# Under free vibration does it vibrate in all the modes

## Main Question or Discussion Point

say a system has 3 modes. under free vibration does it vibrate in all the modes all just one mode? why. also if we apply a either a impulse or pulsating force to the system, in which modes the system is going to vibrate and why? thanks.

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hi,
atoms in a molecule normally vibrate.
One molecule cannot vibrate in all modes perfectly (but it can vibrate in ground state in all modes, something random).
Each mode of vibration has its own energy, i.e. in order to make the atom to vibrate in a particular mode one have to excite the by molecule supplying photon of specific energy. Normally one varies this photon energy from 0 to 1000 cm-1 (0 to 123 meV).
Why you do like this? vibrational properties are fundamental and are important for predicting many properties of the molecule, eg., symmetry properties, bonding, force constant, etc. Moreover, vibration is not dependent on temperature and even at 0 K they will vibrate.

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SpectraCat

Ok, I think there are some English issues here, so let's take this one line at a time
hi,
atoms in a molecule normally vibrate.
No, atoms do not vibrate, molecules vibrate through motion of the atoms about a common center of mass. This motion is determined by the connectivity of the atoms through chemical bonds, which can be thought of like "springs", each with an associated spring constant. Thus, it is usually a good approximation to treat the vibrational motion of molecules as harmonic, at least at low energies.

One molecule cannot vibrate in all modes perfectly (but it can vibrate in ground state in all modes, something random).
I have no idea what that sentence was intended to convey. The vibrational motion of a molecule in the harmonic limit can be expressed in a complete basis set of orthogonal normal modes. Each normal mode can be represented as an independent harmonic oscillator, and so in this limit, it is completely reasonable for all of the modes of the molecule to be excited to any arbitrary vibrational state (i.e. quantum number). For example, in the harmonic limit, for the OP's 3-mode molecule example, you could easily have 1 quantum in each mode, or two quanta in each mode, or 3 quanta in one mode and zero in the others. *In the harmonic limit*, those states would be stable for an indefinite period of time (i.e. they are eigenstates). Of course, in any real molecule, the modes are not completely harmonic, which leads to anharmonic coupling terms that allow energy to flow between the modes over time, this is the well-known phenomenon of intramolecular vibrational redistribution, or IVR.

Each mode also has a finite zero-point energy (ZPE), so even in the vibrational ground state of the molecule (i.e. zero quanta in each mode), there is some vibrational energy in the molecule. However, the ZPE is an intrinsic property of the quantum system and cannot be "accessed" by chemical or physical processes.

Each mode of vibration has its own energy, i.e. in order to make the atom to vibrate in a particular mode one have to excite the by molecule supplying photon of specific energy.
That is not completely true, the normal modes *can* be excited by electromagnetic radiation (i.e. photons), however they can also be excited (or de-excited) by collisions with other molecules. This is called T-V energy transfer, (translation to vibration).

Normally one varies this photon energy from 0 to 1000 cm-1 (0 to 123 meV).
No, the vibrational spectrum of molecules ranges from about 50-100 cm-1 (the terahertz regime), for large amplitude motions of heavy molecules, up to about 4000 cm-1, for stretching vibrations of light, strongly bonded molecules like H2 and HF.

Why you do like this? vibrational properties are fundamental and are important for predicting many properties of the molecule, eg., symmetry properties, bonding, force constant, etc.
Ok, I agree with that completely

Moreover, vibration is not dependent on temperature and even at 0 K they will vibrate.
No, this is not true. Vibration is *completely* dependent on temperature. You can define a Boltzman population for each normal mode that describes the probability that a given quantum state will be populated at a given temperature. These factors increase with temperature, and thus so does the average internal energy for an ensemble of molecules at a given temperature. In fact, all molecules have a thermal dissociation threshold, which is the temperature where the average internal energy of the molecules is large enough to break the weakest chemical bond in the molecule.

Finally, it can be misleading to say that molecules "vibrate" even at zero K. I prefer to say that they have vibrational ZPE, which is completely correct, and does not give the impression that they are somehow moving. They certainly are not "moving" in the classical sense.

thanks Rajini. actually this is a rather simeple question which has nothing to do with atoms. just think about a cantilever beam or any multidegree system. it has multiple natural frequencies. i am out of school not touching dynamic for a long time. now I am doing something related to dynamic. just need someone to help me to clear some concepts.

Hi spectract,
I agree you generalized the reply..
Why dont you say the motion of atom as vibration (like to know)?
Each mode has its own energy, what is wrong in this?
Just think you have a molecule and that molecule vibrates in a specific mode...and that mode has a energy (E). One cannot change this E..that means only with that E you can excite/deexcite that specific mode..
I understood that in Raman/IR you scan the energy range from 7 to 500 meV..
But there are some techniques where people usually scan from 0 to 100 meV (metal-oxide vibrations occurs in this region) and in this techniques the important part is 0 to 10 meV (boson peak appear in this region), moreover some important properties like Debye-Waller factor is decided in this small region.
And
what i mean by temperature dependent is..
provided the molecule doesn't breaks, then for any temperature each mode has its own energy E..of course peak amplitude of the vibrational peak may change.

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Hi, For cantilever beams:
may be construction engineers check the quality and safe factor of the beam..so they need to know all these details..

i may post on a wrong forum. let's put this way, say you have a 2 degree system with 2 spring and two masses. it should have 2 natural frequencies. under free vibration it vibrate in both modes or only one mode. why?

Hmm, you question probably belong to mechanical engg. part.
http://emweb.unl.edu/Mechanics-Pages/Scott-Whitney/325hweb/Beams.htm
from this i understand (?) that when a beam vibrates it can have all possible modes of vibrations..first few (may be first and second mode of vibration) dominates, i.e. produces more displacement of the beam.

SpectraCat

I think the OP got his answer, or posted elsewhere, but I just wanted to answer your questions.

Hi spectract,
I agree you generalized the reply..
Why dont you say the motion of atom as vibration (like to know)?
It is a semantic point ... it makes sense to say that vibrations arise from atomic motion, or that vibrations arise from motions of atoms in molecules (or materials). However, I don't like the term "vibrations of atoms", because that makes it sound like the atoms have an intrinsic *internal* vibration, which they don't. From your posts, it sounds like you have a background in solid state/materials, so in that case I think it could be correct to say that vibrations in materials arise from "vibrations of atoms around their lattice sites", or something like that, where it is clearer what you mean from context.

Each mode has its own energy, what is wrong in this?
Just think you have a molecule and that molecule vibrates in a specific mode...and that mode has a energy (E). One cannot change this E..that means only with that E you can excite/deexcite that specific mode..
Nothing, I wasn't disagreeing with that part of your statement .. only the part about requiring a photon for the excitation. As I say, vibrations can also be excited via collisions of molecules with other molecules (or materials, i.e. surface impacts).

I understood that in Raman/IR you scan the energy range from 7 to 500 meV..
But there are some techniques where people usually scan from 0 to 100 meV (metal-oxide vibrations occurs in this region) and in this techniques the important part is 0 to 10 meV (boson peak appear in this region), moreover some important properties like Debye-Waller factor is decided in this small region.
Fair enough .. I wasn't thinking about solid state vibrational spectroscopy. Since the OP was asking about 3-mode systems, I automatically thought of small molecules, not solid state samples.

what i mean by temperature dependent is..
provided the molecule doesn't breaks, then for any temperature each mode has its own energy E..of course peak amplitude of the vibrational peak may change.
That is only true in the limit of perfectly harmonic vibrations, which never actually occurs either in molecules or in the solid state. With real systems there is a slight anharmonicity, so the "energy" you are talking about (I would call it the fundamental frequency of the mode), decreases with increasing quantum number. Furthermore, there is anharmonic coupling between the modes as I described, and so energy that is initially deposited in a particular mode will "leak out" into other modes in the molecule, generally quite quickly, on the timescale of a few vibrational periods.

Hi, Okay i clearly understand what you said regarding the vibration of atoms/molecules, hmm it makes sense to say always that vibration means it is molecular vibration rather than atoms.
Actually it is clear for me to say it as energy rather than frequency or wavenumber..(just a matter of taste)
Anyways thanks for discussing all these stuffs..
1 eV=8065.5 cm-1 !

When hit by a random impulse, a mechanical system will vibrate in all modes, but in different amplitudes.. Amplitudes for each mode, Ai, will depend on the displacements of the structure right after the impulse.

Here, an impulse is defined as an instantaneous relief from a deformed state.

Consider a string of length, L.
D(x,t) is its vertical deformation in space and time.
if the impulse is:
D(x,0-) = A*sin(pi/L*x)
then it will vibrate only in its 1st mode, with amplitude A.
D(x,t) = A*sin(pi/L*x)*sin(kt)

similarly, for the 2nd mode..:
D(x,0-) = A*sin(pi/L*2x) <-- impulse
D(x,t) = A*sin(pi/L*2x)*sin(2kt) <-- vibration etc...

however, strum the string on a random point and there will be harmonics (oscillations in the higher modes). In this case, 1st mode *generally* has a higher amplitude compared to other modes, but I can't make out the reason behind that. Any ideas?