PHYSICS OF MEDICAL IMAGING
PHYSICS OF MEDICAL IMAGING PHYS 5110
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This 0 page Class Notes was uploaded by Kendrick Wilderman on Sunday November 1, 2015. The Class Notes belongs to PHYS 5110 at Oklahoma State University taught by James Wicksted in Fall. Since its upload, it has received 20 views. For similar materials see /class/232920/phys-5110-oklahoma-state-university in Physics 2 at Oklahoma State University.
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Date Created: 11/01/15
Version 23 HighResolution Transmission Electron Microscopy 187 Version 23 8639Mxmw5 high resolution transmission electron microscopy HRTEM resolve object details smaller than 1nm 10399 m 0 image the interior of the specimen compare e g scanning tunneling microscopy atomic resolution however only at the surface 0 local compare e g X ray diffraction averaging statistical information 0 direct imaging HRTEM images can be gtgtintuitivelyltlt interpretable however severe restrictions apply see below 0 direct imaging of atom arrangements in projection crystals and quasi crystals in particular structural defects interfaces grain boundaries interphase interfaces stacking faults anti phase boundaries inversion domain boundaries crystals dislocations disclinations misfit dislocations however projection gt no individual point defects gt defects need to be parallel to the viewing direction 188 Version 23 0 techniques HRTEM high resolution transmission electron microscopy since 1980ies gtgtsteadyltlt improvement of conventional TEM widespread application focus of interest in this course holography high resolution electron holography is relatively new record amplitude and phase not just intensity of the electron wave in the image plane HAADF high angle angular dark field imaging z contrast new method scanning transmission electron microscopy requires annular HAADF detector HRTEM versus CTEM conventional TEM contrast formation in CTEM sometimes absorption contrast mostly dljf action contrast crystallites grains of different structure different phases or orientation distortions induced by particles dislocations variation of the scattering amplitude by stacking faults grain boundaries 189 CTEM bnthxeld imaging svecumer mmnsuy CTEM darkrfxeld imaging A spcmmer apErturs focal plzm Image mmnsuy Version 23 0 example Co precipitates in CuCo line through particle center bright field image distortion field introduced by the particles plane bending changes diffraction conditions in the matrix gt gtgtcoffee beanltlt contrast 0 principle of CTEM different specimen regions generate Bragg re ections of different in tensity contrast either Bragg re ections or transmitted beam do not con tribute to the image gt consequence interatomic spacings cannot be resolved 191 Version 23 Abbe Theory of Resolution 0 ideal object 9 crystal lattice with period 1 coherent illumination plane waves wavelength A diffraction path difference between waves emitted from neighboring slits must equal an integer number of wavelengths z A diffracted beams make angles p2 with the plane of the lattice and 3mm 0 Abbe only the dif acted beams carry information about the spacing 1 gt to image the lattice the optical system must at least include one dif fracted beam 2 1 192 Version 23 gt generalization to resolve an object under coherent illumination the image forma tion must include at least the first diffraction maximum of the object gt the aperture semi angle oc introduces a limit for the spatial resolution Object aperture semi angle Linse aperture Mnage gt resolution limit 5 smallest distance d that can be resolved 3mm Sinoc 2 5 4 Slnoc gt resolving interatomic distances requires larger aperture than CTEM im aging and interference of at least two different Bragg re ections 193 Version 23 gt transition from CTEM to HRTEM instrumental pre requisites for interference images illumination with a high degree of coherence small source small spread of the wavelength mechanics and electronics sufficiently stable electron lenses with small aberrations spherical and chromatic aberration astigmatism remove objective aperture ultra thin specimen to avoid absorption and inelastic electron scat tering destroys coherence orient the specimen to generate suitable Bragg re ections ideal HRTEM imaging neglecting lens aberrations A B spemmen thin Image IIWtE SIty If a rm contrasH 3 I quot gt high resolution but no contrast 194 Phase C onlnsl Cansxdertkmspemmenplaneeleth ananE lDynaabsarphan gt specimenjust mmdusss smzn Inme wme phase 511fo refrachve mdexe slssusseans psesnnsl x u mnmmnmumm spemmen WVW LIJN WVWN N W s J X2 3 EWM my EH J Wm p z i phdae ahlfmvZ a mum i 4 D 3939D gt Mn ml wumwm mlensny dlfferznce quotcm rasl39 express exxtwave 2 as sum af mudsmmvs W and attered Wave Version 23 phase of 1M x is shifted by 90 versus 1m wa wo4w4 gt intensity I l i Xllz IweX2 lwotxlz gt no contrast 0 must convert locally different phase shifts to locally different intensities contrast l intensity variation 9 if the optics introduces an additional phase shift of the scattered wave by 90 2 2 2 PM Xll Wolxl wslxll lt H 0X 0 problems the 90 phase shift cannot be realized in an ideal manner for TEM there exists no A 4 plate as for light optical microscopy in HRTEM the required phase shift is introduced by the spherical aberration of the objective lens defocusing of the objective lens the image which is obtained by interference of coherent electron waves often has a complex relationship with we the electron wave function at the exit surface of the specimen 39 consequence Correct interpretation of HRTEM images requires a quantita tive understanding of image formation 196 example AlMgAle mferhce HRTEM m lt110gt pmJECan 7 mmmsmpedEMARM 1250 Stuttgart AI MgAIZO a wk earewmckatams 7 A1Mg0 mm Calumns whxequot ar blatkquot7 7 glam szxbxl y af he mns m the spmel7 197 Version 23 ammo 0076 0 HRTEM interference a wave properties a llwave model of electron mandatory 0 exact treatment of electron interference quantum mechanics 0 however simpler wave optics also yields correct results apart from image rotation and adjustments of the wave length A Properties of Fast Electrons Physical Constants rest mass m0 91091 10 31 kg charge Q e 1602 1019 c kinetic energy E e U 1 eV 1602 1019 Nm velocity of light C 29979 108 m s 1 energy at rest E0 m0 C2 511 keV 0511 MeV Planck constant h 66256 10 34 N m s 198 Electrons in Motion property non relativistic relativistic I d p d v d p d Newtonslaw F dt nbdt Fdt t mo m nb mass 1 v2 energy EeU12m mc2E0EmOCZeU 1 v 26 v c 1 1 ve oc1 ty mo 1 E E02 momentum p nbV anE p mvg1IZEE0 E2 h h be de Broglie A wavelength p m ZEEO E 2 0 de Broglie wavelength as a function of the accelerating voltage Alpm1 30 25 20 15 105 o5 classical relativistic 200 400 600 800 10001200 kV 199 Version 23 velocity as a function of the accelerating voltage W01 0 veocity of light 08 06 04 02 39 39260 39466 39 660 39 866 39139039039039139239039039 U39k39 kinetic and potential energy of the electrons on their way through the microscope A W 539 H m a a 5 kineti e Ital energy 5 5 a E a O Q l cathode anode specimen camera specimen electrostatic potential varies gt diffraction free surfaces 9 average positive a inner potential U1 attractive for electrons a acceleration increase of the average velocity 9 wave length it becomes smaller 200 Version 23 fRAMWOffk 0FFR46 770391 Fresnel diffraction a Fraunhofer diffraction plane incident wave diffraction pattern at infinite distance alternatively diffraction pattern at back focal plane of a lens 1P0 object If We j if v 1Pf 3 V 39 1m 0 wave at the exit surface of the specimen we 0 focal plane rays that leave the specimen under the same diffraction an gle 6 intersect in the same point of the focal plane 0 note path difference between neighboring rays 201 Version 23 a phase shift to be taken into account when calculating the amplitude in the back focal plane 0 path difference for scattering of a plane incident wave at two scattering centers P and Q 3 U4 Hidquot ray through P has longer path than ray through Q path difference A59 uO Ar uAr tk k0Ar tq Ar uo u unit vectors q scattering vector l phase pp of the ray through P lags behind compared to phase pQ of the beam through Q phase difference 2n Apg arpp pQ 7Asg Zrcq39 Ar 202 Version 23 ray path for HRTEM see above for a given diffraction angle 6consider phase difference pg between ray through r and ray through origin of the object plane r 0 Ar rxy0 A Pg Pglrl 1 CPlrl 0 p0 focal plane phase of the ray emitted at r 0 with angle 0 pr focal plane phase of the ray emitted at r with angle 0 TEM small wavelength small diffraction angles gt q is approximately parallel to the focal plane 9 9 q Ik k0quot ZkSIn z 0 in the focal plane all rays with the same scattering vector q intersect in a point which is displaced from the origin in the direction of q 0 distance of the intersection point from the focus origin of focal plane f6 q a use q as coordinate vector in the focal plane a vector of length in the focal plane corresponds to q f gt amplitude at location q in the focal plane 1pfq fzper Expi gr d5 Se f er Exp2ni q r dzr 5e Se object plane 203 Version 23 gt weighed sum plane waves with the same scattering vector q and the same wave vector k but different phases 211i q r contribution of each wave is proportional to the amplitude of the wave function a the exit surface of the specimen Fourier transformation The wave function W in the back focal plane corresponds to the Fourier transform of the object wave function we at the exit surface of the specimen 1PfF1Je however this is strictly correct only for a perfect ideal lens real lenses see below wave function in the image plane consider point r x y of the object a conjugate point in the image plane image H Mr M magnification this implies 1 Mr 1Pe Vr amplitude 1p1r39 at point 1quot in the image 204 smee he rmag asah areaxs M2 largenhah the area self he rmage mtensxty must detrease as M2 gt arhphhrde detreases asM sum avereahmbumhs mm aupmhes anhe fatal plane akmg mm aunt he respethve phases phase dxfferente af nexghbanng rays that mtersett at 1mage pamU cansxder ray mtersedmg thh the fatal plane at q away mm he fat us q a snare q u r the path dxfferente can be expressed by the scalar pradr uctqr 14 Version 23 gt the phase shift introduced between the focal plane and the image plane exactly compensates the phase shift introduced between the object plane and the focal plane 211 pg 7As g 2nqr ideal lens different rays emerging from the same point of the object arrive at the image point without relative phase shifts gt amplitude at 1quot in the image plane ir39 V1f1pfq Exp 2ni qrd2q 5r pm Exp 2ni q d2q S f focal plane 0 often one neglects the magnification and the inversion of the image and expresses the image with respect to the coordinate system of the object letM 1 1quot r 2 Mr f w M Exp 2m q r d q sf 9 inverse Fourier transformation The wave function 111 in image plane corresponds to the in verse Fourier transform of the wave function Ipf 1m F3911pr a result for the complete imaging process ideal lens M 1 1quot r 206 Version 23 The wave function Ipi in the image plane corresponds to the inverse Fourier transform of the Fourier transform of the ob ject wave function we 1Pi F 1F1P e ideal lens M 1 1quot r perfect restoration of the object wave function in the image 0 quotrealquot imaging gt finite aperture 9 truncates the Fourier spectrum in the focal plane lens aberrations introduce additional phase shifts a the objective lens shifts the phases of the plane waves of which we is composed in a complex way versus each other gt Ipi does not exactly correspond to we limited coherence a attenuation of the interference diffuse background interpretation of real images requires in depth understanding of Fourier transformation 207 f39OMQfk 7RAIS FORMA7O General Properties 0 definition of the Fourier transform FIX a Fu EXp2ni ux dx 0 inverse Fourier transform F 1Fu5 x EXp Zati uxd U 0 extension to 11 dimensions FfXEFU1ffXEXpZatiU an Rn F1Fu5 X5 fFUEXp Zniqunu Rn convention factor 211 in the exponent sometimes neglected in quantum mechanics for example but this requires to multiply either F or F 1 by factor 2n 1 or each F and F1 1 by 21112 208 Version 23 properties of the Fourier transform real space Fourier space M Fu alfx ng aFu bQu fx Fu 4x a Fu Exp2ci and M F rm zniumu rt 1 zni uquot Fu Exp 2ci ax 6u a cap a Exp2ni au M six Fu39qu u391d tram Gnu iv qx x1dx39ltfggtx Fu GM 209 Della Function defmman af he ammuan actually a dxstnbuh anquot smte n is nut de ned m every argument 2 vxsualxzahan deuamnmm Canespand a anamaldxskxbutxanxn the hme af mm 5th shndard devxahan Version 23 1 1 x 2 6 X Iim Ex a gt0L 04271 p 2 O J 0 under an integral the delta function selects a value of the integrand function f6x adxdx9af6X adx Qa 0 concerning Fourier transforms the following relation is particularly important Exp2ni ux d X 6u Convolution 0 definition of the convolution integral of two functions x and gx f j x39QX X39dx39 substitution y39 x x dx dy reveals that the convolution is symmet ric with respect to exchanging the argument functions mm ffx ygydy gyfx ydy g mm 0 interpretation spread function f with the function obtained by inverting function g 211 Exampl nfa Cnnvnln nn
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