Midterm 1: Difference between revisions
(New page: PHZ 3400 – Midterm Exam (with solution) – March 20, 2009 ==Problem 1== '''Give some examples of spontaneous symmetry breaking. Can molecular magnets, consisting of 12 atoms, display ...) |
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PHZ 3400 – Midterm Exam | PHZ 3400 – Midterm Exam – February 24, 2011 | ||
==Problem 1== | ==Problem 1== | ||
'''Give | '''Describe the difference between a first order and a second order phase transition. Give examples of each type.''' (10 points) | ||
==Problem 2== | ==Problem 2== | ||
''' | '''Give some examples of spontaneous symmetry breaking. How does it depend on the system size? Explain how it can happen, in apparent violation of the ergodicity hypothesis of Boltzmann. According to Boltzmann, what determines the probability of some microscopic configuration (10 points)''' | ||
==Problem 3== | ==Problem 3== | ||
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==Problem 4== | ==Problem 4== | ||
'''The mean-field equation of state for a ferromagnet is given by the expression | '''The mean-field equation of state for a ferromagnet is given by the expression''' | ||
<math>m = \tanh \left( \beta Jz m + \beta h \right)</math>. | <math>m = \tanh \left( \beta Jz m + \beta h \right)</math>. | ||
Consider the system at high temperature, where nonzero magnetization is found only in presence of an external magnetic field. Assume that both the field and the magnetization are small, and determine (linearizing the RHS in m and h) the magnetization as a function of temperature, in presence of a weak field. Calculate the magnetic susceptibility and sketch its inverse as a function of T. Where does it diverge?''' ( | '''Consider the system at high temperature, where nonzero magnetization is found only in presence of an external magnetic field. Assume that both the field and the magnetization are small, and determine (linearizing the RHS in m and h) the magnetization as a function of temperature, in presence of a weak field. Calculate the magnetic susceptibility and sketch its inverse as a function of T. Where does it diverge?''' (30 points) | ||
==Problem 5== | ==Problem 5== | ||
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==Problem 6== | ==Problem 6== | ||
Consider a molecule consisting of two | '''Consider a molecule consisting of two identical atoms. Assume that the ground state energy of each atom is <math>E_o</math>, and that the corresponding hopping element between the two atoms is <math>t(R)</math>. Calculate the energy difference between the bonding and the anti-bonding molecular state for this molecule. How do you expect it to depend on the inter-atomic distance R? What is the energy needed to break the molecule, i.e. the energy of the covalent bond that binds the atoms in the molecule?''' (30 points) | ||
Latest revision as of 10:25, 24 February 2011
PHZ 3400 – Midterm Exam – February 24, 2011
Problem 1
Describe the difference between a first order and a second order phase transition. Give examples of each type. (10 points)
Problem 2
Give some examples of spontaneous symmetry breaking. How does it depend on the system size? Explain how it can happen, in apparent violation of the ergodicity hypothesis of Boltzmann. According to Boltzmann, what determines the probability of some microscopic configuration (10 points)
Problem 3
What determines the Curie temperature of a ferromagnet? How does it depend on the coordination number of the corresponding crystal lattice? Sketch the magnetization as a function of temperature in zero and in finite magnetic field. How does the field affect the transition? (10 points)
Problem 4
The mean-field equation of state for a ferromagnet is given by the expression
.
Consider the system at high temperature, where nonzero magnetization is found only in presence of an external magnetic field. Assume that both the field and the magnetization are small, and determine (linearizing the RHS in m and h) the magnetization as a function of temperature, in presence of a weak field. Calculate the magnetic susceptibility and sketch its inverse as a function of T. Where does it diverge? (30 points)
Problem 5
What is the physical principle behind the fact that atoms act as hard spheres, i.e. they have a “core” impenetrable to other atoms? (10 points)
Problem 6
Consider a molecule consisting of two identical atoms. Assume that the ground state energy of each atom is , and that the corresponding hopping element between the two atoms is . Calculate the energy difference between the bonding and the anti-bonding molecular state for this molecule. How do you expect it to depend on the inter-atomic distance R? What is the energy needed to break the molecule, i.e. the energy of the covalent bond that binds the atoms in the molecule? (30 points)
