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## Main Question or Discussion Point

Introduction
In this article we will discuss the fundamentals of the diffraction grating spectrometer.  The operation of the instrument is based upon the textbook equations for the far-field interference (Fraunhofer case) that results from a plane wave incident on a diffraction grating.   It is rather remarkable how the standard textbook equations can be used to tell most everything one needs to know in order to understand the complete operation of the instrument.  It is hoped that upon reading this article, the reader will have a good understanding of how a diffraction grating spectrometer works.
For a diffraction grating spectrometer, the grating is the dispersive element instead of a prism.  In very simple form, the primary maxima from a diffraction grating for wavelength $\lambda$ are found at angles $\theta$ that satisfy $m \lambda=d \sin{\theta}$, with $m=$ integer.  The result is different...
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## Answers and Replies

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neilparker62
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Thanks - read your article with interest albeit perhaps needing to do some careful study to properly understand all the equations.

In the Insights article I wrote on the Deuterium Lyman Alpha line, the claimed resolution with a "3 metre vacuum grating spectrograph in fifth order" was:

$$L_\alpha(D)=1215.3378\pm0.00025 Å$$

Is this level of resolution achievable with a modern diffraction grating spectrometer and if so why has there been no attempt to repeat Herzberg's measurement (at least not as far as I can make out anyway) ?

Thanks Charles! It's a welcome addition to your knowledge base!

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Thanks - read your article with interest albeit perhaps needing to do some careful study to properly understand all the equations.

In the Insights article I wrote on the Deuterium Lyman Alpha line, the claimed resolution with a "3 metre vacuum grating spectrograph in fifth order" was:

$$L_\alpha(D)=1215.3378\pm0.00025 Å$$

Is this level of resolution achievable with a modern diffraction grating spectrometer and if so why has there been no attempt to repeat Herzberg's measurement (at least not as far as I can make out anyway) ?
That kind of resolution would be very difficult to achieve, and I believe it is well beyond the accuracy of any commercially available instrument. To achieve anywhere near this kind of precision would be quite painstaking. I can see where no one has attempted to repeat it to that level of accuracy. (Edit note: These initial comments may be in error. See post 5, etc.). With a commercially available instrument, one would typically measure something like $\lambda= 1215.3$ angstroms at first order, and you can get additional precision by doing it at fifth order. In general, the accuracy would be limited to the accuracy of the other spectral lines that you use as a standard. $\\$ In the Herzberg measurement, he most likely measured it from first principles=i.e. measuring $\sin{\theta_i}$ and $\sin{\theta_r}$, rather than using other spectral lines. Alternatively, he could have calibrated his spectrometer with another source, such as the iron lines, that are commonly used as a wavelength standard. I don't have access to any iron calibration standard handbook at present, but if I remember correctly, the precision is perhaps $+/- .001$ angstroms. @neilparker62 Perhaps you could try to find some info on the currently available precision of these standards.
When working from first principles, besides measuring $\theta_i$ and $\theta_r$ very accurately, to get tremendous accuracy, you need to know the dimensions of the grating, i.e. the spacing $d$ between the lines of the grating, to very high precision.

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neilparker62
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@neilparker62 Perhaps you could try to find some info on the currently available precision of these standards.
Re-Optimized Energy Levels and Ritz Wavelengths of $^{ 198}$Hg I,
A. Kramida,
J. Res. Natl. Inst. Stand. Technol. 116, 599–619 (2011)
DOI:10.6028/jres.116.008
You can compare and make your own correction for Herzberg’s D measurement. In the H-D-T compilation, I used the values from Saloman. This is the update mentioned in the H-D-T compilation.

See page 610 of above reference for the Mercury standard lines used by Herzberg.

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Thank you @neilparker62 I need to look these over carefully, but a quick comment=on at least one of them, they are using an FTS=Fourier transform spectrometer, which uses a Michelson interferometer based system. In any case, the resolution is quite phenomenal. $\\$ Edit: I also see that another of the measurements uses a Fabry-Perot interferometer. The reader may find this write-up that I previously did of interest: https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/

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sophiecentaur
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they are using an FTS=Fourier transform spectrometer, which uses a Michelson interferometer based system. In any case, the resolution is quite phenomenal.
Frequency measurement and resolution at RF will take you easily to One part in 1011. Pretty damn good eh?

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Frequency measurement and resolution at RF will take you easily to One part in 1011. Pretty damn good eh?
It appears that the FTS Michelson interferometer may have the best (absolute) resolution/precision for determining the wavelength of isolated and very bright and narrow spectral lines, to be used as wavelength calibration standards. Once these "standard" wavelengths are determined, a diffraction grating spectrometer can be used to accurately measure the wavelengths of other sources, including those that have many spectral lines. $\\$ With a diffraction grating spectrometer, when working from first principles, (i.e. working with $d, \theta_i$, and $\theta_r$, and computing $\lambda=\frac{d(\sin{\theta_i}+\sin{\theta_r})}{m}$), the highest (absolute) precision that can be readily achieved may be on the order of +/- .001 angstroms. Using wavelength calibration standards from an FTS, relative precision for the diffraction grating instruments can then be taken to the diffraction limit of resolution, which may be a couple orders of magnitude smaller.

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neilparker62
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In light of the last few posts then, is there any reason in principle why we can't 're-run' Herzberg's measurement with any one - or combination of - the above-mentioned methods ? The impression I have is that this measurement represented some kind of 'pinnacle' in what was achievable by using diffraction grating technology. Thereafter laser-based methods took over leading (for example) to resolution of fine structure lines associated with Hydrogen/Deuterium Balmer alpha. And also to a very accurate measurement of "Ground State Lamb Shift" - the original intent of Herzberg's work. Although (for some reason) this term - ie "Ground State Lamb Shift" - seems to have more or less disappeared from the scientific landscape as of about 1995 - I'm not sure why ?

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neilparker62
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Yes - I was aware of Parthey's measurements. In fact it was only because of comparing these against calculated values (per Dirac energy formula), that I discovered there were discrepancies and hence eventually came to learn firstly about fine structure differences (2p 1/2 - 2p 3/2) , then about the 'original' Lamb shift (2s - 2p 1/2) and so-called 'ground state lamb shift'.

The simplified Dirac energy equation I have used calculates the 1s - 2p 3/2 energy difference which should differ from an accurately measured value mainly on account of QED contributions - ie "ground state Lamb shift". Herzberg sought to obtain a value for this 1s lamb shift by measuring the same transition and subtracting from theory.

In my article on the Deuterium Lyman Alpha line I said I thought it was perhaps a little surprising that if the Hydrogen and Deuterium 1s - 2s transitions could be measured with such precision, why not also the 1s - 2p 3/2 transitions ? Herzberg's 1950s measurement remains the most accurate we have of this particular transition - at least for Deuterium. In his paper he explains why he measured Deuterium and not Hydrogen 1s - 2p 3/2.