When ^4He is cooled below 2.17 K it becomes a \"superfluid\" with unique propert
ID: 930803 • Letter: W
Question
When ^4He is cooled below 2.17 K it becomes a "superfluid" with unique properties such as the viscosity approaching zero. One way to learn about the superfluid environment is to measure the rotational-vibrational spectrum of molecules embedded in the fluid. For example, the spectrum of OCS in a low-temperature ^4He droplet has been reported {Journal of Chemical Physics, 112, 2000, 4485). For OC^32S the authors measured a rotational constant of 0.203 cm^-1 and found that the intensity of the J=01 transition was 1.35 times greater than that of the J= 1 2 transition. Using this information, provide a rough estimate of the temperature of the ^4He droplet.Explanation / Answer
Quantum theory successfully predicts the line spacing in a rotational spectrum. An additional feature of the spectrum is the line intensities. The lines in a rotational spectrum do not all have the same intensity.The maximum absorption coefficient for each line, max, is proportional to the magnitude of the transition moment, T which is given by Equation (7-47), and to the population difference between the initial and final states, n. Since n is the difference in the number of molecules present in the two states per unit volume, it is actually a difference in number density.
max=CTn..............(1)
where C includes constants obtained from a more complete derivation of the interaction of radiation with matter.
The dependence on the number of molecules in the initial state is easy to understand.
Whether absorption or stimulated emission is observed when electromagnetic radiation interacts with a sample depends upon the population difference, n, of the two states involved in the transition. For a rotational transition,
n=nJnJ+1................(2)
where nJ represents the number of molecules in the lower state and nJ+1 represents the number in the upper state per unit volume. If this difference is 0, there will be no net absorption or stimulated emission because they exactly balance. If this difference is positive, absorption will be observed; if it is negative, stimulated emission will be observed.
We can develop an expression for n that uses only the population of the initial state, nJ, and the Boltzmann factor. The Boltzmann factor allows us to calculate the population of a higher state given the population of a lower state, the energy gap between the states and the temperature. Multiply the right-hand side of Equation (2) by nJ/nJ to obtain
n=(1nJ+1/nJ)nJ.....................(3)
Next recognize that the ratio of populations of the states is given by the Boltzmann factor which when substituted into yields
n=(1exp(hJ/kT)/nJ.................(4)
where hJ is the energy difference between the two states. In the rigid rotor model J=2B(J+1) so (4) can be rewritten as
n=(1-exp(2hB(J+1)/kT)/nJ..............(5)
Equation expresses the population difference between the two states involved in a rotational transition in terms of the population of the initial state, the rotational constant for the molecule, B, the temperature of the sample, and the quantum number of the initial state.
To get the number density of molecules present in the initial state involved in the transition, nJ, we multiply the fraction of molecules in the initial state, FJ, by the total number density of molecules in the sample, ntotal.
nJ=FJntotal...............................(6)
The fraction FJ is obtained from the rotational partition function.
FJ=(2J+1)(hBkT)exp(2hB(J+1)/kT)............................(7)
The exponential is the Boltzmann factor that accounts for the thermal population of the energy states. The factor 2J+1 in this equation results from the degeneracy of the energy level. The more states there are at a particular energy, the more molecules will be found with that energy. The (hB/kT) factor results from normalization to make the sum of FJ over all values of J equal to 1. At room temperature and below only the ground vibrational state is occupied; so all the molecules (ntotal) are in the ground vibrational state. Thus the fraction of molecules in each rotational state in the ground vibrational state must add up to 1.
By substuting your values in the above equation you will get your answer. If you need still more assistance let me know.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.