Generation and Absorption of X-rays PDF

Generation and absorption of X-rays. X-ray crystallography The text under the slides was written by Ágnes Nagyné Szabó 2024 This instructional material was prepared for the biophysics lectures held by...

Generation and absorption of X-rays. X-ray crystallography The text under the slides was written by Ágnes Nagyné Szabó 2024 This instructional material was prepared for the biophysics lectures held by the Department of Biophysics and Cell Biology Faculty of Medicine University of Debrecen Hungary https://biophys.med.unideb.hu 2 3 During the lecture, we will get insight into the Bohr's atomic model and the discrete energy levels of bound electrons, which will be necessary to understand the mechanisms of the generation and interaction of characteristic and braking X-rays with matter. In medical diagnostics, braking X-rays are used, the energy of which is typically between 20-200keV. We will learn the attenuation coefficient of materials for X-ray and its role in describing the decrease of X-ray intensity as it passes through a particular material. X-rays interacting with materials can be absorbed by three mechanisms:  photoeffect  Compton-effect,  pair production. Finally, the phenomenon of X-ray diffraction will be discussed in which X-rays are scattered by or reflected from crystals and from the interference pattern thus formed, the structure of the crystal can be determined (this technique is X-ray crystallography). We will get insight into the interpretations of this phenomenon according to Laue and Bragg, and the biological applications of X-ray crystallography. 4 On page 123 of the Medical Biophysics Textbook, Figure II.27. displays a similar electromagnetic spectrum showing the ranges of electromagnetic radiation as a function of frequency (f) and wavelength (λ). Radiations can be grouped in several ways, and different radiations have common properties. One of the most important among such principles is that energy propagates in every kind of radiation. Radiations usually start from some source and then reach the irradiated body after covering a certain distance in which they can be partially or completely absorbed, scattered, or even reflected. From a medical point of view, the electromagnetic spectrum can be divided into two major parts: - non-ionizing radiations (radiowave, microwave, infrared radiation, visible light, low energy ultraviolet radiation with a wavelength longer than 50-100 nm) - ionizing radiations (high-energy ultraviolet radiation with a wavelength of less than 50- 100 nm, X-rays, gamma radiation) Electromagnetic waves do not require an elastic medium for propagation, they also propagate in vacuum. Their propagation speed (c) can be expressed by the formula c    f. Electromagnetic radiations have dual nature, i.e. their wave nature must be used to interpret some phenomena whereas their particle nature is required to explain others. The "particles" of electromagnetic radiation are photons, their energy (E) can be described by the following c formula: E photon  h  f photon  h  , where h is Planck’s constant (h=6.63×10-34 Js) (see  photon 5 Lecture 1: Electromagnetic waves, the dual nature of light for more details). It can be seen from the formula that photons with higher frequencies, i.e. lower wavelengths, have higher energies. The energies of X-ray photons are greater than the energies of ultraviolet photons. The energies of X-ray and gamma ray overlap significantly. The two radiations are separated according to the way they are generated: gamma radiation is generated during radioactive processes or particle collisions, while the generation of X-rays is always related to electron shells or deceleration of electrons. 6 In 1913, with the help of Bohr's atomic model, it was also possible to establish the discrete spectrum of the emission of the hydrogen atom theoretically, for which the Danish physicist Niels Bohr was awarded the Nobel Prize in 1922. According to Bohr's assumption, electrons in an atom can only occupy certain energy orbits (E1, E2,…) around the positively charged nucleus, and he called these states stationary orbits. The velocity and the energy of an electron in a given stationary orbit are constant over time. Electron transitions between stationary orbits of different energies correspond to the absorption or emission of a photon of discrete energy. When an E  h  f energy photon is absorbed, the atom acquires extra energy, it becomes excited, which means that the electron moves from a lower energy state (E1) to a higher energy state (E2). The energy of the absorbed photon covers the energy difference between the two states: E  E photon  h  f  E2  E1. However, the lifetime of the excited state is extremely short, the electron in the higher energy state (E2) returns (relaxes) to the lower energy state (E1), and it radiates the difference between the two energy levels in the form of a photon of the same energy E photon  h  f as required for excitation. The energies absorbed and emitted are equal because the atom does not lose energy between absorption and emission. This is in contrast to the relationship between the absorption and emission energies of more complex molecules (see Jablonski diagram in the lecture on fluorescence). The formula at the bottom of the slide describes the relationship between the wavelengths of the absorbed or the emitted photons during the various transitions (E1 → En or En → E1). The 7 same atoms have the same energy levels, the atoms of different elements may have different energies. Note: The energy required for the transition between stationary orbits is not always provided by photons, for example, in the event of a collision, the electron may obtain sufficient energy to jump to an orbit with a higher energy. Although the Bohr model successfully described the energy levels of the H atom and the spectral lines of all single-electron (“hydrogen-like”) ions, the spectral lines of multi-electron atoms and ions as well as molecules could not be explained; these were solved by the quantum mechanical model. The electrons in an atom do not have enough energy to leave the environment of the nucleus, these electrons are in a bound state and their energy can only be changed discretely. In order to remove bound electrons from the atom, energy must be invested, which must cover the ionization energy of the atom and the kinetic energy of the liberated electrons:  1 2  Ekin  mv   2  The ionization energy of an atom is the smallest energy required to remove an electron from the ground state atom, i.e., to ionize the atom. This energy is equal to the work function. The potential energy of an electron (free electron) at an infinite distance from the nucleus is zero, but that of an electron bound to the nucleus is negative. As electrons are moved away from the nucleus, their potential energy increases (becomes less negative). In general, we can say that the energy of the electrons bound to atoms and molecules can only take discrete values, however the energy of the free electrons can be changed with arbitrary values (for example, in the case of electrons traveling in an X-ray tube). 8 The slide summarizes the most important features of X-rays. X-ray was named after its discoverer, the German physicist Wilhelm Konrad Röntgen, who was awarded the first Nobel Prize in physics in 1901. X-rays are high-energy electromagnetic radiation, the "particles" of which are photons whose energy (E) can be described by the c formula E  h  f  h  , where h is Planck’s constant (h = 6.63 × 10-34 Js), f is the frequency,  c is the velocity of light and λ is the wavelength. The frequency of X-rays is much higher than the frequency of visible light (see the electromagnetic spectrum), therefore the energy of X- ray photons is also higher by orders of magnitude than the energy of photons of visible light. Depending on the energies of the photons, we distinguish between soft and hard X-rays. Soft X-rays contain photons of higher wavelengths and lower energies ( 5keV). Although the energy of gamma photons is higher than that of X-ray photons, there is no really sharp boundary between the two. Characteristic and braking X-rays can be produced in X-ray tubes, which can be used in medicine for both diagnostic and therapeutic purposes. During the interaction of X-rays with a material, it is attenuated by three different mechanisms (photoeffect, Compton-effect, pair production). During the absorption of X-rays, electrons and ions are formed in each of the three different kinds of processes, which can induce further ionizations. For this reason, X-rays are indirectly ionizing radiation. 9 The slide shows the application areas in which X-rays are used in medicine.  The energy of X-rays used in medical diagnostic imaging procedures is usually between 20-150keV and it is suitable for taking two-dimensional or three-dimensional images, depending on the method used (these will be discussed in more details in a later presentation): o two-dimensional technique: conventional X-ray, fluoroscopy (this approach is used during catheterization) o three-dimensional technique: computed tomography (CT)  In radiotherapy, higher-energy X-ray photons (≈10MeV) are used to irradiate cancer patients. 10 In the remainder of the presentation, we will discuss in detail the answers to the two questions above. 11 In terms of their generation mechanism, there are two types of X-rays, characteristic (characteristic of the anode material) and braking (Bremsstrahlung, independent of the anode material) X-rays. Both types of radiations are usually generated simultaneously in an X-ray tube, but on this slide, we will only discuss the mechanism of the generation of characteristic X-rays in more detail.  Imagine a tube with vacuum inside (top figure on the slide). The electrode (cathode) at one end is heated, causing electrons to leave the material (thermal emission). These electrons collide at high speed (high energy) into a positive electrode (anode), made from a metal of large atomic mass, at the opposite side of the tube. The electrons are accelerated by a voltage between the cathode and the anode, causing different wavelengths (different energies) of X-ray photons to emerge from the metal. These X- ray photons include characteristic X-rays with discrete energies depending on the material of the anode and braking X-rays with a continuous spectrum. The X-ray tube produces X- rays only with low efficiency, since most of the energy of the electrons hitting the anode is converted into heat (therefore the anode must be cooled).  What is the velocity / energy of the accelerated electrons? It depends on the accelerating 1 voltage used: eU  mv 2  Ekin , where U is the voltage, e is the charge of the electron. 2  How is the characteristic X-ray photon generated? Imagine a single atom of the anode material (bottom right figure on the slide). If the energy of the electron bombarding the anode is high enough (Ekin), then an electron is kicked out from the lowest-energy K-shell 12  closest to the nucleus, creating a vacancy on the innermost shell. (Electrons hitting the anode can knock out electrons from other shells, see the following figure.) This vacancy is immediately filled by an electron from a higher energy L- or M-shell jumping to the lower energy, innermost K-shell, and the energy difference between the two states is emitted in the form of a characteristic X-ray photon: E photon  hf  EL  EK  What does the energy of the characteristic X-rays depend on? It depends on the electronic structure of the anode material, therefore, characteristic X-ray photons can only have discrete, specific energies that are characteristic of the given element the anode is made of. This is the reason for why it is called characteristic X-ray. 13 In heavy elements, filling of the vacancy on any of the inner shells (K, L) can lead to X-rays. However, if the energy of the accelerated electron is not high enough to create a vacancy on one of the inner shells, characteristic X-ray photons will not be produced since the energy differences between the outer shells are much smaller. The energy of the accelerated electron can be increased by increasing the accelerating voltage applied in the X-ray tube. By convention, the X-ray that results from the transition between the L and K shells (L→ K) is called Kα radiation, the radiation resulting from the M→K transi on is called Kβ radiation. In the same vein, the photons produced by an M→L transition constitute radiation called Lα, etc. In general, the first letter indicates in which shell the transition ends, and the subscript specifies the details of the process (α: the electron is from the adjacent shell, β: the electron is from the second level relative to the shell of the knocked-out electron, etc.). 14 X-rays can also be generated by accelerating charges, since all accelerating (decelerating) charges radiate, and if the acceleration is large enough, radiation in the wavelength range of X-rays, called braking radiation, is generated. The mechanism of this process can be seen in more detail on the slide.  As described in detail for slide 9, the accelerated electrons in the X-ray tube collide with the anode, thereby suddenly slowing down (or even stopping completely) while emitting electromagnetic waves, i.e., braking X-rays.  What happens exactly? Electrons hitting the anode interact with the atomic nuclei of the anode in such a way that they decelerate primarily in the force field of the atomic nucleus, losing their energy either in a single step (the lower right figure on the slide) or in several steps (this is more typical). This is why the spectrum of braking radiation is continuous (see next slide). o A photon with maximum energy (Ephoton,max) is generated when the electron is completely decelerated in a single collision step and loses its total kinetic energy in this c 1 single step (Ekin): E photon ,max  hf max  h  Ee  , kin  mv12  eU. It can be seen from min 2 the formula that the resulting X-ray photon has the highest energy from among the X- ray photons produced in the deceleration process, and thus its frequency (fmax), called the limiting frequency, is the largest. Due to the inverse relationship between the frequency and wavelength of electromagnetic radiation the wavelength of this photon (λmin), called the limiting wavelength, is the shortest. 15  o If the accelerated electron transmits its kinetic energy in several steps (see the drawing on the slide), the energies of all the resulting X-ray photons, and thus their frequencies, will be lower than those produced by an electron decelerated in one step 1 1 (see previous point): E photon1  hf1  mv12  mv2 2 2 2 o The higher the accelerating voltage (U) applied in the X-ray tube, the higher the kinetic energy of the electrons, thus higher energy braking X-ray photons are produced. From the mechanism described in detail above we can see that the energy of braking radiation is independent of the material of the anode, it only depends on the magnitude of the accelerating voltage applied in the X-ray tube, by which the energies of X-ray photons can be adjusted during medical imaging or therapy. Recall that the energy of the characteristic X-ray photons, on the other hand, depends solely on the material of the anode, i.e. on the electronic structure of its atoms. 16 The upper left figure shows the intensities of braking radiation photons of different wavelengths and thus of different energies as a function of their wavelength, which forms a continuous spectrum (the reason is discussed in detail in the previous slide and see the answer to question 1 on this slide). The accelerating voltage was higher in the case of the red spectrum (U=20kV) than in the case of the blue curve (U=10kV), therefore the limiting wavelength (λmin) of X-ray photons is smaller in the case of the red curve while the limiting frequency (fmax) is higher, because applying higher acceleration voltage, higher energy braking radiation photons are generated because the kinetic energy of the accelerated electrons will also be higher (see explanation for question 2). This spectrum contains only braking radiation photons, since the accelerated electrons did not have enough energy to create a vacancy on one of the inner electron shells of atoms in the anode and to produce characteristic X-ray photons. In the lower left figure, in the case of the green curve, the accelerating voltage was higher (U=25kV), and the discrete lines of high-energy Kα- and Kβ-characteristic X-ray photons appeared at low wavelengths, the position of which depend on the anode material. In this case, the braking and characteristic X-rays were generated in the X-ray tube at the same time (see question 3). Only in atoms with high atomic mass (e.g. molybdenum) are characteristic X- ray photons formed because the energy difference between electron orbits is not high enough in lighter elements. In medical diagnostics, lower energy (20-150keV) braking radiation is used. 17 We will discuss the mechanisms leading to the absorption of X-ray photons on the next slides. Such absorption mechanisms lead to a loss of X-ray intensity on the other side of the absorber (e.g. human body) opposite to the X-ray source. Such a lower detected X-ray intensity, which is the consequence of absorption, typically corresponds to bright areas in an X-ray image. 18 When an X-ray or gamma ray passes through a material, its intensity decreases continuously as the number of photons in the beam decreases (see the figure at the bottom of the slide). These radiations are absorbed to varying degrees depending on the density and the atomic number of the material. The attenuation coefficient (µ) is proportional to the density of the irradiated material and to the third power of the (effective) atomic number (although some mechanisms of absorption show a different dependence on the atomic number, see later and in the CT lecture) and it also depends on the energy of the radiation. The higher the attenuation coefficient is, the greater the ability of the material to absorb the radiation passing through it. If the material does not contain only one component (e.g., lead, copper, etc.), but it is a complex material or tissue, then the effective atomic number should be considered, which depends on the atomic number of the components. As the atomic number of chemical elements that make up soft tissues is low (carbon: Z=6, nitrogen: Z=7, oxygen: Z=8, hydrogen: Z=1), the effective atomic number of soft tissues is 7.4, therefore they only slightly absorb X-ray photons (µ~Zeff3), so they are not visible in the x-ray image (dark area). In order to increase the contrast, a contrast agent with high atomic number element is used, usually barium (Z=56) or iodine (Z=53), which accumulates in the organ/tissue to be examined and is able to absorb X-rays to a greater extent, making them visible as white areas in the images. The effective number of bones (Zeff=13.8) is higher due to their calcium content (Z=20), therefore they absorb X-rays to a greater extent than soft tissues, so they appear as white areas in the images. For similar reasons, objects made of high atomic number (Z=82) lead are used to shield X-rays and gamma rays (lead apron, lead crucible). 19 The intensity of X-rays (I) decreases exponentially with the thickness (x) of the absorbing material: I  I 0 e  x , where µ is the attenuation coefficient, its unit is 1 / m. The attenuation coefficient is the reciprocal (1/x) of the distance (x) where the radiation intensity decreases to 37% (e-1). But why?  If the thickness (x) is replaced by the reciprocal of the attenuation coefficient (1/µ) in the 1   formula: I  I 0 e  I 0 e 1  I 0  0.3679 , which means that at a thickness equal to 1/µ, the radiation intensity is reduced to ≈37% of the initial intensity, i.e. 63% was absorbed in the material.  The figure on the left shows the attenuation of the X-ray intensity (I/I0) on a linear scale as a function of the thickness (x) of the absorbing material. Comparing the two curves, we see that black one decreases more steeply than the blue, which means that the material characterized by the black curve has a higher attenuation coefficient, i.e. it can absorb more photons at a given thickness than the material characterized by the blue curve. The attenuation coefficient can also be determined from the figure: read the thickness (x) where I/I0 is 0.37 (the radiation intensity is reduced to 37%) and then take the reciprocal of this thickness (1/x=μ).  In the figure on the right, the logarithm of X-ray intensity [ln (I/I0)] is plotted as a function of the thickness (x) of the absorbing material, and the attenuation coefficient can be calculated from the slope of the obtained line (-µ). The attenuation coefficients of copper and aluminum are also determined by this method in the biophysics laboratory practice. 20  Explanation There is a certain confusion surrounding the nomenclature of scale types. The exponential absorption of X-ray is plotted in a linear plot, having linear X- and Y-scales, in graph A. If the equation shown beside the graph is converted by taking its natural logarithm (denoted by “ln”), the exponential relationship becomes a linear one. Therefore, if the logarithm of the intensity, ln (I/I0), is plotted on a linear Y-scale, the slope of the line will be - (graph B). If the original exponential relationship is plotted on a logarithmic Y scale (graph C), the exponential curve is also converted to a straight line. The plot in part C is called a semilogarithmic plot (having a linear X scale and a logarithmic Y scale). Even though the appearance of the two plots in B and C are similar (both of them are linear), the logarithm of the exponential relationship is plotted on linear scales in part B (and therefore it is not appropriate to call this graph type logarithmic or semilogarithmic), while the original exponential relationship is plotted in a semilogarithmic graph in part C. Despite the fact that the two plots in B and C are both linear, the slope of the line is - only in B (-5/100=20), and therefore graphical determination of the attenuation coefficient from the slope of the line is only possible using the graph shown in B. 21 Through the next few slides, we will learn the mechanisms by which X-rays can be absorbed in a material (e.g., during medical diagnostic imaging or therapeutic treatments). These mechanisms apply not only to X-rays but also to gamma rays. Although gamma radiation is generated by the decay of radioactive nuclei, it is also a high-energy electromagnetic radiation and behaves similarly to X-rays (see the Features of nuclear radiation and its interaction with absorbing material lecture for more detail). In photoeffect (photoelectric absorption), the X-ray photon is absorbed by the material, it imparts all of its energy to the atom, and as a result, an electron in the inner shell is ejected 1 (ionization of the atom): E photon  hf  A  me  ve2. The energy (hf) of the absorbed photon 2 covers the binding energy (A, ionization energy) and the kinetic energy of the liberated electron. It can be seen from the formula that the energy of the absorbed X-ray photon must be greater than the binding energy of the electron on the inner shell. The probability of photoeffect is inversely proportional to the third power of the X-ray energy: 1 interaction probability  3. In general, X-ray photons with an energy less than ~30keV E photon are attenuated primarily by the photoeffect.  What happens with the liberated electron? This liberated photoelectron can cause further ionization in the medium, or even secondary X-rays (braking and characteristic) can be generated in the surrounding tissues until it loses its kinetic energy. For this reason, X-rays and gamma rays are both considered as ionizing radiations, although the rays 22  themselves do not carry a charge, but their absorption results in the formation of high- energy free electrons. The photoeffect is the most important mechanism occurring during a diagnostic X-ray because most of the atoms in the human body have small atomic numbers and the binding energy of electrons on the K-shell is small. Therefore, the photoeffect is easily created and the photoelectron does not get too far (it is absorbed by soft tissues within 1-2mm).  What happens with the atom after the ejection of the photoelectron? Due to the liberated electron, an electron vacancy is generated on the inner shell of the atom, but an electron jumps off the outer shell to fill in the vacancy, and the energy difference between the two states is emitted in the form of a characteristic X-ray photon. This is called secondary radiation (it is generated in the same way as the characteristic X-ray photon in the X-ray tube, which is a primary radiation, and has a much higher energy because the anode has a high atomic number and a high binding energy). If a contrast agent (e.g. barium, iodine) is used, the energy of the secondary radiation will be higher.  For the attenuation coefficient of the photoeffect, the notation τ is usually used instead of µ, which is also proportional to the density of the absorbing material (ρ), the third power of the effective atomic number (Zeff3) and depends on the photon energy of X- rays. 23 The Compton effect occurs when the incident X-ray photon is not completely absorbed in the material, but it is only scattered. In this case, it transfers only part of its energy to a loosely bound electron on the outer shell. The electron is then ejected (Compton electron), the atom is ionized, and the X-ray photon travels with a lower energy (lower frequency, 1 f’scattered), changing its direction (Compton photon): E photon  hf  hf 'scattered  A  me ve2. The   2 energy of the incident X-ray photon (Ephoton=hf) covers the binding energy (A, ionization energy) and the kinetic energy of the electron removed from the outer shell, as well as the energy of the scattered photon (hf’scattered, and f> f’scattered). In general, X-ray photons with energies greater than ~30keV are attenuated primarily by the Compton effect, and Compton scattering plays a role not only in medical diagnostic imaging but also in therapeutic applications.  What happens with the liberated Compton electron? Similar to the photoeffect, it causes further ionization in its environment.  What happens with the scattered Compton photon? The scattered photon still has a relatively high energy (since the original photon only transferred its energy to an electron in the outer shell), therefore it interacts with additional atoms and ionizes them. Finally, it gets absorbed by photoeffect. Until its complete absorption, it has high enough energy to cause radiation danger and deteriorate the image quality.  What happens with the atom after the Compton electron is removed? Since only an electron on the outer shell was liberated, the atom is ionized. 24  The attenuation coefficient corresponding to the Compton effect, usually denoted by σ, is proportional to the density of the absorbing material (ρ), but depends only weakly on the effective atomic number (Zeff) and depends on the photon energy of X-rays. 25 If the energy of the X-ray photon exceeds 1.02MeV, it will interact with a nucleus of the absorbing material and pair production will occur. In this process the X-ray photon disappears and its energy is converted into an electron-positron pair. Since the energy of the X-ray photon (hfmin) must cover the energy corresponding to the rest masses of the resulting particles, the process takes place only above a minimum photon energy, which can be calculated by Einstein's mass-energy equivalence equation: hf min  ( me  me )c  1.02 MeV , where me- and me+ are the rest masses of the electron and   2 the positron, respectively, and c is the velocity of light. Pair production can only occur near a heavy nucleus due to the law of conservation of momentum (details not discussed). If the energy of the X-ray photon is greater than 1.02MeV, the excess energy above that required for creating a resting electron-positron pair is used for the kinetic energy of the electron and the positron. X-rays that have such high energy are called hard X-rays.  What is a positron and what happens with the resulting positron? A positron is the antiparticle of an electron, which means that they have the same properties (e.g. they have the same mass), only their charge is the opposite. Antiparticles are unstable in our material world, so the resulting positron immediately collides with an electron (not the one formed during the pair production) and two gamma photons with the same energy (0.51MeV) moving in opposite directions are formed. This process is called annihilation (see box on page 165 of the textbook). Let us remember this when it comes to PET (Positron Emission Tomography) imaging technique in a few lectures. 26  What happens with the resulting electron? Similar to the other two X-ray absorption mechanisms, the resulting electron can generate ionization or even secondary X-rays. 27 Several methods have been developed to determine the structure of molecules and crystals, but the most powerful of them is X-ray crystallography, which is based on the diffraction and reflection of X-rays on crystals. X-rays primarily interact with the electrons of atoms/molecules, but they do not lose their energy, they are scattered elastically. Recall what we have learned about the interference during the first lecture. At the lattice points of a crystal, there may be atoms, ions, or even molecules that can be considered as scattering centers, since by illuminating the crystal with an X-ray beam, the X- ray photons will be diffracted in all directions. These photons can extinguish (destructive interference, dark spots) or amplify each other (constructive interference, bright intensity maxima). In reality, constructive interference can only be observed in a few preferred directions, which form bright spots in the diffraction image. The phenomenon of X-ray diffraction can only be detected if the distance between the scattering centers is in the order of the wavelength of the X-ray radiation. The crystal structure can be determined from these interference patterns. In order to obtain an X-ray diffraction image, a crystal must be grown from the molecule to be examined, which is usually a rather complicated task. The diffraction pattern was previously recorded on X-ray films, but today a CCD camera sensitized in the X- ray range is used to capture the diffraction pattern rapidly. 28 The figure shows the diffraction of X-rays projected onto a one-dimensional crystal. The scattering centers are located along the z-axis, at a distance of c from each other, to which the X-rays fall perpendicularly, parallel to the x-axis. X-rays are scattered from every scattering center in all directions, however, in most directions this is not detected due to destructive interference. The radiation will be detectable only in those directions in which the condition of constructive interference is fulfilled, i.e. the path difference (Δs) between interfering waves is an integer multiple (l) of the wavelength (λ): s  c  cos   l   , where c is the lattice constant, γ is the angle of diffraction and l=1,2,3,…. In the figure, this path difference is indicated by the thick black lines, which show the distance with which the bottom ray travels more than the top ray before they interfere with each other. The figure shows an example of constructive interference for 2 different angles of diffraction, therefore there are 2 thick black lines. The photon with larger diffraction angle (γ1) has a shorter path difference, which is equal to the wavelength (λ), while the smaller diffraction angle (γ2) corresponds to a longer path difference (longer thick black line), which is twice the wavelength (2λ). 29 The figure shows the diffraction of X-rays on a one-dimensional crystal when the incident X- ray is not perpendicular to the z-axis, i.e. it is inclined. In this case, the path length difference (Δs) consists of two components (Δs1, Δs2) shown in the figure: s  s1  s2  c  (cos  0  cos  )  l   If we want to apply this equation in three dimensions, we get the 4 equations shown on the next slide. 30 Of the 4 equations shown on the slide, the first 3 are the so-called Laue equations, and together the 4 equations describe in three dimensions the directions in which constructive interference is obtained when X-ray photons are scattered at the lattice points. The 4 equations are overdetermined, containing three unknowns (a, b, c), so it has a solution only in certain cases. These special cases can be produced in two ways, but both methods have in common that many combinations of angles are present, and there will be a combination among them where the system of the 4 equations has a solution: 1. Rotating crystal method: during rotation, the crystal will eventually reach such a position in which the conditions of the Laue equations are satisfied. 2. Powdered crystal method: in a powdered sample there are always crystal particles for which the Laue equations are satisfied. 31 According to Bragg, X-rays can not only be scattered from the scattering centers of crystals (Laue’s interpretation), but can also be reflected from the crystal planes, and these reflected rays can also amplify each other. This case can be seen in the figure demonstrated with a two- dimensional crystal. A path difference (Δs, indicated by a red line in the figure) is created between the rays reflected from the parallel planes. The reflected rays produce observable radiation (bright intensity maximum in the diffraction image) if the condition of constructive interference is fulfilled, i.e. the path difference between interfering rays (Δs, red line in the figure) is an integer multiple (l) of the wavelength (λ): s  l   The Bragg equation describes the angles of incidence (α) at which constructive interference occurs with X-rays of a given wavelength: s  2d  cos   l   , where d is the distance between the parallel crystal planes and l=1,2,3,…. If the crystal is irradiated with X-rays of known wavelength (λ) and the direction of the maximum intensities (α) is measured, the distance (d) between the crystal planes can be determined using the Bragg equation. 32 If groups of atoms or molecules are present in lattice points instead of single atoms, scattered rays from adjacent atoms within the molecule can also interact (yellow and blue atoms in the image). A lattice plane drawn across a unit cell is decomposed into several nearby lattice planes corresponding to the lattice planes of each atom that makes up the molecule (blue and red lines in the image, one for the blue atom and the other for the yellow atom). Since the intensity of the reflected rays is influenced by the distance between these lattice planes (the red and blue lines), the internal structure of the molecules can be determined from the X-ray diffraction image. 33 X-ray crystallography can be used to determine the exact spatial arrangement of thousands of atoms in proteins, so it can also be used to explore three-dimensional structures. The figure shows the three-dimensional structure of a potassium ion channel. 34 In order to determine the three-dimensional structure of proteins by X-ray crystallography, the protein must first be synthesized or isolated, purified, and finally crystallized. Knowing their three-dimensional structure, proteins can be used to design drugs more accurately. The slide shows the three-dimensional structure of the epidermal growth factor receptor (EGFR), to which a drug called gefitinib is bound. Gefitinib is able to specifically inhibit EGFR tyrosine kinase activity by binding to the ATP-binding site of the receptor. Because EGF receptor overexpression, observed in many tumors, activates signaling pathways and enhances cell division, such a drug can be used for treating tumors. 35 36