What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1....
Transcript of What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1....
![Page 1: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/1.jpg)
What We Need to Know About ElectronsWhat We Need to Know About Electrons
1. Electrons are moving charged particles therefore they are magnetic.
2. Electrons move in two ways –
a. Spin on their axis – given quantum number S
b. Orbit the nucleus – given quantum number L
Can define a total angular momentum – quantum number J
J = S + L
In simplest case L can be neglected as electron is an orbitally non-degenerate state. Therefore can consider “Spin Only” behaviour.
Spin only is a good approximation for some transition metal ions:
Ni2+ in an octahedral field; high spin Mn2+ or Fe3+; octahedral Cr3+ or Mn4+; Cu2+ and Mn3+ as Jahn-Teller distorted.
Terrible approximation for others – e.g. octahedral Co2+
![Page 2: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/2.jpg)
Curie LawCurie Law
1. As single electrons are magnets, if you place them in a magnetic field they’ll align with the field. However the energy difference between aligned with field and against field is << thermal energy at room temp.
Get random orientation – equal populations of alignment with/against field.
2. As you lower T, energy difference becomes more important and population changes – more align anti-parallel to the field.
3. To explain this behaviour Curie invented a parameter – called “Magnetic Susceptibility”, χ, – which is a measure of how attracted a sample is to a magnetic field. Normally measured as an apparent mass increase. As more electrons align anti-parallel to the field at low temperature, χ increases. In fact χ is inversely proportional to field: this is the Curie Law
1/χ = CTC = “The Curie Constant”
![Page 3: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/3.jpg)
Curie Law PlotsCurie Law Plots
Slope = C
![Page 4: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/4.jpg)
The Curie Constant and Magnetic MomentThe Curie Constant and Magnetic MomentCurie Law: 1/χ = CT
The molecular information is in the slope of the line,
i.e. C – “the Curie constant”. C = χT
Traditionally this is converted into “magnetic moment”, µµ = (8C)1/2 – 8 comes from a bunch of constants
Also: µ = g[J(J+1)]1/2 which becomes
µSO = g [S(S+1)]1/2 if L = 0
For systems where unpaired electrons don’t communicate to each other (magnetically dilute) µ is a good parameter as the Curie Law applies.
For systems where unpaired electrons communicate (magnetically non-dilute) µhas little meaning as the Curie Law is not obeyed:
Reason: 1/χ vs. T is not a straight line, therefore a parameter derived from the slope, C, is meaningless
![Page 5: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/5.jpg)
χχTTC and µ are pretty redundant. If we combine all the previous equations we get:
χT = C = µ2/8 = [g2S(S+1)]/8
This is much better because it relates the measured parameters directly to molecular properties.
If there is more than one spin centre, you simply include that as a parameter giving:
χT = g2 n. S (S+1)8
Where n is the number of centres. If there is more than one type of centre or spin level, it is easy to adjust:
χT = ga2 na. Sa (Sa+1)
8+ gb
2 nb. Sb (Sb+1)8
![Page 6: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/6.jpg)
χχT T –– Simple ExamplesSimple Examples
χT = g2 n. S (S+1)8
Take three systems containing five electrons (assume g = 2):
≡ Anti-ferromagnetic
≡ Ferromagnetic
Spin pairing favoured
Spin pairing disfavoured
Non interacting
χT = 3/8χT = 35/8χT = 15/8
n = 1, S = 1/2n = 1, S = 5/2n = 5, S = ½
One L.S. Fe3+One H.S. Fe3+Five Ti3+
![Page 7: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/7.jpg)
Less Simple Example Less Simple Example –– A Copper Dimer(1)A Copper Dimer(1)
Put unpaired electrons on different metal centres, e.g. two Cu2+
ions. Let them interact. Get two possibilities:
1. Align parallel i.e. S = ½ + ½ = 1
2. Align anti-parallel, i.e. S = ½ - ½ = 0
YOU ALWAYS GET BOTH CASES. QUESTION IS WHICH IS
GROUND STATE
![Page 8: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/8.jpg)
Less Simple Example Less Simple Example –– A Copper Dimer(2)A Copper Dimer(2)
Assume S = 1 and S = 0 have same energy, then use:
χT = ga2 na. Sa (Sa+1)
8+ gb
2 nb. Sb (Sb+1)8
Complication – must allow for degeneracy of states. Degeneracy given by 2S+1. If S = 1 and S = 0 have same energy, three times as many molecules will be in the S = 1 state as S = 0.
χT = 1/4 1.ga2 0 (0+1)8
+ 3 gb2 1 (1+1)
8{ }χT = 3g2/16
If we had two non-interacting S = ½ then
χT = g2 n. S (S+1)8
χT = 3g2/16
At high T often find χT ≈ value for non-interacting spins. This is not because there is no interaction – simply says interaction energy < kT
![Page 9: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/9.jpg)
Less Simple Example Less Simple Example –– A Copper Dimer(3)A Copper Dimer(3)
If S = 0 has lowest energy then:
With S = 0, i.e. χT = 0
At low T will always find χT tends to 0 if S = 0 is the ground state because at very low temperature interaction energy >kT
χT = g2 n. S (S+1)8
S = 0
S = 1 (threefold degenerate)
At low T Boltzmann gives all population down here
χT = ga2 na. Sa (Sa+1)
8+ gb
2 nb. Sb (Sb+1)8
becomes
![Page 10: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/10.jpg)
Less Simple Example Less Simple Example –– A Copper Dimer(4)A Copper Dimer(4)
If S = 1 has lowest energy then:
With S = 1, i.e. χT = g2.1.(2)/8 = g2/4
At low T may find χT tends to g2/4 if S = 1 is the ground state because at very low temperature interaction energy >kT
χT = g2 n. S (S+1)8
S = 1 At low T Boltzmann gives all population down here
S = 0
![Page 11: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/11.jpg)
Less Simple Example Less Simple Example –– A Copper Dimer(5)A Copper Dimer(5)Summarise:
1. For two Cu2 ions if no interaction then χT = 3g2/16 - this is likely to be the limiting value at high temperature
even if there is an interaction
2. At low T the value of χT will tend to 0 if S = 0 is the ground state- this is the case if anti-ferromagnetic exchange is the major
interaction, i.e. aligning neighbouring spins anti-parallel
3. At low T the value of χT will tend to g2/4 if S = 1 is the ground state- this is the case if ferromagnetic exchange is the major
interaction, i.e. aligning neighbouring spins parallel
S = 1
S = 0
Nomenclature
= 2J or J
Sane version:
J negative = AF
J positive = F
![Page 12: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/12.jpg)
Treating Raw DataTreating Raw DataNot said anything about experiment – and not going to (until we have a SQUID).
Experimental data: from a SQUID will normally get a column of temperature vs“long moment” at a specific field (also a whole pile of columns you don‘t need).
3.274861e-0031.003232e+0011.000000e+0027.022155e-0043.206844e+0011.000000e+0024.590345e-0046.241134e+0011.000000e+0023.642680e-0049.020821e+0011.000000e+0022.952116e-0041.198679e+0021.000000e+002
Long Moment (EMU)
Temperature (K)Field (Oe) χ comes from long moment. Very simple equation:
χ = (Long moment/field)(mass of sample/MW)
[You then need to add a diamagnetic correction – calculated from Pascal’s constants (Rubber Handbook and O’Connor review).]
Most common problem – getting MW correct. If there’s solvent and you don’t allow for it, χ will seem low. If you include solvent and it ain’t there, χ will be high.
Diamagnetic correction vital for weakly paramagnetic samples (e.g. Cu2+
complexes); less vital for strongly paramagnetic samples (e.g. Fe3+).
Result – “absolute” values are more likely to be wrong than shape of curves
![Page 13: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/13.jpg)
Plots for AF Copper Plots for AF Copper DimerDimer -- χχTT
0 50 100 150 200 250 300
0.0
0.2
0.4
0.6
0.8
ChiT
T
J = 0J = -10J = -50
J = -100
J = -200
Obvious: decline in χT faster for larger AF J value
Less obvious: lack of maximum makes fitting data imprecise. Also, as χT zero at low T, fit will be very sensitive to impurities.
![Page 14: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/14.jpg)
0 100 200 300
0.00
0.02
Chi
T
Plots for AF Copper Plots for AF Copper DimerDimer -- χχ
J = 0
J = -10
J = -50 J = -100J = -200
Maximum in χ arises because Curie law says χ rises as T falls, while energy levels means S = 1 becoming depopulated.
When fitting for AF system should fit χ vs T first, as the maximum is very sensitive to value of J for moderate J.
(For large J fit will always be approximate).
![Page 15: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/15.jpg)
Plots for F Copper Plots for F Copper DimerDimer -- χχ
Plots shown for J = 0, +10, +20, +50, +200
Virtually indistinguishable!
Plot of χ vs T useless for getting size of J (or even telling F exchange from no interaction).
0 100 200 3000.00
0.02
0.04
0.06
0.08
0.10
Chi
T
![Page 16: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/16.jpg)
Plots for F Copper Plots for F Copper DimerDimer -- χχTT
0 50 100 150 200 250 3000.8
0.9
1.0
1.1
ChiT
T
J = 0
J = 20J = 10
J = 200
J = 50
χT very dependent on J: still no maximum so it will be more difficult to get a precise J cf. moderate negative J value.
![Page 17: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/17.jpg)
Why AF or F for Copper?Why AF or F for Copper?Both F and AF exchange interactions are observed for copper(II). One classic piece of magnetochemistry (due to Hatfield) demonstrated what controls this for hydroxide bridged dimers:
Cu
OH
Cu
OH
J value depends on Cu-O-Cu angle:
Angle > 98.4°, J > 0 i.e. F
Angle < 98.4°, J < 0 i.e. AF
![Page 18: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/18.jpg)
Why AF or F for Copper?Why AF or F for Copper?Easily explained from MO picture:
Single upe on Cu(II) is in dx2-y2 orbital – which has a lobe on the Cu...O vector.
Imagine O uses a p-orbital to bond to each Cu(II) centre.
Cu
OH
Cu
OH
x
y If Cu-O-Cu angle = 90° then:O-atom will bind to one Cu with px orbital and the other with py orbital.
![Page 19: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/19.jpg)
Why AF or F for Copper?Why AF or F for Copper?
Cu
OH
Cu
OH
x
y Look at bonding: get two pairs of bonding/anti-bonding orbitals:
dx2-y2
px, py
Add electrons: 2 from each O orbital + 1 from each dx2-
y2, get 6.
Feed into orbitals gives two upe in ground state, i.e. F exchange
![Page 20: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/20.jpg)
CuOH
Cu
Why AF or F for Copper?Why AF or F for Copper?x
y
Change angle to ca. 180°
Now O-atom only uses one orbital. Energy levels become:
dx2-y2py
Only need include 4 electrons (2 from O and 1 each Cu): gives 0 upe, i.e. AF exchange
![Page 21: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/21.jpg)
Why AF or F for Copper?Why AF or F for Copper?M.O. picture tells us about extremes: doesn’t tell us where F AF transition should occur.
First part (F exchange) is a specific case of the orthogonalityprinciple: if the unpaired electrons are in orthogonal orbitals, then the exchange will be ferromagnetic.
Angle of 98.4° only has importance for planar hydroxide bridged copper(II) dimers.
Not uncommon for assumption to be made that this cross-over applies to other systems – it might, but that will be coincidence.
Does appear to be common that the more obtuse the angle, the stronger the AF exchange found.
![Page 22: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/22.jpg)
Bigger Clusters(1)Bigger Clusters(1)The bigger the cluster the more complicated the magnetic properties.
The basic interpretation (F or AF) is normally fairly simple.
First complication: AF exchange does not always lead to S = 0 ground state
For example – consider a linear trinuclear cage with S spin on each ion:
Ferromagnetic exchange.
SGS = 3S
Anti-ferromagnetic exchange.
SGS = 1S
Ferro’ exchange will always give a larger SGS than AF exchange, but AF exchange can give very large SGS if the shape is correct.
![Page 23: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/23.jpg)
Bigger Clusters(2)Bigger Clusters(2)For example:
Ferromagnetic exchange.
SGS = 7S
Anti-ferromagnetic exchange.
SGS = 5S
Most high spin cages involve AF exchange as the predominant interaction: e.g. all Mn12 derivatives, Fe19, Fe14, Cr12
![Page 24: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/24.jpg)
Bigger Clusters(3)Bigger Clusters(3)Second complexity: more than one exchange path, therefore more than one J value (in general)
Separating the two components can be difficult. Can normally work out sign of stronger interaction.
e.g. have centre-edge (black) and edge-edge (red) interactions
![Page 25: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/25.jpg)
Bigger Clusters(4)Bigger Clusters(4)
Notice: black interaction overcomes red and aligns spins parallel. Does not matter whether red is AF or F.
Suppose black interaction strongly AF.
Suppose red interaction weakly AF.
Result SGS = 5S
![Page 26: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/26.jpg)
Bigger Clusters(5)Bigger Clusters(5)
Problem: ring couples to give Sring = 0, but centre wants to be coupled AF to two different spins. Central spin is “frustrated”.
Result: probably ground state is S.
Suppose red interaction strongly AF.
Suppose black interaction weakly AF.
Result SGS = ?
![Page 27: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/27.jpg)
Examples of Bigger Clusters(1)Examples of Bigger Clusters(1)
Ni12 SGS = 121. χT never falls – sign
that exchange is ferromagnetic.
2. χT close to Curie law (red line) – shows coupling very small
3. Low T χT does not reach value for S = 12: even at 1.8 K excited states are occupied. Confirms point 2.
4. Shape of χT curve more important than low T value in showing F or AF exchange.
![Page 28: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/28.jpg)
Examples of Bigger Clusters(2)Examples of Bigger Clusters(2)Fe10 SGS = 11
1. χT starts below value for 10 S = 5/2 ions. Says exchange large and AF.
2. Low T χT plateaus near value for S = 11. Confirms exchange large.
3. Real problem with sample: working out molecular weight. Crystals heavily solvated, but solvent lost during SQUID measurement.
![Page 29: What We Need to Know About Electronsschnack/... · What We Need to Know About Electrons 1. Electrons are moving charged particles therefore they are magnetic. 2. Electrons move in](https://reader034.fdocuments.us/reader034/viewer/2022051917/60099892ec0e885cb136d5cc/html5/thumbnails/29.jpg)
Examples of Bigger Clusters(3)Examples of Bigger Clusters(3)
Fe9 SGS = unknowableAny attempt to assign spin rapidly becomes impossible – and impossible many times!? ?
?