TEM presentation Final - Florida International...

$: TEM presentation Final - Florida International Universityweb.eng.fiu.edu/wangc/TEM_presentation_Final.pdf · We Control Quality of our image, Diffraction patterns, and analytical$
Presenters:Presenters:Majid Beidaghi

Yamini Parikh

Yin Song

Abirami Dhanabalan

Varun Penmatsa

EMA 6518 EMA 6518 -- Transmission Electron Microscope Transmission Electron Microscope –– Lenses, Apertures &resolutionLenses, Apertures &resolution

•• Why should we learn lenses?Why should we learn lenses?

•• Light optics and electron opticsLight optics and electron optics

- How to draw a ray diagram

- The principle optical elements

- The lens equation

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We Control Quality of our image, Diffraction patterns, and analytical signals by adjusting the lenses and their apertures.

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Understanding of electron lenses will help us to answer such questions as:� Why we see finer details with an electron microscope than with a light microscope?� Why can’t we see as much as detail as we might expect from physics?� Why does TEM have a better depth of field and depth of focus than the light microscope?

Convex glass lens:

It can be used in two ways to control light rays coming through it

- It produces magnified image of the object

- It can focus a parallel beam of light to a point

Electron lenses in TEM are basically doing the same job.


“Transmission Electron Microscopy”, David. B. Williams and C. Barry Carter 4


β

α

Optic axis

How to Draw a Ray Diagram How to Draw a Ray Diagram


Figure 1: Image formation by a convex lens



Figure 2: How to draw a Ray diagram


The Principal Optical Elements The Principal Optical Elements


Figure 3: A complete ray diagram for a finite object


“Transmission Electron Microscopy”, David. B. Williams and C. Barry Carter

The Lens Equation The Lens Equation

u = Distance from the object plane to the lens

V = Distance from the lens to the image plane

f = Focal length

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• Magnification, Demagnification and focus*

• Depth of Focus and Depth of field*

• Apertures and Diaphragms*

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*David. B. Williams and C. Barry Carter, Transmission Electron Microscopy, Basics, Chapter 6.

Magnification/ Demagnification:Magnification/ Demagnification:

It is the process of enlarging something only in

appearance, not in physical size.

NewtonNewton’’s Lens equation for a convex lens:s Lens equation for a convex lens:

MMTT = v / u Where, = v / u Where,

MMTT = Magnification of lens= Magnification of lens

u = Distance between specimen u = Distance between specimen

and Lensand Lens

v= Distance between Lens and v= Distance between Lens and

Image planeImage plane

D = Demagnification = 1 / M

• Strengthening the lens shortens the focal length f.

• A weaker lens (f1) produces a higher magnification of the object than a shorter lens (f2) since the image distance v increases, but the object distance is unchanged. 10


Figure 4: (b) Ray diagram showing

two different image planes

β

α

Optic axis

Figure 1: Ray

diagram

v

u

Figure 4: (a) TEM column 1

Angular magnification MA = α/ βWhere, β = Collection semi-angle of the lens subtended at the object, α = Convergence semi-angle of the lens subtended at the image

1www.nobelprice.org2www.rodenburg.org2

• How can we change the strength of magnetic

lens?

1. Increase the number of turns of copper wire

in the lens,

2. Increase the amount of iron in the lens, either

by enlarging the core or adding the pole

piece,

3. Increase the amount of current applied to the

windings of the lens.

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*David. B. Williams and C. Barry Carter, Transmission Electron Microscopy, Basics, Chapter 6.

In a magnetic field, an electron experiences the Lorentz force F:

F = -e (E + v x B)

|F| = e v B sin(v, B)

E: strength of electric field

B: strength of magnetic field

e/v: charge/velocity of electrons

Figure 4(c): Electromagnetic lens

Focus: Focus: is is also called an image point, It is the point where, the electron rays which are originating from a point on the object, converge .

a) Ray diagram explaining the concepts of overfocus, in which a strong lens focus the rays before the image plane,

b) Underfocus, where a weaker lens focuses after the image plane

c) In Underfocus the convergent rays are more parallel than the equivalent divergent rays at overfocus ( α2 < α1)

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Figure 5: (a) Over-focused , (b)

Focused, (c) Under-focused

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Figure 6: (Top) Underfocused, (

Middle) at focus, ( bottom)

Overfocused

http://nobelprize.org/educational_games/physics

/microscopes/tem/tem.html

TEM Simulator: Magnification and focus TEM Simulator: Magnification and focus

effect:effect:Resolution is the ability to tell two points apart as separate points. If the resolving power of your lens is 2um that means two points that are 2um apart can be seen as separate points. If they are closer together than that, they will blend together into one point.

The magnification is something different-the ability to make an object larger. If the resolving power of a microscope is poor, it will just magnify a blurry object.

They are separate properties.

Do not get confused between Magnification and Do not get confused between Magnification and

Resolution!Resolution!

Focusing simulation

Depth of field: Depth of field: DDobob

Depth of sharpness in Depth of sharpness in objective planeobjective plane“The distance along the axis on both sides of the object plane within which the object can be moved w/o detectable loss of sharpness of the image.”

Depth of focus: Depth of focus: DDimim

Depth of sharpness in Depth of sharpness in imaginary planeimaginary plane“The distance along the axis on both sides of the image plane within which the image appears sharp.”

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Figure 7: Two different rays convergence

( Up ) depth of field, ( Down) Depth of

focus

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Figure 8: (a) Ray diagram showing how a

diaphragm restricts the angular spread of

electrons entering the lens, (b) A

selection of diaphragms: the top and

middle left are upper and lower views

respectively, of a conventional objective

diaphragm

Apertures:• Control the resolution of the image

formed by lens,• The depth of field, • The depth of focus,• The image contrast,• The collection angle of the electron

energy loss spectrometer,• The angular resolution of the diffraction

pattern.

Diaphragms:• Exclusion of high angle scattered

electrons to pass through the aperture,• Prevents the specimen damage due to

stray radiation.

Aperture is the hole in the disk, while the metal surrounding the aperture is called diaphragms.

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Figure 8: (Left) Condenser aperture, (Middle) Objective

aperture, (Right) TEM column with all lenses.

Types of Apertures:

1.Condenser aperture (diaphragm) is used to limit the beam divergence to reduce beam load on the specimen.2.Objective aperture located to the back focal plane to control the image contrast.

How aperture alters the depth of field and depth of focus?

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•• Electron lenses:Electron lenses:

– Pole pieces and coils

– Different kinds of Lenses

– Electron ray paths through magnetic fields

– Image rotation

– Deflecting the beam

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Two parts to make a magnetic electron lens:

• Pole pieces: a cylindrically

symmetrical core of soft magnetic material with a hole drilled through it.

• Coils of copper wire: coils surround each pole piece and when there is a current passing through the coils, a magnetic field is generated in the bore.

Figure 10: Schematic diagram of a magnetic

lens

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Objective lens:Objective lens:

• The objective lens is the most important lens in the whole microscope.

• The objective lens forms an inverted initial

image, which is subsequently magnified.

• In the back focal plane of the objective lens a diffraction pattern is formed. The objective aperture can be inserted here.

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Figure 12: Schematic

diagram of objective lens

Intermediate lens :Intermediate lens :

• The first intermediate lens magnifies the initial image that is formed by the objective lens. The lens can be focused on.

• Second image formed by the objective lens or diffraction pattern formed in the back focal plane of the intermediate lens.

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Figure 13: Intermediate lens

Projector Lens:Projector Lens:

• Magnification in the electron microscope can be varied from hundreds to several hundred thousands of times.

• This is done by varying the strength of the projector and intermediate lenses. Not all lenses will necessarily be used at lower magnifications.

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Figure 14: Projector lens

Resulting Lorentz force FResulting Lorentz force F

q=electron charge,

E=electric fields of strength,

B=magnetic fields of strength,

v= velocity of electron,

r= radial distance of the electron from the optic axis,

m= mass of the electron

V= accelarating voltage of the microscope

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Figure 15: Electron trajectories in a magnetic field

While all these ray equation are approximations, they form the basis of more detailed mathematical models of electron motion through lenses.

The effects of image rotation of TEM is because the image, or diffraction pattern,

rotates on the viewing screen as focusing and changing magnification.

http://www.matter.org.uk/tem/rotations_in_tem.htm

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• This operation is important to the whole process of forming a scanning image.

• The way is to apply an electromagnetic field to tilt or traverse the beam, or an electrostatic field to blank it.

• For small angle of deflection ε =e LB/mv

e= electron charge

L= a length over which the magnetic field acts

over

B= magnetic fields of strength

m= mass of electron

v= velocity of electron

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• Real lenses and their problems:

– Spherical aberrations

– Chromatic aberrations

– Astigmatism

– Coma

– Field curvature

– Distortion

– Anisotropic - coma, distortion, astigmatism

– Axial astigmatism

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• Spherical aberration (SA) is an image imperfection that is due to the spherical lens shape.

• Not a problem with glass lenses

• Disc of minimum confusion results instead of point focus.

• Not correctable for electromagnetic lenses

ab

c

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Figure 16 (a): Spherical aberration of the lens

Figure 16 (b): Longitudinal section of focused beam

•Under corrected spherical aberration: Marginal focus is closer to the lens than the axial focus.•Overcorrected spherical aberration: Marginal focus is located beyond the axial focus the lens.

TThe diameter of the Gaussian image of a point formed by paraxial rays is given

by,

Where Cs is the spherical aberration coefficient.

The radius of the spherical aberration disk in the Gaussian plane under non-paraxial

conditions is given by,

Where β= Maximum semi angle collection of the objective apertures.

The smallest dimension called disk of least confusion has a radius of

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• Reducing β gives a large reduction in the radius of disk of confusion.

• But for optimal resolution we need large β !

• Best compromise is with β = 10-3 radians (= f/500)

• Other way is to reduce the focal length.

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• When different colors of light propagate at different speeds in a medium, the refractive index is wavelength dependent. This phenomenon is known as dispersion.

• Chromatic aberrations : Departures from perfect imaging that are due to dispersion. Chromatic aberrations are only noticed with polychromatic light.

• Seidel aberrations: Monochromatic.

• The objective lens of the TEM bends electrons with lower energy more strongly and hence the electrons from a point in the object once again form a disk image.

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The radius rchr of the disk is given by

rchr =Cc∆Eβ/E0

Where ;Cc is Chromatic aberration coefficient∆E is the energy loss of electronsE0 is the initial beam energyβ is the semi angle of collection of the lens

Can be minimized using Thin specimens

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Figure 17: Chromatic aberration results in electrons with different energies being focused in different planes

y-focus

x-focusy

x

• Astigmatism arises when the lens is more powerful in one plane than in the plane normal to it.

• Cause points to be imaged as short lines, which ‘flip’ through 90 degrees on passing through ‘focus’ (minimal confusion)

• Loss of axial asymmetry

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Figure 18 (A): Schematic of astigmatism of the lens

Figure 18(B) : Diffractograms of carbon foil (a) weak astigmatism (b)stronger astigmatism

Astigmatism arises fromAstigmatism arises from

• inherent geometrical defects in ‘circular’ bore of lens

• inherent in homogeneities in magnetic properties of pole piece

• build-up of contamination on bore of pole-piece and on apertures gives rise to non-conducting deposits which become charged as electron strike them

– hence astigmatism is time-dependent

– cannot be ‘designed out’

– inevitably requires continuous correction

There is a variety of contributions to astigmatism, which distorts the image

by an amount given by,

where ∆f= maximum difference in focus induced by astigmatism

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• With glass optics (as in spectacles) astigmatism is corrected using an additional lens of strength & asymmetry ,opposed to the asymmetry of the basic (eye) lens with electron optics, same principle employed:

– electrostatic stigmator lens apposed to main lens

– strength & direction of its asymmetry user-variable

• The OBJECTIVE lens needs accurate correction

• Correction usually good for 1-2 hours for routine work

•

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Astigmatism simulation

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• The RESOLUTION of the electron lenses

– Theoretical resolution

– Spherical aberration – limited resolution

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⁻ Resolution is the finest detail that can be distinguished in an image

⁻ The resolving power of the microscope is the ability to make points which are closely adjacent in the object

⁻ The minimum distance apart of the adjacent points in the object is the minimum resolvable distance

⁻ The point-to-point resolution of TEM images of a sample is expressed in terms of the phase-contrast transfer function (PCTF)

⁻ PCTF (Phase Contrast Transfer Function) is the function which modulates the phases of the electron diffraction pattern formed in the back focal plane of the objective lens

Theoretical resolutionTheoretical resolution

⁻ If there are no aberrations at all, the resolution of any lens (glass, electromagnetic) is customarily defined in terms of the Rayleigh criterion

⁻ Rayleigh criterion gives us a figure of merit in terms of the eyes ability to distinguish separate images of two self-luminous incoherent point sources

⁻ A single point source will not be imaged as a point, even if no aberrations or astigmation are present

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⁻ The finite size of the lens result in diffraction of the rays at the outermost collection angle of the lens.

⁻ This diffraction results in a point being imaged as a disk (called the airy disk) shown in Fig 19

⁻ Rayleigh stated that if the maximum from one source lies over the first minimum from the other source

⁻ The distance apart of the two incoherent point sources is defined as the theoretical resolution of the lens rth and is given by the radius of the Airy disks

38EMA 6518 EMA 6518 -- Transmission Electron Microscope Transmission Electron Microscope –– Lenses, Apertures &resolutionLenses, Apertures &resolution

Figure 19: (a) The airy disk intensity profile from

Two point sources P1 and P2 defines the resolution

of the lens (b) The two are separated such that the

maximum in the image of P1 overlaps the minimum

in P2. This is Rayleigh criterion

rth – Theoretical resolution

λ – Wavelength

β -- aperture collection semi-angle

Spherical Aberration – Limited Resolution

• The electron microscope has an aberration that cannot easily be corrected: the

spherical aberration

• Spherical aberration is an optical effect observed in an optical device that occurs

due to the increased refraction of light rays when they strike a lens, mirror near its

edge, in comparison with those that strike nearer the center.

• Spherical aberration (rsph) plays a role when the chromatic aberration is negligible (if

the specimen is thin enough)

• Resolution in the object is defined by the combination of the Rayleigh criterion and

the aberration error when rsph has strong dependence on β according to the formula

• Hawkes (1972) described a combination for resolution often known as the figure of

merit

• First consider combination of Rayleigh and spherical aberration disks in quadrature

39EMA 6518 EMA 6518 -- Transmission Electron Microscope Transmission Electron Microscope –– Lenses, Apertures &resolutionLenses, Apertures &resolution

rth – Theoretical resolution

rsph – Spherical aberration

• We can thus relate how r varies with β

• Since the two terms vary differently with the aperture collection semi-angle β, a compromise angle exists when

• From this equation the optimum (compromise) value of β is given by

• If this expression for βopt is substituted into equation, we get a minimum value of r(β)

• Typically, the value for rmin is ~0.25-0.3nm, but the high-resolution instruments have rmin~0.15nm

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Practical Resolution

•Principles of working of electromagnetic lens have been explained via Ray diagrams.

•Lenses are abysmal in performance, demanding small limiting apertures.

•The lens aberrations limit the resolution of the microscope, and requires an optimum aperture to get minimum resolution.

•The small apertures cut down the electron beam intensity, but also give us remarkable depth of focus and depth of field in our images and specimen, respectively

ThankThankThankThankThankThankThankThank

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TEM presentation Final - Florida International...

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