we know from
waves of light: How do they
arise, how do they spread out, are they reflected, refracted,
etc.?Similarity of X-rays and light rays
Replace in Fig. 612 W1 and W2 by a wooden, light proof, closed container with a photographic plate, place your hand on the spot met by the rays and replace the rays from L for a certain exposure interval by X-rays. After development, the plate will display the skeleton of your hand, photographed through the wooden plate, as a shadow image (projected by straight line rays from L) - as well as details of the skeleton as gradations of the shadow. The soft parts of your hand and possibly foreign bodies in it such as a bullet, a splinter, a needle appear as such gradations (important for medical examinations). In contrast, if you replace W1 and W2 by a plate, which has been covered on its lower face with a fluorescence capable layer (paper with a thin layer of barium platino-cyanide or zinc silicate of calcium sulphide ) and view W1, W2 from below (imagine the figure rotated out of its plane by 90º), you see your hand, lying behind the plate and covered by it - its skeleton with its shade details as shadow image on the otherwise brightly glowing X-ray screen.
X-rays allow to look into opaque bodies and photograph their interiors - these two facts demonstrate a few of their most impressive, physical properties and have led to X-ray techniques, medical as well as physical-technical ones. Information about the nature of these rays arose from the discovery of interference phenomena by Laue in 1912: X-rays are electro-magnetic waves like waves of light, but ten thousand times shorter; this difference is the reason for the varying behaviour as these rays on encountering matter.
Since by their nature, X-rays
are waves of light, we seek with them about the same properties,
which we know from
waves of light: How do they
arise, how do they spread out, are they reflected, refracted,
etc.?
We have described previously their formation. Their wave length lies between 75 and 0.1 Å.E. (visible red is about 0.8·104 Å.E., visible violet about 0.4 ·104 Å.E.). They spread along straight lines, which is proved by the sharpness of the shadows. - They are regularly reflected only at lattice planes of crystals, and this is not a pure surface effect, but a space effect, since they penetrate to a certain depth. On other surfaces, they are reflected diffusely.
They are refracted, but only when they
almost graze the refracting surface, and indeed
they are
refracted away from the perpendicular. The regular refraction of
X-rays can also be displayed experimentally in a prism (Larsson and Siegbahn 1924, Fig. 813). The theoretical value of the refractive index of the
rays is several millionths smaller than 1, whence an incident ray
coming from air or vacuum undergoes refraction, and in certain
cases total reflection. At the front face of the prism in Fig. 812, you see a totally reflected ray. -
Also, the presence of dispersion (even anomalous one),
polarization, refraction, interference have been demonstrated by
experiments, all of which lead to the conclusion that every
phenomenon in the field of waves of light has its counterpart in
the field of X-rays, even though the experimental equipment in
1935 was not able to demonstrate all of them, for example, a Doppler effect of double refraction of X-rays.
Absorption and dispersion of X-rays during their passage through matter
By far the most striking property of X-rays is their ability to penetrate substances, which - even as a thin layer - are opaque to rays of light. The intensity of the radiation is weakened in the process and this weakening depends on, apart from the thickness of the irradiated layer, on the chemical nature of the irradiated substance (atomic number of the elements forming it). The weakening is partly due to dispersion by the substance's atoms (diffusion), partly due to absorption by the atoms which are thereby ionized. You measure the total weakening, which is assessed by the total mass weakening coefficient t + s, where t is the coefficient of absorption, s that of dispersion. Each of them is defined by the equation Jx = J0·e-a x, analogous to the previous equation , where now x is the thickness of the layer and a either t or s. You refer the coefficients to the mass unit, whence you employ it in the form t /r , etc., with r denoting the density. In general, the absorption t /r is for the chemical elements proportional to Z3l3, where Z is the atomic number of the element and l the wave length. In the case of ordinary X-ray radiation, sent through an arbitrary substance, m = t + s decreases as the thickness increases and approaches its limiting value asymptotically. Hence, ordinary X-ray radiation appears to be heterogeneous, but yields through continued filtering approximately homogeneous, residual radiation. The absorption equation Jx = J0·e-a x applies strictly only to homogeneous rays. The filtering is frequently employed for the generation of homogeneous radiation (for medical purposes). Hardness is measured by the half value layer of the radiation in aluminium, that is, in the aluminium layer (of piles of aluminium foil) with a thickness so that the intensity of the incident radiation is halved.
Hence the absorption of X-rays manifests itself in two independent of each other effects: The first involves a weakening of the bundle, the second a material change of the irradiated substance (photoelectric effect). X-rays act by this change of the irradiated substance on the photographic plate, evoke fluorescence, ionize gases and act biologically, that is, on the substance of living tissue. Due to their biological action, they are of great importance for the medical sciences such as in dermatology, gynaecology, diseases of the lymphatic glands, the spleen, the marrow of bones, during tuberculosis and other malignant organic changes. Given in too large a dose, the rays can damage the organism badly and for ever, whence there during work with X-rays the operator's own body must be protected against its action as well as against scattered radiation.
However, the weakening of the
bundle is not only due to absorption, but, as stated above, also
due to the dispersion of the radiation - diffusion. It has been
assumed that diffusion, as it corresponds to the electro-magnetic
theory, takes place without a change of the wave length. However,
this is not the case! Even a strictly mono-chromatic bundle of rays splits during diffusion
into two parts: One diffuses unchanged, the other undergoes a lengthening of the waves (reduction in
frequency). It varies in dependence on the angle between the
incident bundle of rays and the rays deviated by
dispersion independently of the wave length between 0.02 and 0.04 Å.-E. (Compton effect, 1923) This fact is of great importance for the
theory of radiation: It supports the light quanta hypothesis.
The image of a hand and its skeleton shows how much less the rays are weakened by the soft parts, all of which contain only elements with low atomic numbers (H, C, N, O with 1, 6, 7,8); in contrast, the bones contain P and Ca with 15 and 20, or there may even be a metal ring on a finger; the higher is the atomic number of the ring's metal, the stronger it sticks out of its environment - copper more than aluminium, platinum more than copper. A wood splinter in the hand produces a hardly visible shadow, a metal needle in any case a much darker one. The shorter are the waves, the more penetrative they are - the harder - ; the longer they are, the more readily they are absorbed by the irradiated substances - they are softer. The X-ray screen and the photographic plate display these difference in hardness: The harder rays yield a brighter, that is, an image richer in differences in brightness, the softer rays a dark and little differentiated image.
Next to their ability to
penetrate, the most striking properties of X-rays are their
ability to generate fluorescence, to act on the photographic
plate and to ionize irradiated air - these three properties
assist in the recognition of the presence of X-rays. Fluorescent
barion platino-cyanide drew Röntgen's 
attention; photo-electric detection is more sensitive,
because the photographic action increases with time. When the
brevity of the exposure time is important, you can raise the
photographic action by means of a screen, similar to a
fluorescent screen, which is covered with calcium wolframate and
on which you place the photographic plate; it fluoresces
violently violet under the action of the X-rays, adds its action
to the photographic one of the X-rays (intensifying screen) and
thereby reduces substantially the required exposure time. -
Ionization by X-rays, which is equally sensitive as the
photographic method, occurs in a chamber, which most often is
filled with a heavy gas (methyl iodide) (Fig. 820). The gas becomes a conductor, which is
proportionate to the radiation input, and displays its
conductivity by an electrometer, connected to the chamber.
Refraction of X-Rays by crystals (Laue 1912)
The most important counterpart
in the physics of X-Ray waves to that of the waves of light is
interference. Since radiation spreads along straight lines and
does not react to magnets, Röntgen
considered it at first to be wave
radiation. The discovery of the polarization of the waves (Barkla 1905) supported cross oscillations; different
experiments to demonstrate refraction of the rays by a slit had
suggested approximately 10,000 times shorter waves than waves of
light (also greater
hardness corresponding to shorter waves); however, accurate
measurement of the wave length and hereby the possibility of a
spectroscopy of X-Ray radiation is due to Laue 1912: "His brilliant idea was the insight, that the
space structure of crystals is equally fitted to the wave length
of the X-rays as the structure of a Rowland-lattice to
the wave length of ordinary light, that is, that one can obtain
the required diffraction apparatus for X-Rays directly out of the
hands of nature - in the form of its master piece - a regularly
grown crystal." (Sommerfeld)
Even the proof of the wave nature of X-rays was only due to him, because a wave motion becomes only convincing as such by demonstration of interference and diffraction phenomena. According to the diffraction formula l=dsin j, the wave length must be smaller than the lattice constant in order for diffraction to occur. If the X-rays are 10,000 times shorter than the waves of light, they demand 10,000 times tighter lattices than the optical ones; the space lattices of crystals lie in these dimensions. Fig. 813 displays the set-up, by which Laue's colleagues Friedrich and Knipping obtained the first Laue diagram. Fig. 814 displays around the perforation point of the primary ray the intersection of the diffracted secondary X-ray with the photographic plate - Laue-interference spots.
Following an elementary presentation, due to Ernst Wagner, we
explain its formation according to Laue's theory by a
cubic lattice (Fig. 815), and
indeed by the interference of the secondary waves within the plane of the image (principally speaking, this is
sufficient). - Two conditions must be met simultaneously, in order for the waves of the
diffracted rays to support each other: The first concerns the waves, which simultaneously leave the neighbouring atoms of the same lattice plane, the second concerns the waves which emit successively from different,
mutually parallel lattice planes . For example, the first applies
to the waves of the rays HS2 and FS1;
their phase difference HJ is (in the figure drawn with a
difference of two wave lengths): HK=2l=asinj. The second applies to the waves of the
rays HS2 and CS2. We will
find their phase difference in the following manner: While the primary wave from the atom H to C advances
by a, the secondary
wave, which arises at H
(in space), reaches the plane of the image at the circle CBF with
radius a in the direction S2 to the
point B. At this instant, the wave CS3,
which forms at C when CA 
HS2, is with respect to HS2
behind by the required phase difference
AB = HB - HA = a - acosj.
The phase
differences HK (of the waves S1 and S2)
and AB (of the waves S2 and S6)
are only equal to each other for the refraction angle j = 90º; in general, HK > AB,
that is, a simultaneous realization of the two
interference conditions for other values of j is excluded. However, it becomes
possible by inclusion of interferences of higher order. when there come to the larger
phase difference HK of a pair of rays a larger number (m)
of whole wave lengths than (n) of the smaller AB
of the other pair of rays, so that HK = m·l and AB = n·l, where m and n are
integers and m > n. In the figure, m
= 2 and n = 1. It shows :
1. The number m and n define definite directions
j of the secondary rays;
2. to each definite direction
j belongs a definite
wave length, which, in addition, becomes proportional to a.
These rules
characterize the Laue effect as it is experimentally
indicated:
1. There only exist definite directions
of refraction (no spectra as in the case of the optical simple
line lattice!),
2. Its own wave length belongs to each direction
.
Accordingly, the Laue effect assumes a continuous form of primary X-ray spectrum - white X-ray radiation - from which the interfering waves select the mono-chromatic range which creates the effect. (If the retina were to react to X-rays, we would see every group of points of the Laue diagram (Fig. 814) mono-chromatically, however, always two different groups in a different colour.)
Laue effect can be explained by reflection at a group of lattice planes (W.H.Bragg and W.L.Bragg).
The Laue effect can also be interpreted by a process other than refraction (W.H.Bragg and W.L.Bragg): The direction of refraction depends only on the numbers n and m; in Fig. 815 (above), the number m = 2 and n = 1, so that AB = and HK = 2l. However, HK = A. Hence, in the shaded triangle ABC - we will call it the phase triangle - the short sides behave like the phase differences l and 2l. Its angle a = j /2, because the perpendicular from H onto the hypotenuse CB bisects the angle j = CHB and CHL = a (as angles with pairwise, mutually perpendicular legs). We now see how the (half) refraction angle and the number in the phase triangle are interrelated:
tan j/2 = AB/(CA=HK) = l/2l = 1/2 = n/m.
The line HL, which bisects the angle j between the primary and secondary rays, has a characteristic location in the space lattice:
CL/CH = tan j /2 = 1/2 = n/m.
Hence the direction of the angle bisector HL in the space lattice is given by the line, which links H to any atom with co-ordinates n·a and m·a; in Fig. 815, where n = 1, m = 2, it is given by EH, since E, referred to H, has the co-ordinates 1·a and 2·a. Hence only such rows of atoms can bisect the refraction angle and you can consider the refracted rays to be reflected in the rows of atoms (better, in the planes of atoms, which are perpendicular to the plane of the drawing and the traces of which represent those rows of atoms.)
Hence all wave lengths can be reflected in one plane of atoms, although with different intensities. How does this fit in with the above established mono-chromatic nature of the rays, refracted according to Laue? Answer: It is not a single plane which reflects, but a large number of parallel equi-distant planes - a part of space. The reflections follow each other at equal intervals in time and only those waves reinforce each other, the period of which agrees with this. Fig. 816 explains the difference in the paths of the rays P1 and P2 of the same bundle, which are reflected successively at the neighbouring planes E1 and E2 (distance d = lattice constant). Let e be the touching angle - glance angle. The phase difference is ab + bc = 2d·sine, if Aa and Ac are perpendiculars (wave fronts) to the rays. The phase difference must be a complete wave length or in the case of nth order the n-fold value: nl = 2d·sine.
This is the fundamental equation of X-ray spectroscopy (W.L.Bragg). It characterizes reflection as mono-chromatic and as selective: At a given angle of incidence, only one definite colour is reflected; conversely: A given wave length is only reflected at definite angles by the group of planes under consideration - otherwise it is allowed to pass. - Bragg's equation is not strictly fulfilled, Ewald's equation agrees completely with measurements.
Reflection is present also in Fig. 816, which presents Laue's concept. P1 is reflected in the plane HE as the ray HS2, in the following plane MC as the ray CS3. AB was the phase difference of both. The distance CB is twice the distance of the planes 2d. Thus again, the phase triangle yields Bragg's equation ( a = j/2 = e): AB = nl = 2d sin j /2.
You can compute l with the equation nl = 2d sin a, when you have measured a and know d, the lattice
distance of the planes employed in the reflection. The angle is
best measured with the Bragg spectrometer (Fig. 820). The distance of the lattice plane d
is obtained in absolute measure from the crystal's density, the Loschmidt number, the atomic weight of the particles forming the
crystal and the structure of the space
lattice . For the
measurement of small wave lengths, you use rock salt and calcite.
The d of the rock salt crystal for the (in spectroscopy
always employed lattice planes, parallel to the cleavage planes)
is 2.814·10-8cm. The intensities during 1., 2. and 3.
order reflections are in the ratios 100:20:7. Hence you employ
the first order reflection, that is, n = 1. Calcite
with d = 3.029·10-8cm is preferred to rock
salt (crystal defects). Larger wave lengths are measured with
crystals with larger distances between the atoms: For quartz
(prism face): d=4.427·10-8cm, gypsum
(cleavage face): d = 7.578·10-8cm, mica
(cleavage face): d = 10.1·10-8cm. You can
also use optical lattices to form refraction
spectra of X-rays, if
they almost touching during incidence - with metal reflection lattices (Compton
50 line/mm) as with glass lattices (Jean Thibaud
1901-1960, 50 - 200 lines). This is important due to the
measurement of the length of X-ray waves in standard units,
without interim crystal dimensions. Optical lattices yield at touching incidence of the rays
refraction spectra of radiations in the range between long wave
X-rays and ultra-violet. On the same plate, Thibaud has obtained ultra-violet lines and X-ray lines of
medium wave length. For the measurement of extremely long X-ray
waves (lines K of oxygen, carbon, boron with 45.5 and
73.5 Å.-E.), certain organic compounds have been employed, in
which the stratification planes (40 - 90 Å.-E.) take the place
of the lattice plane distances in crystals.
The shortest X-ray wave is that of the K-absorption edge of uranium with 0.1048Å.-E. The continuation of X-rays beyond those of their shortest waves are the g-rays in the radiation of radio-active substances, which have a smaller nature to X-rays (Villard 1900). In 1935, the shortest measured waves were those of thorium with 0.052 Å.-E. (Thibaud 1925).
Yet ten times shorter than the waves of g-rays are those of the penetrating (probably cosmic) radiation - Hess-radiation, called after its discoverer -; they were the shortest waves known in 1935. - This radiation comes from outer space into earth's atmosphere; it arises possibly at certain stars as a side effect of the formation of higher atoms from simpler units (Nernst).
The penetrating power of this radiation is extraordinarily large, so that its existence could be confirmed, for example, in the Bodensee ( a lake at the border ot Germany and Swizerland) at a water depth of 230 m with submerged, self registrating eletroscopes (Regener). However, at this depth, the absorbing action of the water had reduced the intensity of the radiation, compared with that at the surface, to 1 - 2 promil. On the other hand, if you rise up from earth's surface, for example, with a balloon equipped with electroscopes, you detect a strong rise in intensity of this radiation. However, this is not surprising - it differs from the case of visible light - because the atmosphere acts on the cosmic radiation about as strongly by absorption as a mercury layer of 760 mm thickness. During travel in balloons. electroscopes have displayed at a height of 9000 m about 50 times the intensity at the ground (Werner Heinrich Julius Kohlhörster 1887-1946), pilot balloons at a height of 26 km 150 times that intensity (Regener).
The discovery and first investigations of cosmic radiation employed highly sensitive electroscopes. However, such measurements are always very difficult, since the intensity of the radiation is extremely low. In more recent times (1935), one has employed the electron counting tube (Hans Geiger 1882-1945), an instrument which reacts much more strongly to cosmic radiation than the most sensitive electroscope. This counter works on the same principle as the a-ray counter: In an about 20 cm long metal tube with 5 cm diameter, a thin isolated wire is stretched axially. Between the wire and the wall acts high electrical tension, by which even the weakest ionization effect in the tube is amplified so much that you can demonstrate with standard measuring equipment, for example, with the fibre electrometer, even the weakest ionization effect in the tube. At the surface of Earth, such a counting tube is all the time penetrated by cosmic rays, which, like every short wave radiation, releases electrons at the tube's wall. Every individual one of these electrons causes, as a result of the multiplication effect discussed above, jerky movement of the fibre electrometer. Due to the cosmic radiation, a counter of the just stated size generates at Earth's surface about 100 such jerks per minute. At a water depth of 200 m, the number of jerks drops to about 20 per minute, while in deep mine shafts, provided any radiation of radio-active substances has been completely eliminated, there will occur no jerks whatsoever.
Diffraction of electron rays. Matter waves
Next to the
diffraction of X-rays, you have electron waves, more exactly:
Rays comprising electrons. Through their ability of becoming
diffracted (by crystals and metal lattices, and by similar
methods for X-rays) electrons manifest themselves as being linked
to waves. 
Broglie has 
predicted 1924
the existence of these waves: Following an idea of Hamilton (1833), he allotted to every moving mass particle, that is,
also to moving electrons, a field of waves. In this sense, you
speak now of matter
waves, even though one
cannot allot to this wave a definite physical nature.
In order to demonstrate how one can arrive at a double-faced mass wave, we recall the photo-effect: If ultra-violet light (for example, a discharge spark) irradiates a metal surface, electrons escape. The exit velocity (Philip Eduard Anton Lenard 1862-1947) - this is the crucial point! - does not depend on the intensity of the incident light, but only on its frequency, and is the larger, the shorter are the incident waves. As long as the frequency remains unchanged, the photo-effect remains constant, however small is the intensity of the incident light. In other words: We can place the source of radiation as far away from the irradiated metal surface without changing the photo effect (only with a decrease of the light intensity, the number of the per second emitted electrons drops). Hence the energy, acting on the surface, cannot have been transferred to it by a wave, for the further away is the irradiated surface from the source of radiation, the smaller is the energy, which the arriving spherical wave has per cm² and can transfer to it. From where takes an electron, which always moves with the same velocity as its predecessor, its kinetic energy, if the distance from the source of light is so large that the intensity of the light almost vanishes? Einstein's light quantum hypothesis answers this question: The energy, emitted by the source of light, does not only remain in time, but also in space concentrated at certain accumulation points - or, in other words: The energy of light does not spread out perfectly uniformly in all directions in infinitely advancing dilution, but it remains always concentrated in certain definite and colour dependant quanta, which fly away at the velocity of light in all directions. (Again in other words: Light consists of corpuscles and their energy in light of frequency n is nh.) A quantum, which encounters a metal, transfers there its energy to an electron, and this energy is always the same, irrespectively of the distance from the source of light. However, the photo-effect indeed depends on a frequency of oscillation, that is, a wave process, and therefore a wave remains linked to the quantum of light. De Broglie transcended this concept of Einstein. He pursued the idea of Hamilton, already referred to, and alloted to every moving mass particle a wave field, although he could not ascribe to this wave a definite physical nature. The motion of the mass particle is linked to the motion of its accompanying wave; according to de Broglie, it is determined by the equation u·v = c², where u is the group velocity of the wave, v the velocity of the mass particle and c the velocity of light in vacuum.
How large is
the energy of the mass wave? First of all, the mass wave 
is a wave. Hence its energy can be
computed, firstly, as wave energy and, secondly, as mass energy. De Broglie equates the results of the two computations. The energy
of the wave is expressed by hn with n the frequency of oscillation and h
Planck's constant. The energy of the mass m
is, according to Einstein, mc². De Broglie writes hn=mc²,
and, since u = nl,
then l = hu/mc²; however, uv
= c, that is, l=h/mv,
whence there corresponds to a mass m moving with the
velocity v the wave length, which equals Planck's constant h, divided by the momentum of the
mass particle.
The essential characteristic of any wave propagation is diffraction. Hence you will also expect diffraction from this group of waves when the mass particle (in its form as a wave group) passes the edge of an obstacle or through a narrow opening. This diffraction has indeed been discovered with electron rays. These are the diffraction phenomena, which, as has been mentioned above, occur side by side with the diffraction of X-Rays. The following reasoning has led to them, starting from the equation l = h/mv. At a velocity as that of the cathode rays, l is of the order of 10-9, of the electrons sent out by glowing wires 10-7, that is, the wave length of soft X-rays. If the hypothesis of de Broglie is justified, you should be able to generate with electron rays, accelerated by a few 100 Volt, X-rays of a few Å.-E.; indeed, this is what happened. One has been able to diffract electron waves at crystals (Fig. 817) as was done by Laue with X-rays, but also on metal foils, cellulose films; one has even succeeded to do so on optical lattices and been able to thus demonstrate the presence of matter waves. Fundamentally speaking, the same is found with rays of protons, helium atoms, etc., which is a direct proof of the association of a wave with the motion of such particles. De Broglie's hypothesis is the origin of the youngest phase (in 1935) of theoretical physics - quantum mechanics, associated up to 1935 with the names of Bohr, Heisenberg, Pascual Jordan
1902-1980 and Schrödinger. The discovery of the diffraction of electron rays and thereby of matter waves forms a side piece to the discovery of the photo-effect: The photo-effect demonstrates that the wave theory is not sufficient for the interpretation of actions of light, that is, that the wave must be complemented by Newton's corpuscular theory - on the other hand, diffraction of electrons by crystals shows that the motion of a material point (in our view the simplest of all physical processes!) must be conceived as being linked to a space filling radiation process. The electron diffraction figure - partner to the Laue-diagram - can even be displayed visibly in the cases of fast electrons (10,000 Volt) and sufficiently large electron flows on a screen (zinc-silicate with Mn addition).
Realization of interference condition for X-rays
The interference
of X-rays is connected with the condition nl = 2dsina. In order for l, d and a 
to
encounter each other according to this condition, you must create
systematically the appropriate conditions. Laue sends for this purpose through the crystal (a foil of
about 0.5 mm thickness) a thin X-ray bundle with all possible wave lengths. Among the rays of the
bundle, there will also be rays with direction, that is, incident
at such angles a
that they fulfil in the presence of such a multitude of wave
lengths Bragg's equation. The rays in the crystal,
which are accordingly diffracted, meet the photographic plate and
generate the Laue-diagram. - Debye and Scherrer (Fig. 818) send
a mono-chromatic ray bundle S through
ta multitude of random oriented micro-crystals (thin rods KP
of finest crystal powder). Among them are also some, which are
oriented so that any lattice plane forms with the incident l the reflection angle a, required by Bragg's equation. At the same angle a, a certain number of tiny crystals
reflect in the multitude around the incident rays in a definite,
and indeed in all of the same (that is, specified by the same
indices) lattice plane, so that there arises around the incident
ray a circular cone of reflected rays with the opening angle 2a. At the same time, there arise with it
co-axial circular cones with other opening angles, formed out of
those rays, which are reflected by other
lattice planes and which have with respect to the incident
ray a phase difference of another order. A photographic plate PP,
perpendicular to the incident ray, would intersect the coaxial
cones in concentric circles; they intersect the cylindrical
mantle, which the film forms about the rod as axis, in more
complicated curves. Fig. 819 shows a record of quartz powder. - Bragg sent a mono-chromatic bundle to a crystal, which he
rotated slowly about an axis in the reflecting plane and which by
the rotation induced all possible angles a of incidence. The text under Fig. 820 states details of the method. It forms
the basis of X-ray spectroscopy.
The discovery of X-ray wave interference has let to two new fields of research: X-ray spectroscopy (more correctly: Spectrometry or spectrography in the X-ray field) and crystal structure analysis - both are very important for the study of the structure of matter: The first for the structure of the atom, the second for the arrangement of atoms in crystals. The basic equipment of X-ray spectroscopy is sketched in Fig. 820. It is similar to the optical spectrometer procedure, which is aimed at the reflection of rays of light, but the telescope and eye are replaced either by an ionization chamber and electrometer or by a photographic plate.
You employ the reflection of a fine bundle of X-rays on a single, strongly reflecting crystal face. Its rotation allows you to change the angle e between 0º and 90º. Thus, the reflecting bundle presents, step by step, separated in space, the wave lengths range from l = 0 to l = 2d as spectrum. For the execution of the photographic method, you place a film around the partitioned circle or place (for a smaller range of angles) a plate tangentially to the circle.
Fig. 821
presents a spectrogram of tungsten, taken with an ionization
chamber and electrometer (as anti-cathode of a Coolidge X-ray tube). The ordinates of the individual points of
the curve are deflections of the electrometer - some of them very
strong like small high prongs, which repeat
themselves
along the curve rhythmically, but become weaker - they correspond
to the 1., 2., 3. order of the locations, which we will later
encounter during photographic recording as spectral lines; their
intensity drops like 100:20:7 over the first three orders. The
spectrogram also displays X-ray radiation, comprising a
continuous spectrum and a superimposed line spectrum. Moreover,
towards the shorter waves, the continuous spectrum stops abruptly
at a certain wave length, towards the longer ones, its intensity
drops gradually (absorption by the glass wall). Next, we discuss
photographic recording.
What does the plate display after you have irradiated it with the aid of the rotating crystal and it has been developed? It shows a more or less darkly shaded band, which reaches the further into the regime of the shorter wave lengths the higher is the tension at the X-ray tube; on the band, it displays as image of the slit several black straight lines which recall Fraunhofer lines (Fig. 807). The band is the continuous spectrum of the impulse "white" radiation, the lines (they correspond to the prongs of the electrometric spectrogram) are the spectral lines of the characteristic radiation of the anti-cathode material; the term characteristic is employed, because the properties only depend on the radiation (wave length, absorption) of the radiator - more exactly: The element, of which the radiator is formed - and are characteristic for this element. This second radiation - every line on its own is homogeneous and follows strictly the absorption equation - is the main object of X-ray spectroscopy. It is a line spectrum, the lines of which form groups (series), clearly separated from each other - the K-, L-, M-, N- series. During their discovery of the first two series, Barkla and Sadler (1908) employed this notation in the centre of the alphabet and thus made allowance for the discovery of other radiations on both sides; they discovered the M-radiation in 1916.
The K-group consists of the 4 lines: Ka1 Ka2 Kb Kg, of which the first two lie close together and are the strongest: Every following group contains more lines. Records of elements lying close to each other (Fig. 822) demonstrate that each group repeats itself from element to element almost with the same structure, except that the lines of a lighter element, compared with those of a heavier one, are displaced towards shorter wave lengths. The K-series is the hardest (measured by the absorption index in aluminium) and has the shortest waves. Every succeeding one is softer and has longer waves; the hardness increases with the atomic number: With rising atomic number, all series shift towards the side of shorter waves (Fig. 822). At least the K- and L-radiations are known for every element. The examination of the especially short and long waves demands conditions, the technical fulfilment of which is very difficult, of the first, because of the extremely high tensions required in the tube, of the second, because they are absorbed already after a few centimetres in air (need for a vacuum spectrograph). For the excitation of the characteristic radiation of an element by X-rays, the primary radiation must be harder than the characteristic one (whence it is also called fluorescence radiation, as analogue to optical fluorescence, for the excitation of which shorter (harder) waves are required). For the excitation by cathode rays, these must have a corresponding velocity.
Every group demands for its formation its own, definite threshold tension value, but the K-group behaves with respect to its threshold value in a characteristic manner: At bypassing the threshold value, there appear immediately all its lines, in the other groups, every individual line has for its generation its own threshold value. According to Einstein's equation , the threshold value of the K-group corresponds to a frequency, which almost corresponds to that of the shortest K-line (you have to know for X-ray spectral analysis the excitation limit in Volts for each element and every group ).
During absorption, the emission lines of the X-ray spectrum do not arise - that is in a complete contrast to the optical spectra! - there only exists continuous absorption. If you examine absorption with a Bragg spectrometer (ionization method) at different wave lengths, you obtain Fig. 823, in which instead of the values of the coefficient m and the wave length l their logarithms have been plotted. The absorption changes linearly with the wave length; it has a discontinuity at a definite wave length, very close to the K-group, and indeed on the side of the shorter waves. In the area of the L-group, you find three such discontinuities, in the M-group five. These limit frequencies characterize a chemical element just as well as the spectral lines and are equally sharp. If you photograph (in the manner described above) the continuous spectrum coming from an anti-cathode, you obtain, if you introduce an absorbing substance into the path of the rays, in the blackening at the absorption limit of the absorbing substance a discontinuous change. The long wave border of the absorption field is called the absorbing edge. In this sense, you speak of the K-, L-, M-edge.
The X-ray spectra of the
elements and their order number in the periodic system (Moseley's law, 1913).
Moseley's law formulates a fundamental regularity within the X-ray spectra: If you plot in a coordinate system the atomic number of the elements along the abscissa and the square root of the just referred to X-ray frequencies along the ordinate axis, the frequencies of homologous spectral lines yield a straight line or at least a curve, which deviates little from a such a line. Fig. 824 summarizes graphically the content of the law: The square root of the reciprocal wave lengths of a line increases proportionally to the atomic number. The great significance of the law is - with the insight that the atomic number of an element determines its position in the periodic system - not its atomic weight, as had been assumed previously. As a result of this insight, the initial sequence had to be changed at several locations (argon-potassium, cobalt-nickel, tellurium-iodine). Also, for the filling of the gaps in the system, the law has been of great help, because you can also compute exactly the wave lengths of an unknown element from those of the neighbours in the periodic system and identify by the measured wave lengths the newly found one as belonging to that gap (this happened during the discovery of the elements 43 and 75 by Noddack and Tacke 1925). Moreover, it is remarkable that the X-ray spectrum is only determined by elements in a substance and does not depend on its chemical bonds. For example, the X-ray spectrum of brass is composed out of those of zinc and copper.
This effect becomes clear through the concept of light quanta. The photo-effect occurs through absorption, the Compton-effect through absorption and - the main point! - simultaneous dispersion. We know dispersion of visible light from the dust particles dancing in the sun's light: The fact that you can at all see the sun's rays as they fall through a small opening in the wall of a dark room is linked to the in all directions directed dispersion of the light at the fine, floating dust particles (Tyndall-effect)*. In the case of the Compton-effect, one is concerned with the dispersion of Compton waves, that is, of waves which are about 10,000 times shorter than the waves of light. If such an X-ray quantum hn encounters a freely moving electron and is dispersed by it, it continues in a direction, which differs from its previous direction. Experiments have shown that it undergoes in the process a change: Its energy hn decreases, that is, its frequency n drops. Transferred to the field of waves, perceptable by the eye, this would mean: The colour of the light changes as a consequence of the dispersion, it shifts towards the red end of the spectrum, in other words: For example, dust particles illuminated with yellow light would appear to an eye, viewing them from the side, to be red (which in the case of the Tyndall-effect is indeed not the case. The Tyndall-light contains exclusively wave lengths, which are also in the primary radiation, although preferably the short waves.
* A Tyndall effect in a cosmic scale is the sky's blue (Rayleigh): Instead of dist particles, you have the molecules of the air. However, the brightness of the dispersed light drops with the decreasing cross-section of the dispersing particles. instead the complete depth of Earth's atmosphere disperses - whence there occurs the huge intensity of the Tyndall effect. Without the Sun's radiation in the air, the sky would also during bright sunshine appear to be black and only the Sun would appear on a dark background as a bright disk.
We must now ask : What has happened to the energy, which the X-ray quantum hn loses during dispersion? Answer: The electron, initially at rest, has received during dispersion an impulse, so that it moves away at a definite, measurable velocity. It has after dispersion kinetic energy which is equal to the X-ray quantum's loss of energy.
You can view this process, in correspondence with the light quanta concept, as an impact of material particles. Like during the impact of a sphere against a sphere at rest, the latter starts to move, while the impacting sphere is diverted from its direction; this is what happens also here. The X-ray quantum impacts against a freely movable electron at rest. In the process, the X-Ray quantum is diverted and the electron accelerated. In the case of impact of an X-ray quantum against the electron, you maintain agreement with the observations, if you compute as if you are dealing with elastic spheres. You allot to the impacting sphere the same energy, which has the X-ray quantum, and set the mass of the sphere at rest equal to that of the electron. Fig. 826 displays the result of the computation for the case, when the X-ray quantum. as it meets the electron. is deflected by 90º. Such a deflection can, according to the elementary laws of impact, - you should think here of billiard balls - only occur when simultaneously the electron moves away in another direction. The arrows hn and hn ' show the direction of the X-ray quantum before and after impact, the arrow E the electron's trajectory. The computation tells also (after application of the energy- and impulse laws) something definite about how the energy distributes itself after the impact between the quantum and the electron. In order to also illustrate by Fig. 826 the energy conditions, the lengths of the arrows are proportional to the energy values of the particles. Since the total energy is not changed by the impact, the sum of the lengths of the arrows hn ' + E must equal that of hn . Naturally, the impact can also occur so that the quantum is deflected by a smaller or larger angle than is shown in the figure. You can draw for each case the corresponding figure for impacting spheres, from which again the dispersion process of the Compton-effect may be derived.
Theoretically speaking, the same process must also take place during the dispersion of visible light by molecules (predicted 1923 by Adolf Smekal 1895-1959), but the recoil of the molecule and the change of the l of the scattered light, computed from the involved quantities n and the mass of a molecule, turns out to be so small that it cannot be measured. However, indeed, there exists such a radiation, which is coupled with the Tyndall-effect; it was unknown until it was discovered by Raman (1928, simultaneously by Landsberg and Mandelstam in Moscow). This radiation opened up a quite new area for spectroscopic exploration of the structure of molecules.
In fact, if you generate the Tyndall-radiation only with the violet end of the spectrum (F1 in Fig. 825), the scattered light contains, although with very small intensity, wave lengths in the green-yellow part of the spectrum. This has been observed over and over again in a large number of cases of carefully cleaned fluids of very different kinds (fatty acids, benzol, water, etc.). This is by no means a fluorescence phenomenon. The radiation of changed wave length does not come into existence as in the case of fluorescence by the primary radiation or one quantum of it being absorbed by a molecule, this utilizes a part of the intake energy elsewhere and emits again the remainder as quantum of light of smaller energy, that is, at smaller frequency. On the contrary, the incident quantum of light is dispersed largely directly and only a smart portion of it passes on into the dispersing molecule, so that eventually again the wave length of the secondary radiation is shifted in the direction of red. In contrast to the Compton-effect, the energy, withdrawn from the quantum, is not converted into kinetic energy; it is taken by it in a different form. One is here concerned with a stimulation of oscillations in the nuclei of the atoms of the molecule. The differences between the wave numbers of the primary lines and the Raman-lines are really identical to the frequencies of the oscillations of the nuclei of the substances under consideration, as they are know from research in the ultra-red.
An example: During irradiation of the blue Hg-line 4358 Å.-E. into organic fluids with a C-H-bond (in chloroform or dichlor-azethylen as well as in benzol, toluol or chlor-benzol), there appears in scattered light at 5000 Å.-E. a line, while it is absent in the case of CCl4 or C2Cl4; the wave number difference 1/4358·10-8 - 1/5000·10-8 = 2940 cm-1 corresponds to the frequency of an ultra-red line at the wave length 3.4m, which on its part is known in the ultra-red spectrum of all these substances and is there ascribed to the C-H bond. Since, in general, the more complicated organic molecules posses a considerable number of nuclear oscillation frequencies, there correspond to every primary line a large number of Raman-lines.
The Raman-effect is important for the insight into the structure of molecules.
