The Radial Distribution Function

From sharp spots to broad halos

The last two guides in this rung settled the central fact about the disordered state: a glass or an amorphous solid has no long-range order — no lattice, and therefore no reciprocal lattice to photograph — yet it keeps real short-range order, definite nearest-neighbour bonds at a definite distance, and even medium-range order beyond that. That raises a sharp practical question. In the diffraction rung, Bragg's law and the reciprocal lattice gave us a crisp machinery for reading a crystal: sharp spots, indexed, unit cell from positions and motif from intensities. Point that same diffractometer at silica glass and the spots are gone. What you get instead is a handful of broad, blurry humps — diffuse halos smeared across a wide range of angle.

Why halos and not spots? Sharp Bragg peaks are the exclusive signature of a reciprocal lattice, and only a periodic crystal owns one; a glass simply has no reciprocal lattice for the spots to land on. But — and this is the whole point — the diffuse pattern is not featureless. A truly structureless material, an ideal gas of atoms with no correlations at all, would scatter into a flat, smooth background with no humps whatsoever. The halos exist precisely because there IS short-range order: nearest neighbours sit at a well-defined separation, and wave interference from those correlated pairs piles scattered intensity up at some angles and thins it at others. The halos are the short-range order talking. We just need a statistical language to hear what they say — and that language is the radial distribution function.

Sitting on an atom: the pair distribution function g(r)

Here is the idea, and it is wonderfully physical. Pick any atom in the glass, sit right on it, and look outward. Ask one question: at distance r from me, how crowded are the other atoms, compared with the material's average density? That ratio is the pair distribution function g(r) — also called the pair correlation function. Read it as a crowd-o-meter. g(r) = 1 means 'exactly average density out here, nothing special'; g(r) greater than 1 means 'atoms like to sit at this distance, it is more crowded than average'; g(r) less than 1 means 'atoms avoid this distance, it is emptier than average.' Because every atom is an equally good place to sit, g(r) is really an average over all of them at once — a statistic, not a snapshot of one frozen arrangement.

Now walk outward and watch g(r) unfold. At very small r it is flat zero: atoms are hard, they cannot overlap or interpenetrate, so no neighbour ever sits closer than a bond length. Then g(r) leaps up into a tall, fairly sharp first peak centred at r1, the nearest-neighbour bond length — your bonded neighbours all cluster there. Past it, g(r) dips into a valley, rises into a broader, lower second peak (the second-neighbour shell), swells into a gentler third, and the wiggles keep damping down. Within a few bond lengths g(r) has flattened to 1 and stays there: far enough away, the glass has forgotten where you are, and every distance just shows average density. That fade of the oscillations to 1 is the mathematical face of 'no long-range order' — in a crystal, by contrast, the peaks would stay sharp and never die, marking exact shell after exact shell out to infinity.

One refinement turns g(r) into the quantity the field actually plots. The number of atoms in a thin spherical shell between r and r + dr is 4 times pi times r^2 times rho_0 times g(r) times dr, where rho_0 is the average number density (atoms per unit volume). The function RDF(r) = 4 pi r^2 rho_0 g(r) is the radial distribution function proper — literally the count of neighbours found at each distance. Notice the 4 pi r^2 shell factor: even out where g(r) has settled to 1, the RDF keeps climbing as r^2, simply because a bigger shell holds more atoms. So g(r) is the crowd-o-meter (density relative to average), and the RDF is g(r) reweighted into an honest headcount per unit distance. The two carry the same information; which one you draw is just a matter of convenience.

RADIAL DISTRIBUTION   g(r) = local density / average density

CRYSTAL (long-range order): sharp spikes at exact shell radii, forever
  g(r)|   |       |         |            |             |
      |   |       |         |            |             |
    1 |---|-------|---------|------------|-------------|---
      +---+-------+---------+------------+-------------+--> r

GLASS / LIQUID (short-range order only): peaks broaden and damp to 1
  g(r)|    __
    2 |   /  \      __
      |  /    \   _/  \__          _____________________
    1 |-/------\-/------\___.---''                        g -> 1
      |/        V           (no correlation: average density)
    0 |__
      +----+---------+----------+-----------+-----------+--> r
         r1        ~2 r1
         ^
  first peak:  centre = nearest-neighbour bond length
               area   = coordination number

Reading the first peak: bond length and coordination number

The first peak is a small gold mine, because three of its features each report a different physical thing. Its CENTRE is the nearest-neighbour bond length — the equilibrium spacing at which your bonded neighbours sit. Its WIDTH measures how much those bonds vary: a spread from static disorder (the glass froze in with a range of slightly different distances, not one exact value) plus thermal vibration blurring each atom about its mean. And its AREA — the number of atoms tucked under the first peak of the RDF — is the coordination number, the average count of nearest neighbours. That is the same coordination number you met for crystals in the close-packing rung (FCC gives 12, diamond-cubic silicon gives 4), now measured for a solid that has no lattice at all.

Put real numbers on it with amorphous silicon, a-Si. Its measured g(r) puts the first peak at r1 = 2.35 angstrom — identical, to within the peak's width, to the Si–Si bond in crystalline diamond-cubic silicon. Integrate the RDF's first peak out to its first minimum and the area comes to about 4. So every silicon atom in the glass still has 4 nearest neighbours arranged in a tetrahedron, exactly as in the perfect crystal: the short-range order is essentially untouched. What has been lost lives farther out. The second-neighbour peak is noticeably broadened, the third is smeared toward the g -> 1 background, and beyond that the curve is flat. That single curve is the entire amorphous state in one picture — short-range order intact, long-range order gone — and it is precisely the target any continuous random network model of a-Si must reproduce, the subject of the next guide.

The structure factor S(Q): what the detector actually sees

Here is an awkward truth: no instrument ever measures g(r) directly. What a detector records is the angular pattern of scattered intensity — those broad halos versus angle. Packaged and normalized, that pattern is the static structure factor S(Q), written as a function of the length of the scattering vector, Q = 4 pi sin theta / lambda. (This Q is just 2 pi times the scattering-vector length s = 2 sin theta / lambda you met in the diffraction rung — same arrow, physicist's units.) For a glass, S(Q) is a smooth curve carrying a few broad humps — the halos again — that oscillate and then settle to S(Q) -> 1 at high Q. That high-Q limit of 1 is the reciprocal-space twin of the g -> 1 limit in real space: both are just the material saying 'no correlations survive out here.'

Now the beautiful part, and it is an old friend in new clothes. g(r) and S(Q) are a Fourier-transform pair — the very same real-space-versus-reciprocal-space duality that turned a crystal into its reciprocal lattice, now doing its work for a material with no lattice. A single sine Fourier transform converts the measured S(Q) into the real-space radial distribution function: schematically, G(r) is proportional to the integral of Q times [S(Q) minus 1] times sin(Q r) over Q. So the experiment lives in Q-space, where all you can see are broad blurry halos, and one transform hands you the r-space picture — sharp bond lengths, a countable coordination shell — that you can actually reason about. This whole workflow, measure the total scattering and Fourier-transform it into g(r), is exactly pair distribution function analysis, the everyday tool for glasses, liquids, and nanostructured matter.

1. Collect the total scattering. Shine X-rays or neutrons on the glass or liquid and record intensity versus angle — a smooth curve of a few broad halos, no sharp peaks anywhere.
2. Convert angle to Q = 4 pi sin theta / lambda, then correct and normalize (subtract background, divide out the atomic scattering) to get the structure factor S(Q).
3. Sine Fourier-transform S(Q) minus 1 into real space to obtain g(r), and multiply by 4 pi r^2 rho_0 to get the RDF, the neighbour count at each distance.
4. Read the first peak: its centre is the nearest-neighbour bond length, and its width is the spread of bond distances (static disorder plus thermal vibration).
5. Integrate the RDF's first peak out to its first minimum — that area is the coordination number, the average number of nearest neighbours.

Two honest closing notes about what S(Q) gives and what it hides. The first, lowest-Q hump — often called the first sharp diffraction peak — is the fingerprint of medium-range order: correlations that reach beyond nearest neighbours, such as the loose network repeat of corner-sharing tetrahedra in silica, which shows up around Q of order 1.5 per angstrom. That is real structure past the first shell, and S(Q) sees it. But now the deep limit: because a glass is isotropic, you only ever measure a spherically averaged, one-dimensional curve. The transform to g(r) throws away every scrap of directional information — bond ANGLES, ring statistics, the actual three-dimensional packing. Genuinely different 3D arrangements can share the very same g(r). So g(r) and S(Q) tightly CONSTRAIN a structure but never uniquely PIN it down — a disordered cousin of the phase problem you met with crystals. (A smaller practical wrinkle: you can only measure Q up to some finite maximum, and truncating the transform there sprinkles small false ripples, called termination ripples, into g(r).)

The referee for every glass model

Because g(r) is what we can actually measure, it becomes the umpire: any structural model of a disordered solid is only credible if it reproduces the observed g(r) and S(Q). The next two guides build the two great model families, and both are judged at this bar. For covalent glasses like silica, Zachariasen's continuous random network takes rigid SiO4 tetrahedra — silicon acts as a network former — and links them corner to corner through shared bridging oxygens into an aperiodic three-dimensional net. The O–Si–O angle inside each tetrahedron stays sharp near 109.5 degrees, but the Si–O–Si angle joining tetrahedra is broadly spread. That combination is exactly what makes g(r) show a razor-sharp first peak yet damped outer peaks. Stir in a network modifier such as Na2O and it snaps some of those bridges, creating non-bridging oxygens that loosen the network — the structural chemistry behind why soda-lime glass melts and works so much more easily than pure silica.

For metallic glasses the winning picture is completely different: random close packing of hard spheres, atoms jammed together like ball bearings poured into a bag and shaken down, with a coordination number near 12 to 13 and a strong flavour of icosahedral local order — but no crystalline plane anywhere. Its g(r) carries a giveaway signature: a characteristically SPLIT second peak, a shoulder-and-hump shape that dense random packings show and simple crystals do not. When you see that split second peak in a measured curve, you are looking straight at a dense random-packed metallic glass rather than a fine-grained crystal — a diagnosis made purely from the shape of g(r), which is guide 5's whole story.

One last question the RDF sets up but does not answer on its own: why does any of this freeze in at all? A liquid cooled fast enough to dodge crystallization becomes a supercooled liquid; as it cools its viscosity rockets, and at the glass transition the atoms can no longer rearrange within the time of the experiment — the whole structure kinetically locks. This is a kinetic arrest, not a thermodynamic phase change: the glass-transition temperature drifts with cooling rate, and the free volume (the little extra room atoms need to shuffle past one another) gets frozen in below it. Metals crystallize so eagerly that the earliest metallic glasses needed rapid quenching of order 10^6 kelvin per second. What tips the odds toward a glass is frustration: when a liquid's favourite local packing — icosahedra, with their forbidden 5-fold symmetry — simply cannot tile space periodically, crystallization is geometrically frustrated, and the melt slides into a glass instead of finding a lattice.