This property of having class limits defined in terms of whole number 

 powers of 2 millimeters led Krumbein (1936) to propose a phi unit scale based 

 on the definition: 



Phi units ( 4>) = - log 2 (diameter in mm) (4-1) 



Phi unit scale is indicated by writing ij) or phi after the numerical 

 value. The phi unit scale is shown in Figure 4-7. Advantages of phi units 

 are 



(1) Limits of Wentvrorth size classes are whole numbers in phi units. 

 These phi limits are the negative value of the exponent, n , in the relation 

 iP" . For example, the sand size class ranges from +4 to -1 in phi units. 



(2) Sand size distributions typically are near lognormal, so that a unit 

 based on the logarithm of the size better emphasizes the small significant 

 differences between the finer particles in the distribution. 



(3) The normal distribution is described by its mean and standard 

 deviation. Since the distribution of sand size is approximately lognormal, 

 individual sand size distributions can be more easily described by units based 

 on the logarithm of the diameter rather than the absolute diameter. Compar- 

 ison with the theoretical lognormal distribution is also a convenient vay of 

 characterizing and comparing the size distribution of different samples. 



Of these three advantages, only (1) is unique to the phi units. The other 

 two, (2) and (3), would be valid for any unit based on the logarithm of size. 



Disadvantages of phi units are 



(1) Phi units increase as absolute size in millimeters decreases. 



(2) Physical appreciation of the size involved is easier when the units 

 are millimeters rather than phi units. 



(3) The median diameter can be easily obtained without phi units. 



(4) Phi units are dimensionless and are not usable in physically related 

 quantities where grain size must have units of length such as grain size, 

 Reynolds number, or relative roughness. 



Size distributions of samples of littoral materials vary widely. 

 Qualitatively, the size distribution of a sample may be characterized (1) by a 

 diameter that is in some way typical of the sample and (2) by the way that the 

 sizes coarser and finer than the typical size are distributed. (Note that 

 size distributions are generally based on weight, rather than number of 

 particles.) 



A size distribution is described qualitatively as weVi sorted if all 

 particles have sizes that are close to the typical size. If the particle 

 sizes are distributed evenly over a wide range of sizes, then the sample is 

 said to be well graded. A well-graded sample is poorly sorted; a well-sorted 

 sample is poorly graded. 



4-14 



