1. 1. A tricritical point is a point in a phase diagram where a line of second-order critical points meets a line of first-order critical points. We can model this by consider a Landau-Ginsburg free energy density of the form fLG = am? + bm' + cm® where we must have c > 0 for stability but a and b can be negative as well as positive. (a) Show that a second-order transition occurs at a=0 when b>0. (b) Taking a>0 and b<0, sketch fLG at a point on the first-order line. Find the first-order line by solving fic(m) = 0 and fLG(m) = 0 simultaneously. (c) Where is the tricritical point in the (a, b) plane? Sketch the phase diagram in %3D %3D that plane, showing the 1st- and 2nd-order lines as well as the tricritcal point.

1. 1. A tricritical point is a point in a phase diagram where a line of second-order critical points meets a line of first-order critical points. We can model this by consider a Landau-Ginsburg free energy density of the form fLG = am? + bm' + cm® where we must have c > 0 for stability but a and b can be negative as well as positive. (a) Show that a second-order transition occurs at a=0 when b>0. (b) Taking a>0 and b<0, sketch fLG at a point on the first-order line. Find the first-order line by solving fic(m) = 0 and fLG(m) = 0 simultaneously. (c) Where is the tricritical point in the (a, b) plane? Sketch the phase diagram in %3D %3D that plane, showing the 1st- and 2nd-order lines as well as the tricritcal point.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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