Johnson noise and absolute temperature

Experimental evidence using Josephson junction devices has suggested that Johnson noise in copper fails to be proportional to absolute temperature below 10 millikelvin. A microscopic theory is presented which gives the Johnson noise temperature $\text{T}{}_{\mathrm{<mtext>J</mtext>}}\text{= ?}{}_0{}^1\text{XT}{}_0$ coth ($\text{XT}{}_0\text{T}$) dX where $\text{T}{}_0\text{=}\text{hν}{}_{\mathrm{<mtext>F</mtext>}}\text{2kN}$. For copper, the calculated T₀ = 3.84 mK agrees closely with the value extracted from experimental data, 3.89 mK. Within a few percent, $\text{T}{}_{\mathrm{<mtext>J</mtext>}}\text{? (}\text{T}{}_0\text{2}\text{)}$ coth ($\text{T}{}_0\text{2T}$), and this adequately fits the available experimental data. ν_F is the fermi velocity and N is the length of the resistor. The Johnson noise parameter “T₀” presumably can measure ν_F along different crystal orientations.