Advanced Characterization methods



Advanced Characterization methods

1. Administration (syllabus, grading , etc.)

Syllabus:

Overview of course

Error analysis

Spectroscopy

Underlying physics – characterization of energy transitions in materials

Optical (absorption and emission, Raman, time dependent)

XPS,

NMR, EPR

Electrochemical

Scattering/diffraction

Fundamental idea – structure determination through scatter of radiation or particles

Optical

X-ray, neutron, electron

Fractionation

Separation of complex mixture according to specific quantity, followed by detection

Mass spec

Chromatography

Imaging

Optical (transmission, fluorescence, confocal, TP)

Electron microscopy (SEM, TEM, STM)

AFM & variations

Direct

Direct measurement of quantity

Electrical

Magnetic

Thermal

|Date |Topic |Assignment |

|10/20 |Intro, error analysis | |

|10/21 |Lab – fluorescence spectroscopy |Lab notebook |

|10/22 |Optical spectroscopy | |

|10/24 |Optical spectroscopy |Choose presentation topic |

|10/27 |Surface spectroscopy (XPS) | |

|10/28 |Lab – XPS | |

|10/29 |NMR | |

|10/31 |EPR, EC, etc. |1st homework |

|11/03 |Intro to scattering/diffraction, XRD | |

|11/04 |Lab – EC detection | |

|11/05 |XRD, E-diffraction, neutron | |

|11/07 |Intro to imaging, optical |1st lab due |

|11/10 |Electron microscopy | |

|11/11 |Lab-DLS | |

|11/12 |Scanning probe microscopy |2nd homework |

|11/14 |Fractionation – MS | |

|11/17 |Mid-term exam | |

|11/18 |Lab – SEM | |

|11/19 |Chromatography | |

|11/21 |Chromatography, cont. | |

|11/24 |Direct measurements – electrical | |

|11/25 |Lab – AFM | |

|11/26 |Magnetic, thermal |3rd homework |

|11/28 |Thanksgiving holiday | |

|12/01 |Elemental analysis | |

|12/02 |No lab – oral presentations | |

|12/03 |On-line vs. lab-based |2nd lab due |

|12/05 | | |

Grading

Labs – 6 labs, notebooks should be kept, two lab reports, will not know beforehand which ones. One report technical, the other less technical. 20%

Three homeworks 15%

One mid term exam 15%

One topic for presentation – three types of presentation (15-30 second, 2-3 minute, 10-15 minute) 15 %

Final 20%

Integrative experience 15%

2. Overview of course

More of a survey course – not in much depth about specific techniques, but will cover a lot of ground

However, will still explore underlying physics of general classes of technique (spectroscopy, scattering) as well as specific examples in some detail

Techniques will be described in terms of:

What exactly is being measured?

What is the underlying physics of the method?

How is the raw data analyzed, what is the end result (after analysis)?

What are the assumptions being made in both the measurement and analysis?

What sample preparation is needed?

What are the (dis)advantages compared to other techniques?

What are the limitations?

Characterization methods may be for determination of:

Composition

Structure

Process

Specific properties (thermal, electrical, etc.)

Can subdivide the course accordingly, but from the underlying physics, this doesn’t make mush sense (and often hard to separate, many techniques cover more than one)

The course will be divided according to some underlying principle of the method, in particular, we’ll cover the following

1. Spectroscopy – dispersion, energy resonant processes, e.g. optical absorption & fluorescence, NMR, ESR, XPS, SIMS, Auger, DSC, rheology.

2. Scattering and diffraction – interaction of radiation with material, not energy dispersion, but relating output to some structure or composition, X-ray, light scattering, electron, neutron diffraction.

3. Fractionation – Subdivide the sample according to some physical property, e.g., mass spec, GC, GPC, HPLC, electrophoresis.

4. Imaging – Create visual representation of some property – optical microscopy, TEM, SEM ,scanning probe.

5. Specific properties & other – electrical, mechanical, BET.

6. Process – measurement of something that changes with some reaction of system, generally more empirical.

For the specific properties, the analysis is straightforward. For example, in a measurement of electrical conductivity the current response to an applied voltage is measured. The assumptions and analysis may need to consider the relation between the actual response we measure (needle deflection) to the underlying quantity.

For others, we are using some raw data to infer other characteristics, which is likely to require both more analysis as well as assumptions about the underlying physics. For example, X-ray diffraction itself might be interesting but the main purpose is to reveal underlying crystal structure.

3. Spectroscopy (spectrometry) – material response vs. input energy

Materials have some energy response determined by composition and structure – the location and strength of these will be unique to the particular sample. Spectroscopy deals with looking at this dependence by applying some energy to a system and looking at the response as a function of this energy.

A. Optical – input is in the form of EM radiation, change in the energy is a change in the photon energy, i.e., the frequency or wavelength (not the irradiance).

Absorption

UV-VIS

IR, FTIR

Emission

Fluorescence

Interaction

Raman

Variations

Nonlinear

Time dependent

Combinations

Background

Interactions of EM radiation with matter

Photons can be absorbed and emitted, but also interact non-resonantly (scatter – all optical interaction with matter can be described in terms of scattering).

Semiclassically, the optical electric field of the radiation interacts with the charges in the material. For now, we consider the interaction with the electrons, although this can easily be generalized. Interaction with the electrons is the main one for optical properties in the ultraviolet and visible. Macroscopically and to first order, the complex polarization vector is the key quantity. Usually, the electric dipole approximation is invoked, which says that the important interaction between material and radiation is simply the interaction of a dipole with the field. The term in the Hamiltonian is

[pic]

This is often much weaker than other effects and can be calculated with perturbation theory. The term that comes out of this is the matrix element of r, which is proportional to the transition dipole moment

[pic]

Note that this is not the (permanent) dipole moment of the ground state – this is a dipole moment of the transition itself. On the macro scale, this is the difference between a permanent and induced polarization.

How is this connected to absorption?

We can wave our hands and use the following argument. The applied (optical) electric field induces a dipole. The energy of the dipole will be proportional to the square of the polarization and so the square of the dipole moment. So energy remove from the beam (& given to the dipole) is proportional to the square of the dipole moment as defined above. The irradiance of the incident beam is the energy per cross sectional area per time, so the absorption (change in this energy per unit cross section) will also be proportional to the square of the dipole moment.

The propagation of EM radiation through material can be characterized by the complex refractive index. The real part is the “usual” refractive index that relates to the speed of light in the material. The complex part is related to the absorption. The infinitesimal change in irradiance is

[pic]

This equation tells us that the relative change will be proportional to a constant (() and the infinitesimal distance the beam travels. Solving it gives

[pic] (Beer’s law)

Where L is the distance traveled through the medium.

Often we want an absorption per unit concentration of the absorbing species. Also, logs to base 10 are more practical for applications. The molar extinction coefficient is defined as

[pic]

where C is the concentration, usually molar (short aside on molar concentrations if needed). The corresponding equation for the irradiance is

[pic]

{Note – be careful about the use of ( and (, since they are defined by different people different ways. Chemists usually use ( to be absorption coefficient divided by the concentration, and occasionally ( is used with the natural log instead of the base 10 log.}

The exponent is the Absorbance, or Optical Density of the sample, and is a unitless quantity.

As we’ll see later, most transitions are not narrow lines, but rather broad peaks. A typical absorption peak in the visible looks like (OHP)

in other words, the absorption coefficient is some function of frequency or wavelength. An integrated absorption coefficient can be defined as

[pic]

where ( is the frequency. A unitless quantity called the oscillator strength can be defined as

[pic]

where me is the electron mass, c is the speed of light, (0 is the free space permittivity, L is the path length and e is the electron charge. Finally, one can show that the relation to the dipole moment defined earlier is

[pic]

for a transition between states n and m.

The transition dipole moment leads to selection rules for optical transitions. If the symmetry of the initial and final states are the same, the transition dipole moment will be zero, and so there will be no interaction with the field. This results in a so-called forbidden transition – in more intuitive terms, transitions are forbidden when charge distribution of ground and excited states does not change symmetry

A forbidden transition will have orders of magnitude weaker absorption than an allowed one.

Why weak and not completely gone?

- have neglected other parts of wave function (nuclear) – remove symmetry – e.g. spin-orbit coupling allowing triplet state excitation

- have used the dipole approximation – higher order terms may allow transition, but are weaker

In semiconductor and insulating crystals, the picture is similar. If the bottom of the conduction band and the top of the valence band are at different values of k, optical absorption or emission can only occur with the absorption or emission of a phonon, in a so-called indirect transition. This can be much weaker, in particular for emission.

Einstein A, B coefficients (detailed derivation will come next semester)

The rate of stimulated emission will be proportional to the input irradiance. The coefficient of proportionality is called the Einstein B-coefficient. The rate of spontaneous emission is constant for an excited state and is referred to the Einstein A-coefficient. These two are not independent, one can use energy conservation to find that

[pic]

Fate of excited states (OHP)

1. Luminescence

2. Inter- and Intra-molecular transfer

3. Quenching

4. Ionization

5. Isomerization

6. Dissociation

7. Direct reaction or charge transfer

We’ll talk more about these when we come to fluorescence emission spectroscopy.

Absorption spectroscopy

UV-VIS-NIR – UV radiation technically runs from about 10 nm (~100 eV) to about 390 nm (~3.2 eV), but most absorption spectroscopy is done in the lower end of energy, in the wavelength range 180 to 390 nm. Visible is defined as between 390 and 780 nm. Infrared is separated into three regions: NIR (780 nm to 3.0 microns) MIR (3.6-6.0 microns) FIR (6 to 15 microns). Sometimes an additional region called the extreme IR is defined (15 to 1 mm) but recently this has received more attention as the so-called THz region.

UV-VIS are usually grouped together because they involve transitions in the outer shell electrons. Not only are these important for identifying materials (composition) but are much more sensitive to surrounding changes such as bonding (structure). Infrared spectroscopy usually involves transitions between vibrational states, with energies corresponding to wavelength ranges from 800 nm to several microns.

UV-VIS – broad, often featureless, so not useful for identification of composition. More likely to be used for process control or as basic property – where something absorbs. Visual aspect of application, e.g., paper, colorants. Photometric titration.

Time scales (excited state lifetimes) – 10-11 to 10-8 sec

Causes of nonzero peak widths:

1. Intrinsic (lifetime, natural) broadening (uncertainty principle – in excited state for finite time, so the specification of it cannot be arbitrarily precise).

2. Vibrational and rotational sublevels

3. Doppler broadening – Doppler shifts due to motion of scatterers, direction and speed are distributions, so shifts have cumulative effect of broadening

4. Interactions with surroundings

Pressure (collisional) broadening – collisions with other scatterers induce shifts in frequency

Interactions with solvent – similar to pressure

Most of this discussion has been relevant to atomic and molecular spectra. Similar effects are also present in bulk materials such as semiconductors. In this case, it is often difficult to measure transmission spectra, so reflection spectra can be taken instead. These will probe the surface of the sample.

In ATR = Attenuated Total Reflection spectroscopy, the sample is in contact with a prism. Light is incident on the sample through the prism, where it is reflected at the prism/sample interface. At this interface, the light penetrates into the sample, even for total reflection. At absorbing wavelengths, this removes energy from the beam, which is measured in the output (reflected) beam spectrum.

Practical application

Spectrometers (spectrophotometers) – dispersive (moving, diode array), FT

Broadband sources, usually cw

Deuterium – UV

Tungsten – Vis

Xe – both

heater (blackbody) – IR

Detectors

PMTs

Photodiodes (arrays)

For UV-VIS, two calibrations are usually required – dark and reference. The dark removes the baseline signal from ambient light, detector noise or electronic offset, and the reference determines the I0 term in Beer’s law. This includes the spectral input of the source, any absorption by the cuvette and solvent, and the spectral response of the detector.

FT spectrometers

Instead of dispersive (either many detectors or moving grating, prism) all light from a broadband source is used. In absorption spectrometers, this is sent through an interferometer (usually Michelson). Changing the length of one of the interferometers arms modulates the output and creates a Fourier transform of the input light.

(see handout)

In an FTIR, this is used to modulate the input broadband IR source. The output in the FT of the source, with the displacement of the arm being the FT variable. This is sent through the material, and the output is FT’ed to give the transmitted light, and then the spectrum is determined.

IR absorption

The energy of photons in the IR region corresponds mostly to transitions between vibrational and rotational states in materials. The optical absorption bands for these transitions are much narrower than typical ones in UV-VIS spectroscopy, and so are much more useful for identification of the constituents of a sample.

The basic physics underlying these bands is easily understood at a simple level using the mass and spring model. The response of a system of masses attached by springs will depend on the excitation (initial conditions), the masses and the spring constants. In this case, the masses are the atoms and the springs the bonds, so the IR spectrum will uniquely determine the constituents and bonds between them, i.e., the molecules in the sample. Since calculation from first principles is difficult/time consuming, most analysis is empirical – bands of known compounds or functional groups have been measured and categorized in the literature.

(OHPs– energy levels, typical IR spectra)

Solid samples: powder samples are usually pressed in KBr pellets – these give very IR transparent samples with little scattering.

Liquids: Cells are usually very narrow, since IR absorption coefficients are generally high. Sodium chloride is a typical material for the cell windows, since it has good transparency in the IR.

Most current IR absorption spectrometers are of the FT type.

(Handout)

There are many advantages to FT-type spectrometers, the main one being that all the light transmitted through the sample is measured by the detector.

Raman Spectroscopy

IR spectra have similar selection rules as for optical spectroscopy – this means that many bands can not be measured with usual IR absorption. Raman spectroscopy (scattering) offers a way around this - since the interactions involve more than one photon, the selection rules are different than in standard absorption spectroscopy. Raman spectroscopy also has the useful advantage of much fewer problems due to water than IR spectroscopy (water gives a large broad signal conventional IR spectroscopy, which can hide other lines of interest).

The Raman effect is a parametric effect in which a photon is “absorbed” and re-emitted, with the energy difference being taken up by or released from a vibrational state of the material. When the final state has higher energy than the initial, the results are the so-called Stokes lines, when the final state has lower energy, they are the anti-Stokes lines. The Stokes lines arise from the final state being a higher vibrational state of the ground state manifold, the anti-Stokes line occurs when the initial state is one of these and the final state a lower energy vibrational level or the ground state. Since these rely on a finite population in these vibrational levels, they are generally much weaker and also temperature dependent.

X-ray absorption. The absorption of higher energy photons, in the X-ray region, usually causes transitions involving inner shell electrons, which are less affected by environment. This can give elemental information, rather than molecular. These electrons are generally ejected from the atom. (OHP of typical spectrum).

Appearance of X-ray absorption – now kinetic energy of electrons released plays a role – the edges are sharp with tails as more KE is given to the electrons.

The absorption can be measured in the usual way, or by monitoring the resulting fluorescence.

The measurement of the emitted electrons constitutes XPS – more on this later.

Photoluminescence

There are a number of possible fates of the excited state atom or molecule in terms of how it loses its energy, usually decaying back to the ground state. We can broadly divide these into two groups of transitions: radiative and non-radiative, depending on whether or not a photon is emitted in the process. (OHP)

The emission of photons in the UV-VIS range is referred to generally as luminescence. When the excitation is by a photon, the process is called photoluminescence (general term) or fluorescence/phosphorescence, depending on the state the emission comes from (and the time scale of the decay). Other processes can excite the material which then emits photons, for example chemical processes (chemiluminescence) biological (bioluminescence), electron injection (electroluminescence), etc.

The spin multiplicity of a state, defined as 2S+1, can yield information on the probability of a radiative transition. The usual stable (lowest energy) state is one where the electron spins are paired, S=0 and the multiplicity 2S+1=1 – this is termed the singlet state. When two electrons are unpaired, S=1, 2S+1=3 and the state is called a triplet state. The ordering of the state in energy is often added as a subscript, i.e., ground state S0, 1st excited singlet state S1, etc. Triplet states start at T1 being the lowest energy triplet state. The T1 state is lower in energy that S1.

Molecules will maintain spin multiplicity during transitions, meaning that transitions between singlet and triplet states are technically forbidden. However, since spin is usually not a pure quantity in a real system, transitions do occur but with low probability. This means that molecules in an excited S1 state may decay into a lower energy T1 state rather than the S0 ground state. This process is called intersystem crossing (ISC). Since this state and the ground state do not have the same spin multiplicity, the probability of decay is also small and the state consequently has a long lifetime (ms to seconds, minutes, hours). This radiative decay, when it does occur, is called phosphorescence, and is responsible for “glow in the dark” objects.

The emission of photons is nearly always from the lowest excited state (Kasha’s rule). In other words, the excited state will decay to the lowest singlet state before radiating. It also will generally emit from the lowest vibrational sublevel of the excited state by nonradiative processes, then emit the photon and decay to the ground state (manifold). The hand-waving argument supporting this is that vibrational relaxation is generally much faster (~femtoseconds) than electronic (~nanoseconds), so that it can happen before the electron has a chance to decay radiatively.

What does this have to do with spectroscopy? In fluorescence emission spectroscopy, the radiative emission due to a S1 to S0 transaction is measured as a function of the wavelength (energy) of the emitted photon. The final state will be a vibrational sublevel of the ground state, and vibrational structure is often seen in fluorescence emission spectra. However, there is also a finite probability that the emission will come from a triplet state – the spectral information will be the same, since all transitions come form one initial state and the final state manifold is the same, but of course the entire spectrum is shifted in this case.

Since the fluorescence emission comes from the bottom of the excited state manifold to the various vibrational sublevels, while the absorption is from the bottom of the g.s. manifold to the sublevels of the excited state, the transitions are not the same. However, in most cases the vibrational spectrum is nearly the same for both states. Even then, though, there is a shift of energy between the peak locations for absorption and emission, known as the Stokes shift. This is most easily explained in a diagram (OHP).

Time dependence of emission

In contrast to absorption, which occurs on the fs time scale, the decay of the excited by emission of photons is usually much longer. The decay time will be dependent on a number of factors, including of course the molecule and transition, but also on the surroundings. Time-resolved fluorescence spectroscopy is a useful technique for determining these. For nanosecond decay times, this can be measured directly in the time domain with PMTs (response times ~nanoseconds) fast photodetectors (ps response times). Often the electronics to process the detector response in the limiting factor in the minimum time that can be measured. For faster decay times, the measurements can be done using nonlinear optical (NLO) techniques. Measurements are also routinely done in the frequency domain by modulating the input light and measuring both the in and out of phase components of the output.

Nonlinear Optical (NLO) techniques

Up to now we considered only the linear response of a material to light, i.e. the first order polarization. Higher order terms exists, and can be used to characterize materials in a number of ways, as well as integral parts of other techniques.

Without going into too much detail, the polarization of a material in response to an optical electric field can be expanded in a power series in the field E. The linear term is the one we’ve been using so far. The next two terms, proportional to the square or cube of the field, are the source terms in second-order and third-order nonlinear optics, respectively. The effects due to them are progressively weak, but with various experimental techniques they can be measured easily.

We’ll mainly discuss second order effects, and then only briefly. In this case, there is a symmetry of the output that can be exploited to gain information about molecular orientation in a material. A polarization in response to the square of the electric field will only be non-zero for a material not having inversion symmetry. In a second order process, because of the quadratic field dependence, the polarization vector will be the same for the field in one direction as for the exact opposite field. Now if we inverted the material, and it had inversion symmetry, the response should be the same, but this can’t be since the polarization vector should be in the opposite direction. Therefore, we need this non-centrosymmetric property for second order NLO.

This can be useful in spectroscopy. Observation of a second order effect in an ensemble of molecules will imply that there is a molecular ordering giving a non-centrosymmetric material on the length scale of the wavelength or above.

Another useful application is for time dependent fluorescence (or other optical) properties. Second order processes allow one to combine two beams to produce a third at a different wavelength. Normally, photons do not interact, so two beams passing through a material will not affect each other. In a second order material, the polarization is proportional to the square of the applied field. If this comes from two different sources, there will be cross terms, giving a resulting polarization depending on both beams. By measuring the effect of this polarization (radiation),

Third order nonlinear spectroscopy

Pump-probe – in this method, excited states are studied by looking at the time dependence of the absorption after the material has been excited. First a very short monochromatic pulse excites the material, then usual absorption spectroscopy is done, but also with a short, broadband pulse at a fixed delay from the first pulse. This gives a view of both the absorption from the excited state as well as the depletion of the ground state (decrease in the usual absorption).

Electron spectroscopy

Instead of just letting the electrons fly off, they can be used to determine the energy levels of the molecule they came from. We’ll concentrate on two of these types of spectroscopy: X-ray Photoelectron Spectroscopy (XPS) and Auger electron spectroscopy. Both are mainly used for surface characterization due to the large absorption or scattering in most materials of the exciting beam (X-rays or electrons) and the emitted electrons.

XPS

In XPS the input energy is from X-rays in the keV region. This ejects electrons from the inner shells of the material being characterized, which are measured spectroscopically (energy). Standard electron spectrometers use a magnetostatic field to bend the electron beam through a semicircle to the detector; the field strength dictates the energy of the electrons needed to stay on this path to reach the detector. (OHP)

Source: For XPS, typically K( lines of Magnesium ((=9.8900 Å or E=1.254 keV) or Aluminum (( = 8.3393Å or E=1.487 keV) (OHP)

A related technique, Ultraviolet Photoelectron Spectroscopy (UPS) uses the same principle, but with the exciting photons in the UV range (10’s of eV).

XPS (and UPS) can give information about composition, but also structure and oxidation state. The reaction is written

A+h((A+*+e-

where A is an atom or molecule and A+* is the electronically excited ion. The electron emitted will have a kinetic energy

Ek=h(-Eb-(

where Eb is the binding energy of the electron and ( is the work function of the material (energy needed for charge to escape surface). The work function can be measured in other ways, and the binding energy is the energy of the atomic or molecular electron orbitals relative to zero at infinite distance from the nucleus. The energy spectrum of the emitted electrons, minus the input photon energy and work function, therefore, gives the electron energy spectrum. The energy of the emitted electrons is in the range 250 to 1500 eV, with binding energies in the 0 t o1250 eV range.

The approximate sampling depth for XPS is d=3(sin(, with values typically 1 to 10 nm. The spatial (cross sectional) resolution down to 10 (m, so that surface imaging is possible at the microscopic level.

Applications of XPS

Low resolution XPS is often used for elemental analysis at surfaces. This is usually a qualitative analysis in the sense that the elements are identified only, but the relative amounts present are not. This limitation stems from the need to calibrate the detectors in terms of the peak heights or peak areas as a function of energy, which is difficult to do accurately.

Chemical shifts: High resolution XPS can be used to investigate molecular environments, because slight shifts in peak locations give information about the chemical environment. This is due to the weak effects of the outer shell electrons on the energy levels of the inner shell electrons. The outer shell electrons are affected by bonding and oxidation state. This can also yield structural information, depending on the electron withdrawing (or donating) abilities of the other atoms bonding to a particular atom. (OHP)

Limitations

Due to the short penetration depth, XPS is mainly limited to being a surface technique. Chemical structure near a surface may not also represent the bulk, for example many metals oxidize, so XPS spectrum are of the oxide and not the metal itself. This can be very useful, though, in characterizing these surface phenomena.

The difficulty of calibrating peak heights also makes this technique difficult for quantitative analysis, as described earlier.

There also can be interference from Auger electrons (see next) although this is easily accounted for.

Finally, some materials are sensitive to high energy irradiation, so there is a danger of either damaging the sample, or worse, changing its chemical composition during measurement without knowing it.

Auger electron spectroscopy (AES)

AES is a related technique based on a similar, two step excitation process. In it, an electron in an atom or molecule is ejected by an incident electron beam or X-ray. There is now a vacant inner orbital state. This excited ion can decay by either an electron dropping down from a higher level with the emission of a photon (usually X-ray – this is X-ray fluorescence) or by the Auger process. In this process, an upper level electron also drops down, but the energy it releases is given to a second electron, which is ejected from the ion. The first process can be written as

A+ e-i (or h()(A+*+e-i‘+e-A

Where e-i is the incident electron before (unprimed) or after (primed) the interaction and e-A is the “first” Auger electron that is ejected from the inner orbitals.

The second process can be either X-ray fluorescence

A+*(A++ h(

or the Auger process

A+*(A+++e-A

where the ion is now doubly ionized and the emitted Auger electron is the one that is spectroscopically analyzed. The advantage with using this process over XPS is that the kinetic energy of the Auger electron is now independent of the original excitation. The kinetic energy will simply be the difference between the energy of relaxation (Eb- Eb‘ where the prime is the initial and unprimed the final energy level) and the energy needed to remove the second electron, Eb‘. Note that these are the binding energies of the electrons, i.e., we are going from zero at infinity to larger values as we go closer to the nucleus (need more energy to remove). The kinetic energy will then be

Ek=(Eb-Eb’) - Eb‘ = Eb-2Eb’

In Auger spectroscopy, the independence of the process on the source energy means that broadband sources can be used. The excitation generally produces both Auger and XPS electrons – they can be differentiated by this dependence on the excitation source. In Auger spectroscopy with broadband sources, the XPS electrons will just be a broad background, but in XPS, the Auger electrons can be accounted for only by using a different source.

NMR spectroscopy

Nuclear magnetic resonance is an extremely useful technique for materials characterization and imaging. The principle is based on the absorption of radio frequency radiation by nuclei in a material. Normally these nuclei would not absorb, but the application of a strong static magnetic field causes a splitting of nuclear energy levels due to nuclear spin. This allows one to vary both the energy levels by changing the static field, as well as scanning the radio frequency to determine the level splitting. The splitting will depend on both the field strength and the nucleus itself, but also on the surrounding electrons, which shield the nucleus from the applied magnetic field.

Theory

A nucleus has a magnetic dipole moment (m proportional to its angular momentum, p:

(m =(p. The constant of proportionality, (, is called the gyromagnetic (or magnetogyric) ratio. This ratio has a unique value for every nucleus and is the final quantity used for the analysis in NMR spectroscopy. The angular momentum vector is proportional to the nuclear spin I: p=(I, giving (m =((I. The interaction of an external field B with this moment gives rise to an interaction energy

U=-(m ·B = -((I·B

The direction of B specifies a direction (B=B0k), so the spins (angular momenta) along this (z-) axis will be quantized with quantum numbers: mz=I, I-1, ... –I, where I is the magnitude of the vector I. The resulting energy term is

U= -((mzB0

In a nucleus with I=½, the two possible values of mz are ±½, giving a splitting of the energy level (E=-((B0

which corresponds to EM absorption at a frequency (=(E/h=-(B0/2(. This is the basic equation for NMR. It says that the frequency of absorption of EM radiation by a nucleus in the presence of an applied magnetostatic field will be proportional to the field strength and the magnetogyric ratio.

For example, for a proton, ( = 2.675(104gauss-1s-1, giving an absorption frequency in Hz of 4.258(103 B0 (gauss) – typical absorption frequencies are in the radio range. By applying either a constant field and scanning the radio frequency or a constant frequency and changing the applied field (usual method), the magnetogyric ratio can be found and the material identified.

Since the energy level difference between the two split levels depends on the applied field, it can be (usually is) small compared to kT at room temperature (kT~40meV). To determine the dynamics of a nuclear spin system under a field plus EM radiation, we need to use statistical mechanics/thermodynamics.

In thermal equilibrium, the proportion of spins in the two states can be calculated using Boltzmann’s equation as

[pic]

The total magnetization will then be

M=((N2-N1) = N(tanh((B/kT)

When the magnetization is no longer in thermal equilibrium, we assume that it approaches the equilibrium value exponentially, with a rate proportional to the deviation from the equilibrium

[pic]

T1 is called the longitudinal relaxation time. If a sample in thermal equilibrium is placed in a field, the magnetization will realign to a new equilibrium value. Integrating the equation above gives:

[pic]

Semiclassically, the torque on the moment (m will be (m × B giving the equation of motion for the moment as

dp/dt = (dI/dt = (m × B

which leads to

d(m/dt = ((m × B

If we rewrite this in terms of the macroscopic magnetization vector, M=((mi where the sum is over all spins in the material, we can write

dM/dt = (M × B

where we’ve assumed only one isotope is important, so the magnetogyric ratios are the same for all species.

Now we also need to include the relaxation term. The result for the z-component of magnetization is

[pic]

Likewise the transverse components of the magnetization will decay to their equilibrium value of zero, if they were not at zero when the field was applied

[pic]

[pic]

where T2 is called the transverse relaxation time. These three equations together describe the motion of the magnetization vector. It will precess about the applied field, depending on its direction when the field was applied, but also relax to an equilibrium alignment with the field. Since Mx and My are perpendicular to the field, no energy flows in or out of the system, so the relaxation will be different than for Mz (hence different relaxation time). T2 can be thought of as the time required for the spins to lose their phase.

{There is a similar description of the resonant absorption and decay of optical radiation by a two level system. In it, the T1 relaxation time is called the population decay time and T2 the phase decay time}

These three equations are called the Bloch equations – from them, we can determine the response of a system to an applied field in terms of line width and intensity.

The longitudinal relaxation is often called the spin-lattice relaxation, after the main mechanism for it: the interaction of the spins with the lattice. The transverse relaxation is called spin-spin relaxation.

Typical values for T1 are on the order of seconds, while T2 values are generally smaller.

As described earlier, the positions of the resonances in the spectra are unique to each species. The electrons will affect these resonances, so information on the bonds can also be extracted from the spectra. These so-called chemical shifts are central to the use of NMR spectroscopy in materials analysis. They can give information not only on bonds themselves, but also their orientations.

Application

Standard NMR is preformed in the time-domain by application of short duration RF pulses, followed by measurement of the relaxation vs. time of the “Free Induction Decay” (FID). By Fourier transforming this data, a spectrum is obtained. Two main types of NMR are proton (H1) and carbon, referring to the specific nuclear moment being measured. Since the nuclear spin of carbon-12 is zero, the line for carbon-13 (~1% of naturally occurring carbon) is used. For quantitative analysis of the material present, the chemical shifts are the quantity of interest, so these shifts are measured for the proton or carbon lines.

Since the splitting is B-field dependent, and calibrating the field exactly is difficult, a reference is usually needed. This is usually accomplished by adding a well characterized material to the sample - Tetramethylsilane, (CH3)4Si, usually referred to as TMS, is the standard for this. Solvents used for samples must also have no strong lines to interfere with the sample signal. This is often achieved by using deuterated solvents for proton NMR.

General point about lines in spectroscopy – if two states are separated by an energy corresponding to frequency (( and if the decay time of the states is on the order of (, the lines will no longer be capable of being resolved when (((~1 (1/2(). One way to think about this is in terms of the uncertainty principle – the energy of states will be only defined to within the inverse lifetime, so if the states are short-lived, the uncertainty in energy means the spectral lines are correspondingly broad and overlap. In this case, only a single broad feature is measured instead of separate lines. Another way to think about this is that over some characteristic time of the measurement, the molecule will rapidly change between states, so the spectrum will be some average of the two states.

Due to the close spacing of the spectral lines in NMR, this consideration is particularly important for these measurements.

Limitations

NMR is limited to nuclei that have nonzero magnetic moments. It is also less sensitive compared to other methods for analysis.

2D NMR

Multiple RF pulses can also be used to reorient dipoles that are losing phase by precession. This allows one to independently determine both T1 and T2 relaxation times. This can help resolving lines that cannot be isolated in standard 1-D NMR spectroscopy.

ESR

Electron Spin Resonance spectroscopy (also know as EPR – electron paramagnetic resonance) is similar to NMR but uses the interaction of an applied field with magnetic moments of unpaired electrons to split energy levels. This is useful for materials such as radicals, metal containing complexes and semiconductors.

In contrast to NMR, this technique measures effects associated with the unpaired electrons in the valence (outer) electron bands of a material, and so it is sensitive to environmental effects. In particular, the interaction of this electron with its own nucleus leads to hyperfine structure in the resonance spectrum, which can yield information about the electron density in relation to the nucleus.

Applications

ESR is extremely sensitive, and since it detects materials with unpaired spins, it can be used to detect the presence of impurities such as metal ions in a polymer or semiconductor.

Mössbauer spectroscopy

Absorption spectroscopy extended to extremely high energies ((-rays, 1019 Hz). Samples are embedded in a crystal lattice. The lines measured here are extremely sharp, and give similar information to that obtained with NMR. The sharpness of the lines is due to 1) long lifetimes for nuclear excited states, so small lifetime broadening, 2) momentum of photon is small since recoil is taken up by lattice – this means the Doppler shift will be small.

Elemental analysis

One final, non-spectroscopic method for analyzing composition of materials is elemental analysis. This is simply using a stoichiometric chemical reaction to produce a product that can be more easily quantified. Some of these final quantification steps are simple, for example gravimetric analysis. In this method, the reaction causes the precipitation of a known product, which is weighed to calculate the total amount. Other techniques include all of those we’ve talked about or will talk about for identifying composition.

Since the reaction must be chosen to react with a particular species, this method is most often used when the presence of a particular species is known or suspected, and the analysis is simply to determine the amount.

4. Scattering and diffraction techniques

As we’ve seen, spectroscopic techniques yield information on energy levels in materials. This has most application for determination of composition or for assessing changes in materials as a function of some processing. While structural information can also be found, it usually involves some assumed models of the relation of the energy levels to the structure.

A more direct characterization of the structure can be obtained from scattering methods. (General use of the term scattering).

Scattering, general

In principle, the propagation of radiation (or other wave-like things) in material can always be discussed purely in terms of scattering. At a molecular or atomic level, an atom sees an incident EM field, polarizes (interacts nonresonantly with the field) and this oscillating dipole radiates in all directions. The sum of the incident field and all the radiating fields is the EM radiation field in the material. For example, the refractive index of a material (speed of light) can be determined from this type of calculation.

Starting out with this simple idea, we assume the dipole moment of the scatterer as

((t)=(0cos((t); where ((t) and (0 may be complex.

The resulting radiation field will be

[pic]

which gives an irradiance

[pic]

Note the inverse square law and the dependence of the irradiance on the fourth power of the frequency.

This scattering is elastic – there is no change in energy (frequency) between the incident and scattered photon.

Now consider a collection of scatterers. The phases from all of these will add up to the resulting field at any point in space. If the scatterers are randomly placed, the scattering at directions other than forward will be random. If their distribution is tenuous enough, the radiation from individual scatterers will arrive at some point away from the forward direction completely incoherent with the radiation from the other scatterers. In this case, we can simply add the irradiances, ignoring phase (interference) effects to determine the total scattered irradiance. For small particles or molecules, this is called Rayleigh scattering.

Since the dipole moment of a particle (a collection of molecules) will go as the volume of the scatterer, the irradiance will have a sixth power dependence on the particle size. This allows one to use static particle scattering for determining particle size in dilute dispersions.

Note that in the forward direction, the scattering from separate scatterers will always be in phase.

Now we’ll start discussing diffraction/interference effects, which use the wave nature of EM radiation (or particles) scattered from a material to understand its structure.

The starting point involves a consideration of a monochromatic plane EM wave.

[pic]

Consider a simple atomic plane in a material. X-rays have wavelengths on the order of sub 1Å, which is the same order of magnitude as the atomic spacing. First consider what happens to the individual atoms. Each experiences an applied EM field, has a polarizability, and so a dipole (& higher order) is induced. This radiates out in all directions. If the radiation is coherent with the input (probably non resonant) there is a fixed phase relation between the input and output waves. Now consider the next plane down. An atom here sees the same EM field, plus some phase difference due to the path length difference. We assume that the phase relation between input and output is the same. Therefore the rays scattered from the two will be in phase if the optical path length is an integral multiple of the wavelength. This leads to the Bragg condition

[pic]

This is the basis of X-ray (neutron, electron) diffraction.

[review of FT]

Reciprocal lattice

Crystal structures are generally specified according to the real-space intercepts of the crystal planes. For example, a plane that intercepts the Cartesian (x,y,z) axes at ha0, kb0, lc0, where a0, b0 and c0 are the unit cell dimensions, is referred to as the hkl plane. As a results, planes in real space correspond to points in this reciprocal space, with the set of these points making up the reciprocal lattice.

The axis vectors of the reciprocal lattice are constructed from the primitive vectors of the crystal lattice:

An arbitrary vector in reciprocal space may be written

G

X-ray diffraction is a map of the reciprocal space.

Under invariant crystal transformations, the Fourier series for the electron density should be invariant. It will be for wave vectors of the form of G above, e.g., the Fourier series of the electron density can be expanded as

For X-ray scattering from two points within the sample, the phase factor will be

Now for the scattering over the entire material, the electron density times this phase factor must be integrated over the sample. This gives

The result of the integration will be effectively zero when the reciprocal lattice vector is not equal to the scattering vector. In this case, the amplitude will equal the sample volume times the electron density coefficient for vector G (V*ng).

Using the fact that the magnitudes of the wave vectors are equal for elastic scattering, we can derive the equation

This is in essence a generalized version of the Bragg condition.

We can subdivide the effects resulting in the X-ray scattering into three groups: scattering by the electrons themselves, scattering from the atoms and scattering by the unit cell.

1) electron

2) Atom

The scattering by the nucleus is much smaller than that from the electrons due to the 1/m factor in the equation above. Therefore, the atomic scattering will be the sum of the scattering by the electrons with the appropriate relative phases determined by the electronic positions within the atom. This calculation gives the atomic scattering factor f, equal to the amplitude of the wave scattered by the atom divided by the amplitude scattered by a single electron. For forward scattering ((=0) where the contribution from all electrons is in phase, the atomic scattering factor is equal to Z, the total number of electrons.

3) Unit cell

Likewise, the scattering from the unit cell will equal the sum of the scattering from individual atoms multiplied by the appropriate form factors. For a unit cell with N atoms in it, this factor is

[pic]

where ui, vi and wi are the positions of the ith atoms.

Multiple planes in crystal – rotate the crystal – at each angle such that the Bragg condition is fulfilled, there will be strong scattering.

This allows a full determination of the crystal structure.

There are several ways to actually perform this measurement. In Laue diffraction, a stationary bulk crystal is irradiated with polychromatic X-rays. The broad band of the incident radiation insures that the Bragg condition will be fulfilled at some angles. The angular distribution of the diffracted waves is measured and using the Bragg equation, the spacing of the molecular planes is calculated. This method has the advantage of allowing the determination of the planes with respect to the crystal habit.

This method has a simple setup of source and detector and is easy to perform, but requires a bulk crystal. This does lead to an additional application, though. By varying the input beam size and observing the scattering intensity, quantitative information about the crystal grain size can be found.

A second method is the Debye-Scherrer powder method. This uses monochromatic X-rays incident on a powder sample – now the random orientation of the powder insures that the Bragg condition will be met. The scattering intensity as a function of the incidence angle is again measured.

In general, the planar spacing alone does not give the crystal structure directly. A number of pacing values must be determined and the ratios considered in order to form a model of the structure. Often the results are compared to known structures.

Sources: same as XPS

(Brehmstrahlung, atomic fluorescence lines, synchrotron)

Structure factor:

Real crystals are never perfect. The following OHP shows what one might expect from various materials. The x-axis is the magnitude of the so-called scattering vector q=2ksin(.

Small angle scattering

We saw the Bragg diffraction condition can be written as

[pic]

where G is a reciprocal vector of the lattice. diffraction at grazing incidence implies incident and scattered wave vectors k that are almost the same. Therefore (k will be small and the observed diffraction will correspond to small reciprocal lattice vectors, or large spatial features in the sample.

Two examples of small angle diffraction techniques are Small angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS). Both of these probe features on the order of 10 to 1000 Å, compared to ................
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