Hello,
I have made question 1 of exam 2019. But I don’t understand why the answer model has two dispersions relations found in 1b. They used the 2 dispersion also in 1d, so I also don’t understand this question.
Could someone maybe explain this? In the pictures below are my answers.
Thanks in advance,
Nadia
The 3th picture is the answer model
what is the question exactly? You found two equation as well right? (with the plus/minus)
I don’t have the first line of the answer model at 1b (with the abs), only the 3th line (square root). Do I also need this solution?
Because by 1d is said this is the optical band and the solution with the square root is the acoustic bound.
the final expression results from a Taylor expansion of the square root for \kappa_1\gg\kappa_2. Note however that the factor 4 is incorrect (it should be a one).
The Taylor expansion gives \sqrt{\kappa_1^2 + \kappa_2^2 + 2\kappa_1\kappa_2\cos(2ka)}\approx \kappa_1 + \kappa_2\cos(2ka).
the absolute value arises because you take the square root of a squared quantity, which can only be positive
Nadia also points out that in the dispersion relation derived in question 1b (and also used for the remainder of the solutions) there is an absolute term written as |k1 + k2exp(2ika)|, but this must be a mistake. This is because in question 1a where you derive the 2x2 matrix with the spring constants, one exponent is exp(-2ika) while the other is exp(2ika). This also changes the entire epxression of the final dispersion relation where - under the square root - you have k1^2 + k2^2 + 2k1k2cos(2ka)
