Consider the Lennard-Jones potential between atoms in a solid material. Find the dependence of the equilibrium postion x0, on the parameters in the potential. Then for small perturbations, δx around x0, determine the restoring force as a function of δx.

Consider the Lennard-Jones potential between atoms in a solid material. Find the dependence of the equilibrium postion x0, on the parameters in the potential. Then for small perturbations, δx around x0, determine the restoring force as a function of δx.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter2: Newtonian Mechanics-single Particle

Section: Chapter Questions

Problem 2.47P: Consider a particle moving in the region x > 0 under the influence of the potential where U0 = 1 J...

See similar textbooks

Related questions

Q: b) Find the commutator relation 1) [Z², L,] 2) [H,x]

A: (1) [ z2, Lz] = [zz,Lz] = z[z,Lz] + [z,Lz]z But, [z,Lz] = 0 then,…

Q: A body of mass m is suspended by a rod of length L that pivots without friction (as shown). The mass…

A: a. Draw the free-body diagram of the mass m.

Q: using moments, how do u compute these

Q: The force F= (yzi + zxj + xyz)/xyz acts on the particle P(x,y,z) which moves in space. (a) Using the…

A: Given data: The force on a particle is: F The potential function associated with force F is: V The…

Q: Asynchronous Activity Consider a particle of mass m moving freely in a conservative freely in a…

Q: m Which of the following correctly expresses the relationsh

Q: 5.10 On dimensional grounds, show that the energy levels of a purely linear potential, V(r) = For,…

A: This question can easily be solved by applying the basic concept of dimensional analysis .

Q: How do you do this question with integrals instead

Q: A hollow and open-ended cylinder of height 2h and radius b is centered on the origin and aligned…

A: Given Height of the cylinder = 2h Radius of the cylinder = b Surface current =K0 ϕ To…

Q: Given the potential energy function U(x) =−x^2 + 4x+ 2 calculate the force F(x) associated with…

Q: The force vector is conservative or non conservative

A: Given that:- F→=(x2yz)i^+(-xyz2)k^ For a conservative force, its curl is zero.

Q: A 0.4 Gram mass is required to stretch a soap film confined in arectangular frame by 2mm downward…

A: Mass of the object m=0.4 g so force applied by the weight of the object F=mg=0.4*9.8*10-3=3.92*10-3N…

Q: What does your result for the potential energy U(x=+L) become in the limit a→0?

A: We want to calculate lima→0qQ8πε0alnL+aL-a=qQ8πε0lima→0lnL+aL-aa

Q: An object undergoes simple harmonic motion. When it momentarily passes through the equilibrium…

Q: Consider a particle of spin s = 3/2. (a) Find the matrices representing the operators S,,S,,Ŝ,,S and…

Q: Prove that when the average is taken with respect to position over one cycle, the average potential…

A: To prove average potential energy is equal to kA26, In order to obtain the average potential energy…

Q: find a potential function for F or determine that F is not conservative. F = xj + yk

A: The given function, F = xj + yk The curl for F to can be determine if the F is conservative or…

Q: Use integration to compute the total force on q in the x direction

A: Given that:- Charge Q = 3.9X10-4 C Let y be the distance from O to dy

Q: Find the frequeeny of omall oscillatica for falmaing systen. K m

Q: There is a particle of mass m in the following potential. x a How many useful modes are there? How?

A: Energy for n nodes is given by : n2π2h22ma2

Q: What are Miller's coefficients for a surface that cuts from the x-coordinate two units and the…

A: The intercepts on x, y and z is given as x = 2 units y = -3 units z = ∞ ( since the surface is…

Q: An ideal sprng Of Go J when Stretched by 5.0cm from epuribrim position. What Š the fariel spring…

A: Energy stored in the spring, Deflection in the spring,

Q: In any problem, if the number of unknowns is greater than the number of equilibrium equations, the…

A: The 1st question is unclear. Please repost with necessary information. We are solving the 2nd…

Q: A root locus of a system is sketched below (x indicates pole and o indicates zero) X pole Img O zero…

A: In this question we are given that a root locus of a system sketched as in the figure. We have to…

Q: What is the proper dipole location, given a uniform E pointing to the right, that will have a net…

A: When an electric dipole is subjected to uniform magnetic field , net force acting on it is zero…

Q: A point particle moves in space under the influence of a force derivable from a generalized…

A: Classical Mechanics

Q: A PN flip-flop has four operations: clear to 0, no change, complement, and set to 1, when inputs P…

A: Given:A PN flip flop has four operations: clear to 0, no change,complement, and set to 1, when…

Q: If a particle of mass m is in a potential that is only a function of coordinates, calculate the…

Q: Problem 3.36. Consider an Einstein solid for which both N and q are much greater than 1. Think of…

A: In this question we have an Einstein solid for which both N and q are much greater than 1.Think of…

Q: Consider the following potential V(z) position is located at ) = 3+4z+3. The equilibrium equilibrium…

Q: When is Q(hat) Hermitian?

A: To explain: An operator can be hermitian on what condition.

Q: A pendulum hangs from a fixed point, but the pendulum itself is a spring with a constant k and…

A: A pendulum which is a spring hangs from fixed point is as shown in the following figure. Since the…

Q: The Hamiltonian of a particle having mass m in one dimension is described by p 1. H = +mox +2µx.…

A: The energy of nth state of harmonic oscillator potential is given by

Q: A certain comer of a room is selected as the origin of a rectangular coordinate system. If a fly is…

Q: Given that, in free space E = Em sin (wt-Bz) ây. Find D, and Ja. Then show that . (V × D)=0.

A: Let E denotes the electric field vector, E0 denotes the electric field magnitude, ω denotes the…

Q: Consider in three-dimensional space a particle that moves at a central potential . Then the equation…

Q: Explain this

Q: Aball of mass m moves along a thread that rotates around one end at an angular velocity ω. The…

Q: xplain categories of equilibrium? efine a dot product of two vectors A and B, here A = Axi + Avi +…

A: if net force force acting on body is zero then object is said to be in equilibrium.

Q: If A = (x²y)î+ (2y z)j+ (xyz2)k, find the curl of the curl of A at the point (-1,1, –2)

A: Write the given values. A→=x2yi^+2y3zj^+xyz2k^x,y,z=-1,1,-2

Q: Given the Hamiltonian: H=p²/2m+ V(x) What is the value of the commutator [H,x]?

Question

Consider the Lennard-Jones potential between atoms in a solid material.

Find the dependence of the equilibrium postion x₀, on the parameters in the potential. Then for small perturbations, δx around x₀, determine the restoring force as a function of δx.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.

Similar questions

Consider a particle moving in the region x > 0 under the influence of the potential where U0 = 1 J and α = 2 m. Plot the potential, find the equilibrium points, and determine whether they are maxima or minima.
Given the potential energy function U(x) =−x^2 + 4x+ 2 calculate the force F(x) associated with U(x), and find the maxima/minima of the potential. Does the potential have a point of stability?
The force F= (yzi + zxj + xyz)/xyz acts on the particle P(x,y,z) which moves in space. (a) Using the relation derived in Prob. 13.79, show that this force is a conservative force. (b) Determine the potential function associated with F.Reference to Problem 13.79:Prove that a force F(x,y,z) is conservative if, and only if, the following relations are satisfied:
Repeat the preceding problem, with E=2xi+3x2k.
Verify that v=8kBTm. Make the same scaling transformation as in the preceding problem.`
Let f(x)= 4xex - sin(5x). Find the third derivative of this function. Note ex is denoted as e^x below. Select one: (12+4x^3)e^x + 125sin(5x) 12e^x + 125cos(5x) not in the list (12+4x)e^x + 125cos(5x) (8+4x)e^x + 25sin(5x)
Calculate the energy, corrected to first order, of a harmonic oscillator with potential:
Explain about How to calculate the properties of a conservative force if you know the corresponding potential-energy function?
Particle of mass m slides without friction on a wedge of angle alpha and mass M that can move without friction on a smooth horizontal surface,as shown in the figure.Treating the constraint of the particle on the wedge by the method of Lagrange multipliers,find the equation of motion for the particle and wedge.Also obtain an expression for the forces of constraint.Calculate the work done in time t by the forces of constraint acting on the particle and on the wedge.what are the constants of motion for the
A point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space. Find the components of the force on the particle in spherical polar coordinates, on the basis of the equation for the components of the generalized force Qj: Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)
A body of mass m is suspended by a rod of length L that pivots without friction (as shown). The mass is slowly lifted along a circular arc to a height h. a. Assuming the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is F = mg sin u. b. Noting that an element of length along the path of the pendulum is ds = L du, evaluate an integral in u to show that the work done in lifting the mass to a height h is mgh.
Prove the Jacobi identity: A × (B × C) + B × (C × A) + C × (A × B) = 0. Hint:Expand each triple product as in equations (3.8) and (3.9).