Substrate for mask blanks and method for manufacturing the same

A substrate for mask blanks having two main surfaces of a first main surface and a second main surface of 152 mm×152 mm square and having a thickness of 6.35 mm, wherein

• when a range of 132 mm×132 mm square centered on an intersection of diagonal lines is defined as a calculation region in each of the first main surface and the second main surface,
• on a substrate surface of the calculation region of at least one of the first main surface and the second main surface,
• flatness of the substrate surface of the calculation region based on a least square plane is 100 nm or less, and a difference (PV) between a highest value and a lowest value of a height of a calculation surface represented by a difference between a shape of the substrate surface before smoothing processing with a 10-mm Gaussian filter (10 mm×10 mm) and a shape after the smoothing processing based on the least square plane is 20 nm or less, wherein
• when, in at least an optional range of 6 mm×6 mm square set in the calculation region of at least one of the first main surface and the second main surface,
• (1) measuring a surface shape within a range of 6 mm×6 mm square with precision finer than a pitch of 10 μm to create a height map;
• (2) creating an x partial differential map obtained by partially differentiating the height map within the range of 6 mm×6 mm in an x direction;
• (3) creating a y partial differential map obtained by partially differentiating the height map within the range of 6 mm×6 mm in a y direction;
• (4) creating a total differential map from the x partial differential map and the y partial differential map;
• and
• (5) calculating a difference (PVXY) between a highest value and a lowest value of the total differential map, and
• the PVXY is 0.1 nm/μm or less at any position in the calculation region.