lecture 6!!! Flashcards
what quantum numbers from last year do we need to think about when we think of the H atom
n
l
ml
we dont need to think of ms bc H only has 1e- and it doesnt rlly matter if its spin up or spin down
what is spin orbit coupling all about
a moving charge can and does generate a magnetic field
the solution to the angular schrodinger equation gives what
l
the angular momentum
it gave us orbitals
and told us that l can have values of 0 –> n-1
but they need to be integers, ,they cant be a decimal plave
what does l and ml tell us about
tells us about the motion of an e-
what is l
the orbitals angular momentum
how does spin affect angular momentum
the more spin,, the more angular momentum
think of it like:
hands stuck out spinning > hands tucked in spinning.
what is angular momentum tho
the amount by which an e- is spinning arounda fixed proton in the middle
what is S
spin angular momentum
what is the value of S and what is s
s is the spin angular momentum
s value = 1/2
it can either spin clockwise or anticlockwise
how many magnetic fields are there inside an atom
theres the orbital movement and the electron spin movement
SO THERES THE SPIN MAGNETIC FIELD AND THE ORBITAL MAGNETIC FIELD
whats cool about the e- spin and orbital magnetic field
they interact with eachother
which magenetic field in an atom is larger
the orbtial magnetic field is larger than the e- spin magnetic field.
how would the 2 magnetic fields like to interact with eachother
they would like to be opposites of one another : low energy arrangement, one spinning one way and the other spinning the other way.
the other arrangement is where theyre both spinning the same way,, this is a high energy arrangement and isnt favoured.
the size of the coupling depends on what
the magnetic field
everything is still only applicable tooo
the H atom
the whole interaction and ml and l is only for H..
so what is spin orbital coupling
the 2 magnetic fields in a atom// H atom: the orbital magnetic field and the e- magnetic field interact with eachother
what is j
the total angular momentum quantim numberrrr
from the orbital(l) and the e- spin (s=1/2)
describes how the l and s magnetic fields interact with eachother
DESCRIBES SPIN - ORBITAL COUPLING
how do we find the j value + describe how and conditions that need to be met
j is the total angular quantum number : so smt to so with the s and l magnetic fields
j = |l+-s|
the lines show that its an absolute value: it means that if the answer comes back negative,, we do the other sign,,, j must be positive
wait so if the j value = | l +- s| when do we add s and when do we subtract s
we add the s to the l value if the orbital and e- spin magnetic fields have a high energy arrangement: aka if theyre both spinning up in terms of the arrows.
we take the s away from the l value if theres a low energy arrangement: aka when the things are spinning in the opposite direction: when one arrow is pointing up and the other arrow is pointing down.
what does j primarily depend on
j depends primarily on the l value
bc the l value can be 0–> ‘n-1’ whereas ‘s’ will always be = 1/2 bc its the electron spin
what is the ground state of a H atom:
- its e- configuration
- its l value
- its s value
- possible j values
1e- only
so 1s^1 is the ground state e- configuration
l = o (bc its 1s)
s = 1/2 bc obvsss its an e-
j = 0 + (1/2)
j = 0 - (1/2)
but the j value is an absolute value,, meaning it has to be positive,, so the only optioon we really have for the total angular momentum quantum number issssss
j = 0 +1/2 = 1/2 or 0.5 if u will 💗 so easy girl u can do this!!!!
if l = 0 ,,, what does this tell ussssssssss
s is the only other magnet
s= 1/2 which is an actual value
l = 0 ,, this isnt an actual value
so if l = 0 this means that there is only the s magnet.
so no 2 magnets = no interaction between the 2 = no spin orbital couplinggg 🤭
the only motion + magnetic field we have is due to the e- spinning ,, not the orbital spinning bc its l = 0. a cirlce which is the same all round.
1st interesting excited state of H
2p1
bc 2s1 is still l=0 and s=1/2
do the whole breakdown of a 2p1 excited state
n = 2 but we can ignore this bc this doesnt twll us about motion 💸
l = 1 = p
s = 1/2 bc its an e- duh
j = |l+- s |
j = 1 + 1/2
j = 1 - 1/2
as they both give positive values j = 1/2 and 3/2
SO THERE ARE 2 MAGNETIC FIELDS BC NOTHING IS EQUAL TO 0,, MEANING THEY INTERACT,, MEANING THERE IS SPIN ORBITAL COUPLING
do the whole breakdown for the 3d1 electron config
n = 3 but it doesnt tll us about motion so we ignore.
l = 2 = d
s = 1/2 bc its an e- duhhh
j = | l +- 1/2 |
j = 2 + 1/2 = 2.5
j = 2 - 1/2 = 1.5
both work as theyre both positive
meaning theres 2 magnetic fields bc theres e- spin motion and orbital motion,, which can interact with eachother to giveeeee SPIN ORBITAL COUPLINGGGGGG
what is meant by fine structure
fine structure describes what a ‘single energy level’ actually is.
describe the fine structure energy graph
y axis = energy
x axis = increasing l value.
increase in ‘n’ as u go up the y axis obvs
s -> p -> d going to the right
the s shells dont split up into high/ low energy bc theres no spin orbital coupling
the p shells split into high and low energy bc p = l = 1 so u can get 1.5 or 0.5 (different energy)
the gaps within the shell in the fine structure energy diagram can be seen,, what does the energy gap depend on thooooo
depends on the j , l and s quantum numbersssss!!!!
the larger j value hasssss what energy
high energy (more unstable)
higher j value bc u added the different magnetic fields meaning they were in the high energy arrangement… both spinning the same way.
how do we get a larger magnetic field
by having a larger atomic number
larger atomic number =
larger nuc charge =
e- is closer to nuc =
e- is going to spin around it faster =
higher magnetic field bc more faster motion!!!!
okay so will H or Cs have a larger magnetic field
the Cs!!!
bc it has a larger atomic mass = larger Z = e- is pulled in more = spins faster around the nuc = more motion = larger magnetic field
= larger gap between the different energy j values. = LARGER SPIN ORBITAL COUPLING VALUE.
allso kinda think of the radius and think that a larger radius will give u a larger magnetttttt. so the j value separation will be largerrr.
higher j value can be written assss
1/2 hcA
A is large for heavy atoms.
lower j value can be written asssss
- hcA
A is smaller for smaller// lighter atoms.
what unit does A have
cm-1
wavenumber vibessssss
how does the value of A change down the group
down a group = heavier the atom = larger the A value.
so u get a larger energy split between the different j values of the same shell.
okay so so far all this is only applicable tooooo
only applicable to the H atommmmm!!
the magnetic fields interacting and the different j values and all this is.
okay so for multielectron atoms what is Y
Y(1,2)
= Y(1) Y(2)
u multiply them together
helium ground state config ,, it has 2e-
1s2
helium excited states
1s1 2s1
1s1 2p1
whats so different between the e- in hydrogen and any other atoms e-
in hydrogen that e- cannot interact with any other e- as there is no other e-.
in multielectron atoms,, the electrons will interactttttttt.
how do we evaluate the total angular momentum when we have a multielectron atom
we use russel saunders coupling !!
the ‘complex’ thing but we’ve got this 🤭
what are the steps involved in russel saunders’ evaluation of j for multielectron atoms
- start with configuration
- l (orbital momentum) = strongest interaction
- s (e- spin momentum)
then the j value + their interactions!!
for a multi electron atom there are many e-, meaning each has its own set of quantum numbers that describe it,, meaning what
each e- in an atom will have its own l value
these group together to give the atoms L value.
okay girl 👩🏫 how what is L again and how do we find it
- for multi electron atoms there are many e- and each will have its own l value (aka angular momentum aka orbital shape)
- so we need to combine these to give ‘L’ which is the l of the whole atom. big L bc lots of little l’s.
to find L:
l1+l2-0,, l1+l2-1,, l1+l2-2,, l1+l2-3,,,,,,,,, until |l1-l2|
,,,, = in steps of 1
L for He ground state
He ground state = 1s2
1s1 and 1s1
gives l1 = 0
gives l1 = 0
bc theyre both in the s shell = l = 0
L = 0+0,,,, |0+0| = 0
if theres no angular momentum in both e- u can make it out of thin air,, so its obs gonna be 0.
so L for He g.s = 0 = S big S bc for the whole atom
if L = 0 what is it called
S big S
bc L = 0 = s orbital
but bc its for the whole atom its capital : S
S state
ground state for the whole He atom is what
its an S state
found using russel saunders
okay how do we find the acc value of S aka the spin of the whole atom
same way as we find the L
but usig 1/2 bc e- can only have spin = 1/2
so [(1/2+1/2)-0], [(1/2+1/2)-1], [(1/2+1/2)-2] ,,,,, |1/2-1/2|
it should acc be s1 + s2 but s will always be 1/2 so do ur thing girl!!
1,0
multiplicity equation
2S + 1
u use the whole atoms spin : S which u find using the ‘in steps of 1’ thing.
if S = 1, 0
2x1+1 = 3 so u have a triplet
2x0+1 = 1 so u have a singlet
singlet S value
s = 0
bc 2S+1
finding J for the whole atom,, notive how we use small j for the e- and big J for the atom : okay so how do we find J for the atom
and what should we not include for He ground state
we use every combo of L and S resultsssss
J = L+S-0 ,, L+S-1,,,,,, |L-S|
L=0,, S=1 is one combo : J=0+1 tilllll |0-1| so 1,,,1 = 1
L=0,, S=0 is another combo
J=0+0 –> |0-0|= 0
do every combooooo of L and S. takes time so dont worry girl .
so possible J values = 1 and 0
He:gs: 1s2: 2 e- in the 1s: u cant use the S=1 bc that means youve added the spins of both e-: u did 1/2 + 1/2 meaning they have the same spin,, which goes against the pauli exclusion principle: so u acc shouldnt invlove S=1 for the combo!! so the J value is only 0 (the J where L=0 and S=0)
aka to find J what equation do we use + when should we not bother with the equation
J = L+S-0, L+S-1, L+S-2,,,,,, |L-S|
u use the values of S and L
u do this whole thing for every possible combination of S and L values!!
we shouldn’t bother when we’re doing the S=1 for He. bc this means the e- go against the pauli exclusion principle!! bc it means they have the same spin.
