stimulated emission causes due to the energy difference between the higher and lower energy level state, but it doesn't depends in the case of spontaneous emission . spontaneous emission causes without any stimulation .In stimulated emission energy transfer is twice the energy transfer of spontaneous emission.

That's why it's called spontaneous emission you cannot control it. But there is a kind of emission which is called stimulated emission emission. People use stimulated emission in lasers and it can be controlled.

stimulated emission causes due to the energy difference between the higher and lower energy level state, but it doesn't depends in the case of spontaneous emission . spontaneous emission causes without any stimulation .In stimulated emission energy transfer is twice the energy transfer of spontaneous emission.

How the nucleus emits radiations spontaneously?

Because the moment of emission the polarization and the propagation direction , the direction of the wave vector is random. Spontaneous emission causes incoherent radiation.