Ball-and-spring motions

The polyad model for acetylene is an example of a hybrid scheme, combining ball-and-spring motion in a two-dimensional configuration space [the two Franck-Condon active modes, the C-C stretch (Q2) and the tram-bend (Q4)] with abstract motion in a state space defined by the three approximate constants of motion (the polyad quantum numbers). This state space is four dimensional the three polyad quantum numbers reduce the accessible dimensionality of state space from the seven internal vibrational degrees of freedom of a linear four-atom molecule to 7 - 3 = 4. 

Even at 0 K, molecules do not stand still. Quantum mechanically, this unexpected behavior can be explained by the existence of a so-called zero-point energy. Therefore, simplifying a molecule by thinking of it as a collection of balls and springs which mediate the forces acting between the atoms is not totally unrealistic, because one can easily imagine how such a mechanical model wobbles aroimd, once activated by an initial force. Consequently, the movement of each atom influences the motion of every other atom within the molecule, resulting in a com-... 

To model quantitatively an IINS spectrum, it is only necessary to obtain the amplitudes of motion of the atoms in the vibrational modes. These can be calculated by a variety of methods, such as the balls-and-springs approach of the Wilson GF matrix method, ab-initio calculations, and molecular dynamics this point expresses what is undoubtedly the greatest strength of IINS spectroscopy. Examples are presented below. 

The ideas that are outlined in a qualitative v e/ above can also be cast into a useful mathematical form for computer calculation. The basic idea is to write down a (fairly simple and approximate)function that gives the energy of the system as a function of the positions (or coordinates) of its atoms. Because the derivative (or gradient) of this function yields the forces for Newf on s equations, such a function is often called a "force field" and because molecules are viewed as being made up of balls and springs (so that quantum effects are ignored), the term "molecular mechanics" is used to represent a concrete, mechanical picture cf molecular motions and energies. 

Construct a kinetic sculpture depicting an organic molecule such as methane or a longer, branched hydrocarbon. Use Styrofoam balls for atoms and springs or toothpicks for bonds between atoms. Color the Styrofoam balls to represent atom types. Find a way to show atomic vibrations and molecular rotations and translations. Your kinetic sculpture should be in constant motion. 

Classically, the nuclei vibrate in die potential V(R), much like two steel balls coimected by a spring which is stretched or compressed and then allowed to vibrate freely. This vibration along the nuclear coordinated is our first example of internal molecular motion. Most of the rest of this section is concerned with different aspects of molecular vibrations in increasingly complicated sittiations. 

Often Newton s laws predict periodic motion in simple systems. For example, a ball supported from the ceiling by a spring, or a pendulum which is not too far from vertical, will oscillate at a constant and predictable rate. If we connect two balls of comparable mass by a spring and stretch the spring, the entire system will oscillate back and forth, or vibrate. 

A 100 g ball is suspended from a spring. At time t = 0 the spring is neither compressed nor extended, and the velocity is 1 m s 1 (going up). The ball goes through one complete oscillation (up and down) in one second. Calculate how far the ball is extended when the motion reaches its upper limit. 

Cooper pairs are loosely bound electron pairs, which move at the same speed but in opposite directions. An analogy can be made to two balls on the ends of a spring, where the balls move together and apart at the same speed but always in opposite directions. Superconductivity arises from highly coordinated motion of all Cooper pairs within a solid, where these electron pairs are all propagating in the same direction—that of the current flow. 

When the ball is set in motion by a force, F, it oscillates under the Hooke s Law extension and compression of the spring, but its motion is retarded by the viscous drag of the surrounding medium. In infrared vibrational spectroscopy, the ball is an atom set in excited motion by the irradiation and there is no surrounding viscous medium. In dynamic mechanical relaxation, the ball is part of a molecule set in motion by an applied stress and other chains form the surrounding medium. 

Then the total Newtonian equation of motion of the ball has the exciting force, F, the acceleration and deceleration of the vibration, the drag of the viscous medium and the Hooke s Law restoring force of the spring. This gives for the force and the restraints acting in opposite directions ... 

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