15. Physics
The project started off as an idea to make an instrument using a simple laser and an endpoint controller with all the necessary software already on-board, to detect and identify elements using a spectrometer.
This was never meant to take so many years. However as time progressed and I learned more, it became apparent the longer the project took, the greater the likelihood of the equipment failing and the more I considered the need to fully comprehend the process and produce my own system in its entirety.
I never started off as a physicist and in yesteryears I would never have considered such a daunting task, but now I have come so far it seems the next stage is to take the project to this new level (so
much for the next project, which was to be a custom exo-skeleton).
This new section is an attempt to understand the processes involved and will hopefully lead into the design of my own LIBS software to replace the PCM-401 endpoint system.
Unfortunately the project will still be hindered by the laborious time-consuming 3-grating Czerny-Turner Sp-275 monochromator (http://en.wikipedia.org/wiki/Monochromator) which would be so much better if it was a single shot, but formidably expensive Echelle:
THE PROCESS OF ATOMIC SIGNATURE
This Wikipedia entry strikes me as fundamental to understanding the physics of LIBS:
'Each element has a unique set of energy levels, and thus the transition from higher to lower energy levels produces X-rays with frequencies that are characteristic to each element.'
http://en.wikipedia.org/wiki/Characteristic_X-ray
Here too, is another website with a similarly revealing piece of the puzzle:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xrayc.html
This site explores the mathematical relationship between atomic spectra and X-rays:
EQUATIONS
SYMBOLS = Lucida Console font; supported symbols from MS Windows character map:
⌠ ⌡ ∫ │ ─ √ φ θ Θ ∂ δ ζ ξ ς λ ψ ω τ µ Ω ∆ Δ ∑ ∂ ∏ π ∞ ∝ Ξ ○ ≠ ³ ² ± ≤ ≥ ∴
NB lucida console only has the upper case φ not the L.C. which looks like 'up' merged together.
R.Savage's endpoint patent (below) uses this symbol. No Wix font supports this character.
As to the mathematics, I found some glimmering starting points:
This site explores the mathematical relationship between atomic spectra and X-rays:
[L2], Handbook of LIBS, Page 90, '3.6.2 Spectral Response Calibration', explains and provides equations for calibration of instrumentation for detection of elements using LIBS.
[L41], Reproducibility of CIGS [Cu(In,Ga)Se2] Thin Film Analysis by LIBS, includes useful equations for LIBS computation.
I assume [316p01] is the original patents by S.C.Technology for their PCM system:
'Apparatus and method for automatically identifying chemical species within a plasma reactor environment', http://www.google.ch/patents/US5014217
It seems they multiply the received spectra against stored spectra from an element, then subtract the received spectra from the result, which reveals the spectral signature.
But looking at their diagrams, this doesn't make sense: If you multiply 0 x waveform you get 0!
It says at that time (1989) they only needed a turnaround of 1Hz, but anticipate it going up to 100Hz.
The following equation defines correlation of the data:
1 T
C (n∆t) = ─ ∑ a(t) b(t±n∆t) n=200,201...1000, where:
ab T t=0
t = 500 data points [there are 2 little ts. Should this be big T?],
t = 1 data point
a(t) = actual spectrum
b(t±n∆t) = library spectra
n has to be λ incrementing 1nm at a time
'Endpoint determination for recess etching to a precise depth', http://www.google.ch/patents/US6650426
There are some equations in this one, but those in the book below are easier to comprehend:
I also found partial equations describing detection of optical emission and intensity in
Plasma Processing XII, Electrochemical Society Inc., Volume 98-4. [L11], page 173,
Endpoint detection methodology:
[Equation to determine species optical emission spectral wavelength]
'Optical emission signal associated with species can be expressed as:
I(λ,t) = Is(λ,t) + nI(λ)*n2(t)...................................(1)'
where:
I(λ,t) = optical emission intensity,
Is(λ,t) = optical emission intensity from species
n1(λ)*n2(t) = noise portion at wavelength λ and time t'
[sic] 'The differentiated spectrum already subtracted any zero order background noise exists in both plasma system and spectrum detection system. The spectrum is then averaged across time:'
[Equation to average spectral wavelength across time τ]
t + τ t + τ t + τ
⌠ ⌠ ⌠
'I = 1 / τ │I(λ,t)dt = 1 / τ(│Is(λ,t)dt + n1(λ)*│n2(t)dt).........(2)'
⌡ ⌡ ⌡
t t t
It goes on to describe an equation for endpoint where the integral value tends to zero, but this is of little interest for elemental detection alone. It adds:
'...signal to noise ratio...increases as integration time τ increases. The trade off of longer integration time is degrading time resolution...'
A later patent, 'Method and system employing optical emission spectroscopy for monitoring and controlling semiconductor fabrication', seems to be more closely tied to the functionality of the
PCM-4xx and provides deeper insight, including multiple flow diagrams of the mathematical process:
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