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 each mode.

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genius_generous answered 1 year ago

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 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 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

Cohesive energy of a crystal is the energetic cost that holds all atoms together in a...

Cohesive energy of a crystal is the energetic cost that holds all atoms together in a crystal. It can be determined both experimentally and theoretically. Explain, based on your knowledge on the quantum mechanical nature of non–covalent interactions, why the following statement is incorrect: Because silicon crystal are bonded together by strong covalent bonds between silicon atoms, non– covalent interactions, in particular, dispersion interactions, play NO role in determining the cohesive energy in silicon crystals.

A certain crystal is cut so that the rows of atoms on its surface are separated...

A certain crystal is cut so that the rows of atoms on its surface are separated by a distance of 0.352 nm. A beam of electrons is accelerated through a potential difference of 175 V and is incident normally on the surface. If all possible diffraction orders could be observed, at what angles (relative to the incident beam) would the diffracted beams be found? (The surface acts as a diffraction grating where maxima occur when d sin? = n?, n...

A mass spectrometer is designed to calculate masses of atoms. A source produces positive ions of...

A mass spectrometer is designed to calculate masses of atoms. A source produces positive ions of mass m and charge q. The ions move with a velocity v to the right, pass through a narrow slit S1 and enter a region between two charged plates. In this region they experience both an electric and a magnetic field. The electric field E is caused by the 2 charged plates; the upper plate is (-) charged and the lower plate is (+)...

Draw the distribution of d electrons for the central atoms in the following molecules using crystal...

Draw the distribution of d electrons for the central atoms in the following molecules using crystal field theory. K3[Cr(C2O4)3] 3H2O (octahedral) K2[Cu(C2O4)2] 2H2O (tertrahedral)

I need part 1, 2, and 3 answered please just google PhET Masses and Springs Simulations...

I need part 1, 2, and 3 answered please just google PhET Masses and Springs Simulations Lab and click the first link. The preview of the lab should be in a box click the arrow/play button in the box and the lab will open 2. Vectors At the base of your screen select ‘Vectors’. This helps in visualizing all vectors at play in oscillation. Select ‘Displacement’, ‘Mass Equilibrium’, ‘Velocity’, ‘Acceleration’, ‘Gravity’ and ‘Spring’. For first run leave Gravity as Earth....

A crystal composed of atoms of one unknown species has experimentally determined values of molar heat...

A crystal composed of atoms of one unknown species has experimentally determined values of molar heat capacity at constant pressure and two different temperatures: Cp (200 K) = 21.573 J.K-1.mol-1 Cp (100 K) = 13.360 J K-1.mol-1 Determine the approximate molar heat capacity at a constant volume of this crystal as a function of temperature over the interval from 50-300 K and plot it with the two experimental points.

In the Einstein model, atoms are treated as independent oscillators. The Debye model, on the other...

In the Einstein model, atoms are treated as independent oscillators. The Debye model, on the other hand, treats atoms as coupled oscillators vibrating collectively, but the collective modes are regarded as independent. Explain the meaning of this independence, and contrast it with that in the Einstein model

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