VOL.5, No. 10 OPTICALEXPERIMIZNTS WITH ELECTRONS. PART 11 OPTICAL EXPERIMENTS WITH ELECTRONS.
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PART I1
L. H. GERMER. BELLTELEPHONE LABORATORIES, NEWYORKCITY
The Refraction of Electron Waves We shall now return t o a further consideration of the regularly reflected beams. Knowing that electrons have a wave-length equal to h/mer, we can plot the m e showing the intensity maxima of the regularly reflected electron beam against wave-length instead of against the square root of the voltage which was used before. Figure 10 shows a replot of Figure 2 after replacing voltage by the equivalent wave-length. The coiirdinates of the curve on the right of Figure 10 are intensity of the regularly reflected beam and reciprocal of wave-length. On this plot selective reflection of x-rays would occur a t equally spaced intervals. (See Figure 3.) The actual wave-lengths a t which these x-ray reflections would occur are indicated by the positions of the vertical arrows. The numbers attached to these arrows are the orders of the various "Bragg reflections." (The surface of the crystal is the same crystallograpliic plane used in the diffraction experiment, shown in the center of Figure 4. For this surface d = 2.03 A,) All the maxima of the electron reflection curve fall to the left of the positions of what appear to be the corresponding x-ray reflections. The displacement &creases toward the higher orders. Now this is the behavior to be expected if the nickel crystal had an index of refraction for electron' ? waves which is diierent from unity. We may think of the vertical arrows in Figure 10 as indicating the positions a t which the maxima of the electron reflection curve would occur if the refractive index were unity, and we can use the displacement of each observed maximum from the position of its corresponding arrow to calculate a value of the refractive index. In this way from the data of Figure 10 we obtain six different values of refractive index, corresponding respectively to six diierent electron wavelengths. I n Figure 11 these six values of refractive index are plotted against the reciprocal of wave-length. This is a dispersion cnrve of our nickel crystal for electron waves. It is of course only tentative, as there are not enough points to establish the curve very accurately and, furthermore, it is only a conjecture that the displacements of Figure 10 arise because of a refractive index different from unity. When data are taken a t an angle of incidence other than 10' a curve showing selectivity of the reflection is obtained which is somewhat similar to the right-hand curve of Figure 10. Data are then available for calculating additional values of refractive index, and Figure 12 shows the dispersion curve with the addition of points obtained a t angles of incidence other than 10'. By taking data a t angles of incidence which are close
together the density of experimental points along this dispersion curve can be made as great as we like. When all of the data of Figure 12 are carefully considered, they indicate
A-1.06 A.
INTENSITY OF REFLECTED BEAM VS. RECl PROCAL OF ELECTRON WAVE-LENGTH
FIGURE10.-The selectivity of electron refledion-a
replot of Figure 2 using wavelengths instead of equivalent voltages.
. strongly that the departures of the intensity maxima of the electron'beams from those of the x-ray beams are properly accounted for by assuming a
I REFRACTIVE INDEX OF NICKEL FOR ELECTRONS
RECIPROCAL OF WAVE-LENGTH
Flcung 11.-Tentative
dispersion curve of the nickel crystal for electron waves (from data taken at 10" incidence).
refractive index giving a dispersion curve of this form. This assumption adequately accounts for the facts, and it seems that no other assumption will account for the fads unless it gives essentially the same law for the
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departures of the electron beam positions from the x-ray beam positions. Furthermore, this dispersion curve does adequately account for the dis-
I,
..
.,
FIGURE 12.-Dispersion curve. Circles indicate data obtained at 10- incidence, and crosses data obtained at angles of incidence other than 10'.
placement of the assemblage of electron ditrraction beams, which was considered previously,' from the corresponding assemblage of x-ray diffraction ., beams.
"I
.)'
REFRACTIVE INDEX V S REUPROCAL OF WAVE-LENGTH
Yl FIGURE 13.-Complete dispersion c
u ~ of e the nickel crystal for electronwaves.
In plotting Figure 12 a number of experimental points were purposely omitted in order not to distract attention from the general course of the THISJouru~r,5, 1041-55 (Sept., 1928).
curve. These missing points are added in Figure 13. At present we are entirely ignorant of the meaning of the surprising loops in this dispersion curve. In our present state of ignorance the break in the continuity of the curve near the wave-length 1.3 A. is certainly suggestive of the optical phenomenon of anomalous dispersion, and these loops are suggestive of double refraction.
. . . . .
I want to return now to the diffraction experiment in which the nickel crystal was bombarded normally by electrons. Now that we know what the refractive index of nickel is for electrons of different wave-lengths, we understand why the assemblage of the electron diffraction beams arising from the aystal was very similar to the theoretical assemblage of x-ray diffraction beams, and why the electron beams were all displaced from the positions of the corresponding x-ray beams. The dispersion curve accounts quantitatively very nicely for these displacements. We are now able to assign Miller indices to each of the electron beams just as Miller indices are assigned to each Laue x-ray diffraction beam arising from a crystal. I shall show some of these electron beams merely as exhibits. Figure 14 shows again the 54-volt diffraction beam. We know now that this is really a (331) reflection according to x-ray terminology. Because the refractive index happens to be 1.13 instead of unity, as it is approximately for x-rays, this reflection occurs a t 50' for a wave-length of 1.67 A. instead of at 44' for a wave-length of 1.49 A,, which are the conditions for the occurrence o$ a (331) reflection of x-rays. The curves either side of the center in Figure 14 give an idea of how selective is the occurrence of this diffraction beam. The disappearance of- the diffraction beam as the wavelength is changed from the (UIJ"..L..~* critical value of 1.67 A. is FIGURE 14.-A replot of Figure 7, designating this nothing like so sudden as the electron diffraction beam by its Miller indices accordina .to the conventional x-ray nomenclature.
disappearance of an x-ray dif.. fraction beam under similar conditions. This must be attributed to the fact that the penetration of electron waves into the metal is very much less than the penetration of x-rays of the same wave-length. In fact we have used the intensities of the beams shown here to calculate the exponential extinction factor of the metal for electrons having a wave-length of 1.67 A. This is a fairly simple calculation in resolving power. It comes out that, of the electrons of this wave-length incident upon a (111) layer of nickel atoms, about 60% go through the layer without accident.
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OPTICALEXPERIMENTS WITH ELECTRONS. PART I1
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Figure 15 shows the occurrence of this same diffraction beam in the second order. The wave-length is 0.778 A. a t which the refractive index is about 45O 1.04, so that in the second order 5 this beam occurs very close to $ m & 44" which would be the angle of occurrence if the index were unity. According t o x-ray a 60° nomenclature this is a (662) refledion. Figure 16 shows again the 65: 75O volt beam which is now properly designated as a (422) reflection. Figure 17 shows this beam in the second order-an 90" (844) reflection. .. Figure 18 shows a (551) reR 15.-The E second order of the electron flection. The curves of this fig. F ~ diffraction beam shown in Figure 14. ure show the selectivity of this
3
REFLECTION
.
beam. All of the diffraction beams show a selectivity of this ' nature. These curves are similar to the curves of Figure 14, and they are typical of the behavior of all the beams (except one, a (442) reflection shown in Figure 21). Figure 19 shows a (622) reflection. This is another second order beam. The first order of this beam is of unusual interest and I shall postpone showing it until a little later (Figure 27). Figure 20 is a (533) reflection. Figure 21 shows development curves of a (442) reflection. This beam is unusual in that it occurs a t a wave-length of 1.19 A., {~~~}REFLE~TIoN which is close to the critical re16.-A replot of Figure 8 with the beam FIGURE classified in the conventional manners. d o n of the dispersion curve.
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JOURNAL OF
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OCTOBER, 1928
The center of this critical region is a t about 1.3 A,, and the curves of this figure show diiraction a t wave-lengths within the critical region. The diffraction beam is extremely weak a t 1.29 A., and much weaker than we would expect a t 1.26 A. This (442) reflection is the only diffraction beam which shows any departure from the gradual approach to a maximum intensity as we come to the right wavelength for the beam. On the short wave-length side-at the right of Figure 21-the gradual weakening of the beam is quite normal. Curves showing this have been omitted. Tentatively. we attribute the weak beam found a t 1.29 A. to diffraction from only the top ..layer of atoms of the crystal. Figure 22 shows the last of these exhibition curves of various electron diffraction beams. It represents a (773) reflection. 17,-The second order of the diffraction FIGURE . . . . . beam shpwn in Figure 16.
It is likely t o occur to any one that we could have used the departure of the assemblage of electron beams from the theoretical assemblage of x-ray beams t o calculate for each electron beam the refractive index, and thus
establish the general course of the dispersion curve without recourse to the reflection experiment. Because of the limited number of diffraction beams we could not, of course, hope to discover the elaborate flourishes of this curve. The attempt to establish even the general trend of the dispersion
curve from the diffraction data was not successful for three different reasons. In the first place the diffraction apparatus was not so accurately built as the reflection apparatus and the 4 positions of electron beams could not he determined with quite the necessary accuracy. All of the diffraction beams which have been shown are quite broad in 6 angle, which was almost entirely due to the poor resolving power . inherent in the geometry of the 7 apparatus. In the second place, in the diffraction apparatus a beam of course changed its position with changing potential. It was thus more difficult to determine accurately the voltage a t which a beam came to
(54REFLECTION
PICURB 20
30'
I
