Original public paper: Physical PDF pages 258–261. All three optional notes and both long questions are covered.

Source: Cambridge examination PDF, physical pages 215–218; printed paper 5.

Original paper: Public original PDF, physical pages 390–393; question pages 391–393.

Original paper: Advanced Quantum Physics, 27 May 2015; physical PDF pages 343–346.

Original solutions to the released 2016 1/AQC paper. ^[ref]

The Paleocene is the earliest Palaeogene epoch, approximately $66$ to $56$ million years ago.^[ref] It follows the Cretaceous and precedes the Eocene. Correlating the same interval between separated sedimentary rocks requires independent stratigraphic evidence; thickness alone is insufficient because deposition rates, erosion and missing intervals differ between sites.

The Eocene is a Palaeogene epoch following the Paleocene, approximately $56$ to $33.9$ million years ago.^[ref] Its strata are used in paleoclimate reconstruction, where oxygen isotope ratios and other material properties must be calibrated against the physical conditions that produced them. The stratigraphic label identifies time, rather than one globally uniform sediment type.

For stationary covariance $C(t)=(k_{B}T/k)e^{-|t|/{\tau }_k}$, the mean over duration $\mathcal{T}$ has variance $2(k_{B}T/k)[{\tau }_k/\mathcal{T}-({\tau }_k/\mathcal{T}{)}^2(1-e^{-\mathcal{T}/{\tau }_k})]$. At $\mathcal{T}\gg {\tau }_k$ this becomes $2(k_{B}T/k){\tau }_k/\mathcal{T}$. Closely sampled positions remain correlated and cannot be counted as independent.

Expanding the Fermi denominator as an alternating exponential series gives $F_{1/2}(0)=(1-2^{-1/2})\zeta (3/2)\simeq 0.765147$. Thus a band with chemical potential pinned exactly at its lower edge has $n=F_{1/2}(0)N_c$, rather than the classical result $N_c$.

For $F_j(\eta )=\Gamma (j+1{)}^{-1}{\int }_0^{\infty }{t}^j/(e^{t-\eta }+1)dt$, the nondegenerate limit is $\eta \ll -1$. Replacing the denominator by $e^{t-\eta }$ givesbecause the remaining integral is $\Gamma (j+1)$. Expanding in powers of fugacity gives $F_j(\eta )=e^{\eta }-e^{2\eta }/2^{j+1}+\cdots$.

Thus a carrier density proportional to $F_{1/2}[(\mu -E_c)/(k_{B}T)]$ reduces to the familiar exponential occupation when the chemical potential lies several thermal energies below the band edge. At $\mu =E_c$, the expansion parameter is one and the classical replacement is not controlled. The normalization of the Normalized complete Fermi-Dirac integral must be stated before prefactors from different conventions are compared.

For equal-mass conduction and valence edges at $0,-E_g$ and fixed $\mu =0$, $n=N(T)F_{1/2}(0)$ and $p=N(T)F_{1/2}[-E_g/(k_{B}T)]$, where $N(T)=2[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}$. Other states pinning the chemical potential can compensate the unequal carrier charges; imposing intrinsic charge neutrality in addition would change the problem.

The Miocene is the older Neogene epoch, approximately $23.04$ to $5.333$ million years ago.^[ref] Dating a rock to this interval places it before the Pliocene; interpreting the environment in which it formed additionally requires observations such as sedimentary rock structure, composition and fossils. An age label and an environmental reconstruction answer different questions.

The Pliocene is the younger Neogene epoch, approximately $5.333$ to $2.58$ million years ago.^[ref] It is included in the long marine oxygen isotope ratio stack used to compare earlier ocean conditions with those of the Pleistocene. Aligning records makes their temporal patterns comparable; it does not turn one isotope measurement into a unique independent temperature measurement.^[ref]

A channel's electrical conductivity weights its density of states by squared velocity and the transport momentum relaxation time. In an elementary spin-conserving $s$–$d$ scattering model, ${\tau }_s^{-1}\propto |V_{sd,s}{|}^2{D}_{d,s}(E_F)$, so available heavy final states can reduce the current carried by light states. Band filling, defects and interfaces affect the asymmetry; total spin population alone does not determine its sign. A density-of-states sketch without scattering matrix elements is not a universal transport prediction.