By Pauling L.

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**Extra info for A New Crystal for Wave-Length Measurements of Soft X-Rays**

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3 Covariant form for the static response In a covariant version of the description of the static response, P , M are combined into a second rank 4-polarization tensor or 4-magnetization tensor. To avoid confusion with Π let us denote it by M µν (x), which is constructed from P , −M in the same way as the Maxwell tensor, F µν , is constructed from E, B. Similarly, D, H may be combined into a second rank tensor, H µν (x) say. One may choose H µν (x) to reduce to ε0 F µν (x) in vacuo and to ε0 F µν (x) + M µν (x) in a medium.

7) The longitudinal tensor may also be written as the outer product of a longitudinal 4-vector with itself: Lµν (k, u) = −Lµ (k, u)Lν (k, u), kα Gαµ (k, u) ku k µ − k 2 uµ Lµ (k, u) = 2 = . 8) The transverse tensor is T µν (k, u) = = k2 1 − (ku)2 −(ku)2 aµν (k, u) + k 2 g µν − (ku)2 µ kµ kν ν µν L (k, u)L (k, u) + g − . 10) k 2 g µν − k µ k ν = (ku)2 Lµν (k, u) + k 2 T µν (k, u). 11) As already noted, the rotatory tensor is Rµν (k, u) = i µνρσ kρ uσ =i [(ku)2 − k 2 ]1/2 µνρσ Lρ (k)uσ . 12) These three tensors are projection-like operators for the longitudinal, transverse and rotatory parts.

19) The magnetic susceptibility of an electron gas is due to intrinsically quantum mechanical eﬀects (Pauli spin paramagnetism and Landau diamagnetism), and is zero in classical theory. 7 Response tensors for simple media The well-known forms for the response tensors for some simple media are written down in this section and used to construct the corresponding 4-tensors. 1 Cold unmagnetized plasma A well-known form for the response functions for a cold unmagnetized plasma with plasma frequency ωp2 is K L (ω) = K T (ω) = 1 − ωp2 /ω 2 .