Martin-Luther-University Halle-Wittenberg Institute of Physics Advanced Practical Lab Course V 11: Electron Diffraction An electron beam conditioned by an electron optical system is diffracted by a polycrystalline, 5 nm thick carbon sample which is fixed on a copper net. The carbon plate is perpendicularly fixed to the optical axis. 1) Determine the de Broglie wavelength as a function of anode voltage U A (G 3 - potential to ground, see poster in the lab) in the interval 5 < U A /kv < 10 in non-relativistic approximation. Give an equation and a plot of your results. 2) Calculate the relative error concerning the wavelength when relativistic effects are considered! 3) Measure the diameter of the most intensive rings and determine the Bragg angles Θ i (see figure 1; D = (127±3) mm). Change the voltage from 5 kv to 10 kv by 1 kv. 4) Using Bragg s equation and considering figure 5 derive a linear function of the radius of the interference rings with the wavelength λ as independent variable (note for small angle cos α = 1 is valid). Determine the net plane distance from the slope! Furthermore make a λ plot of: f ( U A ) =. 2 sinθ 5) Calculate the structure factor for the graphite structure (structure type A9; a 0 = 0.246 nm; c 0 = 0.671 nm (see crystal model in the lab). Furthermore, calculate the net plane distances from the first 10 reflections. Allocate the reflection of 4) to the corresponding reflection. 6) Determine the coordination number in the (0001) plane for the distance a (see graphite model in the lab). The distance of a C-C single bond in the equilibrium state is 0.154 nm, whereas the according distance of a C=C double bond is 0.133 nm. Sketch the binding structure of the graphite model in the (0001) plane, calculate the mean distance of the C- atoms <a> and compare it with a = 0.142 nm. (Note! a and a 0 are not the same) Explain the correlation between the crystal structure of carbon and its physical properties like charge transport, thermal conductivity and mechanical strength.
Questions for testing your knowledge: Explain commonness and differences of X-ray- and electron diffraction! Why must electron diffraction take place in vacuum? In which way does the reciprocal space change when a 2-dimensional instead of a 3- dimensional structure is considered? Equation (5) is the common formula of the kinetic energy. Derive the classical formula of the kinetic energy using this relation! Explain the crystal structure of diamond! Calculate the structure factor of both crystal modifications of carbon, graphite and diamond! Write down the net plane distances as function of Miller s indices! Calculate the number of atoms in the unit cell of graphite! (density ρ = 2,23 g/cm 3, a 0 = 0,246 pm, c 0 = 0,676 pm). Why do physical properties like hardness or conductivity of graphite and diamond differ so much? Explain the phenomena of hybridization from the example of carbon! Find out the correlation between the hybridization and the crystal structure! Explain the term polymorphism in the crystallography! Quote some possibilities to detect electrons or to make them visible! 2
Basics Electron diffraction is a technique to study matter by firing electrons at a sample and observing the resulting interference pattern. Electron diffraction is most frequently used in solid state physics and chemistry to study the crystal structure Interaction of electrons with matter in almost all electron microscopes electrons impinge on the sample as primary electrons (PE) and leave the target as elastically or inelastically scattered electrons. The probability that a single electron will be scattered is described by the so called cross section σ or by the mean path length dn λ. 1 particles per time impinge on the target with a particle density of dt N A[ # / mol] 3 3 n2 = ρ[ g / cm ] [# / cm ], A[ g / mol] (N A Avogadro s number, A molecular mass). dn dn1 The number of interactions per time is described by = n2dxσ, σ has the dimension of dt dt an area. Another way to describe this phenomenon is the model of the mean free path length. λ = 1 Nσ (1) When the sample is very thin no scattering or only one interaction can occur. In the case of thicker sample multiple scattering can take place. Elastic scattering includes all processes with change in direction and no change in energy of the PE, at least not measurable. On the other hand inelastic scattering is a general term including all interactions of the electrons with matter without any energy loss. Unlike other types of radiation used in diffraction studies charged particles interact with matter through the Coulomb forces. The Rutherford scattering leads to an in forward direction distributed profile of the scattered electrons. At an energy of E 0 of the PE the probability p(θ) that an electron is scattered by an angle θ is given by 2 Z p( θ ) (2) 2 4 E0 sin θ (Z atomic number). The probability decreases strongly with the scattering angle θ and with the energy of the electrons E 0. For example, 100 kev PE show a mean free path length λ in gold (Z=79) of about 5 nm but in carbon (Z=6) 150 nm. According to Huygens principle the diffracted electron beam can be considered as the sum of the partial waves, the same approach as in optics or X-ray physics. The electrons are scattered at the Coulomb potential of the atomic cores. Thus, they probe the scattering Coulomb field in a different way. At the diffraction experiment the coherently scattered intensity is measured. The incident electrons can be mathematically described by a plane wave function, the Schrödinger wave Ψ 0 = exp( k 0 r ωt). After the scattering process, the Schrödinger equation has to be solved, the scattered wave is a spherical one Ψ = exp( k S r ωt) r. The integration over all 3 S /
scattering processes delivers the atomic form factor, similar to the procedure at X-ray diffraction. The next step is the integration over all atoms in the unit cell, which yields the structure factor. For the mean calculation of the structure factor and the position of the reflection, the mathematically easier kinematic diffraction is sufficient. On the other hand for a more detailed consideration (e.g., concerning proper intensities) a dynamical theory becomes highly desirable, since this considers dynamic effects like multiple scattering, refraction, the interaction between primary and scattered beam and extinction. One of the most essential differences between a kinematical and a dynamical description is the calculated intensity. It gets proportional to the squared structure factor (kin.), whereas it is given by the absolute value of it in a dynamic scenario. Reciprocal lattice and Ewald construction Ewald construction is a geometric construct used in crystallography to demonstrate the relationship between the wavelength (de Broglie or X-ray), the diffraction angle and the reciprocal lattice. The construction of the reciprocal lattice is made in such a way that: Each point (hkl) in the reciprocal space corresponds to a set of lattice planes in the real space, the direction of the reciprocal space lattice vector corresponds to the normal of the real space net planes and the magnitude of the reciprocal lattice vector is equal to the reciprocal of the net plane spacing 1/d (d net plane spacing) in the real space. The Ewald construction is equivalent to Bragg s law. The relationship is demonstrated by figure 1. Crystal structure is represented by its reciprocal lattice. Incident plane wave (electron beam) pointing onto the crystal has a wave vector with a length 1/λ. The tip of this vector lies in the origin of the reciprocal lattice. The radius of the sphere around the point Figure 1: Ewalds construction. A is 1/λ. All reciprocal lattice points lying on the Ewald sphere fulfil Bragg s condition. 2 d sinθ = nλ (3), (λ - wavelength, d net plane spacing, θ - Bragg angle, n order of diffraction). Experimental devices Electron diffraction tube In the electron tube the electrons are generated by thermionic emission at a tungsten wire. They are collimated by a Wehnelt cylinder (G1) and show a current of about 1 ma. The sample is placed in a distance of 35 mm from the cathode and the tube has a diameter D of (127±3) mm. When reducing the voltage at the Wehnelt cylinder the electron beam is widened and the copper 4
net carrying the sample can be seen. Figure 2: Scheme of the electron gun. Figure 3: Scheme of the electrodes in the electron gun. Figure 4: Scheme of the wiring diagram. The de Broglie wavelength of particles is given by λ=h/p (h Planck s quantum, p momentum). According to the special theory of relativity matter and energy are equivalent given 2 by the relation E = m c. The kinetic energy, i.e. the difference between total energy and rest energy is given by equation (5): 2 2 1 E kin = E m0c = m0c 1, (5) 2 2 1 v c (m 0 rest mass, c light velocity, v - velocity). The momentum takes into account the relativistic mass change. m0v p =. (6) 2 2 1 v c The relativistic expressions for E and p can be combined into the fundamental relativistic energy momentum equation 2 2 4 2 2 E = m0 c + c p, (7) Use this relation to calculate the systematic error of the de Broglie wavelength. 5
Sample A graphite plate serves as sample and is inside the electron tube. Under ambient conditions graphite is the stable modification of carbon. Graphite consists of a layer structure. Each carbon atom possesses a sp 2 orbital hybridisation and is covalently bonded to three other surrounding atoms. The bonding angle is 120 and the flat sheets of carbon are bonded into hexagonal structures. The fourth, pi orbital electron is delocalized across the hexagonal layers and contributes to the conductivity of carbon. The conductivity parallel to the sheets is greater than perpendicular to the sheets. Furthermore, the bond between the layers is weak, van der Waals force, and the layers can slip over each other. For that reason graphite powder is valued for industrial application because its lubricating properties. The coordination number is the number of the next neighbours. It is different for each carbon allotrope. For more information concerning the crystal structure the powder diffraction files of the PDF-data banc of ICDD can be used. Figure 5: Modifications of carbon. These data sheets include structure data and diffraction diagrams of many crystal structures from all over the world. In the case of X-ray diffraction (structure analysis) the data are used as a fingerprint to identify the crystal phases investigated. As an example the sheets of graphite are 6
showed. Figure 6: PDF-data sheets of graphite. For more information concerning the space groups (S.G.) 186 and 194 see the International Tables of Crystallography. The difference between space group 186 and space group 194 is the location of the carbon atoms in the hexagonal layer. Strictly speaking the carbon atoms of the hexagonal layer shift alternatively about an amount of 1/20 of the lattice constant c. Therefore a slight waveness arises, but usually the crystal structure of graphite can be assumed to be that of the space group 194. References: Bergmann/Schaefer: Experimentalphysik VI (Kapitel 2, Kristallstrukturen) Ashcroft/Mermin: Festkörperphysik Goodhew: Elektronenmikroskopie: Grundlagen und Anwendung Vainshtein. Fundamentals of Crystals Hahn: International Tables of Crystallography Volume A 7