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Introduction to Electron Microscopy

by: Keshaun Ferry

Introduction to Electron Microscopy MSE 421

Marketplace > Boise State University > Material Science and Engineering > MSE 421 > Introduction to Electron Microscopy
Keshaun Ferry
GPA 3.99

Frederick Ubic

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Frederick Ubic
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This 47 page Class Notes was uploaded by Keshaun Ferry on Saturday October 3, 2015. The Class Notes belongs to MSE 421 at Boise State University taught by Frederick Ubic in Fall. Since its upload, it has received 11 views. For similar materials see /class/217985/mse-421-boise-state-university in Material Science and Engineering at Boise State University.


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Date Created: 10/03/15
MSE 42152 Introduction to Electron Microscopy IV The Transmission Electron Microscope A The instrument The first transmission electron microscope was invented in 1933 by Max Knoll and Ernst Ruska at the Technical College in Berlin The transmission electron microscope is the electronic cousin of the transmission light microscope a beam of electrons passes through a thin sample followed by a series of lenses forming a highly magnified image of the sample on a screen Knoll and Ruska found that they could focus their electron beam with a magnetic lens that was produced by sending the beam through a current carrying coil Modern transmission electron microscopes usually consist of a beam column that is about 25m tall with a diameter of about 30cm and they are able to achieve a resolution of about 2A however the addition of aberration correctors has more than doubled this performance Source First condenser lens Cl Spot Size First J 390 J source hm Second condenser lens C2 Brightness Condenser aperture Specinen Objective lens Back focalplane Objective aperture First image plane Selected area aperture Intermediate or diffraction lens Second image 11 ZF Projeaor hinge S Screen R Ubic MSE 42152 Introduction to Electron Microscopy 1 Electron gun At the top of the column the electron gunT delivers high energy electrons to the instrument Thermionic guns tungsten or LaBg are the most common types The appropriate electron energy depends on the nature of the specimen and the kind of information required Higher electron energies allow thicker samples to be analysed and due to their smaller wavelengths increase the resolution possible however it is rare now to see TEMs which operate at energies greater than 200 keV The introduction of field emission guns and improvements in lens design have largely made higher energy microscopes unnecessary for high resolution Additionally higher energy electrons cause increasing amounts of damage to samples Biological samples in particular require lower operating voltages 2 Condenser lens system The condenser lens system then acts to control and reduce the diameter of this beam The first condenser C1 lens or spot size is a strong lens which demagnifies the image of the electron source by about X1 100 to give a small point source at the crossover which is more coherent than the large 50 um diameter filament tip The second condenser C2 lens brightness or intensity is a weaker lens which projects the demagnified source image onto the specimen with a magnification of X2 giving an overall demagnification of X150 This lens controls the spread of illumination on the screen The condenser aperture located just below the condenser lenses sometimes between them collimates makes parallel the electron beam and modifies its intensity Three parameters control the operation of the electron gun the accelerating voltage the lament current and hence its temperature and the bias voltage on the Wehnelt cap The filament current controls the filament tip temperature and hence the number of electrons emitted The emission is maximised by saturating the filament ie increasing the filament current until the number of electrons emitted no longer increases Ramping the filament current to saturation is controlled electronically on the JEOL 2100 and should not need to be adjusted by users The gun bias controls the bias resistor setting which controls the current passing between the high voltage system and earth At low bias values the negative potential of the Wehnelt compared to the filament is ineffective therefore the electrons are accelerated towards the anode with relatively little focusing The beam is consequently spread and appears weak on the screen As the bias is increased the focusing action improves so that the effective beam brightness increases however above a certain value the Wehnelt is so negative with respect to the filament that the brightness starts to decrease because electrons are prevented from being emitted from the filament or if they are emitted are repelled back towards the filament The distance between the Wehnelt and the filament is obviously important in determining the point at which the optimum beam brightness is obtained For this reason if the filament has recently been changed a slightly different emission setting may be required l Never try taking one of these through airport security R Ubic MSE 42152 Introduction to Electron Microscopy 3 Specimen chamber The specimen itself sits below the condenser essentially within the field of the objective lens It is held in place by a sample holderspecimen rod eg single tilt double tilt high tilt etc Samples themselves are thin 3mm disks Samples can be translated in the xy plane by up to 2mm less in the z direction but once in position they must be held absolutely still Movements of more than about 1 As depending on the magnification used would result in blurs when trying to record an image this is equivalent to 1mm in four months Tilting about x and sometimes y is also possible and the extent of tilt allowed is governed by both the holder and the geometry of the objective lens pole piece It is also important that the specimen does not appear to move when it is tilted which can only be achieved if the specimen is located on the axis of tilt which is unaffected by z shifts which must intercept the optic axis The eucentric Latin well centred height can normally be found for x tilting but is not generally possible to achieve for y 4 Objective and intermediate lenses Because the objective lens has a large magnification factor the back focal plane is very close to the lens itself In fact we can think of the back focal plane being co incident with the lens in practice it is about 1 mm below the specimen which itself is essentially sitting inside the objective lens Mounted in the back focal plane is the objective aperture the middle aperture on the column The selected area aperture sits in the first image plane below the specimen which is below both the objective lens and the objective aperture The objective aperture filters out beams which come out of the specimen in a particular range of angles The selected area aperture filters out beams which have come from a particular set of positions in the specimen It is important to emphasise that the difference between diffraction mode and image mode is which plane we choose to map onto the phosphor screen By changing the excitation of the first projector lens also called the intermediate lens or diffraction lens we can form either an image or a diffraction pattern which is an image of whatever is going on at the back focal plane however when we form a diffraction pattern we are still looking through the selected area aperture so the diffraction pattern we see on the screen is not really what s at the back focal plane It s the diffraction pattern we would see from the part of the specimen which has been selected by the selected area aperture in the image plane R Ubic MSE 42152 Introduction to Electron Microscopy Source First condemer lens Cl Spot Size First dernagnj ed source gt Second condenser lens C2 Brightness Condenser aperture Specimen Objective lens Back Objective aperture Selected arw aperture Intermediate or diffraction lem Second Projector Lenses Screen diffraction pattern 5 Practical notes The value of the filament saturation will generally decrease with time as the filament material W LaBG etc evaporates making the filament smaller Very low settings of the filament control are often a sign of imminent failure On the JEOL 2100 the optimum emission setting is often about 63 Never operate the microscope above this pre set value Contamination poisoning of the filament is also a potential problem It is very important to maintain a good vacuum in the gun area Contamination causes high voltage break down arcing which is manifest as voltage instability or complete shut down caused by the loss of vacuum as particles of contamination are vaporised In addition a poor vacuum might cause oxidation of a W filament Only ever turn on the electron beam when there is a good vacuum in the column 3 25 x 10 5 Pa Similarly the gun area should be allowed to cool before the column is brought up to atmospheric pressure this should only be done by BSCMC personnel All parts of the gun area should be handled using appropriate gloves There is no guarantee that when the filament has been changed that it points directly down the optic axis Clearly the beam must be on the optic axis to maximise brightness On the JEOL 2100 the beam is centred using the gun tilt controls R Ubic MSE 42152 Introduction to Electron Microscopy 6 Vacuum System Electron microscopes are operated under vacuum for four reasons Because electrons scatter easily the mean free path of electrons at atmospheric pressure is only about 1cm however at 10 6 Pa they can travel about 65m The vacuum acts as an insulator between the anode and cathode filament and in the area around the field emitters thus hindering unwanted electrical discharge in the electron gun The elimination of oxygen around the filament prevents it from being oxidised and eventually burning out Reduced interaction between the electron beam and gas molecules decreases contamination on the sample The SI unit of pressure is the Pascal Pa Unfortunately there are several other units for pressure also in common use For convenience a conversion chait is shown below A further complication is that despite the fact that vacuums necessarily require low pressures we perversely refer to very low pressures as high vacuums latm 1bar I760mmHg 760Torr 101325Pa 14696psi Different levels of vacuum are required for different parts of the microscope The gun might require 10 9 Pa while the specimen can be at 10 6 Pa and the projection chamber and camera can be at 10395 Pa There are three main types of vacuum pump roughing pumps high vacuum pumps which need backing and high vacuum pumps which do not need backing Alternatively vacuums can be categorise vacuums as rough 100 01 Pa low 10 1 10 4 Pa high 10 4 10 7 Pa or ultrahigh lt 10 7 Pa u Roughing Pump Rotary Pump A rotary pump can pump down from atmospheric pressure but can only reach a rather modest vacuum at best about 1 01 Pa They consist of a belt driven eccentrically mounted reciprocating mechanism suck air through an inlet valve into a chamber and expels it through an exit valve Oil is typically used as a lubricant and gas seal although expensive oil free models are also available The high pressure side can operate at atmospheric pressures while the low pressure side is usually limited by the vapour pressure of the pump oil used These pumps are generally reliable inexpensive noisy and dirty R Ubic MSE 42152 Introduction to Electron Microscopy Exhaust Inlet It is used to pump the chamber from air atmospheric pressure when necessary to pump the specimen transfer chamber also from atmospheric pressure and to back the diffusion pump Because it is used to do so many things it is usually attached to a roughing manifold which is a pipe with lots of other pipes coming off it By opening and closing various valves off the roughing manifold the roughing pump can be switched from one role to another b Diffusion Pump The most common type of pump for use in high vacuum applications is the diffusion pump or more properly vapor jet pump Diffusion pumps are one of the oldest and most reliable ways of creating a vacuum down to 1039 Pa at room temperature hlgh vacuum mum mp to prevent coolnu mayhem nq PW mp5 ml vqpcv lets in mechanical I Dump A diffusion pump cannot begin its work with full atmospheric pressure inside the chamber Instead an ancillary mechanical rotary pump or forepump capable of a modest level of pumping first brings the pressure inside the chamber down to about 10391 Pa At this point the diffusion pump takes over to create a vacuum ranging from 10 1 to 10398 Pa Since the diffusion R Ubic MSE 42152 Introduction to Electron Microscopy pump cannot exhaust directly to atmospheric pressure the forepump is used to maintain proper discharge pressure conditions Diffusion pumps use a high velocity stream of molecules to kinetically trap random gas molecules from the vacuum system that blunder into the stream They basically consist of a stainless steel chamber containing vertically stacked cone shaped jet assemblies each of which will support pressure ratios of approximately 101 or greater Typically there are three jet assemblies of diminishing sizes with the largest at the bottom The pressure on the low pressure side is typically around 10 4 Pa or so while the maximum pressure on the high pressure side is typically on the order of around 7 Pa At the base of the chamber is a pool of a specialized type of oil having a low vapour pressure The oil is heated to boiling by an electric heater beneath the oor of the chamber The vaporized oil moves upward and is expelled through the jets in the various assemblies The high energy oil droplets travel downward in the space between the jet assemblies and the chamber wall at speeds up to 335 ms l The droplets may actually exceed the speed of sound but thankfully there is no sonic boom because the molecules in the partial vacuum are too far apart to transmit the sound energy The capture efficiency of the vapour jet depends on its density velocity and molecular weight The high velocity jet collides with gas molecules that happen to enter it due to their thermal motion This typically imparts a downward motion to the molecules and transports them towards the pump outlet creating higher vacuum at the top of the pump chamber which is connected to the microscope column At the base of the chamber the condensed molecules of atmospheric gases are removed by the forepump while the condensed oil begins another cycle Water circulated through coils on the outside of the chamber cool it to prevent thermal runaway and permit operation over long periods of time The first designs of diffusion pumps go back to 1915 when the pump was invented by Irving Langmuir The original diffusion pump uid used was mercury which could withstand elevated temperatures but had the disadvantage of being toxic and volatile Indeed the oil in a modern diffusion pump would contaminate the vacuum in the TEM if oil vapor were to escape into the column To minimize this problem several kinds of synthetic non hydrocarbon oils with low vapour pressures based on silicones esters per uorals and polyphenyl ethers can be used The polyphenyl ether Santovac 5 and the per uorinated polyethers Fomblin and Krytox have been worldwide standards for some time These oils are expensive 1 2 per ml and may eventually crack or char oxidise if subjected to both high temperature and pressure forming a tar like substance that can be extremely difficult to remove Exactly what happens when diffusion pump oil cracks depends on the type of oil in the chamber and the temperature at which it breaks down Breakdowns with hydrocarbon based oils are hardest to clean up The residue is much like tar and is tenacious Silicone based oils leave a residue that is somewhat easier to remove Oils containing per uorals decompose to form uorine compounds that can be extremely toxic and very damaging to the aluminum jet assemblies The least messy and least damaging are the polyphenyl ethers in part because their breakdown temperature is much higher 350 C vs 300 C and in part because polyphenyl ethers tend to decompose into small non toxic molecules such as water and carbon dioxide Thankfully breakdowns involving polyphenyl ethers are rare R Ubic MSE 42152 Introduction to Electron Microscopy The diffusion pump is usually positioned at the bottom of the column near the back It is used to pump the viewing chamber and the photographic lm casement There is usually a large empty volume between the back end of the diffusion pump and the roughing manifold so that when the microscope is being used the roughing pump can be switched off it creates a lot of vibration The empty volume is very slowly filled with the gas being pumped out of the chamber by the diffusion pump From time to time the rotary pump will come on to empty the backing line This is the loud mechanical noise you hear every few hours If the rotary pump or cooling system fails oil can backstream into the microscope In all diffusion pumps a small amount of backstreaming occurs A liquid nitrogen cryotrap is sometimes used to remove oil particles before they can reach the microscope column the JEOL 2100 does not have this feature but for most applications of microscopy the use of polyphenyl ethers makes a cryotrap unnecessary since the high purity of polyphenyl ethers minimizes backstreaming Although they have been replaced in some applications by more advanced designs such as cryopumps or ion pumps vacuum diffusion pumps are still plentiful because they have several advantages they are reliable they are simple in design they run without noise or vibration and they are relatively inexpensive to operate and maintain In fact diffusion pumping is still the most economical means of creating high vacuum environments 6 anbomolecnlar PumpTurbo Pump Turbo pumps are basically like jet engines Multiple stages of rotating blades rotors and xed blades stators which act as axial compressors turbines create a powerful downdraft and force air out of the microscope The blades are designed like airfoils to enhance the ow of gas and the turbines spin very high speed typically 20000 50000 rpm so are more likely to fail than diffusion pumps The typical description of a turbo pump failure is of a brief scream of anguished metal followed by the entire vacuum system being filled with aluminium confetti Given the high speeds involved a foreign particle faulty bearings etc will cause a rather spectacular failure They also do not contain oil but the moving parts can introduce some vibration although the best turbo pumps are very quiet and almost vibration free In addition the tight tolerances required in the machining of the blades means that these pumps are generally expensive 9 Motor J39x f Slillw Mum r R Ubic MSE 421521 Introduction to Electron Microscopy These pumps usually require a rough backing pump although in theory they can operate slowly from atmosphere increasing speed as the pressure is lowered An ultimate vacuum 1 pressure of 10395 Pa is achievab e Rotary di usion and turbo pumps are all exhaust pumps they pull in air from one end and expel it from the other d on PumpIon Getter PumpSputter Ion Pump Like diffusion pumps ion pumps have no moving parts Unlike diffusion pumps ion pumps contain no oil so contamination of the column is not an issue They rely instead on the ionisation process to remove air The sputter ion pump works by ionizing gas that falls within a magnetically con ned cold cathode then reacting these with or burying them in plates of titanium The ion pump consists of two parallel plates 3 of negativelycharged titanium cathode and cylindrical anodes 4 all surrounded by large pennanent magnets l A 5kV potential difference is maintained between the cathode and anode The titanium cathode emits electrons which spiral in the magnetic eld and ionise air molecules which are then attracted to the cathode As titanium is a very reactive metal most species will simply form a compound with a conveniently low vapour pressure effectively removing them from the pumped volume The nonreactive species eg Ar will tend to simply get embedded in the Ti until the power is removed In addition like tiny meteorites these energetic ions sputter Ti atoms from the surface of the cathode which mainly condense on the anode trapping more gas atoms by chemisorption In this way Ti acts as a getter materla The smaller the ionic current between electrodes the lower the vacuum so the pump acts as its wn vacuum gauge A typical ion pump will actually start to function around 10393 Pa or so but this is really abusive Better practice requires that the chamber already be pumped down with a diffusion pump to about 10394 Pa Ultimate vacuum pressures as low as 10391 Pa are possible in a TEM but for SEM work open vacuum system rubber seals students abusing them 10395 to 10397 R Ubic MSE421521 IrLraducziart ta Electart Micrascapy Pa is more common It is common to add ion pumps directly to the stage or gun chamber of a TEM to concentrate their pumping action on these important parts e CryogenidAa39sorption Pumps Cryogenic pumps are also oilrfree and rely on liquid nitrogen to cool molecular sieves with large surface areas The cold surface removed air molecules from ambient pressure down to 10394 Pa nu man was mumsmu Imn till a mum The anticontamination device ACD is a cold trap that immediately surrounds the specimen in most TEMs and acts as a minicryopump trapping volatiles as they are produced from interaction of the beam with the specimen The ACD provides an alternative site to your specimen for condensation of residual contamination in your vacuum This is an important way to keep the internal components of the TEM clean Once the beam is off and the trap warms up the trapped gasses are released and removed via the normal pumping system It is important that the sample be removed before this happens on pumps and crypumps are trapping pumps They keep the air molecules within them and release them when turned a or warmed up respectively f Vacuum Tube Gauge Pirani Gauge A Pirani gauge named for its German inventor Marcello Pirani in 1906 uses a platinum wire in a sealed vacuum tube and a second wire in specimen chamber A voltage of 6712 V is applied to heat the wires A heated metal filament suspended in a gas will lose heat to the gas as its molecules collide with the wire removing heat and accelerating in the process If the gas pressure is reduced the number of molecules present will fall proportionately and the wire will rise in temperature due to the reduced cooling effect So the hotter the wire the better the vacuum The vacuum is measured indirectly by the current which ows through the wire The higher the temperature of the wire the greater its electrical resistance and the less current will ow The difference in current ow between the known vacuum in the closed tube and the unknown vacuum in the instrument gives an indication of the vacuum in the chamber In many systems the wire is maintained at a constant temperature and the current required to achieve this is therefore a measure of the vacuum being studied R Ubic MSE 421521 Introductton to Electron Mtcroscopy Gauge Reference Tube Tube A Pirani gauges are useful for pressures between about 3970 Pa to 001 Pa The thermal conductivity of the gas may effect the readout from the meter and therefore the apparatus may need calibrating before accurate readings are obtainable g Ion Discharge Gauge Penning Gauge Penning gauges are the most sensitive gauges for very low pressures 10391 to 10398 Pa They sense pressure indirectly by bombarding a gas with electrons thermionically emitted from a filament and then measuring the current produced by the positive ions created The ions are attracted to the cathode known as the collector which is biased by several kV with respect to the anode or grid The current in the collector is proportional to the rate ofionization which is a function of the pressure in the s stem Fewer ions will be produced by lower density gases Hence measuring the collector current gives the gas pressure can ntmsm we wrtttlmen ammua va wmmm The calibration of a Penning gauge is diffith and dependent on the nature of the gases being measured which is not always lcnown R th MSE 42152 Introduction to Electron Microscopy 7 Alignment See the latest version of the JEOL 2100 HR TEM Standard Operating Procedure available in the BSCMC B Image Contrast in the TEM TEM images are simply magnified images of the electron intensity on the bottom surface of the specimen and contrast arises only if the intensity varies significantly from one region to another 1 Absorption contrast We have already examined this mechanism whereby samples which are thicker denser or with higher atomic number allow fewer electrons to pass through they absorb and scatter more of them It applies to both amorphous and crystalline specimens and is used extensively by biologists who call it mass thickness contrast a poor name as it ignores the atomic number component 7 especially as they usually stain their samples with a heavy metal to decorate features of interest In crystalline samples this contrast mechanism is usually swamped by others 2 Diffraction Contrast Diffraction contrast is simply a function of the diffraction conditions It is the method most commonly used to study crystal defects like dislocations stacking faults precipitates etc It is the mechanism which explains extinction bend contours and thickness inges 83 4 100 0 R Ubic MSE 42152 Introduction to Electron Microscopy lt Sample gt i i 517 Objective Aperture lt Electron Beam Objective aperture removes Objective aperture removes scattered off axis beams diffracted beams a Perfect crystals If a beam of electrons is incident upon a crystal of thickness t the diffracted intensity for a given re ection can be calculated as 2 I ns1nnts 3 g m where s is the deviation parameter the deviation of the g vector from the Ewald sphere or the distance in reciprocal space from the exact Bragg condition and g is a material constant for a given g called the extinction distance Although the maximum diffracted intensity will always be for s 0 some diffraction will also occur for s i 0 The thinner a crystal is the further it can deviate from the Bragg condition and still diffract This rule which is also true for other forms of diffraction cg XRD can be thought of with reference to the shape and size of the rel rods IIC7 in reciprocal space and their intersections with the Ewald sphere which we have already discussed As the crystal thickness increases the rel rods become shorter and so make fewer intersections with the Ewald sphere 7 there are fewer re ections in such diffraction patterns Absorption and increased inelastic scattering also means such patterns show more diffuse scattering and eventually when the thickness is too great no intensity at all gets through The extinction distance is given by TEVCCOSGB XF 8 g where V6 is the volume of the unit cell GB is the Bragg angle 7 is the electron wavelength and F g is the structure factor The amount of contrast in a specimen the apparent size of a defect the R Ubic MSE 421521 Innadmmn ta Eleman Mlcmxapy n m 1m fnngex 4h 4 quotdetelmmedby ix ln genelol harp Image are only obtolned whenig l mall to few ten olnm Accoldlng to the g 1 Thexe two equatlon above ln oldel to mlmmlxe 3 en muxt be made mall and F alge sandman are only atleled fol lowrlndex Ie ectlom A plot of Ig 39 not peclmen through the Bragg tondltlon Such o rocking cuwe l hown below The yellow Ammt VFFV Intensity of Di racted Beam o at o o 1 Deviation Parameter nm A mu lnge Juxt llke o rocking ouwe The bend or extmctlon contour which xexult o en re ect the m F V Darkflela39 bend cumulus m ijszoa Bend contours can eonly be dlxtmgulxhed from actual uyxtallme defect by mung the peclmen Bend contour wlll xeem to wwp awn the peclmen a lt l tllted and dlffexent x Ublc MSE 421521 Introduction to Electron Microscopy planes are brought into the Bragg condition Actual defects will not appear to move like this although their appearance will change as the specimen is tilted From the equation for Ig above it can be shown that Ig varies periodically with t becoming zero each time ts is an integer A typical wedge shaped specimen shows thickness fringes The figure below demonstrates this effect The yellow curve represents the kinematical theory which predicts that intensity simply rises with thickness The orange curve represents the more realistic dynamical case and the red curve shows the additional effect of absorption Intensity of Diffracted Beam 0 100 200 300 400 Thickness nm If a crystalline specimen is thicker than about one third the extinction distance then there will be appreciable interaction between the electron beams as they travel through it Such interaction renders the kinematical theory inadequate and a dynamical theory is needed The most straightforward form of this theory only considers interactions between the transmitted beam and one diffracted beam defined by the reciprocal lattice vector g The HowieWhelan equations R Ubic MSE 42152 Introduction to Electron Microscopy can be used to describe the amplitude of both the diffracted og and undiffracted 10 beams as a function of z the distance through the crystal dp in in 7 ipo 7tp exp2msz dz in g 3 dPg in in 7 oex 27u39sz 7 dz 0 p p 0 pg The first term arises from the scattering from the transmitted beam and the second from the diffracted beam The amplitude of each wave changes with z due to contributions from the other The possibility of absorption high angle inelastic scattering can be accounted for by replacing l with the complex parameter l iE In order to calculate the intensity the equations must be integrated over the entire thickness to give 1 and pg at the exit surface of the specimen The bright field intensity is then given by potp and the dark field intensity by pgp where indicates the complex conjugate The g RSiH7US392 intensity of the diffracted beam is then gm39 which is exactly the same as the kinematical solution except for the use of the e ective deviation parameter s39 where b Defects A defect which disturbs the crystal planes will locally modify the deviation parameter In this case the Howie Whelan equations can be re written as dp in in ipoip exp2mszgR dz Q Q 3 damp in in 7tpo exp 2niszgR 7tp dz i0 i0 8 where R is the displacement of atoms from their lattice positions due to the defect cg a Burgers vector and g 0 R modifies the product sz When g 0 R 0 or an integer the defect has no effect on the diffracting planes and so it is invisible This condition is called the invisibility criterion and occurs when g is perpendicular to R Stacking faults grain boundaries and phase R Ubic MSE 42152 Introduction to Electron Microscopy boundaries can all be studied in this way The larger g 0 R is the more obviously visible will be the defect Stacking faults will result in defect fringes which are identical in contrast in both bright field and dark field images above the fault but complementary below it If one grain oriented in a two beam condition overlaps another which is not then the first grain can show thickness fringes just as if it were a tapered single crystal which it is Such fringes will be parallel to the intersection of the grain boundary with the surface and can easily be distinguished from stacking fault fringes by dark field imaging in which case only the strongly diffracting grain will appear bright When both crystals are strongly diffracting moir fringes may appear These are common in images of thin crystalline materials deposited on each other where two crystals are diffracting with slightly different values of g or are rotated slightly with respect to each other d2 d 1 a N Parallel Moir Pattern lII39HIIHIHWIHWWW amp N Rotation Moir Patte rn In the case of parallel lattices the net effect is a set of fringes running perpendicular to g with a spacing and in the case of lattices only rotated by an angle 0c the spacing is d D 1 r 2sin a R Ubic MSE 421521 Introduction to Electron Mrcmmpy In general lattice could be rotated and have different rpacingr in which care dldl d dgildldzcorn Contrart can alro arire from dislocations Planer around the dirlocation core are urually dirtorted quite Severely to if a cryrtal ii oriented into a twobeam condition with x U the rt 4 L t t t t i t m i mwo imaging conditionr can be found for which the dirlocation ii invirible gibl then b can be determined by taking the crorr product hglxgz For edge or mixed dirlocationr thir condition it rlightly lerr Straightforward The analyrir for partial dirlocationr ii alro Somewhat more complicated Dulowtim in nickel a BrightField and Dark Fielilmagiug Typically to rimplify TBM analyrir we require only one ret of planer at the Bragg condition to that there it only one ret of reflecting planer To achieve thir it it necerrary to orient the rample in ruch a way that only a ringle diffraction rpot ii excited Thir reflection together with the tranrmitted beam which it the undiffracted intenrity and lS alwayr prerent in diffraction patternr giver ur what it called atwoleam condition MSE 42152 Introduction to Electron Microscopy B 110 g320 111 g 002 g g 113 For a bright eld image the objective aperture is placed around the transmitted beam excluding the diffracted one The vector from the transmitted beam 000 to the diffracted one kid is called g For a darkfield image the incident electron beam is tilted so that the diffracted beam is parallel to the optic axis by using de ectors In fact it is hkl which is brought to the centre and the associated vector is therefore g I I object 7 VA objective lens T objective aperture It Idzlo39lt Bright field Dark field R1 Ubic MSE 42152 Immdmnmr tn Elemun Mlcvuxcupy 111 layering visible Dark Field 3 Phase contrast and hi heresolution irna in Imp L a r c a waves 4 none or 39 k39 39 aperture If spots along a systematic row are allowed through a lattice image is formed Such ima es can be used to show the extent of crystallisation of a grain un film or the habit plane of planar defects If more diffracted beams are allowed to contribute then a strutture rmage can fo ed Interpreting such images is not trivral and requires lmowled e o specimen thickness defocus and TEM resolution itself dependent on c5 of the objective lens R Ublc MSE 421521 Introduction to Electron Microscopy and wavelength To fully understand highresolution structural images a series of images must be obtained and compared to a simulated series of images generated by inputting the likely crystal structure into a sophisticated software package How many of you were told perhaps during your first science lessons at school that atoms are too small to see Indeed with typical diameters of 103910 m atoms were for a long time considered articles of faith by many scientists We cannot see atoms we are later taught because diffraction places a fundamental limit on the resolution of an image Roughly speaking we cannot see anything smaller than the wavelength of the light used to produce the image and since the wavelength of visible light is some 10000 times larger than the typical distance between two atoms we cannot see individual atoms however other forms of electromagnetic radiation have much shorter wavelengths than visible light The xrays used in crystallography for example have wavelengths of less than a nanometre The problem is that it is extremely difficult to focus xrays Luckily quantum mechanics provides an alternative way to view the microscopic world subatomic particles like electrons In diffraction amplitude contrast imaging in general only one beam is used to form the image ie the transmitted beam in BF or a diffracted beam in DF so that any phase relationship between the beams is lost If the transmitted and diffracted beams can be made to recombine thus preserving their amplitudes and phases a periodic fringe pattern lattice image of the diffracting planes is formed by phase interference between the two beams This technique usually requires a large objective aperture In order to resolve atoms it is necessary to have the smallest possible objective lens focal length and aberration coefficients If the diffracted spots from several systematic rows at a zone axis are included in the objective aperture and used to form the image a structure image of individual rows of atoms may be resolved The principle in this case is the same that of the Abbe theory for gratings in optics R Ubic MSE 42152 Introduction to Electron Microscopy Overlapping lt10 gt SADPs o SF2TCl207 showing 020 spacings d 13599 fl of two twin variants sharing a common 15 habit plane Both highresolution lattice and structural imaging are examples of phase imaging Considering the case of just two beams to form the image in order for the effect of recombining two outof phase waves to be Visible in the highresolution electron micrograph HREM the amplitudes of the resultant wave sum of transmitted and diffracted beams must be different to that of the transmitted beam The transmitted beam will have a much stronger intensity and so constitutes the background of the image The greatest change in amplitude and therefore the greatest contrast in the image corresponds to the case where the two beams are perfectly in phase The result is atoms which appear white on a black background A 900 phase shift causes no change in intensity and therefore no contrast A 1800 phase shift results in a sum wave which is lower in amplitude than the transmitted wave yielding black atoms on a white background The microscope introduces some additional phase shifts which complicate this simplistic picture R Ubic MSE 42152 Introduction to Electron Microscopy phase di erence of0 incident scattered sum white atoms dark background t t t t t 0 TE 21c 31c 41c 51c 61c phase shift of90 1nc1dent scattered sum no contrast 0 TE 21c 31c 41c 51c 61c Rt Ubic MSE 42152 Introduction to Electron Microscopy phase shi of180 incident scattered sum dork atoms white background 0 TE 21c 31c 41c 51c 61c A large objective aperture is required to allow the beams to interfere and form the image The image must also be off the objective focus position to utilise Fresnel defocus diffraction To achieve high resolution a large voltage is usually necessary 2 200kV The proper conditions for forming structure images which best represent the structure of the specimen must be chosen by comparing images obtained in the microscope with computed images based on the dynamical theory e g multislice calculations For interpretable images we need 1 very thin about 5nm specimens If they are too thick inelastic scattering degrades the phase contrast information 2 to be at a zone axis so that many beams are available and the crystallographic information is interpretable Only those diffracted beams which correspond to distances within the point resolution of the HREM will contribute to the image 3 precise alignment of the electron beam down the optic axis Also any defects must lie along the beam direction 4 coherent illumination ie LaBG filament or PEG 5 a specific value of objective defocus In order to establish this value the quantitative defocus associated with each click of the objective focus control must be calibrated As the image obtained is a function of a number of variables these must be defined and calibrated in order to interpret the image 1 Sample thickness 2 Objective lens defocus R Ubic MSE 421521 Introduction to Electron Microscopy 3 Microscope parameters like kV Cs Cc etc 4 Number and type of beams included in the objective aperture 5 Beam tilt Interpreting images of regions inevitably of varying thickness and at different values of defocus is complex because the objective lens is imperfect and itself introduces phase shifts into high angle information This problem is formally expressed in terms of the contrast transfer function CTF of the objective lens which is effectively a map of the phase shift including the microscope effects It gives a measure of the atom contrast 1 2 bright 0 2 invisible l 2 dark as a function of atom spacing but a detailed description of the CTF is beyond the scope of this class 4 Sample Preparation Preparing samples for use in the TEM is not a trivial exercise In order to be inserted into the microscope the sample must be a 3mm diameter disk thin enough to be electron transparent o Inorganic samples The most common way of preparing electrically conductive materials like metals is electropolishing or jet polishing The principle of this method is that the specimen is made the anode in an electrolytic cell When a small voltage is applied metal is dissolved from the anode and deposited on the cathode The thin specimen becomes both thinner and smoother and eventually a hole appears in the sample The regions around the hole should be thin enough for viewing in the TEM Automated electropolishers produce 3mm disk samples consisting of relatively thick rims supporting the thin central region Such samples made in this way are often called foils Ceramic samples can be prepared in a number of ways although as they are non conductive electropolishing is not one of them Starting from a bulk material an arbitrarily thin slice can be cut away with a diamond wafering blade Once a thin piece is obtained one side is polished and glued to a glass slide The exposed side is then ground down with SiC paper until it is Z lOOum thick Next a 3mm disk can be ultrasonically machined from it using an ultrasonic drill and some ceramic abrasive powder This disk will then have to be dimpled which involves using polishing paste on a rotating wheel to gently wear a bowl shaped recess into one side of the disk This process reduces the thickness of the sample and makes it ready for the final 7 and slowest 7 part of the procedure ion beam milling If a beam of energetic ions is directed at a solid atoms can be knocked out of the solid in the process known as sputtering The ion guns commonly used in ion beam thinners generate a plasma by stripping electrons from low pressure argon atoms in a high electric field The electric field also accelerates the ions through an aperture in the cathode producing a beam which is directed onto the TEM sample Two guns are usually used so that the sample can be thinned from both sides simultaneously and the sample is rotated to prevent surface roughness from developing Again eventually a hole will appear in the sample and the region around this hole should be thin enough for examination in the TEM R Ubic MSE 421521 Introduction to Electron Microscopy If no ultrasonic drill is available or if time is short it is possible to thin the sample by hand to the point where it starts to fall apart Once the sample starts to fall apart or wear unevenly it is polished gently A 3mm support ring usually copper is glued to the surface and the sample is oated off the glass slide by applying heat The sample is typically so delicate by this point that the excess around the copper ring can simply be broken off and the ceramic copper assembly can be placed in the ion beam thinner with or without dimpling 17 Organic samples The preparation of organic materials for the TEM is very complex and typically involves the use of several very toxic materials The method starts by fixing the sample by soaking it for 30mins to two hours in chemicals like paraformaldehyde glutaraldehyde and CaCl in cacodylate buffer The sample is then washed for 30 60 mins in a CaCl containing cacodylate solution and postfixed for one two hours in osmium tetroxide Potassium ferricyanide can also be added for glycogen and stronger stained lipid membranes The sample is then washed with distilled water treated in the dark for one two hours with aqueous uranyl acetate and washed again at which stage the sample is finally fixed The next step is to dehydrate the sample In order to minimise lipid loss and shrinkage dehydration is done in stages by soaking the sample in solutions of either ethanol or acetone of increasing concentration 10 20 mins soak in 70 ethanol followed by 10 20 mins soak in 90 ethanol followed by 15 30 mins soak in 100 ethanol Afterwards a five minute soak in a 11 mixture of ethanol and propylene oxide followed by a 10 minute soak in pure propylene oxide will complete the dehydration process The fixed and dehydrated sample is then ready to be embedded in epoxy usually consisting of a polymer resin eg araldite CY212 hardener like dodecenylsuccinic anhydride DDSA plus an accelerator like benzyldimethylamine BDMA Over the course of the next few days the sample is embedded with this resin in a suitable mould and left in an oven at 60 C for 48 hours to complete the polymerisation Once complete thin slices can be sliced off the embedded sample with a microtome which is an apparatus used to cut thin sections of a specimen with a knife that is made of either steel glass sapphire or diamond The thin sections are then collected onto a copper support grid coated with carbon if necessary and examined in the TEM R Ubic MSE 421521 Irmadncnan ta Eleman Mlcmxapy HI Electron Di rac on A Principles of Diffraction F diffraction is th n ff t h r A f A L l 1 at sharp shadows are not produced The phenomenon is the result of interference and is most pronounced when the wavelength of the radiation is comparable to the linear dimensions of the obstacle When a light falls on the edge of an object it will not continue in a straight line but will be slightly bent by the contact causing a blur at the edge o he shadow of he object The amount ofbending will be proportional to he wavelength sequence A diffraction grating consists of a regular twor or dareedlmenslonal array of objects or p gs h tt r light a ording to it w ele g h o i l e de t waves in fact they rei f rce one another in so i tions to ce intense spectral colour Diffrac n ays reveal spectral colours in direct sunlight exist on he me eet es and he sklns of some snakes Perhaps the most outstanding natural m u an opal has a regular dareedlmenslonal array of equalrslze spheres about 250 nrn in diameter which produce he diffraction Huygens showed that every point on a wave front may be regarded as a source of spherical t that the wavelets can interfere and this led to a theory of diffraction l Lnterference Lnterference is the term used to describe the interaction of waves These waves are typically bl h h J er aves helm n etc Any two i in i interfere with each other that is the resultant wave is the sum of the first two Constructive 39 s are exactly p when he peaks of one wave coincide with he troughs of the o her When two waves of equal amplitude are exactly half a wavelength 7E rad or 180 out of phase then their resultant wave has zero arnplitude r they cancel out it Ublc Incl MSE 42152 Introduction to Electron Microscopy Interference also occurs between two wave trains moving in the same direction but having different wavelengths or frequencies The resultant effect is a complex wave and a pulsating frequency called a beat results when the wavelengths are slightly different Rl Ubic Ill2 MSE 421521 In admmn ta Eleman Mlcmxopy 2 Flaunhofel Lgtgttz Single Slit Diffraction plane wave that pan through a lextlicted opening emerge a divergent WBVeX When the opening it 18 than one wavelength in diameter the emergent wave it nearly pherical enevel L L r 1 ofthe wave front some rpreailing Occur at the edge of geomemcal shadows 2 r 1m who oemino at t The mlnlma occur at in9 or R a L l a rneanired on the rcreen at Q The approximation for R axxume L gtgt R what i prolected onto the rcreen Q i a pattem rirnilar to the one below The width of the pattern is inversely proportional to th slit with and when a i e 90 which indicate that the central blight band fill the entire screen the width of the lit by accurately rneaniring the pacing of the rninirna on the rcreen Thin Angle degrees Relative Lntennty O 5 10 15 73 Dixtance cm x Ublc HM MSE 42152 Introduction to Electron Microscopy 3 Fraunhofer 1L gtgt a Double Slit Diffraction This experiment is often attributed to Thomas Young who first performed it in 1801 and thus helped prove the wave nature of light A beam of light is first passed through a single slit as above and subsequently passed through a double slit system The two slits have a width o and are separated by a distance d The equation which describes the resultant intensity as a function of angle 9 is l 2 19 Imwj cosZB where0cEsin9 andBEsin9 0c 7 7b The second factor on the right hand side is the diffraction factor while the last factor is the interference factor The result looks similar to single slit diffraction except the intensity is now modulated by the interference factor The minima now occur at dsine m 7t or R Mz onnFg dZXZ m 2 d and the maxima at approximately dsin9m7t or RM mzml d2 m27b2 d where m 0 1 2 3 As with single slit diffraction it is possible to link the pattern on the screen to the microscopic nature of the slits By accurately measuring the spacing of the new maxima or minima on the screen one can calculate the spacing of the slits d By measuring the distance between the unmodulated minima it is also still possible to calculate the width of the slits as before Angle degrees 1 20 0 20 i i i i i i i i i i i L 05 m I x i i single slit 0 g l 700 nm I double slit i c lSum apart m E 0 6 7 m y 230m 7 E 39 2 H r y 7 a r Z c E 04 7 7 a 7 c4 7 c 02 7 c y i 7 7 7 0 in lxlwi ctrN 1A HAMTVMMVM uni 20 15 10 5 0 5 10 15 20 Distance cm R Ubic Ill4 MSE 42152 Introduction to Electron Microscopy 4 Fraunhofer 1L gtgt a Multiple Slit Diffraction The specific cases of single or double slit diffraction can be generalised for the case of an arbitrary number of slits n In this case the equation of intensity becomes sin2 a sin2 n 10 1m 2 2 a sin 6 The greater the number of slits the sharper and more intense are the diffraction maxima The same relationships between d and the maximaminima spacings still hold and become more accurate the more slits that are included Angle degrees 0 3920 3910 10 20 1 l l l l l l l l 7 c r L I c x1 700 nm 1 r r r 7 7 single slit 77 gtO 8 a 7lm I six slits 7 H 7 7 d m 7L 7 723 g 06 n 6 m c H T y c 3 r 5 7 7 N 3 04 I 7 34 t 7 l l 7 02 39I i Q r 7 7 I l 7 0quot Hi l mm l mfh l i n my imlh m mm l iii if 20 15 10 5 0 5 10 15 20 Distance cm The above graphs have been normalised by a factor of ln2 for convenience In reality the intensity of the diffracted maxima are proportional to n2 therefore a diffraction grating consisting of thousands of slits can be used to yield very intense 7 and sharp 7 diffraction maxima 5 Fraunhofer 1L gtgt a Circular Aperture Diffraction When light from a point source passes through a small circular aperture it does not produce a bright dot as an image but rather a diffuse circular disc known as Airy39s disc surrounded by much fainter concentric circular rings R Ubic Ill5 MSE 42152 Introduction to Electron Microscopy iiiii iiii iiiiHHHHHHHHHHi iL05m i 1700nm Q az7 m 5 L E i 37 57 iiiilnniimiHiwwwimwnnnwif 20 15 10 5 O 5 10 15 20 Radial Distance cm m 1 values for Circular aperture Minima Maxima 1 1220 1amp35 z 111M 2 2233 2679 R Q a 3238 369 ER Relative I Imam Edam Relative 931 75 In lens iw lntensiw R 00 42 magma Z tan 9 z sin 9 z 9 For small angles 6 The intensity is given by 1 tho4 11MRL7 2 4 MR L where 11z is a Bessel function of the first kind so the mimirna occur when 11TWRL7x 0 the zeros of the Bessel function of the first kind are 3831706 70155867 etc and the maxima correspond to the maxima in the Bessel expression Ri Ubic Ill6 MSE 42152 Introduction to Electron Microscopy B Quantum Electrodynamics QED We have already mentioned the law of re ection namely that the angle of incidence equals the angle of re ection 9 9f however now it is useful to examine that law more closely Consider the mirror shown in the figure below The source S emits one photon at a time and a detector is located at P Let s calculate the chance that a photon from S reaches the detector at P In order to block the obvious straight path between S and P we ll put a screen at Q between them Light screen Light sourc e Detector c 6 6663 36 0 Angle of re ection Now we would expect that all the light that reaches P to have been re ected off the middle of the mirror where the angle of incidence equals the angle of re ection and that the far ends of the mirror have no role in the re ection In fact there are millions of routes for the photon to go from S to P as shown below Let s look at these Each of these routes will take a different amount of time for the photon to reach P The photon will clearly take longer to travel from for example S to A to P than from S to G to PT Because they ve been travelling for different times each photon will be slightly out of phase with its neighbours If we represent the phase angle by an arrow vector whose direction corresponds to the relative phase of the photon e g up 0quot right 90quot etc then we can show graphically both the time required for each path and the relative phase of the photon upon reaching P l Imagine yourself running from S to P 7 you d hardly dash off to A first R Ubic Ill7 MSE 42152 Introduction to Electron Microscopy Time l l l l l l l l l l l l l A B C D E F G H l J K L M Nl You can think of the arrows as the direction of the hand of a stopwatch when the photon reaches P By adding up all the arrows we arrive at a resultant vector which represents the probability of a photon from S arriving at P It is apparent that the arrows from the ends of the mirror contribute very little to the overall resultant and it is the middle section of the mirror which contributes most This is true because it is in the middle section where the differences between adjacent paths are smallest so the phase differences are also smallest The ends of the mirror could easily be chopped off and the re ection would be virtually unaffected but that does not mean that re ection is not happening there as well We can test this theory by removing most of the mirror and leaving only a small section way out on the left What s left of the mirror is now in the wrong place for re ection you d think We ll divide this section up more finely now 7 fine enough so that there is not much difference in time between adjacent sections A OUT B OUT C Xx a71suvdsx a71swv l G without scratches W with scratches Now we see that some arrows point more or less to the right while others point more or less to the left If we add all the arrows together we again find they form more or less a circle and add up zero 7 no re ection Now if just the parts of the mirror where the arrows are pointing to the left are removed the remaining arrows do indeed add up to a substantial resultant Re ection does take place Such a mirror is called a diffraction grating 1 Actually the probability is the square of the magnitude of this vector R Ubic Ill8 MSE 42 152 I Imodumon to Elecnon Mlcvoscopy If we designed out gtating fol led light say A 700nm it would be no good fol blue light Fot blue we d have to make out cutaway sttips closet togethet because blue light has a highet equency than ed the stopwatch runs fastet howevet by moving the detectot to a diffetent angle the gtating made fol led light now wotks fol blue In fact if you shine white light onto the gtating ed quot N with mange t H 39t I quot A b all the othet colouts ofthe Iainbowl The ttacks of a CD behave this way The nominal ttack sepatation on a CD is L6 m so fol led light of wavelength 700 nm this would give a fitst OIdeI difftaction maximum at about 26 blue is at 14 l Tn irl ntall thi k A I H39 t39 itelf mu ii i a small cote ofneatby space all paths which have Ioughly the same length and so tequite the same time fot the election Ifthe gap is wide then thete ate many possible lineat paths to an offraxis spo all ofwhich cancel out and the elecbon will not be difftacted thetel Ifthe gap is u a too man L few possible too few to cancel out and 39 t th Itonically 39 39 39 39 a mi r behave as 391 diffm H n wa n quot L L 39 miton 39 apart an 39 i iuie i t 39 700 nml A crysml whose atoms ate pethaps no mote than a few Angsboms apart is a natuxal dif action quot phnmns fox L39 L L 39 39 r mn times fastet than Au 39 quot 39 L39 L L 39 39 about 220000 r times fastet than fol visible light at vatious angles mm which can be detetmined the spacings and exact anangement ofthe individual atoms C Geometry of Electron Diffraction 1 f action eomett in the TEM As we have alteady seen electrons although patticles can behave as waves and so can be dif actedl 39 I t acceletated by 39 39 P 394 39 39 by A Mawch 21V1 R szc 11179 MSE 42152 Introduction to Electron Microscopy With this relation it is quite straightforward to show that the wavelength of electrons accelerated by say 200 kV as they are in the JEOL 2100 is 0025 A much smaller than X ray or typical neutron wavelengths If such electrons are diffracted from the faces of the cubic crystal thallium chloride for instance the angle 0 is only 01860 and the total angular de ection of the electron beam is 0370 If a uorescent screen or photographic film is placed a distance L from the crystal the diffracted spot will be displaced from the undeviated beam by a distance R such that R Ltan20 z 20L since for very small 0 tan20 z 20 Combining this relationship with Bragg s law yields RL Md ndzLMR The factor LA is often referred to as the camera constant and is generally calibrated experimentally for each microscope This equation is used to indeX every kind of electron diffraction pattern Electron diffraction differs from X ray diffraction in a number of ways First electrons are much less penetrating than X rays They are easily absorbed by air where they can ionise air molecules which means that the specimen and the recordingdetecting device must all be enclosed in vacuum Transmission electron diffraction patterns can only be obtained with specimens so thin as to be classed as foils metals or films Second electrons are scattered much more intensely than X rays so that even a very thin layer gives a strong diffraction pattern in a short time Third the intensity of electron diffraction decreases with increasing 20 even more rapidly than for XRD which means that the entire observable diffraction pattern is limited to an angular range of about 40 20 We will cover this topic in more detail later on 001 electron diffraction pattern from an fcc crystal The figure below shows the geometry of electron diffraction in the TEM Since the wavelength of electrons is so small 0025 A at 200 kV 6 is very small 0140 for d 5 A and the Bragg equation reduces to 2d6 xi It is also apparent from the figure that tan26 RL which reduces to 26 RL Combining these two equations yields Rd D1 which is the equation used to indeX every electron diffraction pattern R Ubic Ill10 MSE 42152 Introduction to Electron Microscopy 7 Sample d planar spacing Geometry of electron diffraction in a TEM L is the effective 1 2dsin9l camera length T is the transmitted spot D is the diffracted 3 sineze spot R is the distance between T and D and 20 is the Bragg tan29 RL angle The angle 20is very small due to the small wavelength L 9353 of the electrons so Bragg s Law reduces to 2d0 Z xi Additionally it can be seen that ton2 0 RL which reduces to 20 RL Combining these two equations yields Rd E L7 T i 2 The Structure Factor The resultant wave scattered by all the atoms in a unit cell is called the structure factor F because it describes how the atom arrangement affects the scattered beam Mathematically it is the sum of all the waves scattered by the individual atoms n 2mmkvtw F hkl Z fne 1 where f is the atomic scattering factor a function of sine7 tabulated in the International Tables for X Ray Crystallogrhaphy and the summation extends over all the n atoms of the unit cell The t coordinates of the n atom are unvnwn and those of the re ecting plane are kid It is a complex number combining both the amplitude and phase of the resultant wave It can be re written as F i f cos 215th kvn 1w i sin 215th kvn 1w 1 With this definition in mind it is fairly easy to prove that e 631 651 1 62m e41ri 2661a 21 em 6 where nis any integer em 1 where nis aninteger l Recall that e cosx ismx R Ubic Illll MSE 42152 Introduction to Electron Microscopy These simple rules enable us to calculate the structure factors for most re ections however there are certain special cases in which we are required to use the trigonometric functions instead eg if hunkvnlwn i 0 12 1 Now let s look at a real crystal 7 the simple case of NaCl This crystal has face centred symmetry and because it consists of two different ions can be described as two interpenetrating fcc lattices offset from each other by 12 0 0 therefore there are eight ions per unit cell The coordinates are Na 000121201201201212 C1 121212120001200012 FfNanMmfNanntfNae2mfNae2mfCle2mjfCle2ntfCle2mfCle2m F fNa 1e1rihk e1rihl eniklfCle1rihkl 61 e1rik e1ril We can factor out an emmk from the Cl term and obtain F fNa 1en hk enz hl emklfCleLihkl1ernikl 6wlthzgt erm kl Now because em e the signs of the exponents in the Cl term can be changed F fNa 1e1rihk e1rihl e1riklfClenihkl1e1rikl e1rihl e1rikl This expression can be re grouped F 1e1rihk e1rihl e1riklXfNa fCle1rihkl Here the first set of parentheses corresponds to the face centring whereas the second term corresponds to the basis of the unit cell 7 the one Na and one Cl associated with each lattice point As we have already seen in example three the first part vanishes if h k and l are mixed odd and even otherwise it is equal to 4 For unmixed indices all odd or all even F 4fNa fClenihkl 4th fa if h k lis even 4th fa if h k lis odd Introducing a second atom type has not eliminated any re ections allowed in example three but it has decreased the intensity of some For example the 111 re ection now involves the difference rather than the sum of the scattering powers of the two ions This result is independent of which way round the Cl and Na are labelled in the unit cell It is the relative position of the atoms which is important R Ubic Ill12 MSE 42152 Introduction to Electron Microscopy 3 Spot patterns The nature of the diffraction pattern is in indication of the number of grains contributing towards the pattern or the crystallinity of the specimen A single crystal will give rise to a spot pattern Whereas diffraction from several crystals causes several overlapping spot patterns in which all the spots can be seen to lie along Debye rings In the extreme case of hundreds of crystals contributing to the pattern then the resulting spots are so closely spaced that they are no longer identifiable and merge into continuous rings Debye rings An amorphous sample results in no diffraction Single perfect crystal Diffraction pattern is regular array of dots Small number of grains crystals with different orientation Diffraction pattern is discrete spots each lying on a Debye ring Large number of randomly oriented grains Diffraction pattern is continuous Debye rings Amorphous material No distinct diffraction pattern R Ubic III l3 MSE 421521 Introduction to Electron Microscopy 4 The reciprocal lattice The concept of the reciprocal lattice is one of convenience as all spacings in a diffraction pattern are inversely proportional to the corresponding real spacing It is constructed by converting each real lattice plane th into a point at a distance g ldhkl from the origin of the reciprocal lattice dhkl is the real lattice spacing of the th planes along a direction perpendicular to hkl Some planes may not appear in the reciprocal lattice if they are forbidden by the extinction rules for the real lattice crystal symmetry If the real lattice has dimensions do 70 co then the reciprocal lattice s dimensions are o loo 17 2 1H 0 lco and o is normal to b0 and co 7 is normal to do and co and 0 is normal to do and by The reciprocal lattice of an fcc crystal is a bcc one and vice versa Points in the reciprocal lattice are not points in the mathematical sense but have some shape and size associated with them inversely proportional to the dimensions of the crystal or feature being examined shape factor 001 1111 N022 The reciprocal of an 7 fcc real lattice is bcc 4z l l l l l 0 l 010 774777 020 af rm gt 110 gt a 1001 Na fcc bcc 5 The Ewald or Re ecting Sphere The Ewald sphere is another geometrical convenience for calculating when diffraction occurs for some arbitrary planes A hypothetical sphere is centred on the point of electron scattering in the specimen and has a radius of ll Because the 7b of electrons accelerated by say 100 kV is only 0037 A the radius of the sphere is 27 Al The surface of this sphere can easily intersect more than one point in the reciprocal lattice and so many spots can appear in a diffraction pattern The size of the sphere is a major advantage of electron diffraction over say x ray diffraction XRD which can excite only one Bragg re ection at a time The Cu Koc radiation often used in XRD has 7 154056 A with the Ewald sphere radius only 065 Al Thus the radius of the sphere is over 40 times bigger for 100 kV electrons and the volume of the sphere is over 72000 times bigger Using the Ewald sphere construction to predict which diffraction spots will occur is equivalent to the Bragg condition The intersection of the sphere with successive lattice planes results in higher order Laue zones HOLZ The zero order Laue zone ZOLZ is typically used most often for indexing patterns as most information is there and the first order Laue zone FOLZ is often too far away to be imaged R Ubic Ill14 MSE 421521 Introduction to Electron Microscopy Ewald Sphere Ewald sphere construction Incident Beam S OLZ FOLZ Reciprocal ZOLZ Lanice Planes 6 Diffraction Theory There are two sets of theory and equations which explain electron diffraction kinematic theory and dynamical theory The kinematic theory is simpler and relies upon a few approximations and assumptions First the intensity of diffracted spots must be much smaller than the intensity of the transmitted beam Second it assumes that electrons scatter only once in the specimen and do not subsequently interact in any way with each other or undiffracted electrons This is approximately the case for exceptionally thin specimens or when the geometry is far from the Bragg angle The dynamical theory on the other hand accounts for absorption by thick specimens and the multiple interactions of electrons which scatter and can be re scattered The intensity of the diffracted waves can be almost as large as that of the transmitted waves Dynamical scattering makes it much more difficult to calculate the intensity of diffracted spots as events like double diffraction can redistribute diffracted intensity from one spot to another even into forbidden re ections 7 Double Diffraction Double diffraction is a dynamical effect whereby electrons which have been Bragg diffracted by one set of planes now satisfy the Bragg condition for another set of planes and so are diffracted twice The figure below shows two DP s a has been calculated kinematically and b is an actual experimental DP obtained from a JEOL 200CX 200 kV The difference in intensities is obvious and a few re ections which occur in b are not predicted by a For any double diffracted spot there must be a diffraction path which allows it For example the 002 spot which occurs in b is forbidden by the extinction rules and so does not appear in a however since both 1T1 and T11 are allowed and 1T1 T11 002 there is a legitimate route for double diffraction and the 002 spot appears A short visual way of determining if a double diffraction route exists is to overlay the simulation tracing it on acetate helps on the experimental DP and aligning the simulated transmitted spot onto each real diffraction spot If the simulation contains a diffracted spot which coincides with an extra spot on the experimental DP then there is a route for double diffraction Essentially this is the same as turning each diffracted beam into another transmitted beam which is exactly what double diffraction is R Ubic Ill15 MSE 421521 Introduction to Electron Microscopy a a Kinematic simulation and b experimental DP of fcc Ndsz207 with the beam parallel to 110 zone axis 2 110 8 Kikuchi Lines Kikuchi lines first observed by S Kikuchi in 1928 are another consequence of the dynamical theory When electrons are incident on a sample some of the electrons are inelastically scattered they lose energy Since more scattered electrons lose small amounts of energy than large amounts the wavefront of the incident beam is biased in the forward downward direction Many electrons are scattered through an angle 0c and now satisfy the Bragg condition for an arbitrary set of planes These electrons end up at E Fewer electrons scatter through the larger angle B and are subsequently diffracted by the same planes to D Since more electrons are re directed towards E than are lost in going to D the result is a bright line at E and a dark line at D T is the transmitted spot and g is the spot corresponding to the re ecting planes hkl When the sample is oriented exactly at the Bragg condition for hkl D passes through the centre of T and E passes through g The distance between E and g is known as the deViation parameter and is a very accurate way of measuring the angular deViation from the exact Bragg condition R Ubic Ill16 MSE 42152 Inmduczim n7 Elecmm Micmxmpy kLd sgt0 s 0 Bragg slt0 m P o m a In reality the bright and dark lines observed on at DP negatives are hyperbolic intersections of the film with cones Kikuchi or Kossel cones one bright and one dark The excess line is always further away from T than the deficit line since on lt B Kikuchi lines are very useful in orienting the crystal precisely in the TEM When the specimen is tilted the Kikuchi lines move as if they were attached to the bottom of the crystal The various diffracted spots never move they just appear and disappear as the angle changes but Kikuchi lines move with the specimen Following them is a handy way to tilt systematically in order to navigate from one zone axis to another or to find appropriate tworbeam conditions where only one set of planes Bragg djffracts for defect analysis Kikuchi lines are only observable if the specimen is thick enough to generate sufficient intensity of scattered electrons In a ve thin specimen the lines will be too weak to distinguish from the background As thickness increases Kikuchi lines and then bands are observed until nally total absorption occurs and nothing is visible 9 Cmtal Shape Factor Since all distances lengths in reciprocal space are lrealrspace distances the shape of the TEM specimen has an effect on the shape of the reciprocal lattice points A diskrshaped TEM sample is very thin about 100nm in the z direction but has a comparatively large area 5 and y directions Consequently a diskrshaped specimen will produce electrons distributed along a rod R Ubic 111717 MSE 421521 Introduction to Electron Microscopy passing through the reciprocal lattice point The rods called rel rods are thin in x and y but thick in z e z y real space x reciprocal space They are also longer for thinner crystals The fact that reciprocal lattice points are in fact three dimensional rods and not mathematically defined zero dimensional points makes it possible to get diffracted intensity even when the Bragg condition is not exactly satisfied The deviation parameter s is defined as the deviation from the exact Bragg condition where the Ewald sphere cuts the rel rod 10 Indexing Diffraction Patterns The method of indexing diffraction patterns labelling the spots with appropriate hkl s varies according to what is known of the specimen crystallography and diffraction conditions camera length and voltage Typically an experimentalist will have a fairly good idea of what phase or possible phases produced the pattern and a reasonable knowledge of L and xi the product L1 is often referred to as the camera constant In such a case the work is much simplified however if the crystallography is completely unknown the procedure is complex and requires a lot of trial and error To begin indexing a diffraction pattern DP such as the one below a list of R spacings is required A good idea is to put a ruler across the negative and measure the space across an entire systematic row of spots and take an average for the space between each spot The corresponding d spacings can then be calculated as LiR These d spacings can be checked against a list of d spacings for various possible phases A good start is the JCPDS card files which usually contain sufficient crystallographic information to calculate d spacings Since more re ections are possible in electron diffraction by double diffraction than occur by XRD the JCPDS cards may not list all the d spacings observed in the DP and the remaining spacings must be calculated There are a few simple equations for calculating d spacings of planes in various crystal systems In the case of a cubic crystal 1 h2 k2 12 d7 a z Once generic hkl s are found they must be made systematic In the case of the DP below g1 g3 and both correspond to a d spacing of 614 A matching the 111 of fcc Ndsz207 The magnitude of vector g2 corresponds to d 376 A matching 220 of Ndsz207 From the DP it is clear that g1 g3 g2 and this must be the case for the R Ubic Ill18 MSE 421521 Introduction to Electron Microscopy hkl s With this rule the spots are made systematic by indexing g1 as lTT g2 as 2f0 and g3 as 1T 1 In this way g1 g3 lTT 1 T 1 2f0 g2 Every other spot in the pattern is now determined by similar vector addition Diffraction pattern from fCC NdszzO7 Since the camera constant is typically only approximately known the ratios of d spacings should be checked to ensure that they match those calculated from the JCPDS card Even if the absolute value of the d s is slightly off the ratio of d s should be very nearly an exact match As an additional check the angles between each plane should be measured to ensure that they are correct For cubic crystals the angle 4 between two planes two plane normals hjkjl and hzkzlz is given by com 2 lith k1k2 1112 hf k12 112h22 k22 122 11 Zone Axis Identification If the planes in a diffraction pattern have been indexed correctly then the zone axis mnp is defined as the direction which they all have in common The g vectors of all the planes must be perpendicular to this direction ie hkl mnp 0 It is determined by finding two spots hjkjl and hzkzlz which are not on the same systematic row Then in 112 21 n 11112 12h p 2 Itij 1sz A simple way is to write down the plane indices in two columns and strike out the first and last digits in each column Then a simple cross multiplication gives the correct result Sometimes a high index direction will be calculated eg 220 as in this case For crystallographic directions one can legitimately multiply through by any number and still arrive at an equivalent direction Multiplying 220 through by 12 yields 110 which is the lowest index possible for this direction and so is reported as the direction of the zone axis the magnitude of vectors 220 and 110 are different but their directions are crystallographically identical R Ubic Ill19 MSE 421521 Introduction to Electron Microscopy mnp k1 11 hl k1 1 1 1 I I g a Generlc illustration of a 12 12 h2 k2 zone axis b Determination of the zone axis of the diffraction 1 1 1 1 a l i 1 1 111111 220 110 b E pattern in the figure above Of course this is exactly equivalent to taking the cross product of the two g vectors det 1 y kl y kl amp k 1 i11 j 1 1k 11220110 l The Weiss zone law says that for any plane hkl contributing to a DP along a given zone axis mnp then mh nk pl 2 0 More formally this rule takes the form of a dot product g I r 0 kid I mnp 0 Well X mnp cos 9 0 cose 0 9 90 R Ubic Ill20 MSE 421521 Introduction to Electron Microscopy 12 Convergent Beam Electron Microscopy gCBED CBED is a powerful tool for probing the crystallography of a small specimen as well for analysing local strains and thicknesses Conventional diffraction is performed using nearly parallel electron beams parallel illumination and so gives rise to diffraction spots The CBED technique involves focusing the incident illumination down to a fine point and so yields diffraction discs Because of the extreme intensity of the focused incident beam the specimen used must be very stable against the radiation The specimen will become very hot in the region of the fine probe and so cooling is generally required to prevent thermal diffuse scattering from obscuring the fine detail within the discs For the same reason a very high vacuum system is required to prevent dirt mostly carbon from being deposited on the specimen surface Additionally the tiny region examined must be defect free and if crystallographic information is required of uniform thickness Clearly CBED is not a trivial operation and the interpretation of CBED patterns is no less difficult From CBED images it is possible to obtain local specimen thickness useful when cg determining defect densities lattice parameters and crystal structure even a full space group determination Incident Electrons Specimen Diffraction Discs R Ubic Ill21


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