X-Ray diffraction Tutorials
How to Read an XRD Graph in 7 Easy Steps
A Complete Beginner-Friendly Tutorial — From Raw Diffractogram to Phase Identification and Crystallite Size
Series: Crystal Structure Hub | Tutorial: XRD Interpretation | Prerequisites: Lecture 01 — Introduction to Crystal Structure & Lecture 03 — Unit Cell and Lattice Parameters
Reading time: 40 minutes | Includes: 7 step-by-step analysis stages, annotated XRD diagram, Bragg's law worked examples, Scherrer equation, phase identification table, MCQs, key takeaways
SEO Keywords: how to read XRD graph, XRD pattern analysis, X-ray diffraction tutorial, XRD peaks, Bragg's law, d-spacing calculation, phase identification XRD, Scherrer equation crystallite size, powder XRD beginners
Before You Begin — What Is an XRD Graph?
Learning how to read an XRD graph step by step is one of the most valuable skills in materials science — and this tutorial will teach you exactly that. XRD (full form: X-ray diffraction) produces a graph called a diffractogram: a series of sharp peaks arranged along an angle axis that is as unique to a material as a fingerprint is to a person. The technique was first demonstrated by Max von Laue in 1912 and developed into a practical tool by W. H. and W. L. Bragg, whose Nobel Prize-winning work remains the foundation of everything we do in XRD today.
According to the International Union of Crystallography (IUCr), powder X-ray diffraction is the single most widely used technique for characterising polycrystalline materials worldwide, spanning materials science, chemistry, geology, and pharmaceutical research. Yet for newcomers, the XRD graph can look intimidating: unlabelled peaks, unfamiliar axes, cryptic notation. This tutorial removes that intimidation entirely. By following seven logical steps — grounded in the same XRD analysis workflow described in the authoritative textbook by Cullity and Stock — you will be able to pick up any XRD pattern and extract its key information with confidence.
When monochromatic X-rays hit a crystalline material, atoms in the crystal planes scatter them. When the scattered waves from parallel planes interfere constructively — that is, when their path difference is exactly one whole wavelength — they produce a strong, sharp signal called a diffraction peak. The precise angle at which each peak appears depends on the spacing between those crystal planes, which in turn depends on the crystal structure. Different materials have different crystal structures → different plane spacings → different peak positions. That is why each material has its own unique XRD fingerprint. For a beautifully detailed explanation of this wave-interference mechanism, see the IUCr teaching pamphlet on powder diffraction.
Before we begin the seven steps, here is an important orientation: every step in reading an XRD pattern flows from a single master equation — Bragg's Law. Originally derived by W. L. Bragg in 1913 and published in the Proceedings of the Royal Society A, this equation connects what you measure on the graph (the angle θ) to what you want to know about the material (the plane spacing d):
That equation is the bridge between what you measure on the graph and what you want to know about the material. Keep it in mind throughout all seven steps. A free, interactive derivation of Bragg's Law is available at the DoITPoMS Teaching Library at the University of Cambridge — highly recommended as a companion resource.
Step 1 — Orient Yourself: Understand the Axes
Before reading any graph, you must know what the axes represent. An XRD diffractogram always has the same two axes — and understanding them is the foundation of everything else. The conventions used here are internationally standardised and described in full by the IUCr Commission on Powder Diffraction.
The X-Axis: 2θ (Two-Theta), in Degrees
The horizontal axis shows the diffraction angle, labelled 2θ (read as "two theta"). It is always expressed in degrees, and a typical powder XRD scan covers a range from roughly 10° to 90°. The lower limit is set by instrument geometry, and the upper limit by the practical decrease in peak intensity at high angles.
Why is it called 2θ and not simply θ? This is a convention rooted in the geometry of the diffractometer. In Bragg's Law, θ is the angle between the incoming X-ray beam and the surface of the crystal planes. For the detector to collect the diffracted beam symmetrically, it must be positioned at an angle of 2θ from the incoming beam direction. So the instrument records and displays 2θ — you halve it to get θ for calculations. The NIST X-ray Powder Diffraction program provides a clear geometric explanation of this convention alongside its standard reference patterns.
Think of a mirror. If you stand at an angle θ to the mirror surface, the reflected beam bounces away at the same angle θ on the other side — making a total angular separation of 2θ between the incoming and outgoing beams. The XRD detector is like that mirror: it sits at 2θ from the source to catch the diffracted signal. That is why the axis is labelled 2θ, not θ. The DoITPoMS Cambridge illustration of diffractometer geometry makes this geometry immediately clear.
The Y-Axis: Intensity (a.u.)
The vertical axis shows the intensity of the diffracted X-ray signal, recorded by the detector. It is usually labelled in arbitrary units (a.u.) or as raw counts per second. Higher intensity means more X-rays were diffracted at that particular angle — meaning there is a well-developed, regularly spaced set of planes in the material oriented to satisfy Bragg's Law at that angle. The statistical significance of intensity counts in XRD is discussed in detail by the IUCr powder diffraction tutorial series.
The absolute intensity values are not usually what we analyse — they depend on the instrument, detector, and sample preparation. What matters is the relative intensity: which peaks are taller than others, and how wide they are. You will use this in Steps 2 and 7. For guidance on instrument-related intensity corrections, the UCL Powder Diffraction Notes provide an excellent practical introduction.
Step 2 — Identify the Peaks
With the axes understood, scan the diffractogram from left to right and locate all the sharp, narrow rises above the background. Each of these is a diffraction peak, and each one corresponds to a specific family of crystal planes in your material satisfying Bragg's Law. The powder diffraction review by Dinnebier and Billinge (2008) in Powder Diffraction: Theory and Practice describes the full physical origin of each peak in a polycrystalline diffractogram.
What Does a Peak Look Like?
A genuine diffraction peak is characterised by three key features. The shape of a powder diffraction peak is formally described as a pseudo-Voigt profile — a convolution of Gaussian and Lorentzian components — as used in all modern Rietveld refinement software:
| Feature | What it looks like | What it means |
|---|---|---|
| Sharp rise and fall | Symmetric bell shape (pseudo-Voigt: Gaussian + Lorentzian mix) | Coherent diffraction from a well-ordered crystal |
| Narrow width | Typically 0.1°–1° wide at half-maximum | Large, well-crystallised grains → narrow; small nanocrystallites → broader |
| Sits above background | Background is a broad, slowly varying hump from air scatter, amorphous content, or fluorescence | Background ≠ peaks. Peaks are sharp; background drifts over tens of degrees. |
Distinguish Peaks from Background
Many beginners misidentify background bumps as peaks. A useful rule: if it rises and falls within one or two degrees with a clear apex, it is a peak. If it drifts slowly over tens of degrees, it is background. Amorphous materials — glass, polymers, liquids — produce only a broad amorphous hump with no sharp peaks at all. This distinction between crystalline and amorphous XRD signatures is clearly illustrated in the Rigaku powder diffraction application guide.
When first examining a pattern, mark each peak and number them in order from left to right as Peak 1, Peak 2, Peak 3, and so on. This systematic approach prevents you from missing weak peaks. For automated peak searching, free tools such as ICDD JADE or the open-source FullProf Suite use second-derivative algorithms to locate all statistically significant peaks, including weak ones that are easy to miss by eye.
Step 3 — Read the Peak Positions (2θ Values)
Once you have identified your peaks, record the exact position of each peak's apex on the 2θ axis. This is your raw measurement. In modern instruments from manufacturers such as Malvern Panalytical or Bruker D2 PHASER, the diffractometer software automatically fits a mathematical curve to each peak and reports the 2θ position to two or three decimal places (e.g., 44.51°). In older or printed patterns, you read the value directly from the graph using the gridlines.
The 2θ position feeds directly into Bragg's Law to give you the d-spacing. Because the sine function appears in Bragg's Law, a small error in 2θ — even a fraction of a degree — can produce a noticeable error in d. For this reason, NIST Standard Reference Material 660c (LaB₆) is routinely used to calibrate instrument peak positions to sub-0.01° accuracy. This calibration procedure is documented in detail by the IUCr Le Bail refinement tutorial.
From the annotated diagram in Fig. 1, extract the four 2θ values. The accepted reference positions for FCC copper are tabulated in Materials Project entry mp-30 and in ICDD PDF Card #04-0836:
| Peak # | 2θ (degrees) | θ (degrees) | Miller index (hkl) |
|---|---|---|---|
| 1 | 38.2° | 19.1° | (111) |
| 2 | 44.5° | 22.25° | (200) |
| 3 | 64.6° | 32.3° | (220) |
| 4 | 77.6° | 38.8° | (311) |
To go from 2θ to θ, simply divide by 2. You will need θ — not 2θ — when you plug into Bragg's Law in Step 4.
Real diffractometers can have a small systematic angular offset called the zero error — the instrument might report 0.02° or 0.05° higher or lower than the true angle. For highly precise measurements, this zero error is determined using a certified standard material such as NIST SRM 676a (alumina) or silicon powder and subtracted from all readings. The IUCr Quantitative Phase Analysis tutorial explains when and how to apply this correction in practice.
Step 4 — Calculate d-Spacings Using Bragg's Law
Now you convert each 2θ position into a physical quantity that describes the material: the interplanar d-spacing (also called d-value or dhkl). The d-spacing is the perpendicular distance between adjacent parallel planes in the crystal, measured in nanometres (nm) or Ångströms (Å). The d-spacing concept is central to all of crystallography and is rigorously defined in the IUCr Online Dictionary of Crystallography.
The conversion is direct — simply rearrange Bragg's Law for d. The formula is given here in its standard form, consistent with the NIST CODATA values for the CuKα wavelength (λ = 0.154056 nm, cited to six significant figures):
Using CuKα radiation: λ = 0.15406 nm. Below is the full d-spacing table for all four copper peaks. You can verify these against tabulated values at the Materials Project (Cu, mp-30) or at the Crystallography Open Database COD entry 9012960:
| Peak # | 2θ (°) | θ (°) | sinθ | d (nm) |
|---|---|---|---|---|
| 1 | 38.2 | 19.1 | 0.3272 | 0.2353 |
| 2 | 44.5 | 22.25 | 0.3784 | 0.2035 |
| 3 | 64.6 | 32.3 | 0.5348 | 0.1440 |
| 4 | 77.6 | 38.8 | 0.6271 | 0.1228 |
These d-values — 0.2353, 0.2035, 0.1440, and 0.1228 nm — are the crystal's identity card. In Step 5, you will match them to a reference database.
Always keep track of units. If your wavelength λ is in nm (0.15406 nm for CuKα), your d-spacing comes out in nm. If you prefer Å, use λ = 1.5406 Å and d will be in Å. The conversion: 1 nm = 10 Å. The NIST Guide to SI Units documents this and all related unit conventions. Mixing units is the single most common calculation error in XRD analysis.
Step 5 — Identify the Phase Using a Reference Database
You now have a set of d-spacing values. The next task is to match them against a library of known materials to identify which material (or materials) you are looking at. This process is called phase identification — and it is the core analytical function of powder XRD in research and industry. The ICDD Powder Diffraction File (PDF) database is the world's largest and most authoritative repository of powder diffraction reference patterns, with over one million entries as of 2024.
The word phase in crystallography means a material with a specific, well-defined crystal structure. Iron in its BCC form (α-iron) is one phase; iron in its FCC form (γ-iron) is a different phase, even though both are pure iron. XRD distinguishes these phases precisely because they have different crystal structures and therefore different d-spacing sets. The definitive description of crystallographic phase relationships is maintained in the Inorganic Crystal Structure Database (ICSD) at FIZ Karlsruhe, Germany.
The Major Reference Databases
There are several globally recognised databases used for XRD phase identification. As a student, you will most commonly use the first two:
| Database | Access & Link | Best For |
|---|---|---|
| ICDD PDF | Commercial — subscription via most universities | Comprehensive; 1M+ entries; global industry standard |
| COD | Free & open-access at crystallography.net | Ideal for students; 500,000+ verified structures |
| ICSD | Institutional subscription — FIZ Karlsruhe | Inorganic and intermetallic phases; curated data |
| Materials Project | Free at materialsproject.org | Computed + experimental structures; XRD simulator |
| AFLOW | Free at aflowlib.org | High-throughput computed alloy and compound structures |
How to Match d-Spacings to a Phase
The matching process — called search-match — works as follows. Take your three strongest peaks and their d-spacings. Look for a reference entry whose d-spacings match yours within a tolerance of ±0.002–0.005 nm. Modern software such as Bruker DIFFRAC.EVA, HighScore Plus, or the free MATCH! software perform this search automatically and rank the best-fitting reference patterns.
Second, confirm the match by checking that all the observed peaks — not just the top three — agree with the reference pattern, including the relative intensities. A useful open-access guide to the full search-match workflow is provided by the Royal Society of Chemistry in its CrystEngComm tutorial series on powder diffraction.
Our calculated d-spacings (0.2353, 0.2035, 0.1440, 0.1228 nm) match the reference pattern for face-centred cubic (FCC) copper: ICDD Card #04-0836 / COD 9012960, space group Fm3̄m, a = 0.3615 nm. All four peaks are accounted for; no unexplained peaks remain.
If your sample is a mixture of two or more crystalline phases, you will see peaks from all phases superimposed in one pattern. Each set is identified separately. Quantitative ratios are determined by the Rietveld method — the gold standard for quantitative phase analysis (QPA) in powder XRD, originally published by H. M. Rietveld in the Journal of Applied Crystallography (1969).
Step 6 — Determine the Crystal System and Lattice Parameters
Beyond identifying what material it is, you can extract the fundamental geometric description of its unit cell — the six lattice parameters — directly from the d-spacings. This process is called indexing: assigning Miller indices (hkl) to each peak and then solving for the lattice parameters. The concept of Miller indices is explained clearly in the DoITPoMS Cambridge teaching library on Miller indices.
For the cubic system (which includes most common engineering metals such as iron, copper, aluminium, and nickel), the formula connecting d-spacing, Miller indices, and the lattice parameter a is elegantly simple. All seven crystal systems and their d-spacing formulae are rigorously tabulated in the International Tables for Crystallography, the definitive reference published by the IUCr:
Using Peak 2 in Fig. 1, assigned Miller index (200) (h=2, k=0, l=0). The reference value for copper is a = 0.36150 nm, as listed at Materials Project mp-30 and confirmed in the NIST Crystal Lattice Structures database:
Each peak should give the same value of a — this cross-check confirms internal consistency of the indexing. Any deviation signals a non-cubic system or lattice strain in the material.
Indexing Formulae for All Crystal Systems
For lower-symmetry systems, automated indexing programs such as TREOR, DICVOL, or NIST's GSAS-II are routinely used. For the four most common systems in materials science:
| Crystal System | d-Spacing Formula | Example Materials |
|---|---|---|
| Cubic (a = b = c) | 1/d² = (h² + k² + l²) / a² | Cu, Fe, NaCl, Si, Au |
| Tetragonal (a = b ≠ c) | 1/d² = (h² + k²)/a² + l²/c² | TiO₂ (rutile), SnO₂ |
| Hexagonal (a = b ≠ c, γ = 120°) | 1/d² = (4/3)·(h²+hk+k²)/a² + l²/c² | Ti, Co, ZnO, graphite |
| Orthorhombic (a ≠ b ≠ c) | 1/d² = h²/a² + k²/b² + l²/c² | BaTiO₃, orthoferrites |
All 230 space groups and their systematic absences (selection rules that govern which hkl reflections are allowed) are tabulated in the freely accessible IUCr Symmetry Database and in the web-based Bilbao Crystallographic Server — an essential tool for determining whether your peak list is consistent with a proposed space group.
Step 7 — Estimate Crystallite Size Using the Scherrer Equation
So far, you have read the peak positions. Now pay attention to peak width. The width of a diffraction peak carries critical physical information: it tells you how small the crystalline domains in your sample are. This is especially important in nanomaterial research, thin films, and nanoparticle characterisation. The FDA Nanotechnology Characterization Laboratory includes XRD-based crystallite size analysis as part of its standardised nanoparticle characterisation protocol.
The key insight is this: in a perfectly large crystal, diffraction peaks are extremely narrow. As the crystalline domain (called a crystallite) becomes smaller and smaller, the peaks become progressively broader. This phenomenon — called size broadening — arises because peak sharpness requires coherent diffraction from many planes in a row. A detailed quantum-mechanical explanation is given in Warren (1969) X-ray Diffraction and summarised in the IUCr line-broadening analysis tutorial.
Imagine hearing an echo in a canyon. A long canyon (many rock faces in a row) produces a sharp, clear echo. A short canyon (only a few faces) gives a weak, diffuse echo spread over a wider time window. Crystallites behave the same way: a large crystallite produces a sharp, narrow diffraction peak; a nanoscale crystallite produces a broad peak. This analogy is formalised as the coherence length concept in wave physics.
The Scherrer Equation
The relationship between peak width and crystallite size is quantified by the Scherrer equation, originally published by Paul Scherrer in Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1918). It remains one of the most widely cited equations in materials science, with tens of thousands of citations in journals including Journal of Alloys and Compounds and Nature Communications:
The FWHM (β) is read from the diffractogram in degrees (e.g., β = 0.45°). Before using it in the Scherrer equation, convert to radians: β (rad) = β (°) × π/180. Forgetting this conversion overestimates D by a factor of ~57 — one of the most frequent errors in the literature, flagged explicitly in the IUCr line-profile analysis guidelines and discussed in Ingham (2015) in Crystallography Reviews.
How to Measure the FWHM
The full width at half maximum (FWHM) — labelled β in Fig. 1 — is the peak width measured at exactly half its maximum intensity above the background. To measure it manually: (1) draw a horizontal baseline under the peak; (2) find the peak's maximum height above the baseline; (3) mark the half-height level; (4) read the angular width between the two intersection points. For automated measurement, the GSAS-II software from NIST and HighScore Plus fit a pseudo-Voigt profile to each peak and extract the FWHM to four significant figures.
Suppose Peak 2 (the (200) reflection at 2θ = 44.5°) has an FWHM of β = 0.45°. For comparison, a similar calculation is shown in the comprehensive Scherrer review by Ingham (2015):
Important caveat: The Scherrer equation gives a lower-bound estimate. It does not account for microstrain, instrument broadening, or dislocations. For more accurate analysis, use the Williamson-Hall method or the full Rietveld refinement approach.
Real-World Applications of XRD Pattern Reading
The seven steps above are not merely an academic exercise. XRD pattern reading underpins some of the most important analytical capabilities in modern science and engineering. According to a 2023 survey in the Cambridge University Press journal Powder Diffraction, over 60,000 peer-reviewed papers per year now report XRD data as primary characterisation evidence.
Quality Control in Pharmaceutical Manufacturing
Many drugs exist in multiple crystalline forms called polymorphs — the same molecule arranged differently in the solid state. Different polymorphs can have drastically different solubility and bioavailability. The FDA ICH Q6A guidance specifies XRD as an accepted method for polymorph identification and control in pharmaceutical manufacturing. The United States Pharmacopeia (USP Chapter 941) provides the regulatory standard for XRD-based polymorph testing.
Monitoring Phase Transformations in Steels
When steel is heat-treated, its crystal structure can transform from BCC (ferrite) to FCC (austenite) and back. Following these transformations in real time using high-temperature XRD at synchrotron facilities such as the Advanced Photon Source (APS) at Argonne National Laboratory provides precise transformation temperatures and kinetics data that guide industrial heat treatment schedules. The technique is described in the ASM Handbook on Heat Treatment.
Characterising Nanoparticles
In nanotechnology, the Scherrer equation (Step 7) provides a rapid, non-destructive way to estimate nanoparticle size without requiring expensive electron microscopy. A 2021 study in ACS Nano Letters demonstrated XRD-based size determination of gold nanoparticles with results in excellent agreement with TEM measurements. The NIST Materials Measurement Laboratory provides free online tools for Scherrer analysis of nanoparticle XRD patterns.
Detecting Residual Stress in Engineering Components
Residual stresses in manufactured parts slightly shift peak positions and change d-spacings compared to the stress-free reference values. High-precision XRD measurement of these shifts is used routinely in aerospace and automotive engineering, and is standardised in ASTM Standard E915 (Standard Test Method for Verifying the Alignment of X-Ray Diffraction Instrumentation for Residual Stress Measurement). The ASM International publishes detailed case studies of residual stress measurement in turbine blades, welded joints, and automotive crankshafts.
Earth Science and Mineralogy
Geologists routinely use the full seven-step workflow to identify mineral phases in rock samples and soils — from clay mineral identification (phyllosilicates with characteristic basal d-spacings) to the detection of carbonate versus silicate minerals in sedimentary sequences. The Mindat.org mineral database and the Clay Minerals Society provide free XRD reference patterns for thousands of mineral species. The US Geological Survey open-file report on XRD analysis of clay minerals is a widely cited practical guide.
Series Navigation — Crystal Structure Hub
Introduction to Crystal Structure Lecture 02
Lattice and Basis Lecture 03
Unit Cell & Lattice Parameters Current
How to Read XRD Graphs Lecture 04
BCC Crystal Structure Lecture 05
FCC Crystal Structure Lecture 06
HCP Crystal Structure
Packing Fraction and Theoretical Density — Learn how to calculate theoretical density from XRD-derived lattice parameters.
Bragg's Law — Derivation and Applications — Deep dive into the physics behind the master equation used in Steps 3 and 4.
Introduction to Rietveld Refinement — Advanced full-pattern fitting to go beyond the Scherrer estimate.
Ferroelectrics Tutorial Series — Applications of lattice parameter analysis in functional perovskite materials.
Practice Questions
Test your understanding with the following questions. All correct answers are grounded in the content above and verified against sources including the IUCr Teaching Pamphlets and Cambridge DoITPoMS XRD questions.
Frequently Asked Questions — XRD Graph Reading
These are the most-searched questions about reading XRD patterns, answered concisely for quick reference.
What does each peak in an XRD pattern represent?
Each sharp peak in an XRD pattern represents a specific family of parallel crystal planes — identified by Miller indices (hkl) — that satisfies Bragg's Law at that diffraction angle. The position (2θ) of the peak gives the interplanar d-spacing via Bragg's Law. The height (intensity) reflects how many planes are diffracting and the atomic scattering factor. The width (FWHM) tells you the crystallite size — broader peaks mean smaller nano-sized domains.
What does 2θ mean on the x-axis of an XRD graph?
2θ (two-theta) is the diffraction angle between the incident X-ray beam and the diffracted beam, measured in degrees. The XRD detector is always positioned at 2θ from the source, so the instrument records and displays 2θ rather than θ alone. To apply Bragg's Law, divide the measured 2θ by 2 to get θ. A typical powder XRD scan covers 2θ from approximately 10° to 90°.
How do you calculate d-spacing from an XRD pattern?
Calculate the interplanar d-spacing from an XRD pattern using the rearranged Bragg's Law: d = λ / (2 sinθ), where λ is the X-ray wavelength (0.15406 nm for CuKα radiation) and θ = (2θ_measured)/2. For example, a peak at 2θ = 44.5° gives θ = 22.25°, sinθ = 0.3784, and d = 0.15406/(2 × 0.3784) = 0.2035 nm. Each peak in the XRD pattern yields one d-spacing value for that set of crystal planes.
What is the Scherrer equation and how is it used in XRD?
The Scherrer equation estimates crystallite size D from XRD peak broadening: D = Kλ / (β cosθ), where K ≈ 0.94, λ = X-ray wavelength (nm), β = full width at half maximum (FWHM) of the peak in radians (convert: β_rad = β_deg × π/180), and θ = Bragg angle. A broader peak means smaller crystallites. It is widely used in nanoparticle research to estimate particle size non-destructively from a single XRD measurement. Note: the Scherrer equation gives a lower-bound estimate and does not account for microstrain.
What is XRD full form in materials science?
XRD full form is X-Ray Diffraction. In materials science, XRD is an analytical characterisation technique that uses monochromatic X-rays to determine a material's crystal structure, phase composition, lattice parameters, and crystallite size. When X-rays strike a crystalline solid, they diffract at specific angles governed by Bragg's Law (nλ = 2d sinθ), producing a characteristic XRD pattern or diffractogram that acts as a unique fingerprint of the material's crystal structure.
How do you identify a phase from an XRD pattern?
To identify a phase from an XRD pattern: (1) Calculate d-spacings for all peaks using Bragg's Law. (2) Search a reference database — the free Crystallography Open Database (COD) or the ICDD PDF. (3) Find a reference card whose d-spacings match your values within ±0.005 nm. (4) Confirm that all observed peaks — including weak ones — are consistent with the reference. This process is called search-match and can be automated using free software such as MATCH! or GSAS-II.
Why do XRD peaks broaden for nanoparticles?
XRD peaks broaden for nanoparticles because sharp diffraction requires coherent scattering from many parallel crystal planes in a row. In a large crystal, thousands of planes contribute, producing narrow peaks. In a nanocrystallite (typically 1–100 nm), only a few planes are present, so the diffraction condition is met over a wider angular range, broadening the peak. This is quantified by the Scherrer equation: D = Kλ/(β cosθ) — as crystallite size D decreases, peak width β increases. This makes XRD peak broadening one of the fastest ways to characterise nanoparticle size.
What does the y-axis of an XRD graph show?
The y-axis of an XRD graph shows the intensity of the diffracted X-ray signal, usually in arbitrary units (a.u.) or raw detector counts per second. A taller peak means more X-rays were diffracted at that angle — indicating a well-developed set of crystal planes with strong periodic ordering. The absolute intensity values depend on the instrument and sample preparation; it is the relative intensities between peaks, and the peak widths, that carry the most analytical information about crystallinity and crystallite size.
Key Takeaways
Here is a compact summary of the complete seven-step XRD reading workflow, validated against the IUCr Powder Diffraction Tutorial Series:
X-axis = 2θ (degrees); Y-axis = intensity (a.u.). Always orient yourself by understanding the axes before anything else.
Peaks are sharp, narrow features above a gradual background. Count and number every peak systematically from left to right.
Read 2θ peak positions at the apex. Modern instruments report to 0.01° precision. Calibrate using NIST SRM 660c for high-accuracy work.
Convert 2θ to d-spacing via Bragg's Law: d = λ / (2 sinθ). Use λ = 0.15406 nm for CuKα. Verify against the Materials Project.
Match d-spacings to a reference database (ICDD, COD, ICSD) to identify the phase. Confirm all peaks match, not just the strongest.
Index peaks with Miller indices (hkl) and calculate lattice parameters using the crystal-system formula. Cross-check all peaks for internal consistency.
Peak width → crystallite size via the Scherrer equation: D = Kλ/(β cosθ). β must be in radians. Broader peaks = smaller crystallites.
The master equation is Bragg's Law: nλ = 2d sinθ. Every step flows from this single relationship.
References
All references are in IEEE citation style. All sources are peer-reviewed journals, internationally recognised textbooks, or authoritative academic databases.
- B. D. Cullity and S. R. Stock, Elements of X-Ray Diffraction, 3rd ed. Upper Saddle River, NJ, USA: Pearson Prentice Hall, 2001. — Definitive textbook reference for Bragg's Law, d-spacing calculation, and lattice parameter determination. [Publisher page]
- W. D. Callister Jr. and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed. Hoboken, NJ, USA: John Wiley & Sons, 2018, ch. 3. [Wiley page]
- C. Kittel, Introduction to Solid State Physics, 8th ed. Hoboken, NJ, USA: John Wiley & Sons, 2005, ch. 2. [Wiley page]
- P. Scherrer, "Bestimmung der Größe und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen," Nachrichten Ges. Wiss. Göttingen, pp. 98–100, 1918. — Original Scherrer equation publication. [Springer reference]
- International Union of Crystallography (IUCr), "Powder Diffraction," IUCr Teaching Pamphlets. [iucr.org/education/pamphlets]
- A. Jain et al., "The Materials Project: A materials genome approach," APL Mater., vol. 1, no. 1, Art. no. 011002, 2013, doi: 10.1063/1.4812323. [materialsproject.org]
- S. Gražulis et al., "Crystallography Open Database," J. Appl. Cryst., vol. 42, pp. 726–729, 2009. [crystallography.net/cod]
- R. Verma and S. K. Rout, "Frequency-dependent ferro–antiferro phase transition in donor and acceptor doped bismuth sodium titanate ceramics," J. Appl. Phys., vol. 126, no. 9, Art. no. 094103, Sep. 2019, doi: 10.1063/1.5111505
- H. M. Rietveld, "A profile refinement method for nuclear and magnetic structures," J. Appl. Cryst., vol. 2, pp. 65–71, 1969. [IUCr DOI]
- W. L. Bragg, "The diffraction of short electromagnetic waves by a crystal," Proc. Camb. Phil. Soc., vol. 17, pp. 43–57, 1913. [Royal Society full text]
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