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In high pressure physics what is the difference between diamond anvil cells (DAC) and Bridgman cells? My understanding is that they are both forms of anvil cell but in the scientific literature I'm reading they are referred to separately.

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In 1958 DAC was developed in NIST by Weir, Lippincott, Van Valkenburg, and Bunting. This DAC article shows an image of a hand palm sized DAC in the NIST Museum.

The diamond anvil cell (DAC) is a refined device, based on the Bridgman cell. The latter was developed in the beginning of the first half of the 20th century. Your question aims on the updated DAC technique. Diamond anvil cell (DAC) allows optical/ X-Ray experiments. Flat-top should be optical polished to reduce scattering. A Ruby usually is used as a pressure indicator. Source: wiki, under the Creative Commons Attribution-Share Alike 3.0 Unported license

What is the technical distinctive feature?

The Bridgman cell has a quatro-plate geometry for applying force. This first developed technique has the feature to allow rays or wires between a pair of pressure plates and is available in a wide pressure range. Polished diamond flat-tops allow beam transmission through the sample. This is depicted in the wikipedia article. However beam intensity is extenuated due too high Fresnel losses. Refractive index $n_{diamond}\approx 2.4$ is high. Spectroscopic measurement may be influenced by natural lattice defects in diamond lattice.

The refined method, called DAC, allows higher pressure. To archive this the Bridgman anvil made of tungsten-carbon alloy was replaced by a single-crystal diamond. The problem of breaking anvils was solved and a new pressure range was available.

Fun fact

Diamonds used by NIST often were destroyed due to high pressure. The diamonds were former property of smugglers. Government agents confiscated the gem diamonds. They were given for scientific purpose to the researchers.

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