Question
In: Physics
Consider a one-dimensional lattice with a basis of two non-equivalent atoms of masses M_1and M_2. 1....
Consider a one-dimensional lattice with a basis of two non-equivalent atoms of masses M_1and M_2.
1. Find the dispersion relations (ω versus k).
2. Sketch the normal modes in the first Brillouin zone.
3. Show that the ratio of the displacements of the two atoms u/v for the k = 0 optical mode is given by:u/v=-M_2/M_1
Solutions
genius_generous answered 2 years agoRelated Solutions
6. A one-dimensional diatomic crystal contains 2 different atoms with masses ?1 and ?2, with ?1<...
6. A one-dimensional diatomic crystal contains 2 different atoms
with masses ?1 and ?2, with ?1< ?2, in each of its unit cell.
Assume that there is interaction between every nearest neighbor
atoms that can represented by a spring with spring constant ?.
Derive the phonon frequency dispersion relation ?(?) of the
crystal. Indicate which one is the acoustic branch and which one is
the optical branch.
Consider a two-dimensional triangular lattice described by the two primitive vectors (in an orthogonal coordinate system...
Consider a two-dimensional triangular lattice described by the
two primitive vectors (in an orthogonal coordinate system
Find the two primitive lattice vectors describing the reciprocal
lattice. Find the area of the 1st
Brillouin zone and its relation with
the area of the direct lattice unit cell.
Consider a two-dimensional hexagonal lattice: (a) Draw the free electron Fermi surface in the reduced zone...
Consider a two-dimensional hexagonal lattice:
(a) Draw the free electron Fermi surface in the reduced zone
scheme when the lattice points are occupied by atoms with: i. One
valence electron/atom. ii. Two valence electrons/atom.
Consider a two-dimensional hexagonal lattice: (a) Find Fermi wave vector of the free electron circular Fermi...
Consider a two-dimensional hexagonal lattice:
(a) Find Fermi wave vector of the free electron circular Fermi
surface in reciprocal space. (b) Draw the free electron Fermi
surface in the reduced zone scheme when the lattice points are
occupied by atoms with: i. One valence electron/atom. ii. Two
valence electrons/atom
Consider a two-dimensional hexagonal lattice: (a) Find Fermi wave vector of the free electron circular Fermi...
Consider a two-dimensional hexagonal lattice:
(a) Find Fermi wave vector of the free electron circular Fermi
surface in reciprocal space.
(b) Draw the free electron Fermi surface in the reduced zone
scheme when the lattice points are occupied by atoms with:
i. One valence electron/atom.
ii. Two valence electrons/atom.
Masses coupled through springs is a model for the behavior of atoms in a crystal. Consider...
Masses coupled through springs is a model for the behavior of
atoms in a crystal. Consider a one-dimensional crystal, what if we
have a crystal made of two kinds of atoms with different masses?
Consider a one-dimensional lattice made up of alternating large and
small masses, M and m, respectively. Assume periodic boundary
conditions. (a) Calculate the normal modes of this lattice (you
should find two normal modes). (b) Describe the relative motion of
the two types of masses for...
Consider a one-dimensional chain of identical atoms. The springs between them alternate in strength between values...
Consider a one-dimensional chain of identical atoms. The springs
between them alternate in strength between values K1 and K2.
a) Find the vibrational frequencies as a function of wave number
q. Study the low q limit and find the sound velocity.
b) Discuss the physical meaning of the two branches. Sketch the
way the atoms move in both cases!
c) Discuss the dispersion and the normal modes for K1 ≫ K2.
Consider a system of three non-interacting particles confined by a one-dimensional harmonic oscillator potential and in...
Consider a system of three non-interacting particles confined by a
one-dimensional harmonic oscillator potential and in thermal
equilibrium with a total energy of 7/2 ħw.
(a) what are the possible occupation numbers for this system
if the particles are bosons.
(b) what is the most probable energy for a boson picked at
random from this system.
Consider the Russell- Saunders coupling scheme for two non-equivalent electrons in an excited 2p1 3d1 configuration....
Consider the Russell- Saunders coupling scheme for two
non-equivalent electrons in an excited 2p1 3d1 configuration. For
all parts (a) to (c) give your reasoning:
(a) Show that strong coupling between orbital angular momenta (ℓ
ℓ coupling) gives rise to P, D and F terms with L magnitudes of
2½ħ, 6½ħ and 12½ħ.
(b) If coupling between spin momenta (s-s coupling) is
considered in addition to ℓ ℓ coupling, what six terms arise?
(c) If coupling between the orbit and...
1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?)....
1.
Let ? be a finite dimensional vector space with basis {?1,...,??}
and ? ∈ L(?).
Show the following are equivalent:
(a) The matrix for ? is upper triangular. (b) ?(??) ∈
Span(?1,...,??) for all ?.
(c) Span(?1,...,??) is ?-invariant for all ?.
please also explain for (a)->(b) why are all the c
coefficients 0 for all i>k? and why T(vk) in the span of
(v1,.....,vk)? i need help understanding this.
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