Let's consider an excited nucleus emitting one gamma-ray (not cascade etc). Is the direction of gamma-ray emission completely random? In other words, is the probability to detect this gamma equal for any angle?
Will the answer change in case of presence of magnetic field?
1) To define an angle for a nuclear reaction one has to have an orientation for the nucleus. This could be a magnetic moment or a dipole moment.
2) a photon carries away spin 1, i.e. angular momentum, and will leave a nucleus minus that angular momentum. As the word states, angles are involved, and the probability distribution for the gamma ray will in general not be uniform. This can be calculated given the initial state:
There are examples of nuclei whose emission is not isotropic in the presence of a magnetic field. Feynman gave an example of this during a lecture on symmetry in physical laws (vol 1-52).
In particular in 52-7 he mentions an experiment in which the emission of an electron by a cobalt nucleus (Co-60) is asymmetrical with respect to the magnetic axis - more is emitted from one pole than from the other. This was experimental proof of no conservation of parity and led to the Nobel Prize in 1957 for Lee and Yang.
You can find details in http://www.wikipedia.org/wiki/Wu_experiment - it's not exactly gamma emission but the closest example I know of...
Anna and Floris have given excellent descriptions of how the photon emission depends on the orientation of the nucleus, however there is another aspect of the question that I think is worth mentioning.
If you conside an isolated nucleus, e.g. no magnetic field, then it will be in a suerposition of all possible orientations so overall the system is spherically symmetric. When the gamma ray photon is emitted it will be in a superposition of all possible directions preserving this spherical symmetry. When we interact with the photon by detecting it the superposition will collapse into a specific photon momentum state, and therefore a specific momentum state for the nucleus it came from. Since the system was originally spherically symmetric the probability of measuring the photon is the same in all directions.
If you apply a magnetic field then you break the spherical symmetry, typically down to an axial symmetry. In this case the probability of measuring the photon will no longer be spherically symmetric. As Anna and Floris have said, it will be a complicated function of the nuclear structure. However we would still expect the probability of detection to be axially symmetric.
