GREPhysics.NET: GR9277 #23

GR9277 #23

Problem

GREPhysics.NET Official Solution

\prob{23}
The Fermi temperature of Cu is about 80,000 K. Which of the following is most nearly equal to the average speed of a conduction electron in Cu?

2E-2 m/s
2m/s
2E2 m/s
2E4 m/s
2E6 m/s

Statistical Mechanics $\Rightarrow$ }Fermi Temperature

(Much of the stuff I classified as Stat Mech might also be considered Condensed Matter or Solid State Physics. They are classified as thus because the Stat Mech book I mentioned in the booklist on the site $http://grephysics.yosunism.com$ is perhaps the best intro to all this.)

The Fermi velocity is related by $\epsilon_F=k T_F =\frac{1}{2}m v^2\Rightarrow v=\sqrt{\frac{2kT_F}{m}}$ , where $\epsilon_F$ is the fermi energy, and $T_F$ is the Fermi temperature.

One should know by heart the following quantities, $k=1.381E-23$ and $m=9.11E-31$ (but then again, they are also given in the table of constants included with the exam). Plug these numbers into the expression above to find $v$ ,

$v=\sqrt{\frac{2kT_F}{m}}\approx \sqrt{\frac{2\times 1.4 E-23 \times 8E5}{9E-31}}=\sqrt{\frac{2.8 E-23 \times 8E5}{9E-31}}\approx\sqrt{0.3 E8 \times 8E5}=\sqrt{24/10 E13}\approx \sqrt{10^{12}}=10^6,$
the choice that comes closest to this order is choice (E).

Note that the hardest part of this problem is the approximation bit. No calculators allowed. Sadness.

Alternate Solutions

claire
2009-09-12 11:31:59

If you don't want to have to bother with the big numbers for the approximation, you can use your knowledge that for room temperature (~300K) you have kT= $\frac{1}{40}$ eV (this is a useful thing to know). And you know that an electron mass is .5MeV/c $^2$ .

The problem uses 80,000K, but do it for 90,000, since thats easier with the numbers you know.
Then kT~ $\frac{30}{4}$ , so $v_{f}$ = $\sqrt{\frac{30eV\times9\times10^{16}\frac{m^2}{s^2}}{10^6eV}}$ ~ $\sqrt{2.7\times 10^{12}}$ ~ 2 $\times$ 10 $^6$ m/s, option (E).

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Comments

nyuko
2009-10-30 10:10:08

I did this problem in seconds. I just remember the typical Fermi speed for electron in metals is of the order $\10^{6}m/s$
I think there are some typical values for physicists to remember. Fermi speed of electrons in metal is one.

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claire
2009-09-12 11:31:59

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ramparts
2009-08-02 17:36:13

I did this much more quickly by just looking at orders of magnitudes. Ignoring dimensionless constants (which will never change the answer by a factor of 2), temperature depends on the energy divided by the Boltzmann constant, and the energy is roughly kinetic energy, mv^2. So we have

$v \sim \sqrt{\frac{T}{k m_e}}$

Plug in the appropriate factors of 10, and you get $10^{6.5}$ . Excellent.

I have no idea if the physics is right, but the dimensions work out and the answer is right ;)

Albert
2009-11-03 05:05:17

Yes, your method is really bad. Not only your physics is abominable, your math is plain wrong too. You got the formula wrong and that's besides the fact that you did basically the same thing as Yosun, just didn't write the steps. And then got it all mixed up. I wonder how you ever reached the answer (or did you?). Sorry, but it's important that truth be spoken.

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none
2008-10-19 23:01:52

If you write things a bit differently you get $\sqrt{x.y \times 10^{12}} = \sqrt{x.y} \times 10^6$ . The only answer with the right order of magnitude is (E)

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student2008
2008-10-14 09:58:44

Actually, ETS means the average velocity modulus, $\langle v\rangle=\sqrt{\frac{8kT}{\pi m_e}}\approx 1.75\cdot 10^{6}\ \frac{m}{s}$ , which can be obtained using the Maxwell distribution. Though such rigorous formula isn't relevant for the real exam, of course.

student2008
2008-10-14 10:08:31

Even more precise would be $T=\langle \epsilon_{kin}\rangle =3/5 \epsilon_F$ and $\langle v\rangle \approx 1.36\cdot 10^6 \ \frac ms$ , but apparantely it's not for the exam :)

student2008
2008-10-15 00:14:53

Stupid me, all I wrote above is wrong. In fact, the concept of Fermi-gas (and Fermi-energy) of electrons requires $T \ll T_F$ , not $T = \frac35T_F$ . What is relevant here is the low-temperature Fermi distribution of electrons $f(p)\simeq\theta(p_F - p)$ (which is true up to melting temperatures: $T_{melt}\sim 10^3 K \ll 80 000 K$ ), not the Maxwell distribution. However, the problem asks about the mean speed of the $\bf conduction$ electrons. And these are the electrons near the Fermi surface, since only their energy may change (under the not-extremely-high voltage). So, Yosun's solution is right, with this reservation.

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prismofmoonlight
2007-10-31 11:22:40

I think T_F is 8E4, not 8E5 as given in your equation (though the change is not enough to affect the approximation too terribly much).

Thanks for the great site!

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