We consider the Casimir interaction energy between a plane and a sphere of radius R at finite temperature T as a function of the distance of closest approach L. Typical experimental conditions are such that the thermal wavelength ${\lambda }_T=\hslash c/k_{\mathrm{B}}T$ satisfies the condition $L\ll {\lambda }_T\ll R$. We derive the leading correction to the proximity-force approximation valid for such intermediate temperatures by developing the scattering formula in the plane-wave basis. Our analytical result captures the joint effect of the spherical geometry and temperature and is written as a sum of temperature-dependent logarithmic terms. Surprisingly, two of the logarithmic terms arise from the Matsubara zero-frequency contribution.

中文翻译：

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平面和球体之间的卡西米尔相互作用：中温下逼近逼近的校正

我们认为，在有限温度T下，平面与半径为R的球体之间的卡西米尔相互作用能是最接近距离L的函数。典型的实验条件是热波长${\lambda }_{Ť}=\hslash C/{ķ}_{\mathrm{乙}}Ť$ 满足条件 $\mathrm{大号}\ll {\lambda }_{Ť}\ll \mathrm{[R}$。通过在平面波基础上建立散射公式，我们得出了对这种中间温度有效的近似力近似值的领先校正。我们的分析结果捕获了球形几何形状和温度的共同影响，并写为温度相关对数项的总和。令人惊讶的是，两个对数项源自松原零频贡献。