In most metals, the atomic ions form a regular arrangement cal...

Question

In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through this lattice. FIGURE P40.34 is a one-dimensional model of a crystal lattice. The ions have mass m, charge e, and an equilibrium separation b. Suppose this crystal consists of aluminum ions with an equilibrium spacing of 0.30 nm. What are the energies of the four lowest vibrational states of these ions?

Accepted Answer

Hey everyone. So this problem is dealing with quantum mechanics. Let's see what it's asking us consider a one dimensional crystal lattice of copper ions with an equilibrium spacing of 0.25 nanometers. The copper ions have a mass of ma charge of E and are arranged in a regular pattern. We're asked to determine the energies of these ions three lowest vibrational states within the lattice. We are told to consider that the net electric force on the middle charge is given approximately by the equation F is equal to negative E squared multiplied by X divided by D cubed multiplied by pi multiplied by E knot. Where D is the distance between the adjacent charges. Our multiple choice answers in units of electron volt are a 0.080 0.0230 0.078 B 0.080 0.023 and 0.039 C 0.004 0.012 and 0.027 or D 0.0620 0.012 and 0.039. OK. So to solve this problem. We can recognize that ions in a crystal lattice act similarly to a harmonic oscillator. And so energy in a quantum harmonic oscillator is given by the equation E is equal to N plus one half multiplied by H bar multiplied by omega where N is our principal quantum number H bar is the reduced plans constant. And omega is our angular velocity we're asked to solve for the three lowest vibrational states. So that will be when N equals zero N equals one and N equals two. In turn, we can recall that our angular velocity is equal to the square root of K divided by M where K is the spring constant and M is the mass. And then Hooks law tells us that the force is equal to negative K multiplied by X. The force we are given in the problem as F is equal to negative E squared X divided by D cubed multiplied by pi multiplied by epsilon knot. And so if we combine these two equations, we get a spring constant of E squared divided by D cubed multiplied by pi multiplied by epsilon knot where E is the charge of an electron D is the distance between the adjacent charges. And then epsilon knot is our electric constant. And so we can use that to solve for our angular velocity and then finally solve for our energy. So angular velocity will be E squared to the square root of E squared divided by D cubed multiplied by M multiplied by pi multiplied by epsilon knot. And so when we plug in those values, we can recall that the charge of an electron is 1.6 times 10 to the negative 19 poems. That quantity squared divided by that distance between the charges which was given as 0.25 nanometers keeping everything in standard units, we will rewrite that as 0.25 times 10 to the negative 9 m. That quantity cubed multiplied by the mass of the copper ion, which we can recall is 64 U or 64 multiplied by 1.67 times 10 to the negative 27 kg multiplied by pi multiplied by that electric constant, which we can recall is 8.85 times 10 to the negative 12. And that's in units of Newton meters squared per Coolum squared. And so we stretch that square root all the way out plug that in to our calculator, we get an angular velocity of 2.35 times 10 to the 13 radians per second. And so now we have to solve for our energies for when N is equal to zero, for that first level, when N is equal to one and when N is equal to two. So we have zero plus one half is simply one half multiplied by our reduced planks constant, which we can recall is 1.05 times 10 to the negative 34 Jews multiplied by that angular velocity we just saw for 2.35 times 10 to the 13 gradients per second. And then we can recognize that that would give us an answer in joules and all of our answers, multiple choice answers are in electron volts. So we can recall that conversion factor is one electron volt per 1.6 times 10 to the negative 19 joules. And that gives us an energy of 0.008 electrons. And so for our energy level, for energy, for when N equals two, sorry E two, for one N equals one will be very similar. The only term that changes is that first term where it's N plus one half. And so for E two, we will simply have three hats. So let me go back to blue. And then the rest of that equation is the same. And that comes out to 0.023 electron volts. And then very similarly for that third vibrational state, when N equals two, we have five halves multiplied by H bar, multiplied by angular velocity. And then that conversion factor from Juls to electron volts. And that gives us an energy of 0.039 electron volts. And so those are our answers for this problem. And when we look at our multiple choice answers, we can see that, that aligns with answer choice B. So B is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.

Key Concepts

Crystal Lattice

Vibrational States

Quantum Harmonic Oscillator

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