| MadSci Network: Chemistry |
Greetings: The quantum mechanical models for your questions are presented in a clear manner in Chapter 5 of "An Introduction to Lasers and Masers" by Professor A. E. Siegman, McGraw-Hill, 1971, one of the first and best text books in the field. In Chapter 5 the topic of population inversion, negative resistance and negative temperature is discussed in detail with several pictures and charts that nicely explain the phenomena that you ask about. The negative temperature comes about because of the mathematical expressions and not because there is a negative temperature. Temperature has an absolute value. Because the Boltzman population distribution for a two level system (N2 & N1) for a total population Nt, is formulated with expodentials of the form [exp -(x/T) ] the solution for the population difference becomes : (N1-N2) = Nt tanh(hf/2kT). Where h is Planck's constant, k is Boltzmann's constant f is frequency and T is temperature. When (N1-N2) is plotted as a function of (hf/2kT) , N1 approaches Nt for large positive values of (hf/2kT) and N2 approaches Nt for large negative values of (hf/kT) enabling use to evaluate all values of the population difference. Since the tanh (x) function is zero where the axis intersect (hf/2kT) must be zero and negative in the left quadrant. At the center point in the plot N1 = N2. Mathematically we could make h or f or k or T negative to make (hf/ 2kT) negative; however, h, f and k are constants so T is the only physical parameter that we can make negative to follow the tanh(x) function (In electrical engineering, negative frequencies (-f) are often used to enable symmetry). This problem with mathematical models occurs often in science and sometimes there are even discontinuties in the models that do not occur in the real world!. Professor Siegman points out that many of us can understand the concept of negative resistance because we are familiar with electronic amplifiers; however, the concept of negative temperature seems foreign to most of us. The details about the quantum mechanics behind the values of N1 and N2 are beyond presenting in this note; however, I'd like to coment of the concept of model symmetry. When we mathematically model closed physical systems we can pick our measurement reference point any where we want to place them. What we actually are interested in is measuring differences. We could formulate the Boltzmann equations so that T does not become negative but then we will loose the simplicity and symmetry of the model. Often the symmetry of the models not only simplify the mathematics, it gives us better insight to what is happening. Ohms Law is a classic simple linear model where picking a reference point doesn't much matter and symmetry would complicate the model. Ohmm's Law states that Voltage (E) = Current (I) times Resistance (R). E = I * R Suppose I have a simple 12 volt battery (E) connected to a 12 Ohm resistor (R) so that 1.0 ampere of current (I) flows in the circuit. If I put my common voltmeter probe on the negative terminal of the battery all voltages measured would be positive. If placed the common probe on the positive terminal of the battery all voltages measured would be negative. Now suppose that I know nothing about the battery voltage and the network except for my resistor and the current flowing through it. For reasons of symmetry I decide to place the common probe of my volt meter in the middle of the resistor so that I measure + 5 volts on one end of the resistor and - 5 volts on the other end. I could say that in my model there is a negative resistance on one side to cause the negative voltage measurement (E = I* -R) and there is a positive resistance on the other side to get the positive voltage (E = I * +R). In this case symmetry and use of negative resistors unnecessarily complicates the problem; however the differences all work mout the same!. In the nonlinear Boltzmann model, choosing a reference point for symmetry does require a negative temperature; however, in this case the model can then be formulated in a much less complex form with the point N1 = N2 in the center of the graphical plot. Professor Seigman, in his book, also adds other information to this plot to provide additional insight. I hope that this note does not confuse you more than it helps. In any case find Professor Seigman's book if you plan to go on in laser/maser physics or chemistry, you'll enjoy it. Best regards, Your Mad Scientist Adrian Popa
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