Method for producing abrasive grains, method for producing slurry, and method for producing polishing liquid

A production method for abrasive grains, comprising mixing a first liquid, which is an aqueous solution of at least one tetravalent metal element salt, and a second liquid which is a solution of at least one alkali, so as to form a liquid mixture, under conditions such that parameter Z in the following formula (1) is 5.00 or greater, to yield abrasive grains including a hydroxide of the tetravalent metal element, wherein during the mixing the liquid mixture is stirred using a stirring blade, wherein the tetravalent metal element is at least one kind selected from the group consisting of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium and lutetium, and wherein the alkali is selected from the group consisting of ammonia, triethylamine, pyridine, piperidino, pyrrolidine, imidazole, chitosan, potassium hydroxide and sodium hydroxide:

• Z=(1/(ΔpH×k)) ×(N/M)/1000 (1),
• wherein in formula (1), ΔpH represents a variation in pH per minute in a reaction system of the liquid mixture, k represents a reaction temperature coefficient represented by the following formula (2), N represents a cycle count (min−1) represented by the following formula (3), and M represents a substitution count (min−1) represented by the following formula (5);
• k=2((T−20)/10) (2),
• wherein in formula (2), T represents a temperature (° C.) of the reaction system;
• N=(u ×S)/Q (3),
• wherein in formula (3), u represents a linear speed (m/min) represented by the following formula (4), for the stirring blade stirring the liquid mixture obtained by mixing the first liquid and the second liquid, S represents an area (m2) of the stirring blade, and Q represents a liquid volume (m3) of the liquid mixture;
• u=2π×R×r (4),
• wherein in formula (4), R represents a rotational speed (min−1) of the stirring blade, and r represents a radius of rotation (m) of the stirring blade;
• and
• M=v/Q (5),
• wherein in formula (5), v represents a mixing rate (m3/min) of the first liquid and second liquid, and Q represents a liquid volume (m3) of the liquid mixture, and
• wherein:
• the ΔpH is in the range of 0.001 to 5.00,
• the cycle count N is in the range of 1.00 min−1 to 200.00 min−1,
• the substitution count M is in the range of 1.0 ×10−5 min-1 to 1.0 min−1,
• the linear speed u is in the range of 5.00 m/min to 200.00 m/min,
• the mixing rate v is in the range of 1.00 ×10−7 m3/min to 1.00 ×10−2 m3/min, and
• the rotational speed R is in the range of 30 min−1 to 1000 min−1.