The mass density of copper is 8920 kg m^-3,and a cooper atom has mass 63.5 U_ A) Calculate the number density of copper: State the equations used and/or explain the...

(a) Show that the density of states at the Fermi energy is given by $$\begin{aligned} N\left(E_{\mathrm{F}}\right) &=\frac{(4)\left(3^{1 / 3}\right)\left(\pi^{2 / 3}\right) m n^{1 / 3}}{h^{2}} \\ &=\left(4.11 \times 10^{18} \mathrm{m}^{-2} \mathrm{eV}^{-1}\right) n^{1 / 3} \end{aligned}$$ in which $n$ is the number density of conduction electrons. (b) Calculate $N\left(E_{\mathrm{F}}\right)$ for copper, which is a monovalent metal with molar mass 63.54 $\mathrm{g} / \mathrm{mol}$ and density 8.96 $\mathrm{g} / \mathrm{cm}^{3} .(\mathrm{c})$ Verify your calculation with the curve of Fig. $41-6,$ recalling that $E_{\mathrm{F}}=7.0 \mathrm{eV}$ for copper.

In this problem on the topic of solid state physics, we're told that in the metal, the density of states can be written as a E. To the power half where is a constant Andy measured from the bottom of the conduction band. We want to first show that the total number of states is 2/3 a E. F. To the power 3/2. We then want to find the fraction of conduction electrons that is within Katie of the Fermi energy and then evaluate this fraction for copper at 300 Kelvin. So firstly the number of states N he is equal to the integral From 0 to the Fermi Energy of G D E, Which is the integral from zero to E. F. Of a two behalf D E, which is a times two thirds E to the power 3/2, evaluated from zero to the funny energy, which gives to a over three times the fermi energy E F. To the power 3/2 as required. Yeah. Now in part B, we want to find the fraction of the conduction electrons within Katie, of the farming energy. So the number of electrons in this range and prime is the integral from E f minus Katie to the fermi energy ef of E to the half D E. And so this is equal to To a to a over three into EF. to the power 3/2 minus E F minus K T. To the power 3/2. And so we can write this as To a over three into EF. to the power 3/2 minus E F. To the power 3/2 into one minus Katie of E F. All to the Power 3/2. Because for most metals, Katie is much less than the fermi energy E F, we have one minus Katie over E F. Or to the 3/2 can be approximated as one minus 3/2. K T E F. To the power half. And so therefore in prime is equal to to a over three into E. F. To the power 3/2 minus E F. To the power three over to plus 3/2 Katie E. S. To the power half. And so in prime is then a okay tee times E. F. To the power half. And so the fraction of Katie of the fraction within Katie of Ef we'll call it F. Is then in prime over in which is a K T times the square root of E F, divided by To a over three times E F. To the power 3/2. And so we get this fraction to be three, Kati divided by two E F. Yeah, Next for Part C. We want to find this fraction for copper at 300 Calvin. So for copper we have the funny energy ef at the 100 Calvin to be 7.0 for electron volts. And so this fraction F is three into zero point zero 25 85 electron volts, divided by two into 7.4 electron volts, Which gives us the fraction 0.00 55.

The energy level occupied by an electron energy is given by easy call to this is for about a feasible to e f for me energy plus Katie Times Ellen off par probability minus one This is seven Electron world Eyes, the farming energy plus a boatman constant, which is 8.62 times 10 to the power minus five electoral World Times the temperature reaches 1000 Kelvin, this time's Ellen off, probably we have is a 0.9 a minus one here This will give us energy off 6.821 Electron World photo part B The states, then will be the density of states will be See Time's energy one or two. She's a constant which we know is 6.81 times 10 to the power 27 times the energy 56.81 electron world one or two which we just obtained. This won't give us 1.77 times 10 to the power 28 Ah, part meter cube elect for electronic world then for apart see off the problem The identity off the occupied state with energy is given by n Lord E is equal to probability off off with energy E times The density off the state This is 0.900 times. Ah, the identity of state. We obtained 1.77 times gentle power 28. This will give us a 1.59 times dental power 28 firm Major Cube up our electron world.

For this problem a density of folk abide state and nod. Um, he is given by density off estates times the probability of United States with energy e This is given by C times E one or one over two are times so one or two years in the exponents so e energy minus e f the f the Fermi Energy divided by Katie plus plus one minus one in power where C is ah eight Swerved off to buy mass Ah, three over two divided by itch three though substituting your values for ah electron mass in the point constant we obtained the constant c two B 1.62 times 10 to the power 56 kg, uh three or two for Joel tree or 2nd 3 in this skin beard in a electron world six point 81 times 10 to the power 27 For me to cube Electron world three minus three or two For about a it energy E is equal to fall Electron, Walt. Then we can right the occupy density off the states, uh, will be substituting your values. So this will be 6.821 times 10 to the power to any seven per meter. So that is minus three times electron world minus three over two times a four electron world one or two this divided by, um, exponential energy four electron world minus seven. Electron world divided by a bolt man Constant. The times for the temperature 2000 Calvin plus one Here. Um so this will view us. This is will give us They're not the occupied state number off. Provide state. They're not to be 1.36 times 10 to the power 28 farm Nature cube elect for electron worlds pulled apart to be for energy for energy of e is equal to are 6.75 electron wall. We use the same procedure that we did for about A and substitute the value for the tea to be a 1000 Calvin. We attend the occupy density off the states and not to be 1.68 times 10 to the power 28 per meter cube electro but electron worlds Similarly, for a party for energy equal to seven electron world, the value for and lord is mine for 01 times 10 to the power 27 per meter queue for electron world body At energy, he's equal to 7.25 across the world. The value off a nod we'll turn here is 9.56 times 10 to the power 26. Well part you need to cube electron world butter lip from you Then for a part e at energy nine Electron world So energy we have here is a nine electron worlds. We obtained the value and Lord and no number off occupied state to be 1.71 times 10 to the power 18 for the energy nine electron world So this is the power 18 for me to pube for electron won't

Okay. In this question, we have to show that The term the rest to the 4 -2 X. Is equal to. All right, 0.1 foreign electron. And he raised to the power minus two X. is equal to. Then raised to the power -38 Fire protection. I thought putting the value of by putting value of eggs the minus E. And I am into the expression expiration date is He -2 X. Big Gate. So first we will find for the electron so far electron. This expression can be written as he managed to X. He is equal to he minus two into en route to em into e minus of the into X divide by H. So this become into -2 uh into an route and wrote to Multiply by nine x 11. 9111 to drive by 10 Raised to the Power -31. Multiplied by 1.6. multiplied by 10. Raised to the power -19 into To multiply by 10. Raised to the power of Afghanistan on divide by Divide by 6.6 three multiplied by 10 raised to the power minus 34 Divide by two by so this become a minus two. Find 05 So this is equal to 0.1. And for proton The award Trump can be written as -2. My areas to the power minus two X. Is equal to He raised the board -2 into one route to um be into I'll be minus yeah are divided by age into X. So this can be written as He -2 into into or to multiply by 1.67 multiplied by 10 raised to the power 4 -27. I want to play by 1.6 multiply by 10 days to the goal -19. Be worried by 6.63 multiply by 10. The rest to go on -34, divide by two by into To multiply by 10, raised to the power-. But then so this is able to minus 87 x nine and that this becomes turn the race to the bottom of my nest did.