---
title: "NMR Free Induction Spectrometer Interactive Lab | Sandbox Physics"
description: "Turn a fading voltage into a spectrum. Tip an ensemble with a radio-frequency pulse. Retain both quadratures of the induced voltage, transform those records into a spectrum, and estimate the signed frequency offset and effective transverse decay."
canonical_url: "https://sandboxphysics.com/labs/nmr-free-induction-spectrometer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/nmr-free-induction-spectrometer/"
last_modified: "2026-10-18"
---

NUCLEAR INDUCTION / FREQUENCY

# NMR Free Induction Spectrometer

Tip an ensemble with a radio-frequency pulse. Retain both quadratures of the induced voltage, transform those records into a spectrum, and estimate the signed frequency offset and effective transverse decay.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Ideal single-resonance ensemble with instantaneous broadband pulses, exponential intrinsic decay and a Lorentzian static field distribution. Calibrated quadrature receiver, white Gaussian voltage noise and complete reset between repetitions. The inferred effective decay cannot separate intrinsic relaxation from static broadening. Synthetic data; no molecule or historical apparatus is fitted.

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 / AVERAGED RECEIVER VOLTAGES

## After the pulse, listen.

Equilibrium reset $\theta_x$ Receiver blanking Signed quadratures

One fully reset acquisition contains 256 complex samples, every 0.4 milliseconds after a 0.4-millisecond dead time. Repeats are averaged sample by sample. The receiver resolves the slow beat against the RF reference, not the fast carrier.

Ready. Predict, then acquire.

RAW VOLTAGE RECORD / SEPARATE SOURCE EXPECTATION

### Two quadratures. 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 comes from those voltages.

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

A uniformly spaced record is needed. Rectangular window, no zero filling. Magnitude and finite-window width are not a fitted absorption line or an independent relaxation measurement.

INFERENCE / SOURCE PARAMETERS EXCLUDED

Inferred effective transverse time

$\text{—}$

Inferred signed frequency offset

$\text{—}$

Inferred initial voltage magnitude

$\text{—}$

Retained complex samples

$0$

Voltage residual RMS

$\text{—}$

Global bounded search with profiled amplitude. Weak-information and boundary checks are diagnostic heuristics, not parameter confidence intervals. A synthetic recovery does not independently establish the physical law.

Latest raw observations

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

SAVED COMPARISON / MODEL BOUNDARY

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](https://sandboxphysics.com/labs/nmr-inversion-recovery/) [Read the magnetic-resonance evidence](https://sandboxphysics.com/stories/quantum-evidence/magnetic-resonance-beams-absorption-induction/) [TeachSpin original experimental guide](https://home.sandiego.edu/~severn/p480w/PNMR_Concept_Tour.pdf) [Manufacturer instrument manual](https://physics.wm.edu/~inovikova/phys352/PHYS352-Manuals-Instructions/Pulsed-NMR/nmr_company.pdf) [Bruker transmit/receive probe manual](https://2210pc.chem.uic.edu/nmr/downloads/bruker/en-US/pdf/z31339.pdf) [NIST reference gyromagnetic ratio](https://physics.nist.gov/cuu/Constants/Table/allascii.txt)

Physics tutorial

## NMR Free Induction Spectrometer

**Background** Tip an ensemble with a radio-frequency pulse. Retain both quadratures of the induced voltage, transform those records into a spectrum, and estimate the signed frequency offset and effective transverse decay.

**Why it matters** A constant is inferred through a calibrated response and retained observations.

Start with the essentials

- **Focus question**: Turn a fading voltage into a spectrum.
- **One-sentence intuition**: Keep source assumptions separate from estimates made from records.

### Core mathematical model

#### A transverse ensemble induces a voltage

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

A Lorentzian distribution of static offsets has an exponential characteristic function. This special distribution gives additive decay rates. The complex amplitude is unknown to the estimator.

#### Transform the retained voltages

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

The displayed spectrum uses the actual finite, uniformly sampled record with a rectangular window. Its nonnegative magnitude loses phase; the signed frequency axis retains rotation direction. Partial records have coarser bins.

#### Infer from both voltage channels

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

Profile out the complex amplitude at every candidate frequency and time. The source decay, broadening, phase and flip angle never enter this fit.

### Common difficulties

#### A simulation supplies its law

**Typical misconception** Recovering a source parameter independently confirms the law.

**Better mental model** These synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

### Run the experiment

1. 01

   #### Acquire before transforming

   Acquire the reference pulse and inspect both raw voltage channels and the finite-window spectrum.

   **What to observe:** The spectrum appears only after retaining a uniform record. It is not a separately drawn ideal line.

2. 02

   #### Distinguish broadening from lifetime

   Save the record, then broaden the field without changing the intrinsic source time.

   **What to observe:** The effective time shortens. A single free decay cannot identify which process caused its width.

3. 03

   #### Reverse the beat

   Choose the negative-frequency preset, then remove transverse preparation.

   **What to observe:** Complex sampling distinguishes frequency signs. With no coherent signal, the weak-information rule withholds the estimate.

Physical conclusion

Ideal single-resonance ensemble with instantaneous broadband pulses, exponential intrinsic decay and a Lorentzian static field distribution. Calibrated quadrature receiver, white Gaussian voltage noise and complete reset between repetitions. The inferred effective decay cannot separate intrinsic relaxation from static broadening. Synthetic data; no molecule or historical apparatus is fitted.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)