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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 537
rearranged to give cos (kπ/[N + 1]) = (E − α)/(2β). �ese two substi-
tutions are used in the expression for the derivative
dE
dk
= − 2βπ
N + 1
sin( kπ
N + 1
) = − 2βπ
N + 1
[1 − cos2 ( kπ
N + 1
)]
1/2
= − 2βπ
N + 1
⎡⎢⎢⎢⎢⎣
1 − (E − α
2β
)
2⎤⎥⎥⎥⎥⎦
1/2
Density of states ρ(E) is de�ned as ρ(E) = dk/dE, thus
ρ(E) = dk
dE
= 1
dE/dk
= − (N + 1)/2βπ
[1 − ( E−α
2β )
2
]
1/2
(b) As E → α ± 2β, E − α → ±2β, and thus (E − α)/2β → ±1. It follows that
ρ(E)→ − (N + 1)/2βπ
[1 − (±1)2]1/2
=∞
�us the density of states increases towards the edges of the bands in a
one-dimensional metallic conductor.
P15C.6 �e wavefunction for a ring of N atoms must be such that it has the same value
at the (hypothetical) atomwith index (N+1) as it does for the atomwith index
1 because, in a ring these two atoms are the same.�is can only be achieved if
the wavefunction varies in such a way that it goes through a whole number of
cycles around the ring.
�e wavefunction for a line of atoms is not constrained in this way, so if a linear
chain were to have its ends ‘connected’ to form a ring, some wavefunctions of
the linear chain would not satisfy requirements necessary to be a wavefunction
for a ring of atoms.
�e energy levels of a line of atoms are non-degenerate, but for a ring all of the
levels, apart from the one with k = 0, are doubly degenerate. �is degeneracy
can be associated with the possibility that the electron can travel clockwise or
anti-clockwise around the ring. For a chain, no such net motion is possible.
P15C.8 (a) Assuming that the ionic radius of Ca+ is close to that of K+, the radius-
ratio rule can be used to predict the lattice structure of CaCl.�e radius
ratio is γ = 138 pm/181 pm = 0.762; this is > 0.732, therefore eightfold
coordination, as in CsCl, is predicted and the appropriate value of the
Madelung constant (Table 15C.3 on page 662) is A = 1.763.
�e Born–Mayer equation, [15C.5–662], is used to estimate the lattice
enthalpy since when zero-point contributions to potential energy are ne-
glected then lattice enthalpy is equal to the negative of Ep,min. In this
expression, z i is the charge number of ion i, d is the distance between