of would have tunneled. The current of tunneling electrons at each instance is therefore proportional to divided by , which is the time derivative of , . The time scale of the measurement in STM is many orders of magnitude larger than the typical femtosecond time scale of electron processes in materials, and is large. The fraction part of the formula is a fast oscillating function of that rapidly decays away from the central peak where . In other words, the most probable tunneling process, by far, is the elastic one, in which the electron's energy is conserved. The fraction, as written above, is a representation of the delta function, so . Solid-state systems are commonly described in terms of continuous rather than discrete energy levels. The term can be thought of as the density of states of the tip at energy , giving . The number of energy levels in the sample between the energies and is . When occupied, these levels are spin-degenerate (except in a few special classes of materials) and contain charge of either spin. With the sample biased to voltage , tunneling can occur only between states whose occupancies, given for each electrode by the Fermi–Dirac distribution , are not the same, that is, when either one or the other is occupied, but not both. That will be for all energies for which is not zero. For example, an electron will tunnel from energy level in the sample into energy level in the tip (), an electron at in the sample will find unoccupied states in the tip at (), and so will be for all energies in-between. The tunneling current is therefore the sum of little contributions over all these energies of the product of three factors: representing available electrons, for those that are allowed to tunnel, and the probability factor for those that will actually tunnel. . Typical experiments are run at a liquid helium temperature (around 4 K) at which the Fermi level cut-off of the electron population is less than a millielectronvolt wide. The allowed energies are only those between the two step-like Fermi levels, and the integral becomes . When the bias is small, it is reasonable to assume that the electron wave functions and, consequently, the tunneling matrix element do not change significantly in the narrow range of energies. Then the tunneling current is simply the convolution of the