ENERGY MIGRATION AND THE PHOTOSYNTHETIC UNIT 1287 



and transfer are equal) is reduced, compared with Perrin's estimate, by 

 a factor of [A'v/riAvy]'^': 



(32.5) do = i\/27)[A'v/t(Auy]'^' 



Here, A'j^ is the range of overlapping fluorescence and absorption frequen- 

 cies, Av the band width and i the average duration of excitation. With 

 A'j//Av = 0.1, ^ = 5 X 10-9 second and Av ^ 1.5 X 10l^ Forster calculated 

 (Iq o:=i 7.5 m/x, corresponding to about 20 molecular diameters. An im- 

 proved, quantum-mechanical calculation leads (Forster 1948, 1951) to the 

 same order of magnitude for the distance over which excitation energy can 

 be transmitted. The frequency of energy transfer is, according to this 

 calculation, proportional to the inverse sixth power of the distance between 

 molecules, and therefore directly proportional to the sq^iare of concentra- 

 tion. The following equation was obtained by Forster (1948) as a substi- 

 tute for equation (32.5) : 



(32.6) rfo -^ SicTl/STSim'^ul 



where t ^ average life-time of excitation, c = velocity of light, n = index 

 of refraction of the medium, A'"' = number of molecules in a millimole; 

 vo, the frequency of the band (average of the frequencies of the absorption 

 and fluorescence peaks) ; and T^ the integral : 



(32.4) n = 2.302y^" a,«2. 



.a. 



which is a measure of the overlapping of the fluorescence and the absorption 

 spectra (a^ being the decimal molar extinction coefficient at frequency i'). 

 Applying this equation to chlorophyll a in ether (/i = 1.35) Forster ob- 

 tained do = 8.0 mju as the distance at wliich energy transfer becomes equally 

 probable with re-emission. This is the average intermolecular distance 

 at 7.7 X 10—* m./l. Treating chlorophyll grana as a homogeneous system 

 0.1 ilf in chlorophyU (cf. Vol. I, page 411), i. e., neglecting the probable 

 orderliness of the arrangement, Forster calculated that about 10'' transfers 

 can occur during the excitation period of 3 X 10^^ sec. 



According to the equation AE ^ hv (where v is the frequency of the 

 energy exchange) this rate of exchange should cause a change, AE, in the 

 energy of the excited state, equivalent to about 10 cm.-^ [10V(3 X 10"* 

 X 3 X 10 1°) = 10]. This corresponds to a shift of the absorption band, 

 leading to this state, by about 0.4 /x. It will be noted that, with 10* trans- 

 fers per second, 30 to 300 molecular vibrations (with periods between 10~^* 

 and 10-^^ sec.) wifl be possible per visit; consequently, the band shape — 

 determined by the couphng of electronic excitation and intermolecular vi- 

 brations (according to the Franck-Condon mechanism) — wiU be preserved 

 more or less unchanged. 



