X-ray Version of Davisson-Germer Experiment

. Application of unitcell deﬁned boundary conditions leads to the question on the formation of standing waves within unitcell. We design and propose X-ray version of Davisson-Germer experiment as the answer. In the proposed experiment, a tunable synchrotron beam re-places the variable de Broglie wavelength of incident electron beam. Among the two series of Davisson-Germer peaks from original experiments available in literature, ﬁrst series demonstrates standing wave description and the second series demonstrates running wave description. Both running waves and standing waves cannot simultaneously exist within the unitcell and the proposed experiment alone can resolve between the two. The experiment can be conducted on a macromolecular crystal also.

tion. Both running waves and standing waves cannot simultaneously exist within the unitcell and the proposed experiment alone can resolve between the two. The experiment can be conducted on a macromolecular crystal also. 1 Introduction.
Currently existing Cartesian description of crystal diffraction, including in text books, has no mention of standing waves within the unitcell. On a stretched string, the fixed end points as boundaries give rise to formation of standing waves. Likewise, the fixed boundary planes of the unitcell presents possibility for the formation of standing waves. To investigate the possibility, we take recourse to the X-ray version of the historic Davisson-Germer(DG henceforth) experiment. DG experiment 1 demonstrated de Broglie hypothesis 2,3,4 and wave-particle duality of electrons 7,8,9 .
The objective of the X-ray version of DG experiment being proposed in this article is to discover whether or not standing waves exist within the unitcell in crystal diffraction.
2 Bragg planes and DG planes.
A series of DG peaks can be measured from any hkl set of Bragg planes. A given hkl set of Bragg planes defines a superset of DG planes for which the interplanar spacings are given by, where nh = n * h, nk = n * k, nl = n * l and n is the order of the DG peak counted from left to right on the √ V − I plot, where I is diffraction intensity. The Bragg angle at which the six DG peaks are measured is given by, We note that the Bragg angle remains invariant for all the six DG peaks while the incident λ varies from peak to peak.
3 The historic DG experiment.
We need a series of six DG peaks to investigate the formation of standing waves within crystal unitcell. We have two such series from the original experiments by Davisson in the √ V − I plot, where V is accelerating voltage and I is intensity, shown 6 in figure 1.
We now investigate whether it is the running wave description or the standing wave description that can explain all the six DG peaks recorded in the experiment.
Running wave description. Following de Broglie relation, kinetic energy T contained in one wavelength of a running wave is given by, in units of eV or joules, where m is mass of electron and h is Planck constant. In a crystal diffraction condition, integer multiplier n to λ can also be viewed as multiplier to inter-planar spacing. Hence, eq. (4) can also be written with n 2 in the numerator as, From eq. (5), kinetic energy T in eV is equal to accelerating potential V in volts given by, We call eq. (6) as running wave kinetic energy relation or simply running wave relation. From Bragg's law given λ = 1.65 Å and eq. (6), the calculated √ V peaks for n = 1, 2, 3, · · · are given by, We now show that formation of standing wave radiation field within unitcell successfully explains all the six Davisson-Gemer peaks and thus demonstrates existence of standing waves within the unitcell in crystal diffraction.
Standing wave description. A standing wave does not transport energy, but stores kinetic energy in its anti-nodes. Because two opposite traveling waves of equal wavelength constitute a standing wave, kinetic energy stored in one wavelength of a standing wave is equal to kinetic energy contained in two wavelengths of the constituent running waves. Thus, with 2λ in place of λ in the denominator in eq. (5), kinetic energy T in one wavelength of a standing wave is given by, in units of eV or joules. From eq. (8), the kinetic energy T in eV is equal to the accelerating potential V in volts given by, V = 37.7 n 2 (λ in Å) 2 ; √ V = 6.14 n λ in Å (9) We call eq. (9) as standing wave kinetic energy relation or simply standing wave relation. From Bragg's law given λ = 1.65 Å and eq. (6), the calculated √ V peaks for n = 2, 3, 4, · · · are given by, The peaks in eq. (10), calculated from standing wave description, are in perfect agreement with all the six DG peaks listed in eq. (3) recorded in the experiment, except a little aberration in calculated √ V only for the last peak. Higher accelerating potentials correspond to higher energy electrons and hence the aberration may be due to onset of Comptom effect 12 .
The excellent agreement between theoretical and experimental values of √ V for all the six historic DG peaks shown in figure 1 demonstrates the formation of standing wave radiation field within Nickel crystal unitcell. Thus, standing wave description alone can explain all the six peaks recorded in DG experiment. Thus, DG experiment becomes direct experimental proof for the formation of standing wave radiation field within unitcell in crystal diffraction.
Incident beam in DG experiment is an electron beam, which is particle radiation beam 1 .
Applicability of Karle-Hauptman triple phase tangent formula based probabilistic direct methods 13 to both X-ray crystal diffraction and neutron crystal diffraction demonstrates that the phenomenon of diffraction in both is the same. Hence, presence of standing waves in one and absence of standing waves in the other is not possible. Thus, formation of standing waves within nickel crystal unitcell in electron diffraction in DG experiment implies formation of standing waves in the unitcell in X-ray crystal diffraction also. The harmonic standing wave anti-nodes within unitcell act as secondary sources and re-radiate incident energy into the diffraction intensities.
4 Prediction of all the six peaks in a DG peak series.
In section 3, we have seen that eq. (6) predicts only first, third and fifth peaks and eq. Only when the parameters are within allowed range, Bragg's law derived λ = 1.65Å agrees with the λ dB = 1.67Å of the first DG peak and the agreement between the two values becomes the proof for wave-particle duality. The as an allowed range for θ hkl−Bragg in the experiment. For all DG peaks measured from {hkl} planes in a given crystal, the Bragg angle sin θ hkl−Bragg remains invariant at a value within allowed range and only accelerating V varies from peak to peak.
Validity of d = 0.91 Å in the proof for wave-particle duality. For a given crystal, Xray diffraction experiment measures Bragg angle for a diffracted beam and Bragg's law gives d = λ/(2 * sin θ Bragg ) as the interplanar spacing of Bragg planes for the diffracted beam. We note that we have no need to know even the unitcell dimensions of the crystal in determining the interplanar spacing.
Among the three parameters in Bragg's law, incident λ is known and cannot be doubted. Therefore, the validity of d from an X-ray diffraction experiment cannot be doubted or disputed.
For nickel crystal, X-ray diffraction experiments established that d = 0.91Å is an interplanar distance for some Bragg planes within the crystal 10 . D&G used d = 0.91Å as an interplanar spacing within nickel crystal and Bragg angle from DG experiment in calculating the λ for the first DG peak. We specifically note that it is the value d = 0.91Å from X-ray diffraction experiments that provided the proof for wave particle duality.
The X-ray diffraction experiment on nickel crystal prior to DG experiment may have been conducted with Cu K α radiation. With Cu K α incident X-rays at λ = 1.5406Å diffraction in- Again, the first measured peak at √ V = 8.0 that corresponds to de Broglie λ = 1.54Å is indicative that D&G were trying to closely follow X-ray diffraction experiment on nickel crystal with X-rays of λ from Cu K α source. The second series of peaks were reported to have been measured from The second series of DG peaks is less reliable due to following reasons.
1). For d 111 = 2.03 Å in nickel crystal 7 , the allowed range of θ hkl−Bragg from eq. (11) is We note that the Bragg angle of 80 • used in measuring the DG peaks is not within the allowed range defined by the operable V in a DG experiment. Only when all the parameters in the experiment are within allowed range, the de Broglie λ for the first measurable DG peak and Bragg's law derived λ become equal in obtaining the proof for wave-particle duality of electron.
Bragg's law derived λ = 4.0Å is outside the allowed range of λ dB in measuring the second series of DG peaks.
2). D&G claimed that the first peak with Bragg's law derived λ = 4.0Å corresponds to √ V = 3.1 which is too low and hence is not measurable. If a DG experiment cannot measure the largest d 111 peak as the first peak, it is like an X-ray diffraction experiment that cannot measure the {111} or similar lower order intensities and can measure only the higher order intensities. Any given crystal has innumerably many sets of hkl planes, in principle, and we can choose any set of planes for all the parameters are within allowed range in the DG experiment. If D&G had simply chosen a d hkl in nickel crystal such that the Bragg's law calculated λ becomes λ = 1.54 corresponding to de Broglie λ of the first experimental √ V = 8.0 peak, the predicted values from eq. (9) would have been correct without uncertainty for √ V = 8 peak between third and second experimental peaks reported by D&G 7 . When a peak is not measurable, it is not a valid DG peak.
Hence, counting √ V = 3.1 as the first DG peak led to the uncertainty to D&G in identifying the first experimentally measured peak for √ V = 8.0 7 either as third peak or as second peak 7 .
3). In the expression for kinetic energy in a DG experiment, it is whether 2λ or λ in the denominator that respectively determines whether it is standing waves or running waves within the unitcell. This feature of the expression for kinetic energy cannot be changed by changing the remaining two parameters in Bragg's law. At an accelerating V defined incident λ dB in the experiment, d hkl and θ hkl−Bragg determine each other for diffraction to occur. D&G claimed 7 that the running wave description itself could explain all the six peaks by merely changing d hkl and θ hkl−Bragg in the experiment. Our analysis contends that such a reasoning is not acceptable. Hence, the second series of DG peaks is less reliable. Only either one of the descriptions is possible for both first and second series. Which means only either one of the two series of DG peaks is acceptable as valid data from the experiment.
Because of the above five reasons, we need to perform the x-ray version of the DG experiment to discover which one of the two peaks is experimentally correct.
6 The X-ray version of DG experiment.
The historic DG experiment with incident electron beam demonstrated wave-particle duality of electrons. Likewise, the objective of X-ray version of DG experiment is to demonstrate the proof for formation of standing waves within crystal unitcell.
D&G reported that the running wave description could not predict all six peaks in the first series 6 because of the experimental errors in measuring the peaks 7 . The X-ray version of DG experiment does not involve vacuum tube and voltage monitoring electronics and thus avoids experimental errors. Hence measurement from X-ray version of DG experiment is extremely precise and reliable. Our question now is not the proof of wave-particle duality. Our question now is whether or not standing waves exist within crystal unitcell. To address our question, we need to know which one of the two series of DG peaks in the manuscript is reliable and acceptable. One way is to repeat the experiment with nickel crystal. But nickel crystal gives only back scattered beams from surface scattering and hence may not be effective for the objective of our experiment.
and in standing wave description, The striking agreement between the digits in eq. (6), eq. (9) and eq. (14), eq. (16) demonstrates that we can reach conclusive on the objective. We note that the X-ray version of DG experiment is devoid of the uncertainties and experimental errors involved in vacuum tube, surface diffraction, LEED experiments. We note that the objective of the experiment is to confirm whether the experiment locates all six peaks or only odd integer count peaks only and therefore we are not concerned about the accuracy in the intensities of the peaks. Hence, we do not need explanation from dynamical theory of crystal diffraction.
The choice of d hkl to measure the DG peaks from is arbitrary and hence we can choose a strong reflection in the more reliable mid-angle region of the diffraction space. Only precaution we must exercise is that the predicted and experimental λ values must be in agreement for the first measurable DG peaks for the measured series of DG peaks to be valid and acceptable. We can conduct the experiment on a protein crystal also.
If eq. (14) predicts all six peaks, we do not have standing waves within unitcell. If eq. (16) predicts all six peaks, we have the proof for the formation of standing waves within unitcell.

Conclusions.
We propose the X-ray version of DG experiment to avoid the experimental errors reported by Davisson and Germer in measuring the first series. A strong Bragg reflection implies stronger Davisson-Germer peaks. We can measure a series of six Davisson-Germer peaks from any set of Bragg planes. Thus, X-ray version of Davisson-Germer experiment with tunable incident beam from a sychrotron resolves the question on the formation of standing waves within the unitcell.
The experiment is very important in that the discovery of the presence of standing waves will have transformational consequences in the description of crystal diffraction. Hence, the experiment is as important to quantum crystallography as the phtoelectric effect is to quantum mechanics.