Photophysics, Photobiology and Photosynthesis
This chapter aims to start with basic principles of photophysics and its relation to both photosynthesis as well as rate equations. Photophysics as the physical background behind effects like absorption, emission and energy transfer, the mathematical description as an idealized projection into a formalism which is, once settled up, pure logic and inerrable and photosynthesis as the biochemical process and its biological implementation form a triangle between reality, the abstraction into concepts and the mathematical modelling which go hand in hand to make us understand photosynthesis. The chapters 2.1 and 2.2 are kept quite mathematical as they introduce some background that is needed to calculate electromagnetic waves or transition probabilities. Readers who are more interested in the biological aspects of this book can proceed with chapter 2.3
Light Induced State Dynamics
Funfzig Jahre angestrengten Nachdenkens haben mich der Antwort auf die Frage “Was sind Lichtquanten?” nicht naher gebracht. Heute bilden sich Hinz und Kunz ein, es zu wissen. Aber da tauschen sie sich.
Albert Einstein (1879-1955), in a letter to M. Besso, 1951
In the following, we show some selected theoretical concepts from classical electrodynamics and quantum mechanics used to describe light and the interaction between light and matter. This will be complemented by an approach to thermodynamics based on rate equations. The concept of rate equations was motivated and shortly introduced in chapter 1.3. It is necessary to briefly show the underlying physical concept from which the probabilities of energy transfer, and thus the values of the “rates” in the rate equation models, are calculated. Therefore, the theory of excitation energy transfer will be introduced.
For me it was a key element during my studies to understand that everything in classical electrodynamics can be derived from the basic equations found by James Clerk Maxwell in 1864. In spite of students learning the complicated optical laws describing diffraction and interference like the Fresnel equations for the reflection and transmission of light on surfaces, just to name one example, everything can be derived from the basic Maxwell equations. However, it is surely not pragmatic to do this in every single case and often it is just necessary to know some special laws themselves. However the ability to use such basic equations as a tool for deriving any law in optics and electrodynamics is a key element in understanding the beauty of nature as it is described by equations like those. The first time dependent Maxwell equation in vacuum (eq. 11) tells us that in vacuum there are no direct sources for electric fields (as there exist no charges which form the source of electric monopols in matter, the divergence of the electric field is always zero):
The temporal change of the magnetic field induces rotation of the electric field (eq. 12):
There is no source of the magnetic field (i.e. there exist no magnetic monopols, eq. 13)
The temporal change of the electric field induces rotation of the magnetic field (i.e. currents are the sources of magnetic rotation. Eq. 14)
From these four time dependent Maxwell equations in vacuum describing the dynamics of electric fields E(j, f) and magnetic fields B(t, f) one can for example derive a wave equation e.g. for the electric field with help of eqs. 11, 12, and 14:
The general solution of eq. 15 is an arbitrary linear combination of plane waves which depend on space coordinate r and time t:
Interestingly Maxwell derived this wave equation and its solution in 1864 and he realized that he had proved mathematically the existence of electromagnetic waves propagating through electromagnetic fields. In eq. 15 there is the explicite velocity of these waves “c” he could calculate from known parameters as about 310.000 km/s. Maxwell wrote in his publication: “This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.” I found statements like this astonishing as Maxwell surely somewhat better understood the nature of light than his colleagues might have understood during this time. The approach he chose was pure mathematics but the output of his calculations drew new pictures he was able to interpret.
In eq. 16 к denotes the wave vector which depends on the frequency a and -E(tw) is the frequency dependent electric field amplitude.
Equation 16 solves 15 with the dispersion relation of the vacuum
Eq. 17 introduces the most important property of light quanta, i.e. the constant group velocity in vacuum which is the product of wavelength Я
and frequency v as |k and co = 2nv
The coordinate system can be shifted arbitrarily and the observation can be fixed to one spatial point. If the point x = y = z = 0 is chosen then the solution 16 simplifies to
which is the Fourier transformed of Ё(а>) . Due to the Fourier Theorem, eq. 19 and therefore especially eq. 16 represent any desired solution for the electric field ё(^ denoted in eq. 19 if it is periodic in time or vanishes in infinite time. If we assume a simplified solution Ё(®) = Ё05(а-(о0} with the Dirac Delta function S(co-0) for a fixed
frequency co0 then Ё(^ = Ё0ещ* becomes an infinitely oscillating function with a constant frequency and vice versa.
This already accounts for an uncertainty relation since any fixed frequency needs to be expanded as an infinite plane wave whereas a fully localized function could only be composed by an infinite number of wavelengths according to the Fourier integral representation of the solution of the wave function as given by eq. 16.
In other words: It is just not possible to get an idea of the wavelength, frequency and direction of a wave if we look at a single point only, or in the particle picture: Any absolutely localized particle has an absolutely uncertain momentum. Interestingly the Fourier transformation between frequency and time or wave vector and space immediately introduces an uncertainty principle for the corresponding two canonical variables.
A pure delta-distribution in the frequency domain does not exist physically. In general in the frequency domain as well as in the time domain we find wave packages with a certain distribution for the time the wave function is present or the frequency interval that it is composed from. Both are connected via a Fourier relation as given by eq. 16 and therefore show indirect proportional to one another as it is stated by Heisenberg’s uncertainty principle. For example, a laser pulse can be described by a temporal Gaussian and a spectral Gaussian. If the pulse should be short in time it is necessary to add a large number of frequencies to achieve destructive interference in all areas except the short pulse duration. The pulse becomes white. Vice versa, it is not possible to make short monochromatic pulses. General consequences of Heisenberg’s uncertainty principle directly follow, as pointed out here, from the Fourier transformation that results from the mathematical structure of a general solution to our wave equation 15.
If we continue our classical view (which, as pointed out, results in wave packages and an uncertainty relation - the basic principle of the quantization of light fields) we tend to focus on the intensity of the laser pulse in a generalized point of view:
The time dependent energy density u(t) (i.e. the energy per space unit) in the electric field denotes to
Eq. 21 introduces the spatial photon density n(t) of the photons in the pulse with the expected frequency ©0 (assuming a fixed frequency as a simplification of an ultrasharp continuous wave laser) equation 20 must be equal to
The photon density n(t) is a probability measure for the transfer of electrons by optical excitation from one state into another. We find that n(t) for the incoming photons in time is proportional to the square of the electric field as calculated from the Maxwell equations 11-14:
From equation 21 we can calculate the time dependent intensity distribution of the laser pulse employing the continuity equation:
and with help of 22 we find the photon current (photons per area and time) in analogy to equation 23
which fullfills the relation I — jphot-h d>0.
Our description focuses on the time dependent evolution of an average electric field, intensity and photon density propagating with the laser pulse. It delivers the relation between a number of photons per square and time unit jphot as it is necessary to activate discrete events that can be described by rate equations like absorption and emission of photons, or the photoeffect and the electric field amplitude E0 which is the basic entity for the formulation of electromagnetic phenomena in form of the Maxwell equations 11-14.
With equations 22-24 one can formulate the photon stream that causes the state changes per time and area unit directly from a given laser intensity and derive this formulation from the Maxwell equations.