W2.2 핵의 크기와 스핀(Nuclear size and Spin)

[커세라] 입자물리학 입문(Particle Physics: an Introduction)

1주: 물질과 힘 그리고 측정(Matter and forces, measuring and counting)
    W1.0 환영(Welcome)
    W1.1 물질(Matter)
    W1.2 힘(Forces)
        W1.2a 자연단위(Natural units)
        W1.2b 특수 상대론과 4-벡터(Special relativity and four-vector)
        W1.2c 가상입자(Virtual Particles)
    W1.3 확률과 단면(Probability and cross section)
        W1.3a 광자 빔의 감쇄(Attenuation of a photon beam)
    W1.4 러더포드 실험(Rutherford experiment)
        W1.4a 러더포드 단면(Rutherford cross section)
        W1.4b 산란율 계산(Counting rate Rutherford)
    W1.5 양자 산란(Quantum scattering)
    W1.6 러더포드 실험 실습(Rutherford experiment in practice)
    W1.7 1주 평가문제(Graded quiz for module 1)

2주: 핵 물리학(Nuclear Physics)
    W2.1 핵 질량 및 결합 에너지(Nuclear mass and binding energy)

W2.2 핵의 크기와 스핀(Nuclear size and Spin)/동영상/영문자막/슬라이드

[2.2-1]---------------------------------------------------------------------


During this second module, we deal with nuclear physics and its applications. In this second video, we will summarize what is known about the size and the spin of nuclei. The goals for you are

- To know how nuclear size is measured and what the results are.
- To know general facts about the spin of nuclei.
- To be able to describe the valley of stability of nuclei as a function of the number of protons and neutrons.

이번 강의의 목표,

- 핵의 크기 측정 방법과 측정 결과
- 핵의 스핀에 대한 이해
- 핵의 안정곡선(valley of stability)을 양성자와 중성자의 수에 따른 함수로 알아보기

[2.2-2]---------------------------------------------------------------------


- The size of a subatomic object must be carefully defined. In a quantum system, it is given by the root mean square of the eigenvalue of the coordinate operator in its ground state.

원자 이하의 크기를 재려면 먼저 조심스러운 정의가 필요하다. 양자 체제(quantum system)에서는 입자가 기저상태(ground state)일때 좌표 연산자(coordinate operator)의 고유치(eigenvalue)를 평균자승제곱근(root mean square)하여 정한다.

* ground state, 입자가 외부로부터 에너지를 흡수하여 흥분되면 고유 크기보다 커짐
* coordinate operator, 크기를 재려는 것이므로 당연히 좌표가 등장. 하지만 통계 '연산자'
* eigenvalue, 좌표 연산자는 시공간의 행렬로 주어질 것인데 크기는 스칼라(고유치)
* the root mean square, 통계 처리. 크기 스칼라는 음수가 없으므로 제곱하여 구함.

- For an atom, this is the root mean square of the radial position of the electron, which is farthest away from the nucleus. This distance can be calculated because we know perfectly the binding electromagnetic force and because it is defined with respect to a static reference point, the position of the nucleus.

원자(atom)의 경우 핵에서 멀리 떨어진 전자(구름)의 구형위치(radial position)을 자승평균으로 구함. 이 거리는 전자기력(electromagnetic force)에 의해 묶여있는 크기로서 정확히 계산해 낼 수 있다.

- For the nucleus, we do not have a simple description of these actions. So we must interpret the results of experiments, which probe the distribution of nucleons inside the nucleus.

핵(nucleus)의 경우 핵력(nuclear force)이 지배하고 있어 쉽게 계산 할 수 없음. 핵내의 핵자들의 배열(distribution of nucleons) 구조를 밝힐 수 있는 실험이 필요함.

- It will be unwise to use hadronic probes to do so because they are sensitive to the nuclear force. High energy electrons, on the contrary, can penetrate inside the nucleus and their scattering maps out the charge distribution of the target.

이 핵자들의 구조를 밝히는 실험에 핵력에 민감한 강입자류(hadronic probe)를 사용하는 것은 바람직하지 않음. 고 에너지 전자(electron) 역시 핵을 그냥 통과해 버리므로 역시 바람직 하지 않음.

- The cross section for a point-like target without spin is given by the Mott formula displayed here.

스핀(spin)을 감안하지 않은 점으로 간주한 목표입자(point-like target)에 대한 단면(cross section, dσ/dΩ)은 위의 그림에 보인 것처럼 모트 공식(Mott formula). 산란각의 사인 네제곱분의 일, 1/(sin θ/2)^4.

-  If the charge instead is distributed according to a volume density ρ(x), leaving the total charge intact, the cross section will be reduced by form factor F(q), which is a function of the momentum transfer q.

전하가 총 전하량에는 변화가 없이 체적밀도(volume density) ρ(x)를 따라 분포한다면 단면(cross-section)은 전하량에 따른 크기를 함수 F(q)로 나타낼 수 있을 것임. (단면을 단순히 전하량의 함수로 나타냄)

[2.2-3]---------------------------------------------------------------------


(?)

- For a static target, F(q) is the Fourier transform of the spatial charge distribution.
(?) 정 전하분포(static charge distribution) F(q)는 공간상의 전하 분포로 퓨리어 변환(근사식)으로 계산

- For small momentum transfers, we can develop the form factor in a Taylor series.
(?) 운동량 전달은 테일러 급수(Taylor series)를 활용하여 전개

- If the distribution is spherically symmetric, the terms with an odd power drop out. The dominant second term is proportional to the mean square radius, <r^2> of the charge distribution. It can thus serve as a size estimator for the nucleus.

(?) (공간상에 분포하는 전하의 영향범위를 감안)구형 대칭성(spherically symmetric)을 고려하면 퓨리어 변환 중 거리 r의 홀수제곱 항은 사라지고, 거리의 제곱 평균 <r^2>이 핵의 크기를 평가 할 수 있는 인자로 남는다.

- For an exponential charge distribution, the form factor takes what is called a dipolar form.

(?) 지수형 전하분포(exponential charge distribution)를 크기측정 요인(form factor)으로 삼는데 이를 쌍극형(dipolar form)이라 함.

- The dependence of the electron-nucleus cross section on the momentum transfer for small angle scattering at high electron energies is thus used to measure the size of the nucleus.

(!) 고 에너지 전자를 입사 입자로 한 산란실험을 통해 얻은 작은 산란각을 가지고 운동량 전달을 계산하여 전자와 핵이 결합된(원자) 단면을 계산 할 수 있다. 이 (원자의)단면은 결국 핵의 크기를 측정 하는데 이용된다.

[2.2-4]---------------------------------------------------------------------


(?)

- Scattering experiments establish a simple relation between the radius of a nucleus R and the number of nucleons A. R ~ A^(1/3)

산란실험으로 핵의 반경과 핵자의 숫자의 관계를 얻었다. 핵의 반경 R은 핵자의 수 A의 세제곱근에 근사. R ~ A^(1/3)

- The radius R is proportional to the cube root of the number of nucleons A, with a universal proportionality constant of 1.2 femtometers. Nuclei are thus indeed small
compared to the atomic size.

핵의 반경 R은 핵자의 수의 세제곱근에 비례하는데 비레상수는 1.2 펨토미터, 1.2x10^(-15)m. 핵은 실제로 원자의 크기보다 아주 작다. 원자 반지름은 원자핵에서 가장 바깥 궤도의 전자까지의 거리로 수소원자의 크기는 25 피코메터, 25x10^(-9)m

- The nuclear volume is proportional to A. This corresponds to densely packed and
incompressible nucleons, which do not fuse.

핵의 부피는 핵자의 개수에 비례(당연히!). 이는 융합되지 않고 밀집된 핵자들에 해당함.

- The mass density of nuclear matter is of the order of 10^14 grams per cm^3.

핵물질의 질량밀도의 규모는 세제곱 센티미터당 10의 14승 그램. 10^14 grams per cm^3.

[2.2-5]---------------------------------------------------------------------


(?)

Nuclear spin is the sum of the spin S of all individual nucleons and the relative angular momentum, L.

- Protons and neutrons are fermions with spin one-half.

- Like in atoms, the nuclear angular momentum L follows an integer quantum number.

- The total should therefore be a half integer number if A is odd, an integer if A is even.

- Indeed, all nuclei with both N and Z even have nuclear spin 0. Heavy nuclei have rather small nuclear spin in their ground state. We conclude that neutrons and protons tend to arrange in pairs of opposite spin direction.

[2.2-6]---------------------------------------------------------------------


(...........)

Every charged particle has a magnetic dipole moment associated with its spin, including the nuclei.

The order of magnitude for an electron is given by the Bohr magneton, µ_B.

The nuclear magneton, e/2m_p, is three orders of magnitude smaller due to the larger proton mass.

- The gyromagnetic factor g measures the ratio between the angular momentum and the magnetic moment.

- For a point charge, g is about 2 with small deviation of order 10^-3 for electrons and
muons as we will see in module 4.

- The magnetic moments of proton and neutron are considerably different, +2.79 and -1.91 nuclear magnetons, respectively. This is the first indication of a charged substructure of the nucleons. In fact, since the neutron has zero net charge, it must contain charged particles.

- All nuclei have measured magnetic moments between minus three and ten nuclear magnetons, thus, relatively small ones. This is a consequence of the spin pairings of nucleons, leading to a limited total nuclear angular momentum.

[2.2-7]---------------------------------------------------------------------


(............)

- Most nuclei and their isotopes are unstable. Stable nuclei are found in a narrow band, in the N-Z diagram which is called the valley of stability.

- The valley has the following shape.

    > For lighter nuclei with atomic mass less than 40, the number of neutrons equals
the number of protons.

    > For heavy nuclei with atomic mass bigger than 40, the number of neutrons is about 1.7 times the number of protons.

- This indicates that for heavy nuclei, the charge density, and thus the Coulomb repulsion must be diluted by additional neutrons.

- The decay of unstable nuclei is the source of nuclear radioactivity.

    > Alpha radioactivity is the emission of He-4 nuclei or so-called alpha particles.

    > Beta radioactivity is the emission of electrons or positrons together with neutrinos.

    > Gamma radioactivity is the emission of photons.

- Nuclei with a surplus of neutrons can be stabilized by converting a neutron into a proton. And those with the surplus of protons can convert a proton into a neutron. These are isobar decays of type beta plus or minus.

- Heavy nuclei often decay into a pair of lighter nuclei. This corresponds to spontaneous fission, often by emitting a He-4 nucleus. This is the alpha decay.

- This decay is normally related to an excited state of the daughter nucleus. And they're often followed by a gamma decay towards the ground states.

- We will enter into more detail on these processes in the fourth and fifth video of this module.

[2.2-8]---------------------------------------------------------------------


(결론만이라도 이해? 아니면 외울까?)

So let us summarize what we have learned about the nuclear force.
핵력에 관해 배운 내용을 요약해보면 다음과 같다.

- It has a very short range limited to the nuclear size.
  핵력의 영향 범위는 매우 좁은 핵의 크기에 국한 된다. 약 10^(-14) m

- The binding energy per nucleon it leads to is independent of the size of the nucleus. The nucleon thus interacts only with its nearest neighbors.
  (핵을 구성하는) 핵자당 결합 에너지는 핵의 크기와는 상관 없이 핵자들 끼리 작동한다.

- The nuclear force is attractive and much stronger than the Coulomb repulsion between protons. But it must also have a repulsive component at distances compatible to the size of the nucleon, which is about 1 femtometer. This is due to the existence of quarks inside the nucleon. The repulsive component is necessary to prevent the fusion among nucleons.

  핵력(nuclear force)은 양성자들 사이의 응집력으로 쿨롱 반발력을 능가한다. 하지만 핵력도 핵자들의 크기를 벗어나면 반발력을 띈다. 그 범위는 1 펨토미터(femto-meter), 10^(-15)m 다. 핵력이 반발력으로 발현되는 이유는 핵자(nucleon) 내부에 쿼크(quark)가 존재하기 때문이다. 핵 융합(nuclear fusion)을 어렵게 하는 요인이 바로 이 핵력의 반발력이다.

- QCD(quantum chromodynamics) is a quantum field theory which describes the interaction among colored particles, so quarks inside hadrons, via the exchange of equally colored gluons. But the strong force acts only inside hadrons, which thus have a zero net color.

  QCD(quantum chromodynamics)는 색입자(colored particle)사이에 작용하는 상호작용을 설명하는 양자장 이론이다. 즉, 강입자(양성자)내의 쿼크는 글루온을 통해 힘을 전달한다. 이런 강력(strong force)는 오직 강입자(hadron)내에서만 작동한다. 강입자 전체의 색전하(net color) 값은 0이다.

- Nucleons cannot exchange gluons. It is by the exchange of colorless objects like mesons that they bind together.

- This makes the nucleus a complex multi-body object, which is difficult to understand. The appropriate theoretical methods are effective field theories like Chiral Perturbation Theory or a numerical calculation like Lattice QCD. All of this goes beyond the scope of this course.


In the next video, we will rather concentrate on much simpler models, which describe the gross features of nuclei.

[W2.2 연습문제]--------------------------------------------------------------

W2.1 핵 질량 및 결합 에너지(Nuclear mass and binding energy)

[커세라] 입자물리학 입문(Particle Physics: an Introduction)

1주: 물질과 힘 그리고 측정(Matter and forces, measuring and counting)
    W1.0 환영(Welcome)
    W1.1 물질(Matter)
    W1.2 힘(Forces)
        W1.2a 자연단위(Natural units)
        W1.2b 특수 상대론과 4-벡터(Special relativity and four-vector)
        W1.2c 가상입자(Virtual Particles)
    W1.3 확률과 단면(Probability and cross section)
        W1.3a 광자 빔의 감쇄(Attenuation of a photon beam)
    W1.4 러더포드 실험(Rutherford experiment)
        W1.4a 러더포드 단면(Rutherford cross section)
        W1.4b 산란율 계산(Counting rate Rutherford)
    W1.5 양자 산란(Quantum scattering)
    W1.6 러더포드 실험 실습(Rutherford experiment in practice)
    W1.7 1주 평가문제(Graded quiz for module 1)

2주: 핵 물리학(Nuclear Physics)
W2.1 핵 질량 및 결합 에너지(Nuclear mass and binding energy)/동영상/영문자막/슬라이드

[2.1-1] -------------------------------------------------------------



During this module, we'll deal with nuclear physics and its applications.

둘째주 강의의 목표는 핵물리(nuclear physics)와 그 응용을 다뤄본다

At the end of the module, we will visit the Tokamak of the Swiss Institute of Technology in Lausanne. And the Beznau Nuclear Power Plant which is the oldest one still in operation.

로잔에 있는 스위스 기술원(Swiss Institute of Technology)의 토카막(Tokamak)을 방문해보자. 그리고 가장 오래됀 핵발전 시설중 여전히 가동중인 베즈노(Beznau power plant)를 방문한다.



This is pretty much a self-contained module, if your main interest is nuclear physics, you will be well-served. You will also notice that it is somewhat longer than other modules, so just take your time to digest the contents without pressure.

이번주 강의분은 자습 과정이 많이 포함돼어 있다. 핵물리학에 관심이 있다면 도움이 될 것이다. 다른 주에 비해 분량이 좀 많아 보이지만 부담갖지말고 요약해서 보도록 하자.

In this first video, we will review what is known about the mass of nuclei. The goals for you are the following.

이번주 첫 강의를 마치면 배우게 될 것들,

- To know the nomenclature of atomic nuclei and their periodic system.
  원자핵의 분류(명명, nomenclature)과 주기율표(peoridic system)

- To be able to qualitatively describe the mass and binding energy of nuclei.
  핵의 질량과 결합 에너지를 정량적으로 이해

[2.1-2] -------------------------------------------------------------



Experiments of the Rutherford type demonstrate the existence of a positively charged nucleus, which is four orders of magnitude smaller than the size of the atom. These experiments only require to understand electromagnetic interactions between the project and the target.

러더포드 실험 유형(산란실험)으로 알 수 있는 것은 크기가 원자의 만분의 일(four order of magnitude) 가량되는 크기로 양(전기)으로 하전된 핵이 존재한다는 점. 이런 류의 산란실험(scattering)은 단지 입사입자(projectile)와 목표입자(target) 사이에 전자기적 작용(electromagnetic interaction) 정도를 이해할 수 있다.

Scattering experiments can also yield information about nuclear properties. And thus establish a catalog of the properties of the nuclear interaction that holds together protons and neutrons inside the nucleus.

산란실험을 통해 핵의 특성에 관한 정보가 드러나는데, 그를 토대로 양성자(proton)와 중성자(neutron)가 결합된 핵(nucleus)의 특성을 토대로 목록을 만들어 봤다.


One must not confuse this nuclear force with the strong force introduced in the first module and more extensively discussed in module number five.

이 (양성자와 중성자들을 묶어놓은)핵력(nuclear force)은 앞서 선보였던 강력(strong force)와 다른 것이다. 강력에 대해서는 5주째에 자세히 다루겠다.

The strong force binds together quarks inside hadrons by gluon exchange. It does not permit quarks to leave the hadrons.

강력(the strong force)은 강입자(hadron)내부에서 글루온(gluon)의 (에너지)전환으로 쿼크(quark)들을 묶어놓는 힘이다. 즉, 쿼크들이 강입자 밖으로 빠져 나가지 못하게 잡아둔다.



So hadrons in general and nucleons in particular, do not carry a net color charge. Thus gluons cannot bind protons and neutrons to form a nucleus.

일반적인 강입자(hadron)와 특별한 핵자(nucleon)는 총색전하(net color charge)를 갖지 않는다. 따라서 양성자(proton)와 중성자(neutron)를 묶어 핵을 형성하는데 글루온이 역활을 하는 것이 아니다.

The nuclear force is more like a long distance residue of the strong force in that it resembles the well-known Van der Waals force, which is a residue of the electromagnetic interaction which acts between electrically neutral molecules.

핵력(the nuclear force)은 강력(the strong force)처럼 작용 범위가 넓은데 마치 전기적으로 중성인 분자들 사이에 전자기 작용을 설명하는 반 데르 발스 힘(Van der Waals force)과 비슷하다.

[2.1-3] -------------------------------------------------------------



Let us compare some basic properties of atoms, and the electromagnetic force on one side, to nuclei, and the nuclear force on the other side.

전자기력(electromagnetic force)이 지배하는 원자(atom)와 핵력(nuclear force)이 우세한 핵(nuclei)을 비교해본다.

The electromagnetic force is responsible for holding atoms together. Its properties are well known, classically, and rather easy to extrapolate to quantum distances. The study of atomic spectra indeed gave rise to quantum mechanics, which qualitatively and quantitatively explains many phenomena of condensed matter physics. The fine structure constant, the electromagnetic coupling constant, is a small number, α≊1/137, which makes perturbative calculations feasible.

전자기력(The electromagnetic force)은 원자를 묶어놓는다. 전자기력은 양자 거리에서 벗어난 규모에서 고전적으로 잘 알려져 있다. 사실 원자 스펙트럼(E=ħν)을 연구하는 과정에서 양자역학의 계기가 되었다. 이는 미세물질(아원자 물리학) 물리학(fine matter physics)의 여러 현상들을 잘 설명한다. 미세구조상수(fine structure constant) 혹은 전자기 결합상수(Coupling constant)가 α≊1/137 가량으로 매우 작아서 섭동이론(Perturvation theory) 적 계산이 유효하다.

The nuclear force, on the other hand, must be much stronger, since it wins over the Coulomb repulsion between tightly-packed protons. It must be of short range, since it doesn't make itself felt outside the nuclear volume. It has no classical analog. Only experimental results can help to understand its properties. One thus uses experiments as a guide towards empirical models of the nucleus and of the nuclear force. We will come back to the relation between experiments and models as we go along.

핵력(The nuclear force)은 한편 쿨롱 반반력을 넘어 양성자들을 서로 가깝게 묶어둘수 있을 정도로 훨씬 강력한 힘이다. 아주 좁은 범위에서 작동하며 핵 이외의 영역에 미치지 못한다. 핵력은 고전물리 설명될 수 없고 오직 실험적으로 그 특성을 이해할 수 있다.

[2.1-4] -------------------------------------------------------------



Let us first summarize how we identify and denote nuclei.

- 'Z', the nuclear charge, which is equal to the atomic number in the periodic table. It is given by the number of protons in the nucleus.

Z는 핵의 전하값으로 양성자의 수(number of proton)로 결정된다. 주기율표(ptable.com)의 원자번호(atomic number)에 해당한다.

- 'A' is the number of nucleons, the sum of protons and neutrons, it is also called the mass number.

A는 핵자(nucleons)의 수로 양성자와 중성자를 더한 값이다. 핵의 질량수(mass number)라고 하기도 한다.

- Nuclei are thus completely identified by their electric charge and the number of nucleons. The name we give them identifies Z and usually is supplemented by A. When we say carbon 14, that means a nucleus with Z=6 and A=14.

핵(Nucleus)은 하전량과 핵자의 수에 따라 분류된다. 즉, 원자는 이름에 Z 와 A 수가 따라 붙는다. 예를 들어 탄소14라고 한다면 핵에 Z=6이고 A=14라는 뜻이다.

- Evidently, the number of neutrons is then the difference between the mass number and the atomic number, and is denoted by N.

중성자(neutrons)의 수는 질량수와 원자번호에서 구할 수 있는데 N으로 표기한다. 질량수와 원자번호의 차이가 바로 중성자 수다.

- The chemical properties of the elements are determined by the electron cloud surrounding the nucleus. The periodic table is thus organized according to Z.

원소의 화학적 특성은 핵을 둘러싼 전자구름에 의해 결정된다. 주기율표는 Z로 구성한다. 원자는 하전수(양성자의 수)인 Z에 따라 구분된다.

- Nuclei with the same number of protons, but with a different number of neutrons, are called isotopes. They have very similar chemical properties but not necessarily similar nuclear properties.

핵이 동일한 개수의 양성자를 가졌으나 중성자의 수가 다른 경우를 동위원소(isotope)라고 한다. 이 원소들은 화학적 특성이 비슷하나 핵의 특성이 꼭 같다고 할 수는 없다.

Examples are uranium 235 and uranium 238, which have both 92 protons, but a different number of neutrons and thus different stability properties. Another example is hydrogen, deuterium, and tritium which all have one proton and zero, one, or two neutrons.

예를 들어 우라늄 235와 우라늄 238은 모두 양성자가 92 이지만 중성자의 수가 서로 다르며 그에따라 안정 특성이 다르다(핵이 쉽게 붕괴될 수 있음). 또다른 예로 수소가 있는데, 중수소() 삼중수소()로 불리우는 경우다. 각각 모두 한개의 양성자를 가지고 있지만 중성자의 수가 없거나 한개 혹은 두개를 가지고 있는 경우다.

- Nuclei can also be excited to higher states, keeping the number of protons and neutrons constant. These are called resonances or isomers.

핵이 동일한 양성자와 중성자를 가졌더라도 활성(be excited)상태에 있을 수 있는데 이를 이성질체(resonances or isomers)라고 한다.(양성자와 중성자의 배열, 결합이 다름)

- Nuclei with the same number of nucleons but the different number of protons are called isobars. They have roughly the same nuclear mass. Examples are carbon 12 and boron 12, with both have 12 nucleons.

중성자가 동일하면서 양성자가 다른 경우를 동중원소(isobar)라고 부른다.

[2.1-5] -------------------------------------------------------------



- Naively one might assume that the mass of nuclei is simply given by the sum of the masses of protons and neutrons they contain. But in reality, this mass is, of course, diminished by the binding energy between the nucleons.

핵의 질량(nuclear mass, A)를 단순히 양성자(Proton)와 중성자(Neutron)의 합(A=N+Z)이라고 하지만 엄밀히 말하자면 틀리다. 실제 핵의 질량에는 핵자들의 질량과 그들 사이의 결합 에너지(binding energy)를 포함하고 있다.



- The mass deficit ∆M must always be negative so that the nucleus is in a bound state. We call it the binding energy. It has been measured for practically all nuclei.

핵자(nucleons)의 질량(양성자와 중성자의 질량을 같게 놓은 핵의 질량)과 실제 핵의 질량, (Z*m_p) + (N*m_n) 의 차 ∆M은 0보다 작은데 이로인해 핵이 정상상태에 머무르게 한다.

- The absolute value of the binding energy is the energy required to decompose the nucleus into separate nucleons.

이 질량차 ∆M의 절대값을 결합 에너지라 한다. 역으로 핵자들을 분리하는데 필요한 에너지이기도 하다.



- The binding energy per nucleon, ∆M/A, is the energy required to separate the average nucleon from its nucleus. It is much smaller than the mass of a nucleon p and n. The mass of the nucleus is thus indeed dominated by the mass of its constituents.

핵자 당 결합 에너지 ∆M/A 는 핵에서 핵자 하나를 떼어내는 에너지라 할 수 있다. 이 에너지는 핵자인 p (양성자) 와 n (중성자)의 질량보다 아주 작다. 핵의 질량은 대부분 이들 핵자들의 질량이 차지한다.

- For the nucleons themselves the situation is very different. The total mass of the quarks inside the nucleon is only around 1% of the nucleon mass. It is their binding energy which dominates the nucleon mass.

핵자 하나하나의 질량을 따져보면 좀 다르다. 핵자의 내부를 구성하는 쿼크의 무게는 핵자의 무게에 1%에 지나지 않는다. 결합 에너지가 핵자 무게의 상당부분을 차지한다.

[2.1-6] -------------------------------------------------------------



When one analyzes the dependence of the binding energy, ∆M per nucleon A, on the mass number A one notices the following.

질량수 A와 핵자당 결합 에너지 ∆M/A의 상관관계를 보자.

- For A less than 20 on the left side of this graph, ∆M/A oscillates but rises rapidly with increasing A.

A가 20 이하인 구간에 놓인 원소들인 경우 ∆M/A가 들쭉날쭉 하지만 급격히 증가한다.

- For A between 20 and 60, ∆M/A saturates. For A about 60, it has a broad maximum. That is, the iron group formed by nickel, iron, and cobalt with about 9 MeV per nucleon of binding energy.

A가 20에서 60 사이의 구간에 놓인 원소들인 경우 ∆M/A는 최대치에 포화된다. 핵자의 결합 에너지가 최고인 '금속 계열(iron group)'의 원소들이다. 니켈, 철, 코벌트 등의 원소들이 있다. A가 60즈음에서 최대가 되는데 이런 금속계열 원소의 핵자당 결합 에너지는 9 MeV에 이른다.

- For A larger than 60, the binding energy per nucleon decreases slowly.

질량수 A가 60을 넘는 원소의 경우 결합 에너지는 다소 감소하지만 여전히 매우 높음을 유지한다.

The general mean is about 8 MeV per nucleon. The kinetic energy of nucleons inside the nucleus must thus be relatively small. Otherwise, they would not stay bound. The velocities of bound nucleons are thus non-relativistic.

핵자당 결합 에너지는 평균 약 8 MeV다.  핵 내부의 핵자들의 운동에너지는 단단히 결합되어 있으므로 상대적으로 작다. 묶인 핵자들의 속도는 따라서 비-상대론적이다.

[2.1-7] -------------------------------------------------------------



The binding energy corresponds indeed to a wavelength of nucleons inside the nucleus. This wavelength is less than two Fermi of the order of the nucleus' size itself. It is thus plausible that the nucleus can contain nucleons with the maximum kinetic energy of about 8 MeV. Or a maximum momentum of about 120 MeV.

If, on the other hand, the nucleus would contain electrons, they would be relativistic and their wavelength would be 2.5 x 10^-12 centimeters, much bigger than the nucleus size. The nucleus can thus not contain bound electrons. They need a much larger volume to be contained. This is obviously in agreement with the findings of Geiger and Marsden, that we have discussed in the previous module.

In the next video, we will talk about the size and the spin of nuclei.

[연습문제]-----------------------------------------------------------

W1.7 1주 평가문제(Graded quiz for module 1)

[커세라] 입자물리학 입문(Particle Physics: an Introduction)

1주: 물질과 힘 그리고 측정(Matter and forces, measuring and counting)
    W1.0 환영(Welcome)
    W1.1 물질(Matter)
    W1.2 힘(Forces)
        W1.2a 자연단위(Natural units)
        W1.2b 특수 상대론과 4-벡터(Special relativity and four-vector)
        W1.2c 가상입자(Virtual Particles)
    W1.3 확률과 단면(Probability and cross section)
        W1.3a 광자 빔의 감쇄(Attenuation of a photon beam)
    W1.4 러더포드 실험(Rutherford experiment)
        W1.4a 러더포드 단면(Rutherford cross section)
        W1.4b 산란율 계산(Counting rate Rutherford)
    W1.5 양자 산란(Quantum scattering)
    W1.6 러더포드 실험 실습(Rutherford experiment in practice)

W1.7 1주 평가문제(Graded quiz for module 1)

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W1.6 러더포드 실험 실습(Rutherford experiment in practice)

[커세라] 입자물리학 입문(Particle Physics: an Introduction)

1주: 물질과 힘 그리고 측정(Matter and forces, measuring and counting)
    W1.0 환영(Welcome)
    W1.1 물질(Matter)
    W1.2 힘(Forces)
        W1.2a 자연단위(Natural units)
        W1.2b 특수 상대론과 4-벡터(Special relativity and four-vector)
        W1.2c 가상입자(Virtual Particles)
    W1.3 확률과 단면(Probability and cross section)
        W1.3a 광자 빔의 감쇄(Attenuation of a photon beam)
    W1.4 러더포드 실험(Rutherford experiment)
        W1.4a 러더포드 단면(Rutherford cross section)
        W1.4b 산란율 계산(Counting rate Rutherford)
    W1.5 양자 산란(Quantum scattering)

W1.6 러더포드 실험 실습(Rutherford experiment in practice)/동영상/영문자막



>> To conclude the first module of this course, we visit the lab course on nuclear physics at University of Geneva, to see how one does an experiment of Rutherford. I present Dr. Alessandro Bravar, a senior lecturer in our department, who is responsible for this lab course. He is also an active researcher, like all our collaborators, who is part of our neutrino physics group and also has his own experiment, which he prepares to measure the detailed properties of muons. It will take place in a few years at the Paul Scherrer Institute in Villigen, near Zurich. So in front of us we have a typical set-up for a Rutherford experiment. Alessandro, can you please explain the ingredients of this set-up and the way it works?

첫주 과정의 마지막 편은 제네바 대학교의 핵물리 실험실을 방문하여 실제로 학생들이 러더포드 실험 실습을 어떻게 하는지 살펴보겠습니다. 이 실험은 우리 학과의 전임교수이신 알레산드로 브라바 박사님이 담당하고 계십니다. 중성미자 물리 그룹에 참여 하면서 뮤온의 특성을 정밀하게 측정하는 일을 합니다.

>> Okay. Here, we try to repeat the Rutherford measurements, but with modern equipment and a technique that looks a bit like the experiments that we make today to study, for example, the electron structure. So we have, in our case, an alpha emitter, which emits alpha particles sent to a target, and we are detecting alpha particles scattered by the target.

러더포드 실험을 재현하는데 현대 장비입니다. 알파 입자를 목표에 조사하여 산된된 알파 입자를 검출하는 원리는 같죠.

>> Indeed, it was not Rutherford, who did it first, it was Geiger and Marsden who have implemented his idea.

사실 이 실험장치를 생각해낸 사람은 러더포드가 아니라 가이거와 마스덴이죠.

>> But it was Rutherford who interpreted the observations of Geiger and Marsden. So to make this experiment, we have to work in a vacuum. This will require to open this box in a moment, to show all the ingredients.

하지만 가이거와 마스덴의 실험을 해석한 사람이 러더포드 입니다.

The reason that one works in vacuum, is that we use alpha particles and alpha particles are already stopped by an air layer of about ten centimeters. So that the alpha particles can propagate from the source to the target and our detector, we need to remove all the air that otherwise would stop the particles.

산란 실험장치는 진공용기 안에 있는데 알파 입자가 공기의 입자들과 반응하지 않도록 진공 조건하에서 실험을 해야 합니다.

We have therefore put in in this box with a pump that pumps the air; it makes the noise you hear now. What we will do now is to stop the pump and open the cover to see all the ingredients of the experiment.

>> Let's go. You open the valve?

>> Yes, first thing, open the valve and let the air in, otherwise we cannot lift the cover.

>> We hear the faint sound of air entering the vacuum chamber. I think this is it.

>> Not yet. We have to wait a while. Here we go. So we open the vacuum chamber and the three main components of the experiment can be seen.



Here we have a source of americium that emits alpha particles. Alpha particles travel to a target, here we use a gold foil with a thickness of 1 micron. Alpha particles interact with the gold nuclei and are scattered by the gold nuclei in different directions. To detect these alpha particles, we use a silicon detector, a diode placed in the tube here.

알파 입자 방출기(AM-2)는 아메리시움을 사용합니다. 알파 입자가 두께 1 마이크론의 금 박박으로 향해 조사되죠. 금 핵과 상호작용(쿨롱 력 반발)하여 산란되어 여러각도로 날아가죠. 산란된 알파 입자는 실리콘 다이오드에서 감지됩니다.

We will see the silicon detector in a little more detail. Thank you Martin. That's our silicon detector. The central part we see here is the active part of the detector, while the rest serves to define the acceptance of the detector. Behind it there is a small connector to measure the electrical signal generated by alpha particles, which are completely absorbed in this detector.

>> The signal is then propagated.

>> Yes, it goes through this cable and an amplifier we have here, because the charge left by an alpha particle in the silicon detector is only about a few femto-coulombs and we need to amplify this signal for detection.

>> Then the experiment is to measure the count rate, the rate of alpha particles, which have been deflected, depending on the angle that the detector made with incident alpha beam. The target may be varied, it may be gold of different thicknesses, and can also be made of other metals, such as iron here, 3 microns of iron, nickel, a thin foil 3 microns thick, or different thicknesses of gold, as here, a gold foil of half a micron.

One varies two things: the target, that is to say the Z and the thickness Δx of the target in the direction of the beam, and one varies the angle. How many different angles does one typically measure in an experiment?

>> About a dozen different angles One puts the detector on both sides of the alpha beam to ensure, that we make a symmetric measurement relative to the direction of incident alpha beam.

>> Obviously, the count rate varies very quickly with the angle. This is the famous 1/sin^4(θ/2) at work. Then, the count rate varies between what and what, roughly?

산란율은 각도에 따라 크게 차이가 나겠지요? 잘 알려진 대로 1/(sin(θ/2))^4 가 잘 맞습니까?

>> If you make the measurement in the direction of the incident beam, you measure, in principle, all the alpha particles from the source, those that were scattered to very small angle, but also those, which have not interacted in target. And thus the count rate, in this case, is roughly a few kilohertz. So we can measure very quickly and acquire enough events in a very short period.

만일 알파 입자가 입사하는 각도에 정면에서 측정 하면 수 킬로 헬쯔(초당 수천개)의 알파입자가 감지 됩니다. 이 경우 알파 입자는 산란되지 않고 통과한 입자들이 포함되죠.

>> This also gives us the intensity of the incident beam.

>> Exactly. Then, to measure the count rate at different scattering angles, we move the silicon detector up to about 30 degrees, on déplace le détecteur à silicium sometimes 45 degrees, relative to the axis of alpha particles. It is clear that the count rate drops dramatically as we vary the angle.

>> What is the 45 degree count rate, roughly?

그럼 45도 각도로 감지기를 놓으면요?

>> It's very small, one event per minute so we need to measure over several days to accumulate sufficient statistics.

감지되는 입자수는 아주 작아지죠. 분당 한개 꼴이 되기도 합니다. 통계적으로 의미있는 측정을 하려면 몇일씩 걸리기도 합니다.

>> On the contrary, at low angles, it is a lot higher, right?

>> A lot higher. As I said earlier, this is about a few kilohertz at zero angle, and at small angles the counting rate is still high.

>> Okay. Then, students spend about a day to do this experiment, or a day and a night perhaps?

그럼 학생들이 날밤 세워 실험 하나요?

>> Well, if we measure angles in the direction of the particle, the measurement can be done in a few hours. While when we measure at fairly large angles, we leave the experiment running a whole week. Students spend the day to start the experiment, check that all the ingredients are working properly, and afterwards the measurement can be taken, as I said, in a week.

각도를 바꿔가며 측정을 하는데 보통 수시간씩 걸립니다. 각도를 높히면 일주일씩 측정하기도 합니다.

So, I put the sensor at zero angle to have a significant count rate and see things in real time. To run the experiment, you need to put all back into vacuum. So we will close the vacuum chamber and switch on the pump again.

그래서 실험을 보여주기 위해 검출기 각도를 0에 놓죠(알파 입자 방출기와 정면으로 향함).



>> So, once the chamber is closed, the internal pressure drops relatively quickly, and it is also possible to apply the bias voltage to the silicon diode that is used for counting the passage of particles. Here, you use which bias voltage?

>> Here we work with 24V, this is due to the thickness. Also, one does not need to deplete the detector completely, as alpha particles stop within the first few microns of detector thickness.

>> If the first microns are depleted, this is perfectly adequate for accumulating the signal. So here, we see the signal of alpha particles passing, the typical signal curve shown by a silicon detector; this is the voltage as a function of time.

>> As I said before, we have a very, very low signal, a very, very low charge, some femto-coulombs. So we need a first amplifier to amplify the signal about 10’000 times, the signal seen here, so this is a charge amplifier. That is why we see a signal that rises very quickly, with a much longer decay time.

앞서도 말했지만 검출되는 신호는 아주아주 약합니다. 수 펨토-쿨롱쯤 되요. 그래서 증폭기를 써서 약 1만배가량 증폭 시킵니다. 증폭된 신호의 모습은 이렇습니다. 전하를 띈 알파 입자가 검출기를 급격히 충전 시켰다가 서서히 방전되는 모습이죠.



>> So the method of measurement would trigger a counter once the voltage exceeds a certain threshold, and count the rate of these events to measure the interaction rate.

그럼 알파 입자를 세는 방법은 검출기에 지정된 전압이상으로 올라올때 마다 그 횟수를 세는군요.

>> Here, actually, we do something a little more sophisticated. Instead of only measuring the count rate, we measure the charge deposited by the alpha particle in the detector, to be sure that the detected pulse is generated by an alpha particle. To do this, we need to go through a second amplifier that will shape the signal as seen here. In principle, it converts the signal to almost Gaussian shape, so it has a slower rise, and afterwards we use a charge-to-digital converter. This transforms the signal into something that can be used with a computer.

>> That is to say that we measure the dE/dx of the particles, which they leave in the thickness of the detector.

>> Once we have converted the charge to a binary number, a computer is used to store the data, i.e. the charge deposited by alpha particles, and we obtain an energy spectrum of the alpha particles.

>> Okay. Because they are completely absorbed, it is a kind of miniature calorimeter. But at the same time, one obviously saves the count rate that will give us the Rutherford cross section.

>> Exactly. So we must also record the time we used to measure, because it gives us the count rate. We record a number of pulses and divide by the measurement time.

W1.5 양자 산란(Quantum scattering)

[커세라] 입자물리학 입문(Particle Physics: an Introduction)

1주: 물질과 힘 그리고 측정(Matter and forces, measuring and counting)
    W1.0 환영(Welcome)
    W1.1 물질(Matter)
    W1.2 힘(Forces)
        W1.2a 자연단위(Natural units)
        W1.2b 특수 상대론과 4-벡터(Special relativity and four-vector)
        W1.2c 가상입자(Virtual Particles)
    W1.3 확률과 단면(Probability and cross section)
        W1.3a 광자 빔의 감쇄(Attenuation of a photon beam)
    W1.4 러더포드 실험(Rutherford experiment)
        W1.4a 러더포드 단면(Rutherford cross section)
        W1.4b 산란율 계산(Counting rate Rutherford)

W1.5 양자 산란(Quantum scattering)/동영상/영문자막/슬라이드



In this first module, we're in the process of introducing objects studied by particle physics, namely matter, forces and space-time. And in this context we must obviously discuss scattering processes.

In this fifth video, we will show how to approach the scattering processes between particles in a quantum way. The goals for you are;

- to identify the conceptual differences between the classical approach that we've used up to now, and the quantum evolution of a system.

이제까지 입자의 산란에 대해 고전적인 방식(운동에너지와 쿨롱 력)으로 접근해 봤다. 이제 양자학(quantum physics)으로 시각을 옮겨보기로 한다.

- to know how to draw a Feynman diagram for simple scattering processes and explain its ingredients.

산란의 과정을 쉽게 설명하는 파인먼 도(Feynman diagram)에 대해 배우고 (입자의 세계를) 어떻게 설명하는지 살펴보자.



In video 1.4 Mercedes has calculated the differential cross section for Coulomb scattering off a static target, which is reproduced here. By lucky coincidence, the classical result is still valid in a relativistic quantum context.

앞서 1.4편에서 (이해하기 어려운 과목을 알아듣기 어려운 영어로 강의하심) 메르세데스 선생이 정지한 (무거운)목표 입자의 (고전적인) 쿨롱 산란으로 '단면'을 설명했다. 다행 스럽게도 상대론적 양자론(relativistic quantum context)에서도 이 고전적인 결과는 유효하다.

The reasons are the following.

그 이유는 다음과 같다.

- The result of quantum theory contains no factor of ħ. This means that the artificial limit ħ going to zero, ħ⟶0, which normally takes us back to the classical result will not change the answer that we obtained.

양자론으로 얻은 결과에도 ħ 인자는 포함되지 않는다. 이는 극한 ħ⟶0 일때 고전적 접근 법으로 얻은 결과와 같다는 뚯이다.

- The classical result is also valid in the relativistic regime already. This fact is not surprising, Maxwell's equations are valid for relativistic velocities. After all, they also they describe electromagnetic wave including the motion of photons, which moves at the speed of light.

고전적 결과는 상대론적 영역에서도 이미 잘 작동한다. 이는 맥스웰 방정식이 상대론적 속도에서도 잘 적용된다는 사실은 놀라울 것이 없다. 무엇보다도 맥스웰 방정식은 빛의 속도로 움직이는 광자의 전자기 파동을 잘 기술 한다.

- We also do not need to take into account nuclear interactions between projectile and target, since the alpha particle will never penetrate into the target nucleus, even for head-on collision. Only the electromagnetic force acts outside the nuclear volume.

입사 입자와 목표 입자 사이의 핵반응을 고려할 필요가 없다. 입사하는 알파 입자가 정면 충돌이라 해도 핵을 뚫고 지나가지 못하기 때문이다. 전자기력이 핵의 외부 주변에서도 강력히 작동하기 때문이다.

But until now, we have discussed the scattering process in a language adapted to classical physics. Now, we will change toolkit to discuss the quantum approach.



While the classical results stays valid in the quantum regime, the interpretation is totally different. Particles may well be point-like but they move like a probability wave.

(입자의 산란은 강체의 탄성 충돌로 보고 뉴튼 역학 운동에너지로 푼)고전적 결과가 양자론과도 일치하지만 그 해석은 완전히 다르다. 양자론의 영역에서는 점입자(point-like particle)인 된시에 확률 파동(probability wave)의 성격을 더 띄고 있다고 본다.

- Because of Heisenberg's principle, the impact perimeter b has no more role in our consideration. We must use a scenario of Huygen to understand how scattering works. This means that the target particle will be the origin of a new scattered wave, which will add to the incoming wave of the projectile.

- The classical trajectory is replaced by the wave function ψ(x), as we've mentioned earlier. It contains the particle aspect, energy and momentum, (E, p) and the wave expect, frequency and wave number, (ω,k) of the kinematics at the same time.

- Before the scattering, the projectile is described by a plane wave in the x direction in our little example. Here we have also normalized the amplitude to one for simplicity.

- Huygens' principle says that a small fraction f of the incident amplitude will be spherically re-emitted by the target. The factor 1/r in the scattered amplitude is necessary. It ensures that the number of scattered particles, ψ*ψ, remains independent of distance.

- The remaining amplitude (1-f) continues as an unperturbed plane wave.



As a consequence of the wave approach we must also reinterpret the cross section in terms of intensities of incoming and scattered wave.

- This means that we now look for a relation between the cross section and the scattered amplitude f(θ,φ).

- The square of the second term is proportional to the number of particles scattered into a volume subtended by a solid angle dΩ, and the radial thickness dr. It represents the scattered flux.

- For a normalized incident wave the particle density is ρ=1. Thus the incident flux is simply ρ times velocity or the velocity itself. I = ρv = v

- The ratio between the scattered and the incoming flux is precisely the cross section.

- We find a simple result, f is the probability amplitude for our scattering process. And σ = f^2 is the scattering probability itself.

- The amplitude f is calculable if we know the potential generated by the target, just like in the classical sense.



To visualize reactions between particles and calculate their probability, we use a very useful tool which is called Feynman diagrams.

- They represent lines of propagation of particles in coordinates E and p. Not in space-time but in momentum space.

- They also represent the vertices of an interaction, that is the points where a force particle is emitted or absorbed to transmit energy and momentum.

- Finally they represent the virtual particles living between two vertices. Those have the properties of their real counter part, but not the mass of a free particle.

- Feynman diagrams are a useful visualization of a reaction but also a prescription for
calculating their probability amplitude. One does that by applying what is called Feynman rules.

- At each vertex, energy-momentum and quantum numbers are rigorously conserved.

- The virtual particle transfers energy momentum from one of these particles to the other one, thus a force act between the projectile and the target.

- Details on how to construct a Feynman diagram will be discussed in module 4 when we talk about electromagnetic interactions in a quantum way.



- For each type of elementary interaction there is a spin 1 boson transmitting the force as we have explained before. They are collectively called gauge bosons:

    > The photon transmits electromagnetic interactions.
    > The W and Z bosons transmit the two forms of weak interactions.
    > The eight different gluons are responsible for transmitting strong interactions.

- To absorb or emit one of these bosons, a particle must carry the required type of charge. In our jargon we say that the boson couples to a given charge. The charge is thus a coupling constant.

    > It needs electrical charge Q to couple to the photon. Q has 1 component.
    > It needs weak isospin T and T_3 to couple to W and Z. This charge has 2 components.
    > It needs color charge R, G or B to couple to gluons. This charge has 3 components.

- The probability amplitude to emit or absorb a gauge boson is proportional to the charge of the particle. The probability is thus proportional to the square of the charge as we found for Rutherford scattering already.



- In the Feynman diagram at each vertex energy and momentum are conserved. But do not forget that virtual particles which relate two vertices do not necessarily have the mass of their real counterpart nor even a real number as their mass.

- At each vertex charges as well as baryon and lepton number are conserved. Flavor is conserved by electromagnetic and strong interactions as we stated before but not by the weak one. Charge weak interactions transmitted by a W boson in fact change a particle flavor.

- Here are a few examples of allowed and forbidden vertices.

    > In this top left diagram, neither the photon nor the Z boson can change the flavor of the lepton. The electron must thus stay an electron.



    > In the second top diagram from the left, conservation of charge requires that the emitted W boson has a negative charge. We can only emit a negative boson. This transforms the electron into a neutrino.



    > On the right-hand diagram, we show a diagram that does simply not exist. Since the electron does not carry color charge, it cannot emit or absorb a gluon so this diagram is simply non-existant.



    > On the bottom left you find a u quark that emits a photon or a Z. Neither of these two particles can change the flavor of the quark such that it must stay a u quark and cannot become any other quark.



    > In the middle diagram, the flavor changes. The u quark becomes a d quark by emitting a W. And if you make the balance of the charges you start out with a plus two-third charge and you come out with a minus one-third charge quark. So you must emit a W^+ to conserve charge.



    > The right-most diagram is a diagram of strong interactions. The incoming red d quark emits a gluon and becomes a green d quark. And in order to conserve color the quantum numbers carried by the gluon must be red, anti-green. On the other hand, the gluon is not able to change the flavor of the particles such that the d quark must indeed stay at d quark.



In the next video we will visit the laboratory of the nuclear physics course at University of Geneva to see how our students go about to measure the Rutherford cross-section.