TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Process: The step of separating the components other than the ceria-zirconia solid solution crystal is carried out by using an acid. Preferred Components: The amount of zirconium oxide or zirconium oxide and cerium oxide is 1-99 (preferably 3-97) mol.%. The amount of (I) or (I) and boron oxide is 10-70 mol.%. The amorphous material is in the form of flake or fiber. The ceria-zirconia solid solution crystals have a cubic crystal structure. The amount of (I), or cerium oxide, zirconium oxide, (I) and boron oxide is 0.01-20 mol.%. The ceria-zirconia solid solution crystal fine particles have: an average primary particle size of 5-200 nm; and a specific surface area of at least 20 m2/g. EXAMPLE - Zirconium oxide (1 mol.%), cerium oxide (19 mol.%), a metal oxide compound of formula (RCO3), which was strontium- (13.3 mol.%) and barium oxide (13.3 mol.%), and boron oxide (53.3 mol.%) were dryly mixed and pulverized to obtain a raw material mixture. The raw material mixture obtained was put in a crucible with a nozzle, made of platinum containing 20 mass% of rhodium, and heated in an electric furnace with a heating element made of molybdenum silicide at 1500 degrees C in air for one hour to be completely melted. Then, the glass melt was dropped while heating the bottom end of the nozzle by the electric furnace, and droplets were made to pass between twin rollers with a diameter of 15 cm rotating at 300 revolutions per minute so that the droplets were quenched at 1x 105 degrees C/second to obtain a solid material in flake form. The flakes thus obtained showed a brownish-red color and were a transparent amorphous material. The flakes were heated at 820 degrees C for 8 hours to precipitate ceria-zirconia solid solution crystals. Then, the flakes after the crystallization treatment were shaken and stirred in a 1 mole/l of acetic acid solution at 70 degrees C for at least 6 hours to leach soluble materials. A liquid after leaching was subjected to centrifugal separation, and a supernatant was discarded. After this operation was repeated five times, washing with water was carried out five times, followed by drying to obtain fine particles having a particle size of 5-200 nm.    R = Mg, Ca, Sr or Ba.