X-ray diffraction, SEM and TEM, DSC and TGA, tensile, hardness, fatigue and fracture testing, ASTM methods, polymer, ceramic and thin-film processing, and materials databases.

Materials Characterization & Processing

## Why it matters

X-ray diffraction (XRD) is the first instrument most materials labs reach for. In about half an hour, a powder pattern tells you which crystalline phases are present, their lattice parameters, whether a film is textured, and roughly how small the crystallites are. When a new alloy, catalyst or battery cathode comes out of a furnace, "run XRD" is usually step one.

## The core idea: Bragg's law

A crystal is a stack of atomic planes with spacing `d`. X-rays reflected from neighbouring planes add up (constructive interference) only when the extra path length is a whole number of wavelengths:

```text
n * lambda = 2 * d * sin(theta)
```

- `lambda` is the X-ray wavelength. Most lab diffractometers use copper K-alpha1, `lambda = 1.5406 Å`.
- `theta` is the angle between the beam and the planes. The instrument scans and reports **2-theta**, so always halve it first.
- `n` is the order. In practice we use `n = 1` and treat higher orders as reflections from planes with a smaller `d` (for example (222) instead of second-order (111)).

So every peak position becomes a d-spacing: `d = lambda / (2 sin(theta))`.

## From d-spacings to a crystal structure

For a cubic crystal with lattice parameter `a`, the spacing of the (hkl) planes is

```text
d = a / sqrt(h^2 + k^2 + l^2)
```

Combine this with Bragg's law and square it: `sin^2(theta) = (lambda^2 / 4a^2) * N`, where `N = h^2 + k^2 + l^2`. The ratio of `sin^2(theta)` between peaks is therefore a ratio of integers. Which integers appear depends on the lattice, because of systematic absences (structure-factor rules):

| lattice | allowed reflections | N sequence |
|---|---|---|
| simple cubic | all hkl | 1, 2, 3, 4, 5, 6, 8 ... |
| body-centred (BCC) | h + k + l even | 2, 4, 6, 8, 10, 12 ... |
| face-centred (FCC) | h, k, l all odd or all even | 3, 4, 8, 11, 12, 16 ... |

## Worked example: indexing copper

A copper powder scanned with Cu K-alpha gives peaks at 2-theta = 43.30, 50.43, 74.13, 89.93 and 95.14 degrees.

1. First peak: `theta = 21.65°`, `sin^2(theta) = 0.1361`.
2. Divide every `sin^2(theta)` by the first: 1, 1.333, 2.669, 3.669, 4.003.
3. Those ratios are not integers, so multiply by 3: 3, 4, 8, 11, 12. That is the FCC sequence, so the peaks are (111), (200), (220), (311) and (222).
4. Lattice parameter from (111): `d = 1.5406 / (2 * 0.3689) = 2.0879 Å`, so `a = 2.0879 * sqrt(3) = 3.616 Å`. Averaging all five peaks gives about 3.615 Å, which matches the reference value for copper.

If multiplying by 2 had given 2, 4, 6, 8 the sample would be BCC; if the raw ratios were already 1, 2, 3, 4 it would be simple cubic. Non-cubic phases need more parameters and are indexed with software.

## Phase identification in practice

Most of the time you do not index by hand. You compare the pattern against a reference database: the ICDD Powder Diffraction File (commercial), or free sources such as the Crystallography Open Database (https://www.crystallography.net). Software such as GSAS-II (free) or vendor packages match peak positions and relative intensities, and **Rietveld refinement** fits the whole pattern to refine lattice parameters and phase fractions.

Two more numbers come out of peak shapes:

- **Crystallite size** from peak broadening, using the Scherrer equation `tau = K * lambda / (beta * cos(theta))`, with `K` about 0.9 and `beta` the width in radians after subtracting the instrument's own broadening. It is an estimate, reliable roughly below 100 nm.
- **Strain and texture**: peak shifts point to residual strain or a composition change; unexpected intensity ratios usually mean preferred orientation (texture), common in films and pressed powders.

## Common mistakes

- **Using 2-theta in Bragg's law** instead of theta. Your d-spacings will be about half as large and nothing will index.
- **Forgetting degrees vs radians** in code (`math.sin` expects radians).
- **Ignoring the K-alpha2 shoulder** at high angles, or the sample displacement error, which shifts every peak and biases `a`.
- **Trusting one peak.** Use several peaks, and the high-angle ones give the most precise lattice parameters.
- **Over-reading intensities** from a textured sample as if it were a random powder.

## Safety and training

Diffractometers contain an X-ray source. Modern enclosed units are interlocked, but you still need your institution's radiation safety training and instrument sign-off before you run one, and you never defeat an interlock. In this course you analyse data only.

## Recap

- Bragg's law turns each 2-theta peak into a d-spacing: halve 2-theta first.
- For cubic crystals, `sin^2(theta)` ratios are integer ratios that reveal SC, BCC or FCC.
- Real phase ID matches against databases; peak width and shape give size, strain and texture.

## Try it

Open the **Do** tab to write a Bragg's-law helper that indexes a measured copper pattern and computes its lattice parameter.


Use Bragg's law to turn XRD peak positions into d-spacings, index a cubic pattern and find the lattice parameter.

X-ray diffraction

What is X-ray Diffraction?

Diffraction Lecture 17: Indexing Diffraction Patterns of Cubic Crystals

## Why it matters

Most materials properties come from microstructure: grain size, phases, porosity, cracks, interfaces and defects. Optical microscopes stop at roughly 200 nm resolution because of the wavelength of light. Electron microscopes go far smaller, and choosing the right one (and the right detector) decides whether you answer your question in an afternoon or waste a week on sample preparation.

## Why electrons

An electron's wavelength depends on its accelerating voltage. At 200 kV it is about 2.5 pm, thousands of times shorter than visible light. In practice, resolution is limited by lens aberrations, not wavelength, but electron microscopes still resolve features from tens of nanometres (a typical SEM) down to individual atomic columns (an aberration-corrected TEM).

Electrons also interact strongly with matter, which gives several signals at once: secondary electrons, backscattered electrons, characteristic X-rays and diffracted beams. Each detector answers a different question.

## SEM: scanning electron microscopy

The SEM scans a focused beam (usually 1 to 30 kV) across the surface of a bulk sample and collects what comes back.

- **Secondary electrons (SE)** come from the top few nanometres. They show **topography**: fracture surfaces, particles, coatings, corrosion.
- **Backscattered electrons (BSE)** scale with atomic number. Heavier phases look brighter, so BSE shows **composition contrast** between phases on a polished section.
- **EDS (energy-dispersive X-ray spectroscopy)** reads characteristic X-rays to give elemental composition and maps. It is semi-quantitative unless calibrated with standards, and light elements (B, C, N, O) are hard.
- **EBSD (electron backscatter diffraction)** maps crystal orientation and grain boundaries on a well-polished surface.

Samples can be millimetres to centimetres across. Non-conductive samples (polymers, ceramics) charge up, so they are coated with a few nanometres of carbon or gold, or imaged at low voltage or in a low-vacuum mode.

## TEM: transmission electron microscopy

The TEM sends a high-energy beam (often 80 to 300 kV) **through** a sample thin enough to transmit electrons, usually below about 100 nm.

- **Bright-field and dark-field imaging** show dislocations, precipitates, grain boundaries and stacking faults.
- **Selected-area electron diffraction (SAED)** gives the crystal structure and orientation of a region a few hundred nanometres across.
- **High-resolution TEM** and **STEM** (scanning TEM) image lattice fringes and atomic columns; STEM with EDS or EELS maps composition at the nanoscale.

The cost is sample preparation: electropolishing, ion milling or a focused ion beam (FIB) lift-out to cut a thin lamella from a specific spot. Preparation can itself create artefacts (ion damage, amorphous layers), and a TEM sees a tiny volume, so you must check that it is representative.

## Choosing a technique

| question | first choice |
|---|---|
| Why did this part fracture? (fatigue striations, dimples, cleavage) | SEM, secondary electrons |
| Which phases are present on this polished section, and where? | SEM, BSE plus EDS maps |
| Grain size and texture over a large area | SEM with EBSD |
| Are there nanoscale precipitates or dislocations inside grains? | TEM bright/dark field |
| What is the crystal structure of a 50 nm particle? | TEM with SAED |
| Is this 5 nm interface layer sharp, and what is in it? | STEM with EDS or EELS |

A good rule: start with the cheapest technique that can answer the question. Light microscopy and SEM survey large areas; TEM confirms a specific nanoscale hypothesis. Combine imaging with bulk techniques such as XRD so a few images do not stand in for the whole sample.

## Worked example

A steel bolt failed in service. You want to know whether it failed by fatigue and whether inclusions started the crack.

1. Photograph the fracture surface, then look at it in the SEM with SE imaging: beach marks and striations point to fatigue; dimples point to ductile overload.
2. Find the origin and use EDS on any particle there: manganese and sulfur together suggest a MnS inclusion.
3. Section and polish a sample near the origin and use BSE or EBSD to look at the microstructure.
4. Only if you need nanoscale detail (for example, precipitates that embrittled the steel) cut a FIB lamella for TEM.

## Common mistakes

- **Reading EDS numbers as exact.** Without standards, light elements and small peaks are uncertain, and the X-ray signal comes from about a micrometre deep, not just the surface you see.
- **Charging artefacts** on uncoated insulators, which look like bright streaks and drift.
- **Imaging one pretty spot** and reporting it as typical. Image several regions at known magnifications and include scale bars.
- **Forgetting beam damage**: polymers and some oxides change under the beam.

## Safety and training

Electron microscopes use high voltages, vacuum systems, and (for FIB and sample preparation) chemicals, cryogens and ion sources. You use them only after your facility's training and sign-off, under its procedures. This lesson builds the judgement you bring to that training.

## Recap

- SEM images surfaces of bulk samples: SE for topography, BSE for composition contrast, EDS for elements, EBSD for orientation.
- TEM looks through thin samples for internal defects, diffraction and atomic-scale structure, at the cost of preparation.
- Pick the cheapest technique that answers the question, and check that images are representative.


Compare scanning and transmission electron microscopy and choose the right imaging technique for a materials question.

SEM and TEM

Electron Microscopy (TEM and SEM)

How Electron Microscopes Work: SEM vs TEM Principles, Electron Beams & Resolution

## Why it matters

Thermal analysis answers practical questions fast. Is this polymer batch the same as the last one? What is its glass transition? How crystalline is it? How much filler or moisture does it contain, and at what temperature does it start to decompose? Two instruments do most of this work: **DSC** (differential scanning calorimetry) and **TGA** (thermogravimetric analysis).

## DSC: heat flow

A DSC heats a small sample (typically 2 to 10 mg in a crimped pan) and an empty reference pan at a controlled rate, often 10 K/min, and records the **difference in heat flow** needed to keep both on the same temperature program. Events show up as features on that curve:

- **Glass transition (Tg):** a step in the baseline, because heat capacity increases when an amorphous region becomes rubbery. Report the midpoint and the method you used.
- **Melting (Tm):** an endothermic peak. Its area is the **enthalpy of fusion**.
- **Crystallization (Tc) and cold crystallization:** exothermic peaks on cooling or heating.
- **Curing** of thermosets: a broad exotherm; residual cure shows up as a smaller exotherm in a second run.

Check your instrument's sign convention: some software plots endotherms up, others down. Always label the axis "endo up" or "exo up".

### Peak area to enthalpy

DSC software integrates the peak above a baseline. Doing it yourself makes the units clear. If heat flow `q` is in mW and you integrate over temperature in K, you get mW·K. Divide by the heating rate in K/s to get mW·s, which is mJ:

```text
energy (mJ) = area (mW*K) * 60 / rate (K/min)
enthalpy (J/g) = energy (mJ) / mass (mg)
```

**Worked example.** A 5.0 mg polyethylene sample at 10 K/min shows a melting peak with an area of 125 mW·K above the baseline. Energy = 125 × 60 / 10 = 750 mJ, so `dH_m = 750 / 5.0 = 150 J/g`. Using about 293 J/g for 100% crystalline polyethylene (a literature value; sources vary by a few percent), crystallinity is `150 / 293 = 51%`.

### The first and second heat

The first heating run reflects the sample's **thermal history** (how it was processed and stored). A standard approach is heat, cool at a controlled rate, then heat again: the second heat compares materials on an equal footing. Report both when processing history matters.

## TGA: mass

A TGA weighs a sample continuously while heating it, in nitrogen (to see decomposition) or air (to burn off organics). The curve shows **mass percent vs temperature**; steps are mass losses. The derivative curve (DTG) turns each step into a peak, which makes overlapping steps easier to separate.

Typical uses:

- Moisture and volatile content (loss below about 150 °C).
- Polymer decomposition temperature (often reported as the temperature at 5% loss).
- Filler or ash content: the residue after burning off the polymer in air, for example glass fiber content in a composite.

### Worked example: calcium oxalate monohydrate

Calcium oxalate monohydrate (CaC2O4·H2O, 146.11 g/mol) is a common TGA check standard because it loses mass in three clean steps:

| step | what leaves | theoretical loss |
|---|---|---|
| about 100 to 250 °C | H2O (18.02 g/mol) | 18.02 / 146.11 = 12.3% |
| about 400 to 550 °C | CO (28.01 g/mol) | 19.2% |
| about 600 to 800 °C | CO2 (44.01 g/mol) | 30.1% |

The residue is CaO, about 38.4% of the starting mass. If your instrument measures 12.0% for the first step, the 0.3% difference is something to explain: calibration, buoyancy, or a partly dehydrated standard. Exact step temperatures shift with heating rate and atmosphere.

## Reading curves in Python

Exported data are just columns of temperature and signal. With numpy, `np.interp` reads the mass at any temperature, `np.argmax` finds a peak, and a trapezoid sum integrates a peak above a straight baseline drawn between two chosen temperatures. The baseline choice matters: report the limits you integrated between.

## Common mistakes

- **Mixing up endo up and exo down**, and calling a melt a crystallization.
- **Comparing first-heat curves** of samples with different processing histories.
- **Forgetting units**: integrating over time vs temperature without converting by the heating rate.
- **Too heavy or poorly contacting samples**, which smear peaks and lag the temperature.
- **Ignoring atmosphere** in TGA: the same polymer decomposes differently in air and nitrogen.

## Safety and training

These instruments run to hundreds of degrees (TGA furnaces to 1000 °C or more), use purge gases, and decomposition can release toxic products that must be vented. Run samples only after your lab's training, following its procedures and safety data sheets. In this course you work with data.

## Recap

- DSC measures heat flow: Tg is a step, melting is an endotherm, and peak area gives enthalpy and crystallinity.
- TGA measures mass: each step is a loss you can match to chemistry, and the residue gives filler or ash content.
- Integrate with a clear baseline, mind units and sign conventions, and record thermal history.

## Try it

Open the **Do** tab to compute TGA mass-loss steps and a DSC melting enthalpy and crystallinity from synthetic curves with numpy.


Read DSC and TGA curves: glass transitions, melting enthalpy, crystallinity and decomposition mass-loss steps.

Thermal analysis

Back to Basics: Differential Scanning Calorimetry

Back to Basics: Thermogravimetric Analysis (TGA)

Identify phases and lattice parameters with X-ray diffraction, choose between SEM and TEM, and read DSC and TGA curves.