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Free induction decay

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In Fourier transform nuclear magnetic resonance spectroscopy, free induction decay (FID) is the observable nuclear magnetic resonance (NMR) signal generated by non-equilibrium nuclear spin magnetization precessing about the magnetic field (conventionally along z). This non-equilibrium magnetization can be created generally by applying a pulse of radio-frequency close to the Larmor frequency of the nuclear spins.

If the magnetization vector has a non-zero component in the XY plane, then the precessing magnetisation will induce a corresponding oscillating voltage in a detection coil surrounding the sample.^[1] This time-domain signal (a sinusoid) is typically digitised and then Fourier transformed in order to obtain a frequency spectrum of the NMR signal i.e. the NMR spectrum.^[2]

The duration of the NMR signal is ultimately limited by T₂ relaxation, but mutual interference of the different NMR frequencies present also causes the signal to be damped more quickly. When NMR frequencies are well-resolved, as is typically the case in the NMR of samples in solution, the overall decay of the FID is relaxation-limited and the FID is approximately exponential (with the time constant T₂ changed, indicated by T₂^*).^{[citation needed]} FID durations will then be of the order of seconds for nuclei such as ¹H.

Particularly if a limited number of frequency components are present, the FID may be analysed directly for quantitative determinations of physical properties, such as hydrogen content in aviation fuel, solid and liquid ratio in dairy products (time-domain NMR).^[3]

Advances in the development of quantum-scale sensors, particularly NV centres, have enabled the observation of the FID of single nuclei.^[4] When measuring the precession of a single nucleus, quantum mechanical measurement back action has to be considered. In this special case, also the measurement itself contributes to the decay as predicted by quantum mechanics.

References

[edit]

^ Joseph P. Hornak. "The Basics of MRI". Rochester Institute of Technology. Chapter 4: NMR SPECTROSCOPY.
^ Duer, Melinda J. Introduction to Solid-State NMR Spectroscopy. Blackwell Publishing, 2004, p. 43-58.
^ H. Todt, G. Guthausen, W. Burk, D. Schmalbein, and A. Kamlowski. Water/moisture and fat analysis by time-domain NMR. Food Chemistry 96, 3 p. 436-440 (2006) doi: 10.1016/j.foodchem.2005.04.032
^ K. S. Cujia, J. M. Boss, K. Herb, J. Zopes, and C. L. Degen. Tracking the precession of single nuclear spins by weak measurements. Nature 571, 230-233 (2019) doi:10.1038/s41586-019-1334-9

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Free induction decay (FID) is the time-domain nuclear magnetic resonance (NMR) signal generated by the precession and dephasing of transverse magnetization in a sample following the application of a radiofrequency (RF) pulse in a magnetic field.^[1] This decaying oscillatory signal, induced in a detection coil, captures the raw data from excited nuclear spins relaxing toward equilibrium.^[2] In NMR spectroscopy and magnetic resonance imaging (MRI), FID is produced by a short, intense RF pulse—typically 90°—that tips the net magnetization from the longitudinal axis into the transverse plane, where it precesses at the Larmor frequency specific to each nucleus.^[3] The pulse excites all resonant nuclei simultaneously, enabling efficient data acquisition compared to older continuous-wave methods.^[2] As the spins dephase due to interactions like spin-spin relaxation (T₂) and magnetic field inhomogeneities (contributing to T₂*), the FID manifests as an exponentially decaying sine wave.^[1] The importance of FID lies in its role as the primary observable in pulsed Fourier transform NMR, where multiple FIDs are often averaged to enhance sensitivity, particularly for low-abundance nuclei like ¹³C.^[2] Subsequent Fourier transformation converts the time-domain FID into a frequency-domain spectrum, revealing chemical shifts, coupling constants, and other structural information essential for molecular analysis.^[3] In MRI, FID forms the basis for basic imaging sequences, though it is often modified with gradients or refocusing pulses to mitigate rapid decay and encode spatial information.^[1]

NMR Fundamentals

Spin magnetization

Nuclear spins are intrinsic quantum mechanical properties of atomic nuclei, arising from the combined angular momenta of protons and neutrons within the nucleus. For spin-1/2 nuclei, such as the proton (^1H), the spin angular momentum operator

\mathbf{I}

has eigenvalues corresponding to projections

m_I = \pm 1/2

along a quantization axis, with magnitude

\sqrt{I(I+1)}\hbar = \sqrt{3/4}\hbar

I(I+1)

ℏ=3/4

ℏ.^[4] These nuclei possess a magnetic moment $\boldsymbol{\mu} = \gamma \hbar \mathbf{I}$ , where $\gamma$ is the gyromagnetic ratio, a nucleus-specific constant that relates the magnetic moment to the angular momentum; for protons, $\gamma \approx 2.675 \times 10^8$ rad s^{-1} T^{-1}).^[4] In the presence of a static external magnetic field $\mathbf{B_0}$ aligned along the z-axis, the Zeeman interaction splits the degenerate spin states into two energy levels: the lower energy state ( $m_I = +1/2$ ) with energy $E_- = -\frac{1}{2} \gamma \hbar B_0$ and the higher energy state ( $m_I = -1/2$ ) with $E_+ = +\frac{1}{2} \gamma \hbar B_0$ , resulting in an energy splitting $\Delta E = \gamma \hbar B_0 = \hbar \omega_0$ , where $\omega_0 = \gamma B_0$ is the Larmor frequency.^[4] At thermal equilibrium, the populations of these states follow the Boltzmann distribution, with the ratio of upper to lower state populations $N_+/N_- = \exp(-\Delta E / kT)$ , where $k$ is Boltzmann's constant and $T$ is temperature; for typical NMR conditions (e.g., $B_0 = 11.7$ T, $T = 298$ K), this yields a small excess population in the lower state, on the order of parts per million.^[4] This population imbalance produces a net longitudinal magnetization $M_z$ along the direction of $\mathbf{B_0}$ , given by $M_z = N \gamma^2 \hbar^2 B_0 / (4 kT)$ for a sample of $N$ spin-1/2 nuclei, where the transverse components average to zero due to random orientations.^[4] Classically, the ensemble of nuclear magnetic moments can be described by a macroscopic magnetization vector $\mathbf{M}$ , which at equilibrium aligns with $\mathbf{B_0}$ as $\mathbf{M_0} = (0, 0, M_z)$ .^[5] The torque equation $\frac{d\mathbf{M}}{dt} = \gamma \mathbf{M} \times \mathbf{B_0}$ shows that, in the absence of relaxation, a displaced magnetization vector precesses around $\mathbf{B_0}$ at the Larmor frequency $\omega_0 = \gamma B_0$ , producing circular motion of the transverse components in the xy-plane while preserving the magnitude.^[5] The resonance condition for NMR occurs when an applied radiofrequency field matches $\omega_0$ , allowing energy absorption to induce transitions between the split levels, though the equilibrium alignment itself persists without perturbation.^[4] This classical vector model, introduced by Bloch, provides an intuitive framework for understanding the coherent behavior of spins prior to excitation.^[5]

Radiofrequency excitation

In nuclear magnetic resonance (NMR), radiofrequency (RF) excitation initiates free induction decay (FID) by applying a short burst of oscillating magnetic field, denoted as

\mathbf{B}_1

, which is orthogonal to the static magnetic field

\mathbf{B}_0

. This

\mathbf{B}_1

field rotates in the plane perpendicular to

\mathbf{B}_0

at a frequency matching the Larmor frequency

\omega_0 = \gamma B_0

, where

\gamma

is the gyromagnetic ratio of the nucleus, effectively perturbing the equilibrium magnetization aligned along

\mathbf{B}_0

.^[6]^[7] The extent of this perturbation is quantified by the flip angle

\theta = \gamma B_1 \tau

, where

\tau

is the duration of the RF pulse and

B_1

is the amplitude of the RF field. A 90° flip angle (

\theta = \pi/2

) is commonly used to tip the longitudinal magnetization

M_z

fully into the transverse plane, maximizing the initial transverse magnetization

M_{xy}

that generates the FID signal, as

M_{xy} = M_0 \sin \theta

and

M_z = M_0 \cos \theta

, with

M_0

being the equilibrium magnetization.^[7]^[8] In the rotating frame at frequency

\omega_0

, the Bloch equations simplify the description of this process, modeling the magnetization dynamics as:

\frac{dM_x}{dt} = \gamma (M_y B_{1z} - M_z B_{1y}) - \frac{M_x}{T_2},

\frac{dM_y}{dt} = \gamma (M_z B_{1x} - M_x B_{1z}) - \frac{M_y}{T_2},

\frac{dM_z}{dt} = \gamma (M_x B_{1y} - M_y B_{1x}) - \frac{M_z - M_0}{T_1},

where

T_1

and

T_2

account for relaxation (though negligible during short pulses). Here, the effective field during the pulse is

\mathbf{B}_{eff} = (B_{1x}, B_{1y}, \Delta B_z)

, with

\Delta B_z

as any off-resonance offset, causing the magnetization vector to precess around

\mathbf{B}_{eff}

and tip from the z-axis toward the xy-plane. For on-resonance conditions (

\Delta B_z = 0

), a constant

\mathbf{B}_1

along the x-axis results in a uniform rotation by angle

\theta

.^[7]^[8]^[6] RF pulses are classified as hard or soft based on their power and duration, which determine the excitation bandwidth—the range of Larmor frequencies effectively excited. Hard pulses are short (typically 5–20 μs) and high-power, producing a broad, nearly rectangular excitation profile with bandwidth

\Delta \omega \approx 1/\tau \gg \gamma B_Q

(where

B_Q

represents interaction widths like quadrupolar splitting), suitable for uniform excitation across the spectral range. In contrast, soft pulses are longer (hundreds of μs to ms) and lower-power, often shaped (e.g., Gaussian or sinc), yielding a narrow bandwidth

\Delta \omega \approx 1/\tau

for selective excitation of specific resonances, minimizing off-resonance effects but requiring careful calibration to avoid incomplete tipping.^[9]^[10]

Signal Generation

Transverse magnetization

In nuclear magnetic resonance (NMR), transverse magnetization arises when a radiofrequency (RF) pulse is applied to the equilibrium longitudinal magnetization vector

\mathbf{M_0}

aligned along the static magnetic field direction. The RF pulse rotates

\mathbf{M_0}

by a flip angle

\theta

, producing a transverse component

M_{xy}

with initial amplitude

M_0 \sin \theta

. For a standard 90° pulse (

\theta = 90^\circ

\sin \theta = 1

, so

M_{xy}

achieves its maximum value of

M_0

, fully tipping the magnetization into the xy-plane perpendicular to the static field.^[11] Following the RF pulse, the transverse magnetization

M_{xy}

undergoes free precession in the xy-plane at the Larmor frequency

\omega_0 = \gamma B_0

, where

\gamma

is the gyromagnetic ratio and

B_0

is the static magnetic field strength. This precessing magnetization acts as a time-varying magnetic field, inducing an electromotive force (EMF) in the nearby receiver coil according to Faraday's law of electromagnetic induction. The resulting NMR signal

S(t)

is proportional to the time derivative of the magnetic flux, specifically

S(t) \propto \frac{d M_{xy}}{dt}

, which captures the oscillatory nature of the precession.^[12] In the ideal case, assuming no external gradients or inhomogeneities beyond intrinsic effects, the free induction decay (FID) manifests as a damped sinusoidal oscillation. The transverse magnetization evolves as

M_{xy}(t) = M_0 e^{-t/T_2^*} \cos(\omega_0 t + \phi),

where

T_2^*

represents the effective transverse relaxation time characterizing the decay envelope, and the phase

\phi

encodes the initial orientation. The phase

\phi

is determined by the RF pulse phase relative to the receiver reference and the spin system's initial alignment conditions.^[2]

Precession and phase coherence

In nuclear magnetic resonance (NMR), the precession of nuclear spins around the static magnetic field

B_0

is influenced by the molecular environment. In isotropic environments, such as liquids where rapid tumbling averages molecular orientations, all equivalent spins of a given nucleus precess at a uniform Larmor frequency determined by the isotropic chemical shift, leading to narrow spectral lines.^[13] In contrast, anisotropic environments, like those in solids or oriented samples, introduce orientation-dependent interactions, such as chemical shift anisotropy (CSA), causing the precession frequency to vary with the angle between the molecular axis and

B_0

, resulting in broader powder patterns that reflect structural details.^[14] This variation arises from the tensorial nature of the shielding, with principal components differing by up to 200 ppm for aromatic carbons due to electronic hybridization effects.^[14] Spin isochromats represent coherent packets of spins sharing identical precession frequencies, grouped by factors like chemical equivalence or local field uniformity. Immediately following radiofrequency excitation that tips the net magnetization into the transverse plane—as described in the generation of transverse magnetization—these isochromats start in phase lock, producing a maximum signal amplitude in the free induction decay (FID).^[15] This initial coherence persists briefly as the isochromats precess synchronously around

B_0

, but frequency dispersion among different isochromats leads to gradual fanning out of their phases, modulating the FID envelope.^[15] In the classical vector model, the observable FID signal is the vector sum of these individual isochromat magnetizations, each evolving as

M_{xy} = M_0 e^{-i \omega t}

, where

\omega

is the specific precession frequency.^[16] The primary source of frequency dispersion among isochromats in FID is the chemical shift, which arises from differences in local electronic shielding. The precession angular frequency for a nucleus is given by

\omega = \gamma B_0 \left(1 + \frac{\delta}{10^6}\right)

, where

\gamma

is the gyromagnetic ratio,

B_0

is the applied field, and

\delta

is the chemical shift in parts per million (ppm) relative to a reference standard like tetramethylsilane.^[13] This offset, typically ranging from 0 to 10 ppm for protons, translates to frequency differences of hundreds of Hz at high fields (e.g., 600 MHz spectrometers), causing isochromats from distinct chemical sites to evolve at slightly different rates and initiate phase divergence.^[13] In the vector model, coherence pathways describe the evolution of transverse magnetization during FID and related sequences. For FID, the pathway involves zero-quantum preparation followed by single-quantum coherence (-1 order), where the net transverse vector precesses freely, accumulating phase based on offset frequencies without refocusing.^[16] In spin echo sequences, a 180° refocusing pulse inverts the phase accrual of each isochromat, reversing the fanning and restoring coherence at the echo time, as the vectors reconverge along the transverse axis.^[15] This refocusing highlights reversible phase evolution due to static frequency offsets, distinguishing it from inherent spin dynamics, and is fundamental for resolving chemical shift dispersion in spectra.^[16]

Decay Mechanisms

T2 relaxation

T2 relaxation, also known as spin-spin relaxation, refers to the intrinsic process by which the transverse magnetization of nuclear spins decays irreversibly over time due to local magnetic field fluctuations caused by dipole-dipole interactions between spins and random molecular motions.^[17] These interactions lead to a loss of phase coherence among the spins in the transverse plane, contributing to the exponential decay of the free induction decay (FID) signal.^[17] In the phenomenological description provided by the Bloch equations, the time evolution of the transverse magnetization component

M_{xy}

is governed by the term

\frac{dM_{xy}}{dt} = -\frac{M_{xy}}{T_2},

where

T_2

is the spin-spin relaxation time constant.^[6] This differs from the longitudinal relaxation time

T_1

, which describes the recovery of the magnetization along the static magnetic field direction through energy exchange with the lattice, whereas

T_2

arises from spin-spin couplings without net energy transfer.^[6] The role of T2 in the FID is evident in the signal's envelope, which follows an exponential decay of the form

e^{-t/T_2}

, reflecting the intrinsic broadening of the transverse magnetization.^[6] This intrinsic decay contributes to the overall observed FID decay time

T_2^*

, which is shorter due to additional reversible dephasing effects. To isolate and measure T2 accurately, the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence is employed, applying a series of 180° refocusing pulses to minimize extrinsic dephasing while allowing the intrinsic T2 process to dominate the echo amplitudes.^[18]

Dephasing effects

In nuclear magnetic resonance (NMR), the observed decay of the free induction decay (FID) signal is characterized by the effective transverse relaxation time

T_2^*

, which encompasses both intrinsic spin-spin relaxation and extrinsic dephasing effects. This is mathematically expressed as

\frac{1}{T_2^*} = \frac{1}{T_2} + \frac{1}{T_2'}

, where

T_2

represents the true transverse relaxation time due to molecular interactions, and

T_2'

accounts for the additional dephasing arising from reversible inhomogeneities in the local magnetic field. The contribution from

T_2'

typically dominates in high-field systems, resulting in

T_2^*

being significantly shorter than

T_2

, often by factors of 2–10 depending on the experimental setup. The primary sources of dephasing that contribute to

T_2'

include static and dynamic variations in the main magnetic field

B_0

. Static dephasing arises from fixed field inhomogeneities, such as imperfect shimming of the magnet, which causes spins in different spatial locations to precess at slightly different frequencies, leading to rapid loss of phase coherence across the sample. Susceptibility differences between tissues or materials, for instance at air-tissue interfaces or due to paramagnetic ions like deoxyhemoglobin, induce microscopic field gradients that exacerbate this effect within voxels. Dynamic dephasing, in contrast, involves time-varying field perturbations, such as those from molecular diffusion through existing gradients, where spins sample varying environments and accumulate irreversible phase shifts.^[19] These dephasing mechanisms are reversible under certain conditions, distinguishing them from the intrinsic

T_2

processes. In particular, static dephasing due to

B_0

inhomogeneities and susceptibility can be refocused using a spin-echo pulse sequence, originally demonstrated by Erwin L. Hahn in 1950. In the Hahn echo technique, a 90° radiofrequency pulse is followed by a 180° refocusing pulse after a time

\tau

, which inverts the phase evolution of spins, causing them to rephase at time

2\tau

and form an echo that recovers signal lost to static dephasing while remaining sensitive to dynamic effects like diffusion. This refocusing enables isolation of the true

T_2

decay from extrinsic contributions, improving spectral resolution in applications such as magnetic resonance imaging.

Detection and Analysis

Time-domain acquisition

In nuclear magnetic resonance (NMR) spectroscopy, the free induction decay (FID) signal is detected using receiver coils designed to sense the time-varying magnetic field produced by the precessing transverse magnetization, denoted as

M_{xy}

. Solenoid coils, which consist of a cylindrical array of loops, provide uniform sensitivity across a sample volume and are particularly effective for high-resolution studies of homogeneous samples due to their ability to generate and detect a homogeneous

B_1

field.^[20] Surface coils, in contrast, employ a partial loop geometry tuned to the Larmor frequency and offer high signal-to-noise ratio (SNR) for superficial or localized regions, though their sensitivity decreases rapidly with distance from the coil, limiting their use to samples close to the coil surface.^[20] Both coil types induce an electromotive force (EMF) proportional to the rate of change of magnetic flux from

M_{xy}

, with optimal performance achieved by matching coil size to sample dimensions to maximize sensitivity.^[21] To accurately capture the oscillatory nature of the FID without introducing artifacts, quadrature detection is employed, utilizing two orthogonal phase-sensitive receiver channels referenced at 0° and 90° relative to the carrier frequency. This setup demodulates the signal into real (in-phase) and imaginary (quadrature) components, effectively doubling the usable spectral bandwidth and distinguishing positive from negative frequencies to prevent image artifacts that would otherwise fold into the spectrum.^[22] The two channels are balanced to ensure equal gain and phase, with any imbalances corrected via phase cycling techniques such as CYCLOPS to eliminate ghost peaks arising from quadrature mismatch.^[22] Sampling of the analog FID signal adheres to the Nyquist-Shannon theorem, requiring a minimum sampling rate of twice the maximum frequency component within the desired bandwidth (BW) to avoid aliasing. The dwell time, defined as the interval between consecutive samples, is thus set to

\frac{1}{2 \times \mathrm{BW}}

, ensuring faithful digital representation of the signal's frequency content up to the spectrometer's spectral width.^[23] For typical proton NMR at 400 MHz with a BW of 5 kHz, this yields a dwell time of approximately 100 μs.^[23] Prior to digitization, the weak FID signal (often in the microvolt range) undergoes pre-amplification to boost its amplitude while minimizing noise addition, typically using low-noise amplifiers integrated near the probe.^[24] The amplified signal is then converted to digital form via an analog-to-digital converter (ADC), commonly 16- to 24-bit resolution, which must accommodate the signal's dynamic range—spanning orders of magnitude from strong solvent peaks to weak analyte resonances—without clipping or quantization noise overwhelming subtle features. Receiver gain is adjusted to optimize this range, ensuring the largest signals approach but do not exceed the ADC's full scale while preserving detectability of the smallest ones.^[25]

Fourier transformation

The Fourier transformation converts the time-domain free induction decay (FID) signal, which captures the evolving transverse magnetization, into a frequency-domain spectrum that reveals chemical shifts and coupling patterns in nuclear magnetic resonance (NMR) spectroscopy. This process is essential for interpreting the FID as a superposition of precessing spin frequencies, transforming the decaying oscillatory signal into absorption and dispersion lineshapes. The mathematical foundation relies on the Fourier transform, where the spectrum

S(\omega)

is obtained from the FID

S(t)

via the integral

S(\omega) = \int_{-\infty}^{\infty} S(t) e^{-i \omega t} \, dt

. In practice, since the FID is digitally sampled as a finite set of discrete points, the discrete Fourier transform (DFT) is applied, often efficiently computed using the fast Fourier transform (FFT) algorithm to reduce computational complexity from

O(N^2)

O(N \log N)

, where

N

is the number of data points. This seminal application of Fourier transform techniques to NMR was introduced by Ernst and Anderson in 1966, enabling high-resolution spectra from transient signals.^[26] To mitigate artifacts from the finite duration of the FID, such as sinc-like oscillations due to abrupt truncation, apodization functions are multiplied with the time-domain data prior to transformation. These window functions taper the signal edges, trading off resolution for improved signal-to-noise ratio (SNR) or vice versa; for instance, an exponential apodization

e^{-t / T}

enhances sensitivity by emphasizing early, high-SNR portions of the FID while broadening lines, whereas a Gaussian function provides a smoother decay with better resolution preservation. Common examples include the Lorentzian line broadening for mimicking natural decay and Hamming or Blackman-Harris windows to suppress side lobes in the frequency domain. The choice depends on the experiment's goals, with exponential functions often preferred in routine 1D NMR for optimal SNR in short FIDs.^[27] Post-transformation processing includes zero-filling, where the FID is padded with zeros to increase the digital resolution of the spectrum without altering the underlying frequencies, effectively interpolating points for smoother peak shapes—typically extending the data to 2-4 times the original length for adequate visualization. Phase correction then aligns the real and imaginary components to yield purely absorptive lineshapes, correcting for delays in receiver timing or filter effects; this involves zero-order (constant) and first-order (linear) adjustments, often parameterized as

\phi_0

and

\phi_1

, applied via

S'(\omega) = S(\omega) e^{i (\phi_0 + \phi_1 \omega / \omega_{\max})}

. Automated or manual optimization ensures symmetric, positive peaks essential for quantitative analysis.^[28]^[29] The inherent linewidth of spectral peaks in the transformed domain is directly tied to the FID's decay characteristics, with the full width at half maximum (FWHM) given by

\Delta \nu = \frac{1}{\pi T_2^*}

, where

T_2^*

incorporates both intrinsic spin-spin relaxation and extrinsic dephasing. This relation quantifies how faster time-domain decay—due to shorter

T_2^*

—results in broader frequency-domain lines, limiting resolution; for example, a

T_2^*

of 1 second yields a linewidth of approximately 0.32 Hz, establishing the fundamental limit for distinguishing closely spaced resonances.^[30]

Applications

Spectroscopy techniques

Free induction decay (FID) forms the cornerstone of nuclear magnetic resonance (NMR) spectroscopy techniques, enabling the acquisition of high-resolution spectra for chemical structure elucidation. The transition from continuous-wave (CW) NMR, dominant in the 1950s and 1960s, to pulsed Fourier transform (FT) NMR in the 1970s revolutionized the field by dramatically improving sensitivity and acquisition speed. This shift, pioneered by Richard R. Ernst and Weston A. Anderson, replaced slow frequency-sweeping CW methods with rapid radiofrequency pulses that excite all resonances simultaneously, capturing the entire FID in seconds rather than minutes.^[31] In one-dimensional (1D) NMR, pulsed FT methods rely on the FID as the primary signal source, where multiple acquisitions of the decaying FID are averaged to enhance the signal-to-noise ratio (SNR). The SNR improves proportionally to the square root of the number of scans (√N), allowing detection of low-abundance nuclei like ¹³C that were impractical in CW setups. This averaging, combined with the broadband excitation, made routine 1D proton and carbon spectroscopy feasible for organic and biochemical analysis. Multidimensional NMR extends FID-based detection to resolve complex mixtures by correlating spin interactions across multiple dimensions. In two-dimensional correlated spectroscopy (2D COSY), increments in the evolution time generate a series of FIDs, whose Fourier transforms yield a 2D spectrum with diagonal peaks representing 1D resonances and off-diagonal cross-peaks indicating coupled protons. First implemented by Ernst's group, COSY reveals J-coupling connectivity, essential for assigning structures in molecules like peptides and natural products. Solid-state NMR addresses challenges in rigid, non-soluble samples where FIDs decay rapidly due to strong anisotropic interactions. Magic-angle spinning (MAS), spinning the sample at approximately 54.74° relative to the magnetic field, averages out chemical shift anisotropy and dipolar couplings, extending FID coherence and enabling high-resolution spectra comparable to solution NMR. Introduced by E. Raymond Andrew and independently by I. J. Lowe, MAS facilitates applications in materials science, such as characterizing polymers and catalysts, by mitigating dephasing in FIDs from immobile nuclei.

Magnetic resonance imaging

In magnetic resonance imaging (MRI), free induction decay (FID) serves as the primary signal source for spatial encoding, enabling the reconstruction of anatomical and functional images through the application of magnetic field gradients. These gradients, typically denoted as

G_x

G_y

, and

G_z

, impose linear variations in the magnetic field across the imaging volume, causing spins at different spatial positions to precess at distinct Larmor frequencies during FID acquisition. This frequency variation allows for the encoding of positional information directly into the FID signal, which is sampled over time following an excitation radiofrequency (RF) pulse.^[32] The sampled FID data fills k-space, the Fourier domain representation of the image, where each point corresponds to a specific spatial frequency component. After RF excitation, the FID is acquired under the influence of readout gradients, tracing a trajectory through k-space that captures the transverse magnetization's evolution. Subsequent application of the inverse Fourier transform converts this k-space data into the spatial image, providing contrast based on tissue properties such as proton density and relaxation times. This process fundamentally relies on the coherent decay of the FID to encode and resolve spatial details without mechanical movement.^[32]^[33] A cornerstone method for this encoding is spin-warp imaging, which systematically applies gradients to select and encode slices and in-plane positions. Slice selection occurs via the

G_z

gradient paired with a shaped RF pulse, exciting a thin slab of tissue where the pulse bandwidth determines the slice thickness, typically on the order of 1-10 mm. Phase encoding then involves stepwise increments of the

G_y

gradient duration or amplitude before each FID acquisition, introducing a position-dependent phase shift across the sample; this is repeated for 128 or 256 steps to sample multiple lines in k-space. Finally, frequency encoding employs the

G_x

readout gradient during FID sampling, converting spatial positions into frequency differences that are resolved in the Fourier transform, yielding a two-dimensional image with resolutions down to sub-millimeter scales.^[34]^[32] The inherent T2* decay of the FID during acquisition contributes to contrast in MRI, particularly in gradient-echo sequences where no refocusing pulses are used, leading to T2*-weighted images sensitive to magnetic field inhomogeneities. This weighting arises from the exponential decay of the FID signal, governed by the effective transverse relaxation time T2*, which is shorter than the intrinsic T2 due to dephasing factors. In blood oxygenation level-dependent (BOLD) functional MRI, these T2*-weighted contrasts detect neural activity-induced changes in deoxyhemoglobin concentration, as deoxyhemoglobin's paramagnetic properties shorten T2* and attenuate the FID signal; activation increases oxygenated blood flow, reducing this effect and enhancing signal in active brain regions.^[32]

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