The XRD pattern of Cs_xTi_2−x/2Mg_x/2O₄ (x = 0.70) (Figure 2(a)) was indexed on the basis of an orthorhombic cell of a = 0.3824 (2) nm, b = 1.704 (3) nm, and c = 0.2929 (1) nm (Table 1). The lattice constants of the sample are in good agreement with those prepared by Reid et al. (a = 0.3821 nm, b = 1.7040 nm and c = 0.2981 nm) [9] .

The XRD pattern of the Li⁺ exchange product is shown in Figure 2(b). The pattern was indexed as a single phase with orthorhombic lattice constants of a = 0.378 nm, b = 1.72 nm, and c = 0.292 nm (Table 1). The lattice constants of a and c were almost unchanged. This shows that the host layer of Cs_xTi_2−x/2Mg_x/2O₄ (x = 0.70) is maintained through the Li⁺ exchange. The interlayer spacing which

corresponds to b/2, increased from 0.852 nm to 0.860 nm. The TGA curve of the product (Figure 3(a)) shows a weight loss from 20˚C to 200˚C corresponding to the dehydration of interlayer water. The composition was estimated to be Li_0.53H_0.13Cs_0.14Ti_1.65Mg_0.30O₄∙0.92H₂O by chemical analysis and weight loss. It was found that 14% of the Mg in the titanate was leached out in solution during the ion exchange. England et al. [10] also studied the Li⁺ exchange product and estimated the composition to be Li_0.33Cs_0.37Ti_1.65Mg_0.35O₄∙0.72H₂O by the amount of Cs released into solution, determined by photometric analyses and weight loss from TG analysis. They did not analyze the content of Mg in their Li⁺-exchanged product.

The Li⁺-exchange product was heated at 180˚C for 1 h in a vacuum (Figure 2(e)). The XRD pattern was indexed as a single phase with orthorhombic lattice constants of a = 0.371 nm, b = 0.662 nm, and c = 0.300 nm (Table 1). In this case, the lattice constant of b corresponds to the interlayer spacing. The interlayer spacing decreased from 0.860 nm to 0.662 nm because of dehydration of the interlayer water. The dehydrated product of Li_0.53H_0.13Cs_0.14Ti_1.65Mg_0.30O₄ was evaluated for its uses as cathodes in both lithium and sodium batteries.

The XRD pattern of the Na⁺ exchange product showed that the product was a mixture of two phases with the interlayer spacing of d = 1.14 nm and d = 0.89 nm. This product was heated at 40˚C for 1 h. Figure 2(c) shows the XRD pattern of the heated product. The pattern was indexed as a single phase with the orthorhombic lattice constants (Table 1) where the 0.110-nm phase disappeared. These constants show that the host layer of Cs_xTi_2−x/2Mg_x/2O₄ (x = 0.70) is

also maintained through the Na⁺ exchange. The TGA curve (Figure 3(b)) shows two steps of weight loss: 20˚C - 100˚C and 100˚C - 200˚C. Both steps correspond to the dehydration of the interlayer water. The composition was estimated to be Na_0.56H_0.14Ti_1.65Mg_0.35O₄∙1.1H₂O (Table 1). England et al. [10] also studied the Na⁺ exchange product and determined the composition to be Na_0.70Ti_1.65Mg_0.35O₄・0.70H₂O.

The XRD pattern of the H⁺ exchange product is shown in Figure 2(d). The pattern was indexed as a single phase with the orthorhombic lattice constants of a = 0.378 nm, b = 1.77 nm, and c = 0.298 nm (Table 1). This also shows that the host layer of Cs_xTi_2−x/2Mg_x/2O₄ (x = 0.70) is maintained through the H⁺ exchange. The TGA curve (Figure 3(c)) shows two steps of weight loss: 20˚C - 150˚C and 150˚C - 450˚C. The former weight loss corresponds to the dehydration of the interlayer water, and the latter corresponds to dehydration of the decomposition due to the combination of the exchanged H⁺ with the O²⁻ of the host layer. The composition was estimated to be H_0.99Cs_0.07Ti_1.65Mg_0.17O₄∙1.2 H₂O (Table 1). It was found that 51% of Mg in the titanate was leached out in solution during the ion exchange. England et al. [10] studied the H⁺ exchange product and estimated the composition to be H_0.65Cs_0.05Ti_1.65Mg_0.35O₄・0.7H₂O.

Figure 4 shows the discharge-charge curves of the Li/ Li_0.53H_0.13Cs_0.14Ti_1.65Mg_0.30O₄ cell. The cell voltage decreased rapidly from the rest potential of 3.1 V to 2.0 V and then decreased slowly to the cutoff voltage of 1.0 V. The discharge capacity was 151 mAh/g for the first cycle. The amount of Li⁺ intercalated in this process was 0.98 for the formula unit. The discharge potentials of Ti⁴⁺/Ti³⁺ in a Li_4/3Ti_5/3O₄ spinel oxide is reported to be 1.55 V, with the insertion of Li⁺ in the three-dimensional spinel framework [11] . The Li/Li_0.53H_0.13Cs_0.14Ti_1.65Mg_0.30O₄ cell exhibited almost the same voltage as shown in the figure, so we can conclude that the discharge process corresponds to the intercalation of Li⁺ into the vacant space of the interlayer and the reduction of Ti⁴⁺ to Ti³⁺ in the lepidocrocite structure.

The first discharge and charge capacities were 151 mAh/g and 114 mAh/g, respectively. The amounts of Li⁺ intercalated and deintercalated were 0.98 and 0.74 of the formula unit, respectively. At the 10th cycle, the cell exhibited 73% (110 mAh/g) of the first discharge capacity and 83% (95 mAh/g) of the first charge capacity. At the 20th cycle, the cell exhibited 70% (105 mAh/g) of the first discharge capacity and 82 % (93 mAh/g) of the first charge capacity.

Figure 5 shows the discharge-charge curves of Na/Li_0.53H_0.13Cs_0.14Ti_1.65Mg_0.30O₄ cell. The cell voltage decreased rapidly from the rest potential of 2.7 V to 1.8 V and then decreased slowly to the cutoff voltage of 0.70 V. The discharge capacity was 91 mAh/g for the first cycle. The amount of Na⁺ intercalated in this process was 0.59 for the formula unit. Recently, we reported that Li₂Ti₅O₁₁ derived by ion-exchange reaction from the layer-structured titanate Cs₂Ti₅O₁₁ exhibited discharge-charge capacities of 120 and 100 mAh, respectively, for the first cycle in a rechargeable sodium battery in the voltage range 0.70 - 4.0 V [8] . The discharge potential of Ti⁴⁺/Ti³⁺ in the layer-structured Li₂Ti₅O₁₁ was approximately 1.2 V with the insertion of Na⁺. This shows that the discharge process of the Na/Li_0.53H_0.13Cs_0.14Ti_1.65Mg_0.30O₄ cell corresponds to the intercalation of Na⁺ and the reduction of Ti⁴⁺ to Ti³⁺ in the lepidocrocite structure.

The first discharge and charge capacities were 91 mAh/g and 77 mAh/g, respectively. The amounts of Na⁺ intercalated and deintercalated were 0.59 and 0.50 of the formula unit, respectively. At the 10th cycle, the cell exhibited 64% (58 mAh/g) of the first discharge capacity and 73% (56 mAh/g) of the first charge capacity. At the 20th cycle, the cell exhibited 38% (35 mAh/g) of the first discharge capacity and 44% (34 mAh/g) of the first charge capacity.

The lower discharge-charge capacity of the sodium battery compared with that of the lithium battery may be attributed to the difference in ionic volume of Na⁺ and Li⁺. The larger volume of Na⁺ as compared with Li⁺ has a disadvantage in the intercalation into the vacant space of the interlayer.