International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 2.9, p. 129

## Section 2.9.6. Resolution in real space

G. S. Smitha and C. F. Majkrzakb

aManuel Lujan Jr Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, and bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

### 2.9.6. Resolution in real space

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From Fig. 2.9.2.3, the period δQ of the reflectivity oscillation (in the region where the Born approximation becomes valid, sufficiently far away from the critical angle) is inversely proportional to the thickness t of the film. That is, 2π/(δQ) = t. Consequently, in order to be able to resolve reflectivity oscillations for a film of thickness t, the instrumental Q resolution ΔQ [from equation (2.9.5.1)] must be approximately 2π/t or smaller. With sufficiently good instrumental resolution, even the thickness of a film with non-abrupt interfaces can be accurately determined, as demonstrated by the hypothetical case depicted in Fig. 2.9.6.1 (where the instrumental resolution is taken to be perfect): an overall film-thickness difference of 2 Å (between 42 and 40 Å films) is clearly resolved at a Q of about 0.2 Å−1. In practice, differences even less than this can be distinguished. Note, however, that to `see' more detailed features in the scattering-density profile (such as the oscillation on top of the plateau shown for the long-dash profile in the inset of Fig. 2.9.6.1), other than the overall film thickness, it can be necessary to make reflectivity measurements at values of Q corresponding to 2π/(characteristic dimension of the feature).

 Figure 2.9.6.1 | top | pdf |Calculated neutron reflectivity curves corresponding to the three density profiles in the inset (i.e. solid line in density profile corresponds to solid line in reflectivity plot). Note that in the density plots the solid and long-dash curves coincide except at the oscillation on top of the plateau, whereas the solid and short-dashed curves coincide except at the trailing edge between 30 and 42 Å.