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Extracting the pair distribution function of liquids and liquid-vapor surfaces by grazing incidence x-ray diffraction mode
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Image of FIG. 1.
FIG. 1.

Setup for measuring bulk structure factor of a liquid in reflection mode. The incident beam with a wave vector hits the surface at a fixed angle with respect to the liquid surface. For bulk measurements, is kept above the critical angle for total external reflection. The scattered beam is collected at an angle with respect to the surface and at an angle with respect to the axis in the plane.

Image of FIG. 2.
FIG. 2.

The coherent form factor (dashed line) and the total incoherent Compton scattering (dotted line) as given in Ref. 23. The solid line shows the sum of the incoherent and coherent scattering .

Image of FIG. 3.
FIG. 3.

(a) Scattered intensities vs momentum transfer for various incident-beam angles at . The background at all values is at the level, in the same units as shown in the figure. (b) Same data after normalization by the effective volume of scattering . All the data at different incident angles collapse to a single master curve without any fitting parameters. The data are also corrected by the polarization factor. This is up to a scale factor.

Image of FIG. 4.
FIG. 4.

(a) (at ) obtained after scaling the data shown in Fig. 3 (circles), the best fit (solid line) and a second fit (dashed line) with different PDF but within the uncertainty range. (b) Two PDFs used to calculate the best fit for the shown in (a). (c) calculations extended to large values using the two PDFs shown in (b) showing the high values of are almost identical.

Image of FIG. 5.
FIG. 5.

Raw GIXD data after background subtraction above and below the critical incident angle for total reflectivity as indicated. Solid lines are the best fit as discussed in the text. Vertical dashed-dotted lines indicate main peak positions of the bulk and surface structure factor.

Image of FIG. 6.
FIG. 6.

A comparison of the PDFs from our study and previous studies as indicated. The two PDFs of our study are the same as those shown in Fig. 4(b).

Image of FIG. 7.
FIG. 7.

Illustration of a side view of the beam footprint (upper panel). As the beam penetrates the bulk, the center of the footprint is shifted along . The top view of the beam footprint (, middle panel) shows the cross section with the footprint of an outgoing beam at angle . The lower panel shows a footprint of the incident beam at a finite value.

Image of FIG. 8.
FIG. 8.

Attenuation length for an external incident beam (solid line) and for an internal incident beam (dashed line) at the vapor/water interface, calculated by Eq. (A3) for a x-ray beam. The two curves converge at angles larger than the critical angle for total reflection. (Arrow indicates the location of the critical angle.)

Image of FIG. 9.
FIG. 9.

Effective volume of scattering using Eq. (A2) at various angles of incident beam and . The calculation is for a beam and slit parameters , , .

Image of FIG. 10.
FIG. 10.

Illustrations for attenuation factors and the integration range of Eq. (B3).

Image of FIG. 11.
FIG. 11.

(a) data from Ref. 14 (circles). Solid and dashed lines are the best fits using the method described in Sec. III. (b) Two extracted PDFs from (a) that fit the data equally well. Despite the higher range of the measured values, possible uncertainty in height, width, and position of the first peak in is evident.


Generic image for table
Table I.

Parameters that generate the best-fit calculated structure factor using Eq. (7).


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Extracting the pair distribution function of liquids and liquid-vapor surfaces by grazing incidence x-ray diffraction mode