19 Magnetic Properties of Atoms.srt

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欢迎大家，这是凝聚态课程的第十九讲
welcome back everyone it's a nineteenth
lecture of the condensed matter course

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快到第六周结束了，我知道很难继续下去了，是你
nearing the end of sixth week I know
it's hard to keep keep going on it's you

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知道六周结束时每个人都很累，但是我们真的越来越
know the end of six week everyone gets
pretty tired but we're really getting

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现在已经接近本课程的结尾了，所以您可以通过打开电源知道
pretty close to the end of this course
now so it's you know power on through so

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当我们上次离开讨论磁学时，我们决定
when we left off last time we were
talking about magnetism we decided a

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一次开始思考磁性的好地方是一次原子
good place to start thinking about
magnetism was one atom at a time so

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真正考虑原子磁性
really thinking about atomic magnetism

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磁性，我们已经写下哈密顿量，实际上是
magnetism and we had written down the
Hamiltonian actually the Hamiltonian for

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一个磁场中的一个原子中有一个电子，我们决定将其写入
one electron in one atom in a magnetic
field and we decided we would write it

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作为零磁场下电子的哈密顿量
as the Hamiltonian for that electron in
zero magnetic field that's just the

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吸引原子核的动能加上两个项，第一个项是
kinetic energy in the attraction to the
nucleus plus two terms the first term is

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玻尔磁子磁场点成轨道角动量加G自旋
Bohr Magneton magnetic field dotted into
orbital angular momentum plus G spin

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角动量和第二项平方超过2m平方，其中a是
angular momentum and the second term a
squared over 2m a squared where a is the

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向量势，其del cross a等于B现在，这两个术语都具有名称
vector potential its del cross a equals
B okay now these two terms have names

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这里的第一个术语称为顺磁术语，这里的这个术语
this first term over here is known as
the paramagnetic term and this term here

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被称为反磁性术语，您可能会猜到这是
is known as the diamagnetic term and as
you might guess it's this term that's

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这个术语通常负责修复磁性
typically responsible repairing
magnetism in this term which is

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通常负责反磁性，这个术语实际上是相当
typically responsible for diamagnetism
and this term it's actually fairly

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很明显为什么这个词要负责权力
obvious why it is that this term is
going to be responsible for power

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磁场通常只是磁场与磁场的耦合
magnetism generally this is just
coupling of the magnetic field to the

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旋转或覆盖电子的动量，现在可以了
spin or overlying the mentum of the
electron and the okay now it's a little

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由于减号而复杂，但自旋要订购
complicated because of the minus signs
but the spin wants to order the

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上覆动量都想指向B相对，使得B的负能量
overlying momentum both want to point
opposite B such that the neg energy of

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这将变为负数，如果自旋Orval动量与B相反， 
this becomes negative and if the spin
Orval a momentum is opposite B that

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表示磁矩与B方向相同，因为
means the magnetic moment is the same
direction as B because the charge of

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电子为负，所以力矩指向与角动量相反的方向
electron is negative so the moment
points opposite of the angular momentum

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所以无论如何还是为了降低能量
so anyway to or in order to lower the
energy the

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 B相对的B线和/或Belanger精神相对的B线旋转，因此
spins line up opposite B and/or Belanger
mentum lines up opposite B and therefore

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当所有在同一方向上排列的那一刻是B并且你有
the moment all lines up in the same
direction is B and you have

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现在您可能会注意到这两个术语的顺磁性是
paramagnetism now one thing you may
notice about these two terms is this

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功率磁铁的磁项在B中是线性的，而该项在b中是二次的
power magnet magnetic term is linear in
B whereas this term is quadratic in b

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和b，因此一般来说，小的是线性的，反过来是功率的磁
and b and so generally for small be the
linear in turn be the power magnetic

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只要没有特别好的理由，这个术语将占主导地位
term is going to dominate as long as
there is not a particularly good reason

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为什么功率磁项将被抑制或以某种方式甚至为零，那么
why the power magnetic term is going to
be suppressed or somehow even zero then

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反磁术语是无关紧要的，我们只需要担心功率
the diamagnetic term is irrelevant and
we just have to worry about the power

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具有磁性的术语，我们会发现该术语为0或被抑制的情况
magnetic term we will find occasions
where this term is 0 or is suppressed

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然后反磁项将有机会主导物理学，但我们
and then a diamagnetic term will have a
chance to dominate the physics but we're

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首先要专注于这一点，它将变得更加重要
going to focus on this one first
figuring it's going to be more important

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所以让我们用这个学期，我们要做的就是
so let's a take this term and what we're
going to want to do is we're going to

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现在想稍微简化一下，我们可以做的一件事就是可以将它们加在一起
want to simplify it a little bit now one
thing we can do is we can add together

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轨道角动量的角动量求和
the angular momentum of the orbital
angular momentum to get the total

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角动量J是L加Sigma，但您会注意到，这不是我们
angular momentum J is L plus Sigma but
you'll notice that's not what we

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实际上在哈密顿方程式中，我们有L加G Sigma现在您可能已经了解到
actually have in the Hamiltonian we have
L plus G Sigma now you may have learned

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在您或您应该已经在原子物理学课程中学习到
in your or you should have learned in
your atomic physics course that under

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您知道J的大小L的大小
conditions where you know the magnitude
of J the magnitude of L the magnitude of

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 Sigma您可以用一些有效的G因子G重写整个术语
Sigma you can rewrite that entire term
in terms of some effective G factor G

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有效的称为Al和AG因子，通常用上标j 
effective known as Al and AG factor
usually written with a superscript j mu

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 B BJ并且有一些公式可以让您获得G为有效G 
B BJ and there's some formula that
allows you to get the G the effective G

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从G和J的量级到L的量级的因子
factor from G and the magnitude of J and
the magnitude of L in the magnitude

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西格玛（Sigma），如果您知道这些，则该公式不是本课程的一部分，但我相信
Sigma if you know these that formula is
not part of this course but I believe

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这是您另一门课程的一部分，所以如果您要参加，您可能应该知道
it's part of your other course so you
probably should know it if you're taking

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您将不会被要求在原子物理学课程中推导它
the atomic physics course you won't be
asked to derive it in this course you

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在其他过程中，可能会被要求以任何比率得出道德是您
may be asked to derive in the other
course any rate the moral is that you

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可以总结一下，您可以将轨道和自旋角动量加在一起
can just sum you can put together the
orbital and spin angular momenta and

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在您知道JL和Sigma是什么的情况下，您只需想到
under the conditions that you know what
JL and Sigma are you can just think of

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这是一个在磁场中重新定向的单个角动量矢量， 
this as a single angular momentum vector
that reorients in a magnetic field and

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我们的条款可以重置可以总结
the
our term can be reset can be summarized

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正因为如此，这是哈密顿主义下的一个相当简单的术语
as as this so it's a fairly simple term
in the Hamiltonian under those

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我们可以将其视为单个总角动量的条件
conditions where we can just think of it
as a single total angular momentum being

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通过磁场重新定向，但实际上这个故事要多一些
reoriented by the magnetic field but in
fact the story is a little more

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比这复杂，比这更复杂，因为我们有
complicated than this and it's more
complicated than this because we have

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原子中有很多电子原子中不仅有单个电子
many electrons in an atom many an
electrons in atom not just a single

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电子，这意味着我们必须跟踪整个角动量
electron and that means we have to keep
track of the entire angular momentum

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大写L是轨道角的所有电子的总和
capital L that's the sum over all the
electrons of the orbital angular

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每个电子的动量，然后是每个电子的总自旋角动量
momentum of each electron and then the
total spin angular momentum of each of

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电子，当然总的角动量就是总的
the electrons and of course the total
angular momentum altogether is the total

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轨道加上总旋转，您可能会从冒险中获悉
orbital plus the total spin and as you
probably learned from your adventures in

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 clebsch-gordon系数，您可以使用非常复杂的方法进行相加
clebsch-gordon coefficients that you can
have very complicated ways of adding

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将角动量（例如两个自旋1/2扫描）相加在一起
together angular momenta for example to
two spin 1/2 scan add together to make a

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单重态或三重态，然后事情会变得非常混乱，如果您有20 
singlet or triplet and then things get
really messy and if you have 20

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您原子中的电子可以加在一起，也可以通过两种疯狂的方式加在一起
electrons in your atom they can add
together and also two crazy ways and so

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我们需要一些东西来指导我们弄清楚这些电子是怎么回事
we need something to guide us to figure
out how it is that these electron

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角矩将要加起来，我们必须指导我们的第一件事是
angular momenta are going to add up
first thing that we have to guide us is

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原子的壳结构，所以让我们回想起
the shell structure of the atom so let's
remember back to the beginning of the

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术语壳结构，我们有两个定律帮助我们解决了第一个问题： 
term shell structure we had two laws
that helped us out the first was the

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 Aufbau原理，基本上是弓形，我相信在德国意味着建筑
Aufbau principle which basically a bow
which in german i believe means building

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或构造它意味着您一次填充一次炮弹，然后第二次填充
up or construction it means you fill up
shells one at a time and the second

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我们所遵循的原则是玛德隆的玛德隆规则，该规则告诉我们
principle we had was madelung's
madelung's rule which told us in which

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我们应该开始填充填充壳，这样可以帮助我们
order we should start filling filling
shells so that's going to help us out a

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很多，对我们有很大帮助的原因是， 
lot and the reason that's going to help
us out a lot is because there's an

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填充壳的重要原理
important principle that a filled shell

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 l等于s等于j等于零填充时完全没有角动量
has l equals s equals j equals zero no
angular momentum at all in a filled

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壳为什么这么好，我们使每个轨道都充满了自旋和每个轨道
shell why is that well we filled every
orbital with a spin up and every orbital

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向下旋转，则总旋转为零，向上旋转的次数等于
with spin down so the total spin is zero
the number of ups is equal to the number

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关于ELL的表现如何，我们进行了自旋
of down
how about ELLs well we spin we've filled

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所有lz +状态，我们将所有lz-状态均等地填充到填充的外壳中
all the lz+ states and we equally filled
all the lz- states in in a filled shell

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所以L Z彼此抵消，Lu最终退出的总数为零
so the L Z's all cancel each other and
the total Lu end up getting out is zero

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所以填充的壳根本没有角动量，这意味着我们仅
so a filled shell is has no angular
momentum at all and that means we only

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只看填充壳的部分好看填充壳的部分
only look at part filled shells look at
part filled shells okay so that's going

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对我们有很大帮助，但您仍然可以将一部分装满三个
to help us out a lot but still you can
have a part filled shell with three four

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里面有五个六七个电子和一个部分充满的劳特壳（如果是） 
five six seven electrons in it and a
partially filled laut shell if it's a

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大壳，您可以知道部分填充的壳中有13 14个电子，因此
big shell you can have you know 13 14
electrons in a partially filled shell so

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幸运的是，这仍然可能非常复杂。 
this could still be pretty complicated
fortunately we have a set of rules that

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帮助我们弄清楚这些有角度的Mendte是如何相加的，这些是已知的
helps us figure out how these angular
Mendte add together and these are known

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匈奴继弗雷德里克·洪德之后匈奴的统治
as huns rules
Hun's rules after frederique hund and he

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早在1925年的第一年就将它们写下来了
wrote them down actually first 1925 a
very long time ago back in the early

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量子力学的日子，人们正在学习
days of quantum mechanics just when
people were learning about shells of

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现在我可能不知道Zots上的原子在德语中是什么意思
atoms now I may not know what on Zots
means in German I may not know what

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 alpha在德语中是指，但我知道引擎盖在德语中是指狗，所以无论如何
alpha means in German but I know that a
hood means dog in German so whatever

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这是值得的，这是狗的规则，但嗯，所以规则1规则1电子自旋
that's worth these are the dogs rules
but um so rule 1 rule 1 electron spins

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如果可以的话，将一条直线旋转一条直线，换句话说，您可以最大化所有
spins a line a line if they can if they
can in other words you maximize s all

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对，所以这实际上是一个非常好的规则，因为它有点像是
right so this is actually a very nice
rule because it's kind of like being a

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铁磁体中的所有自旋线都可以使您最大化
ferromagnet in a ferromagnet you get all
your spins to a line and you maximize

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总的自旋不是铁磁体，因为我们只是在说
the total spin it's not really a
ferromagnet because we're just talking

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这里只有一个原子，所以它只是有限数量的电子，而不是整个
about a single atom here so it's just a
finite number of electrons not a whole

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你知道整个固体电子很大，但至少具有相同的定性

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物理学所以你能理解为什么电子自旋会想要
physics so can you understand why it is
that the electron spins would want to

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好吧，好吧，为什么为什么要把规则1做好第一件事不是因为
align well ok so why why rule 1
well the first thing it is not because

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波兰pol相互作用的偶极-偶极相互作用
of Dai Poland i pol interactions
dipole-dipole interactions between the

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自旋为什么不可以呢自旋确实有偶极矩和偶极矩
spins why not yes the spins do have
dipole moment and the dipole moments

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会相互作用，但如果您尝试计算偶极子相互作用
will interact but if you try to
calculate the dipole dipole interaction

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两个偶极子之间的强度，其偶极矩为玻尔磁子
strength between two dipoles whose
dipole moment is a Bohr Magneton in a

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单个原子太小了，它是如此之小，您可以完全将其扔掉
single atom it's insanely small it's so
small you can just completely throw it

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完全无关紧要，因此保持它重要不是因为这个
out totally irrelevant so it's not
because of this that's important to keep

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请记住，这是由于库仑相互作用所致。 
in mind what it is due to is due to
Coulomb interaction due to Coulomb let's

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看看我们是否能弄清楚为什么库仑相互作用会关心电子
see if we can figure out why Coulomb
interaction would care if the electron

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自旋对齐或对齐不正确，因此如果打开大多数固态
spins were aligned or not aligned well
so if you open up most solid state

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物理书籍或大多数形式的物理书籍，通常会告诉您一些
physics books or most physics books of
any sort they usually tell you some

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关于为什么电子自旋想要对准以及为什么是这样的故事
story about why it is that electron
spins want to align and why it is that

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在这方面服从人的角色，故事在质量上是对的
one's role is obeyed in this respect and
the story is sort of qualitatively right

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但这有很多问题，所以这是一个警告，我想我给了
but it has a lot of things wrong with it
so this is a warning I think I gave a

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00:10:03,240 --> 00:10:07,050
本书中的行为有很多警告，因此您可以阅读我们将要详细介绍的内容。 
lot of caveats in the act in the in the
book so you can read the details we'll

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00:10:07,050 --> 00:10:10,589
先讲讲这个故事我先讲童话
go through this a little bit tell the
story first I'll tell the fairy tale

124
00:10:10,589 --> 00:10:14,040
首先，我将向您解释为什么童话故事不完全正确
first then I'll explain to you why the
fairy tale isn't exactly right to have a

125
00:10:14,040 --> 00:10:17,970
更好地了解实际发生的事情，以便弄清楚为什么会这样
better idea of what actually is going on
so in order to figure out why it is that

126
00:10:17,970 --> 00:10:21,839
库仑相互作用很重要，让我们回想一下您是否具有两个波函数
Coulomb interaction cares let's recall
if you have a wavefunction for two

127
00:10:21,839 --> 00:10:27,180
波动函数的电子应分解成轨道部分
electrons that wave function should be
decomposed into the orbital part which

128
00:10:27,180 --> 00:10:32,839
取决于两个原子的位置对不起两个电子和自旋部分
depends on the position of the two atoms
sorry two electrons and the spin part

129
00:10:32,839 --> 00:10:38,550
现在取决于两个电子的自旋
which depends on the spins of the two
electrons now the overall wave function

130
00:10:38,550 --> 00:10:46,319
必须是反对称的，因为电子是费米子，所以这告诉我们
must be anti symmetric because electrons
are fermions and so that tells us that

131
00:10:46,319 --> 00:10:53,550
如果Chi自旋等于使两个自旋指向相同的方向（如向上） 
if Chi spin equals having both spins
pointing in the same direction like up

132
00:10:53,550 --> 00:10:58,829
如果自旋对齐，则上下或同时指向侧面，这是
up down down or both pointing sideways
if the spins are aligned then this is

133
00:10:58,829 --> 00:11:04,440
这是对称的然后是轨道部分
this is then symmetric symmetric then
the orbital part

134
00:11:04,440 --> 00:11:13,350
叹息球是反对称的，这意味着如果我们将叹息轨道写为a 
sigh orb is anti-symmetric what that
means is if we write sigh orbital as a

135
00:11:13,350 --> 00:11:22,680
 R 1减去R 2的函数，我们取R 1至R 2，波动函数必须为0 
function of R 1 minus R 2 we take R 1 to
R 2 the wave function has to go to 0

136
00:11:22,680 --> 00:11:27,810
因为如果采用反对称函数，然后采用
because the if you take an
anti-symmetric function and you take its

137
00:11:27,810 --> 00:11:31,620
函数的参数必须为0，反对称必须为0 
argument to 0 the function must go to 0
anything anti-symmetric has to go

138
00:11:31,620 --> 00:11:39,060
在0处通过0表示什么，这意味着电子不能
through 0 at 0 so what does that mean
what that means is that electrons cannot

139
00:11:39,060 --> 00:11:44,630
彼此靠近不能靠近
get close to each other cannot get close

140
00:11:46,040 --> 00:11:54,390
彼此靠近，您可能会认为库仑相互作用会在乎
close to each other and you might think
that the Coulomb interaction would care

141
00:11:54,390 --> 00:11:57,630
关于这一点，的确是通常讲的故事，所以这个故事
about that and indeed that's the story
that's usually told so the story goes

142
00:11:57,630 --> 00:12:01,980
如果自旋对齐，则有点像这样，因为
kind of like this if the spins are
aligned then because of the symmetry of

143
00:12:01,980 --> 00:12:06,060
波函数使电子无法彼此靠近，这使得
the wave function the electrons can't
get close to each other and that makes

144
00:12:06,060 --> 00:12:09,180
库仑互动很高兴，因为库仑互动不希望
the Coulomb interaction happy because
the Coulomb interaction doesn't want the

145
00:12:09,180 --> 00:12:12,570
电子彼此靠近，因此能量自然降低
electrons to get close to each other
either so the energy is naturally lower

146
00:12:12,570 --> 00:12:17,130
因为电子彼此之间保持更远的距离，所以
because the electrons are staying
farther apart from each other ok it's

147
00:12:17,130 --> 00:12:21,320
几乎是正确的，但是它不是真的正确，并且它不是真的正确的原因是
almost right but it's not really right
and the reason it's not really right is

148
00:12:21,320 --> 00:12:34,190
因为更重要更重要的是电子原子核
because really more important more
important is the electron nucleus

149
00:12:34,190 --> 00:12:42,030
原子核原子核不是电子，而是一种非常
nucleus nucleus not the electron
electron and it's it's sort of a very

150
00:12:42,030 --> 00:12:46,680
电子核与电子之间的细微差异是因为
subtle difference between electron
nucleus and electron electron because

151
00:12:46,680 --> 00:12:50,490
到最后都是库仑的互动，但让我尝试解释一下
it's all Coulomb interaction at the end
of the day but let me try to explain

152
00:12:50,490 --> 00:12:58,380
我的意思是，所以让我们想象一下，我们在这里有一个核，然后我们有两个
what I mean by this so let's imagine we
have a nucleus here and then we have two

153
00:12:58,380 --> 00:13:02,610
自旋的自旋是反对齐的，所以因为自旋是反对齐的，所以它们
spins who spins are anti aligned so
because the spins are anti aligned they

154
00:13:02,610 --> 00:13:07,200
可以彼此靠近，特别是一个电子可以进入
can get close to each other and in
particular one electron can get in

155
00:13:07,200 --> 00:13:11,520
原子核中另一个电子之间的关系现在可以记得我们去的时候
between the other electron in the
nucleus ok now remember when we went

156
00:13:11,520 --> 00:13:15,240
回来，我们谈到了电负性和电离能
back and we talked about
electronegativity and ionization energy

157
00:13:15,240 --> 00:13:18,209
当一个电子可以进入
when one electron can get inside the
orbit of the

158
00:13:18,209 --> 00:13:23,459
其他的它可以屏蔽核，可以屏蔽来自核的电子， 
other it can screen the nucleus it can
screen the electron from the nucleus and

159
00:13:23,459 --> 00:13:28,079
使有效核电荷看起来更小，结果是
make the effective nuclear charge look
smaller as a result the electron on the

160
00:13:28,079 --> 00:13:37,170
外面的约束力弱，因为它看到的有效核电荷较小
outside is weakly bound because it sees
a smaller effective nuclear charge on

161
00:13:37,170 --> 00:13:41,759
另一方面，如果对齐电子的自旋，则电子将无法
the other hand if you align the spins of
the electrons then the electrons can't

162
00:13:41,759 --> 00:13:45,119
彼此靠近，尤其是这个电子不能
get close to each other and in
particular this electron cannot get

163
00:13:45,119 --> 00:13:51,689
在这个电子轨道内部，它无法从原子核中屏蔽掉这个电子
inside this electrons orbit and it can't
screen this electron from the nucleus so

164
00:13:51,689 --> 00:14:01,949
结果，电子被牢固地束缚，这是更多的原因
as a result the electrons are strongly
bound and this is a more of the reason

165
00:14:01,949 --> 00:14:08,220
为什么将两个电子自旋对齐会降低能量
of why it is that the it's lower energy
to have the two electron spins aligned

166
00:14:08,220 --> 00:14:11,249
因为然后两个电子牢固地结合到原子核上
because then the two electrons are
strongly bound to the nucleus and that

167
00:14:11,249 --> 00:14:16,889
降低他们的精力，所以这是hons第一个统治的诚实原因
lowers their energy okay so that's more
of the honest reason why hons first rule

168
00:14:16,889 --> 00:14:22,769
适用，所以我们甚至可以概括匈奴
applies okay
so let's we can even generalize huns

169
00:14:22,769 --> 00:14:27,139
如果我们想要的话，可以从原子到分子滚动，假设我们有一个分子
roll-up from atoms to molecules if we
want so suppose we have a molecule

170
00:14:27,139 --> 00:14:35,249
分子假设我们有一个带有两个正核的分子，说两个
molecules suppose we have a molecule
with two positive nuclei and say two

171
00:14:35,249 --> 00:14:41,160
如果两个电子自旋对齐，则这里的电子完全相同。 
electrons up here it's exactly the same
physics if the two electron spins align

172
00:14:41,160 --> 00:14:45,480
那么电子必须彼此分开而不能屏蔽
then the electrons have to stay apart
from each other and they can't screen

173
00:14:45,480 --> 00:14:50,369
互相从核中旋转，而自旋反对齐，则它们可以
each other from the nucleus whereas the
spins are anti aligned then they can get

174
00:14:50,369 --> 00:14:54,809
彼此之间，他们可以筛选核，最后结合
inside of each other and they can screen
the nucleus and in the end the binding

175
00:14:54,809 --> 00:15:01,709
较弱，因此洪德法则希望旋转是反对齐洪德法则仍然
is weaker so hund's rule wants the spins
to be anti align the hund rule still

176
00:15:01,709 --> 00:15:06,470
想要旋转甚至在分子中对齐
wants spins to align even in a molecule

177
00:15:07,519 --> 00:15:12,809
但这必须与其他事物竞争，而另一事物必须
but that has to compete with something
else and the other thing that has to

178
00:15:12,809 --> 00:15:23,670
与之竞争的是共价键，债券希望事前对齐，事前对齐
compete with is a covalent bonding bond
wants ante aligned ante aligned what's

179
00:15:23,670 --> 00:15:27,749
我们称我们在两个轨道上拥有共价键时学到的东西
we call what we learned about covalent
bonding when we had orbitals on two

180
00:15:27,749 --> 00:15:32,010
原子都具有能量e而不是当原子彼此靠近时它们形成
atoms both with energy e not when the
atoms come close together they form

181
00:15:32,010 --> 00:15:36,270
一个键合和反键合的轨道，我们想将两个电子都放入
a bonding and anti-bonding orbital and
we want to put both electrons in the

182
00:15:36,270 --> 00:15:40,710
反结合轨道，我们必须反对准它们的自旋以形成单线态， 
anti-bonding orbital we have to anti
align their spins to make a singlet and

183
00:15:40,710 --> 00:15:44,850
使它们都处于较低的能量状态，因此Hunza和
put them both in a lower energy state so
there's a competition between Hunza and

184
00:15:44,850 --> 00:15:49,110
分子是共价键物理与洪德规则之间的竞争
molecules is a competition between
covalent bonding physics and hunds rules

185
00:15:49,110 --> 00:15:52,950
物理学中的一个想要旋转以对齐其中一个想要旋转以
physics one of them wants the spins to
align one of them wants to spins to

186
00:15:52,950 --> 00:15:56,010
反对齐，所以要弄清楚一个分子要复杂得多
anti-align so it's a lot more
complicated in a molecule to figure out

187
00:15:56,010 --> 00:16:00,330
如果旋转对齐，或者如果不对齐，那么您应该知道那两个
if spins align or if they don't so when
you should just be aware that those two

188
00:16:00,330 --> 00:16:05,280
互相竞争好吧，这是一个人的第一法则
are competing with each other okay all
right so that's one's first rule it

189
00:16:05,280 --> 00:16:08,940
告诉我们，如果有壳的话，自旋会尽量对齐
tells us as the spins will try to align
if they can so if you have a shell

190
00:16:08,940 --> 00:16:13,680
小于1/2填充，您可以将所有电子指向相同
that's less than 1/2 filled you can put
all of the electrons pointing the same

191
00:16:13,680 --> 00:16:17,280
方向当您充满时，您必须开始将电子放入
direction when you get to have filled
you have to start putting electrons in

192
00:16:17,280 --> 00:16:20,490
向相反的方向，因为您已经用所有
in the opposite direction because you've
filled all the states with all the

193
00:16:20,490 --> 00:16:23,970
电子方向相同，但在一定程度上可以
electrons in the same direction but to
the extent that they can the electrons

194
00:16:23,970 --> 00:16:29,550
现在我们将始终尝试统一，说我们仍然必须弄清楚会发生什么
will always try to align now that said
we still have to figure out what happens

195
00:16:29,550 --> 00:16:36,270
到轨道的自由度，所以规则2非常相似，基本上是L 
to the orbital degree of freedom so rule
2 is very similar which is basically L

196
00:16:36,270 --> 00:16:49,500
被最大化以服从规则1到规则1，这更重要，所以让我们
is maximized to subject to rule 1 to
rule 1 which is more important so let's

197
00:16:49,500 --> 00:16:53,670
看看我们是否可以举一个例子，这可能是解释这个问题的最简单方法
see if we can do an example of this
probably the easiest way to explain this

198
00:16:53,670 --> 00:16:59,130
是为了展示它是如何工作的，所以让我们考虑一个亚当的舞台
is to show how it works so let's
consider a proscenium Adam this is

199
00:16:59,130 --> 00:17:08,190
原子数原子数是59邪恶的59，在56中等于填充
atomic number atomic number is 59
evil's 59 which in 56 equals filled

200
00:17:08,190 --> 00:17:12,150
炮弹，我认为它充满了炮弹，持续了6秒
shells
I think it's filled shells up to 6 s

201
00:17:12,150 --> 00:17:21,660
实际上，这意味着我们已经充满了炮弹加上3个电子，所以59等于
actually so that means we have filled
shells plus 3 electrons ok so 59 equals

202
00:17:21,660 --> 00:17:32,820
充满的壳加上F中的3个电子壳中的4f中只有3个电子
filled shells plus 3 electrons in the F
shell 3 electrons in 4f in well just F

203
00:17:32,820 --> 00:17:45,690
外壳F为l等于3记住s PDFG，所以f在0 1 2 3 F为l等于3 
shell F is l equals 3 remember s P D F G
so f at 0 1 2 3 F is l equals 3

204
00:17:45,690 --> 00:17:52,710
所以我们可以写出所有赢得FL座位等于3的轨道2 
so we can write out all the orbitals won
the seat for F L equals 3 there's 2 L

205
00:17:52,710 --> 00:17:58,140
从负3 LZ到正1的轨道等于负3到正3，所以我们写
plus 1 orbitals going from minus 3 LZ
equals minus 3 to plus 3 so let's write

206
00:17:58,140 --> 00:18:05,880
他们全部出来，所以LZ等于我们有负3负2负1 0 1 2 3负3负
them all out so LZ equals we have minus
3 minus 2 minus 1 0 1 2 3 minus 3 minus

207
00:18:05,880 --> 00:18:13,950
 2减1为0 1 2 3，我们现在必须在这些轨道中放入3个电子
2 minus 1 is 0 1 2 3 and we have to put
3 electrons in those orbitals now the

208
00:18:13,950 --> 00:18:17,910
第一法则hons第一法则说，我们放入的电子必须有它们的
first rule hons first rule says that the
electrons we put in have to have their

209
00:18:17,910 --> 00:18:22,380
旋转对齐，所以我们知道我们必须对齐旋转，然后
spins aligned so that's so we know we
have to have aligned spins and then what

210
00:18:22,380 --> 00:18:26,400
我们想要做的就是我们要最大化l在满足所有
we want to do is we want to maximize l
subject to the condition that all the

211
00:18:26,400 --> 00:18:31,020
自旋是对齐的，所以我们可以考虑让我们尝试最大化LZ，我们会做
spins are aligned so we can think well
let's try to maximize LZ and we would do

212
00:18:31,020 --> 00:18:35,610
通过将电子尽可能地移到最右边，我们将得到
that by putting the electrons as far to
the right as possible and we would get

213
00:18:35,610 --> 00:18:40,350
 LZ等于6 3加2加1现在我们也可以尝试将它们放到最左边
LZ equals 6 3 plus 2 plus 1 now we can
also try to put them as far to the left

214
00:18:40,350 --> 00:18:44,400
尽可能得到LZ的最小值
as possible and we get the smallest
value of LZ possible which would be

215
00:18:44,400 --> 00:18:49,950
减6减3减2减1好的，所以如果我们能得到的最大LZ是
minus 6 minus 3 minus 2 minus 1 okay so
this if the largest LZ we can get is

216
00:18:49,950 --> 00:18:56,450
加6，我们可以得到的最小LZ为负6，这意味着我们有L 
plus 6 and the smallest LZ we can get is
minus 6 it means that the L we have l

217
00:18:56,450 --> 00:19:03,150
在这种情况下等于6好吧，现在我们也可以
equals 6
in this case ok good now we can also

218
00:19:03,150 --> 00:19:08,900
立即写下s是三分之二3旋转所有对齐的s等于
write down s immediately is three-halves
3 spins all aligned is s equals

219
00:19:08,900 --> 00:19:13,140
三半，这就是洪德定律和洪德定律的运作方式
three-halves so that's how hund's rule
works and hund's rule

220
00:19:13,140 --> 00:19:16,560
第二条规则也是由库仑相互作用驱动的，但是没有
second rule here is also driven by
Coulomb interaction but there aren't

221
00:19:16,560 --> 00:19:22,200
很好的卡通图片来解释为什么在HoN中L最大化
very good cartoon pictures to explain
why it is that L is maximized in HoN

222
00:19:22,200 --> 00:19:26,730
第二个角色，所以我很抱歉，这只是一个观察结果
second role so I apologize about that
it's just a you know it's an observation

223
00:19:26,730 --> 00:19:30,780
那个引擎盖造了，没有人真正提出许多非常简单的解释
that hood made and no one's really come
up with many very simple explanations

224
00:19:30,780 --> 00:19:35,940
在这一点上，嗯，所以这告诉了我们几乎所有我们需要知道的内容，但是
for it at this point um ok so this tells
us almost everything we need to know but

225
00:19:35,940 --> 00:19:42,000
我们仍然有一个问题，即L如何与s对齐或反对齐，所以我们有
we still have the question of how does L
align or anti align with s so we have

226
00:19:42,000 --> 00:19:56,940
规则3第三个规则上的规则3，L平行于s或L减去
rule 3 rule 3 on this third rule which
is either L is parallel to s or L minus

227
00:19:56,940 --> 00:20:02,370
 L与s um和平行
L is parallel to s
um and the way it goes is that J

228
00:20:02,370 --> 00:20:11,190
 J的绝对值是绝对L加上或减去绝对s（如果选择）-如果
absolute value of J is absolute L plus
or minus absolute s where we choose - if

229
00:20:11,190 --> 00:20:21,270
我们有少于一半的填充壳少于一半的填充壳，这意味着
we have if less than half filled shell
less than half filled meaning that the

230
00:20:21,270 --> 00:20:25,620
以其角动量和轨道角反旋转
spins anti aligned with their angular
momentum and with their orbital angular

231
00:20:25,620 --> 00:20:42,390
一半，一半以上就可以了，所以这是第三个规则
man half it's more than half filled okay
so that's one's third rule which tells

232
00:20:42,390 --> 00:20:46,590
您如何了解轨道和旋转角度Mendte如何彼此对齐或
you how the orbital and spin angular
Mendte either aligned with each other or

233
00:20:46,590 --> 00:20:52,170
现在彼此反对齐，所以在这种情况下，例如我们这里的主席团
anti aligned with each other now so for
example in this case here Presidium we

234
00:20:52,170 --> 00:20:57,510
壳的填充少于一半，所以J等于6减去我们选择的三分之二
have less than a half filled shell so J
equals 6 minus three-halves we choose

235
00:20:57,510 --> 00:21:01,230
负号，因为它小于填充的一半，这使我们的J等于9 
the minus sign because it's less than
half filled and that gives us J equals 9

236
00:21:01,230 --> 00:21:05,340
一半，所以我们知道关于主席团的所有信息，我们知道
halves okay so we know everything there
is to know about Presidium we know what

237
00:21:05,340 --> 00:21:10,500
 L是我们知道s在哪里，我们知道从狩猎卷中知道J是什么，所以为什么它是
it L is we know where s is we know what
J is from hunt rolls okay so why it's

238
00:21:10,500 --> 00:21:15,930
狩猎第三条法则是正确的狩猎第三条法则实际上与
hunts third rule true well hunts third
rule is actually a little bit unlike the

239
00:21:15,930 --> 00:21:20,580
另外两个是因为它不是由交互驱动的，而是由驱动的
other two because it's not driven by
interactions it's actually driven by

240
00:21:20,580 --> 00:21:28,020
它是由规则三驱动的，它来自自旋轨道的规则三
something else it's driven by rule three
is from rule three from spin orbit

241
00:21:28,020 --> 00:21:33,210
耦合自旋轨道实际上是相对论效应自旋轨道耦合
coupling spin orbit which is actually a
relativistic effect spin orbit coupling

242
00:21:33,210 --> 00:21:38,190
实际上，基本上就是当我们写下哈密顿量时
and in fact it is just it's basically
that when we wrote down the Hamiltonian

243
00:21:38,190 --> 00:21:43,380
我们省略了一个小术语，即自旋轨道项，它取下一个项，因此
we left out a small term the spin orbit
term which takes the following term so

244
00:21:43,380 --> 00:21:47,100
我将其写为Delta EIJ，这是我们忽略的哈密顿量的一小部分， 
I'll write it as Delta EIJ a small piece
of the Hamiltonian we left out which we

245
00:21:47,100 --> 00:21:52,470
可以写成所有电子轨道角动量的α和
can write as alpha sum over all the
electrons orbital angular momentum of

246
00:21:52,470 --> 00:21:57,180
电子点缀着电子的自旋角动量就可以了
the electron dotted with the spin
angular momentum of the electron okay

247
00:21:57,180 --> 00:22:03,540
现在汉密尔顿方程中的alpha大于零，所以我的意思是
now in this Hamiltonian alpha is greater
than zero and so I mean you can sort of

248
00:22:03,540 --> 00:22:07,110
看到这将使轨道角动量和自旋
see that this is going to make the
orbital angular momentum and the spin

249
00:22:07,110 --> 00:22:11,940
角动量计数器对齐以使其尽可能负
angular momentum counter align to make
this as negative as possible you're

250
00:22:11,940 --> 00:22:14,010
要旋转的是贝尔兰加精神
going to have to
the spin is up the or Berlanga mentum

251
00:22:14,010 --> 00:22:17,429
想下来等等，但您可能会很好奇
wants to be down and so forth but you
might wonder well what about this

252
00:22:17,429 --> 00:22:20,429
业务大约一半装满了贝壳，不到一半超过一半
business about half filled shells in
less than a half thousand more than half

253
00:22:20,429 --> 00:22:24,090
充满贝壳，似乎有点奇怪，事实证明那一点
filled shells that seems a little bit
strange it turns out that that little

254
00:22:24,090 --> 00:22:27,390
奇怪的清单实际上只是来自簿记，您只需要小心
strange list is actually just from
bookkeeping you just have to be careful

255
00:22:27,390 --> 00:22:31,620
关于簿记，每个进入的电子总是跟随
about the bookkeeping every electron
that goes in is always following the

256
00:22:31,620 --> 00:22:35,910
同样的规则遵循hunds规则1遵循hunds规则2，然后尝试使其
same rule follow hunds rule 1 follow
hunds rule 2 and then try to make this

257
00:22:35,910 --> 00:22:40,049
尽可能小，如果您遵循该规则，您将始终得到
as small as possible and if you if you
follow that rule you will always get

258
00:22:40,049 --> 00:22:46,380
洪德法则3让我们一秒钟看看它是如何工作的
hund's rule 3 let's see how that works
for a second let's consider a case where

259
00:22:46,380 --> 00:22:52,950
我们不知道D外壳，所以在广告外壳中，我们的LZ等于-2减1， 
we have I don't know a D shell so in a d
shell we have L Z equals -2 minus 1 and

260
00:22:52,950 --> 00:23:00,000
 1 2 3 LZ等于-2减去1 0 1 2这些就是可能性
1 2 3 L Z equals -2 minus 1 0 1 2 those
are the possibilities let's put one

261
00:23:00,000 --> 00:23:04,530
如果你在里面放一个电子的话，它就很简单
electron in if you put one electron in
it while ok it's pretty simple it wants

262
00:23:04,530 --> 00:23:09,360
最大化其自旋角动量，因此自旋井只有一种选择
to maximize its spin angular momentum so
the spin well it only has one choice you

263
00:23:09,360 --> 00:23:15,900
旋转1/2，这样很容易，并且它想最大化其轨道角动量为
spin 1/2 so that's easy and it wants to
maximize its orbital angular momentum as

264
00:23:15,900 --> 00:23:21,690
好，所以我们可以把它放在这里或现在的电子路径上
well so we can put this the the electron
way over here or way over here now if

265
00:23:21,690 --> 00:23:28,980
如果要使旋转降低，则希望将ll点Sigma最小化
you want to minimize the ll dot Sigma if
we put it if we make the spin spin down

266
00:23:28,980 --> 00:23:34,890
那么您将其放在此处，以使Sigma Sigma Z等于负1/2 
then you would put it over here so that
Sigma Sigma Z equals minus 1/2

267
00:23:34,890 --> 00:23:40,799
但LZ等于加1/2，所以Sigma点L的乘积为负，这就是我们
but LZ equals plus 1/2 so the product of
Sigma dot L is negative which is what we

268
00:23:40,799 --> 00:23:44,549
希望我们希望使该术语尽可能地否定，然后确定
want we want to make that term as
negative as possible ok and then sure

269
00:23:44,549 --> 00:23:53,160
这两个JZ的总和是LZ减去Sigma Z，对不起LZ加Sigma Z是
enough the sum of these two JZ is L Z
minus Sigma Z sorry LZ plus Sigma Z is

270
00:23:53,160 --> 00:23:58,500
三分之二，并声称Weesa的壳不到一半，所以J 
three-halves and as claimed
Weesa less than a half filled shell so J

271
00:23:58,500 --> 00:24:06,480
是2和1/2的差额，所以基本上可以告诉你
is the difference of 2 and 1/2 okay so
that works basically just telling you

272
00:24:06,480 --> 00:24:12,780
这样，当您添加时，可能有1/2小于一半的填充壳旋转
that when you add you might have 1/2
less than half filled shell the spin and

273
00:24:12,780 --> 00:24:16,500
轨道指向相反的方向，所以自旋轨道是
the orbit went to be pointing the
opposite direction so spin orbit is

274
00:24:16,500 --> 00:24:20,820
负面的，这很容易，现在当我们进入半填充的外壳时会发生什么
negative that's easy ok now what happens
when we get to a half-filled shell when

275
00:24:20,820 --> 00:24:25,810
我们得到一个半满的壳LZ，我们等于负
we get to a half-filled shell LZ we have
equals minus

276
00:24:25,810 --> 00:24:32,560
二减一零一二当我们半满时，我们现在拥有这五个状态
two minus one zero one two we have these
five states now once we're a half-filled

277
00:24:32,560 --> 00:24:38,050
外壳，我们必须将所有五个旋转对齐，因为我们要对齐所有
shell we have to put in all five spins
aligned because we want to align all the

278
00:24:38,050 --> 00:24:43,780
自旋，所以在这里，我们等于五个一半，五个自旋都指向
spins so here we have s equals five
halves five spins all pointed in the

279
00:24:43,780 --> 00:24:49,960
方向相同，但L等于零，因为我们已经填补了所有负数
same direction but L here equals zero
because we've filled all the negative

280
00:24:49,960 --> 00:24:54,340
状态和所有加号状态都还好，所以这没问题
states and all the plus states as well
okay so here it's no problem

281
00:24:54,340 --> 00:24:59,740
我的意思是，匈奴人第三卷甚至都不适用，因为L为零，所以旋转
I mean Huns third roll doesn't even
apply because because L is zero so spin

282
00:24:59,740 --> 00:25:08,830
就是你的绝对J，只是旋转角度，所以我们
is just your absolute J's it's just it's
just that it's just a spin angular so we

283
00:25:08,830 --> 00:25:14,020
现在不必担心L，我们再添加一个电子，这样当您
don't have to worry about about L now
let's add one more electron so when you

284
00:25:14,020 --> 00:25:23,140
再加上一个电子LZ等于5减1减2减1 0 1 2 
add one more electron LZ equals go minus
1 out of 5 minus 2 minus 1 0 1 2 we put

285
00:25:23,140 --> 00:25:28,210
所有这些家伙都旋转下来，我们再增加一个电子，那一个电子会去哪里
all these guys spin down we add one more
electron where does that one electron go

286
00:25:28,210 --> 00:25:33,220
好吧，它仍然想要满足这个条件，它想要自己旋转
well ok it still wants to satisfy this
condition that it wants its own spin

287
00:25:33,220 --> 00:25:37,870
角动量指向其轨道角动量的相反方向，因此它必须走
angular momentum to point opposite its
orbital angular momentum so it has to go

288
00:25:37,870 --> 00:25:42,940
然后旋转起来，所以我们必须把它放在这里，所以我们放入的最后一个电子
and spin up so we have to put it over
here so this last electron we put in has

289
00:25:42,940 --> 00:25:48,970
 Sigma Z等于正负1/2，但LZ等于负2，这仍然让它高兴
Sigma Z equals plus 1/2 but LZ equals
minus 2 and that still makes it happy

290
00:25:48,970 --> 00:25:54,070
因为Sigma Z指向LZ的对面，但现在注意这里发生了什么
because Sigma Z is pointing opposite LZ
but now notice what's going on here the

291
00:25:54,070 --> 00:26:01,150
总SZ总SZ实际上仍然是负数实际上是负数
total SZ the total SZ is actually still
negative is actually negative what it's

292
00:26:01,150 --> 00:26:07,480
负2对，所以我放入的最后一个电子指向总数
negative 2 right so the last electron
that I put in was pointing up the total

293
00:26:07,480 --> 00:26:15,010
自旋实际上仍然指向下方，即使l LZ在这里
spin is actually still pointing down so
and even though the l LZ is here is

294
00:26:15,010 --> 00:26:19,840
减去2，因为一半填充的半壳的LZ总数等于0 
minus 2 the total because the half the
half filled shell had LZ total equals 0

295
00:26:19,840 --> 00:26:24,160
我们添加了一个电子，该电子朝上，但总自旋角
we added one electron that electron is
pointing up but the total spin angular

296
00:26:24,160 --> 00:26:28,720
动量仍然指向下方，因此总旋转角度
momentum is still pointing down
so here the the total spin angular

297
00:26:28,720 --> 00:26:31,480
总轨道角动量中的动量现在指向相同
mentum in the total orbital angular
momentum are now pointing the same

298
00:26:31,480 --> 00:26:34,540
方向，我们仍然遵循相同的规则
direction and we were still just
following the same rule that each

299
00:26:34,540 --> 00:26:39,309
电子想要抵消其自旋和轨道角动量的对准
electron wants to counter align his spin
and his orbital angular momentum

300
00:26:39,309 --> 00:26:42,460
彼此，换句话说，我们来的原因是我们来到簿记
each other so in other words the reason
we came we came to this bookkeeping

301
00:26:42,460 --> 00:26:46,899
问题是因为当我们有一个半满的外壳时，我们有s，它不是
problem is because when we have a
half-filled shell we have s which is non

302
00:26:46,899 --> 00:26:53,259
零，但是当我们再增加一个电子时，L就是零
zero but L is zero when we add one more
electron the net spin of the whole

303
00:26:53,259 --> 00:26:57,190
系统指向那个旋转的相反方向
system is pointing in the opposite
direction of the spin of that one

304
00:26:57,190 --> 00:27:01,299
我们添加的电子，所以这就是为什么第三个规则在
electron we added so that's why one's
third rule has this complication at

305
00:27:01,299 --> 00:27:07,600
一半的外壳是透明的，希望可以，好吧，对不起，我的意思是我
half-filled shell is that clear yeah
hopefully okay all right sorry I mean I

306
00:27:07,600 --> 00:27:11,980
知道这很令人困惑，但是生活有时就是这样，所以我们必须处理
know it's confusing but but that's the
way life is sometimes so we have to deal

307
00:27:11,980 --> 00:27:19,990
这样就可以了，所以鉴于我们知道LS和J，我们可以写下我们的
with it okay so given that we know LS
and J we can then write down our

308
00:27:19,990 --> 00:27:25,600
哈密​​顿量，我们能做的第一件事是我们可以做L加GS做同样的事情
Hamiltonian well first thing we can do
is we can take L plus GS do the same

309
00:27:25,600 --> 00:27:32,320
 λG因子技巧，并将其重写为一些G有效时间J，其中J为L 
lambda G factor trick and rewrite that
as some G effective times J where J is L

310
00:27:32,320 --> 00:27:39,039
再加上其他，然后我们可以针对这些角度写下有效的哈密顿量
Plus else and then we can write down our
effective Hamiltonian for these angular

311
00:27:39,039 --> 00:27:49,480
 Momenta是G有效的J mu B BJ，我们可以用
momenta which is G effective J mu B BJ
and we can do statistical mechanics with

312
00:27:49,480 --> 00:27:54,549
这种汉密尔顿式的，所以提醒您这是如何工作的
this kind of Hamiltonian so to remind
you how this works this is something

313
00:27:54,549 --> 00:28:01,960
您去年所做的，所以举一个非常简单的例子，让我们举J 
that you did last year so example take a
really simple example let's take J

314
00:28:01,960 --> 00:28:09,070
等于1/2且G因子为2，因此自旋可以指向整体
equals 1/2 and a G factor of 2 so a spin
can point up with this this overall

315
00:28:09,070 --> 00:28:13,509
角质子能指向上或指向下，两种可能的能量是
angle mentum can point up or can point
down and the two possible energies are

316
00:28:13,509 --> 00:28:19,990
然后加上和减去玻尔磁子乘以磁场，我们可以写出
then plus and minus Bohr Magneton times
magnetic field we can then write the

317
00:28:19,990 --> 00:28:23,830
此系统的分区函数，是的两个状态的总和
partition function for this system which
is the sum over the two states of the of

318
00:28:23,830 --> 00:28:31,269
角动量，即使是负beta u也等于B加e等于正beta mu D等于
the angular momentum even the minus beta
u be B plus e to the plus beta mu D be

319
00:28:31,269 --> 00:28:34,869
从分区功能，我希望这一切看起来都很熟悉，我们可以编写免费的
from the partition function I hope this
all looks familiar we can write the free

320
00:28:34,869 --> 00:28:44,139
能量KBT log Z我们可以写出磁矩磁矩是猜测
energy KBT log Z we can write the
magnetic moment magnetic moment is guess

321
00:28:44,139 --> 00:28:46,809
带有减号的地方在这个地方加入这个减号，我得到一个减号
with the minus sign join this minus sign
here somewhere I get there's a minus

322
00:28:46,809 --> 00:28:51,649
缺少能量的迹象，我们可以写磁
sign missing
energy we can then write the magnetic

323
00:28:51,649 --> 00:28:59,510
现在的DF DB，我相信您去年就这样做了，如果没有，您会
moment DF DB which I believe you did
this last year and if you didn't you'll

324
00:28:59,510 --> 00:29:08,330
今年再做一次beta UVB，它具有针对小B小B的转换
do it this year again beta UVB it has
this transform for small B small B this

325
00:29:08,330 --> 00:29:16,720
变成βμB B平方乘以磁场，然后磁化
becomes just beta mu B squared times
magnetic field then the magnetization

326
00:29:16,720 --> 00:29:26,090
大写m是每体积的力矩每体积的力矩是每次旋转的力矩
Capital m which is moment per volume
moment per volume is the moment per spin

327
00:29:26,090 --> 00:29:32,139
 M小M乘Rho自旋密度
M small M times Rho the density of spins

328
00:29:32,409 --> 00:29:41,929
旋转的密度，然后我们可以得到磁化率Chi 
density of spins and then we can get the
susceptibility Chi which is by

329
00:29:41,929 --> 00:29:48,919
定义mu而不是DM D在B等于零时取，如果通过
definition mu not DM D be taken at B
equals zero and if you go through that

330
00:29:48,919 --> 00:29:56,529
达到这个极限，您将得到Rho mu naught mu B平方超过KBT 
take that limit you get Rho mu naught mu
B squared over KBT

331
00:29:56,529 --> 00:30:01,970
希望从去年开始看起来很熟悉，这就是居里的
which hopefully maybe looks familiar
from last year this is known as Curie's

332
00:30:01,970 --> 00:30:07,940
居里定律实际上，居里定律是任何时候
law Curie law in fact more generally
Curie's law is any time when the

333
00:30:07,940 --> 00:30:13,730
磁化率在T上是一个常数，这确实很重要
susceptibility is some constant over T
which indeed this is so the important

334
00:30:13,730 --> 00:30:18,380
关于居里定律的敏感性要注意的是
thing to notice about this Curie law
susceptibility is that it diverges at

335
00:30:18,380 --> 00:30:22,100
低温，随着温度的升高，磁化率会发散
low temperature you have a divergent
susceptibility as temperature goes to

336
00:30:22,100 --> 00:30:25,250
零，这实际上是正确的，也很有意义
zero and this is actually correct and
it's meaningful

337
00:30:25,250 --> 00:30:29,299
磁化率在零温度下发散的原因是
the reason the susceptibility diverges
it to at zero temperature is because

338
00:30:29,299 --> 00:30:33,769
温度为零，系统始终处于最低能量状态，因此如果
it's zero temperature the system is
always in the lowest energy State so if

339
00:30:33,769 --> 00:30:38,149
你有一束自旋，无论磁场有多小，磁场都很小
you have a bunch of spins and you have a
tiny magnetic field no matter how tiny

340
00:30:38,149 --> 00:30:42,200
磁场是一个最低能级
that magnetic field there's a lowest
energy state where the moment is aligned

341
00:30:42,200 --> 00:30:47,029
磁场，所以即使磁场很小
with the magnetic field so even with an
arbitrarily small magnetic field the

342
00:30:47,029 --> 00:30:49,880
时刻都对齐并指向该方向，这意味着您随后
moments all align and point that
direction which means you then have an

343
00:30:49,880 --> 00:30:55,039
无限的磁化率，因为它只需要一个极小的磁场即可
infinite susceptibility because it takes
only an infinitesimal magnetic field to

344
00:30:55,039 --> 00:30:58,940
将所有旋转方向定为相同方向，这样实际上
orient all the spins in the same
direction set clear so this is actually

345
00:30:58,940 --> 00:31:02,690
当温度升高时，有意义的陈述开始旋转
a meaningful statement when you
the temperature then the spins start

346
00:31:02,690 --> 00:31:05,750
整个地方都在波动，它们并没有全部指向
fluctuating all over the place and they
don't all point in the direction of the

347
00:31:05,750 --> 00:31:09,110
磁场，我的意思是，它的方向可能有点模糊
magnetic field I mean it may be point
sort of vaguely in the back direction of

348
00:31:09,110 --> 00:31:12,650
磁场，但并非完全沿磁场方向
magnetic field but not all exactly in
the direction of the magnetic field so

349
00:31:12,650 --> 00:31:16,640
现在，随着温度的升高，磁化率下降
the susceptibility drops as you go to
higher temperature now more generally

350
00:31:16,640 --> 00:31:22,880
比这个简单的旋转1/2计算首先暴饮四转
than this this simple spin 1/2
calculation for first binge a four spin

351
00:31:22,880 --> 00:31:29,870
一般而言，J遵循相同的故事，只是我们从分区开始
J in general we follow the same same
story except we start with a partition

352
00:31:29,870 --> 00:31:38,360
在JZ上求和的函数等于负J 2加J e至负beta 
function which is sum over J Z equals
minus J 2 plus J e to the minus beta the

353
00:31:38,360 --> 00:31:51,020
我认为状态J的自旋能量是J G，对J有效，然后是mu bb jz 
energy of the spin in state J which is J
G effective for J then mu b b jz i think

354
00:31:51,020 --> 00:31:57,080
我在这里暗示而没有指出b指向Z 
i implied here without without stating
it that that b is pointing in the Z

355
00:31:57,080 --> 00:32:01,400
应用于Z方向，否则我们将使用
direction that we apply to be in the Z
direction otherwise we would use a

356
00:32:01,400 --> 00:32:04,730
不同的轴，如果我们将B放在X方向上，它将是J， 
different axis and it would be J if we
put B in the X direction we would just

357
00:32:04,730 --> 00:32:10,120
使用不同的基础，我们将JX减去J来计算-J对此感到抱歉
use a different basis where we count J X
from minus J - J sorry about that

358
00:32:10,120 --> 00:32:14,720
好的，那么您要进行完全相同的操作，代数是一点
ok then you go through exactly the same
manipulations the algebra is a little

359
00:32:14,720 --> 00:32:19,610
比起其他一些困难，您获得的功能比
bit more different difficult you get out
some more complicated functioning than

360
00:32:19,610 --> 00:32:23,870
在ang上，这是一个称为Bruin函数的函数，我认为这有些混乱
at ang it's a function known as a Bruin
function it's some messy thing I think

361
00:32:23,870 --> 00:32:27,800
您去年在统计机修课程中计算出它有一些双曲线正弦
you calculated last year in your stat
mech course it has some hyperbolic sines

362
00:32:27,800 --> 00:32:31,790
和其中的双曲余弦比较混乱，但是出现了相同的故事
and hyperbolic cosines in it a little
more messy but the same story comes out

363
00:32:31,790 --> 00:32:34,580
在一天结束时在一天结束时，您仍然会得到居里定律
at the end of the day at the end of the
day you still get a Curie law in the

364
00:32:34,580 --> 00:32:38,060
出于相同的原因，它是一个常数除以T 
sense that it's a constant divided by T
for exactly the same reason an

365
00:32:38,060 --> 00:32:43,580
现在，无穷小场将所有矩定向在同一方向上
infinitesimal field will orient all the
moments in the same direction okay now

366
00:32:43,580 --> 00:32:48,020
一路上实际上要注意的一件事是非常重要的一点
one thing actually to note along the way
which is quite important it's a little

367
00:32:48,020 --> 00:32:53,270
顺便说一句是，该分区函数是B的函数
bit of an aside is to note that this
partition function is a function of B

368
00:32:53,270 --> 00:32:57,440
仅在T上它不是b NT的功能可以
over T only it's not a function of b NT
separately ok

369
00:32:57,440 --> 00:33:04,970
这实际上很重要，因为这意味着熵也将是
and that's actually important because it
means that the the entropy will also be

370
00:33:04,970 --> 00:33:11,360
 b相对于t的函数，并且它在我不知道的时候就被注意到了
a function of b over t only and that it
was noticed way back in I don't know

371
00:33:11,360 --> 00:33:15,140
 1900年代初期，我们的英雄之一是彼得·德拜（Peter Debye） 
early 1900s by one of our hero is Peter
Debye

372
00:33:15,140 --> 00:33:19,550
在这个课程中已经出现了几次，他注意到
has shown up several times already in
this course and he noticed that that

373
00:33:19,550 --> 00:33:23,510
表示您可以使用系统进行以下实验
would mean that you could do the
following experiment you take a system

374
00:33:23,510 --> 00:33:27,200
在给定温度下的磁场中，如果绝热地减少
in a magnetic field at a given
temperature if you adiabatically reduce

375
00:33:27,200 --> 00:33:32,390
磁场绝热意味着熵必须保持固定
the magnetic field adiabatic means that
the entropy has to stay fixed the

376
00:33:32,390 --> 00:33:36,380
绝热降低磁场温度必须成比例变化
adiabatically reduce the magnetic field
the temperature must change proportional

377
00:33:36,380 --> 00:33:41,110
对磁场，这就是所谓的绝热退磁
to magnetic field this is what's known
as adiabatic demagnetization

378
00:33:44,440 --> 00:33:54,680
去磁，这实际上是制作冰箱的好方法
demagnetization and it's actually a
really good way to make a refrigerator a

379
00:33:54,680 --> 00:33:59,180
非常强大的冰箱，您需要在某些温度下使用系统
very powerful refrigerator you take a
system at some temperature in some

380
00:33:59,180 --> 00:34:02,690
磁场，您可以减小磁场并保持熵
magnetic field you reduce the magnetic
field and in order to keep the entropy

381
00:34:02,690 --> 00:34:07,460
固定绝热地改变事物，温度必须成比例地降低
fixed changing things adiabatically the
temperature must reduce proportionally

382
00:34:07,460 --> 00:34:10,940
到磁场，这就是人们建立世界上一些最
to the magnetic field this is the way
people build some of the world's most

383
00:34:10,940 --> 00:34:14,659
功能强大的冰箱可降到Millikelvin甚至Micro Kelvin 
powerful refrigerators that get down to
millikelvin or even micro Kelvin

384
00:34:14,659 --> 00:34:20,270
温度，因此了解和最终掌握技术是一件好事
temperature so it's a good thing to know
about and eventually the technique

385
00:34:20,270 --> 00:34:25,280
当我们简单的近似于
breaks down and it basically breaks down
when our simple approximation of just

386
00:34:25,280 --> 00:34:28,790
独立旋转，此分区功能最终会中断
independent spins and this partition
function being right eventually breaks

387
00:34:28,790 --> 00:34:33,110
在非常低的温度下，我们将继续讨论下一个
down at very low temperatures we'll come
to that and next lecture the one after I

388
00:34:33,110 --> 00:34:40,070
想一想，所以这基本上就是关于功率磁化的故事
think all right so this is basically the
story of power magnetism one atom at a

389
00:34:40,070 --> 00:34:44,300
时间，但是当您一次开始将许多原子放在一起时，事情可能会发生
time but when you start putting together
many atoms at a time things can be

390
00:34:44,300 --> 00:34:47,690
不同的是，这个词在浓缩物质界周围
different there's this phrase that goes
around the condensed matter community

391
00:34:47,690 --> 00:34:51,080
每当您将很多东西放到一起时，更多的就是不同的
more is different whenever you put lots
of things together you can get different

392
00:34:51,080 --> 00:34:56,630
物理学，所以我仍然会思考一次或一次旋转
physics so I'm still going to think what
in some sense one spin at a time or one

393
00:34:56,630 --> 00:35:00,560
电子一次一次一个原子，但是当我们开始将其放入
electron at a time one atom at a time
but when we start putting it in the

394
00:35:00,560 --> 00:35:05,120
一堆其他原子的环境事物可以开始改变，所以可以
environment of a bunch of other atoms
things can start changing so what can be

395
00:35:05,120 --> 00:35:11,270
不同的地方出了什么问题，固体的不同之处
different what goes wrong and what is
different in the solid what different in

396
00:35:11,270 --> 00:35:17,420
我们可以写下几件事
a solid different in a solid okay a
couple things we can write down

397
00:35:17,420 --> 00:35:24,800
立即我们可以进行电子跳跃电子跳跃，如果
immediately one we can have electron
hopping electron hopping and if

398
00:35:24,800 --> 00:35:29,079
电子从一个原子跳到另一个原子，我们期望得到
electrons hop from one atom to the next
we expect to get

399
00:35:29,079 --> 00:35:34,839
带物理，我们希望获得电子带物理带，如果您有一个
band physics we expect to get electron
band physics bands and if you have a

400
00:35:34,839 --> 00:35:39,660
部分填充的乐队，你有一个金属管和物理，你可以得到一个金属， 
partially filled band you have a metal
tube and physics you can get a metal and

401
00:35:39,660 --> 00:35:45,520
如果您在金属中，则期望具有多顺磁性，因此会产生包利
if you're in metal you expect to have
poly paramagnetism and therefore pauli

402
00:35:45,520 --> 00:35:52,990
顺磁性，你会得到费米表面，部分顺磁性，如果你有
paramagnetism you get a Fermi surface
and partly paramagnetism and if you have

403
00:35:52,990 --> 00:35:57,880
功率功率磁化率九倍能力减小了一倍。 
power power magnetism the susceptibility
Sept ability is smaller by a factor of

404
00:35:57,880 --> 00:36:04,450
大约超过EF的KBT，其原因是您仍然尝试翻转
roughly KBT over EF and the reason for
this is that you you still try to flip

405
00:36:04,450 --> 00:36:08,619
施加磁场时会过度旋转，但必须服从宝力
over spins when you apply a magnetic
field but you have to obey the Pauli

406
00:36:08,619 --> 00:36:12,309
排他性原则，这可以防止您将其中的大部分
exclusion principle and that prevents
you from flipping over a lot of them

407
00:36:12,309 --> 00:36:15,849
只有一小部分会在磁场中翻转，所以
it's only a small number of them will
flip over in a magnetic field okay so

408
00:36:15,849 --> 00:36:19,390
如果电子从一个跳到另一个，那可能是不同的一件事
that's one thing that can be different
if the electrons are hopping from one

409
00:36:19,390 --> 00:36:22,869
原子到另一个原子，那么您可能会期望具有金属和pauli顺磁性
atom to another then you might expect to
have a metal and pauli paramagnetism

410
00:36:22,869 --> 00:36:30,309
磁化率小得多，所以您可能会问我们什么时候不跳频
much smaller susceptibility so you might
ask well when when do we have no hopping

411
00:36:30,309 --> 00:36:40,240
什么时候没有跳就没有跳，因为你知道为什么会
when is there no hopping is there no
hopping because you know why why would

412
00:36:40,240 --> 00:36:44,200
他们从来没有跳过，为什么电子不会从一个原子跳到
they ever not hop why would the
electrons ever not hop from one atom to

413
00:36:44,200 --> 00:36:48,970
下一口井基本上来自于当我们
the next well basically this comes from
interactions us when there's strong

414
00:36:48,970 --> 00:36:56,260
互动，我们在谈论mod之前就已经讨论过了
interactions and we discussed this
before when we talked about mod

415
00:36:56,260 --> 00:37:03,400
绝缘子我会提醒您，什么是Mod绝缘子？ 
insulators I'll remind you what a mod
insulators is Mott insulator is when you

416
00:37:03,400 --> 00:37:07,839
每个位点只有一个电子，由于库仑相互作用，它可以防止
have one electron in each site and due
to the Coulomb interaction it prevents

417
00:37:07,839 --> 00:37:10,750
您不会在该站点上拥有两个电子，那么您就有了
you from ever having two electrons on
the site and then you have this idea of

418
00:37:10,750 --> 00:37:15,130
电子的交通阻塞，每个原子上有一个电子，没有人能
a traffic jam of electrons where there's
one electron on each atom and no one can

419
00:37:15,130 --> 00:37:17,440
跳到他的邻居，因为他们都是因为已经有人
jump to his neighbors because they're
all because there's already someone

420
00:37:17,440 --> 00:37:21,490
坐在那里，每个人都冻结了，但是只有一个电子在冻结
sitting there so everyone's frozen but
it's frozen with only one electron on

421
00:37:21,490 --> 00:37:25,210
每个位点，因此您可以担心电子自旋或磁极以下
each site and so you can worry about
that electron spin or or below mag

422
00:37:25,210 --> 00:37:28,720
角动量，可以重新定向，它非常好
angular momentum and that can be
reoriented and it makes a very nice

423
00:37:28,720 --> 00:37:32,920
这样的功率磁铁，使这些绝缘子的高度至少在
power magnet that way so these menthe
insulators at height well at least at a

424
00:37:32,920 --> 00:37:36,250
高温，我们将在下一个讲座中在低温下讨论它们
high temperature we'll talk about them
at low temperature in the next lecture

425
00:37:36,250 --> 00:37:40,360
不久后，Magnus产生了相当不错的力量
long after make a fairly good power
Magnus

426
00:37:40,360 --> 00:37:46,560
这就是第一点和第二点的不同点
so that's point one what can be
different in a solid point two of what

427
00:37:46,560 --> 00:37:51,790
固然可以说环境可以事关环境可以
can be different as a solid is that the
environment can matter environment can

428
00:37:51,790 --> 00:38:01,420
问题，并举例说明为什么可以假设您有
matter and to give you an example of why
that can be let's imagine that you have

429
00:38:01,420 --> 00:38:08,320
长的四分之一晶胞中的一个原子，比它的宽高得多，并且
an atom in a long tetr-a g''l unit cell
which is much taller than it is wide and

430
00:38:08,320 --> 00:38:12,730
您可能会想到，由于
you might imagine that in tetragonal
unit cell like this due to the

431
00:38:12,730 --> 00:38:19,660
轨道角动量可能想要向上或指向的环境
environment the orbital angular momentum
might want to be pointing up or pointing

432
00:38:19,660 --> 00:38:30,910
直接向下可能需要L指向上方或L指向下方而不是
down directly might want want L pointing
up or L pointing down and not in the

433
00:38:30,910 --> 00:38:35,230
所以即使它是F电子，可能指向LZ也等于
middle so even if it was an F electron
which could point in LZ equals you know

434
00:38:35,230 --> 00:38:40,180
从负3到正3一直是因为环境
all the way from minus 3 to plus 3 it's
still because of the environment it

435
00:38:40,180 --> 00:38:44,380
可能更喜欢直接指向上方或下方，而不是指向
might prefer to point directly up or
directly down and not anywhere in the

436
00:38:44,380 --> 00:38:47,650
中间，这可以改变整个故事，这就是所谓的
middle and that can change the story
quite a bit this is what's known as

437
00:38:47,650 --> 00:38:56,040
晶体场分裂场分裂
crystal field splitting field splitting

438
00:38:56,460 --> 00:39:02,770
原子环境倾向于确定允许的角矩的位置
where the environment of an atom tends
to fix which angular momenta are allowed

439
00:39:02,770 --> 00:39:09,070
哪些不被允许，以及在这种情况下发生的一件事
and which ones are not allowed ok and
one thing that happens in this case

440
00:39:09,070 --> 00:39:13,360
尤其重要的是有时候您会遇到这样的情况
which is particularly important is
sometimes you can get a situation where

441
00:39:13,360 --> 00:39:23,530
晶体场实际上是相当普遍的力L基本上为0 
a crystal field it's actually quite
common forces L to basically be 0 make

442
00:39:23,530 --> 00:39:29,260
 L的某种叠加，迫使它们为0，在这种情况下， 
some sort of superposition of L's that
force them to be 0 in which case which

443
00:39:29,260 --> 00:39:35,230
意味着总的J就是自旋s，而L没有贡献，所以
implies that the total J is just then
the spin s and L doesn't contribute so

444
00:39:35,230 --> 00:39:39,250
即使您有F电子或部分填充的F壳，其中有很多
even if you had F electrons or partially
filled F shell where there's lots of

445
00:39:39,250 --> 00:39:43,150
电子，您可能会想像它会因休斯角色而具有非零的L 
electrons and you might imagine that it
would have a nonzero L by huns role

446
00:39:43,150 --> 00:39:47,950
由于原子L的环境，可以强迫其为零，然后您
because of the environment of the atom L
can be forced to be zero and then you

447
00:39:47,950 --> 00:39:50,560
只会获得自旋角动量，而不会获得轨道角动量
would only get the spin angular momentum
and not the orbital angular momentum

448
00:39:50,560 --> 00:39:58,000
这是典型的过渡金属，她告诉您所谓的过渡
this is typical of transition metals
she tell you what's called of transition

449
00:39:58,000 --> 00:40:09,520
金属，例如铬um，这就是众所周知的名字
metals things like chromium um this is
known as give it a name it's known as

450
00:40:09,520 --> 00:40:18,069
当L基本上由于环境而从问题中除去时淬灭ml 
quenching ml when L is basically removed
from the problem due to the environment

451
00:40:18,069 --> 00:40:26,789
原子，这是过渡金属的典型特征，但不会发生不会发生
of the atom it's typical of transition
metals but does not occur does not occur

452
00:40:26,789 --> 00:40:34,020
稀土元素会发生稀土元素或镧系元素和act系元素
occur for rare earths for rare earth
rare earth or lanthanides and actinides

453
00:40:34,020 --> 00:40:42,220
地球原子，其原因有些微妙，但实际上并非如此
earth atoms and the reason for this is a
little subtle but actually not not that

454
00:40:42,220 --> 00:40:47,020
复杂的原因是，在过渡金属中，部分
complicated the reason for this is that
in a transition metal the partially

455
00:40:47,020 --> 00:40:54,549
填充的外壳是3D外壳，所以在这里我们有一个原子，可能是
filled shell is say the 3d shell so here
we have an atom and it's maybe the

456
00:40:54,549 --> 00:40:59,380
填充最远的外壳是4s，部分填充的外壳是3d外壳
farthest out shell that's filled is 4s
the partially filled shell is a 3d shell

457
00:40:59,380 --> 00:41:04,390
像这样的3d外壳，而3d外壳非常靠近
like this 3d shell like this and the 3d
shell is very close to the outer edge of

458
00:41:04,390 --> 00:41:14,890
过渡金属中的原子是过渡金属，因此部分
the atom in the transition metal this is
transition metal and so the partially

459
00:41:14,890 --> 00:41:18,819
填充的外壳强烈看到原子的环境，因为
filled shell sees the environment of the
atom quite strongly because the

460
00:41:18,819 --> 00:41:23,099
电子非常靠近原子的边界，而在稀土中
electrons are very close to the boundary
of the atom whereas in the rare earths

461
00:41:23,099 --> 00:41:30,579
地球在哪里，你可以拥有最外壳的东西
where our Earth's you can have something
where the outermost shell is something

462
00:41:30,579 --> 00:41:36,220
像是6s，部分填充的外壳是4f，因此非常容易受到保护
like 6s and the partially filled shell
is 4f and so it's very protected from

463
00:41:36,220 --> 00:41:39,579
环境，因为部分填充的壳半径要小得多
the environment because the partially
filled shell it's a much smaller radius

464
00:41:39,579 --> 00:41:44,230
而不是整个原子，所以基本上对于部分填充的壳
than the atom as a whole so basically
the for the partially filled shell

465
00:41:44,230 --> 00:41:48,130
甚至看不到环境，因为它与环境隔离开来
doesn't even see the environment because
it's so shielded from the environment by

466
00:41:48,130 --> 00:41:53,970
原子中较大的轨道，所以或多或少地解释了为什么它是
the larger orbitals in the atom so it's
more or less explains why it is that the

467
00:41:53,970 --> 00:42:00,160
稀土没有角质素的淬灭，而过渡
rare earths don't have quenching of
angular mentum whereas the transition

468
00:42:00,160 --> 00:42:04,059
金属做的很好，这就是我们或多或少要说的一切
metals do all right so this is
everything we have to say more or less

469
00:42:04,059 --> 00:42:09,550
关于力量'而不是ISM回到第一个方程式
about power
'not ISM back up to the first equation

470
00:42:09,550 --> 00:42:13,300
我们今天在此处写在板上，并担心反磁性
we wrote on the board here today and
worried a little bit about diamagnetism

471
00:42:13,300 --> 00:42:26,290
所以什么时候可以得到抗磁性，哪里有抗磁性
so when can we get diamagnetism when is
there diamagnetism well there a couple

472
00:42:26,290 --> 00:42:31,900
您如何获得反磁性的第一种经典案例
ways that you can get diamagnetism the
first sort of classic case of when you

473
00:42:31,900 --> 00:42:36,550
当您恰好有一个
get diamagnetism is when j equals zero
when you just so happen to have a

474
00:42:36,550 --> 00:42:39,310
没有角动量来重新定向的情况
situation where there's no angular
momentum to reorient there's no

475
00:42:39,310 --> 00:42:44,680
顺磁性，因为没有J，例如，如果您有一个
paramagnetism because there's no no J
and for example example if you have a

476
00:42:44,680 --> 00:42:51,609
填充的壳，像稀有气体，L等于s等于J等于零， 
filled shell like a noble gas that would
have L equals s equals J equals zero and

477
00:42:51,609 --> 00:42:55,150
您无法重新定向任何磁矩，因为没有磁矩
you can't reorient any magnetic moment
because there is no magnetic moment to

478
00:42:55,150 --> 00:42:58,390
重新定向，这样，在这种情况下，您基本上可以在
reorient so in that case you can
basically just throw out this term in

479
00:42:58,390 --> 00:43:01,750
哈密​​顿量剩下的唯一东西就是哈密顿量的反磁项
the Hamiltonian the only thing left is
the diamagnetic term of the hamiltonian

480
00:43:01,750 --> 00:43:08,260
还有可能导致反磁性的情况，我还应该提到
another case where you can get
diamagnetism I should also mention that

481
00:43:08,260 --> 00:43:12,490
不仅可以填充壳，还可以填充分子轨道
not only you can have filled shell you
can also have filled molecular orbitals

482
00:43:12,490 --> 00:43:20,589
您知道类似的分子轨道，就像我在我们
molecular orbitals which you know
similar like I mentioned this when we

483
00:43:20,589 --> 00:43:24,670
谈到范德华键时，您有一个惰性原子如氮- 
talked about van der Waals bonding you
have an inert atom like a nitrogen -

484
00:43:24,670 --> 00:43:29,020
对不起像氮气这样的惰性分子-您可以将其视为
sorry an inert molecule like a nitrogen
- molecule which you can think of as

485
00:43:29,020 --> 00:43:34,030
只是一堆填充的分子轨道，所以它的净J等于0 
being just a bunch of a filled molecular
orbitals so it has a net J equals 0 at

486
00:43:34,030 --> 00:43:38,589
分子水平，因此在这种情况下，实际上没有任何方向可以调整
the molecular level and so there's
really nothing to reorient in the case

487
00:43:38,589 --> 00:43:44,230
氮转化为分子，因为所有分子轨道都被填充
of a nitrogen to molecule because all
the molecular orbitals are filled sort

488
00:43:44,230 --> 00:43:49,150
类似于具有一个轨道壳的较大原子， 
of analogous to to a larger atom with
orbitals that are orbital shells that

489
00:43:49,150 --> 00:43:53,980
充满了另一种情况，你可以得到反磁性
are filled there's another situation
where you can get diamagnetism which is

490
00:43:53,980 --> 00:44:05,560
当您患有肺泡炎时，您的顺磁性一直很弱
when you have when when you have pauli
paramagnetism is weak it's always weak

491
00:44:05,560 --> 00:44:13,839
因为它的Chi较小，是KBT在EF上面写下的， 
because it's Chi is smaller by wrote
this down above by KBT over EF which is

492
00:44:13,839 --> 00:44:20,110
这是一个很大的因素，因此，如果您有铜之类的金属
a pretty big factor so if you have a
metal like copper or something like that

493
00:44:20,110 --> 00:44:25,000
您会期望它是一个顺磁体，因为它具有旋转的方向，但是
you would expect it to be a paramagnet
because it has spins to reorient but

494
00:44:25,000 --> 00:44:28,630
由于多排除原则，它们的方向不太理想，您
because of the poly exclusion principle
they don't reorient very well and you

495
00:44:28,630 --> 00:44:32,380
在这种情况下，反磁性项具有
only get a very small susceptibility in
which case the diamagnetic term has a

496
00:44:32,380 --> 00:44:37,120
有机会与顺磁性术语竞争，所以这是一个挑战
chance to compete with the paramagnetic
term and so here's a challenge you

497
00:44:37,120 --> 00:44:43,300
应该问你的导师是铜的反磁铁还是顺磁性，除非
should ask your tutors is copper a
diamagnet or a para magnet and unless

498
00:44:43,300 --> 00:44:46,570
他们去年我们在辅导这门课，在这种情况下，学生问他们
they we're tutoring this course last
year in which case students asked them

499
00:44:46,570 --> 00:44:50,170
去年是同样的问题，所以四个新导师大多数都会把它弄错
the same question last year so four new
tutors most of them will get it wrong

500
00:44:50,170 --> 00:44:54,010
大多数人会说它是金属，所以它是多对位磁铁， 
most people will say that well it's a
metal so it's a poly para magnet and

501
00:44:54,010 --> 00:44:58,870
的确具有泡利顺磁性物理学，但是它是如此的弱以至于
indeed it has the pauli paramagnetism
physics but that's so weak that the

502
00:44:58,870 --> 00:45:02,680
反磁术语实际上有机会与之竞争，实际上
diamagnetic term here actually has a
chance to compete with it and in fact

503
00:45:02,680 --> 00:45:07,510
对于铜，反磁性反而足够令人惊讶，所以您
for copper it turns out to be
diamagnetic surprisingly enough so you

504
00:45:07,510 --> 00:45:11,740
当仅具有功率磁性的金属可以得到时
can with when metals which have only
power power magnetism you can get

505
00:45:11,740 --> 00:45:16,330
反磁性赢得了挑战，也挑战了您的导师，看看有多少人
diamagnetism winning over also challenge
your tutors with that see how many get

506
00:45:16,330 --> 00:45:21,130
很好，很快，我想我们可以在五分钟内完成
it right all right so very quickly I
think we can do this in five minutes we

507
00:45:21,130 --> 00:45:24,550
可以计算出该反磁性项的影响，因此
can calculate the effect of this
diamagnetic term so the part of the

508
00:45:24,550 --> 00:45:31,600
我们感兴趣的哈密顿量是e平方超过2m的平方，我们将
Hamiltonian we're interested in is e
squared over 2m a squared and we'll put

509
00:45:31,600 --> 00:45:39,970
 V在Z方向上，所以B等于BZ hat，所以我们可以写一个矢量势
V in the Z direction so B equals B Z hat
and so we can write a vector potential

510
00:45:39,970 --> 00:45:48,310
一半的V越过R，这样del越过a等于B，则B越过2 
one-half V cross R so that del cross a
equals B and that will then be B over 2

511
00:45:48,310 --> 00:45:55,330
我想我们会写Y帽子X减去X帽子Y 
I guess we'll write Y hat X minus X hat
Y you need enough so this term in the

512
00:45:55,330 --> 00:46:01,060
然后，我们感兴趣的哈密顿量在8 m B上平方e×B平方x平方
Hamiltonian we're interested in is then
e squared over 8 m B squared x squared

513
00:46:01,060 --> 00:46:07,930
加y平方，则该项在哈密顿量中的期望能量
plus y squared then the expectation
energy of this term in the Hamiltonian

514
00:46:07,930 --> 00:46:12,010
我们在零温度下这样做很方便
which we're doing this at zero
temperature which is convenient to do

515
00:46:12,010 --> 00:46:15,400
那就是能量的变化与自由能的变化相同
that's the change in the energy is the
same as the change in the free energy a

516
00:46:15,400 --> 00:46:21,970
平方超过8 MB平方期望x平方加上y平方
squared over 8 M B squared
expectation of x squared plus y squared

517
00:46:21,970 --> 00:46:31,870
因此磁矩为D减去预期能量DV 
and so the magnetic moment is then minus
D expectation of energy DV

518
00:46:31,870 --> 00:46:38,800
然后-B e在x平方加y平方的4m上平方
which is then - B e squared over 4m
expectation of x squared plus y squared

519
00:46:38,800 --> 00:46:42,480
现在记得在上次演讲中我提到的那个
now remember in the last lecture I
mentioned to you that

520
00:46:42,480 --> 00:46:47,560
反磁性有点像镜片定律，就像一圈电线，如果你
diamagnetism is a little bit like lenses
law it's like a loop of wire and if you

521
00:46:47,560 --> 00:46:50,440
只有一圈电线，你从电流中得到的磁矩量
only have a loop of wire the amount of
magnetic moment you get from a current

522
00:46:50,440 --> 00:46:54,940
绕过那圈电线与它的面积成正比，实际上
going around that loop of wire is
proportional to its area and indeed this

523
00:46:54,940 --> 00:46:59,920
 x平方加y平方的期望值是
expectation of x squared plus y squared
is the area enclosed by the orbital of

524
00:46:59,920 --> 00:47:03,940
电子，因此非常类似于镜头定律，它实际上是在测量
the electron so it's quite analogous to
lenses law it's really measuring the

525
00:47:03,940 --> 00:47:09,790
电流以与绕射定律给您的方式相同的方式绕过原子
currents going around the the atom in
the same way that lenses law gives you a

526
00:47:09,790 --> 00:47:14,400
与绕线环流动的电流相关的磁铁
magnet associated with the current
running around the loop of wire

527
00:47:14,400 --> 00:47:19,480
在球形原子中，计算x平方加y平方并不难
calculating x squared plus y squared
isn't so difficult in a spherical atom

528
00:47:19,480 --> 00:47:25,840
因此，如果我们有一个球形原子，那么这些天大多数原子都是球形的，我们
so if we have a spherical atom most
atoms are spherical these days atom we

529
00:47:25,840 --> 00:47:30,130
可以这样写：x平方加y平方的期望的三分之二
can write that as then two-thirds of
expectation of x squared plus y squared

530
00:47:30,130 --> 00:47:37,060
再加上Z平方或R平方期望值的三分之二，因此我们可以
plus Z squared or two-thirds of
expectation of R squared and so we can

531
00:47:37,060 --> 00:47:44,110
然后重写每个原子的磁矩，这里减去-6 m的平方
then rewrite the magnetic moment per
atom here is minus b e squared over 6 m

532
00:47:44,110 --> 00:47:48,880
那6是因为将三分之二与1/4的时间相加
that 6 comes from putting together the
two-thirds with the 1/4 their times

533
00:47:48,880 --> 00:47:55,870
 R的平方期望，然后磁化强度是Rho的M倍，其中Rho 
expectation of R squared and then the
magnetization is M times Rho where Rho

534
00:47:55,870 --> 00:48:04,480
是整个系统中电子的电子密度， 
is the density of electrons of electrons
in in the entire system and that gives

535
00:48:04,480 --> 00:48:08,350
我们很好，就磁场而言， 
us well we differentiate that with that
with respect to the magnetic field

536
00:48:08,350 --> 00:48:19,900
我们将在此处获得等于M的零，然后减去M的零。 
we'll get here chi equals nu nought the
M DB which is then minus mu naught e

537
00:48:19,900 --> 00:48:27,400
 R平方的原子期望值的Rho平方除以质量的6倍
squared Rho size of the atom expectation
of R squared divided by 6 times the mass

538
00:48:27,400 --> 00:48:33,520
电子，这就是所谓的拉莫尔结果拉莫尔结果拉莫尔或
of the electron and this is known as the
larmor result larmor result larmor or

539
00:48:33,520 --> 00:48:38,250
有时拉莫尔较大的面包车长沙板
sometimes larmor larger van long Shaban

540
00:48:38,580 --> 00:48:43,710
反磁性的对不起，这是拉莫尔或朗伊人
diamagnetic
sorry this is larmor or Longy man

541
00:48:43,710 --> 00:48:49,680
反磁性，剩下的两分钟就是现在的期望
diamagnetism and that's the expectation
of this now in the remaining two minutes

542
00:48:49,680 --> 00:48:54,240
我要告诉你一些不可考量的东西，这个计算是
I'm going to tell you something that's
not examinable this this calculation is

543
00:48:54,240 --> 00:48:59,790
考试的非常标准的方法是计算原子或某些系统的反磁性
very standard for exams calculate the
diamagnetism of an atom or some system

544
00:48:59,790 --> 00:49:04,380
原子有不可检验的东西，那就是我们是
of atoms there's something that's non
examinable which is that we were a

545
00:49:04,380 --> 00:49:09,210
抛弃顺磁性这个词有点太快了
little bit too fast in throwing away the
paramagnetic term the reason we were too

546
00:49:09,210 --> 00:49:13,920
很快，我们就知道了，我们最初说的很清楚，功率磁项是
fast you know we were we we initially
said well the power magnetic term is

547
00:49:13,920 --> 00:49:18,720
 in的线性是B的二次方，因此我们将处理该项
linear in in be the diamagnet term is
quadratic in B so we'll treat this term

548
00:49:18,720 --> 00:49:22,860
首先，然后仅当此术语碰巧很小但要处理此问题时
first and then only treat this one if
this term happens to be small but

549
00:49:22,860 --> 00:49:25,800
实际上，我们应该多加注意，不要担心
actually we should be a little more
careful that we shouldn't worry about

550
00:49:25,800 --> 00:49:31,290
这个词也是二阶的，如果我们对它仔细的话
this term at second order as well that
if we were careful about it we would

551
00:49:31,290 --> 00:49:36,300
还必须写一些看起来像二阶扰动的术语
have to also write some term that looked
like this second order perturbation

552
00:49:36,300 --> 00:49:48,780
结果且不等于n亩BB点L的0加GS 0的平方，这也是
result and not equal to 0 of n mu B B
dot L plus GS 0 squared this is also

553
00:49:48,780 --> 00:49:57,480
磁场E的二次方-这个术语显然是负的
quadratic in magnetic field E - e n this
term is manifestly negative the sign of

554
00:49:57,480 --> 00:50:02,880
二阶摄动理论显然是负的，所以它将
the second order perturbation theory is
manifestly negative and so it would

555
00:50:02,880 --> 00:50:10,640
实际上给了我们顺磁性的贡献
actually give us a paramagnetic
contribution power mag contribution

556
00:50:10,640 --> 00:50:19,140
在诺贝尔奖获得者之后，被称为范·弗莱克（Van Vleck）的强磁力范·弗莱克（Van Vleck） 
known as Van Vleck power magnetism
Van Vleck after Nobel laureate who

557
00:50:19,140 --> 00:50:27,239
发现它往往很小，但尤其是在以下情况下会发生
discovered it it tends to be small but
it can occur in particular in cases when

558
00:50:27,239 --> 00:50:37,170
 J等于零，但l和s不等于零，所以我们有J或L等于s但J 
J equals zero but l and s are not equal
to zero so we have J or L equals s but J

559
00:50:37,170 --> 00:50:42,690
等于L减去s等于零，它可能发生的原因是，如果您
equals L minus s equals zero and the
reason it can occur is because if you

560
00:50:42,690 --> 00:50:48,390
记得当我们回到这里时，我们说过我们可以在
remember when we went back here and we
said that we could replace this term in

561
00:50:48,390 --> 00:50:55,079
哈密​​顿量与J这两个项之和实际上是我们预先设定的
the Hamiltonian with J the sum of these
two terms we were actually presupposing

562
00:50:55,079 --> 00:50:59,609
 L的大小，s的大小和J的大小是固定的，但是
that the magnitude of L the magnitude of
s and the magnitude of J were fixed but

563
00:50:59,609 --> 00:51:05,549
实际上，虽然旋转L＆S的位置不固定，但您可以在x'处获得激励x' 
in fact you can have excitation x' where
you unfix though is where you rotate L&S

564
00:51:05,549 --> 00:51:09,809
彼此远离并改变激发态下j的值
away from each other and change the
value of j in the excited states and you

565
00:51:09,809 --> 00:51:13,859
可以得到这个的非零贡献，这样就不会发生
can get a nonzero contribution of this
then so this it doesn't tend to happen

566
00:51:13,859 --> 00:51:19,019
对于稀有气体，因为稀有气体的L和s均为0，但存在
for noble gases because noble gases you
have L and s both being 0 but there are

567
00:51:19,019 --> 00:51:22,739
元素周期表中的一些元素，我认为它少于1/2 
some elements on the periodic table
which are I think it's one less than 1/2

568
00:51:22,739 --> 00:51:26,789
根据洪德定律，在组合中L恰好等于s的填充壳为
a filled shell where L happens to equal
s in the combination by hund's rule is

569
00:51:26,789 --> 00:51:31,019
零，这个项实际上可以出来，给你一个顺磁性的贡献
zero and this term can actually come out
and give you a paramagnetic contribution

570
00:51:31,019 --> 00:51:34,680
也是不可考的，不用担心，这只适合您
as well that's not examinable don't
worry about it it's just for your

571
00:51:34,680 --> 00:51:40,430
一般的教育，我们明天再谈磁性
general edification we'll talk more
about magnetism tomorrow see you then

572
00:51:41,609 --> 00:51:43,670
您
you