What are electron microscopes? Lecture Module #14 Introduction to - - PowerPoint PPT Presentation

Lecture Module #14 Introduction to Electron Microscopy

An overview

What are electron microscopes?

• Scientific instruments that use a focused beam of

electrons to examine objects on a very fine scale.

What is electron microscopy?

• Electron microscopy is the science and technology of

using an electron beam to form a magnified image.

• Advantages:

– The use of electrons rather than light provides a nearly 1000 fold increase in resolving power (i.e., ability to focus fine details) over light.

• Disadvantages:

– High cost – Time commitment – Small areas of analysis

Magnification

• Refers to how large an object can be made (and still

resolved).

• Advantages:

– You get higher magnification with electron microscopes techniques than you can with light. – Smaller wavelength higher resolving power.

• bject
• f

size image

• f

size ion magnificat =

Resolution is:

• The closest distance between two points that can

clearly be resolved as separate entities through the microscope.

d1

ro

Intensity Distance

Consider light as EM energy transmitted as a wave motion. We can consider light as a series of ripples impinging upon an obstacle.

Abbe relationship NA n d r

• λ

α λ 61 . sin 61 . 2

1

= = =

λ = wavelength of illuminant α = semi-angle n = index of refraction NA = numerical aperature

Resolution is defined by:

↓

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

Depth of Field

• A measure of how much of the object that we are

looking at remains in focus at the same time.

• Advantages:

– You get higher depth of field with (many) electron microscopy techniques than you do with light. – WHY?

Important Terms

• Depth of Field

– Height above and below the plane of focus that an image remains sharp. – DOF is a function of magnification, α, and probe size

Plane of focused image α BEAM Scan Region of image in focus DOF

SEM SEM TEM TEM EPMA EPMA

Other instruments we have or are getting

• DualBeam Focused Ion Beam (DB-FIB)

– Nano-machining platform

• X-ray photoelectron spectrometer / Auger (XPS/Auger)
• ???

What do we get out of electron microscopes?

• Topography

– Surface features of an object. “How it looks.”

• Morphology

– Size and shape of particles making up object.

• Composition

– Relative amount of elements and compounds making up the

• bject.
• Structure

– Crystallography. How atoms are arranged in the object

Comparison of Selected Characteristics of Light and Electron Microscopes

FEATURE Light Microscope SEM TEM

Uses Surface morphology and sections (1-40 µm) Surface morphology Sections (40-150 nm)

• r small particles on

thin membranes Source of Illumination Visible light High-speed electrons High-speed electrons Best resolution 200 nm 3-6 nm 0.2 nm Magnification range 10-1,000× 20-150,000× 500-500,000× Depth of field 0.002-0.05 nm (NA=1.5) 0.003-1 mm 0.004-0.006 mm (NA=10-3) Lens type Glass Electromagnetic Electromagnetic Image ray- formation spot On eye by lens On CRT by scanning device On phosphorescent screen by lens Information generated Phases Reflectivity Topography Composition Crystal orientation Crystal structure Crystal orientation Defects Composition Limiting Factor Wavelength of light Brighness, signal/noise ratio, emission volume Lens quality

Adapted from: Scanning and Transmission Electron Microscopy, S. L. Flegler, J.W. Heckman Jr., K.L. Klomparens, Oxford University Press, New York, 1993.

All of this information is related to properties.

Mechanical, Physical, Thermal, Optical, Electrical, etc…

structure properties processing performance

History of electron microscopes

• Developed due to limitations of light optical

microscopes (LOMs)

– LOM: ~1000x magnification; 0.2 µm (200 nm) resolution

• Transmission Electron Microscope (TEM) was

developed first.

– M. Knoll and E. Ruska, 1931 – Patterned “exactly” like a LOM. Uses electrons rather than light.

• Scanning Electron Microscope (SEM)

– 1942

Goodhew, Humphreys & Beanland Electron Microscopy and Microanalysis, 3rd Edition Taylor & Francis, London, 2001 70X 300X 1400X 2800X

Electron microscopy provides significant advantages over light

• ptical microscopy in

terms of resolution and depth of field

How do electron microscopes work?

• Form a stream of electrons in the electron source

(thermionic emission) and accelerate them towards the specimen using a positive electrical potential.

• Use apertures and magnetic lenses to focus the

stream into a thin focused monochromatic beam

• Focus the beam onto the sample using another

magnetic lens.

• Interactions occur inside the irradiated area of the
• sample. Collect results of interactions in a suitable

detector and transform them into an image (or whatever you are interested in).

The Electron Gun

• The electron gun provides an intense beam of high

energy electrons.

• There are two main types of gun. The thermionic

gun, which is the most commonly used, and the field emission gun.

• Electrons are generated and emitted from a filament

by thermionic emission (W or LaB6) or from a sharp tip by field emission (single crystal of W).

• The electrons are accelerated through a potential

difference.

Electron Source (i.e., electron gun)

Thermionic electron gun most widely used (~$70K-$150K) Electrons are emitted from a heated filament and accelerated towards an anode

Wehnelt cap

(negative potential)

W Filament Anode Plate

(positive potential) Electron Beam Space charge Crossover

– 10-1000 kV + Ground Bias resistor Wehnelt cap

(negative potential)

W Filament Anode Plate

(positive potential) Electron Beam Space charge Crossover

– 10-1000 kV + Ground Bias resistor Filament Made from a high Tmp material with a low work function (φ) in

• rder to emit as many electrons

as possible. The work function is the energy needed by an electron to overcome the barrier that prevents it from leaking out

• f the atom. φW = 4.5 eV, φLaB6 =

3.0 eV. Wehnelt A biased grid with a potential that is a few hundred volts different than the filament (cathode). This helps to accelerate the electrons and causes their paths to cross over. Crossover The effective source of illumination for the

• microscope. The size

is critical for high resolution applications. Anode Positively charged metal plate at earth potential with a hole in

• it. It accelerates the electron

beam to the high tension potential.

Operation of thermionic gun

• Apply a positive electrical potential to the anode
• Heat the cathode (filament) until a stream of electrons is

produced

– >2700 K for W

• Apply a negative electric potential to the Wehnelt

– electrons are repelled by the Wehnelt towards the optic axis

• Electrons accumulate within the region between the filament tip

and the Wehnelt. This is known as the space charge.

• Electrons near the hole exit the gun and move down the column

to the target (in this case the sample) for imaging.

Other Types of Electron Sources (i.e., electron guns)

Lanthanum hexaboride (LaB6) Higher brightness than W. More $$$ (~$500K) Field emission source Highest brightness. Even more $$$ (~$750K) ADVANTAGES Provide higher current density → Higher brightness

http://www.matter.org.uk/tem/electron_gun/electron_sources.htm

Brightness, B, is the beam current density per unit solid angle. B = jcπβ2 jc = current density β = convergence angle

The function of the electron gun is: To provide an intense beam of high energy electrons There are two main types of gun. The thermionic gun, which is the most commonly used, and the field emission gun.

• This is what makes electron microscopy possible.
• Electrons strike the sample leading to a variety of reactions.
• Use results of reactions to form image or generate other

information.

Beam-Specimen Interaction

Incident high kV beam of electrons Secondary e- (SE) Characteristic x-rays Visible light Backscattered e- (BSE) Auger e- Inelastically scattered e- Elastically scattered e- Bremsstrahlung x-rays (noise) Direct (transmitted) beam Absorbed e- e- - hole pairs

Bulk (SEM) Foil (TEM)

Interaction Volume

Interaction Volume

• Represents the region penetrated by electrons

– Signals must escape the sample to be detected

Secondary electrons Backscattered electrons X-rays Incident electron beam

~1 nm – Auger electrons ~5–50 nm Secondary electrons ~300 nm Backscattered electrons ~50 nm TEM specimen thickness ~1.5 µm X-rays ~1.0 µm X-ray resolution

Z = 29 (Cu), Accelerating voltage = 20kV

INCIDENT BEAM

Different signals come from different depths in the target (sample)

• X-rays:

– most escape – sampling volume ≈ interaction volume

• Backscattered electrons:

– originate further from the incident beam (some fraction of a µm)

• Secondary electrons:

– escape from a region slightly larger than the incident beam (several µm) – yield the best spatial resolution

Interaction Volume Backscattered electrons (BSE)

• Formation

– Caused when incident electrons collide with an atom in a specimen that is nearly normal to the path of the incident beam. – Incident electron is scattered backward (“reflected”).

• Use

– Imaging and diffraction analysis in the SEM. – Production varies with atomic number (Z). – Higher Z elements appear brighter than lower Z elements. – Differentiate parts of specimen having different atomic number Backscattered electrons are not as numerous as others. However, they generally carry higher energies than other types of electrons.

Secondary Electrons (SE)

• Formation

– Caused when an incident electron “knocks” and inner shell electron (e.g., k-shell) out of its site. – This causes a slight energy loss and path change in the incident electron and ionization of the electron in the specimen. – The ionized electron leaves the atom with a small kinetic energy (~5 eV)

• Use

– IMAGING! – Production is related to topography. Due to low energy, only SE near the surface can exit the sample. – Any change in topography that is larger than the sampling depth will change the yield of SE. More abundant than other types of electrons. They are electrons that escape the specimen with energies below ~50eV

Auger Electrons (AE)

• Formation

– De-energization of the atom after a secondary electron is produced. – During SE production, an inner shell electron is emitted from the atom leaving a vacancy. – Higher energy electrons from the same atom can fall into the lower energy hole. This creates an energy surplus in the atom which is corrected by emission of an outer shell (low energy) electron

• Use

– AE have characteristic energies that are unique to each element from which they are emitted. – Collect and sort AE according to energy to determine composition. – AE have very low energy and are emitted from near surface regions.

X-rays

• Formation

– Same as AE. Difference is that the electron that fills the inner shell emits energy to balance the total energy of the atom.

• Use

– X-rays will have characteristic energies that are unique to the element(s) from which it originated. – Collect and sort signals according to energy to yield compositional information. – Energy Dispersive X-ray Spectroscopy (EDS) Foundation of XPS (X-ray photoelectron spectroscopy). XPS can be used to determine the “state” of an atom and to identify chemical compounds.

Transmitted electrons

• Can be used to determine:

– thickness – crystallographic orientation – atomic arrangements – phases present – etc.

• Foundation for Transmission Electron Microscopy (TEM)

Specimen Interaction Volume

• Volume where electron beam - specimen interactions
• ccur.
• Depends upon:

– Z of material being examined

• higher Z materials absorb more electrons and have smaller

interaction volume

– Accelerating Voltage

• higher voltages penetrate further into the specimen and

generate larger interaction volumes

– Angle of incidence of electron beam

• larger angle leads to a smaller interaction volume

~1 nm – Auger electrons ~5–50 nm Secondary electrons ~300 nm Backscattered electrons ~50 nm TEM specimen thickness ~1.5 µm X-rays ??? X-ray resolution

Z = 29 (Cu), Accelerating voltage = 20kV

INCIDENT BEAM 10 Å - 30 Å – Auger electrons ~100 Å Secondary electrons <1µm - 2µm Backscattered electrons ~5 µm X-rays ~1.0 µm X-ray resolution

Z = 28 (Ni), Accelerating voltage = 20kV

INCIDENT BEAM ~50 nm TEM specimen thickness

• Interaction volume is larger for materials that have

lower atomic numbers and for higher incident beam energies!

~50 nm

Increasing atomic number (Z) Increasing incident energy (E0)

Z1 Z2 Z3 Z1 < Z2 < Z3

“Instruments of the trade”

• Primary Instruments

– Transmission Electron Microscope (TEM) – Scanning Electron Microscope (SEM)

• Variants

– Electron Probe Microanalyzer (EPMA) – Scanning Transmission Electron Microscope (STEM) – Environmental SEM (ESEM) – High Resolution TEM (HRTEM) – High Voltage TEM (HVTEM) – etc...

TEM

• Patterned after transmission light optical microscopes
• Yield Following Information:

– Morphology

• Size shape and arrangement of particles, precipitates, etc.

– Crystallographic information

• Atomic arrangement

– Compositional Information

• If proper detector is present

The TEM is analogous to the transmission LOM

Source Condenser system Stage Objective lens Final imaging system Observation and recording system

x y z

+ Tilt & rotation Vacuum column Apertures

x y z

1 in 1020 electrons are collected <1 in 106 photons are collected

TEM LOM

Phosphor screen; film; digital recording system Eyepiece or TV screen; film; digital recording system

Light source Electron gun

Components of the TEM

• Source – filament plus anode plates with

applied accelerating voltage.

• Condenser Lenses – electromagnetic lenses

adjusted by lens currents not position.

• Specimen Stage – allows translations and tilts.
• Objective Lens – usually <50X.
• Imaging System – multiple electromagnetic

lenses below the objective: set magnification, focal plane (image vs. diffraction pattern).

• Observation – fluorescent screen, plate film,

CCD camera.

x y z

+ Tilt & rotation Vacuum column 1 in 1020 electrons are collected

Phosphor screen; film; digital recording system

Electron gun

Adapted from B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

Condensor Lenses

ELECTROMAGNETIC lenses focus the electron beam to as small a spot as is

• possible. They are

equivalent to convex lenses in optical lens systems.

Light Source

OPTICAL LENS

Cu coils Soft Fe pole pieces

N S N S

Electron Source

MAGNETIC LENS

Adapted from B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

Electron Lenses

• Use electrostatic or electromagnetic fields to focus

beams of charged electrons.

• ELECTROMAGNETIC – Most lenses are of this type.

Consist of Cu wire coils around soft Fe cores. Sometimes an Fe pole-piece is used to “shape” the field.

• ELECTROSTATIC – Unusual. Only common

example is the Wehnelt aperture in the electron gun.

Electrons in Magnetic Fields

• A. Electrons moving through a perpendicular magnetic field experience

perpendicular forces.

• B. Electrons moving parallel to a magnetic field are unaffected.
• C. Electrons moving nearly parallel to a magnetic field take a helical

path around the magnetic field direction (i.e., the microscope axis).

electron Magnetic field Force on electron is

• ut of plane

No Force Electrons follow a helical path unaffected Nearly aligned electron

A B C

Adapted from B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

Trajectories in Electromagnetic Lenses

• When you adjust the magnification (and the focal length),

you modify the lens strength by adjusting the current in the electromagnetic lens coils.

• Since the magnetic field is changed,

so are the helical trajectories.

• This ultimately leads to image

rotation which must be corrected for or calibrated.

Rotated image Object Cu coils Soft Fe pole pieces

N S N S

Image is inverted and rotated Electron Source

MAGNETIC LENS

Beam Tilting and Translation

• The electron beam can be positioned for fine

measurements (spot modes) or scanning (SEM, STEM)

BEAM TRANSLATION BEAM TILTING

APPARENT ORIGIN APPARENT ORIGIN Upper scan coil Lower scan coil

Condenser Alignment

• In TEM it is important to center the condenser aperture about

the optical axis.

• If the aperture is off center, the beam is displaced away from

the axis when the condenser lens is focused.

• You can try condenser alignment at:

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

Condenser Astigmatism

• Astigmatism in the condenser lenses distorts the beam to an

elliptical shape either side of focus and prevents the beam being fully focused. It is corrected by applying two

• rthogonal correction fields in the x and y directions.
• You can try stigmator adjustment at:

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

Objective Aperture

• Depending upon where you position the objective aperture

and whether or not you tilt the beam, a bright field image, a dark field (i.e., scattered) image, or a diffraction pattern can be collected.

• You can look at this in more detail at:

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

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) AND http://www.matter.org.uk/tem/stem_images.htm

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

BF TEM Micrograph and SADP

Grain structure Crystallographic information [111] zone axis

NiAl-1.0Hf

250 nm

1hr NiAl-1.0Hf

BF TEM Micrographs

Precipitate distribution Defect structures

4hr NiAl-1.0Hf

100 nm

β′

Defect Structures

Dislocations imaged in NiAl-0.5Zr single crystals deformed at elevated temperatures.

Nano-scale X-ray Analysis

β′ -NiAl EDX spectrum

β′ -precipitate

β′

HAADF (Z-contrast imaging) analysis

2 1

Ni2AlHf Hf rich region

• Hf segregates to GB’s in as-deposited & annealed coatings.
• Hf precipitates as β´-Ni2AlHf phase after annealing.

NiAl-1.0Hf NiAl-1.0Hf

As-deposited Annealed 1000°C/1h

20 nm 200 nm

How does a TEM work?

• Pass a focussed beam of electrons through a thin foil
• As beam passes through sample, it is scattered
• Project the transmitted (scattered) beam onto a

phosphor screen to form an enlarged image

• Modes:

– Bright Field/Dark Field modes for visualization of structure and defects – Selected Area Diffraction/Convergent Beam Diffraction for crystallographic information

Electron Beam Thin Foil < 500 nm thick

Electron Gun Anode Plate First Condenser Lens (C1) Second Condenser Lens (C2) Condenser Aperture Sample Objective Lens Objective Aperture Selected Area Aperture First Intermediate Lens Second Intermediate Lens Projector Lens Phosphor Screen Used to magnify image.

Schematic Representation

• f optics in a

TEM

Technical Details

• Produce a stream of monochromatic electrons in the electron gun
• Focus the stream into a small coherent beam using C1 and C2

– C1 determines the “spot size” (i.e., size of electron probe) – C2 changes intensity or brightness

• Use condenser aperture to restrict the beam
• Part of the beam is transmitted through the sample
• Focus transmitted portion using the objective lens to form an image
• Objective and selected area apertures are used to restrict the beam further

– allows examination of diffraction from specific atoms, crystals, features… – SAD, CBD

• Enlarge image with intermediate and projector lenses

SEM

• Patterned after reflecting light optical microscopes
• Yield Following Information:

– Topography

• Surface features of an object. Detectable features limited to a

few nanometers depth.

– Morphology

• Size shape and arrangement of particles, precipitates, etc

– Compositional Information

• Elements and compounds the sample is composed of

– Crystallographic information

• Possible using new techniques (OIM/BKD)

x y z

Vacuum column

x y z

TEM SEM

Electron gun Condenser system Stage Objective lens Final imaging system Observation and recording system Probe lens Stage Scan coils

Time base Signal amplifier CRT

Condenser lens Also tilts and rotates Also tilts and rotates

In the SEM you use secondary signals to acquire images.

Electron gun Detector and processing system

Components of an SEM

• Source:

– same as TEM but lower V

• Condenser:

– same as TEM

• Scan Coils:

– raster the probe

• Probe Lens:

– lens that forms a spot at the specimen surface

• Detector & Processing System:

– collects signals such as X-rays and electrons as a function of time and position. – Provides digital images for real-time viewing, processing, and storage.

x y z Probe lens Stage Scan coils

Time base Signal amplifier CRT

Condenser lens Also tilts and rotates Electron gun Detector and processing system

Schematic Representation

• f the optics in

an SEM

Electron Gun Anode Plate First Condenser Lens (C1) Second Condenser Lens (C2) Condenser Aperture Sample Objective (probe) Lens Objective Aperture Scan Coils Detector

SEM: Technical Details

• Produce a stream of monochromatic electrons in the electron gun
• Focus the stream using the first condenser lens

– Coarse probe current knob

• The beam is constricted by the condenser aperture (eliminates high-angle electrons)
• Second condenser lens is used to form electrons into a thin, tight, coherent beam.

– Use fine probe current knob

• Use objective aperture to limit beam (i.e., eliminate high-angle electrons)
• Scan coils raster the beam across the sample, dwelling on the points for a

predetermined period of time (selected using scan speed)

• Final objective lens focuses beam on desired region.
• When beam strikes the sample, interactions occur. We detect what comes out of the

sample.

Secondary electron mode

Could get additional information in backscattered or mixed mode

10 µm

Substrate IDZ Coating γ′ γ γ+γ′ γ+γ′

40 50 60 70 80 10 20 30 40 10 20 30 40

• 15
• 10
• 5

5 10 15 20 25

0.0 0.2 0.4 0.6 0.8 1.0

γ' Substrate γ + γ'

Composition (at%)

Ni-AD Ni-Annealed

Ni

Al-AD Al-Annealed

Al

Pt-AD Pt-Annealed

Pt

Hf-AD Hf-Annealed

Hf

Distance (µm) Figure 4.8. Composition profiles of the major elements measured by WDS in an EPMA.

X-ray Analysis in the SEM EDS or WDS

Elemental diffusion profile NiAlPtHf coatings on CMSX-4 substrates for a temperature of 1150°C and times from (a) 20 hrs and (b) 100 hrs.

Al O Ni Pt Hf Al Ni O Pt Hf

Al2O3 + ???

???

• Change magnification by changing the scan area
• Focus by changing the objective lens current
• The probe current/spot size are controlled by the

condenser lens current

Useful things to remember about SEM operation

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

Microanalysis in Electron Microscopy

• Characteristic X-rays are always generated by

interactions between the incident electron beam and the sample. They constitute a fingerprint of the local chemistry.

• Collect X-ray signal to determine local chemistry
• Common Techniques:

– Wavelength-Dispersive Spectrometry/Spectroscopy (WDS) – Energy-Dispersive Spectrometry/Spectroscopy (EDS)

WDS

• Yields best discrimination of emitted X-ray signal
• Use a series of bent crystals to cover the range of

wavelengths of interest

• Scan wavelengths within each range by rotating the

crystal and moving the detector while keeping the position of the crystal fixed.

• X-rays are collected from the sample at a fixed angle.

The angle at the collecting crystal will vary with 2θ and the diameter of the focusing circle will change

Specimen X-ray Detector Receiving slits Bent crystal Radius = 2R R Focusing Circle

WDS WDS continued

• All peaks are scanned sequentially. You cannot

record more than one characteristic line at once unless there are multiple spectrometers available.

• EPMA!
• Data is collected sequentially. Takes more time but

the results are much more precise than EDS.

Resolution of M lines in a WDS spectrum of a superalloy (Oxford Instruments)

EDS vs. WDS

• Pulse height is recorded by the detector. It is related

to the energy of the photon responsible for the pulse.

• Solid-state detectors are generally used.
• EDS is faster than WDS
• Problems with EDS:

– Poor discrimination or energy resolution. WDS systems are much better, particularly when characteristic lines from different elements overlap. – Need a windowless or thin window detector to pick up light elements. – WDS is more quantitative

The BSE sampling volume is large which limits resolution

Multiple elements of BSE detector “Overlapping shadows” Reduced resolution SE image Single segment of BSE detector

Can use BSE signal in conjunction with SE signal to yield enhanced topographical information

Images

• Topography and atomic number influence signal.
• Backscatter coefficient:

– η = -0.254 + 0.016Z - 1.86x10-4Z2 + 8.3x10-7Z3

• Atomic number contrast:

– C = (η1 - η2)/ η1 TOPOGRAPHIC CONTRAST TOPOGRAPHIC CONTRAST Signals from 2 detectors

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)