Introduction:
When a photon of frequency ν strikes an electron initially at rest, it undergoes scattering and loses half of its energy. The scattered photon moves at an angle φ, and the electron recoils at an angle θ. In this explanation, we will analyze the conservation of momentum and energy to determine the relationship between φ and θ.

Conservation of Energy:
The energy of a photon can be calculated using the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency. Since the photon loses half of its energy, the energy after scattering is E/2.

Conservation of Momentum:
The momentum of a photon can be calculated as p = hf/c, where p is the momentum and c is the speed of light. Since momentum is conserved, the momentum before scattering is equal to the momentum after scattering.

Photon Scattering:
When a photon scatters, it changes its direction and loses energy. The change in momentum of the photon can be calculated using the equation Δp = 2p sin(φ/2), where Δp is the change in momentum and φ is the scattering angle. Since momentum is conserved, the change in momentum of the electron can be calculated as Δp_e = -Δp.

Electron Recoil:
The electron recoils in response to the scattering of the photon. The momentum of the recoiling electron can be calculated as p_e = m*v_e, where p_e is the momentum of the electron, m is the mass of the electron, and v_e is the velocity of the electron. The change in momentum of the electron can be calculated as Δp_e = p_e - 0, since the electron was initially at rest.

Relationship between φ and θ:
Since momentum is conserved, Δp_e = -Δp. By equating the expressions for Δp_e and Δp, we get m*v_e = -2p*sin(φ/2). Rearranging this equation gives v_e = -2p*sin(φ/2)/m. The angle θ can be calculated using the equation tan(θ) = v_e/(c+v_e). Substituting the value of v_e in this equation gives tan(θ) = -2p*sin(φ/2)/(mc+2p*sin(φ/2)). From this equation, we can see that θ depends on φ and other variables such as the mass of the electron and the momentum of the photon.

Conclusion:
In summary, when a photon of frequency ν strikes an electron initially at rest and scatters at an angle φ while losing half of its energy, the electron recoils at an angle θ. The relationship between φ and θ can be determined by analyzing the conservation of momentum and energy. The angle θ depends on φ, the mass of the electron, and the momentum of the photon.