Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Advanced Physics Homework Help
Triangular lattice and criticality
Reply to thread
Message
[QUOTE="CAF123, post: 5442844, member: 419343"] [h2]Homework Statement [/h2] Consider a two dimensional triangular lattice, each point of which can either contain a particle of a gas or be empty. The Hamiltonian characterising the system is defined in terms of the particle occupation numbers ##\left\{n_i \right\}_{i=1,...,N}## which can either be 0 or 1. N is the total number of points on the lattice. The Hamiltonian is $$H = \mu \sum_i n_i - J \sum_{\langle i j \rangle} n_i n_j$$ with ##\mu >0## the chemical potential and ##J \geq 0## an effective interparticle interaction. a) In the limit J=0, find the partition function, the free energy and the average occupation number. b) Using a mean field variational approach, the average occupation number is $$\langle n \rangle = \frac{1}{1+ \exp(3J\beta - 6 \beta J \langle n \rangle)}$$ Expand this equation around ##\langle n \rangle = 1/2## and find the critical value of J for which the mean field predicts a phase transition between a particle poor phase and a particle rich one. [h2]Homework Equations[/h2] $$Z = \sum_{\left\{n\right\}} e^{- \beta H}$$ [h2]The Attempt at a Solution[/h2] a) In the first part I was getting ##Z = (2\cosh \beta \mu)^N##, ## F = -N/\beta \,\, \ln(2 \cosh \beta \mu)## and ##\langle n \rangle = -N \tanh \beta \mu## b) Write the equation as $$ \langle n \rangle = (1 + \exp 3J \beta (1-2 \langle n \rangle))^{-1}$$ Write $$\exp 3J \beta (1-2 \langle n \rangle) = \exp(3J \beta) \exp(-6J \beta \langle n \rangle)$$ and expand around ##\langle n \rangle = 1/2## so set ##\langle n \rangle = 1/2 + \eta## and get $$ \exp(3J \beta) \exp(-6J \beta( 1/2 + \eta)) \approx (1-6J \beta \eta)$$ Inputting this into the equation for ##\langle n \rangle##, and replacing ##\eta = \langle n \rangle - 1/2## I get a quadratic equation for ##\langle n \rangle##? Should this happen and if so, how to interpret the two solutions? I think J is that value at which either side of it ##\langle n \rangle## goes from being less than 1/2 to greater than it. (I haven't found such an equation yet either) Thanks! [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Advanced Physics Homework Help
Triangular lattice and criticality
Back
Top