NOVELTY - Improving the smoothness of the surface of the resultant composite material, which has a functional group on its surface and which exhibits enhanced adhesion to a matrix.    USE - The vapor grown fine carbon fiber is useful for an electrically conductive material, a secondary battery, and a gas occlusion material. It is also useful as a filler material for improving electrical conductivity and thermal conductivity, as an electron emission material for producing field emission displays, as a medium for sorption of hydrogen, methane or various other gases, and a material employed in, e.g. transparent electrodes and electromagnetic wave shielding materials.    ADVANTAGE - The inventive vapor grown fine carbon fiber exhibits excellent adhesion to a matrix formed of, e.g. resin, ceramic or metal, which can be uniformly dispersed in the matrix, and which has a low aspect ratio.    DETAILED DESCRIPTION - A vapor grown fine carbon fiber, each fiber filament comprises, in its interior, a hollow space extending along the fiber filament, and having a multi-layer structure, an outer diameter of 2 to 500 nm, and an aspect ratio of 1-100. The fiber filament comprises a cut portion on its surface along the hollow space.    INDEPENDENT CLAIMS are also included for:    (a) a fine carbon mixture comprising a fine carbon fiber as above, in an amount of 5-80 mass% on the basis of the entirety of the carbon fiber mixture;    (b) a method for producing a fine carbon fiber, comprising wet-grinding vapor grown carbon fiber containing branched vapor grown carbon fiber in the presence of water and/or an organic solvent, each fiber filament of the carbon fiber comprising, in its interior, a hollow space extending along the fiber filament, and having a multi-layer structure, an outer diameter of 2-500 nm, and an aspect ratio of at least 10;    (c) a fine powder composition comprising a resin; and    (d) a secondary battery comprising, as an electrode material, and the fine carbon fiber above.