NMR Signal Properties & Data Processing

COURSE#1022: Biochemical Applications of NMR Spectroscopy http://www.bioc.aecom.yu.edu/labs/girvlab/nmr/course/ NMR Signal Properties & Data Processing LAST UPDATE: 1/13/2012

Reading Selected Readings (any of the following): Evans, pp 2-13 Teng, Chapter 1, pp. 12-17 Cavanagh, Fairbrother, Palmer, & Skelton (1st Ed), pp. 14-16 and 100-126 Derome, pp 9-29 Sanders and Hunter, 10-55 Keeler, chapters 4 and 5 Online Sources Content from the following online sources were used for some slides in this presentation: http://tonga.usip.edu/gmoyna/nmr_lectures http://www.oci.unizh.ch/group.pages/zerbe/notes.html http://www.cis.rit.edu/htbooks/nmr/

Quantum Mechanics picture: Overview of Basic Principles some nuclei pocess spin which is quantized in a magnetic field, B o, the spins have discrete energy states and populate the energy levels according to Boltzmann distribution transitions between energy levels are stimulated by photons with frequency: = B o or = B o /2 where and is the Larmor frequency in radians/s and Hz, respectively The quantum-mechanical selection rule states that only transitions with m=±1 are allowed.

Overview of Basic Principles Classical Mechanics picture (vector model): When sample is placed in a magnetic field, nuclear spins align with or against the field and a net magnetization, M o, is created within the sample. the bulk magnetization precess around the magnetic field axis at its Larmor frequency an RF pulse is sent to probe coil which creates a magnetic field (B 1 field) along some axis in transverse plane. The B 1 field places a torque on M o and tips the nuclear magnetization from z-axis into transverse plane (according to right-hand rule). shut off RF and detect magnetization in transverse plane - magnetization will oscillate at its Larmor frequency,, and return back to z-axis due to relaxation. This gives rise to an FID a time dependent voltage induced in probe coil Fourier transform (FT) of FID gives Larmor frequency of nuclei in sample. FT

The NMR Spectrum The NMR spectrum is information rich because it consists of a signal for each detected atom in the molecule and each signal has the following properties: position or chemical shift (related to its larmor frequency) for each type nuclei which is sensitive to its chemical environment (CH proton will be different than OH proton) intensity which is proportional to # of nuclei of that type (for example, a methyl CH3 group proton signal would be 3 times as intense as a CH proton signal) width which is sensitive to overall size of molecule and the dynamics or motion of the individual atoms splittings or couplings which is sensitive to # of bonded nuclei So the signals within an NMR spectrum depend on the surrounding environment (i.e. structure) and dynamics of each dectected nucleus in the molecule 1 H NMR spectrum of an organic compound 1 H NMR spectrum of a protein

The RF Pulse To detect the magnetization, we must tip it into the transverse plane by exciting the spins with a radiofrequency that corresponds to its Larmor frequency. An NMR sample may contain many different magnetization components, each with its own Larmor frequency so we must excite the sample with many different radiofrequencies at the same time. An RF pulse (also called the B 1 field) is a poly-chromatic source of radiofrequency and it covers a broad band of frequencies. The covered band width is proportional to the inverse of the pulse duration. Short (hard) pulses are required for uniform excitation of large bandwidths (non-selective excitation). Long (soft) pulses lead to selective excitation. Hard pulses have lengths of 5-20 μs and can excite over a wide range of frequencies (>10 khz), whereas soft pulses may last for 1-100 ms and excite only a small range of frequencies (<100 Hz): Left: Excitation profile of a hard pulse, right: Ex. profile of a soft (long) pulse

Effect of RF Pulse on Equilibrium Magnetization Spins respond to an RF pulse in such a way as to cause the net magnetization vector to rotate about the direction of the applied B 1 field. In the laboratory frame the equilibrium magnetization spirals down around the Z axis to the XY plane. In the rotating frame (to be discussed) a pulse about the Y axis will cause the magnetization to tip down toward X. The rotation angle (tip angle) depends on the length of time the field is on,, and its magnitude B 1 : = B 1 Apply pulse along one axis (eg. y-axis) which tips I z through flip angle a towards x-axis (according to Right-hand rule): z z y I y y x x using right-hand rule: I z I z cos + I x sin for I y RF pulse I z I z cos -I y sin for I x RF pulse

Some Useful Pulses The most commonly used pulse is the 90 degree pulse ( /2), because it puts as much magnetization as possible in the <xy> plane (more signal can be detected by the instrument): z z M o / 2 x x y y M xy Also important is the 180 degree pulse ( ) pulse, which has the effect of inverting the populations of the spin system (to be discussed later): z z M o x x y With control of the spectrometer we can basically obtain any pulse width we want and flip angle we want. y -M o

SIDE NOTE: only ½ of the B 1 field interacts with the magnetization. The B 1 field is linearly polarized along the axis of RF pulse. This B 1 field can be decomposed into two circularly rotating fields rotating in opposite directions along the z-axis: Only the field rotating in the same sense as the magnetic moment will interact with the magnetization and tip it; the counter-rotating field does not influence the magnetization and can be ignored.

The Time Domain NMR Signal Free Precession An NMR sample may contain many different magnetization components, each with its own Larmor frequency. These magnetization components are associated with the nuclear spin configurations joined by an allowed transition line in the energy level diagram. Based on the number of allowed absorptions due to chemical shifts and spin-spin couplings of the different nuclei in a molecule, an NMR spectrum may contain many different frequency lines. After the RF pulse, each magnetization component will rotate in the transverse plane according to its own Larmor frequency ( ): y y y y ti z x x x x This is a rotation around z-axis with angle given by t: I x I x cos t + I y sin t

The RF Transmitter and Receiver Coil An RF coil creates the B 1 field which rotates the net magnetization during an RF pulse (excitation). It also detects the transverse magnetization as it precesses in the XY plane (detection). Most RF coils on NMR spectrometers are of the saddle coil design and act as the transmitter of the B 1 field and receiver of RF energy from the sample. You may find one or more RF coils in a probe depending on how many different types of nuclei are manipulated during a given experiment. Each of these RF coils must resonate, that is they must efficiently store energy, at the Larmor frequency of the nucleus being examined with the NMR spectrometer. All NMR coils are composed of inductive elements and capacitive elements and the resonant frequency,, of an RF coil is determined by the inductance (L) and capacitance (C) of the circuit: TUNING and MATCHING PROBE (to be discussed in Lab) involves changing L and C of coil using knobs until desired frequency is obtained. For example: detecting 1H will require a certain L & C that is different than what is used to detect 13C

As the magnetization rotates in the transverse plane, it generates a fluctuating magnetic field which induces a time dependent voltage (the FID) in the receiver coil the time dependence is determined by the Larmor frequency. If the magnetization starts along the x-axis (eg. if a 90 degree pulse is applied along the y-axis), then the time dependence would look as follows: I x I y t t view along x-axis view along y-axis We will see later that both the x and y components of the magnetization are detected using a process called quadrature detection. It is convenient to express I x and I y as real and imaginary parts of I, the total magnetization: I(t) = Icos t + iisin t = Iexp(i t) The detected time domain signal is complex with real and imaginary parts corresponding to x and y components of the signal.

Return to Equilibrium Relaxation The magnetization does not precess infinitely in the transverse plane but returns back to the equilibrium state by a process called relaxation. We ll see the physics that govern relaxation in a later lecture. Two different time-constants describe this behavior: the re-establishment of the equilibrium α/β state distribution (T 1 ) dephasing of the transverse component (destruction of the coherent state, T 2 ). The T 2 value characterizes the exponential decay of the signal in the receiver coil the shorter T 2 is, the faster the signal decays:

The precessing spins return to the z-axis. Instead of moving circularly in the transverse plane, they follow a spiral trajectory until they have reached their initial position aligned with the z-axis and the time dependence of total magnetization becomes: I(t) = Iexp(i t)exp(-t/t 2 ) = Iexp(i t - t/t 2 ) governed by T 2 governed by T 2 governed by T 1 FIGURE left: Trajectory of the magnetization, right: individual x,y,z component

Signal Detection The Carrier Frequency In order to be stored in a computer, the analog signal that comes from the coil must be converted into a number which is done by the analog-to-digital converter (ADC) and is known as signal digitization or signal sampling. The resulting FID is a time-domain signal with numbers which represent signal intensity vs. time [f(t)]. However, before digitization, the signal is mixed with a reference frequency (called the carrier frequency) to produce a frequency which is the difference between the two the resultant frequency is an audio frequency which is easier to detect and handle than a radio frequency. In the rotating frame of reference, a frame of reference that rotates at the carrier frequency, ref relative to the laboratory frame, the vector would appear to rotate at a frequency equal to the difference between and the Larmor frequency ( o ) and the rotation frame frequency ( ref ): rotating frame = o - ref

The +/- Frequency Convention Transverse magnetization vectors rotating faster than the rotating frame of reference are said to be rotating at a positive frequency relative to the rotating frame (+ ). Vectors rotating slower than the rotating frame are said to be rotating at a negative frequency relative to the rotating frame (- ). -10 Hz = -10hz +10 Hz 0 Hz = +10hz 0 Hz

Quadrature Detection Can we discriminate between frequencies which only differ in their sign; the answer is yes, but only provided we use a method known as quadrature detection. The figure below illustrates the problem. The left side shows a vector precessing with a positive offset frequency (in the rotating frame); the x component is a cosine wave and the y component is a sine wave. The right side shows the evolution of a vector with a negative frequency. The x component remains the same but the y component has changed sign when compared with evolution at a positive frequency. In order to distinguish between positive and negative frequencies we need to know both the x and y components of the magnetization and thus detect two RF signals that are 90 degrees out of phase. Evolution of the magnetization over time; the magnetization starts out along the x axis.

Signal Digitization In order to be stored in a computer, the analog signal that comes from the coil must be converted into a number which is done by the analog-to-digital converter (ADC) and is known as signal digitization or signal sampling. The resulting FID is a time-domain signal with numbers that represent signal intensity vs. time [f(t)]. The signal is sampled stroboscopically and neighboring data points are separated by the dwell time (DW). Left:Rotating spin with its position at certain time intervals 1-6 are marked. Right: Corresponding signal in the receiver coil.

Spectral Sweep Width and the Nyquist Theorem The Nyquist theorem says that to reproduce an analog signal of frequency f, the sampling rate of the ADC is required to be at least 2f. Since the highest frequency of the spectrum is SW (in Hz), the dwell time (in s) must be set to: DW = 1/(2SW) When quadrature detection is being used, the dwell time becomes: DW = 1/SW because we can discriminate between positive and negative frequencies. So, for example, if the desired SW=10000 Hz, then the DW must be set to 100 s. NOTE: signals that have a frequency larger than SW, will appear folded or aliased in the spectrum and their position will be incorrect (Figure 3). Figure 2: Basic parameter for specifying the acquired spectral region Figure 1: Left: Sampled data points. Right: Black: Sampled data points. Red: Missing sampling point for a frequency of twice the nyquist value.. Figure 3: Illustration of the concept of folding. In spectrum (a) the peak (shown in grey) is at a higher frequency than the maximum set by the Nyquist condition. In practice, such a peak would appear in the position shown in (b).

The Fourier Transformation The analog signal that comes from the coil is converted into a number by the analog-to-digital converter (ADC). The resulting FID is a time-domain signal with numbers which represent signal intensity vs. time [f(t)]. Each time point is complex it has two values which represent the I x and I y components of magnetization that are the result of quadrature detection. This data is converted into a spectrum that represents signal intensity vs. frequency [f(ω)] by use of the Fourier Transformation (FT). The Fourier theorem states that every periodic function may be decomposed into a series of sine- and cosine functions of different frequencies and the Fourier transform is defined by the integral: The complex Fourier transform can be decomposed into a cosine transform and a sine transform that yield the real and imaginary parts of the signals, respectively and are stored in separate memory locations. The cosines give absorptive lineshapes while sines give dispersive lineshapes: A(ω) is an absorption mode lorentzian lineshape centred at ω = Ω D(ω) is a dispersion mode lorentzian lineshape centred at ω = Ω

Discrete Fourier Transformation Note that in FT-NMR the signal is sampled as discrete points. Hence, the discrete FT has to be utilized, which transforms a N-point time series consisting of values d k into a N-point spectrum: The discrete FT is implemented in form of the very fast Cooley-Tukey algorithm. The consequence of using this algorithm is, that the number of points the spectrum has must be a power of 2 (2 n ). So will often see (where SI is # of points in FID to collect): SI=8192 SI=16384 SI=32768 SI=65536, etc.

Results of Different Functions after Fourier Transformation in Frequency Space This figure summarizes the FT of some important functions (the so called Fourier pairs): (A) The FT of a cosine-wave gives two delta functions at the appropriate frequency. (B) The FT of a decaying exponential gives a lorentzian function with a characteristic shape. (C) The FT of a properly shimmed sample containing a single frequency gives two signals at the appropriate frequency with lorentzian lineshape. In fact the FT of (C) can be thought of a convolution of (A) and (B). (D) The FT of a step function gives a sinc function The sinc function has characteristic wiggles outside the central frequency band. (E) A FID that has not decayed properly can be thought of as a convolution of a step function with an exponential which is again convoluted with a cosine function. The FT gives a signal at the appropriate frequency but containing the wiggles arising from the sinc function. (F) A gaussian function yields a gaussian function after FT.

Phasing Purely absorptive signals have a much narrower base than a dispersive signal, so that differentiation of peaks very close to each other is easier if all the signals in the spectrum have purely absorptive line shape. The phase of the signals after Fourier Transformation depends on the first point of the FID. If the oscillation of the signal of interest is purely cosine-modulated, the signal will have pure absorptive line shape. If the signal starts as a pure sine-modulated oscillation, the line shape will be purely dispersive (see Figure 1). Ideally, collection of a purely cosine-modulated signal would occur if one turned on the receiver immediately after the pulse and start digitizing the signal. However, in a real experiment the signal has to travel through cables before it is digitized which causes a delay and there is also a protection delay (called DE on Bruker instruments), to wait for the pulse ring-down (see Figure 2) otherwise, the detector would be overloaded by the RF pulse. Therefore, a phase correction is needed to yield purely absorptive signals. Figure 1: The relative phase of the magnetization with respect to the receiver coil determines the phase of the signal after Fourier transformation Figure 2: A) Ideal FID. B) Introduction of the pre-scan delay to enable pulse power ringdown.

Phasing, cont. There are two different types of phase-correction. The zero-order phase correction (PH0) applies the phase change to all signals in the same way. The first order phase correction (PH1) applies a phase change, whose amount increases linearly with the distance to the reference signal (see figure below). Top: The zero order phase correction changes the signal phase for all signals by the same amount. Lower: For the first order phase correction the applied correction depends on the frequency difference to the reference signal (marked by an arrow). In practice what happens is that we Fourier transform the FID and look at the spectrum. If the signals look out of phase and have some dispersive appearance, we then adjust the phase until the spectrum appears to be in pure absorption mode usually this adjustment is made by turning a knob or by a click and drag operation with the mouse or most modern spectrometers can do this automatically. The whole process is called phasing the spectrum and is something we have to check each time we record a spectrum.

Resolution of the Spectrum The spectral resolution is a measure of how well we can resolve or separate signals within a spectrum. The spectral resolution that can be obtained from a spectrum is determined by two factors: the natural line width of the signal at half-height (LW 1/2 ) which is related to the transverse relaxation time or T 2 : 1/2 or LW 1/2 (Hz) = 1/ T 2 Fast transverse relaxation may also be caused because the B o magnetic field is inhomogenous (inhomogenous broadening or T 2* ) recall that as B o changes, changes so that a range of B o s within a sample will cause a range of s. Slowly decaying FIDs lead to narrow lines (left), rapidly decaying ones to broad lines (right).

Resolution of the Spectrum, cont. the number of points which are used to digitize the signal. The digital resolution (in Hz/point) of the FID collected can be calculated as: Digital resolution = SW (Hz) / TD = 1/AQ (s) where TD is the number of complex points acquired and AQ is the acquisition time for each FID. It is often useful to append zeros to the FID to add additional resolution (called zerofilling). Adding zeros once (that means making SI = 2*TD where SI is the number of complex points processed) gives a significant improvement in resolution but adding more zeros gives cosmetic effects but does not add more information. Effect of zero-filling Effect of zero-filling Effect of increasing number of data points, TD, of FID.

Sensitivity Enhancement of the Spectrum by Apodization The resolution is determined by the transverse relaxation time T 2 of the spins. A long transverse relaxation time leads to a slowly decaying FID and fast relaxation leads to a quickly decaying FID (Figure A). Therefore, the resolution is determined by the amount of signal which remains towards the end of the FID. The sensitivity (signal-to-noise, S/N) of a spectrum is determined by the amount of signal at the beginning of the FID and the amount of noise collected throughout the FID. Once the signal has decayed, only noise is being sampled by the spectrometer. If too many data points (TD) are sampled, the spectrum will become excessively noisy (Figure B). There are a number of so-called window or apodization functions that can be multiplied with the FID in order to enhance the beginning or end of the FID to improve either S/N or resolution, respectively (Figure C). Most manipulations that improve resolution lead to a loss in S/N and vice versa. The goal is to achieve a balance between resolution and sensitivity. Figure A Figure B Figure C

Window Functions (A) raw spectrum after FT. (B) Multiplication with exponential,lb=5. (C) as (B), but LB=1. (D) sinebell. (E) 45 degree shifted sine-bell. (F) 90 degree shifted sine-bell.

Window Functions, cont. Exponential multiplication: The line broadening constant LB can be positive (sensitivity enhancement) or negative (resolution enhancement). A lorentzian line of width W will have a width of W + LB Hz after apodization. It can be shown that the best SNR is obtained by applying a weighting function which matches the linewidth in the original spectrum such a weighting function is called a matched filter. So, for example, if the linewidth is 2 Hz in the original spectrum, applying an additional line broadening of 2 Hz will give the optimum S/N ratio: Lorentz-to-Gauss transformation: This a combination of exponential and gaussian multiplication. The factor LB determines the broadening, whereas GB determines the center of the maximum of the gaussian curve. When GB is set 0.33, the maximum of the function occurs after 1/3 of the acquisition time. By this multiplication, the Lorentzian lineshape is converted into a gaussian lineshape, which has a narrower base Sine-Bell apodization: Sine-bell apodization is frequently used in 2D NMR processing. A pure sinebell corresponds to the first half-lobe of a sine-wave and multiplies the FID such that it brought to zero towards the end. However, such a function also sets the first data points to zero and hence leads to severe loss of S/N and introduces negative wiggles at the bases of the signals in addition to a strong improvement in resolution. Shifting the sine-bell by 90 degrees (a cosine-bell) means that the function used starts of the maximum of the first lobe and extends to zero towards the end of the FID. This is usually used for FIDs that have maximum signal in the first data points (NOESY, HSQC). A pure sine-bell is used for such experiment in which the FID does not have maximum signal in the first points (such as COSY) or for improving magnitude data (HMBC)

Window Functions, cont. Different window functions: A) exponential multiplication with LB=1,3,5Hz. B) Lorentz-Gaussian transformation with LB=-5 and GB =0.1, 0.3, 0.5 Hz. C) Sine-bell shifted by 0, π/2, π/4, π/8. D) Sine-bell squared shifted by 0, π/2, π/4, π/8.

FID Truncation If we stop recording the signal before it has fully decayed the FID is said to be truncated. As is shown in the figure below, a truncated FID leads to oscillations around the base of the peak; these are usually called sinc wiggles or truncation artifacts the name arises as the peak shape is related to a sinc function. Clearly these oscillations are undesirable as they may obscure nearby weaker peaks. To remove, one can either increase the acquisition time or apply a decaying weighting function (line broadening) to the FID so as to force the signal to go to zero at the end. Unfortunately, the later will have the side effects of broadening the lines and reducing the S/N. Illustration of how truncation leads to artifacts (called sinc wiggles) in the spectrum. The FID on the left has been recorded for sufficient time that it has decayed almost the zero; the corresponding spectrum shows the expected lineshape. However, if data recording is stopped before the signal has fully decayed the corresponding spectra show oscillations around the base of the peak.

Sensitivity Enhancement of the Spectrum by Signal Averaging The signal-to-noise (S/N) of a spectral peak is the ratio of the average height of the peak to the standard deviation of the noise height in the baseline. To correctly interpret an NMR spectrum, it is usually important to see all signal features clearly which requires a spectrum with sufficient S/N (the weakest signal must at least be above the noise!). The signal to noise ratio may be improved by performing signal averaging. Signal averaging is the collection and averaging together of multiple spectra. The signals are present in each of the averaged spectra so their contribution to the resultant spectrum add. Noise is random so it does not add, but begins to cancel as the number of spectra averaged increases. The signalto-noise improvement from signal averaging is proportional to the square root of the number of scans (NS) averaged. S/N NS 1/2 NS NS 1/2 1 1.00 8 2.83 16 4.00 80 8.94 800 28.28 Spectrum with NS=1 Spectrum with NS=800 Other ways to enhance S/N: use more sample (double sample saves 4x in time for same S/N) use higher field, B o (S/N~B o 3/2 ) use CryoProbe (reduces thermal noise in coil)

Other Data Processing Topics Baseline Correction (BC): Baselines that are not flat can be a problem in FT-NMR. Firstly, spectra with non-flat baselines give wrong integrals. Secondly, in spectra with baseline roll small signals may not be recognized. Thirdly, those spectra are difficult to phase. Furthermore, peak-picking routines need a threshold for minimal signal intensity and such peak-picking for tiny signals is not possible when the baseline is not flat. There are many reasons for bad baselines. In general, baseline distortions are caused by distortions of the first points of the FID. Baseline correction routines work on the Fourier transformed data and can help flatten the baseline of the spectrum. NOTE: generally baseline correction can t help flatten rolling baselines that occur when datapoints are clipped because the receiver gain is set too high (to be discussed in lab). Linear Prediction (LP): For exponential decaying signals that are sampled at constant spacing (as is usually the case for NMR) new data points can be predicted and added to the FID. The reason behind this is that one can assume that new data points can be represented as a fixed linear combination of immediately preceding values. LP provides a means of fitting a time series to a number of decaying sinusoids. Provided that S/N is good the number of points can be doubled with LP (or the measuring time can be halved to achieve the same resolution). 2D spectra (A) 240 points in F1. (B) 64 points in F1. (C) LP to 128 points. (D) mirror -image LP to 240 points.

;zg ;avance-version ;1D sequence Putting It All Together: The Basic 1D FT-NMR Experiment Basic 1D pulse Bruker Pulse Sequence: #include <Avance.incl> 1 ze 2 d1 p1 ph1 go=2 ph31 wr #0 exit ph1=0 2 2 0 1 3 3 1 ph31=0 2 2 0 1 3 3 1 =300.0015MHz =300.001MHz =300.000MHz =299.9988MHz ;pl1 : f1 channel - power level for pulse (default) ;p1 : f1 channel - high power pulse ;d1 : relaxation delay; 1-5 * T1 ref =300.000MHz (Hz) 1,500 1,000 0-1,200

Parameters for a 1D FT-NMR Experiment typical acquisition parameters for a 1D NMR experiment: pulprog zg basic 90 pulse sequence TD 16k number of data points (time domain size) NS 8 number of scans DS 0 number of dummy scans SW 6000 Hz sweep width in hertz O1p 5ppm center of spectrum D1 1.5 delay parameter in seconds P1 9.5 pulse length in microseconds PL1-6 pulse power level in terms of db typical processing parameters for a 1D NMR experiment: SI 8k real spectrum size, si = td/2 (no zero fill) LB 0.30 exponential weighing factor The ASED and EDA displays on a Bruker lists the acquisition parameters for a given experiment

Old School: Continuous Wave NMR In continuous wave NMR (CW-NMR), the sample is continuously irradiated with a frequency while the magnetic field is varied and the spectrum is a recording of which magnetic fields caused the sample to absorb RF (happens when the Larmor frequency condition is met like in FT-NMR = B o The main disadvantages of CW-NMR are: time consuming takes 100-1000 times longer to record a scan relative to FT- NMR the data is in frequency domain so time domain manipulations such as filtering, apodization and linear prediction cannot be readily applied to the data B o time B o