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Chapter 7: Q 7.37 (page 293)

Prove that the peak of the Planck spectrum is at x = 2.82.

Short Answer

Expert verified

Hence proved that plank's spectrum is at x = 2.82.

Step by step solution

01

Given information

The peak of the Planck spectrum is at e = 2.82.

02

Explanation

Set the derivative of x3/ex-1with respect to x equals to zero to get the maximum, then solve for:

∂∂xx3ex-1=03x2ex-1-x3exex-12=03x2ex-1-x3ex=03x2ex-3x2-x3ex=03ex-3-xex=0

Using matlab to solve the above equation: The code is:

Therefore,x=2.8214

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Most popular questions from this chapter

An atomic nucleus can be crudely modeled as a gas of nucleons with a number density of 0.18fm-3(where 1fm=10-15m). Because nucleons come in two different types (protons and neutrons), each with spin 1/2, each spatial wavefunction can hold four nucleons. Calculate the Fermi energy of this system, in MeV. Also calculate the Fermi temperature, and comment on the result.

Consider a free Fermi gas in two dimensions, confined to a square area A=L2•

(a) Find the Fermi energy (in terms of Nand A), and show that the average energy of the particles is ∈F2.

(b) Derive a formula for the density of states. You should find that it is a constant, independent of ∈.

(c) Explain how the chemical potential of this system should behave as a function of temperature, both when role="math" localid="1650186338941" kT≪∈Fand when Tis much higher.

(d) Because g∈is a constant for this system, it is possible to carry out the integral 7.53 for the number of particles analytically. Do so, and solve for μas a function of N. Show that the resulting formula has the expected qualitative behavior.

(e) Show that in the high-temperature limit, kT≫∈F, the chemical potential of this system is the same as that of an ordinary ideal gas.

Show that when a system is in thermal and diffusive equilibrium with a reservoir, the average number of particles in the system is

N—=kTZ∂Z∂μ

where the partial derivative is taken at fixed temperature and volume. Show also that the mean square number of particles is

N2¯=(kT)2Z∂2Z∂μ2

Use these results to show that the standard deviation of Nis

σN=kT∂N—/∂μ,

in analogy with Problem6.18Finally, apply this formula to an ideal gas, to obtain a simple expression forσNin terms ofN¯Discuss your result briefly.

Problem 7.67. In the first achievement of Bose-Einstein condensation with atomic hydrogen, a gas of approximately 2×1010atoms was trapped and cooled until its peak density was1.8×1014atoms/cm3. Calculate the condensation temperature for this system, and compare to the measured value of50μK.

Repeat the previous problem, taking into account the two independent spin states of the electron. Now the system has two "occupied" states, one with the electron in each spin configuration. However, the chemical potential of the electron gas is also slightly different. Show that the ratio of probabilities is the same as before: The spin degeneracy cancels out of the saha equation.
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