-cooled to 15-25 K Reduces thermal noise leading to increased S/N (Signal is not enhanced but noise is reduced) 23

The Probe The probe is that part of the NMR spectrometer system, which is used to transmit RF signals to the sample and receive the emitted RF signal from the sample The RF coils are constructed in such a manner that they wrap around the sample. Usually there will be two layers of the coil. The inner coil is used for 1 H and the outer coil for other nuclei. In many of the probes, the coils can be tuned to multiple nuclei The probe has temperature probes to monitor the temperature around the sample 24

The RF coil Saddle coil 25

The RF coil Important points to consider regarding the RF coil B1 inhomogenity The applied B1 field is not usually uniform across the sample. The outer Edges of the sample experience less B1 than middle of the sample This causes loss of sensitivity. Cryogenic probes have not-so-good B1 inhomogenity Filling factor The ratio of the volume occupied by the sample visible to the coil to the volume occupied by the coil. Higher the filling factor better is the sensitivity. Quality factor (Q-factor) The Q-factor is indicator of the sensitivity of the probe. Lower the Resistance of the coil, higher is the Q-factor.

The RF coil Tuning the RF coil to the right frequency RF coils used in NMR spectrometers need to be tuned for the specific sample being studied. An RF coil has a bandwidth or specific range of frequencies at which it resonates. When you place a sample in an RF coil, the conductivity and dielectric constant of the sample affect the resonance frequency. Furthermore, because the coil will not be efficiently detecting the signal, your signalto-noise ratio will be poor.

The RF coil Probe diameter (sample volumes) Probes are usually addressed by the diameter of the RF coil present. The usual RF-coil diameter is 5 mm (hence the probe is called 5mm probe). Typical volumes required for a 5 mm probe are 250 L. However, if one is limited by the amount of the sample (not the concentration), then RF coils with smaller diameter can be used. E.g.: 3 mm, 1.7 mm and 1 mm. The typical volume required for 1mm probe is ~20-30 L. These RF coils are also called micro-coil Smaller coil volumes are also preferable for high salt samples, as high conductivity increases the resistance of the RF coil causing decrease in S/N If one is limited by the concentration and not by the amount of sample, then higher volume probes should be used.

Shimming magnetic field is not uniform magnetic field is uniform molecules in different positions will experience different field strengths molecules in different positions will experience same field strength

Shimming The process of shimming involves adjusting the current in shim coils till a homogenous B 0 is obtained across the sample Adjust Shims (Eg. Z1 and Z2)

The shim coils (This indicates how the current varies along x, y and z) Z 0 Z 4 YZ 2

Gradient coils The gradient coils produce the gradients in the B o magnetic field needed for performing gradient enhanced spectroscopy, diffusion measurements, and NMR microscopy. The gradient coils are room temperature coils (i.e. do not require cooling with cryogens to operate) which, because of their configuration, create the desired gradient.

The ADC ADC is Analog to Digital Converter, a device which digitizes an analog signal FID: (Analog) time FID: (Digital) t time The FID is sampled at regular intervals ( t) Dwell time Fourier transformed The Dynamic range of ADCs is measured in bits. Modern day spectrometers have 16 bit ADCs

The ADC The FIDs have to be sampled at regular intervals. The frequency of sampling: 1/ t is called the sampling rate or sampling frequency time t >= 1/2f The sampling frequency is determined according to the Nyquist rule: If f is the highest frequency to be observed in the spectrum, then the sampling rate has to be atleast: 1/2f This means that at least 2 points are needed per oscillation

-1.0 Aliasing Frequency=X FID: Sampling=1/X FID: (Digital) time 1.0 Frequency=0 Undersampling Causes Aliasing

Only one point Per oscillation Aliasing Actual frequency Observed/apparent frequency

Intensity Quadrature Detection FID time FT cos( t)*e -t/t 2 -

Intensity Intensity Quadrature Detection Complex FID time time cos( t)*e -t/t 2 Real sin( t)*e -t/t 2 Imaginary S(t) = cos(wt) + i*sin(wt) FT

Quadrature Detection

Nuclear Spin Relaxation If the magnetization state of the system is disturbed from equilibrium, they tend to go towards their equilibrium state. This is called Relaxation z z y Non-equilibrium x Equilibrium z Return to Equilibrium: Relaxation Fanning out of magnetic moments (called de-phasing ) T 2 Relaxation z y x Return of net magnetization in the +ve z direction T 1 Relaxation 40

Nuclear Spin Relaxation 41

Using the equation: T1 Relaxation

Tranverse relaxation T2 Transverse relaxation time T2 controls the rate of decay of the magnetization in the XY plane & is responsible for the line-width of the observed NMR signal T2 is measured using the popular CPMG (spin-echo) sequence

T2 measurement I = I 0 e - /T2

One Dimensional (1D) NMR spectroscopy: Data acquisition and processing 45

Pulse sequence in NMR Every NMR experiment has an associated pulse sequence Pulse sequence consists of a series of RF pulses preceded or followed by periods where no RF pulse is applied. This period is called a delay. The delays and the RF pulses applied are appropriate for the given experiment. During the final delay in the pulse sequence, the signal is acquired Pulse sequence is like a blue-print of a NMR experiment. Every details of the NMR experiment is embedded in the pulse sequence 46

1D Pulse sequence Experiment 90º pulse (t) equilibration detection of signals 47

1D Pulse sequence e.g. A pulse sequence for one dimensional NMR experiment Relaxation delay (seconds) RF-pulse (duration: a few μs) Signal (FID) is acquired (duration: ms to sec) The experiment is repeated a number of times (scans) 48

Observing the NMR signal Each nuclei in a molecule will have its characteristic resonance frequency ( 0 ) depending on the chemical shift. Then how do we apply RF irradiation to all nuclei? This is done using a RF pulse. An RF pulse is a short burst of RF irradiation applied for a short time (micro-seconds) and switched-off. A short RF pulse can simultaneously excite ALL the nuclei in the molecule irrespective of their chemical shifts ~ s The RF pulse is applied at a frequency which is usually the centre of the spectrum of frequencies in the molecule. (i.e., if the molecule has nuclei resonating between 0 and 10 ppm, the RF pulse will applied with frequency 5 ppm) B 1

R.F. Pulses Hard Pulse Non-selective Usually rectangular Excitation range ~ 1/pulse-width Power of pulse = 1/4 pulse-width Soft-pulse Selective on a particular range of frequency Frequency range selected depends on the pulse width and the offset where the pulse is applied Rectangular Easier to apply But excitation profile not clean Shaped (Gaussian, Sinc etc..) Amplitude/Phase modulated Cleaner excitation

R.F. Pulses How much off-resonance =90 implies on resonance Phase

R.F. Pulses The trajectory of Magnetization under the action of any pulse can be Computed using the Bloch Equations -> Useful for evaluating excitation profile

R.F. Pulses Bloch Equations

R.F. Pulses: Excitation Profile Hard 90 0, 180 0 pulses 90 0 pulse 180 0 pulse Shaped pulse Are used for Achieving wider Or selective excitation

Shaped RF Pulses Amplitude Modulation Phase modulation

Shaped RF Pulses Gaussian SEDUCE Half- Gaussian RSNOB Re-Burp Sinc

Relaxation delay between scans Relaxation delay is the delay before the first pulse is applied. This delay should be sufficiently long for the system to reach equilibrium Relaxation delay The value of this delay is ideally 5*T 1 but is usually set to 1.25*T 1. Typically, relaxation delay is in 1-10 seconds. T1 usually decreases with size of the molecule. Hence, for smaller size molecules, longer delay is required (> 5 seconds)

Acquisition time Acquisition time is the duration time for which FID is acquired. This is different from measurement time which is the total duration of the experiment. Acquisition time Acquisition time depends on the T 2 of the sample. The longer the T 2 the FID can be acquired for a longer time. This will increase resolution. Data should not be acquired beyond ~3.0*T 2. By this time the signal has decayed to zero and only noise is collected. T 2 for small molecules is in the range of 100 ms to 300 ms. 58

Sensitivity of a NMR Signal The overall sensitivity of a NMR experiment/spectrum depends on a number of factors. The sensitivity is usually measured as the Signal-to-noise ratio (S/N). The factors that determine the S/N are: 1. Sample concentration (S/N increases linearly with conc.) 2. Temperature (S/N can increase or decrease with temp) 3. The magnetic field strength (S/N increases as B 3/2 0 ) 4. The type of nucleus being observed (S/N increases with the of the nucleus) 5. The type of probe being used (cryogenic probes have high sensitivity) 6. The measurement time used to record the data (S/N T) S/N

Processing a NMR spectrum The Raw FID acquired in a 1D spectrum has to be processed before it can be analyzed Following steps are carried out before it can be analyzed: (1) Applying a window function (or apodization) (2) Zero filling the data (3) Fourier transformation (4) Phase correction (5) Baseline correction

Processing a NMR spectrum 1. Applying a window or weighting function The idea is to emphasize certain portion of the FID at the cost of other region of the spectrum e.g. Give more weighting to initial region of the FID which has more sensitivity Or give more weighting to the later portion of the FID which carried more information on resolution. Weighting /window function can either serve to enhance the signal to noise or to increase the resolution. The most commonly used window functions are: 1. Exponential (also called line-broadening) 2. Shifted sine bell 3. Gaussian window function 61

Applying a window or weighting function The window function of the suitable form is simply multiplied with the FID. In the above exponential window function, the initial part of the FID is given more weightage and hence the sensitivity (S/N) improves 62

Applying a window or weighting function Resolution enhancement 63

Data Processing FID Fourier transform (after zero-filling > twice) No Apodization Exponential Apodization (lb) Line is broadened (max sensitivity) Shifted Cosine 90 0 (ssb=2) 70 (ssb=2.5) Optimal for resolution And sensitivity Gaussian High resolution low sensitivity

Applying a window or weighting function for resolution enhancement 65

Applying a window or weighting function The window function is also used for dealing with truncated artifacts To avoid these sinc wiggles, the FID is multiplied by a window function so that the Last point goes to zero. * FT 66

Linear Prediction (Before Fourier transform) Raw data/fid Spectrum (after FT) Full FID Truncated FID Linear predicted Twice Linear predicted Full