Family of growth fractals with continuously tunable chemical dimension

We introduce a new class of statistical growth fractals which is of interest because the chemical dimension d, is continuously tunable. We also study other exponents characterising these fractals. There has been considerable recent interest in uncovering the fashion in which the familiar laws of physics are modified for fractal objects, in part because of the large number of important realisations of fractals in nature [l-81. It has only recently become appreciated that the physics of fractals is determined by more than just ‘the’ fractal dimension df which describes how the cluster mass N scales with the cluster radius R, N Rdf. ( l a ) Several additional fractal dimensions have recently been found to be of use. One of these is the chemical dimension d, that describes how the cluster mass scales within a chemical distance 1 (the chemical distance is the length of the shortest path on the fractal connecting two sites) [9-141, N I d ! . (1b) It is of course very important to seek relations among the various new fractal dimensions-indeed, it was the search for relations among critical exponents that led to the discovery of scaling laws 20 years ago and eventually to the development of the renormalisation group. Here we study the relation between the two exponents d, and d, by introducing a new family of cluster growth models in which d, can be varied in a controlled fashion over a wide range of values. The clusters are grown by the following procedure. First we place a seed particle at the origin of a d-dimensional lattice of coordination number z. At step 1 = 1, we randomly choose a certain number B( l ) of the z neighbours and occupy these sites. These sites constitute the first shell, and clearly have chemical distance 1 = 1 from the origin. The remaining z -B( l ) sites will be regarded as ‘blocked’ for the duration of the growth process. At step 1 = 2, we consider z2 available (unblocked and unoccupied) neighbours of the first shell. We randomly occupy B(2) of these, and block the 0305-4470/85/171103 + 05$02.25 @ 1985 The Institute of Physics L1103 L1104 Letter to the Editor 0 0 0.0 0 0.0 0 0 . x O A O 0 0 A x o 0 . A ~ 0 o x o x x la) I61 IC) Figure 1. Schematic illustration of how a cluster grows for E ( I) = cl" with c = 2 and a = 0.5, where E ( / ) is the number of growing sites at shell /, and a is the tunable parameter. In figure l (a) , at step I = 0, the seed particle (A) is at the origin of a square lattice with four perimeter sites (0). In figure l (b) , at step I = 1, we choose B(1) ( = 2 ~ 1 ' . ~ = 2 ) sites (0) randomly out of four perimeter sites of figure l (a ) and occupy them. The remaining sites ( X ) are blocked forever. Five new perimeter sites are created. In figure l(c), at step 1 = 2, B(2) (=2 x ~ O . ' = 3 ) sites are added to randomly chosen perimeter sites (0) of figure l(b) and the rest are blocked ( x ) . Remaining shells grow similarly. remainder. This process is continued until a cluster with a total of I, chemical shells has been created, with B( I ) I" ( 2 a ) occupied sites in chemical shell I ; here a is the tunable parameter?. The total cluster mass after I shells have been added is given by N(1) = B(Z') I d ! , l '= 1 dl= CY + 1 . (2c) A schematic illustration of our cluster model is shown in figure 1, and a typical cluster is shown in figure 2. We begin by considering how the Pythagorean distance R scales with the chemical distance 1. From equations ( l a , b ) R d f I d : so R I s , ( 3 a ) Figure 2. A typical cluster grown on a square lattice with d, = 1.2. t The case a = 0 was considered recently [ 151. letter to the Editor L1105 Table 1. Data for i; = d;:,,, d i , d, and fd, in two dimensions. is increasing and d, is decreasing as dl increases, but d: has a maximum. The fracton dimension d, is also monotonically increasing. dl ;=dit , , dl, d w td, 0.5 f 0.03 1 .o 0.67 f 0.02 2.0 3.0f 0.1 1.2 0.73 f 0.03 2.2 f 0.1 3.0f 0.1 0.55 i0 .03 1.4 0.80*0.03 2.37f0.1 2.9 i 0.1 0.60 f 0.03 1.64 0.88 i 0.03 2.30i0.1 2 . 6 i 0 . 1 0.7 i 0.03 2 .4 i0 .1 0.77 f 0.03 1.8 0.93 f 0.03 2.29i0.1 T8ble 2. Data for 5 in d = 3 , 4 and 5.