Inverse centimeter as an energy unit

[khansson]

Hopefully this is a quick question. I'm looking at writing some code for spectroscopy calculations. Oftentimes, physicists use cm^-1 and a unit of energy (of an atomic state, for example ). Is it possible to do everything is S.I. (or isq) but have cm also be defined as a energy^-1 type as well as a length type?

[mpusz]

I am afraid that I do not know enough about this domain. I was not aware that it is possible to measure energy with cm^-1. I assume that it is similar to Natural Units where some constants are made 1. This is totally possible in the library but may not be compatible with the SI or ISQ. Everything depends on the implementation details. You can find some ideas in the following example: https://github.com/mpusz/mp-units/blob/master/example/total_energy.cpp.

[mpusz]

OK, I did some reading, and I think I know now what you are asking for. To answer your question directly, it is impossible for the same SI unit to measure both length and energy quantities, as the SI is explicit about what those units describe.

However, the library provides Faster than Lightspeed Constants that allow you to be physically correct and not impact the runtime performance or the precision of the result. For example, you can write:

constexpr auto c = 1 * si::si2019::speed_of_light_in_vacuum;
constexpr auto h = 1 * si::si2019::planck_constant;

auto wavenumber = 1200. * isq::wavenumber[1 / cm];
QuantityOf<isq::energy> auto energy = wavenumber * h * c;

std::cout << "Wavenumber: " << wavenumber << "\n";
std::cout << "Energy: " << energy << "\n";
std::cout << "Energy in eV: " << energy[eV] << "\n";

The above prints:

Wavenumber: 1200 1/cm
Energy: 1200 c h/cm
Energy in eV: 0.148781 eV

Try it here: https://godbolt.org/z/5Wx8918TK.

As you can see, the original value of the energy is probably what you expect. The only difference is that the unit is not 1/cm but c h/cm.

The above is the only solution to make your computations be compatible with other SI-based equations.

However, if you do not care about the SI, you can define your own system of units where you can explicitly state that you are using 1/cm to measure both length and energy (assuming c = 1 and h = 1). You can build it on top of the ISQ as it is done here: https://github.com/mpusz/mp-units/blob/master/src/systems/natural/include/mp-units/systems/natural/natural.h, or you can even provide your own underlying system of quantities.

The downside of the above approach is that such a system will not be compatible with the SI, and in case you need to use both, you can expect some issues.

I hope this answers your question and potentially helps implement the best solution for your needs. Please let me know if I can help you more with it.

[mpusz]

Please note that I've just added the following example: https://github.com/mpusz/mp-units/blob/master/example/spectroscopy_units.cpp. It provides the following output:

| Energy          | Wavenumber      | Frequency       | Temperature     | Wavelength      |
| --------------- | --------------- | --------------- | --------------- | --------------- |
| 1 eV            | 1 eV c⁻¹ h⁻¹    | 1 eV/h          | 1 eV/k          | 1 c h/eV        |
| 1 c h/cm        | 1 1/cm          | 1 c/cm          | 1 c h cm⁻¹ k⁻¹  | 1 cm            |
| 1 THz h         | 1 THz/c         | 1 THz           | 1 THz h/k       | 1 c/THz         |
| 1 K k           | 1 K k c⁻¹ h⁻¹   | 1 K k/h         | 1 K             | 1 c h K⁻¹ k⁻¹   |
| 1 c h/µm        | 1 1/µm          | 1 c/µm          | 1 c h µm⁻¹ k⁻¹  | 1 µm            |
| --------------- | --------------- | --------------- | --------------- | --------------- |
| 1 eV            | 8065.54 1/cm    | 241.799 THz     | 11604.5 K       | 1.23984 µm      |
| 0.000123984 eV  | 1 1/cm          | 0.0299792 THz   | 1.43878 K       | 10000 µm        |
| 0.00413567 eV   | 33.3564 1/cm    | 1 THz           | 47.9924 K       | 299.792 µm      |
| 8.61733e-05 eV  | 0.695035 1/cm   | 0.0208366 THz   | 1 K             | 14387.8 µm      |
| 1.23984 eV      | 10000 1/cm      | 299.792 THz     | 14387.8 K       | 1 µm            |

[khansson]

Cool, this is lovely, thank you. The more I think about it, this problem is less a concern for the internals of our code. The problem may simply be a matter of getting a bunch of datafiles which are given in wavenumbers to convert nicely to eV.

That conversion will need some some sort of function such as

isq::energy ParseString(std::string &InputData); // Some string input like "100 eV" or "3000 1/cm"

but that function needs a runtime if statement to determine if its really looking at a wavenumber or an energy, correct?

Something like (forgive my pseudo code)

isq::energy ParseString(std::string &InputData) // Some string input like "100 eV" or "3000 1/cm"
{
     std::string_view unitString = GetUnitString(InputData); // whatever that function ends up being
     double value = GetValue(InputData); // Unknown function

     bool IsEnergy = IsUnitStringAnEnergyUnit( unitString) ; //Another yet to be determined function

     if (IsEnergy)
     {
          isq::energy ToReturn{value};
          return ToReturn;
     }
     else
     {
         isq::wavenumber ToConvert{value};
         return isq::energy(h * c / ToConvert); //assume h and c are defined somewhere
     }
}

or is there a nice trick in the library that I missed?

[mpusz]

I am not sure if I understand the problem correctly. You said that you measure the energy in 1/cm (which is fine if you assume c=1 and h=1). So the question is if the text input provides energy in either eV or 1/cm, or maybe the input is either energy[eV] or wavenumber[1/cm]? I think that the former has more sense. Typically you have to know with what quantity type you are dealing with on input, and it should be consistent (i.e. energy). In this case, you need a simple function that will return one preferred unit of choice:

quantity<isq::energy[eV]> ParseString(const std::string& InputData)
{
  double value = get_value;
  if(is_ev)
    return value * eV;
  else
    return value * (h * c / cm);
}

The latter case is more tricky. If you do not know what kind of quantity you deal with on input, then you will have to return something like std::variant<quantity<isq::energy[eV]>, quantity<isq::wavenumber[1/cm]>> from the function, which will be much harder to handle in the rest of the code.

[i-ky]

You said that you measure the energy in 1/cm

I think this is jargon. It is not energy, that is measured in 1/cm, it is some other quantity, like energy scaled by Plank's constant and speed of light (in order to make conversion between energy and wavelength easier). So when someone says "energy is x 1/cm" it actually means an "energy a photon would have if its wavelength was 1/x cm".