Can a Photon be Absorbed Without the Exact Energy Level?

[NaiveTay]

In my education of QM, I've heard countless times how energy is quantized by Planck's Constant, and how radiation is only emitted and absorbed in these discrete steps. Recently I've heard that that's not the full picture of energy, and I was hoping you could draw some clarity for me.

In reading Feynman's lectures, he mentions the quantization formula E = hf, where Planck's Constant, h, is multiplied by the frequency of radiation. He then says that though the constant is a certain step size, since the frequency of light can be anything whatsoever, then the energy of a photon can have any value whatsoever.
This is all good and well, but my question is along the lines of: if radiation from atoms is only ever created and only ever absorbed at very specific energy levels (because of the finite shells for an electron to jump), then what happens to light that doesn't exactly "fit the bill"? If radiation can be created by other means outside of electron jumps, then what are the chances this radiation will be the exact amount required for an electron jump of any of the finite number of elements?
Or, perhaps more encompassing, if a photon's energy is affected by collisions, gravity, and other interactions, then how does even a single photon maintain the precise energy needed to be reabsorbed? Clearly my picture is a little skewed.

My guess would be that when people say that energy can only be absorbed in particular amounts, it's not that there's only certain kinds of photons that can be absorbed atomically, but instead that atoms can only absorb a set amount but they reflect the excess energy as a lower frequency photon(s).
Is this correct? What are the limitations of absorption for photon's that excessively "fit the bill"? Obviously the only criteria for atomic absorption isn't just having the required amount of energy, otherwise all light with sufficient energy would be partially absorbed and the remainder reflected. Certain materials have certain colors, which wouldn't be the case (as far as I can tell) if there were really no limitations for absorption other than a photon having adequate energy. But, equally, it doesn't make sense to me to say that there are exact absorption energies which must be matched exactly by the energy of a photon (so that the photon is completely absorbed, with no excess energy remaining) if the energy of a photon is dynamic and can scale an infinite range.

Summarized: if a photon can have any energy level, and if that energy is continually nudged by interaction, how can a photon ever have the exact energy any electron jump demands? And if the photon's energy doesn't exactly have to match to be absorbed, then what are the limitations that allow for colors to be isolated in their reflections?

 

[ZapperZ]

I don't quite understand the issue here, so I'm going to shoot out a few points:

1. If a photon does not have the right energy to be absorbed by a photon, it doesn't get absorbed. It will simply passes through the atom without any interaction. This is what we see in absorption spectrum measurements... a continuous spectrum, with only several "dark lines" signifying the frequencies that happen to match the allowed absorption.

2. Just because a broadband photon energies/frequencies can be produced, doesn't mean that a particular system must have all of the same broadband absorption. I can produce a wide band of frequencies of photons from an undulator at a synchrotron light source. However, this is a different system than an isolated atom with discrete energy levels. So why would one absorb the same way as the other is transmitting?

There has been at least one other similar thread on this:
https://www.physicsforums.com/threads/energy-levels-of-electrons.866359/

Zz.

 

[Dale]

This is a good question. I will move it to the quantum mechanics section where the best experts will see it easier.

I am not an expert, but my understanding is that even though energy is conserved it is uncertain. The faster a transition happens the more uncertain the energy. So emission and absorption spectra are not pristine spikes, but rather broad peaks where a range of energies will be absorbed or emitted characteristic of the time scale of the interaction.

 

[Buzz Bloom]

Hi Naive:

Another factor that broadens the width of a line is the velocity difference between the emitting particle and the measuring device due to Doppler effects.

Regards,
Buzz

 

[NaiveTay]

This is a good question. I will move it to the quantum mechanics section where the best experts will see it easier.

I am not an expert, but my understanding is that even though energy is conserved it is uncertain. The faster a transition happens the more uncertain the energy. So emission and absorption spectra are not pristine spikes, but rather broad peaks where a range of energies will be absorbed or emitted characteristic of the time scale of the interaction.

Thank you for moving this to the proper section, I'm still very new here so I appreciate the hand.

Ask a quantum mechanical question, get a quantum mechanical answer. It seems my fault in logic was in trying to place a classical picture on a quantum system (as is still so easy for me to do without realizing). If I'm hearing you correctly, the required absorption energy for a certain spectral line doesn't have to be exactly matched by the incoming photon energy in order to cause the electron to jump to a higher orbit. There's an intrinsic... variability, or "uncertainty" in all quantum systems which allows for a range of possibilities.
That must mean, as you mentioned, that the spectral lines aren't so much sharp lines as bell-shaped probability curves, where photons closer to the precise absorption energy have a higher probability of being absorbed, but they don't have to match exactly. I'd be curious to understand the falloff of these spectral lines if this is the case.
Would you say that's on the right track? I find your explanation quite interesting and satisfying, so thank you for taking the time to enlighten me.

 

[mfb]

If I'm hearing you correctly, the required absorption energy for a certain spectral line doesn't have to be exactly matched by the incoming photon energy in order to cause the electron to jump to a higher orbit. There's an intrinsic... variability, or "uncertainty" in all quantum systems which allows for a range of possibilities.

Right.

That must mean, as you mentioned, that the spectral lines aren't so much sharp lines as bell-shaped probability curves, where photons closer to the precise absorption energy have a higher probability of being absorbed, but they don't have to match exactly. I'd be curious to understand the falloff of these spectral lines if this is the case.

Perfectly motionless atoms show a Breit-Wigner distribution (which has various names). In most experimental setups, thermal motion of the particles is more relevant, which leads to a Gaussian distribution. Collisions with other atoms can be relevant as well.

Usually you need lasers and/or very sensitive diffraction gratings to note those effects. Shorter-living excited states lead to larger energy uncertainties and broader lines. This is not limited to spectroscopy, it applies to all short-living states, including particles. If you measure the mass of short-living particles, you won't always get exactly the same result. For particles, this is called "decay width".

 

[stevendaryl]

Putting aside the broadening of the energy levels, it's not completely clear to me whether the rule that an atom can only absorb discrete amounts of energy is a rigorous result, or whether it's just the dominant effect.

If you model the interaction between an atom and the electromagnetic field by a time-dependent potential, considered as a perturbation to the hamiltonian (using non-relativistic quantum mechanics), then the first-order amplitude for transitions from state $|n\rangle$ to state $|m\rangle$ is given by:

$T_{nm} = \frac{-i}{\hbar} e^{-i \omega_m t} \int^t dt' V_{mn}(t') e^{i \omega_{mn} t'}$ (see equation 8.42 in http://web.mst.edu/~parris/QuantumTwo/Class_Notes/TDPT.pdf) where $\omega_m$ is the frequency corresponding to $|m\rangle$ and $\omega_{mn}$ is the difference in frequencies, and $V_{mn}$ is the matrix element of the interaction potential: $\langle m|V(t)|n\rangle$

This transition is time-dependent. If you let the interaction perturbation last forever, then the integration becomes a Fourier transform, and you only get a nonzero result for the Fourier component of $V_{mn}$ with frequency $\omega_{mn}$. However, there are two buts to this:

1. If the perturbation doesn't last forever, then there will be a nonzero amplitude even if the potential doesn't have frequency $\omega_{mn}$
2. This is only the first-order effect, so getting zero here doesn't imply that the full transition amplitude is zero.

So this makes me think that "atoms can only absorb energy at discrete frequencies" is more of a rule of thumb than a rigorous result.

 

[ZapperZ]

Putting aside the broadening of the energy levels, it's not completely clear to me whether the rule that an atom can only absorb discrete amounts of energy is a rigorous result, or whether it's just the dominant effect.

If you model the interaction between an atom and the electromagnetic field by a time-dependent potential, considered as a perturbation to the hamiltonian (using non-relativistic quantum mechanics), then the first-order amplitude for transitions from state $|n\rangle$ to state $|m\rangle$ is given by:

$T_{nm} = \frac{-i}{\hbar} e^{-i \omega_m t} \int^t dt' V_{mn}(t') e^{i \omega_{mn} t'}$ (see equation 8.42 in http://web.mst.edu/~parris/QuantumTwo/Class_Notes/TDPT.pdf) where $\omega_m$ is the frequency corresponding to $|m\rangle$ and $\omega_{mn}$ is the difference in frequencies, and $V_{mn}$ is the matrix element of the interaction potential: $\langle m|V(t)|n\rangle$

This transition is time-dependent. If you let the interaction perturbation last forever, then the integration becomes a Fourier transform, and you only get a nonzero result for the Fourier component of $V_{mn}$ with frequency $\omega_{mn}$. However, there are two buts to this:

1. If the perturbation doesn't last forever, then there will be a nonzero amplitude even if the potential doesn't have frequency $\omega_{mn}$
2. This is only the first-order effect, so getting zero here doesn't imply that the full transition amplitude is zero.

So this makes me think that "atoms can only absorb energy at discrete frequencies" is more of a rule of thumb than a rigorous result.

But you can't just say that and then turn a blind eye on the numerous absorption spectra results! Bad theorists do that! How does what you have described above fit in with all of the experiments and analysis that have been done?

Zz.

 

[stevendaryl]

But you can't just say that and then turn a blind eye on the numerous absorption spectra results! Bad theorists do that!

I was just asking whether it was a rule of thumb, or a rigorous result. I don't understand why you want to be belligerent about it.

There are two different issues: (1) What do the experiments show? (2) How (and if) the experimental results are explained by the theory. Yelling doesn't seem an appropriate. I consider your post to be a violation of the standards of civility for Physics Forums.

 

[stevendaryl]

I was just asking whether it was a rule of thumb, or a rigorous result. I don't understand why you want to be belligerent about it.

There are two different issues: (1) What do the experiments show? (2) How (and if) the experimental results are explained by the theory. Yelling doesn't seem an appropriate. I consider your post to be a violation of the standards of civility for Physics Forums.

I'm going to have to make myself calm down a little, rather than impulsively putting ZapperZ into my "ignore" list. But I am very irritated by the incivility.

 

[ZapperZ]

I was just asking whether it was a rule of thumb, or a rigorous result. I don't understand why you want to be belligerent about it.

There are two different issues: (1) What do the experiments show? (2) How (and if) the experimental results are explained by the theory. Yelling doesn't seem an appropriate. I consider your post to be a violation of the standards of civility for Physics Forums.

You are right. I was being a bit "belligerent" in my response. But I find it rather incredulous that you would present such an analysis, but never once mention anything about all these absorption spectra. I am sure you will agree that, no matter what you come up with, it has to be compared against any and all experiments that have been performed. And in this area, there have been a huge body of experimental evidence. UV-VIS experiments, for example, are so common, I have undergraduates doing this as part of their advanced labs.

Now, considering that absorption spectra in atomic gas show discrete absorption spectrum, and that these are then used the same way as emission spectrum to identify elements, I do not see it as being simply a mere "rule of thumb", unless you also think that emission spectra are also a rule-of-thumb.

Zz.

 

[mfb]

If the perturbation has a finite time, it does not have a well-defined single frequency.

 

[stevendaryl]

You are right. I was being a bit "belligerent" in my response. But I find it rather incredulous that you would present such an analysis, but never once mention anything about all these absorption spectra.

I was not giving a course on the subject. I was asking a question, and giving some background to my question. It's not usually required that a post on a topic contain a survey of everything known about that topic.

 

[NaiveTay]

But you can't just say that and then turn a blind eye on the numerous absorption spectra results! Bad theorists do that! How does what you have described above fit in with all of the experiments and analysis that have been done?

Zz.

Experimental results are always limited by the rudimentary measuring apparatus we have available, so I don't see anything wrong with conjecturing beyond our current limitations to experiment, as long as it's clear it's just a conjecture. Now I understand that if an idea does not fit with direct observational results, it is wrong, but our examination of spectral lines is limited to a certain degree of accuracy, and to say that we have them completely understood is saying more than the experiments have shown. But that might not be what you're saying, so my apologies if experiments have proven that it cannot be just "a rule of thumb".
That's not to say I understand spectral lines or have anywhere near your expertise in this area. I'd be very interested to know how accurately we know the energy range of these spectral lines. However, brushing off stevendaryl as a "bad theorist" seems like a truncation of curious reasoning. We all gain an understanding from discussing things openly and kindly, and being harsh just shuts off the flow of ideas with a wall of egotism, and I find that intolerable.Back to the topic at hand. You all have a deeper knowledge of spectral lines than I do, and it seems I'm missing something because from my viewpoint, you all disagree. Yes, there's a width to a spectral line, so clearly there's a range of permitted energies, but then I read "...atomic gas show discrete absorption spectrum", and I'm uncertain all over again. The meaning of "discrete" seems to be different to us. A discrete quantity to me would be something like pi, an exact numeric value with unlimited precision. Now if the absorption energy must "discretely" be matched by the radiation energy, there must be something else going on because the emission of a "discrete" wavelength of radiation is instantly varied due to many factors, e.g., gravitational influences. So that "discrete" quantity is changed, and because there's an infinite range of these "discrete" quantities on any numeric scale, it would be impossible for any photon to ever have any "discretely" predefined amount of energy. Clearly that's bullocks, so I return to my original question about the range of permitted absorption energies. What is the falloff of spectral lines? In other words, if the bands have a width to their dark portion where radiation within that range is completely absorbed, then is there also a partially darkened area on either side of these bands where the radiation has a probability of being absorbed? That's the question I'd really like answered.

 

[mfb]

"...atomic gas show discrete absorption spectrum"

Within the width of the line, which is often so small that you don't care about it. Just a sloppy use of words. The opposite would be a continuous absorption spectrum where relevant absorption happens over a large frequency/wavelength range.

In other words, if the bands have a width to their dark portion where radiation within that range is completely absorbed, then is there also a partially darkened area on either side of these bands where the radiation has a probability of being absorbed?

There is no frequency with 100% absorption. The strongest absorption is in the center of the line, towards the sides absorption goes down.

 

[vanhees71]

Well, the first-order time-dependent perturbation-theory analysis in the dipole approximation explains a lot but for sure not all phenomena of photon scattering on atoms. You can have as well elastic and inelastic scattering of single photons or two-photon absorption from a classical em. wave (coherent state in the quantum treatment of the em. field) etc. etc. It's a rich field covered in books on atomic physics, e.g., H. Friedrich, Theoretical Atomic Physics, Springer (2006)

For the most simple standard analysis of the photoeffect (absorption of a single "photon" out of a classical em. wave), see my Insights article

https://www.physicsforums.com/insights/sins-physics-didactics/

 

[stevendaryl]

Well, the first-order time-dependent perturbation-theory analysis in the dipole approximation explains a lot but for sure not all phenomena of photon scattering on atoms. You can have as well elastic and inelastic scattering of single photons or two-photon absorption from a classical em. wave (coherent state in the quantum treatment of the em. field) etc. etc. It's a rich field covered in books on atomic physics, e.g., H. Friedrich, Theoretical Atomic Physics, Springer (2006)

For the most simple standard analysis of the photoeffect (absorption of a single "photon" out of a classical em. wave), see my Insights article

https://www.physicsforums.com/insights/sins-physics-didactics/

The original poster's question can be interpreted (I think) as a question about inelastic scattering of photons. Is it possible for an electron in a bound state with energy $E_n$ to interact with a photon of energy $\hbar \omega$, with the result being that the electron transitions to state $E_m$ and a photon with energy $\hbar \omega'$ is produced? (Where $\hbar \omega' + E_m = \hbar \omega + E_n$) That would be sort of like Compton scattering, but involving bound electron states.

 

[Dale]

considering that absorption spectra in atomic gas show discrete absorption spectrum

But they don't show discrete absorption spectra if you look close enough. That is the point of the OPs question and stevendaryl's response. It is important to keep the theory relevant to measurements, but careful measurements also support the question and the reaponse. Stevendaryl is not turning a blind eye, but rather giving a closer scrutiny than you see in undergraduate laboratory exercises.

 

[mfb]

The original poster's question can be interpreted (I think) as a question about inelastic scattering of photons. Is it possible for an electron in a bound state with energy $E_n$ to interact with a photon of energy $\hbar \omega$, with the result being that the electron transitions to state $E_m$ and a photon with energy $\hbar \omega'$ is produced? (Where $\hbar \omega' + E_m = \hbar \omega + E_n$) That would be sort of like Compton scattering, but involving bound electron states.

This is called Raman scattering.

 

[stevendaryl]

But they don't show discrete absorption spectra if you look close enough. That is the point of the OPs question and stevendaryl's response. It is important to keep the theory relevant to measurements, but careful measurements also support the question and the reaponse. Stevendaryl is not turning a blind eye, but rather giving a closer scrutiny than you see in undergraduate laboratory exercises.

Thanks for the words of support.

 

[NaiveTay]

Within the width of the line, which is often so small that you don't care about it.

Thank you, that's more or less what I wanted to know.

Making sure I understand you correctly, within the confines of a spectral line, a photon within that range of energy has a probability of being absorbed, but outside of that spectral line the probability drops completely to zero?
That would suggest to me that there are other energy factors at play (like the electron momentum relative to the radiation?) that can handle a small range of energy difference on either side of the center of a spectral line. So even if a photon has a frequency dead center on a spectral line, it's still not certain it will be absorbed because of those "other factors".
But if that were the case where the spectral line width is just a matter of something else making up for the lack of energy or accepting the excess energy, then it again comes back to an exact quantum of energy being required for any particular collision with an electron. At some point, there must be a range of energies a photon can have so that with the exact same collision, a photon can have a bit more energy and still be absorbed, and the excess would be handled some other way.

Now, if this is allowed from quantum mechanics, then isn't it also a possibility that a photon with an energy outside of the visible spectral line might still be absorbed by the same means of probability as a photon within the range of the line's frequency range?
Does the probability really go to zero at some set distance from the center of a spectral line?

 

[mfb]

Making sure I understand you correctly, within the confines of a spectral line, a photon within that range of energy has a probability of being absorbed, but outside of that spectral line the probability drops completely to zero?

There is no "exactly zero", but the probability gets so small that it is negligible in most cases.

(like the electron momentum relative to the radiation?)

That does not make sense.

But if that were the case where the spectral line width is just a matter of something else making up for the lack of energy or accepting the excess energy

It is not.

 

[ZapperZ]

But they don't show discrete absorption spectra if you look close enough. That is the point of the OPs question and stevendaryl's response. It is important to keep the theory relevant to measurements, but careful measurements also support the question and the reaponse. Stevendaryl is not turning a blind eye, but rather giving a closer scrutiny than you see in undergraduate laboratory exercises.

I don't get it. Are you referring to the "broadening" of the line spectra that have been alluded to? Or are you saying that, in principle, atomic absorption spectra do not show any sign of discrete absorption?

Let's do a specific example. Looks at the absorption spectrum show here:

http://faculty.sdmiramar.edu/fgarces/labmatters/instruments/aa/AAS_Instrument/AASInstruments.htm

Depending on a number of things, the width of each of the peaks can change. But this does not change the principle that each of the peaks signifies an electronic transition from a specific state to a specific state. Now, is this in dispute?

What is the origin of the broadening? As stated on that page, there can be a number of sources, but these are part of the experimental consideration. For example, when you do such an experiment, you just don't do an absorption of one photon with one atom. You do an absorption of a lot of photons with a lot of atoms! So already there is a statistical variance to the result since this is often done at a finite, non-zero temperature. And even if this is done, in principle, at 0K, you will STILL get a broadening because of the finite resolution of the instrument.

But this has nothing to do with the principle involved, i.e. the fact that the spectrum shows features of discrete values of absorption. You can see the difference here when you compare to broadband absorption from neutral density filters! See Fig. 2 from

http://www.lightboxkit.com/Features_Filters.html

where you don't see any transmission over a broad range! Now this, I argue, is a sign that you don't have discrete absorption.

I honestly don't understand what is the issue here. Are we arguing that the physics that has been used to interpret absorption spectra is wrong? That there really is NO discrete absorption? Or are we trying to understand the peak width of these absorption lines and somehow deduce that these are indications that the absorption is not discrete?

Zz.

 

[Dale]

Are you referring to the "broadening" of the line spectra that have been alluded to?

Yes. That is what the OP is asking about.

Depending on a number of things, the width of each of the peaks can change. But this does not change the principle that each of the peaks signifies an electronic transition from a specific state to a specific state. Now, is this in dispute?

No, that is not in dispute. The OP was confused by taking the idea to its extreme. If only an exact match could be absorbed then the probability of absorption should be 0 since the probability of any single real number is 0.

 

[ZapperZ]

Yes. That is what the OP is asking about.

No, that is not in dispute. The OP was confused by taking the idea to its extreme. If only an exact match could be absorbed then the probability of absorption should be 0 since the probability of any single real number is 0.

Then I don't understand your dispute of my statement arguing that these are discrete spectrum of absorption just because there are line broadening in the data. The physics is clear, and the broadening is consistent with the physics AND the experimental method. Without such discrete absorption lines, you will not have such distinct peaks, and will have a more broadband spectrum.

Zz.

 

[Dale]

I don't understand your dispute of my statement arguing that these are discrete spectrum of absorption just because there are line broadening in the data

The point is that the OP was unaware that there is any broadening.

 

[zonde]

But if that were the case where the spectral line width is just a matter of something else making up for the lack of energy or accepting the excess energy, then it again comes back to an exact quantum of energy being required for any particular collision with an electron. At some point, there must be a range of energies a photon can have so that with the exact same collision, a photon can have a bit more energy and still be absorbed, and the excess would be handled some other way.

I found this entry in HyperPhysics about natural line width. As I understand, mismatch between photon energy and electron transition energy will lead to shorter average lifetime of excited state (probably unless excess energy is dumped as say inter-particle vibration).

 

[vanhees71]

The point is that the OP was unaware that there is any broadening.

Of course, any absorption process also implies the corresponding "time reversed" emission process since the electromagnetic interaction is time-reversal invariant. So any excited state of the atom has a finite width as soon as you quantize also the electromagnetic field and thus allow spontaneous emission in the more complete picture beyond the semiclassical one (semiclassical means in this case to treat the electrons in the atom as quanta and the electromagnetic field as classical field, as you start in QM 101). This is the "natural line width". On top, of course, if the atom is not in isolation but in the medium you have in addition collisional broadening. So the atomic states all have a finite width due to the collisions. So strictly speaking the absorption lines are in fact (usually narrow) peaks of finite width. Nothing one must get very excited about. It's just quantum mechanics ;-)).

 

[houlahound]

Wouldn't the uncertainty principal ensure line broadening via energy/time product being finite valued?

 

[DrDu]

Wouldn't the uncertainty principal ensure line broadening via energy/time product being finite valued?

That's exactly the point. The finite lifetime of the excited states explains the line broadening a la Breit Wigner. From a theoreticians point of view, this blurs the distinction between discrete line spectra and continuous absorption. In practice, one will speak of a line spectrum as long as the inverse lifetime is smaller than the spacing between the states.

 

[vanhees71]

To the contrary, the widths of the lines in the line spectrum must be small compared the distance between the states (in energy), which means the states must be long-lived enough. You have $\tau=1/\Gamma$ (in natural units).

 

[DrDu]

Isn't that exactly what I said?

 

[vanhees71]

Sorry, I misread your statement by overreading "inverse". Sorry.

 

[NaiveTay]

There is no "exactly zero", but the probability gets so small that it is negligible in most cases.

Thanks for clarifying, that makes sense. I guess I misread your other post "within the width of the line." Sorry, that's what had me confused.A spectral line has an inherent width explained by the uncertainty principle. Further broadening occurs in our methods of measuring these lines. That's the gist I've gotten from this discussion.
Much thanks to you all for your patience and for making it more enjoyable than scavenging the internet for reliable information. Cheers.

 

[votingmachine]

A spectral line has an inherent width explained by the uncertainty principle. Further broadening occurs in our methods of measuring these lines. That's the gist I've gotten from this discussion.
Much thanks to you all for your patience and for making it more enjoyable than scavenging the internet for reliable information. Cheers.

The transition is an electron from a specific state to another specific state, and that energy difference may be very spectrally tight. But few things are isolated. Anything in the environment around that atom where the transition is occurring may alter the energy of that transition. The easiest to see is the doppler effect in a gas molecules may be moving forward or backward relative to the incident light. But a crystal imperfection may also distort the environment around an atom. Temperature will increase vibrations, or collisions, or just varying inter-atomic distances. Those effects can alter the energy levels of the transition.

I used to measure DNA concentrations in solution. The standard methods use the absorbance at 260 nm. At that wavelength, the purine rings and the pyrimidine rings all have some absorption. But the nucleotides are stacked in a double helix, one on top of the other. And the helix can have strain as it is usually twisted either more or less than the lowest strain geometry. The absorption is highly accurate for determining concentration. But the peak at 260 nm is quite broad. The DNA molecules are structurally complex. And even as a cold solution, there are thermal motions.

I will just guess, but I would assume the tightest spectral lines would be from something very cold, and very simple. Based on that assumption, I googled for "liquid helium absorption spectrum". One link I found has a comment about how sharp the absorption bands they found at 4.2-degrees-K.

http://www.nrcresearchpress.com/doi/abs/10.1139/p58-160?journalCode=cjp#.VzK0z2Zfslk