To determine the lattice parameter \(a\) for a simple cubic crystal, we use Bragg's Law, which gives the condition for constructive interference from crystal planes. The law is expressed as:

\(n\lambda = 2d\sin\theta\)

Where:

• \(n\) is the order of reflection,
• \(λ\) is the wavelength of the X-ray,
• \(d\) is the interplanar spacing,
• \(θ\) is the angle of incidence (in this case, \(45^\circ\)).

For simple cubic crystals, the interplanar spacing \(d\) for the plane with Miller indices (hkl) is given by:

\(d = \frac{a}{\sqrt{h^2 + k^2 + l^2}}\)

For the (110) plane, \(h = 1\), \(k = 1\), and \(l = 0\), thus:

\(d = \frac{a}{\sqrt{1^2 + 1^2 + 0^2}} = \frac{a}{\sqrt{2}}\)

Inserting \(d\) into Bragg's equation:

\(n\lambda = 2\left(\frac{a}{\sqrt{2}}\right)\sin\theta\)

Given that \(n = 1\) and \(θ = 45^\circ\), we know: \(\sin 45^\circ = \frac{\sqrt{2}}{2}\)

Thus, the equation simplifies to:

\(\lambda = \sqrt{2}a \cdot \frac{\sqrt{2}}{2}\)

\(\lambda = a\)

Given \(\lambda = 1.54 \ \text{Å}\), it follows that \(a = 1.54 \ \text{Å}\).

Therefore, the lattice parameter \(a\) is 1.54 Å.

The correct answer is: 1.54 Å.