Nuclei and units
Only spin-½ nuclei make clean qubits: two levels, no quadrupole. This is the table the simulator uses, with the conventions that connect a circuit to a spectrum.
Spin-½ nuclei
| Isotope | Name | γ/2π (MHz/T) | ν₀ at 11.74 T (MHz) | Natural abundance (%) | Reference (0 ppm) |
|---|---|---|---|---|---|
| ¹H | Proton | 42.5775 | 499.86 | 99.985 | TMS (Si(CH₃)₄) |
| ¹⁹F | Fluorine-19 | 40.0780 | 470.52 | 100 | CFCl₃ |
| ³He | Helium-3 | -32.4340 | 380.78 | 0.000137 | He gas |
| ²⁰⁵Tl | Thallium-205 | 24.9749 | 293.21 | 70.48 | TlNO₃ aq |
| ²⁰³Tl | Thallium-203 | 24.7316 | 290.35 | 29.52 | TlNO₃ aq |
| ³¹P | Phosphorus-31 | 17.2515 | 202.53 | 100 | 85% H₃PO₄ |
| ¹¹⁹Sn | Tin-119 | -15.9660 | 187.44 | 8.59 | Me₄Sn |
| ¹¹⁷Sn | Tin-117 | -15.2610 | 179.16 | 7.68 | Me₄Sn |
| ¹¹⁵Sn | Tin-115 | -14.0077 | 164.45 | 0.34 | Me₄Sn |
| ¹²⁵Te | Tellurium-125 | -13.5454 | 159.02 | 7.07 | Me₂Te |
| ¹²⁹Xe | Xenon-129 | -11.8604 | 139.24 | 26.44 | XeOF₄ |
| ¹²³Te | Tellurium-123 | -11.2349 | 131.90 | 0.89 | Me₂Te |
| ¹³C | Carbon-13 | 10.7084 | 125.72 | 1.109 | TMS (Si(CH₃)₄) |
| ¹¹³Cd | Cadmium-113 | -9.4871 | 111.38 | 12.22 | Me₂Cd |
| ¹⁹⁵Pt | Platinum-195 | 9.2922 | 109.09 | 33.83 | Na₂PtCl₆ aq |
| ¹¹¹Cd | Cadmium-111 | -9.0692 | 106.47 | 12.8 | Me₂Cd |
| ²⁰⁷Pb | Lead-207 | 8.8817 | 104.27 | 22.1 | Me₄Pb |
| ²⁹Si | Silicon-29 | -8.4655 | 99.38 | 4.685 | TMS (Si(CH₃)₄) |
| ⁷⁷Se | Selenium-77 | 8.1573 | 95.77 | 7.63 | Me₂Se |
| ¹⁹⁹Hg | Mercury-199 | 7.7123 | 90.54 | 16.87 | Me₂Hg |
| ¹⁷¹Yb | Ytterbium-171 | 7.5261 | 88.36 | 14.28 | Yb(C₅Me₅)₂ |
| ¹⁵N | Nitrogen-15 | -4.3172 | 50.68 | 0.366 | CH₃NO₂ |
| ¹⁶⁹Tm | Thulium-169 | -3.5310 | 41.45 | 100 | Tm(NO₃)₃ |
| ⁸⁹Y | Yttrium-89 | -2.0949 | 24.59 | 100 | Y(NO₃)₃ aq |
| ¹⁰⁹Ag | Silver-109 | -1.9924 | 23.39 | 48.161 | AgNO₃ aq |
| ¹⁸³W | Tungsten-183 | 1.7957 | 21.08 | 14.31 | Na₂WO₄ aq |
| ¹⁰⁷Ag | Silver-107 | -1.7331 | 20.35 | 51.839 | AgNO₃ aq |
| ⁵⁷Fe | Iron-57 | 1.3816 | 16.22 | 2.119 | Fe(CO)₅ |
| ¹⁰³Rh | Rhodium-103 | -1.3477 | 15.82 | 100 | Rh(acac)₃ |
| ¹⁸⁷Os | Osmium-187 | 0.9856 | 11.57 | 1.96 | OsO₄ |
A negative γ means the spin precesses the other way; frequencies on the tab use |γ|. Natural abundance is what limits a sample: a ¹³C qubit at 1.1% needs an enriched molecule, which is why the library's carbon samples are labelled ¹³C.
Frequencies
| Quantity | Formula | Example at 11.74 T |
|---|---|---|
| Reference frequency ν₀ | |γ/2π| · B₀ | ¹H 499.9 MHz, ¹³C 125.7 MHz |
| Resonance | ν₀ (1 + δ · 10⁻⁶) | chloroform ¹H at 7.26 ppm: ν₀ + 3.63 kHz |
| Offset in Hz | δ · ν₀ (ν₀ in MHz) | 7.26 × 499.9 = 3629 Hz |
| Doublet lines | offset ± J/2 | 3629 ± 107.5 Hz for J = 215 Hz |
| Line width | 1 / (π T₂) | T₂ = 0.3 s gives 1.06 Hz |
| Hard pulse length | θ / (γ B₁) | 90° at 25 kHz nutation: 10 µs |
Conventions
- |0⟩ is the spin state along +B₀ (α, m = +½). A 90° readout pulse turns it into a positive absorptive line; |1⟩ gives an inverted line.
- A coupling partner in |0⟩ moves a line by +J/2, a partner in |1⟩ by −J/2.
- Angles in Qubi rotation gates are multiples of π; the NMR functions take degrees and convert.
- Times are seconds, frequencies hertz, fields tesla, B₁ millitesla. 1 mT of B₁ is 42.6 kHz of proton nutation.
- The circuit state is a pseudo-pure state. A real ensemble at 11.74 T and room temperature carries a proton polarisation of about 4 × 10⁻⁵, which scales every signal but changes nothing else.