Crystallography and Drug Polymorphs — Why the Same Molecule Behaves Differently
Bragg's law is on every solid-state syllabus, usually applied to sodium chloride or a simple metal. Its most demanding real application is in the solid form of medicines, where the same molecule can crystallise in more than one packing arrangement, and where those arrangements dissolve at different rates. That phenomenon is polymorphism, and it is the reason a pharmaceutical company will spend months characterising a crystal before it ever formulates a tablet.
This article uses two formulas you already have — Bragg's law and the unit-cell density relation — to show how two forms of one compound are distinguished, and then sets out carefully what those measurements do and do not prove.
The formulas
| Symbol | Meaning | Unit |
|---|---|---|
| λ | Wavelength of the X-rays used. Copper Kα radiation, common in powder work, is 1.5406 Å | Å |
| d | Spacing between one family of parallel lattice planes | Å |
| θ | Bragg angle. Instruments report 2θ, so always halve it first | degrees |
| n | Order of the reflection; taken as 1 in routine indexing | — |
| Z | Number of formula units in one unit cell | — |
| M | Molar mass of the compound | g/mol |
| NA | Avogadro constant, 6.022 × 1023 mol−1 | mol−1 |
| Vcell | Volume of the unit cell (for an orthorhombic cell, a·b·c) | cm3 in this formula; 1 Å3 = 10−24 cm3 |
What polymorphism actually is
A polymorph is a different crystalline arrangement of the same chemical substance. The molecules are identical; only their packing, orientation and intermolecular contacts in the lattice differ. Because those contacts set the lattice energy, the two forms are genuinely different solids with different physical properties, even though a chemical analysis cannot tell them apart.
| Property | Same or different between polymorphs? |
|---|---|
| Molecular formula and connectivity | Identical — that is the definition |
| Solution NMR, mass spectrum, elemental analysis | Identical (they measure the dissolved or vaporised molecule) |
| Melting point, enthalpy of fusion | Different |
| Crystal density | Different |
| Solubility and dissolution rate | Different — often substantially |
| Powder X-ray diffraction pattern | Different — this is the primary identification method |
| Mechanical behaviour on compression | Different, which affects tablet manufacture |
The dissolution point is the one with consequences. A metastable form has a higher free energy than the stable form, so it dissolves faster and reaches a higher transient concentration. For a drug whose absorption is limited by how fast it dissolves, changing form changes how much reaches the bloodstream. That is why the solid form is a controlled specification, not an incidental detail — and why a batch that unexpectedly converts to a different form during storage is a serious manufacturing problem.
Two related solid forms are worth naming, because exam questions often mix them up. Solvates and hydrates include solvent or water molecules in the lattice, so they are not polymorphs of the pure compound — the composition differs. An amorphous solid has no long-range order at all; it gives no sharp diffraction peaks and typically dissolves fastest of all, but it is thermodynamically unstable and tends to crystallise over time.
Worked example 1 — a d-spacing from a low-angle peak
Copper Kα radiation, λ = 1.5406 Å. A powder pattern shows a strong peak at 2θ = 12.50°.
Step 1 — halve the angle: θ = 12.50 ÷ 2 = 6.25°
Step 2 — sin θ = sin 6.25° = 0.10887
Step 3 — Bragg's law rearranged: d = nλ ÷ (2 sin θ) = 1.5406 ÷ (2 × 0.10887)
d = 1.5406 ÷ 0.21773 = 7.076 Å
Sanity check: molecular crystals have large cells, so d-spacings of several ångström at low 2θ are exactly what you expect. A value of 0.7 Å or 70 Å would mean an arithmetic slip.
Worked example 2 — the second form, and why the peak moved
A second batch of the same compound, run on the same instrument with the same radiation, shows its strongest low-angle peak at 2θ = 9.80° instead.
θ = 4.90°, sin θ = 0.085417
d = 1.5406 ÷ (2 × 0.085417) = 1.5406 ÷ 0.170834 = 9.018 Å
Two samples that are chemically identical are giving lattice plane spacings of 7.076 Å and 9.018 Å. The molecules are packing differently: this is the diffraction evidence for two polymorphs.
Notice the direction of the effect, because it is a standard exam trap. In Bragg's law d and sin θ are inversely related, so a peak at a lower angle corresponds to a larger spacing. Students frequently assert the opposite.
Worked example 3 — crystal density from the unit cell
Suppose a structure refinement on one form of a C13H18O2 compound gives an orthorhombic cell with a = 7.10 Å, b = 12.40 Å, c = 9.80 Å and Z = 4. (These cell dimensions are illustrative numbers for the arithmetic, not a real published structure.)
Molar mass, from C = 12.011, H = 1.008, O = 15.999:
C: 13 × 12.011 = 156.143
H: 18 × 1.008 = 18.144
O: 2 × 15.999 = 31.998
M = 156.143 + 18.144 + 31.998 = 206.285 g/mol
Cell volume: V = 7.10 × 12.40 × 9.80
7.10 × 12.40 = 88.04; 88.04 × 9.80 = 862.79 Å3
In cm3: 862.79 × 10−24 = 8.6279 × 10−22 cm3
Density:
numerator Z·M = 4 × 206.285 = 825.14
denominator NA·V = 6.022 × 1023 × 8.6279 × 10−22 =
6.022 × 8.6279 × 101 = 519.58
ρ = 825.14 ÷ 519.58 = 1.588 g/cm3
A value near 1.2–1.6 g/cm3 is typical for an organic molecular crystal, so this is self-consistent. If the same compound in a second form gave a different cell volume with the same Z, its density would differ — and the denser form is usually, though not always, the more stable one.
How forms are told apart in practice
| Technique | What it tells you about the solid form |
|---|---|
| Powder X-ray diffraction (PXRD) | The routine fingerprint. Peak positions identify the form; it is the primary method for confirming which polymorph a batch is |
| Single-crystal X-ray diffraction | The full three-dimensional structure — atom positions, packing, hydrogen bonding. Needs a good single crystal |
| Differential scanning calorimetry (DSC) | Melting points, enthalpies, and solid–solid transitions between forms on heating |
| Thermogravimetric analysis (TGA) | Mass loss on heating — distinguishes a hydrate or solvate from a true polymorph |
| Infrared and Raman spectroscopy | Sensitive to differences in intermolecular contacts, especially hydrogen bonding, in the solid state |
| Solid-state NMR | Distinguishes crystallographically distinct environments; works on mixtures and on poorly crystalline material |
Which form appears is controlled by how it is crystallised: solvent, degree of supersaturation, cooling rate, stirring, presence of seed crystals and even surfaces in the vessel. A common pattern is that a fast crystallisation gives a metastable form which slowly converts to the stable one on storage — a reminder that crystallisation is often under kinetic rather than thermodynamic control, exactly as in reaction chemistry.
Where the simple formula stops being valid
- Bragg's law gives spacings, not a structure. Peak positions give d values, and that is all. Determining where the atoms are needs the peak intensities and a full refinement. Treating a d-spacing calculation as "solving the structure" is a serious overclaim.
- Relative intensities in a powder pattern are not reliable identifiers. Needle- and plate-shaped crystals settle preferentially in the sample holder, so preferred orientation can make a peak strong in one preparation and weak in another. Match on positions; treat intensities with caution.
- Peak positions shift with temperature and with wavelength. Thermal expansion changes d, and 2θ depends on λ. A pattern compared against a reference taken with different radiation must be converted to d-spacings first — comparing raw 2θ values across wavelengths is meaningless.
- Amorphous material gives a broad halo, not peaks. PXRD is therefore poor at detecting small amounts of amorphous content, and quantifying a mixture of forms from peak areas alone has real limits.
- Similar patterns do not prove identity. Related forms can have patterns that overlap heavily at low resolution. Confirmation normally uses more than one technique, which is why the table above lists six.
- The density formula assumes you know Z. Z comes from the structure solution, not from the cell dimensions. An assumed Z gives a density that is wrong by a whole-number factor — the tell-tale sign of the error.
- Watch the unit conversion. Å3 to cm3 is a factor of 10−24, not 10−8. Getting this wrong is the single commonest failure in the density calculation.
Which solid form of a medicine is used is a regulated formulation decision made by qualified professionals. Nothing here is guidance about any medicine.
Where this appears in exams
| Exam | Typical question |
|---|---|
| IIT-JAM | Bragg's law substitution; unit cell types; density from cell parameters and Z |
| CUET-PG | Definitions of polymorphism, isomorphism, allotropy and amorphous solids |
| GATE | Indexing simple patterns, systematic absences, and structure–property reasoning |
| CSIR-NET | Reciprocal lattice, structure factor, symmetry and space groups; solid-state characterisation methods |
Do the trigonometry and the powers of ten carefully. These calculations fail on sin θ and on the Å3 → cm3 conversion far more often than on the physics. There is no dedicated diffraction tool in the suite, so the honest link is to the scientific calculator that a plain visit opens — it has the trigonometric and exponent keys these two formulas need.
Open the ABC Chemistry Calculator Suite →Preparing for IIT-JAM, GATE, CSIR-NET or CUET-PG? ABC Chemistry runs dedicated competitive-exam batches at its coaching centre and online for students across India — details at abcchemistry.in.