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Chapter 10: Q54E (page 470)

Section 10.6 notes that as causes of resistance, ionic vibrations give way to lattice imperfections at around 10 K. A typical spring constant between atoms in a solid is or order of magnitude103鈥�Nm and typical spacing is nominally10-10鈥�m . Estimate how much the vibrating atoms locations might deviate, as a function of their normal separations, at 10 K.

Short Answer

Expert verified

The vibrational distance is only 0.5%of the nominal separation.

Step by step solution

01

 Step 1: Determine the formulas

Consider the formulato determine the thermal energy as:

E=knT 鈥�.. (1)

Here, knis Boltzmann constant and the temperature is T.

Consider the expression for the vibrational distance as:

x=2(蝉辫谤颈苍驳鈥塸辞迟别苍迟颈补濒鈥塭苍别谤驳测)魏

02

Determine the number of vibrating atoms locations that might deviate.

From equation (1) solve for the thermal energy as:

ET=(1.38脳10鈭�23鈥�JK)(10鈥块)=1.38脳10鈭�22鈥塉

Solve for the expression for the vibrational energy as:

x=2(1.38脳103)103鈥�Nm=5脳10鈭�23鈥尘

Solve for the ratio of the distance as:

xa=5脳10鈭�2310鈭�10=0.5%

Therefore, the vibrational distance is0.5% only of the nominal separation.

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