Weak equivalence principle check for non-barionic matter using eclipsing spectrometric binaries. No evidence for dark matter.

 

Abstract.

Weak equivalence principle (the bodies are gravitating equally per inertial mass irrespective of the chemical composition) was confirmed for barionic matter with very high accuracy. However, a priory it is not clear, how to check weak equivalence principle for the mixture of barionic and non-barionic matter (light is inside the ordinary matter). For example, how fast would the sphere full of photons fall in the Earth gravity field? The experiment is not possible on Earth. However, such verification is possible for stars using the observational data on binary stars. In this article the analysis of the mass-luminosity was made for similar stars forming binary versus different stars forming binary and the slopes were found the same with accuracy of 6%. That would be the accuracy of confirmation of the equivalence principle for non-barionic matter (actually a mixture of barionic and non-barionic matter with around 0.14% of non-barionic matter ratio). While some violations of weak equivalence principle are still possible (the idea of strong gravitation of slow light) the scale of such violations is clearly well below the level expected for explanation of dark matter.

Introduction.

In order to check the weak equivalence principle for non-barionic matter, it would be necessary to find the object where such form of energy would be present in great amount. The only such object which is relatively easy to find is a star.  Indeed, the star should burn some matter and transform it into the light. The light can not leave star instantly and trapped inside for many thousands of years (possibly millions of years), slowly diffusing toward chromosphere. During such a process the light is absorbed and re-emitted again, and during the short life time the photons are gravitating independently of the surroundings and thus may be considered as the non-barionic matter trapped inside the barionic matter. If the light would gravitate differently, the obtained additional pulse would contribute back to barionic matter at re-absorption, thus making the overall gravitation of the mixture different from pure barionic matter. The total mass loss due to the thermonuclear synthesis in the star is around 1.4% of initial mass and the shortest lifetime for largest known stars is around 10 million years. Therefore, on average around 0.14% of total mass is emanating from the large star per million of years and assuming the light is trapped inside for around 1 million years too, the total energy kept inside the star as photons of all kinds (non-baryonic matter) would be around 0.14% of its barionic mass. 

              The idea is to use the data on binary stars and to compare the mass-luminosity curve for the stars with close masses and the mass-luminosity curve for the stars with as much difference in mass as possible.

There are many binary stars which are visible as double stars with resolved period and axis and ratio of inertial masses (through measurements of the velocities of stars). Many parameters of such stars are published in Internet.

The usual formula applied to the stars from the third Kepler Law:

T²=4π²*a³/[G(m₁+m₂)]         (1)

Here T is the period of rotation of one star around the second one, a is semi-axis, m₁ and m₂ are masses of the stars (assuming gravitational mass is equal to inertial mass) and G is gravitational constant.

However, the light theoretically may have much higher gravitational pull compare to the inertial mass from E=mc² relation (it is assumed that the inertial mass of light being emitted and reabsorbed inside star is still according to E=mc², as it was proved by Einstein himself). The presence of slow light may modify the gravitational pull, making it much stronger for the star which has more trapped light (and other non-baryonic matter). While the exact amount of trapped light is difficult to calculate (not much is known about the light content of the interior of fully ionized plasma), it is obvious that this amount is correlated with luminosity of the star - the higher the luminosity, the higher the amount of trapped light and the higher the additional gravitational pull on the star (the higher the deviation between the gravitational and inertial mass).

In the derivation of the formula (1) the gravitational masses are always comes as a product [1]:

F=G*M₁*M₂/r²

Here M₁ and M₂ are gravitational masses. Assuming the added pull is proportional to luminosity which is proportional to mass (whether gravitational or inertial), it is possible to assume:

F=G*K₁*K₂*m₁*m₂/r²

Here K₁ and K₂ are multiplicity coefficients, the value of K may be especially high to ultra-bright star (because due to very short life time the ultra- bright star should emit more light per second and as a consequence has more light “on hold”, ready to be emitted but so far trapped inside). If weak equivalence principle hold, K=1. It is important that both coefficients for binaries are always a product.

The modified third Kepler Law:

T²=4π²*a³/[G*K₁*K₂*(m₁+m₂)]

Here m₁ and m₂ are inertial masses. When K₁=K₂=1, the third Kepler Law for baryonic matter is obtained.

To determine the masses from the observation of binaries we need: T, a, and ratio of masses m₁/m₂=n. Since the ratio of masses is determined through the Doppler shift of spectra of stars, it is a ratio of inertial masses. We have two equations for masses m₁, m₂

G*K₁*K₂*(m₁+m₂)=4π²*a³/T²

m₁/m₂=n

Then:

m₂=4π²*a³/[G*T²*K₁*K₂*(n+1)]

m₁=4π²*a³*n/[G*T²*K₁*K₂*(n+1)]

Suppose we decided to determine the inertial masses from the visual binaries with two distinct masses m₁>>m₂ taken in different combinations.   How it would influence the mass-luminosity correlation?

It is possible to show that for very strong effect (K is large) the slope of mass-luminosity curve will depend upon the choice of stars in pair (Kepler third law is not valid any more).

 Lets  consider three cases:

1.Binary m₁ and m₁

2.Binary m₂ and m₂

3.Binary m₁ and m₂

In the first case the value of m1 is (because n=1)

m₁=m₁(old)/[K₁*K₁], here m₁(old)=4π²*a³/[G*T²*2]

Here m₁(old) is real inertial mass. K₁ is large and the value of m₁ is shifted strongly toward smaller mass compare to real inertial mass.

In the second case the value of m₂ (n is equal to 1)

m₂=m₂(old)/[K₂*K₂]

If K₂ is smaller (closer to 1)  the mass of smaller star will be actually equal to inertial mass

In the third case the value of m₁ is

m₁=m₁(old)/[K₁*K₂], m₁(old)=m₁(old)=4π²*a³*n/[G*T²*(n+1)]

Since both coefficients K₁ and K₂ are here, one is small and one is big, the shift down compare to the real inertial mass is smaller compare to the case of the big equal masses, but still present.

m₂=m₂(old)/[K₁*K₂]

The smaller mass is becoming too small for this type of star, well below the real inertial mass for smaller star.

This idea may be immediately checked. If the mass-luminosity curve is plotted using first only stars with close masses, it will be compressed  toward y-axis because of K₁*K₁ and K₂*K₂ coefficients along the x-axis (the slope will be larger). If the same curve is plotted using the stars with different masses  the slope will be smaller. In addition since the same stars now would be in pairs with different masses the scattering will be much larger (the same star like Sun in pair with another Sun-like star would give almost the inertial mass, but in pair with blue giant  a much smaller mass, thus creating additional to the experimental error scattering). In [2] this idea was checked for visual binaries from publication, which is 70 years old. The results showed that indeed the slope for the mass-luminosity curve was higher for close masses.

The results were checked with the help of visual binaries using the modern data from Wikipedia. The slope for the close masses was higher again. However, the most prominent effect is expected for the ultra bright stars with masses 30-100 of Sun mass. For them the percentage of trapped light should be tens of thousands times more compare to Sun and smaller stars (because the total amount of light trapped inside is inversely correlated with life time of star and ultra bright stars are very short lived).

In this case the only way to verify the idea it to use data on spectroscopic binaries. According to [1] the sum of masses is determined by the formula:

m₁+m₂=[P/(2*π*G)]*[(V₁+V₂)³/Sin³(i)]

and ratio of masses is determined through the ratio of velocities: m₁/m₂=V₂/V₁

Here P is the period of binary, G is gravitational constant, V₁ and V₂ are semi-amplitudes of velocities (they marked K₁ and K₂ in Wikipedia articles on binaries), Sin(i) is the sin of the angle between the axis of the rotation and Earth-binary direction. For very important subset of spectroscopic binaries called eclipsing spectroscopic binaries both stars are eclipsing each other thus guarantee that the angle i is close to 90 degrees and that allowed determination of masses of such stars using the known astronometric data. I used binaries: 1 Persei, Theta 1 Orioni 3, Prismis 24-1, NGC 3603-A1, CD Crucis for the brightest stars with close masses and WR22, LY Aurigae, AO Cassiopei for the largest stars with different masses. For the smaller masses the stars from the visual binaries were used (except for stars smaller than Sun). The results are below:

With accuracy of 6% the slopes are the same. Intercept on both curves put on zero. Therefore, the expected from the preliminary results higher slope for the close masses is not confirmed for the ultra bright stars (where the effect should be the largest). While the weak equivalence principle still may be violated due to stronger gravitation of slow light (the error is rather large), the effect on rotation of Galaxy is negligible and by no means may be responsible for the explanation of large scale phenomena like dark matter.

 

                 

References.

1.      https://www.astro.caltech.edu/~george/ay20/Ay20-Lec4x.pdf

2.      https://vixra.org/pdf/2005.0250v1.pdf

3.       

Supernova mechanism of transfer of thermonuclear energy into the energy of accelerated rotation in galaxy. Violation of weak equivalence principle for the supernova remnants.

In addition to the mechanisms responsible for transformation of the thermonuclear energy into the energy of the rotation of the stars in galaxy [1-4] another important mechanism would be the supernova explosion (and, to the lesser degree, nova explosions and other types of star explosion). The overall idea is that during such event a lot of energy is transformed into the light, neutrinos and ultra-relativistic matter like cosmic rays. All of them will be moving with speed of light and have twice the cold barionic matter gravitational pull [1,5]. 

Indeed, the total energy release of supernova would be 2*10exp(44) Joules. The remnants of the star will be moving away with the velocity of 20000 km/h (5.6*10exp(3) m/s). Assuming the shed mass is equal to 10 masses of Sun, which would be 2*10exp(31) kg, it means that the energy stuck in relatively cold barionic matter is only 6.3*10exp(38) Joules, and the rest of the released energy would be in light, neutrinos, cosmic rays and other ultra-relativistic matter. The associated inertial mass would be (from 

E=mc² formula) equal to 2*10exp(27) kg. This effective mass will be gravitating twice as strong as cold barionic matter. Age of Milky way is 13.51 billions of years and supernova comes every 30 years, thus making the total number of supernovas equal to 4.5*10exp(8). The total inertial mass of created ultra-relativistic particles (light and neutrinos and cosmic rays combined) would be 9*10exp(35) kg. The total mass of stars in Milky way is 50 billions  Suns [6] (dark matter excluded) and total inertial mass thus estimated is 10exp(41) kg.

The ratio of the double gravitating matter created by supernovas to the total mass of the stars is not high - only 10exp(-6). However, evaluation of this amount may change a lot if other explosions are taken into the consideration. For example nova explosions take place 2500 more frequently compare to the supernova. If the similar release of ultra-relativistic particles takes place this means up to 0.25% of doubly gravitating matter was created during the Milky Way lifetime. 

From different ideas of the presence of the violating weak equivalence principle matter (light inside the stars, ultra-relativistic matter inside the stars, formation of ultra-relativistic matter during violent explosion of the stars) it seems that the accelerated rotation of galaxies (dark matter problem) may be solved rather by accurate accounting for all those small contributions, rather than by postulating of the predominant presence of invisible interacting only gravitationally particles.

References.

1. http://tipikin.blogspot.com/2019/10/stars-are-full-of-trapped-light-may.html

2. http://tipikin.blogspot.com/2019/12/light-matter-attraction-as-driving.html

3. http://tipikin.blogspot.com/2019/09/accelerated-rotation-of-star-because-of.html

4. http://tipikin.blogspot.com/2020/06/gravitation-of-ultra-relativistic.html

5.  D.Fargion "Deflection of Massive Neutrinos by Gravitational Fields" // Lettere al Nouvo Cimento, Vol.31, No 2, 1981

https://www.researchgate.net/publication/

227245454_Deflection_of_massive_neutrinos_by_gravitational_fields

6. https://en.wikipedia.org/wiki/Milky_Way

Gravitation of ultra-relativistic barionic matter. Implications for weak equivalence principle.

Previously it was found that the non-barionic matter inside the stars will be gravitating much stronger per  mc² compare to the barionic matter (2*n² stronger, where n is the effective refraction coefficient for the light in the fully ionized plasma) [1]. For the high-energy gamma-quanta the effective gravitation is only 2 times stronger (this is because of the general relativity prediction  for the deviation of the light beam near the sun being twice the Newton value - the experimental fact confirming general theory of relativity).

However, any barionic matter composed of ordinary particles (electrons, protons, neutrons, neutrinos) which is the main composition of the star (fully ionized plasma) when heated to tens of million degrees (Suns core) will be closer to relativistic compare to cold barionic matter. Ultra-relativistic matter will have the energy-pulse relation E=pc (here E-energy, p=pulse, c- speed of light) exactly like photons and thus it should behave exactly like photons and gravitate like photons (twice the Newton value). 

Indeed, accurate calculations based on general relativity were performed few decades ago and confirmed, that the ultra-relativistic particle  will be deflected by the gravity exactly like photons (very small difference is present, of course, because they do have rest mass) [2].

However, the star is almost entirely composed from very hot matter (fully ionized plasma at temperature of ten of millions degrees). Application of the formula output procedure described in [1] to the ultra-relativistic particle (it is exactly like photons but the refraction coefficient is 1, the particle is moving with ~ speed of light) will lead to the increase of the gravitational force experienced by the star as a whole. 

Exactly how big is this contribution? The rest energy of the proton is 1.5*10exp(-10) J and for electron it is 8.2*10exp(-14) J. Assuming the kinetic energy of the electron or proton is still governed by the Boltzman formula E=3/2*kT, the kinetic energy of the particles inside the star core would be 2.1*10exp(-16) J - still to small to claim the particles are ultra-relativistic.

In this case the gravitational properties of the star barionic matter will be between the Newton limit (cold barionic matter) and Einstein limit (ultra-relativistic barionic matter). Application of the formula from [2] for the intermediate region lead to the following approximation: for the slow particles the gravity increase would be proportional to 1+v²/c²=1+mv²/mc²=1+2E_k/E_o

For the protons this means the change in gravity of only 1.4*10exp(-6). That would be too little to explain the presence of dark matter (most probably another, non-barionic matter responsible, see [1, 3,4], but clearly means that weak equivalence principle is not valid for star considered as a whole. 

References.

1. http://tipikin.blogspot.com/2019/10/stars-are-full-of-trapped-light-may.html

2. D.Fargion "Deflection of Massive Neutrinos by Gravitational Fields" // Lettere al Nouvo Cimento, Vol.31, No 2, 1981

https://www.researchgate.net/publication/

227245454_Deflection_of_massive_neutrinos_by_gravitational_fields

3. http://tipikin.blogspot.com/2019/12/light-matter-attraction-as-driving.html

4. http://tipikin.blogspot.com/2019/09/accelerated-rotation-of-star-because-of.html

5. 

Increase of the yield of the elementary particles created at collider through manipulations with quantum vacuum. A chemical approach to the elementary particle physics.

The modern accelerators reached complexity level enough for more intricate manipulations with generated elementary particles. In modern approach the beams are interacting and the results of the strike is analyzed. Essentially this is close to the way the free radicals were created in the chemical experiments decades ago: powerful laser strike breaks away chemical bonds with generation of assembly of unstable chemical products (ions, free radicals, ion-radicals) which are to be analyzed spectroscopically. 

Researchers quickly realized, that in some cases the concentration of the unstable particles may be greatly increased if they quickly frozen inside some neutral matrix (liquid helium, liquid neon, liquid argon or liquid nitrogen). Such matrix isolation at helium temperature prevents any mobility and thus allows accumulation of the unstable products in high concentrations.

Such approach needs, of course, some manipulations with the molecular beams: not simply strike them with laser pulse, but do in the presence of the cold medium, so that the products created are delivered to the isolation matrix fast enough.

The only "isolation matrix" available in high-energy physics is quark-gluon plasma. So the idea of the future accelerator would be the cross-accelerator: in the reaction chamber the first beam prepares the spot of quark-gluon plasma and the second, more powerful beam will generate the elementary particles inside such a spot in the correct time: just before adronisation (cooling) of the quark-gluon plasma.

Why would new particles be created:

1.The quark-gluon plasma is kind of primordial quantum vacuum - the type of the vacuum existing soon after the big bang. Thus all four forces are closer to each other compare to modern quantum vacuum and the yield of the exotic particles responsible for unification of forces (including gravitational force) is expected to be higher.

2.Despite all those exotic particles are expected to be extremely short-lived, they must have at least some time in modern quantum vacuum. So if they are created just before adronisation of the quark-gluon plasma, they are to be cooled to modern quantum vacuum temperatures before they decay (obviously the decay time in the primordial quantum vacuum also is shorter compare to the modern quantum vacuum because of its temperature).

3.Possibly any interaction with parts of the quark-gluon adronising parts may accelerate cooling thus increasing the life-time (pretty much  like weak complexes between the highly reactive free radical and neutral molecule in the isolating matrix spread the wavefunction thus lowering the energy).

4.Future computers will be powerful enough to see the signs of the new particles in the mess created by the adronized quark-gluon plasma. Undoubtedly the interpretation of the results will be much more difficult.

Such experiments may seems to be far future for now, but in design of the future chambers it may be necessary already now to add the additional input window for the future crossed path of the preparation beam. It is expected that by the time the next generation of the accelerators is ready the concept of complex manipulation with the quantum vacuum (preheating of the quantum vacuum in the place of the particle generation) will be already developed to the level of possible implementation.

The particular particles of the interest from author point of view would be antigravitational Higgs boson [1]. The absence of the symmetry between matter and antimatter is not understood, but presumably the absence of the symmetry between matter and gravitational antimatter will be even worse (the gravitational force was split from the unified force first). So even inside the primordial quantum vacuum the probability of generation of any antigravitational particle (like antigravitational Higgs boson [1]) is really small. Correspondingly the direct observation of such antigravitational particle in the modern quantum vacuum is highly improbable (too long waiting time to see any event). Hopefully the idea of the preheated quantum vacuum  may increase the probability of the creation of such elementary particle and thus improve the chances of its detection in reasonable time.

References.

1.http://tipikin.blogspot.com/2019/12/quantum-vacuum-application-to-gravity.html

Multiple energies photons - may they be seen by two photon absorption spectroscopy?

Two photon absorption spectroscopy is well known non-linear optic phenomena [1]. In this phenomena the virtual level is present which allows to absorb the second photon and thus excite the state with energy equal to two times the energy of the original quantum. The corresponding two-photon excited fluorescence is well known (and now three-photon excited fluorescence is known well). The most important observation connected with multiple quanta absorption is the non-linear dependence on power: this allows easily distinguish it from other phenomena.
The fundamental hypothesis outlined in the previous post [2] concerning the harmonics of de Broglie waves may be also stated for the photon itself. Indeed, the initial hypothesis of Plank concerning the photons [3] was the energy of photons itself is:  E=n*h*ν [3]. From observation of photoelectric effect Einstein deduced the more commonly known rule:  E=h*ν, which eventually lead to the development of the quantum electrodynamics and numerous discoveries. 
However, any mathematical expression is only the approximation to the natural law, and the idea of the photons having energy of only  E=h*ν may be very successful but not finally correct. Indeed, the double energy photon (with  E=2*h*ν) may be so rare that virtually non-observable and thus making the Einstein idea so exceptionally great fit to the natural law that it looks absolute. The double energy photons may easily decay into two ordinary photons or mutate into the photon with double frequency.
Even the probability of the existence of such photons would be governed by the usual Boltzman rule (from [2]):
the population of double energy photons would be exp[-hν/(kT)] less compare to the usual photon. That value for relatively small energy infrared photons (1064 nm wavelength, 1.17 eV ordinary photon energy) would be at room temperature only 2.35*10exp(-20). This means that even such photons exist, they are so rare that virtually non-observable.
However, the two photon excited fluorescence is a convenient way to check theirs presence. Indeed, in addition to the quadratic in power term of such fluorescence (due to virtual levels creation [1]) an extremely small linear in power fluorescence is predicted. This fluorescence is so weak that it is necessary to consider the background created by usual thermal excitation (with some non-zero probability the same excited level may be reached by the usual thermal excitation according to the Boltzman formula P/Po=exp[-E/(kT)].
The trick to subtract background is that the two photon fluorescence is a resonant phenomena. It means that for the deviation of the wavelength from the resonant value it will quickly disappear. Since for the linear phenomena search the laser should not be powerful (to prevent observation of the more common quadratic in power two photon fluorescence) it may be with tuned frequency and thus allowing to observe the linear in power resonant phenomenon. When the frequency of the laser deviates from the frequency necessary for two quanta fluorescence (this frequency may be obtained from the quadratic term of the induced fluorescence) the observed linear term should quickly disappear (and the thermal background stay the same). 
In summary: the observation of the linear term in the two photon fluorescence is predicted due to hypothetical existence of double energy photons (from Planks rule  E=n*h*ν [3]), the phenomenon would be really weak (20 orders of magnitude weaker compare to one-photon fluorescence at least for infrared photons) but resonant in photon frequency.



References.
1. https://en.wikipedia.org/wiki/Two-photon_absorption
2. https://tipikin.blogspot.com/2020/05/quantization-rule-and-harmonics-of.html
3. http://web.phys.ntnu.no/~stovneng/TFY4165_2013/BlackbodyRadiation.pdf

Quantization rule and harmonics of matter waves.

When modern scientist is recalling the quantization rule for photons, usually the famous E=h*ν is recalled. However, in the original Planck's derivation the more general rule of quantization was assumed: E=n*h*ν [1]. A similar rule of quantization was assumed by Nield Bohr concerning the orbital moment. The values n=2,3,4 .. are responsible for the excited states of the quantum system.
In quantum electrodynamics the quantization rule for electromagnetic field is similar to oscillator:
E=n*h*ν +0.5*h*ν 
At the same time the wavelength of the de Broglie wave has only one value: λ=h/p=h/(m*v)  not λ=n*h/p, where n is a number 1,2,3..., p is the pulse of the particle (applicable for any particle), m - rest mass and v - velocity of the particle (applicable only for non-relativistic case). 
Since the matter waves are not really easy to investigate, it may happened that the de Broglie wavelength also follows the most general rule with the presence of the harmonics:
λ=n*h/p,
But they were simply overlooked and careful experiment would be necessary to discover them.
For de Broglie wave it is would not be easy to find presence of such harmonics, because during the interference experiment they would generate the maxima and minima, which coincide with maxima and minima of the main matter wave. If harmonic is present in the miniscule amount it will lead to some hardly observable effect. 
Let's consider for example the case of only one added harmonic. Let the first harmonic has maximum of 1 and decays as exp(-0.1*m), where m is the interference band number, m=0,1,2,3,4... In this case the amplitudes of the interference bands would be: Io=1, exp(-0.1), exp(-0.2), exp(-0.3), exp(-0.4)=1, 0.905, 0.819, 0.741, 0.670...
Second harmonic of de Broglie wave would have the amplitude of 0.01 and decays with distance away from the center according to the same law exp(-0.1*m), here m is the interference band for the second harmonic, which would coincide with a certain band for first harmonic (because wavelength is exactly 2 times larger). The amplitude would be I=0.01, 0.01*exp(-0.1), 0.01*exp(-0.2)… =0.01, 0.00905, 0.00819 ...
The sum of the amplitudes would be:
1.01; exp(-0.1); exp(-0.2)+0.01*exp(-0.1); exp(-0.3); exp(-0.4)+0.01*exp(-0.2); …..=
1.01; 0.905; 0.828; 0.741; 0.679; …..
In order to distinguish the case of the one and multiple harmonics the ratio of the amplitudes of the consecutive bands may be calculated: bamd1/band0; band2/band1, band3/band2.....
For exactly one de Broglie wavelength that would be monotonic function:
0.905, 0.905, 0.905 …. (constant in this example, because the chosen decay function was exponential)
For the sum of harmonics it would be: ratios are:
0.896; 0.915; 0.895; 0.916 …. - non-monotonic function, the superposition of monotonic function and ao*Cos(pi*m), where m is the interference band number.
The third and higher harmonics will add more "waviness" to the smooth function.
How to estimate the amplitude of the harmonics in matter wave? The idea of evaluation is inferred from the reciprocity principle: the particle is both matter and wave [2]. Assuming the matter wave is something real (similar to a photon, but permanently "attached" to the particle), the probability of the excitation of the second energy level would be similar to the idea proposed by Plank [1]:
population of each next level would follow Boltzmann rule [3]:
-log(N_i/N)~E_i/kT
But what is the expected energy of the initial de Broglie wave? Hypothesizing that  the particle is both matter and wave it is possible to speculate about this value.
Since de Broglie wave is "attached" to the particle, the only velocity it may have is equals to the velocity of particle v. From the general rule connecting velocity, wavelength and frequency of the wave it follows:

v=λ*ν or λ=v/ν
here v is the velocity of the particle, λ is the wavelength, ν is the frequency of the wave.
Substituting λ  into the formula for non-relativistic de Broglie wave:
λ=h/(m*v)   and v/ν=h/(m*v)   
which may be transformed as follows:
mv²=hν  or mv²/2=hν/2
For non-relativistic particle the energy of de Broglie wave can not be larger than the full kinetic energy of the particles and quantized as oscillator (quite reasonable idea, because the zero energy of electromagnetic field has the same value). Assuming the next harmonic will have the energy according to the Planks rule (or quantum electrodynamic rule, similar to oscillator), the difference in energy between two levels for de Broglie wave would be double the kinetic energy of the non-relativistic particle (for relativistic particle the quantization rule is simpler and coincides with the quantization rule for photons).
Than the ratio of the amplitude of the second harmonic of de Broglie wave to the initial amplitude would be equal (from Boltzmann rule): 

I/I_o=exp(-2E_k/kT)
where Ek is the kinetic energy of the non-relativistic particle. For example for electron with possible to reach energy of 0.1 eV, observed at the room temperature (300 K, at this temperature de Broglie wave is still well resolved since 0.1 eV > kT), the ratio would be:

I/I_o=exp(-2E_k/kT)=4.4*10exp(-4)
Which is small, but at numerous averaging is possible to reach and to discover.


References.
1. http://web.phys.ntnu.no/~stovneng/TFY4165_2013/BlackbodyRadiation.pdf
2. https://tipikin.blogspot.com/2019/09/the-possible-way-to-search-for-new.html
3. https://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution

Quantization of the gravitational dipole

At the present time gravity is considered by many scientists as the under-investigated  force of nature due to  its weakness. It is interesting to investigate the hypothetical possibility of the associated with the mass of the elementary particle gravitational dipole. That would be analogous to electric dipole for the electric field and it would reflect the non-uniform distribution of the mass inside the elementary particle.
Since all the physical values which have the dimensions of energy*time (J*s) are quantized (the Plank constant), it would be interesting to see, what physical values may be quantized, too. For example, production m*v*r (mass*velocity*radius is quantized and this is orbital moment). Investigation of quantization of this moment lead to all modern quantum mechanics, started by Niels Bohr. However, the same production may be considered as production of m*r (gravitational dipole, similar to electric dipole  q*r - charge times distance) and velocity v of the particle.
Following Niels Bohr steps, it is possible to suppose that such value is quantized too:
(m*r)*v=n*h
here h is Plank's constant.
It may be rewritten as follows:
m*r=n*h/v,
Since h/(m*v) is the de Broglie wavelength λ
m*r=n*m*(h/m*v)=n*m*λ
It means that the mass of the particle is "spread" in the space as de Broglie wavelength. Here n is the number 1,2,3,4...
The most important consequence of the hypothesis - the gravitational dipole moment is not equal to zero! This is a conclusion similar to de Broglie wavelength - it must exist, due to quantum mechanics the particle can not be represented as a point, therefore the dipole moment can not be zero under any circumstances.
Lets estimate the additional gravitational force due to the gravitational dipole moment of the particle. Lets consider the ball with mass M as the second body (the first body has mass m). The gravitational dipole force between the dipole and spherical mass M is:
Fd=m*r*grad(Eg)
here Fd is the force acting onto the dipole m*r, grad(Eg) is the gradient of the gravitational field, what is equal for the spherical mass to:
grad(Eg)=d/dr(M*G/(R*R))=2*M*G/(R*R*R)
Here G is the gravitational constant, M is the mass of the second body, R is the distance between the centers of the attracting masses. Then the gravitational dipole force may be written as:
Fd=n*m*λ*(2GM/R³)=2n*(λ/R)*(GmM/R²)=2n*(λ/R)*Fg
Here Fg is the classical gravitational force between two spherical masses separated by distance R between centers. For the multiple harmonics of the gravitational dipole  force (n>1) and for very slow electron (de Broglie wavelength is high) it may be comparable with gravitational force and measurable relatively easily - the electrons will be split into several beams. The gravitational force is not quantizied and the same for all electrons, but the dipole gravitational forced is different depending upon n.  
Here comes the different problem discussed in another blog - de Broglie wavelength is unique and not quantizied according to Bohr rule. It may happened that there is no "excited" states for gravitational dipole, only the lowest state exist (n=1).
For the ultraslow electron with the temperature of 1 micro-Kelvin (reachable now in some experiments) the velocity of electron would be 6.7 m/s and de Broglie wavelength is 0.1 mm. For the second body with radius of 0.1 m the ratio of dipole gravitational force to gravitational force is 2*0.0001/0.1=0.002.
The accuracy of the direct measurements of gravitational force today (Kavendish experiment) is much higher than 0.2%, thus making such measurements quite possible. 
Additional alleviation may be from the shape of the second mass - the gradient of the gravitational force, similar to the gradient of the electric field, will be much stronger near the sharp edges, so the manipulation with different shapes of the second body with mass M will allow to amplify the gravitational dipole force while keeping the classical gravitational force the same.