SN.2 · Bragg diffraction and electron-gas heat capacity

GRE · GRE Subject Test · GRE 物理 · 知识点 45

45.1

Bragg diffraction 布拉格衍射 and electron-gas heat capacity

Rotating a crystal in an X-ray beam maps its atomic planes, because constructive reflection happens only at discrete angles.

Prerequisites: 8, 36.

Apply the Bragg condition 布拉格条件 to find plane spacings or diffraction angles
Estimate free-electron Fermi energy 费米能 and Fermi temperature
Distinguish electronic and lattice heat-capacity contributions

词汇训练

English	中文	拼音
Bragg condition/bræɡ kənˈdɪʃn/	布拉格条件	bù lā gé tiáo jiàn
Fermi energy/ˈfɜːmi ˈenədʒi/	费米能	fèi mǐ néng

45.2

Apply the Bragg condition

Bragg reflection from parallel planes spaced d interferes constructively when 2d sinθ=nλ, with θ measured from the plane surface (the glancing angle), not from the normal. First order n=1 gives the smallest angle. For λ=0.154 nm and a peak at θ=20°, d=0.154/(2 sin20°)≈0.225 nm. If 2d<λ no order exists; the sine must stay at or below 1.

45.3

Read the lattice constant

In a cubic crystal the (hkl) plane spacing is d=a/sqrt(h²+k²+l²) with lattice constant a; the (110) planes give a/√2. A measured Bragg angle therefore measures the lattice constant. Systematic absences from the basis (for example body-centred lattices missing h+k+l odd) carry structure information beyond the spacing formula.

45.4

Fill the electron sea

The free-electron model packs N conduction electrons into states up to the Fermi wavevector k_F=(3π²n)^(1/3) for number density n, giving E_F=$\hbar$²k_F²/(2m). For n=8.5×10²⁸ /m³, k_F≈1.36×10¹⁰ /m, E_F≈7.05 eV and T_F=E_F/k_B≈8.2×10⁴ K. Since T_F far exceeds room temperature, only a fraction of order T/T_F of the electrons are thermally excited.

45.5

Separate the heat capacities

Metals therefore have two low-temperature heat-capacity terms: an electronic term γT linear in T and a lattice (Debye) term βT³. Plotting C/T against T² gives intercept γ and slope β. At room temperature the lattice part approaches the classical Dulong–Petit value and dominates; the electronic term matters at low T, where the T³ lattice term collapses faster.

45.6

Worked method

Bragg diffraction uses angle theta from the crystal plane, not from its normal.

$$2d\sin\theta=n\lambda,\qquad d=\lambda/(2\sin\theta)=(0.154\,\mathrm{nm})/[2\sin(20^\circ)]=0.225\,\mathrm{nm}.$$

Low-temperature metal heat capacity has separate electronic and lattice terms:

$$C=\gamma T+\beta T^3,\qquad C/T=\gamma+\beta T^2.$$

A plot of C/T against T squared has intercept gamma and slope beta. A straight line identifies this model's regime.

Bragg diffraction and electron-gas heat capacity: GRE original diagram — Bragg diffraction and electron-gas heat capacity: original GRE teaching diagram.

词汇训练

English	中文	拼音
Bragg diffraction	布拉格衍射	bù lā gé yǎn shè

45.7

Check conditions and vocabulary

Measuring θ from the plane normal instead of the plane, or assigning the whole low-T heat capacity to electrons. The sine condition and the two-term decomposition are where these calculations fail.

Bragg condition: Constructive reflection from lattice planes when 2d sinθ=nλ, with θ measured from the plane.

Fermi energy: The highest occupied electron energy at zero temperature in the free-electron model.