Dear Pooja,

1. Of course You are talking about the phase velocity.

2. Interaction between electric field and electrons in matter causes significant changes in EM wave propagation. Solving Maxwell equations for plane wave in matter gives the equations for the phase velocity of light in matter. It turns out that it can be greater than c for X-rays.

3. Refractive index can be defined as the ratio of light velocities in vacuum and material, that's why it can be less than 1.

The permittivity (and hence index) asymptotes to 1 as frequency goes to infinity. So I believe your premise is wrong. Perhaps you are thinking of the speed of electrons being potentially greater than c in glass (or water); this gives rise to Cerenkov radiation.

Dear Pooja,

1. Of course You are talking about the phase velocity.

2. Interaction between electric field and electrons in matter causes significant changes in EM wave propagation. Solving Maxwell equations for plane wave in matter gives the equations for the phase velocity of light in matter. It turns out that it can be greater than c for X-rays.

3. Refractive index can be defined as the ratio of light velocities in vacuum and material, that's why it can be less than 1.

I fully agree with Marek, check available textbooks- As far as I remember, in J.D. Jackson's "Classical Electrodynamics" and also in Jens-Als Nielsons / Des McMorrows "Elements of Modern X-ray Physics" there are related treatments! Regards to all, Dirk

These are very good references Dirk. I also recommend James R.W., (1967), The Optical Principles of Diffraction of X-rays, Cornell University Press, where you will find detailed discussions on the x-ray propagation in matter.

As I know the refractive index for X-rays or neutrons is a complex number. n=1-delta-ibeta. Delta and Beta small positive numbers so the refractive index is less than unity. This means that the x-rays or neutrons are totally reflected from the surface and penetrate a small distance into the sample.

Delta and beta (in your notation) go asymptotically to zero with energy. If you have detectors that are parallel to a constant phase front they will light up at the same time. The energy or information speed is still c/n where n goes to one at high energy.

Cherenkov effect is well known. Cherenkov, Tamm and Frank were granted Nobel Prize in Physics for discovery of the effect and it's explanation.

John, the speed of information is not c/n, that is the phase velocity. You'r right it approaches unity asymptotically, but from below 1! (There is thus a zero crossing at some finite energy). The speed of information is given by the group velocity instead, and this indeed is always smaller than c.

Edit: "zero crossing" is nonsense here, of course. Relevant is a crossing from n>1 to n

Good point Kai. I remember a paper in which 12 different t speeds were defined. Still don't know why the original question was posed for x-rays except as an example of f -> infinity. Cheers.

The picture and notions of macroscopic electrodynamics in particular refractive index for continuous media is marginal for x-rays cause for x-rays any atomic/molecular medium is not so "continuous". In the limit which respectable commenters discuss above any medium is seen as nearly point-like scatteres, Drude limit for an effective n ~ 1-1/2*(f_plasma/f)^2 shows n

Cesar makes an important point. Naively, if you just take the real part of the index of refraction, phase velocity is smaller than c in the visible and larger than c for X-rays, which means that group velocity has to be larger than c at some point in the UV. Near core hole resonances, group velocity will sometimes even become negatve. These apparent paradoxes are resolved when you look at how the imaginary part of n changes wavepacket propagation -- information is still nicely constrained to travel slower than c.

The keyword then is anomalous dispersion

My reply is not directly on the velocity of X-rays in glass and in other materials or in empty space. After doing 8 fundamental discoveries and spending 25 years it became possible to provide evidences from solar spectra that X-rays travel faster than EUV in space. I was able to show that the starting time is same for different wavelengths arising from a solar flare, but X-rays arrived fast. In genral short wavelengths arrived fast.

Excerpts of the abstract:

"the author succeeded in explaining how the solar spectral findings provide direct evidences on superluminal velocities of GOES X-ray and 13.5 nm Bharat Radiation emissions, when 33.5 nm EUV emission is considered travelling at velocity of light c. Among X-ray wavelengths, the short wavelength 7.0 nm X-rays traveled faster than 9.4 nm X-rays, while X-rays go at superluminal velocities. Among Bharat radiation wavelengths, short wavelengths showed fast travel, while Bharat Radiation goes at superluminal velocities as compared to 33.5 EUV emission".

I appreciate if possible to read the paper and comment:

M. A. Padmanabha Rao,

Discovery of superluminal velocities of X-rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, IOSR Journal of Applied Physics (IOSR-JAP), .Volume 4, Issue 4 (Sep. - Oct. 2013), PP 08-14, DOI: 10.9790/4861-0440814

http://www.iosrjournals.org/iosr-jap/papers/Vol4-issue4/B0440814.pdf?id=3522

Quite interesting.

Rajeev.

Interesting & informative discussions.

KKMaurya

As late unforgettable Chairman Mao Tsedung said: "Let flourish all flowers..."

As unforgettable Chairman Mao said: "Let flourish all flowers..."

Pooja Agnihotri said velocity of X-rays in glass and in other materials greater than its velocity in empty space. Truly speaking none has experimentally measured and confirmed velocity of X-rays in glass and in other materials greater than its velocity in empty space. I can provide answer only on velocity of X-rays in empty space. Astrophysicists from last 50 years have measured solar spectra that provided experimental evidences on superluminal velocity of X-rays and Bharat radiation in empty space. 

WHY SCIENTISTS ARE UNABLE TO INTERPRET EINSTEIN'S FORMULA E=mc^2 AS X-RAYS AND BHARAT RADIATION GO AT SUPERLUMINAL VELOCITIES?

For understanding the experimental discovery by astrophysicists from University of Colorado on X-rays and Bharat Radiation travelling at superluminal velocities, one has to first understand the three papers published.

Before understanding the following paper, please go through the three papers listed afterwards:

M. A. PADMANABHA RAO’S 9TH PHYSICS DISCOVERY

M. A. Padmanabha Rao,

Discovery of superluminal velocities of X-rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, IOSR Journal of Applied Physics (IOSR-JAP), .Volume 4, Issue 4 (Sep. - Oct. 2013), PP 08-14, DOI: 10.9790/4861-0440814

http://www.iosrjournals.org/iosr-jap/papers/Vol4-issue4/B0440814.pdf?id=3522

FIRST TRY TO UNDERSTAND THE FOLLOWING PAPERS:

M. A. PADMANABHA RAO’S SIX FUNDAMENTAL PHYSICS DISCOVERIES ARE CLAIMED IN THE FOLLOWING PAPER:

M.A.Padmanabha Rao

UV dominant optical emission newly detected from radioisotopes and XRF sources,

Braz. J. Phy., 40, no 1, 38-46,2010.

http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332010000100007

Also view to understand what are these discoveries:

Discovery, Volume 4, Number 10, April 2013

http://www.discovery.org.in/PDF_Files/d_20130402.pdf

LATEST DISCOVERIES IN SOLAR PHYSICS IN 2013

M. A. PADMANABHA RAO’S 7TH PHYSICS DISCOVERY

1. M.A. Padmanabha Rao,

Discovery of Sun’s Bharat Radiation emission causing Extreme Ultraviolet (EUV) and UV dominant optical radiation,

IOSR Journal of Applied Physics (IOSR-JAP), Volume 3, Issue 2 (Mar. – Apr. 2013), PP 56-60, DOI: 10.9790/4861-0325660

http://www.iosrjournals.org/iosr-jap/papers/Vol3-issue2/H0325660.pdf

M. A. PADMANABHA RAO’S 8TH PHYSICS DISCOVERY

2. M.A. Padmanabha Rao,

Discovery of Self-Sustained 235-U Fission Causing Sunlight by Padmanabha Rao Effect,

IOSR Journal of Applied Physics (IOSR-JAP), Volume 4, Issue 2 (Jul. – Aug. 2013), PP 06-24, DOI: 10.9790/4861-0420624

http://www.iosrjournals.org/iosr-jap/papers/Vol4-issue2/B0420624.pdf

I feel the answers to some of these fundamental and basic questions can all be found in technical and scientific literature. You can just imagine how many brilliant scientists and scholars have been working in a particular physics field in the entire globe and the enormous amount of publications and excellent reference books already available. The other day I wished to look for latest methods for determining the velocity of light and x-Ray's. It was amazing to see from a website, college students in US doing this as a routine practical work, using lasers of various colours, using time-of -flight technique and other methods. Similarly for velocity of x-Ray's. The values obtained are all same, i.e., equal to c. There are highly qualified experts available in advanced countries to answer also other related questions.

Another simple example: Take the case of a radionuclide emitting beta and gamma radiation. If we see very old science books, we can see UV Radiation is produced by ionisation and excitation of atoms by (spectrum of) beta Ray's and electrons produced by photo and Compton electrons by gamma or x-Ray's in a medium. Even nitrogen gas glows when exposed to beta, low energy x-radiation. When we use a sensitive PM tube detector, it will measure all these UV and visible light components from source housings, intervening air, the PM tube glass and other PM electrodes, etc. Many years ago, we (I am the first author) published a paper in a reputed journal (Applied radiation and isotopes) wherein I described a wide range diagnostic x-Ray dosimeter using a commercially available small area photovoltaic cell/op amp integral assembly. Actually, this system was meant  to be used as a UV/light detector but I found this device has excellent response for low energy X-rays as well. I have also theoretically derived its x-Ray spectral response in this paper. But, in all humility, I will not claim any credit for this finding because similar researches are being done by many many scientists throughout the world. The very feeble light produced in the above experiments can be efficienly measured by employing the photon counting technique(used in astronomy to measure feeble light emitted by stars). We have another paper (again I am the first author) describing a TLD reader using photon counting technique (published long ago in another reputed journal, Medical Physics) for very low radiation measurements.

So my humble suggestion to all is: Read all relevant literature, books and publications in reputed journals; also internal reports from advanced laboratories, if available. This should be the solid base on which we should further work in any field. This has been my experience after working for 55 years in various scientific fields both in India and abroad.

Dr.Sankaran Ananthanarayanan

Retired Senior Scientist, BARC, Mumbai & Ex-Visiting Professor, UCLA, USA (Age: 79)

Note: My other publications on a variety of physics and instrumentation topics can be had from me.

FURTHER EVIDENCES IN 2015 ON LOW WAVELENGTHS GOING FAST AND REACHING MORE DISTANCE IN SPACE.. 

Pooja Agnihotri,  

In 2013, I have reported that X-rays and Bharat Radiation go at superluminal velocities. In clear words, X-rays travel faster than Bharat Radiation, which in turn goes faster than 335A in EUV Band.  

Ref  M.A.Padmanabha Rao,  Discovery of superluminal velocities of X-­rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, IOSR Journal of Applied Physics (IOSR­JAP), .Volume 4, Issue 4 (Sep. ­ Oct. 2013), PP 08­14, DOI: 10.9790/4861­0440814, 

http://www.iosrjournals.org/iosr¬jap/papers/Vol4¬ issue4/B0440814.pdf?id=3522                 

FURTHER ADVANCES:

In 2015, I have succeeded in interpreting how atmospheric temperatures are caused above Earth and other planets by solar emissions produced as a result of  235 Uranium fission taking place on Sun's core surface. The measured  atmospheric temperatures at different heights above Earth by previous researchers could be possible to interpret with unprecedented detail on the basis that low wavelengths go fast and reach more distance. Please refer Table in my paper published in Dec 2015 mentioned below that reveals the following facts.

1. Infrared radiation goes slow and reach least distance from Sun up to around 500 km height above Earth.

2. Visible light with further low wavelengths go fast and reach from 500 to 110 km height above Earth.

3. UV with further low wavelengths go fast and reach from 110 to 100 km height above Earth.

4. Extreme UV (EUV) with further low wavelengths go fast and reach from 100 to 90 km height above Earth.

5. Bharat Radiation with further low wavelengths go fast and reach from 90 to 84 km height above Earth.

6. Solar beta, gamma and X-ray with further low wavelengths go fast and reach

from 84 to 54 km height above Earth.

Breakthrough research in atmospheric physics & planetary temperatures

1. M.A.Padmanabha Rao, All the Sunlight that Earth Receives is not directly from Sun, International Journal of Innovative Research in Science, Engineering and Technology, Vol. 4, Issue 11, November 2015,  DOI>10.15680/IJIRSET.2015.0411050 http://www.ijirset.com/upload/2015/november/50_6_All_the.pdf

2. M.A.Padmanabha Rao, Discovery of Padmanabha Rao Effect controlling planetary temperatures, International Journal of Innovative Research in Science, Engineering and Technology, Vol. 4, Issue 12, December 2015, , DOI>10.15680/IJIRSET.2015.0412129 http://www.ijirset.com/upload/2015/december/129_33_Discovery.pdf  

 All my publications are available in researchgate.                                                                   

Sankaran Ananthanarayanan, YOU WISH TO DOWNPLAY THE BREAKTHROUGHS PUBLISHED ALREADY IN 2013.  IN THE PROCESS, YOUR IGNORANCE OF THE SUBJECT AND BASIC RADIATION PHYSICS IS EXPOSED.

YOUR FIRST PARA: In 2013, I am the first scientist, who reported evidences on superluminal velocities of solar X-rays and Bharat Radiation from measurements of solar spectra reported by Woods et al from University of Colorado. (Title of my paper: Discovery of superluminal velocities of X-rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, 2013.(For full paper refer researchgate). You said college students in USA feel X-rays go with the speed of light.  What they feel is totally irrelevant in the discussion here. However, my breakthrough research work can help them as useful reference.

YOUR SECOND PARA:  Your arguments on luminescence is totally irrelevant to the question asked.  

YOU ARE POOR IN RADIATION PHYSICS:                                                                 What I have made was three experimental physics discoveries.

Discovery 1: UV dominant optical emission from radiochemicals like 131-I. Discovery 2: UV dominant optical emission from XRF sources present as salts. Discovery 3: UV dominant optical emission from metallic 57Co & Cu XRF source etc.  

Ref: UV dominant optical emission newly detected from radioisotopes and XRF sources, Braz. J. Phy., 40, no 1, 38­46, 2010. http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103­97332010000100007

FIRST AND BEST REVIEW OF THE ABOVE RESEARCH PAPER

Margaret West,*a Andrew T. Ellis,b Philip J. Potts,d Christina Streli,c Christine Vanhoof,e Dariusz Wegrzynekf and Peter Wobrauschekc, Atomic spectrometry update-X-ray fluorescence spectrometry, J. Anal. At. Spectrom., 2011, 26, 1919.

DOI: 10.1039/c1ja90038b http://pubs.rsc.org | doi:10.1039/C1JA90038B

Words of citation: The phenomenon of optical emission predominantly in the UV, which accompanies the emission of X-rays, gamma rays, and beta radiation from radioisotope sources and X-ray tubes was investigated by Rao. It was the first work in which the emission of UV radiation was confirmed experimentally and a possible explanation for the mechanism of the UV emission was given by the author.

Already my explanation of UV dominant optical emission from radioisotopes and XRF sources was cited in the J. Anal. At. Spectrom in 2011. What I have discovered was optical emission and what you are trying to explain how luminescence is produced by interaction of ionizing radiation. What you said is totally irrelevant comment to the question, "We see very old science books, we can see UV Radiation is produced by ionisation and excitation of atoms by (spectrum of) beta Ray's and electrons produced by photo and Compton electrons by gamma or x-Ray's in a medium.You are trying to downplay the work already published by irrelevant arguments, confusing the readers".  

I am ashamed to find a scientist from renowned Bhabha Atomic Research Centre saying that "UV Radiation is produced by ionisation"  Please quote the literature or apologize to researchgate and me for your lose comments here. Utter confusion is what you sad, "excitation of atoms by (spectrum of) beta Ray's and electrons produced by photo and Compton electrons by gamma or x-Ray's".  What do you mean by "beta Ray's and electrons produced by photo and Compton electrons". SUGGESTION: Please note you are creating a very poor image for you and your reputed institution BARC.  

Ms Puja Agnihotri,  You have asked, "Why is the velocity of X-rays in glass and in other materials greater than its velocity in empty space"?  I did not come across any reference on experimental finding on Velocity of  X-rays in empty space, other than my research work. Can you please quote references for X-ray velocities in glass and in vacuum? Then we can discuss further.

The following two references may be read for velocity of x-rays and gamma rays in air:

1. M.R.Cleland and P.S.Jastram. The velocity of gamma rays in air. Phys.Rev. 84, 271 (Published 15 October 1951).

The value obtained in this paper is  (2.983±0.015) x 100000 kilometers/sec (same as c)

2. E.Zolotoyabko and J.P.Quintana. Measurement of the speed of x-rays. J.Synchrotron.Rad. 9 (2002) 60-64.

The value obtained converged to the speed of light in vacuum, i.e., c.

It may be noted that the above are two totally different experiments.

Also according to Maxwell's equations, which form the foundation of classical electrodynamics, 

(c)2  = 1/(µ0𝟄0) where µ0 and 𝟄0 are the permeability and permittivity of free space.

It is to be noted that Maxwell discovered a speed equal to the speed of light from a purely theoretical argument based on experimental determinations of forces between currents in wires and forces between electrostatic charges. 

There are college physics experiments using lasers and time of flight techniques to measure the speed of light. One may also do a simple chocolate/microwave oven layman experiment to find the speed of microwaves. All these prove that the velocity of all electromagnetic radiations is equal to c. See several websites on this topic.

Re: the main question raised by Pooja Agnihotri, there are other relevant discussions elsewhere in this ResearchGate and Quora. e.g., pl see the answers to the question raised by Amol Singh of Raja Ramanna Centre for Advanced Technology.

I hope the above is helpful.

While I am still waiting for references that I asked from Ms Puja Agnihotri, I would like to respond to the answer given by Dr Sankaran Ananthanarayanan. I saw the two references mentioned by him from Internet. The first publication being very old (1951), experimental data on velocity of gamma rays is not available. I have glanced the other paper (2002) dealing with velocity of X-rays. (i) These researchers did their best in laboratory conditions. However, when distance travelled by gamma rays and X-rays is limited to few meters, the accuracy of results always suffers. (ii) Another limitation in their studies is the following. While gamma velocity was measured, they did not measure simultaneously light or any other wavelengths to know both arrived at the same time.  Similarly Velocity of X-rays alone was measured without measuring simultaneously light or any other wavelengths to know both arrived at the same time. 

In the research work that I published in 2013 and 2015, I have provided the first and most plausible interpretation of  solar spectra measured by others. For example, a comparison of  arrival of X-rays with that of Bharat Radiation in the measurements of solar spectra reported by Woods et al unfolded that X-rays arrived fast (Sept 2013).

Ref:   M.A.Padmanabha Rao, Discovery of superluminal velocities of X-¬rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, IOSR Journal of Applied Physics (IOSR¬JAP), .Volume 4, Issue 4 (Sep. ¬ Oct. 2013), PP 08¬14, DOI: 10.9790/4861¬0440814, http://www.iosrjournals.org/iosr¬jap/papers/Vol4¬ issue4/B0440814.pdf?id=3522 

Notably, they all started at the same time from 235-Uranium fission taking place on Sun's core surface (july, 2013). 

Ref:  M.A.Padmanabha Rao, Discovery of Self ¬Sustained 235¬U Fission Causing Sunlight by Padmanabha Rao Effect, IOSR Journal of Applied Physics (IOSR¬JAP), Volume 4, Issue 2 (Jul. – Aug. 2013), PP 06¬24, DOI: 10.9790/4861¬0420624 http://www.iosrjournals.org/iosr-jap/papers/Vol4¬issue2/B0420624.pdf

Early arrival of X-rays was interpreted by me as due to fast travel of X-rays over Bharat Radiation. In nutshell, my research paper reported that low wavelengths go fast (Sept 2013).  In the current study, the distance travelled in space is in several thousands of kilometers from Sun to Satellite, so accuracy in the measurements  of solar spectra is more and statistically reliable. In 2015, I have reported further progress: Low wavelengths go fast and reach more distance.

References are already provided by me in an earlier comment on the same topic. 

http://www.iosrjournals.org/iosr¬jap/papers/Vol3¬issue2/H0325660.pdf

http://www.iosrjournals.org/iosr-jap/papers/Vol4¬issue2/B0420624.pdf

http://www.iosrjournals.org/iosr¬jap/papers/Vol4¬ issue4/B0440814.pdf?id=3522

This is just an explanatory addition to the very correct three reasons given by Dr Marek.The refractive index of x-rays is slightly less than 1 means an x-ray entering a piece of glass from air will be bent away from the normal, unlike a ray of light which will be bent toward the normal. The equation n = c/v(p) [n = refractive index] in this case indicates correctly, that the phase velocity of x-rays in glass and in other materials is greater than its velocity in empty space. It should be stressed here that this does not indicate any superluminal information or energy transfer. More details about phase and group velocity can be had from Wikipedia or Optics text books.

As to why the refractive index is smaller than unity in the high frequency limit, it is correct that this can be looked up in any good textbook covering optical properties or x-ray optics. As an "oldie but goldie" I'd mention F. Wooten's "optical properties of solids" which has helped me a lot quite a few years back. It is not a modern treatment but nicely discusses the refractive index and dielectric function.

The essentials can be summarised as follows: The interaction of e/m radiation with matter can largely be understood as the electric field of an e/m plane wave causing a forced oscillation of the charges. In this context the solid is seen as a "collection" of (damped) harmonic oscillators (the basic idea behind linear response theory). Each and every oscillator does two things when exposed to an e/m plane wave (assumed to be a classical electrodyn. field for simplicity):

a) it oscillates with amplitude and phase relation according to its response function.

b) since it oscillates, there are accelerated charges which emit radiation (locally with dipolar emission characteristics). The total travelling e/m field is the sum of the incoming/exciting radiation and the reemitted radiation. The computational task that was mentioned before consists of doing this superposition correctly. It is doable but took me a while to do it myself for plane waves. The result is - again - a plane wave but, as a result, the wavelength of the radiation (~spatial distance between zero crossings of the total {= superimposed} electrical field) is different. Whether this resulting wavelength is larger or smaller than the "original" one explicitly depends on the phase difference between excitation and re-emission (it also depends on the amplitude, though).

Since there is a frequency dependent amplitude and also a frequency dependent phase lag between exciting and re-emitted radiation, the superposition changes as a function of frequency.

Now there exists an upper energy scale for the dielectric oscillations in a solid: the highest frequency oscillator with relatively important oscillator strength occurs on the scale of the binding energy of the 1s electrons (K shell, excitation to excited bound states). Depending on the element, this varies greatly (H: 13.6 eV; Fe:~7keV; Au: ~81keV; U: ~115keV). When we go significantly above this energy, then essentially ALL oscillators in the solid have a phase shift of pi. Under these circumstances it turns out that the resultant wavelength is larger than the original one and hence the refractive index smaller than unity. As the frequency is further increased, the oscillation amplitudes are greatly reduced and the total e/m wave converges to the original one.

This does not mean that it is necessary to have a photon energy larger than the K shell apsorption edge to obtain n

 Sankaran Ananthanarayanan actually made the nice link between phase velocity and refraction. That this actually works is shown in the documents linked below. Refractive lenses for hard x-rays are - so to speak - anti-lenses when compared to the visible optical range. [If you cannot access the first document, alternatively do an image web search on "refractive x-ray lens"]. This effect is being used to focus hard x-rays from synchrotron radiation sources (and then do e.g. x-ray tomography of foamy materials). Just because the deviation of the refractive index from unity is really small, it takes a long array of such "hollow lenses" to actually get the beam focused.

http://iopscience.iop.org/article/10.1088/0022-3727/38/10A/042

https://staff.aist.go.jp/t-tomie/X-ray-lens.pdf

Ms Puja Agnihotri,  WRONG QUESTION:  You asked a question referring to wrong experimental results or conclusions. You have asked two questions. I am referring to the first question: Why is the velocity of X-rays in glass and in other materials greater than its velocity in empty space? I am very sure the velocity of X-rays in glass and in other materials cannot be greater than its velocity in empty space.  When X-rays pass through any medium, energy loss takes place that would be more than in empty space. It is because medium acts like friction, in simple words.  Suppose when X-rays pass through thin sheet of aluminium or lead they get mostly absorbed, because they lose energy very fast. Instead, travelling at greater velocity means, X-rays penetrate more deep into aluminium or lead than in empty space.  If  any researchers reported this, the inferences drawn are totally wrong.  

MY CONCLUSION: X-rays cannot go faster in glass and in other materials than in empty space. 

M.A. Padmanabha Rao, this has been addressed in a number of responses:

• the phase velocity is given by the refractive index via c0/n (could also be given by \frac{\omega}{k})  [w/k]
• this phase velocity is greater than c0 whenever n
  1 votes 1 thanks

Regarding Ms Puja's second question, 

Wikipedia says, "For X-rays the index of refraction is slightly smaller than unity.[1]. https://en.wikipedia.org/wiki/X-ray_optics. 

Generally refractive is given for light as :Speed of light in air divided by Speed of light in medium. 

Excerpt from Wikipedia:  "lenses for visible light are made of transparent materials that can have a refractive index substantially larger than 1".  

That means speed of light in air is more than the speed of light in medium. That is why, light travels more distance in air than in glass. 

REGARDING X-RAYS

Refractive index for X-rays in glass is said to be less than one.  If that is the case speed of X-rays in air should be less than that in glass.  If speed of X-rays in glass or any other media like aluminium or lead is more than in air, X-rays should penetrate more deeply in glass, aluminium and lead than in air.  

IN REALITY THIS DOES NOT HAPPEN. Therefore, in my view the definition of refractive index applied to light should not be applied to X-rays. 

https://en.wikipedia.org/wiki/X-ray_optics

Dear M.A. Padmanabha Rao, I am sorry to say that here you are mistaken. The commonly used wording "speed of light" is not stringent enough to satisfy the present discussion. Phase and group velocities are, in general, different and have different significance. Nevertheless, both are frequently referred to as the "speed of light".

Attenuation, on the other hand, is yet another topic, although not unrelated due to the Kramers Kronig relations. But there is no direct relation between the strength of attenuation (and, equivalently, the numerical value of the absorption coefficient) and the phase or group velocities. All the optical material properties of homogeneous media can be adequately (though phenomenologically) described by the complex dielectric function (or, equivalently, the complex refractive index). Their real and imaginary parts are interrelated through the Kramers Kronig relations. These are integral relations involving integration over all frequencies.

I very warmly recommend to consult F. Wooten's "optical properties of solids", which is well written for this context (alternatively, any other good text on the subject).

I am certainly benefited by the deep analysis of the subject by Dr Kai Fauth. He has unearthed the wealth of knowledge available in the literature to share with us. Thank you very much, Dr Kai Fauth. I completely agree with you.

Dr Kai Fauth, I glanced over some websites regarding refractive x-ray lenses which shows how difficult it is to focus x-rays. Thanks for the summary you had already given. Also, it is more difficult (I hope you will agree) to reflect x-rays, except at a glancing angle.

Elsewhere, i had raised a hard question a few days back which no one has so far answered. Because of the difficulty of making table top x-ray lasers generating picosecond pulses(however, this seems to be a possibility in the not-too-distant future) similar to visible light lasers, combined with the intricacy of making x-ray reflectors, it is not easy to use this method to determine the velocity of x-rays using time-of-flight technique. However, ultrafast photodiodes to receive the reflected x-rays at the same point of generation of x-ray pulses are now available. Let us see the further progress in this regard. REGARDS