Example: Compute the wavelength of a 1 [kg] block moving at 1000 [m/s].

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1 Example: Calculate the energy required to excite the hydrogen electron from level n = 1 to level n = 2. Also calculate the wavelength of light that must be absorbed by a hydrogen atom in its ground state to reach this excited state. Using Equation, with Z = 1 we have E 1 = x J(1 2 /1 2 ) = x J E 2 = x J(1 2 /2 2 ) = x J E = E 2 - E 1 = ( x J) ( x J) = x J Example: Calculate the energy required to remove the electron from a hydrogen atom in its ground state. Removing the electron from a hydrogen atom in its ground state corresponds to taking the electron from n initial = 1 to n final =. Thus, E = x J = x J The energy required to remove the electron from a hydrogen atom in its ground state is x J. Example: What is the wavelength of the line in the visible spectrum corresponding to n 1 = 2 and n 2 = 4? Example: Compute the wavelength of a 1 [kg] block moving at 1000 [m/s]. 1

2 Example: Compute the wavelength of an electron (m=9.1x10-31 [kg]) moving at 1x10 7 [m/s]. Example : Calculate the wavelength of light, in nm, of light with a frequency of 3.52 x s -1. c m/s m s nm m m 852 nm Example :Calculate the energy (in joules) of a photon with a wavelength of nm Example Photons in a pale blue light have a wavelength of 500 nm. (The symbol nm is defined as a m.) What is the energy of this photon? 2

3 Direction Indices - Example Determine direction indices of the given vector. Origin coordinates are (3/4, 0, 1/4). Emergence coordinates are (1/4, 1/2, 1/2). Subtracting origin coordinates from emergence coordinates, Multiply by 4 to convert all fractions to integers Therefore, the direction indices are 3

4 1. What is the coordination number in the: i. face-centred cubic (fcc) structure, ii. hexagonal closed-packed(hcp) structure, iii. body-centred cubic (bcc) structure? 2. How many atoms per unit cell are there in the: i. face-centred cubic structure, ii. hexagonal closed-packed structure, iii. body-centred cubic structure? 3. Calculate the packing fraction for the structure: i. face-centred cubic structure and ii. body-centred cubic Miller Indices - Examples 1. Compute the Miller Indices for a plane intersecting at x= ¼, y=1, and x=1/2, i. Graph the plane and determine the axis intersects of a surface with the Miller Index (013). 2. Explain the meaning of {100} and it s importance. 1. ) 2,1,4( 4

5 2. Intersects at y = 1, z = 1/3, plane does not intersect the x-axis 3. } 111{ is a short way of referring to 6 different planes. These indices all refer to the same lattice as viewed from different points of reference defined by the axis Plot the plane (110) The reciprocals are (1,, ) The intercepts are x=1, y=-1 and z= (parallel to z axis) Crystallographic Planes 5

6 6

7 In the cubic system, a plane and a direction with the same indices are orthogonal 7

8 and finally multiply by a common (factor) denominator. Which is 6, to obtain the miller indices (3 4 6). Exercise 8

9 Ex: Determine the Miller Indices of a plane which is parallel to x-axis and cuts intercepts of 2 and 1/2, respectively along y and z axes. Therefore the required Miller indices of the plane (014) Ex: Determine the M. I. of a plane theat makes intercepts of 2Å, 3 Å, 4 Å on the coordinate axes of an orthorhombic crystal with a:b:c = 4:3:2 Ex:For the intercepts x, y, and, z with values of 3,1, and 2 respectively, find the Miller indices. Intercepts: x=3, y= 1, and z=2 y Take the reciprocal of the intercepts h = 1/x, k=1/y, and l=1/z h=1/3 k = 1/1 l = ½. 9

10 Multiply h, k, and l by 6 to find the smallest integer. (hkl) = (263) 1. Determine the Miller indices of the plane in the figure below. 10

11 Consider a metal with an FCC structure and an atomic weight of When monochromatic x-radiation having a wavelength of nm is focused on the crystal, the angle of diffraction (2θ) for the (311) set of planes in this metal occurs at 71.2 degrees (for the first order reflection n=1) a. Calculate the interplanar spacing for this set of planes. b. Calculate the lattice parameter for this metal. c. Calculate the density of the metal (units of g/cm3) Example: We could then use 2θ values for any of the peaks to calculate the interplanar spacing and thus the lattice parameter. Picking peak 8: 2θ = or θ = d 400 2sin sin(29.71) a 0 d 400 h 2 k 2 l 2 ( )(4)

12 Example Determine the indices for the planes shown in the hexagonal unit cells below: 12