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by: Carmela Kilback


Carmela Kilback
GPA 3.92


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Class Notes
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This 10 page Class Notes was uploaded by Carmela Kilback on Wednesday September 9, 2015. The Class Notes belongs to CHEM 484 at University of Washington taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/192588/chem-484-university-of-washington in Chemistry at University of Washington.




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Date Created: 09/09/15
CHEM 484 Lecture Notes Weeks 3 and 4 1 Determination of Crystal Structure The exact arrangement of atomsions on a solid surface can be directly visualized using scanning tunneling microscopy STM for conductive substrate or an atomic force microscopy AFM for either conductive or nonconductive substrate For both techniques We can only see the atoms or ions sitting on the surface We can not see those atoms or ions underneath the topmost layer Atomsions can also be seen using highresolution transmission electron microscopy HRTEM The most Widely used techniques for the determination of solid structure are based on diffraction With Xrays electrons or neutrons in particular Xray 2 Xrays Xrays Were discovered by Rontgen in 1895 They are electromagnetic Waves With very short Wavelengths 110 A They are produced by bombarding the surface a metal target eg copper With high energy electrons gt30 kV The inner core electrons eg 1s of the metal Will be knocked out by the incident electrons Filling of the empty inner orbit by electrons on the outer orbital results in Xray radiations For copper the 2pgt1s transition corresponds to Kw radiation Which has a Wavelength of 15418 A The 3sgt1s transition corresponds to K radiation Which has aWavelength of 13922 A In a diffraction study the monochromatic Xray can be obtained by passing the light source through a metal With atomic number 21 to filter out K radiation Ejeckd E letAron EA Incident eleclron beam Energy E F chasegrime Xrays E E EX K 0 Energyloss eledron E EAE AE gtE The typical design of an Xray tube lnlnnslly Arbllllry Unlls BERYLWM Tp iif GLASS II N DOW I Af I E COOLING ELECTRONS E TARGET x SCHEMATIC CROSS z 3 a WAT R 3quot E T0 TRANSFORMER FOCUSING CUP VACUUM SECTION OF AN XRAY TUBE RAYSr cummnsn unes 50 w cannnuum I I a 04 as 03 Mun L3 La la 14 L Wavalanalh X Commons and characteristic radiation for copper 3 Scattering of Xrays by an Atom or Ion Xray can be described as an electromagnetic wave with oscillating electric eld which will set the electrons in an atom or ion to oscillate at the same frequency The oscillating dipole will emit radiation at the same frequency as the incident Xray elastic scattering and in phase or coherent with the incident Xray but in all directions The intensity of the radiation scattered coherently by point source electrons is given by the Thomson equation 1p 1 cos2292 Where Ip is the scattered intensity at any point P and 29 is the angle between the directions of the incident beam and the diffracted beam passing through F From this equation it can be seen that the diffracted beam is most intense when parallel or antiparallel to the incident beam This equation is also known as the polarization factor which has to be taken into account when one tries to analyze the intensity of diffraction spots For a real atom or ion there are many electrons in it so the intensity of the scattered wave will be the sum of all radiations coming from all electrons within the atom ion The scattering factor or form factor of an atom is proportional to the total number of electrons in the atomion or the atomic number Z As a result very light atoms such as H do not contribute significantly to the intensity of diffracted beam and it is very difficult to find their locations in a crystal This also explains why Xrays can penetrate very deeply through soft tissues and how Xray images are formed Also Be Z4 and lithium Z3 borate glasses are commonly used as the window materials for Xrays while Pbbased glasses are used to block Xrays Based on this model each atom or ion in a crystalline lattice acts as a secondary point source of Xray The radiations from all these atomsions need to be added together to give the intensity of diffraction spot at any point The intensity of the spot strongly depends on the phase difference between the radiations coming from different atomsions By analyzing this dependence we can figure out the spacing between atoms and thus determine the structure of a crystal 4 Interference of Electromagnetic Waves An electromagnetic wave can be expressed as E Easinoat 4 where n is the angular frequency 0321ru and I is the phase When two waves of the same frequency propagate along the same direction they will interfere to each other The intensity of combined wave will depend on the phase difference between the two waves when Ad n 211 with n being an integer we have constructive interference and the two waves are said in phase and when Ad n 11 with n being an integer we have destructive interference and the two waves are said out of phase A typical device based on such a phenomenon is the Michelson interferometer where the output intensity depends on the difference between the two arms of the device 5 Bragg s Law of Xray Diffraction The dlffracuon of eray from a ervstal ean also be easlly understood tn terrns of Bragg39s Lavv Tu Ml m all u crystalanth er 4 h llerrtlm the sarne dstanee before belng baek at the surface The dstanee traveled depends on the separahon of the layers and the angle at whlch the X1131 enteredthe rnatenal For thts vvave to be In phase vvrth the vvave whlch re eeted from the surface lt needs to have traveled a whole number of wavelengths Whlle mslde the rnatenal Bragg expressed ths m an equatlon now known as Bragg s Law a rugg s law nl2dsin6 t re A ts the wavelength ml the rays 0 ts the angle between the mndent lays and the snrtnee ol the ervstnl d ts the spuclng between levers et ntdms and nanstroetwe tnterlerenee news when n ts an Inleger whale nnmhert When n ls anlnteger l 2 3 ete the re eeted vvaves from dlfferent layers are perfecdyln phase are not tn phase and wlll erther be rnlssrng or fatnt Denvatron ofBragg39s Lavv Note that hdslne so the path length dlfference betvveen these tvvo beans are 2dsrne The phase dlfference wlll be A 2dsln627r7L When th5 phase dlfference ls equal to n21t vvrth n belng 2dsin921t7t n21 or 2dhk1sin9 n 7 n is the diffraction order In general we only need to consider the first order diffraction nl as diffractions of high orders can be considered as the first order diffractions from planes with an interplaner spacing of dhkln For example the second order diffraction from 100 planes can be considered as the first order diffraction from 200 planes 6 Use of Bragg Equation For an orthorhombic lattice the interplanar spacing can be expressed as ldhklz 112 12b2 12 So for any different combination of h k and l we may have a different value for the dspacing and thus different diffraction angles which can be used as finger prints to identify a solid h k l inn 9 l 0 0 dloo 0 l 0 d010 0 0 l d001 l l 0 duo l 0 I dun 0 l I don l l 1 dm 7 Powder Xray Diffraction PXRD httpwwwmatterorgulddiffractionxraypowderimethodhtm Powder Xray diffraction PXRD is perhaps the most widely used technique for characterizing solid materials As the name suggests the sample is in a powdery form consisting of ne grains in the form of single crystallites The term powder really means that the crytalline domains are randomly oriented in the sample When a 2D diffraction pattern is recorded it shows concentric rings of scattering peaks corresponding to the various dspacings in the crystalline lattice The positions and the intensities of the peaks can be used for identifying the underlying structure or phase of a solid material For example the diffraction lines of graphite would be different from diamond even though they both are made of carbon atoms This phase identification is important because the material properties are highly dependent on the structure just think of graphite and diamond If a monochromatic Xray beam is directed at a single crystal then only a few diffracted beams that are allowed by the Bragg s law may result A sample of some hundreds of crystallites ie a powder sample show that the diffracted beams form continuous cones A circle of film is used to record the diffraction pattern as shown Each cone intersects the film giving diffraction lines The lines are seen as arcs on the film For every set of crystal planes by chance one or more crystals will be in the correct orientation to give the correct angle to satisfy Bragg s equation Every crystal plane is thus capable of diffraction Each diffraction line is made up of a large number of small spots each from a separate crystal Each spot is so small as to give the appearance of a continuous line If the crystal is not ground finely enough the diffraction lines appear speckled This arrangement is achieved practically in the DebyeScherrer camera g F After the film strip is removed from the Debye camera after exposure it is developed and fixed Each diffraction line can be measured from its position on the strip of film From the results it is possible to associate the sample With a particular type of crystal structure and also to determine a value for its lattice parameters Nowadays the PXRD pattern of a solid material is commonly recorded using a diffractometer eg the Geiger diffractometer Where the intensities and position 29 of diffracted Xray beams are directly recorded using a detector moving about the sample see below I I I I I I 8 Repeatdistnme13 X E 6 g E 4 I 2 k u n I I I I M 10 20 30 4D 50 60 7 Z lhela 8 The Application of PXRD PXRD is an indispensable tool in the study of solid state materials The PXRD pattern can serve as the fingerprint to identify the phase and structure of a solid material In general it can be used to identify an unknown material verify the purity of a solid product follow the status of a solid state synthesis determinerefine the lattice parameters of a solid determine the crystallite size of a sample from the peak Width See Chapter 2 of the text book JCPDS database 9 The Texturing Effect A tur onentatron along eertatn latuee planes one ean vlew the textured state of a materral tvpreallv m the form of thm lms as an rntermedate state m between a eompletelv randomly onented fabncatlon proeess e g rolhng of thm sheet metal deposrtron ete and may affect the matenal properues by mtroduemg struetural anlsotxopy xture measurement ls also referred to as a pole gure as rt ls olten plotted m polar eoordnates eonsrsung of the alt and rotatron angles wth respeet to a glven ervtallographre onentatron A pole gure ls measured at a xed seattenng angle eonstant d spaemg and or P J lmmm w lHu Vr V db 7 or elevatron graphs wth zero angle m the eenter 5 here ls textunng effeet the ratro of dlffractlon peak sample 3 ary x drip dlffracnon peak whlle a powder sample made of lrregular pam les wlll show both 111 and 200 peaks wlth e roughly the same mueme Pole Fluure Measurement 10 The Scherrer Formula Ap soweean dlffracuon peak However thrs type of peak broademng ls negllglble when the ervstalhte slze ls larger than 200 nm Crystalhte slze ls a measure of the slze of a eoherentlv dlffractlng domam A A aggregates the same thmg as partrele slze He thlEh ls ealled the Seherrer formula KMB eose where t ls the averaged dlmenslon of ervstalhtes K ls the Seherrer eonstant somewhat arbrtrarv value that falls m the range 0 874 0 rtrs usually assumed to be 1 M5 the wavelength oerray and B ls the mtegral breadth of a re eeuon m radlans 29 loeated at 29 B ls 0 B cal t B3B2 11 Reciprocal Lattice httpwwwchembiouoguelphcaeducmatchm729recipvladhtm Direct Lattice or Real Space For a crystal lattice or direct lattice it is in the real space that is the position of each lattice point can be measured in the unit of length Ruvwuavbwc where R is a vector pointing from the origin to the lattice point u v w a b and c are the three axes of the unit cell and are also vectors n l 39 Lattice or Fourier Space kSpace Spaec For each real lattice we can always de ne a reciprocal lattice according to the following rules a 211bxcvol b 2ncxavol c 211aXbvol where vol is the volume of the parallelepiped unit cell associated with the real lattice vol a39bxc a b and c are the three axes of the unit cell associated with the reciprocal lattice and are also vectors It is clear that a is perpendicular to both b and c b is perpendicular to both a and c and c is perpendicular to both b and a In the reciprocal lattice the lattice points can be written as thl ha kbquot 10 thl is perpendicular to the hkl plane of the real lattice In general we have Ruvw 39 thl 211m where m is an integer It is not dif cult to verify that dth Z lthIl


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