Spin Precession Simulator — Larmor Precession and NMR/MRI Basics
Visualize Larmor precession of a magnetic moment in a static field. Adjust B_0, gamma, T_1 and T_2 to see why protons precess at 63.85 MHz in a 1.5T MRI scanner.
Parameters
Static field B_0
T
Gyromagnetic ratio gamma (×10⁷ rad/s/T)
×10⁷
Longitudinal relaxation T_1
s
Transverse relaxation T_2
s
Default gamma = 26.75×10⁷ rad/s/T is the proton (¹H) value. The physical constraint T_2 ≤ T_1 is clamped automatically.
Results
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Larmor frequency f_L
—
Larmor angular freq omega_L
—
Longitudinal T_1
—
Transverse T_2
Spin precession and relaxation curves
Top = xy projection of magnetization M (precession circle decays with T_2); Bottom = M_z(t) longitudinal recovery and M_xy(t) transverse envelope
Theory & Key Formulas
A magnetic moment mu in a static field B_0 precesses about the field axis at a constant angular speed driven by the torque d mu/dt = gamma mu × B_0.
Larmor angular frequency and frequency. gamma is the gyromagnetic ratio:
Free precession after a 90 degree pulse (transverse signal decays with T_2):
$$M_x(t) = M_0\cos(\omega_L t)\,e^{-t/T_2}$$
Saturation recovery of the longitudinal magnetization (time constant T_1):
$$M_z(t) = M_0\left(1 - e^{-t/T_1}\right)$$
A proton (gamma = 2.675×10⁸ rad/s/T) in a 1.5T MRI gives a Larmor frequency of about 63.85 MHz.
What is the Spin Precession Simulator?
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"Spin precession" is not a familiar term. Is this what powers MRI machines?
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It is literally the heart of MRI. Roughly, an atomic nucleus acts like a tiny magnet, and when you place it in a strong field it starts tracing a cone around the field axis. That is Larmor precession. The picture is just like a spinning top precessing under gravity: the field B_0 plays the role of gravity, and the gyromagnetic ratio gamma plays the role of the top's geometry. Raise B_0 on the slider and the Larmor frequency f_L card scales linearly with it.
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The default 1.5T gives 63.85 MHz. What exactly is "spinning" at 63 MHz?
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It means a proton (hydrogen nucleus) sitting in a 1.5T field precesses about 63.85 million times every second. An MRI scanner sends a radio pulse at exactly that frequency to make the protons resonate. When the pulse stops, the protons emit a radio signal at the same frequency, and the scanner picks it up to build an image. Try setting B_0 to 3T on the simulator — the Larmor frequency jumps to about 127 MHz. That higher frequency is why 3T scanners give stronger signal.
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The T_1 and T_2 sliders stretch and squeeze the lower curves. What is the difference?
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T_1 is the time for spins to realign with the field direction, while T_2 is the time for the precessing magnetization in the xy plane to lose phase coherence. White matter and grey matter in the brain have different T_1 and T_2, so different MRI sequences make one bright and the other dark. That is exactly how MRI contrast works. If you try to push T_2 above T_1 the tool clamps it, because losing phase coherence faster than reaching equilibrium is physically impossible.
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I have heard fat and water differ a lot. How different are their relaxation times?
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Good question. At 1.5T fat has T_1 around 0.3 s and T_2 around 0.08 s — both short. Pure water has T_1 around 4 s and T_2 around 2 s — much longer. In a T_1-weighted image (short TR) fat looks bright and water looks dark, while in a T_2-weighted image (long TE) water looks bright and fat looks darker. Slide T_1 from 0.3 to 4.0 in the simulator and watch how the blue recovery curve changes — that is exactly the contrast mechanism you see in clinical scans.
FAQ
gamma is the ratio of the nuclear magnetic moment mu to the angular momentum L (gamma = mu/L) and is a fixed constant for each nucleus. The proton has a large g-factor of about 5.586, giving the large value gamma = 2.675×10⁸ rad/s/T. This is the main reason hydrogen is the nucleus of choice for MRI. ¹³C has gamma = 6.73×10⁷ and ³¹P has 1.08×10⁸, spanning roughly a decade across nuclei. The Larmor frequency therefore changes with the nucleus even at the same B_0, and multinuclear NMR simply retunes the receiver to match.
The T_1 process (longitudinal relaxation) involves energy exchange with the lattice and necessarily also dephases the spins. The T_2 process (transverse relaxation) can additionally come from pure spin-spin dephasing without energy loss. So T_2 relaxation is at least as fast as T_1 relaxation, which gives the constraint T_2 ≤ T_1 in time-constant terms. In most biological tissues T_2 is a few times smaller than T_1, while in pure water both are nearly equal. In real MRI imaging T_2* (T-two star), the apparent T_2 including field inhomogeneity, often dominates signal decay.
An RF pulse at the Larmor frequency drives Rabi oscillation about the B_1 axis. A 90 degree pulse is timed to tip the magnetization fully into the xy plane; a 180 degree pulse flips it to the negative z direction. After a 90 degree pulse the precessing magnetization in xy produces the FID (free induction decay) signal. 180 degree pulses are used in spin-echo sequences to refocus T_2* dephasing. Every MRI imaging sequence is built from combinations of these basic pulses.
Even for the same nucleus, the surrounding electrons produce a small shielding field that depends on the molecular environment. The Larmor frequency therefore shifts by a tiny amount (parts per million, ppm), and this chemical shift is essential for determining molecular structure. In the ¹H NMR of ethanol CH₃CH₂OH, for example, the CH₃, CH₂ and OH protons give peaks at distinct positions, letting chemists read off the structure. In MRI the same effect produces the fat shift artifact, suppressed by fat-saturation sequences.
Real-world applications
Medical imaging (MRI): The largest application by far. Proton Larmor precession is exploited to non-invasively map the distribution of water in the body, with no ionizing radiation. MRI provides soft-tissue contrast inaccessible to X-ray or CT and is now central to diagnosing brain tumors, spinal injuries, cartilage damage, and myocardial infarction. 1.5T is the clinical workhorse, 3T is increasingly common for high resolution, and 7T systems serve research hospitals.
NMR spectroscopy for chemical structure: In organic chemistry, pharmacology and biochemistry, NMR is indispensable for determining molecular structures. Chemical shifts and J-couplings reveal connectivity between atoms, confirming new drug candidates and natural product structures. Protein NMR, which solves three-dimensional protein structures in solution, is one of the two pillars of structural biology alongside X-ray crystallography.
Solid-state physics and materials science: Solid-state NMR (MAS-NMR) is a powerful probe of local structure in glass, ceramics and battery materials, providing atomic-scale information even in non-crystalline samples. In lithium-ion battery research, Li NMR tracks structural changes in cathode materials during charge and discharge cycles.
Quantum computing and atomic clocks: NMR quantum computers were among the earliest physical implementations of quantum information processing, using nuclear or electron spin precession as qubits. CPT (coherent population trapping) atomic clocks also rely on precise measurement of atomic spin precession.
Common misconceptions and pitfalls
The most common misconception is to think that "the Larmor frequency is the speed at which the nucleus itself spins." In reality the Larmor frequency is the quantum-mechanically defined angular frequency of precession of the nuclear magnetic moment, distinct from any rotation of the nucleus on its own axis. Saying that a proton in 1.5T precesses at 63.85 MHz means the magnetization vector traces about 63.85 million circuits per second around the field axis. The precession circle in the top panel of this simulator is the intuitive classical (Bloch) picture of that motion.
Another frequent error is to treat T_1 and T_2 as freely independent parameters. Physically T_2 must satisfy T_2 ≤ T_1, because any T_1 process also contributes to dephasing. The simulator therefore clamps T_2 back to T_1 if you try to exceed it. In biological tissues T_2 is often 10–30 percent of T_1, while in liquids with fast molecular motion such as pure water the two are close (T_1 ≈ T_2 ≈ several seconds). MRI sequence parameters (TE and TR) are chosen specifically to exploit these tissue-dependent T_1/T_2 differences.
Finally, remember that this simulator visualizes only the classical Bloch free precession and does not include quantum effects, RF pulses, or gradient fields. Real MRI imaging adds spatial encoding by gradient fields, selective excitation pulses, k-space sampling, and Fourier reconstruction. Use this tool as an introduction to the starting point of NMR and MRI — Larmor precession and relaxation — and study textbooks or clinical training for the full pulse-sequence and imaging pipeline.