Spin-exchange optical pumping from the laser to the breath-hold: build-up in the cell, the flow trade, decay in the bag, and the flip-angle budget that spends what is left.
Teaching note·version 1.0··not peer reviewed
A toy model built to give intuition for how a continuous-flow SEOP polarizer behaves, not a design tool and not a substitute for measuring your own system. The rate coefficients are literature values and the structure of the model is standard, but several terms are deliberately simplified and the absolute polarization it predicts should be read as order-of-magnitude. Everything it assumes, and everything it leaves out, is listed under assumptions and limits.
Laser light polarizes rubidium vapor, rubidium hands angular momentum to 129Xe through collisions, and xenon accumulates polarization as it flows down the cell. Three rates compete: optical pumping ROP builds Rb polarization, spin destruction ΓSD tears it down, and spin exchange γSE moves it to xenon. Nearly every knob on a real polarizer changes more than one of the three, which is why the optima below are optima rather than monotonic trends.
The simulator integrates along the optical axis. At each slice it computes the local pumping rate from the surviving photon flux, sets the Rb polarization from the balance of pumping and destruction, removes the photons that Rb actually absorbed, and advances the xenon polarization by the time the gas spent in that slice. Downstream panels then take the polarization the cell delivers and spend it: cryogenic accumulation, decay in the bag, and the non-renewable magnetization budget of the scan.
Along the cell: surviving laser flux (orange), local Rb polarization (blue), and the xenon polarization the flowing gas has accumulated by that point (green). When the cell is optically thick the orange curve collapses in the first few centimetres and the blue curve falls with it, so the back of the cell contributes almost nothing.
Polarization per atom and polarized atoms per hour pull in opposite directions, but they do not pull in the same shape. Slow the flow down and each atom spends longer in contact with polarized rubidium, so polarization climbs toward the ceiling rubidium sets, but you make almost no gas. Speed it up and output climbs and then saturates: a cell can only polarize atoms at the rate γSE allows, and every atom pushed through beyond that leaves underpolarized. There is no peak in the output curve to find, only a knee past which you are giving up polarization for almost nothing in return.
Cell temperature behaves differently, and it does have a true interior optimum you can overshoot. More rubidium means faster spin exchange, right up until the vapor is thick enough to absorb the laser in the first few centimetres. Past that point the back of the cell holds rubidium that is dense but unpolarized, and because spin exchange runs both ways, that rubidium actively drags the xenon back down on its way out. Set the temperature slider past the optimum and watch the green curve in panel 1 rise and then fall again before the gas reaches the exit.
Both plots use the cell settings from panel 1, sweeping one variable at a time; the dot marks where you currently sit. Output is litres of xenon per hour weighted by polarization, so 1.0 means one litre per hour at 100 % or two litres at 50 %. The two shapes are not the same: output saturates with flow, while temperature has a genuine optimum with a cliff on the far side of it.
Polarization is non-renewable from the moment the gas leaves the cell. Xenon is frozen out to separate it from the buffer gases, thawed into a bag, carried to the magnet, and inhaled, and every one of those steps costs magnetization that nothing will restore.
Polarization against wall-clock time from cell exit. The shaded stretch is cryogenic accumulation, where the batch is a mix of gas frozen early and gas frozen late; the drop at thaw is the mechanical loss; the slope after it is bag T1; the steep final segment is the breath-hold, where alveolar oxygen dominates.
In proton imaging, longitudinal magnetization recovers between excitations and a flip angle is a signal-to-noise choice. Here there is no recovery, so a flip angle is a spending decision: each pulse removes (1 − cos α) of what remains, permanently. A constant angle spends fastest at the start and leaves the late views starved; the variable-angle schedule αn = arctan(1/√(N − n)) is built so every view returns the same signal and the last pulse is a 90° that takes what is left.
Signal per view across the acquisition, including T1 decay in the lung during the breath-hold. Constant angle in blue, the constant-signal variable schedule in purple. Watch what happens to the k-space weighting when the blue curve decays inside a single acquisition: that is a point-spread-function problem, not just an SNR problem.
| Quantity | Value | Source and caveat |
|---|---|---|
| Rb vapor pressure (liquid) | log10 Ptorr = 7.193 − 4040/T |
Steck, Rubidium 87 D Line Data. Assumes a saturated vapor in equilibrium with liquid Rb, which overestimates density in a cell that has lost its rubidium or is running a cold spot. |
| Rb D1 line | 794.98 nm f = 0.342 |
Steck. Oscillator strength combined with πrec = 2.654×10−2 cm2·Hz for the integrated cross section. |
| Pressure broadening | 18 GHz/amagat | Lumped single value. Real coefficients differ by collision partner (He and N2 near 18, Xe closer to 19 GHz/amagat FWHM), and the model ignores the accompanying line shift. |
| Rb spin destruction | Xe 5.2×10−15 N2 1.0×10−17 He 1.1×10−18 Rb 4.2×10−13 cm3/s |
Typical values from the Walker and Happer review. Reported values scatter by tens of percent between measurements, and the Rb–Xe figure already folds in a van der Waals contribution that is itself density dependent. |
| Rb–Xe spin exchange | binary 3.7×10−16 cm3/s vdW 1.1×10−15 / namagat |
The binary term is a literature value. The van der Waals term is phenomenological: it carries the correct 1/n third-body scaling and is normalized to give an effective coefficient near 1.5×10−15 cm3/s at one amagat, but it is an interpolation, not a first-principles molecular calculation. |
| Standard litre | 0 °C, 1 atm | Flow in SLM is converted to cell conditions by number density. Mass-flow controllers are not all referenced to the same standard temperature, so check yours before comparing. |
The shapes are the point and they are trustworthy: the flow optimum, the temperature optimum, the collapse of mean Rb polarization once the cell goes optically thick, the trade between polarization and throughput as xenon fraction rises, and the way a constant flip angle weights k-space. The absolute numbers are not. Expect the predicted polarization to sit optimistically high against a real machine, since every loss mechanism in the list above is either absent or compressed into a slider. If you want numbers you can quote, measure your own polarizer against a calibrated thermal reference.
Related pages: hyperpolarized imaging and SEOP · dissolved-phase peaks · diffusion and ADC