Three dimensional, particle-in-cell, fully electromagnetic simulations of electron plasma wake field acceleration in the blow out regime are presented. Earlier results are extended by (i) studying the effect of longitudinal density gradient; (ii) avoiding use of co-moving simulation box; (iii) inclusion of ion motion; and (iv) studying fully electromagnetic plasma wake fields. It is established that injecting driving and trailing electron bunches into a positive density gradient of ten-fold increasing density over 10 cm long Lithium vapor plasma, results in spatially more compact and three times larger, compared to the uniform density case, electric fields ($-6.4 times 10^{10}$ V/m), leading to acceleration of the trailing bunch up to 24.4 GeV (starting from initial 20.4 GeV), with an energy transfer efficiencies from leading to trailing bunch of 75 percent. In the uniform density case $-2.5 times 10^{10}$ V/m wake is created leading to acceleration of the trailing bunch up to 22.4 GeV, with an energy transfer efficiencies of 65 percent. It is also established that injecting the electron bunches into a negative density gradient of ten-fold decreasing density over 10 cm long plasma, results in spatially more spread and two-and-half smaller electric fields ($-1.0 times 10^{10}$ V/m), leading to a weaker acceleration of the trailing bunch up to 21.4 GeV, with an energy transfer efficiencies of 45 percent. Inclusion of ion motions into consideration shows that in the plasma wake ion number density can increase over few times the background value. It is also shown that transverse electromagnetic fields in plasma wake are of the same order as the longitudinal (electrostatic) ones.
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