In Au (111) single crystalline as shown in attached file, is there a difference the way electrons are excited when they are excited from perpendicular and diagonal orientation as shown in the figure (attached file). Any help would be much appreciated. 
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Yes, the direction matters.
Take a look at this electronic band structure diagram for gold.
If there was only one spatial dimension (like a wire), the diagram would just have energy vs momentum. In higher dimensions, instead the momentum along certain directions is used to try to "flatten" the band structure onto a diagram we can easily visualize. Gold has a fcc (face centered cubic) crystal structure (which turns out to have a bcc lattice in momentum space), so the letters used (by convention) are $\Gamma$ is the center, and $X,W,K,L,U$ are all symmetry points on the crystal structure (well technically, the first Brillouin Zone in momentum space).
This has a nice description as well as images relating the "real space lattice" and the "momentum space lattice". britneyspears.ac/physics/crystals/wcrystals.htm
$K$ is the line you drew in the direction connecting nearest neighbors.
The other direction you drew I believe doesn't correspond to a nice point on the cube. It would be a direction somewhere between $L$ and $U$ (which unfortunately isn't included in the structure diagram I listed, but I haven't been able to find a diagram that includes those directions as well).
