= Solution
{c}
Without the <Peierls distortion>, adjacent <chemical atoms> are separated by $a/2$. The <primitive unit cell> contains one <chemical atom> and contributes one <electron>. A <energy band> with two <spin> states accommodates two <electrons> per <primitive unit cell>, so the first <energy band> is half filled. Its true first <Brillouin zone> extends to $\pm2\pi/a$, while the <Fermi points> are at $\pm\pi/a$.
Describing the same chain with a doubled <unit cell> of length $a$ merely folds the half-filled <energy band> into two branches that touch at those <Fermi points>. The unbroken <lattice translation> by $a/2$ prevents an artificial <band gap> there. A <Peierls distortion> breaks that smaller <lattice translation> and supplies the missing <Fourier coefficient> at $2k_F=G$; the lower folded <energy band> can then be completely filled and separated from the upper <energy band>. Thus the zero-<band gap> result at $\delta=0$ agrees with the original one-<electron> <band filling>.
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