Berman

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At the lowest level of approximation α(T) can be considered a constant, in which case integration yields, or . Truncation to first order in the expansion of the logarithmic terms results in the expression . Strictly, this truncation leads to a slightly varying thermal expansion coefficient because re-differentiation of the truncated equation leads to , which means that . Berman (1988) proposed a simple extension to accommodate non-linear thermal expansion: 



Differentiation yields . Given the small changes in volume with temperature, this is approximately . The parameter α0 is the thermal expansion coefficient at Tref. However, this equation is not valid for low temperatures because it predicts a finite value for α(T) at absolute zero except for the special case of .