The occupation of orbitals

Now consider the electronic structure of a solid formed from atoms each able to con tribute one electron (for example, the alkali metals). There are N atomic orbitals and therefore N molecular orbitals packed into an apparently continuous band. There are Nelectrons to accommodate. At T = 0, only the lowest N molecular orbitals are occupied (Fig. 20.53), and the HOMO is called the Fermi level. However, unlike in molecules, there are empty orbitals very close in energy to the Fermi level, so it requires hardly any energy to excite the uppermost electrons. Some of the electrons are therefore very mobile and give rise to electrical conductivity.

At temperatures above absolute zero, electrons can be excited by the thermal motion of the atoms. The population, P, of the orbitals is given by the Fermi–Dirac distribution, a version of the Boltzmann distribution that takes into account the effect of the Pauli principle:

P=

The quantity µ is the chemical potential, which in this context is the energy of the level for which P =  (note that the chemical potential decreases as the temperature increases). The chemical potential in eqn 20.23 has the dimensions of energy, not energy per mole.

The shape of the Fermi–Dirac distribution is shown in Fig. 20.54. For energies well above µ, the 1 in the denominator can be neglected, and then P≈e^{−(E−µ)/kT .} The population now resembles a Boltzmann distribution, decaying exponentially with increasing energy. The higher the temperature, the longer the exponential tail. The electrical conductivity of a metallic solid decreases with increasing temperature even though more electrons are excited into empty orbitals. This apparent paradox is resolved by noting that the increase in temperature causes more vigorous thermal motion of the atoms, so collisions between the moving electrons and an atom are more likely. That is, the electrons are scattered out of their paths through the solid, and are less efficient at transporting charge.

Fig. 20.53 When N electrons occupy a band of Norbitals, it is only half full and the electrons near the Fermi level (the top of the filled levels) are mobile.

Fig. 20.54 The Fermi–Dirac distribution, which gives the population of the levels at a temperature T. The high-energy tail decays exponentially towards zero. The curves are labelled with the value of µ/kT. The pale green region shows the occupation of levels at T=0.

اخر الاخبار

اخبار العتبة العباسية المقدسة

قسم التطوير يقيم دورة عن أخلاقيات المهنة للمنتسبين الجدد

تتضمن زيارة المراقد المقدسة في كربلاء.. معهد القرآن الكريم النسوي ينظم سفرة دينية لطالباته

المجمع العلمي يحتفي بذكرى ولادة الإمامين الباقر والهادي (عليهما السلام)

قسم الشؤون الفكرية يطلق فعاليات الأسبوع الثالث من برنامج البذرة الصالحة

المزيد

الآخبار الصحية

ثورة طبية.. الذكاء الاصطناعي يكشف سرطان الكلى في وقت قياسي

لتعزيز الطاقة والتغلب على التعب.. عليك بـ7 طرق بسيطة

لتحسين الهضم والتحكم في الشهية.. ما أفضل وقت لتناول التمر؟

خبراء يكشفون تأثير الاستحمام في الظلام على جودة النوم !

المزيد

الآخبار التكنلوجية

إزالة ثاني أكسيد الكربون من الغلاف الجوي.. ما دور المحيطات والبحار؟

نظائر الهيليوم مقياس لاكتشاف رواسب الذهب.. كيف ذلك؟

الصين تطلق أول ناقلة نفط خام ذكية في العالم تعمل بالميثانول

الزمن على المريخ أسرع من الأرض.. دراسة علمية تكشف الفوارق

المزيد

مواضيع ذات صلة

Light emission by solid-state lasers and light-emitting diodes

Light absorption by molecular solids, metallic conductors, and semiconductors

Insulators and semiconductors

The formation of bands

Molecular solids and covalent networks

Energetics

Structure

Close packing

Structure refinement

The phase problem

Bragg’s law

X-ray diffraction