Transmission Electron Microscope

27,869 views

Published on

In depth understanding of working principle of Transmission Electron Microscope, Electron Diffraction, Indexing of SAED pattern.

Published in: Technology

Transmission Electron Microscope

  1. 1. Transmission Electron Microscopy and Electron Diffraction
  2. 2. The Philips CM200 transmission electron microscope Accelerating voltages is 200 kV Can achieve resolution upto 2 Angstroms .
  3. 4. Why Electron Microscope? <ul><li>Light Microscopes are limited by the physics of light to 500x or 1000x magnification and a resolution of 0.2 micrometers. </li></ul><ul><li>In the early 1930's there was a scientific desire to see the fine details of the interior structures of organic cells (nucleus, mitochondria...etc.). </li></ul><ul><li>This required 10,000x plus magnification which was just not possible using Light Microscopes. </li></ul>
  4. 5. LM, resolving power ~0.25µm, maximum (useful) magnification is about 250µm/0.25µm = 1000X. Any magnification above this value represents empty magnification TEM at 60,000 volts has a resolving power of about 0.0025 nm. Maximum useful magnification of about 100 million times!!!
  5. 6. COMPARISION OF LIGHT AND ELECTRON MICROSCOPE Optical glass lens, Small depth of Field, lower magnification, do not Require vacuum, Low price. Magnetic lens, Large depth of field, Higher magnification and better Resolution, Operates in HIGH vacuum, Price tag. LIGHT MICROSCOPE ELECTRON MICROSCOPE
  6. 7. Electron Microscopy <ul><li>What are electron microscopes? </li></ul><ul><li>Electron Microscopes are scientific instruments that use a beam of highly energetic electrons to examine objects on a very fine scale which yield the following information: </li></ul><ul><li>1. Topography : </li></ul><ul><li>The surface features of an object (hardness, reflectivity...etc.) </li></ul><ul><li>2. Morphology: </li></ul><ul><li>The shape and size of the particles(ductility, strength, reactivity...etc.) </li></ul><ul><li>3. Composition: </li></ul><ul><li>The elements and compounds that the object is composed of and the relative amounts of them. </li></ul><ul><li>4. Crystallographic Information: </li></ul><ul><li>How the atoms are arranged in the object. </li></ul>
  7. 8. SPECIMEN INTERACTION IN ELECTRON MICROSCOPY SPECIMEN INTERACTION VOLUME FOR VARIOUS REACTIONS REACTIONS ON THE TOP SIDE ARE UTILIZED FOR EXAMINING THICK OR BULK SPECIMENS (SEM) RECTIONS ON THE BOTTOM SIDE ARE EXAMINED IN THIN OR FOIL SPECIMEN (TEM ) VARIOUS REACTIONS CAN OCCUR WHEN ENERGETIC ELECTRONS STRIKE THE SAMPLE
  8. 9. THIN SPECIMEN INTERACTIONS REACTION PRODUCT SOURCE UTILIZATON UNSCATTERED ELECTRONS INCIDENT ELECTRONS TRANS- MITTED(NO DEFLECTON FROM THE ORIGINALPATH) THROUGH THE SPECIMEN WITHOUT ANY INTERACTION UNSCATTERED ELECTRON INTENSITY IS INVERSELY PROPORTIONAL TO THE SPECIMEN THICKNESS. THICKER PORTION OF THE SPECIMEN WILL APPEAR DARKER AND CONVERSE IS ALSO TRUE. ELASTICALLY SCATTERED ELECTRONS INCIDENT ELECTRONS SCATTERED(DEFLECTED FROM THE ORIGINAL PATH) BY THE ATOMS IN THE SPECIMEN IN AN ELASTIC FASHION (NO LOSS OF ENERGY) FOLLOW BRAGG’S LAW. SIMILAR ANGLE SCATTERING OF THE ELECTRONS FROM THE PLANE OF SAME ATOMIC SPACING FORM PATTERN OF SPOTS WHICH YIELDS INFOR- MATION ABOUT THE ORIENTATION, ATOMIC ARRANGEMENTS AND PHASES PRESENT . INELASTICALLY SCATTERED ELECTRONS ELECTRONS INTERACT WITH THE SPECIMEN ATOM IN AN INELASTIC FASHION (BY LOOSING ENERGY DURING INTERACTION) <ul><li>UTILIZED IN TWO WAYS: </li></ul><ul><li>KIKUCHI BANDS: BANDS OF ALTER- </li></ul><ul><li>NATING DARK AND BRIGHT LINES RELATED </li></ul><ul><li>TO THE ATOMIC SPACING OF THE SPECIMEN. </li></ul><ul><li>2 . ELECTRON ENERGY LOSS SPECTROSCOPY : </li></ul><ul><li>LOSS OF ENERGY ARE UNIQUE TO </li></ul><ul><li>EACH BONDING STATE OF EACH ELEMENT. </li></ul>
  9. 10. Transmission Electron Microscopy TEM is a unique tool in characterization of materials crystal structure and microstructure simultaneously by diffraction and imaging techniques.
  10. 11. TEM is analogous to a Slide Projector
  11. 12. Transmission Electron Microscopy <ul><li>In a conventional transmission electron microscope, a thin specimen is irradiated with an electron beam of uniform current density. </li></ul><ul><li>Electrons are emitted from the electron gun and illuminate the specimen through a two or three stage condenser lens system. </li></ul><ul><li>Objective lens provides the formation of either image or diffraction pattern of the specimen. </li></ul><ul><li>The electron intensity distribution behind the specimen is magnified with a three or four stage lens system and viewed on a fluorescent screen. The image can be recorded by direct exposure of a photographic emulsion or an image plate or digitally by a CCD camera. </li></ul>
  12. 14. The acceleration voltage of up to date routine instruments is 120 to 200 kV. Medium-voltage instruments work at 200-500 kV to provide a better transmission and resolution, and in high voltage electron microscopy (HVEM) the acceleration voltage is in the range 500 kV to 3 MV. Acceleration voltage determines the velocity, wavelength and hence the resolution (ability to distinguish the neighbouring microstructural features) of the microscope
  13. 15. Depending on the aim of the investigation and configuration of the microscope, transmission electron microscopy categorized as: <ul><li>Conventional Transmission Electron Microscopy </li></ul><ul><li>  </li></ul><ul><li>High Resolution Electron Microscopy   </li></ul><ul><li>Analytical Electron Microscopy   </li></ul><ul><li>Energy-Filtering Electron Microscopy   </li></ul><ul><li>High Voltage Electron Microscopy   </li></ul><ul><li>Dedicated Scanning Transmission Electron Microscopy </li></ul>
  14. 16. IMAGING <ul><li>The image of the specimen in conventional microscopy, , is formed selectively allowing only the transmitted beam (Bright Field Imaging) or one of the diffracted beams (Dark Field Imaging) down to the microscope column by means of an aperture. </li></ul><ul><li>The origin of the image contrast is the variation of intensities of transmitted and diffracted beams due to the differences in diffraction conditions depending on the microstructural features on the electron path. </li></ul>
  15. 17. BRIGHT FIELD IMAGING ALLOWING TRNSMITTED BEAM
  16. 18. DARK FIELD IMAGING ALLOWING DIFFRACTED BEAM
  17. 19. Bright-field TEM micrographs of the as-prepared ZnO powders after annealing for 1 h at various temperatures: a 300 . C, b 400 . C and c 500 . C, respectively.
  18. 20. DIFFRACTION <ul><li>Electrons of 0.072 Angstrom wavelength at 100 kV </li></ul><ul><li>excitation transmitted through about 0.1 micrometer thin foil specimen are diffracted according to Bragg's Law, forming a diffraction pattern (consisting of a transmitted and diffracted beam spots). </li></ul><ul><li>Although diffraction phenomena is a complex interactions of charged electrons with the periodic potential field of the lattice, Bragg's Law or Laue Conditions are sufficient approximations for usual practical applications. </li></ul><ul><li>A diffraction pattern is, in the simplest sense, a Fourier transform of the periodic crystal lattice, giving us information on the periodicities in the lattice, and hence the atomic positions.      </li></ul>
  19. 21. Some fancy Diffraction Patterns
  20. 22. BASIC DESIGN OF TRANSMISSION ELECTRON MICROSCOPE Evacuated metal cylinder within which are aligned, one under another: 1. Tungsten filament (the cathode) 2. A Metal plate with central aperture (the anode) 3. A number of magnetic lenses 4. A Fluorescent screen 5. A photographic plate
  21. 23. DESIGN OF TRANSMISSION ELECTRON MICROSCOPE A simplified ray diagram of a TEM consists of an electron source, condenser lens with aperture, specimen, objective lens with aperture, projector lens and fluorescent screen .
  22. 24. In actuality a modern TEM consists of many more components including a dual condenser system, stigmators, deflector coils, and a combination of intermediate and dual projector lens
  23. 25. Electron Gun <ul><li>Electron beam is generated in the electron gun. Two basic types of guns are used: </li></ul><ul><li>1. Thermionic Gun: </li></ul><ul><li>Based on two types of filaments: Tungsten(W) and Lanthanum-Hexaboride(LaB6). </li></ul><ul><li>2. Field Emission Gun(FEG): </li></ul><ul><li>Employs either a thermally assisted cold field emitter or Schottky emitter. </li></ul>
  24. 27. Functioning of the Thermionic Gun <ul><li>An positive electrical potential is applied to the anode. </li></ul><ul><li>The filament (cathode) is heated until a stream of electrons is produced . </li></ul><ul><li>A negative electrical potential (~500 V) is applied to the Whenelt Cap. </li></ul><ul><li>A collection of electrons occurs in the space between the filament tip and Whenelt Cap. This collection is called a space charge. </li></ul><ul><li>Those electrons at the bottom of the space charge (nearest to the anode) can exit the gun area through the small (<1 mm) hole in the Whenelt Cap . </li></ul><ul><li>These electrons then move down the column to be later used in imaging </li></ul>
  25. 29. Field Emission Gun (FEG) <ul><li>In recent years cold field-emission and thermally-assisted field emission guns have become increasingly common. </li></ul><ul><li>In these a very fine point on the pointed </li></ul><ul><li>filament is formed. </li></ul><ul><li>Electrons are emitted by tunnelling through </li></ul><ul><li>the potential barrier at the tip surface when </li></ul><ul><li>a very high potential field gradient is formed </li></ul><ul><li>at the surface. </li></ul>
  26. 30. FEG requires a different gun design as well as much better vacuum in the gun area (~10e-8 Pa instead of the ~10e-5 Pa). Consists of a small single-crystal tungsten needle that is put in a strong extraction voltage (2-5 kV). In the case of a cold FEG, the needle is so sharp that electrons are extracted directly from the tip. For the Schottky FEG a broader tip is used which has a surface layer of zirconia (ZrO2). The zirconia lowers the work function of the tungsten and can use the broader tip . Unlike the thermionic gun, the FEG does not produce a small cross-over directly below the emitter, but the electron trajectories seemingly originate inside the tip itself, forming a virtual source of electrons for the microscope.
  27. 31. Electron Optics Elements <ul><li>Lense: Focus(or defocus) the beam on the specimen and change the magnification. </li></ul><ul><li>Deflection Coil: Shift or tilt the beam. </li></ul><ul><li>Stigmators: Correct the lenses. Ideally lenses are round and symmetrical but in practice there are small deviation which is corrected by the stigmators. </li></ul>
  28. 32. Magnetic Lenses Electron Optics Elements
  29. 33. MAGNETIC LENSES 1.Coil of several thousand turns of wire through which a current of less than or equal to one amp is passed --- creates a magnetic field. 2.. Electrons are deflected by magnetic field 3. To concentrate field further a soft iron pole piece is inserted into the bore of the objective lens. 5. To focus an electron beam onto a given plane the current through the coils must be set to a precise value. . current – beam focus closer to lens current – beam focus further from lens
  30. 34. Depth of Field: the range of distance at the specimen parallel to the illuminating beam in which the object appears to be in focus. Depth of Focus: the range of distance at the image plane (i.e. the eyepiece, camera, or photographic plate) in which a well focussed object appears to be in focus.
  31. 35. Illuminates the specimen. Relatively weak lens. Longer focal length than objective or projector lens. May bring electron beam into focus directly upon specimen, above the specimen (over focusing) or below the specimen (under focusing). CONDENSER LENS
  32. 36. As magnification increases the condenser lens must be adjusted to properly illuminate the specimen. When the lens is brought to its smallest spot the beam is said to be at the crossover point
  33. 37. Holey Formvar is used to critically adjust the stigmation of a TEM. When the beam is under or over focused on the specimen a Fresnel fringe becomes visible due to the effects of diffraction around the edges of the hole. When this Fresnel fringe is evenly distributed then the beam is said to be stigmated
  34. 38. OBJECTIVE LENS <ul><li>Strong lens </li></ul><ul><li>Highly concentrated magnetic field and short focal length.Causes electron beam, which has passed through specimen, to focus at a point a few mm below specimen. </li></ul><ul><li>Magnification of image produced a short distance below focused point. </li></ul>
  35. 39. Projector Lens <ul><li>Magnification produced by projector lens dependent on current passing through the coil of the lens (ie increase current spreads beam further = higher mag.) </li></ul><ul><li>Projector lens has great depth of focus (several meters). Therefore distance at which fluorescent screen or photographic plate are placed is not critical. </li></ul>
  36. 40. Total magnification in the TEM is a combination of the magnification from the objective lens times the magnification of the intermediate lens times the magnification of the projector lens. Each of which is capable of approximately 100X. M ob X M int X M proj = Total Mag
  37. 41. In older TEMs functions such as gun and beam alignment were accomplished by physically moving components in the column. Today they are achieved by use of electromagnetic deflection coils that are positioned throughout the column Deflection Coils
  38. 42. Using the deflection coils the beam can be shifted so that the focused beam is centered in the back focal plane of the lens and tilted so that the beam is centered on the specimen.
  39. 43. Fluorescent Screen <ul><li>Fluorescence: Property of emitting radiation under the influence of electromagnetic or electron beam bombardment. </li></ul><ul><li>In the TEM, screen coated with a material in the visible range, eg zinc sulphide, is installed beneath the projector lens in the path of the electron beam . </li></ul><ul><li>Screen emits visible light when bombarded with electrons. </li></ul><ul><li>The resolution of the fluorescent screen is limited to 70-100µm by the grain size of the fluorescent material and by light scattering within this material. </li></ul>
  40. 44. Vacuum System <ul><li>Electron can’t travel more than a few angstrom without colliding with gas molecules. </li></ul><ul><li>Distance between photographic plate and electron gun is approximately 1 meter. </li></ul><ul><li>Electron gun must be evacuated (10 -4 torr). </li></ul><ul><li>Two types of vacuum pump are used </li></ul><ul><li>Rotary (mechanical) fore-pump. </li></ul><ul><li>Diffusion pump (Oil or Mercury) </li></ul>
  41. 45. BASICS OF ELECTRON DIFFRACTION <ul><li>No absorption </li></ul><ul><li>No thermal vibration </li></ul><ul><li>No multiple scattering </li></ul><ul><li>Distance between crystal and film >> λ the wavelength of electron </li></ul><ul><li>Large object (macroscopic single crystal) </li></ul>Simplifications:
  42. 46. The de Broglie wavelength of electron is given by or Diffraction intensity in a given direction is the sum over contribution from all location of the specimen taking into account their relative phases. Bragg’s conditions for constructive interference: In vector form: The scattering vector(not necessarily a Lattice vector) = Reciprocal lattice vector
  43. 47. ELECTRON DIFFRACTION UNDER THE CONDITION OF TEM <ul><li>“ wavelength  interatomic distances </li></ul><ul><li>transmission ”  specimen thin along </li></ul><ul><li>the viewing direction </li></ul><ul><li>consequence of limited thickness: “relaxation” of the diffraction condition </li></ul><ul><li>S  g  REL along the viewing direction ( z -axis) </li></ul>Special conditions for TEM:
  44. 48. ELECTRON DIFFRACTION UNDER THE CONDITION OF TEM • Consider a row of N unit cell along the z-axis • The topmost unit cell scattered with amplitude F 0 [s] • S=s-s 0 is the scattering vector • Scattered wave from the next unit cell below is Where c is the lattice vector along the z-axis Final scattered wave intensity is given by Where I 0 [s] = It has sharp maximum for S.c = l an integer Lattice factor
  45. 49. EXCITATION ERROR Assume that the diffraction condition is not exactly satisfied : Where s is the excitation error g is a lattice translation of the REL, satisfying the 3rd Laue equation : s is not a lattice translation of the REL, and we assume that This yields, the diffraction intensity TEM thin foil: Small extension in the z-direction, large extension in the x And y direction. Thus 1 st and 2 nd Laue equation satisfies exactly, Excitation error only have z-component, thus
  46. 50. This yields following expression of intensity First minimum of I occurs at ± 1/t. Thus diffraction Occurs although S is not a RLV. I[s] >0 is represented by ‘rods’ parallel to z* in every REL point (REL rods). Since s is small And the scalar product, N.c.s =
  47. 52. Ewald Construction for Electron Diffraction in TEM 1. Replace REL points by REL rods! 2. Direction of the rods: normal to the plane of the foil (parallel to z *) 3. Length of the rods: ± 1/ t Special Ewald construction for diffraction of electrons at thin foils: • special conditions: 1. Very small wavelength (compared to X-ray diffraction, for example)  Ewald sphere has a very large radius 1/  2. REL points  “rods” normal to the TEM foil
  48. 53. Specimen, transmitted and diffracted beam forming the diffraction pattern. Also see the Ewald sphere construction in Reciprocal Space.
  49. 54. CONSEQUENCES: 1. Even for a “sharp” wavelength,  const., a TEM foil generates diffracted beams, irrespective of the orientation of the foil versus the primary beam! 2. Very small Bragg angles ( 1°) 3. Reflecting planes are approximately parallel to the primary beam 4. = Reduces the Bragg condition to 5. Ewald sphere “flat” in the angular region of ±1° around the direction of the primary beam 6. Curvature of the sphere is negligible compared to length ±1/ t of the REL rods Consequence: Example: Diffraction pattern of f.c.c lattice in <hkl> direction (hkl) plane of b.c.c lattice and vise versa.
  50. 55. In this case incident beam direction B [100] in an Aluminum (f,c.c), single crystal specimen. Transmitted beam is marked as T and the arrangement of the diffracted beams D around the transmitted beam is the characteristic of the four fold symmetry of the [100] cube axis of Aluminum. EXAMPLE OF DIFFRACTION PATTERN
  51. 56. SINGLE CRYSTAL DIFFRACTION PATTERN Single crystal are most ordered (lattice type such as f.c.c, b.c.c, s.c etc.) among the three structures. Electron beam passing through a single crystal will produce a pattern of spots. Type of crystal structure (f.c.c., b.c.c.) and the &quot;lattice parameter&quot; ( i.e. , the distance between adjacent planes) can be determined . Also, the orientation of the single crystal can be determined: if the single crystal is turned or flipped, the spot diffraction pattern will rotate around the centre beam spot in a predictable way.
  52. 57. DIFFRACTION FROM POLYCRYSTALLINE MATERIALS <ul><li>Polycrystalline materials are made up of many tiny single crystal. </li></ul><ul><li>Not ordered, single crystal grains in a polycrystal have random distribution of all possible orientations. </li></ul><ul><li>Diffraction patterns will therefore will look like superposition of single crystal spot pattern: a series of concentric ring resulting from many spots very close together at various rotation around the central spot. </li></ul>From the diffraction rings type of crystal structure and the &quot;lattice parameter“ can be determined. One cannot determine the orientation of a polycrystal, since there is no single orientation and flipping or turning the polycrystal will yield the same ring pattern .
  53. 59. Contribution of Inelastic Scattering <ul><li>Conventional high-energy electron diffraction: elastic scattering </li></ul><ul><li>But in thick enough specimen: also inelastic scattering </li></ul><ul><li>Inelastically scattered electrons: </li></ul><ul><li>1. Travel in all directions </li></ul><ul><li>2. D istribution peaks in forward direction </li></ul><ul><li>3. G rey background around central spot </li></ul>
  54. 60. Bragg Reflection of Inelastically Scattered Electrons <ul><li>Inelastically scattered electrons can subsequently be diffracted </li></ul><ul><li>But only if they are now traveling at the Bragg angle, to a set ( hi) of lattice planes </li></ul><ul><li>Consider (hi) inclined by angle versus primary beam </li></ul><ul><li>Bragg reflection can occur with two sets of inelastically scattered electrons at </li></ul><ul><li>Result: intensity changes in the background </li></ul>
  55. 62. Kikuchi Lines <ul><li>Diffraction of inelastically-scattered electrons: </li></ul><ul><li>In all directions for which </li></ul><ul><li>In 3D (from Laue equations) we know </li></ul><ul><li>Diffracted electrons will form a cone , not a beam </li></ul><ul><li>Intersection of cones with viewing plane: </li></ul><ul><li>hyperbola, not spots! </li></ul><ul><li>Usually: </li></ul><ul><li>Camera length (magnification) >>radius of curvature </li></ul><ul><li>Only small sections visible straight lines </li></ul><ul><li>KIKUCHI LINES </li></ul>
  56. 65. Features of Kikuchi Lines <ul><li>Kikuchi lines belong to particular lattice planes h i a i and can be indexed. </li></ul><ul><li>Spacing , distance of diffraction spot from center spot </li></ul><ul><li>Mirror line in the center between excess and deficiency line trace of planes </li></ul><ul><li>Specimen tilt lines rotate as if “attached” to specimen </li></ul><ul><li>Position sensitive to small specimen tilts </li></ul><ul><li>Adjust crystal orientation and excitation error </li></ul><ul><li>Accuracy: ± 0 . 1 0 </li></ul><ul><li>C ompare accuracy using spot intensities: ± 2 0 </li></ul>
  57. 72. TYPES OF DIFFRACTION PATTERNS AND THEIR USE TYPE OCCURENCE USE RING PATTERN POLYCRYSTALL- INE AND AMOR- PH0US SPECIMEN Identification of phases and determination of grain size of polycrystalline sample produced by electro-deposition or CVD SPOT PATTERN SINGLE CRYSTAL <ul><li>Magnified images of the planar section of RL </li></ul><ul><li>pace perpendicular to the beam direction and </li></ul><ul><li>can be used to determine </li></ul><ul><li>Specimen orientation in the microscope </li></ul><ul><li>Tilt axis by tilting about the bright spot </li></ul><ul><li>Orientation relationship between phases </li></ul><ul><li>Precipitates, twin etc. can be identified </li></ul><ul><li>Quick survey of orientation and hence the </li></ul><ul><li>diffraction vectors. </li></ul>KIKUCHI LINES INELASTIC SCATT- ERING BY A THICK SPECIMEN. SADP FROM SING- LE CRYSTAL <ul><li>Accurate crystal orientation </li></ul><ul><li>Incident beam direction </li></ul><ul><li>Define the sense of tilt </li></ul><ul><li>Crystal symmetry </li></ul>
  58. 73. Camera Length INDEXING DIFFRACTION PATTERNS is the distance between and screen and the crystal (without imaging lens). With imaging lenses effective L is considered. From the figure: Using the reduced Bragg condition: and the above relation we have,
  59. 74. DETERMINATION OF LATTICE PLANE SPACING AND INDEXING DIFFRACTION PATTERNS Ring Patterns: Radius of each ring is characteristics of inter-planar spacing. Steps are: <ul><li>Measure the diameter of the ring. </li></ul><ul><li>Convert the distances into interplanar spacing using </li></ul>3. Use ASTM index to identify the phases. 4. If the material is known, one can measure the ratio’s of the square of the diameters of the outer ring to that of the 1 st or 2 nd (lower index) ring. 5. Check the ratio’s against a table of ratio’s for the crystal structure of interest. Spot Patterns: 1. Measure the distances R1, R2,… of the diffracted spots from the central spot. These distances correspond to the interplanar spacing of the reflecting planes and hence related to the camera length L. 2. The angles between the lines drawn from the central spot to the diffracted spots (h1k1l1), (h2k2l2) are the angle between the planes.
  60. 76. INDEXING DIFFRACTION PATTERNS General Method: 1. Measure R of the fundamental reflection. 2. Calculate the corresponding plane spacing d hkl . 3. Index the reflections with (hkl). In a good approximation primary beam is parallel to the reflecting planes. That is primary beam corresponds to the zone axis of the reflecting plane. Addition Rule: If a zone includes the plane h i and k i , then it also includes the plane (h i + k i ). For the diffraction pattern of a single crystal this implies that after indexing two non-collinear fundamental reflections, the indices of the entire pattern follow from vector addition.
  61. 79. DETERMINATION OF BEAM DIRECTION The zone axis [uvw]=Z=B can be determined by using the relations , u=k1l2-k2l1 v=l1h2-l2h1 w=h1k2-h2k1 Where the spot (h1l1k1) is positioned counter clockwise around the central spot relative to the spot (h2k2l2). If the spots are expressed in terms of reciprocal lattice vector g1 and g2 then B= g1 g2

×