Pulsed Fourier Transform NR The rotating frame of reference The NR Eperiment. The Rotating Frame of Reference. When we perform a NR eperiment we disturb the equilibrium state of the sstem and then monitor the response of the sstem to the disturbance. As a result of the absorption of radiofrequenc energ nuclear spins jump from a lower energ level to a higher one. 90 180 The absorption of energ at this stage is a dominant process due to the population ecess in the lower level at equilibrium. Transitions in the opposite direction correspond to an emission of energ. The probabilit of the spontaneous emission is ver small. Instead, the return of the spin sstem to its equilibrium state is driven b the presence of lattice through a process known as relaation. Each transition is associated with a reversal of the spin orientation (spin flip). When more than two energ levels are present (I 1) onl transitions in which the magnetic quantum number m changes b 1 (single quantum transitions) are allowed: m = 1. The resonance condition: L = = ( / 2) The frequenc of the eternal disturbance (electromagnetic radiation) in NR eperiment is chosen so that to match the Larmor frequenc L. As a source of the eternal disturbance an oscillating magnetic field B 1, along the ais, with frequenc (/2) is used. A linear oscillation can be converted into a superposition of two components rotating in opposite directions:
2B 1 B 1 (l) B 1 (r) Y Y X -2B 1 X In the rotating frame () representation the ais is parallel to the Z ais of the laborator frame ( is aligned along +Z direction) and the and aes are allowed to rotate about the ais at the Larmor frequenc. We then assign + direction to B 1 vector. Z Y X The applied B 1 field is still composed of two counter-rotating vectors: the component vector that rotates in the same direction as the frame appears to be stationar, whereas the other component rotates at frequenc 2 0. As in the laborator frame, + direction defines the orientation of the net magnetiation vector,, at equilibrium, although in the rotating frame the -induced precession of the individual magnetiations is switched off.
On switching on B 1 field the net magnetiation,, starts precessing along B 1 in the plane and is tipped toward the ais. B 1 The B 1 field is applied as a pulse of duration t p, which usuall lasts for a few microseconds. The angle (known as the pulse angle) through which the magnetiation is tipped from the ais increases with the amplitude of B 1 and with the length of time, t p, for which the pulse is applied: rad = rad sec -1 G -1 B 1 (G) t p (sec) The transverse magnetiation is greatest immediatel after a 90 pulse and is ero for 180 pulse. is the component of interest, since the receiver coil is aligned along and as a consequence a signal (free induction deca) proportional to is induced in the receiver coil. The Free Induction Deca If the magnetiation is tipped from the ais to ais b 90 pulse it would remain indefinitel in the plane. In the laborator frame, would precess about at a constant frequenc. If we align the receiver coil about the laborator Y ais, the precessing magnetiation will induce in the coil an oscillating voltage which we can detect on an oscilloscope in a form of free induction deca (FID):
free of the influence of the radiofrequenc field, induced in the coil, and decaing back to equilibrium due to relaation processes. Z Y X L Because the pulse produces a wide range of frequencies (proportional to t -1 p ) of almost equal amplitude, the nuclei throughout the entire chemical shift range are all effectivel tipped b the same angle. Each chemicall shifted nucleus ma be considered to belong to a frame rotating at the corresponding Larmor frequenc. In order to understand what happens after the pulse, it is necessar to assign the frame to a single frequenc, S, the value at the centre of the frequenc band (referred to as carrier frequenc). S In NR eperiment the difference between the Larmor frequenc L and the carrier frequenc S is detected: = L - S The Larmor frequenc for a given tpe of the nucleus slightl differs from S (off resonance conditions) due to chemical shielding effects.
The evolution of the signal seen on the oscilloscope The sample is ecited with a 90 pulse along the ais and the magnetiation along the ais is observed under off-resonance conditions: B 1 B 1 t p = /2 Under the off-resonance condition, the magnetiation is rotating with respect to the frame of observation with offset frequenc A A t t A t
The free-induction deca of protons in water, associated with a 90 pulse along -, and observation along + in the rotating frame: Under on resonance conditions, we would observe a decaing ero-frequenc signal: A t In the case of two or more resonances in the spectrum, the magnetiation vectors move in rotating frame at different frequencies i and produce a signal which is the sum of the components: Fourier Transformation The aim of Fourier transformation isto differentiate frequencies present in a comple waveform and to determine intensities corresponding to each frequenc (in analog a with an ordinar prism: a comple wave, white light, is dispersed into its components, the spectrum).
Pressure variations produced b a tuning fork (top) and a musical instrument (bottom). Fourier s theorem: an reasonabl well behaved periodic function ma be generated b the superposition of a sufficient number of sine and cosine functions: F(t) = (A n sin nt + B n cos nt) where = 2 / T = 2. The coefficients A n and B n indicate the amplitude of the nth harmonic function. The process of determining these coefficients is called Fourier analsis. A square wave signal : F(t)= +1 between t=0 and t= F(t)= -1 between t= and t=2 4 F(t) 1 sin t + 3 sin3 t + 1 5 sin5 t + The period T of function F is equal to 2 sec and = 2 / T = 1 rad/sec.
The superposition of (a) three; (b) ten, and (c) one hundred harmonic terms:
The results of Fourier analsis of a function can be presented b its spectrum, in which relative amplitudes of the components in frequenc domain are displaed. The square wave has onl odd harmonics whose strengths decreases monotonicall as the frequenc increases: f 0 3f 0 5f 0 7f 0 This representation allows to assess the contribution of low and high frequencies. The envelope of the rf pulse transmitted to the sample in the probe is close to the shape of the positive half of the square wave function. If the pulse generator is switched on for a short period then the sample is ecited with a band of frequencies in the range 0 ± 1/t p, where t p is the duration of the pulse. A tpical pulse of duration 10 s gives an effective range of the order of 10 5 H, which is adequate for the ecitation of all the protons in a sample. In the case of the spectral analsis, it is convenient to deal with the corresponding integrals: f ( ) F( t)ep( i2 t) dt F( t) f ( )ep( i 2 t) d {ep(i 2t) = cos(2t) + i sin(2t)} This mathematical procedure is known as Fourier transformation, which relates time and frequenc domains.
Re[ f ( )] F( t) cos( 2t) dt Im[ f ( )] F( t) sin( 2t) dt Absorption mode Dispersion mode In a single line NR spectrum the time interval between successive maima in the FID is 1/, where is the difference in frequenc between the carrier frequenc S and the resonance frequenc i of the nuclei. Time (s) 1/ Frequenc (H)
Single Pulse Eperiment Pulse Initial Dela Dead Time Data Acquisition and Storage Relaation Dela Probe Pre-amp Pulses Receiver Detector Voltage A D C Binar numbers to computer Transmitter Continuos reference ADC analog-to-digital converter. ADC resolution the number of bits used in the binar representation (usuall 12-bit resolution if the receiver has been adjusted so that the maimum signal just fills the ADC, then no signal less than 2 11-1 of the largest signal (i.e. 1/2047) should be detectable [dnamic range 2047:1]) Signal Averaging The single pulse eperiment is repeated for NS times until the desired signal-to-noise ratio is achieved. For best sensitivit there is an optimum combination of the tip angle, the T 1 relaation time, and the time between pulses, given b Ernst s equation: cos = ep(t rep / T 1 )
NR fits the definition of a detector-noise-limited : the process of time-averaging a large number of FID s results in a theoretical improvement in signal-to-noise ratio b a factor equal to NS. Field frequenc locking is used in order to ensure that the relation of the spectrometer frequenc and the Larmor frequencies of the nuclei does not change during signal acquisition. Usuall the frequenc of a deuterium resonance of the solvent is compared to a fied frequenc derived from the master clock used to generate the spectrometer frequenc.