Frequency from nuclear induction

NMR Free Induction Spectrometer

What can the fading voltage after a radio-frequency pulse tell you about the nuclear spins? How far their precession frequency sits from the reference, and how fast they fall out of step. The tipped magnetization precesses and induces a voltage in the coil. Two channels a quarter cycle apart give the offset with its sign, and a Fourier transform turns the record into a spectrum peaked there. The decay mixes true relaxation with an uneven field, and one decay cannot tell them apart.

Tip an ensemble of nuclear spins with a radio-frequency pulse and record both quadrature channels of the voltage it induces. Transform the recording into a spectrum, then estimate the frequency offset, including its sign, and the effective transverse decay.

Tip the magnetization, then read the coil voltage

What can the fading voltage after a radio-frequency pulse tell you about the nuclear spins?

Source intrinsic transverse time · excluded from fit

T_2/\mathrm{ms}

25

Source static frequency half-width

w/\mathrm{Hz}

5

Source frequency offset · excluded from fit

\Delta f/\mathrm{Hz}

120

Source receiver phase · excluded from fit

\phi/{}^\circ

25

Hard-pulse flip angle

\theta/{}^\circ

90

Calibrated single-repeat voltage noise

\sigma/\mathrm{\mu V}

5

Independent repeats averaged

n

4

Time after pulse

t/\mathrm{ms}

20

Moving the time probe retains the record. Other settings start a new acquisition; a saved comparison is retained.

The spins form an ideal ensemble with a single resonance. Pulses are instantaneous and broadband, the intrinsic decay is exponential, and the static field spread is Lorentzian. The quadrature receiver is calibrated, the voltage noise is white and Gaussian, and the spins reset completely between repetitions. The fitted effective decay cannot separate intrinsic relaxation from static broadening. The data are synthetic and are not fitted to any molecule or historical apparatus.

Vertical static field; horizontal transmit/receive coil. A switch isolates the receiver during the pulse, then routes the induced voltage to a phase-referenced quadrature receiver. Cable motion is schematic. $B_0=0.35\,\mathrm{T},\quad f_{\rm ref}\simeq14.902\,\mathrm{MHz}$

Pulse sequence and averaged receiver voltages

After the pulse, listen.

Equilibrium reset

\theta_x

Receiver blankingSigned quadratures

Each acquisition starts from a full reset and records 256 complex samples, one every 0.4 milliseconds, after a 0.4-millisecond dead time. Repeats are averaged sample by sample. The receiver picks up the slow beat against the RF reference; it does not follow the fast carrier.

Ready. Predict, then acquire.

Raw voltages, with the source expectation shown separately

Two quadrature channels keep the sign

$t/\mathrm{ms},\quad I,Q\;(\mathrm{\mu V})$

Teal: in-phase voltage. Gold: quadrature voltage. Dots: retained observations. Thin lines: source expectations. Purple dashes: fit using records only. Gray: saved observations.

Finite-window transform of the record

This spectrum is computed from the voltages above.

$f/\mathrm{Hz},\quad |S_k|/\mathrm{\mu V}$

The transform needs evenly spaced samples. It uses a rectangular window and no zero filling. The magnitude and the width set by the finite window are not a fitted absorption line or an independent relaxation measurement.

Inference without the source parameters

Inferred effective transverse time

\text{—}

Inferred signed frequency offset

\text{—}

Inferred initial voltage magnitude

\text{—}

Retained complex samples

0

Voltage residual RMS

\text{—}

The fit is a global search within bounds, with the amplitude profiled out. The weak-information and boundary checks are rough diagnostics, not confidence intervals on the parameters. Recovering the inputs from synthetic data does not independently prove the physical law.

Latest raw observations

Index	$t/\mathrm{ms}$	$I/\mathrm{\mu V}$	$Q/\mathrm{\mu V}$

Saved comparison and model limits

Single noninteracting resonance; no molecular chemical shifts, coupling multiplets, diffusion, radiation damping, coil loading or realistic pulse bandwidth. Gain and phase are calibrated; no drift or digitizer quantization. Instantaneous hard-pulse approximation; the Cauchy static-offset model is not a computed field map.

Next: inversion recovery Read the magnetic-resonance evidence TeachSpin original experimental guide Manufacturer instrument manual Bruker transmit/receive probe manual NIST reference gyromagnetic ratio

Start with the essentials

Focus question: What can the fading voltage after a radio-frequency pulse tell you about the nuclear spins?
One-sentence intuition: Keep what the simulation assumes about the source apart from what you estimate from the recorded data.

Core mathematical model

The free-induction signal and its decay rate

\begin{aligned}s(t)&=A e^{-t/T_2^*}e^{i(2\pi\Delta f t+\phi)}\\\frac1{T_2^*}&=\frac1{T_2}+2\pi w\end{aligned}

The static frequency offsets follow a Lorentzian distribution, whose characteristic function is an exponential. That special choice is why the two decay rates simply add. The estimator does not know the complex amplitude.

Discrete Fourier transform of the recorded voltages

S_k=\frac1N\sum_{j=0}^{N-1}(I_j+iQ_j)e^{-2\pi i k j/N}

The spectrum on screen is computed from the actual record, which is finite, evenly sampled and cut off by a rectangular window. Plotting only the magnitude loses the phase, but the sign of the frequency still shows which way the signal rotates. A partial record gives coarser frequency bins.

Fitting both voltage channels at once

\underset{a\in\mathbb C,\,f,\,T}{\operatorname{minimize}}\;\sum_j\frac{|I_j+iQ_j-ae^{-t_j/T}e^{2\pi i f t_j}|^2}{\sigma_j^2}

For each trial frequency and decay time, the best complex amplitude is solved for directly and removed. The fit is never given the source decay, broadening, phase or flip angle.

Common difficulties

A simulation already contains its law

Typical misconceptionGetting a source parameter back out of the data independently confirms the law.

Better mental modelThe synthetic data test a measurement and inference procedure. They are not historical observations, they give no confidence interval for the parameter, and they are not external scientific certification.

Run the experiment

01
Acquire before transforming
Acquire with the Reference pulse preset. Look at both raw voltage channels, then at the spectrum of the finite record.
What to observe: The spectrum appears only once an evenly sampled record exists, because it is computed from that record.
02
Distinguish broadening from lifetime
Save the record. Then increase the field broadening and leave the intrinsic source time unchanged.
What to observe: The effective decay time gets shorter. From one free decay alone, you cannot tell which process made the line wide.
03
Reverse the beat
Choose the Reverse frequency preset. Then choose No transverse signal.
What to observe: Sampling both channels tells positive and negative frequency offsets apart. With no coherent signal, the data carry too little information and no estimate is reported.

NMR Free Induction Spectrometer

After the pulse, listen.

Two quadrature channels keep the sign

This spectrum is computed from the voltages above.