In a physics lab, you measure the vibrational- rotational spectrum of potassium bromide (KBr). The estimated separation between absorption peaks is
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- For some materials, the band structure of the conduction band around k=0 can be represented by: ħ²k² 2ma² =(1-27 ²4²) Give an expression for the effective mass of the electron at this region. E (k) =arrow_forwardIn solid KCI the smallest distance between the centers of a. potassium ion and a chloride ion is 314 pm. Calculate the length of the edge of the unit cell and the density of KCI, assuming it has the same structure as sodium chloride.arrow_forwardIn vibrational spectroscopy, the fundamental band refers to a transition from the n = 0 state to n = 1, and the first overtone would be from the n = 0 state to n = 2. For carbon monoxide (CO), the fundamental occurs at 2143.4 cm-¹ and the first overtone at 4259.6 cm-¹. (a) For an anharmonic oscillator, the energy levels can be written (in wavenumbers) as 2 En ) ²³ = ve (n + ¹) - xeve (n +. + Find the values of ve and xeve from the fundamental and overtone transitions in CO. Report your results in units of cm-¹. (b) Use the values found for ve and xeve in part (a) to estimate the number of bound energy levels in CO. (Hint: For an anharmonic oscillator this number is determined by the point at which the spacing between consecutive levels goes to zero.) (c) If CO could be modeled as a Morse oscillator, and the values of ve and xeve provide energy levels consistent with the Morse potential what would be the value of the parameter D in units of cm-¹? How does this compare with the observed…arrow_forward
- The angular frequency of vibrations in a one-dimensional monatomic crystal are given by where w (k) = 4K m sin k = ka 2 2π λ is the wavenumber of the vibration pattern in terms of its wavelength >, the spring constant between neighbouring atoms, and m the mass of each atom. (i) Show that the frequency does not change if one shifts 2π k→ k +. a (ii) Briefly explain the physical reason for this periodicity.arrow_forwardIn this question, we will consider the vibration of the diatomic molecule, H2, found in an eigenstate whose wave function is 1/4 1 p(x) V48 \T where a = ku/ħ. k is the force constant and u is the reduced mass. The vibrational frequency of H2 is 4401 cm-'. а) How many nodes are present in p(x)? b) What is the quantum number that characterizes p(x)? c) Calculate the vibrational energy of the H2 molecule when it is described by the eigenfunction p(x).arrow_forwardConsider a Face Centered Cubic (FCC) lattice structured Nickel crystal. We are looking to find the surface energy of the new surface that is formed after it is sliced at the (100) plane. a- Find the value of R as function of the lattice constant a. 4R Oa = 2R Oa = 4R Oa = = 2/2R V2 Find the area A11 of (111) surface as function of R. 04R? O16R? O8R? OR? How many atoms lie on the plane (111) within the unit cell? N111 = atoms within the unit cell Find the number of atoms per unit surface area. 2 2 R2 8R2 16R? 4R? Which of the following represents the expression of the surface energt? ON BEPAarrow_forward
- Q#04. (a) Calculate the number of atoms per unit area in (100), (110) and (111) planes of in bcc crystal with the lattice parameter of 2.5 angstrom.arrow_forwardAssume the distance between the protons in the H2 molecule is 0.750 x 10-10 m. (a) Find the energy of the first excited rotational state, with J = 1. (b) Find the wavelength of radiation emitted in the transition from J = 1 to J = 0.arrow_forwardThe elements A and B crystallize in bcc and fcc structures, respectively. Suppose that (i) the crystals of A and B have the same density, and (ii) the nearest-neighbor distances in these crystals are equal. Calculate the ratio MA/MB, where MA and MB denote the masses of the A and B atoms, respectively.arrow_forward
- Calculate the radius of a nickel atom in cm, given that Ni has an FCC crystal structure, a density of 7.982 g/cm³, and an atomic weight of 58.69 g/mol.arrow_forwardOne model for the potential energy of a two-atom molecule, where the atoms are separated by a distance r, is U(r) = Uo[(¹) ¹2 – ( )²] where ro = 0.8 nm and U₁ = 6.1 eV. Note: 1 eV = 1.6 × 10-19 J. Some helpful units: [Force] = eV/nm [Energy] = eV [distance] = nm Equilibrium Distance What is the distance between the atoms when the molecule is in stable equilibrium? Click here for a hint T'eq Hint: Hint: Hint: Hint: Hint: Hint: Force If the distance between the atoms increases from equilibrium by r₁ = 0.35 nm, then what is the force from one atom on the other associated with this potential energy? (Enter your answer as postive if they repel each other, and negative if they attract.) Fr(req+r₁) Hint: Hint: 0.89105934nm Kinetic Energy Hint: The atoms are oscillating back and forth. The maximum separation of the atoms is r₂ = 2 nm. What is the kinetic energy of the atoms when they are separated by the equilibrium distance? Click here for a hint K(req) Hint: Hint: = -1.288eV/nm 3.99eVarrow_forwardA hypothetical NH molecule makes a rotational-level transition from l = 3 to l= 1 and gives off a photon of wavelength 1.780 nm in doing so. What is the separation between the two atoms in this molecule if we model them as point masses? (The mass of hydrogen is 1.67 * 10-27 kg, and the mass of nitrogen is 2.33 * 10-26 kg).arrow_forward
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