Fittingwaves into boxes 45kkkyandkkky+δkkkyandwithz-componentbetweenkkzandkkz+δkkz.Callthis number
g(kkx,kkky,kkz)δkkxδkkkyδkkz.Since the states are evenly spread ink-space with the spacing
(π/a),the requirednumberis
g(kkx,kkky,kkz)δkkxδkkkyδkkz=(a/π)^3 δkkxδkkkyδkkz (4.3)Thefunctiongsodefinedisa‘densityofstates’ink.Its constant value simplysets
out that the states are uniformlyspread ink-space. However, this is a significant and
useful idea. The number of states in any group can be evaluated from the volume in
k-space occupiedbythegroup, together with(4.3).
For example, to discuss the equilibrium properties ofgases we shall make extensive
use of another density of states functiong(k),which relates to the (scalar) magnitude
ofkonly,irrespective ofitsdirection. Itsdefinitionisthatg(k)δkisthe number of
k-states with values of(scalar)kbetweenkandk+δk.Its form may be derived from
(4.3)from anintegration over angles. Since thedensity ofstatesinkis constant, the
answerissimply
g(k)δk=(a/π)^3 ·appropriate volumeink-space=(a/π)^3 ·( 4 πk^2 δk/ 8 )=V/( 2 π)^3 · 4 πk^2 δk (4.4)Inthisderivation thefactor 4πk^2 δkarises as thevolume ofasphericalshellofradius
kand thicknessδk.The^18 factor comes since we onlyrequire the^18 of the shell for
whichkkx,kkkyandkkzare all positive (i.e. the positive octant).
Equation (4.4)isthe mostimportant resultofthis section. Beforediscussinghow
itis usedin statisticalphysics, we maynote severalpoints.
The dependence on the box volumeVin (4.4) is always correct. Our outline proof
relatedtoacubeofsidea.However, the resultis truefor abox ofany shape. Itis
easyto verifythatit worksforacuboid–tryit! –butitis not at alleasytodothe
mathematics for an arbitrary shape of box. However, a physicist knows that only
thevolume (andnot theshape) ofa container ofgasisfoundtoinfluenceitsbulk
thermodynamic properties, so perhaps one should not be surprised at this result.
Rather more interestingly, the result (4.4) remains valid if we define differently
what we meanbyabox. Above we adoptedstandingwave-boundaryconditions,
i.e.ψ =0 at the box edge. However, this boundarycondition is not always
appropriate, just because it gives standing waves as its normal modes. In a realistic
gas thereis some scatteringbetween particles, usuallystrongenoughthat the
particles are scattered manytimes in a passage across the box. Therefore it is not
always helpful to picture the particles ‘rattling about’in a stationary standing-wave
state. Thisis particularlytrueindiscussingtransport properties orflow properties
of the gas. Rather one wishes a definition of a box which gives travelling waves
for the normalmodes, andthisisachievedmathematicallybyadopting‘periodic
boundaryconditions’foracuboidalbox.