Introduction:
Thermal neutrons are neutrons with energies corresponding to the thermal energy of surrounding atoms, which is typically around 0.025 eV. These neutrons are often used for structural determination of materials due to their unique properties. In this question, we need to determine the typical lattice spacing of a material by considering the given information about the energy and velocity of thermal neutrons.

Given:
- Energy of thermal neutrons (E) = 300 k (k = Boltzmann constant = 1.38 × 10^-23 J/K)
- Velocity of thermal neutrons (v) = 0.02 s (s = speed of light = 3 × 10^8 m/s)

Formula:
The de Broglie wavelength (λ) of a particle is given by the equation: λ = h/p, where h is the Planck's constant and p is the momentum of the particle.

Solution:
Step 1: Calculate the momentum of thermal neutrons
The momentum (p) of a particle can be calculated using the equation: p = mv, where m is the mass of the particle and v is its velocity.

The mass of a neutron is approximately 1.675 × 10^-27 kg.

p = (1.675 × 10^-27 kg)(0.02 s)
p = 3.35 × 10^-29 kg·m/s

Step 2: Calculate the de Broglie wavelength of thermal neutrons
Using the formula λ = h/p, where h is the Planck's constant (6.626 × 10^-34 J·s), we can calculate the de Broglie wavelength of the thermal neutrons.

λ = (6.626 × 10^-34 J·s)/(3.35 × 10^-29 kg·m/s)
λ = 1.98 × 10^-5 m

Step 3: Determine the typical lattice spacing
The de Broglie wavelength of a particle is related to the typical lattice spacing (d) of a material through the Bragg's Law equation: nλ = 2d sin(θ), where n is an integer representing the order of diffraction and θ is the angle of incidence.

Since the question does not provide any information about the angle of incidence or the order of diffraction, we cannot directly determine the typical lattice spacing.

However, we can make an estimation based on the options provided:

(a) 0.01 nm: This is a very small lattice spacing and is unlikely for most materials.
(b) 0.05 nm: This is a reasonable lattice spacing for many materials.
(c) 0.1 nm: This is a relatively large lattice spacing and is less common.
(d) 0.15 nm: This is an even larger lattice spacing and is less likely.

Based on these estimations, option (b) 0.05 nm seems to be the most reasonable choice for the typical lattice spacing of a material.

Conclusion:
Based on the given information, the typical lattice spacing of a material is estimated to be approximately 0.05 nm.