the scattering angle 2 theta

When a crystal diffracts a beam, the diffracted ray leaves at an angle to the original direction — it has been deflected. The scattering angle, written 2 theta ('two-theta'), is that total angle of deflection: the angle between where the beam was going and where it ends up after diffracting. It is, quite simply, the quantity a diffractometer measures directly, the horizontal axis of almost every powder diffraction chart.

Why two-theta and not just theta? Because of the Bragg geometry. The Bragg angle theta is measured between the beam and the reflecting planes, and the beam is deflected by theta on the way in and another theta on the way out, for a total turn of 2 theta. So the detector, which sees the deflected beam, sits at twice the Bragg angle. A reflection whose Bragg angle is 15 degrees appears at 2 theta = 30 degrees on the chart. Keeping this factor of two straight is essential: quoting a peak at the wrong one of theta or 2 theta corrupts every spacing you calculate from it.

Because 2 theta is what you actually record, converting a pattern to physical spacings means halving it first: take 2 theta, get theta, then use Bragg's law d = lambda / (2 sin theta). Small 2 theta corresponds to large plane spacings (peaks near the start of the chart), and large 2 theta to small spacings (peaks pushed out to the right). The full swing runs from 0 to 180 degrees in principle, though the useful high-angle end is limited by the fall-off of the atomic scattering factor and the Debye-Waller factor.