Theoretical Foundations | Ruby Pressure Calculator

1. Ruby Calibration Principle

The ruby calibration method is based on the pressure dependence of the R₁ and R₂ fluorescence line shifts in ruby (Al₂O₃:Cr³⁺). This technique, initially developed by Mao and Bell (1978), exploits the quasi-linear relationship between pressure and Raman peak shift.

Physical Mechanism

Hydrostatic pressure application alters the crystal field around Cr³⁺ ions, modifying electronic energy levels and consequently the emission wavelength.

$$ \lambda_p = \lambda_0 \left(1 + \frac{aP}{b}\right)^{1/a} $$

Fundamental ruby calibration equation

Calibration Equation Parameters
Symbol	Description	Value	Unit	Uncertainty
$ \lambda_0 $	Zero-pressure wavelength	694.22	nm	±0.03 nm
$ a $	Pressure exponent	7.665	dimensionless	±0.008
$ b $	Pressure coefficient	1904	GPa	±12 GPa

Experimental Considerations

Equation is valid for hydrostatic or quasi-hydrostatic conditions
Room temperature (20-25°C)
Pressure range: 0-500 GPa
Thermal dependence: ~0.007 nm/°C (should be compensated in non-isothermal experiments)

2. Complete Equation System

Pressure → Wavenumber

$$ \lambda_p = \lambda_0 \left(1 + \frac{aP}{b}\right)^{1/a} $$ $$ \nu = \left(\frac{1}{\lambda_{exc}} - \frac{1}{\lambda_p}\right) \times 10^7 $$

Where:

$ \nu $: Raman wavenumber (cm⁻¹)
$ \lambda_{exc} $: Excitation laser wavelength (nm)

Wavenumber → Pressure

$$ \lambda_p = \frac{1}{\frac{1}{\lambda_{exc}} - \frac{\nu}{10^7}} $$ $$ P = \frac{b}{a} \left[\left(\frac{\lambda_p}{\lambda_0}\right)^a - 1\right] $$

Mathematical Derivation

Starting from the equation of state:

$$ \frac{\Delta\lambda}{\lambda_0} = \left(1 + \frac{aP}{b}\right)^{1/a} - 1 $$

For small pressures (P < 10 GPa), linear approximation:

$$ \Delta\lambda \approx \frac{\lambda_0}{a} \ln\left(1 + \frac{aP}{b}\right) $$

3. Assumptions and Limitations

Hydrostaticity

Calibration assumes hydrostatic conditions. In non-hydrostatic media, deviations up to 5% may occur.

Thermal Effects

Temperature variations affect $ \lambda_0 $. Thermal correction is not implemented in the application.

Ruby Homogeneity

Impurities or crystal defects may alter pressure response.

Validity Range

Above 150 GPa, the equation may underestimate actual pressure by ~3-7%.

Comparison with Other Standards

Method	Range (GPa)	Accuracy	Advantages
Ruby (R₁)	0-500	±0.5%	Non-destructive, high resolution
XRD	0-300	±1%	Direct, absolute
Diamond	0-200	±2%	For high temperatures

4. Bibliographic References

Mao, H.K., Xu, J., & Bell, P.M. (1986). Calibration of the ruby pressure gauge to 800 kbar under quasi-hydrostatic conditions. Journal of Geophysical Research, 91(B5), 4673-4676. DOI: 10.1029/JB091iB05p04673
Dewaele, A., et al. (2004). Quasihydrostatic equation of state of iron above 2 Mbar. Physical Review Letters, 93(21), 215504.
Holzapfel, W.B. (2003). Refinements in the ruby pressure scale. Journal of Applied Physics, 93(3), 1813-1818.