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# ifft2

2-D inverse fast Fourier transform

## Syntax

``X = ifft2(Y)``
``X = ifft2(Y,m,n)``
``X = ifft2(___,symflag)``

## Description

example

````X = ifft2(Y)` returns the two-dimensional discrete inverse Fourier transform of a matrix using a fast Fourier transform algorithm. If `Y` is a multidimensional array, then `ifft2` takes the 2-D inverse transform of each dimension higher than 2. The output `X` is the same size as `Y`.```

example

````X = ifft2(Y,m,n)` truncates `Y` or pads `Y` with trailing zeros to form an `m`-by-`n` matrix before computing the inverse transform. `X` is also `m`-by-`n`. If `Y` is a multidimensional array, then `ifft2` shapes the first two dimensions of `Y` according to `m` and `n`.```

example

````X = ifft2(___,symflag)` specifies the symmetry of `Y`. For example, `ifft2(Y,'symmetric')` treats `Y` as conjugate symmetric.```

## Examples

collapse all

You can use the `ifft2` function to convert 2-D signals sampled in frequency to signals sampled in time or space. The `ifft2` function also allows you to control the size of the transform.

Create a 3-by-3 matrix and compute its Fourier transform.

`X = magic(3)`
```X = 3×3 8 1 6 3 5 7 4 9 2 ```
`Y = fft2(X)`
```Y = 3×3 complex 45.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 13.5000 + 7.7942i 0.0000 - 5.1962i 0.0000 - 0.0000i 0.0000 + 5.1962i 13.5000 - 7.7942i ```

Take the inverse transform of `Y`, which is the same as the original matrix `X`, up to round-off error.

`ifft2(Y)`
```ans = 3×3 8.0000 1.0000 6.0000 3.0000 5.0000 7.0000 4.0000 9.0000 2.0000 ```

Pad both dimensions of `Y` with trailing zeros so that the transform has size 8-by-8.

```Z = ifft2(Y,8,8); size(Z)```
```ans = 1×2 8 8 ```

For nearly conjugate symmetric matrices, you can compute the inverse Fourier transform faster by specifying the `'symmetric'` option, which also ensures that the output is real.

Compute the 2-D inverse Fourier transform of a nearly conjugate symmetric matrix.

```Y = [3+1e-15*i 5; 5 3]; X = ifft2(Y,'symmetric')```
```X = 2×2 4 0 0 -1 ```

## Input Arguments

collapse all

Input array, specified as a matrix or a multidimensional array. If `Y` is of type `single`, then `ifft2` natively computes in single precision, and `X` is also of type `single`. Otherwise, `X` is returned as type `double`.

Data Types: `double` | `single` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32` | `logical`
Complex Number Support: Yes

Number of inverse transform rows, specified as a positive integer scalar.

Data Types: `double` | `single` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32` | `logical`

Number of inverse transform columns, specified as a positive integer scalar.

Data Types: `double` | `single` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32` | `logical`

Symmetry type, specified as `'nonsymmetric'` or `'symmetric'`. When `Y` is not exactly conjugate symmetric due to round-off error, `ifft2(Y,'symmetric')` treats `Y` as if it were conjugate symmetric. For more information on conjugate symmetry, see Algorithms.

collapse all

### 2-D Inverse Fourier Transform

This formula defines the discrete inverse Fourier transform X of an m-by-n matrix Y:

`${X}_{p,q}=\frac{1}{m}\sum _{j=1}^{m}\frac{1}{n}\sum _{k=1}^{n}{\omega }_{m}^{\left(j-1\right)\left(p-1\right)}{\omega }_{n}^{\left(k-1\right)\left(q-1\right)}{Y}_{j,k}$`

ωm and ωn are complex roots of unity:

`$\begin{array}{l}{\omega }_{m}={e}^{2\pi i/m}\\ {\omega }_{n}={e}^{2\pi i/n}\end{array}$`

i is the imaginary unit. p runs from 1 to m and q runs from 1 to n.

## Algorithms

• The `ifft2` function tests whether the vectors in a matrix `Y` are conjugate symmetric in both dimensions. A vector `v` is conjugate symmetric when the ith element satisfies ```v(i) = conj(v([1,end:-1:2]))```. If the vectors in `Y` are conjugate symmetric in both dimensions, then the inverse transform computation is faster and the output is real.


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Problems and Solutions from CMI Entrance 2022.  Learn More

# B.Math 2007 Objective Paper| Problems & Solutions

Here are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2007.

Problem 1 :

The number of ways of going up $7$ steps if we take one or two steps at a time is

(A) $19$ ;
(B) $20$;
(C) $21$ ;
(D) $22$ .

Problem 2 :

Consider the surface defined by $x^{2}+2 y^{2}-5 z^{2}=0$. If we cut the surface by the plane given by the equation $x=z,$ then we obtain a

(A) hyperbola;
(B) circle;
(C) parabola;
(D) pair of straight lines.

Problem 3 :

Let $a, b$ be real numbers. The number of real solutions of the system of equations $x+y=a$ and $x y=b$ is

(A) at most $1$ ;
(B) at most $2$;
(C) at least $1$;
(D) at least $2$.

Problem 4 :

If a fair coin is tossed $100$ times, then the probability of getting at least one head is

(A) $\frac{100}{2^{100}}$;
(B) $\frac{99}{100}$;
(C) $1-\frac{1}{100 !}$;
(D) $1-\frac{1}{2^{100}}$.

Problem 5 :

Let $f(x)$ be a degree five polynomial with real coefficients. Then the number of real roots of $f$ must be

(A) $1$;
(B) $2$ or $4$;
(C) $1$ or $3$ or $5$ ;
(D) none of the above.

Problem 6 :

The number of ways in which $3$ girls and $2$ boys can sit on a bench so that no two girls are adjacent is

(A) $6$ ;
(B) $12$ ;
(C) $32$ ;
(D) $120$ .

Problem 7 :

Let $R_{n}=2+\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}\left(n\right.$ square root signs). Then $\lim _{n \rightarrow \infty} R_{n}$ equals

(A) $4$;
(B) $8$;
(C) $16$ ;
(D) $e^{2}$.

Problem 8 :

Let $a_{n}$ be the sequence whose $n$ th term is the sum of the digits of the natural number $9 n$. For example, $a_{1}=9, a_{11}=18$ etc. The minimum $m$ such that $a_{m}=81$ is

(A) $110111112$ ;
(B) $119111113$ ;
(C) $111111111$;
(D) none of the above.

Problem 9 :

$\lim _{n \rightarrow \infty}\left[2 \log (3 n)-\log \left(n^{2}+1\right)\right]$

(A) is $0$ ;
(B) is $2 \log 3$;
(C) is $4 \log 6$;
(D) does not exist.

Problem 10 :

Let $S=\{x \in \mathbb{R}|1 \leq| x \mid \leq 100\}$ be a subset of the real line. Let $M$ be a non-empty subset of $\bar{S}$ such that for all $x, y$ in $M,$ their product $x y$ is also in $M .$ Then $M$ can have

(A) only one element;
(B) at most $2$ elements;
(C) more than $2$ but only finitely many elements;
(D) infinitely many elements.

Problem 11:

An astronaut lands on a planet and meets a native of the planet. She asks the native "How many days do you have in your year?" He answers "It is the sum of the souares of three consecutive natural numbers but it is also the sum of the sauares of the next two numbers". The answer to the astronaut's question is

(A) $365$;
(B) $1095$ ;
(C) $30000$ ;
(D) $10^{10}$.

Problem 12 :

Let $a_{1}=2$ and for all natural number $n$, define $a_{n+1}=a_{n}\left(a_{n}+1\right) .$ Then, as $n \rightarrow \infty$, the number of prime factors of $a_{n}$

(A) goes to infinity;
(B) goes to a finite limit;
(C) oscillates boundedly;
(D) oscillates unboundedly.

Problem 13 :

Suppose that the equation $a x^{2}+b x+c=0$ has a rational solution. If $a, b, c$ are integers then

(A) at least one of $a, b, c$ is even;
(B) all of $a, b, c$ are even;
(C) at most one of $a, b, c$ is odd;
(D) all of $a, b, c$ are odd.

Problem 14 :

Let $S={1,2,3,4}$. The number of functions $f: S \rightarrow S$ such that $f(i) \leq 2 i$ for all $i \in S$ is

(A) $32$ ;
(B) $64$;
(C) $128$;
(D) $256$ .

Problem 15 :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=x^{2}-\frac{x^{2}}{1+x^{2}}$. Then

(A) $f$ is one-one but not onto;
(B) $f$ is onto but not one-one;
(C) $f$ is both one-one and onto;
(D) $f$ is neither one-one nor onto.

Problem 16 :

Define the sequence $\{a_{n}\}$ by $a_{1}=1, a_{2}=\frac{e}{2}, a_{3}=\frac{e^{2}}{4}, a_{4}=\frac{e^{3}}{8}, \ldots$ Then
$\lim_ {n \rightarrow \infty} a_{n}$ is

(A) $0$;
(B) $1$;
(C) $e^{e}$;
(D) infinite.

Problem 17 :

Let $C$ be the circle of radius 1 around 0 in the complex plane and $z_{0}$ be a fixed point on $C$. Then the number of ordered pairs $\left(z_{1}, z_{2}\right)$ of points on $C$ such that $z_{0}+z_{1}+z_{2}=0$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 18 :

The number of real solutions of $e^{x}+x^{2}=\sin x$ is

(A) $0$;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 19 :

The set of complex numbers $z$ such that $|z+1| \leq|z-1|$ is the half plane

(A) of complex numbers that lie above the real axis;
(B) of complex numbers that lie below the real axis;
(C) of complex numbers that lie left of the imaginary axis;
(D) of complex numbers that lie right of the imaginary axis.

Problem 20 :

$\lim _{x \rightarrow 0} \cos (\sin x)$ is

(A) $-1$ ;
(B) $0$;
(C) $1 / 2$;
(D) $1$ .

Problem 21 :

The number of rational roots of the polynomial $x^{3}-3 x-1$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $3$.

Problem 22 :

$\sum_{n=1}^{\infty} \frac{n^{2}}{n !}$ equals

(A) $e$;
(B) $2 e$;
(C) $e^{2}$;
(D) $\infty$.

Problem 23 :

Let $n>1$ be a natural number and let $A=\left(\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right) .$ Then

(A) $A^{n}=I d$;
(B) $A^{n^{2}+1}=I d$;
(C) $A^{n^{4}+1}=I d$;
(D) none of these numbers.

Problem 24 :

The number of ways of breaking a stick of length $n>1$ into $n$ pieces of unit length (at each step break one of the pieces with length $>1$ into two pieces of integer lengths) is

(A) $(n-1) !$;
(B) $n !-1$;
(C) $2^{n-2}$;
(D) $2^{n-1}-1$.

Problem 25 :

Let $A B C$ be a right angled triangle in the plane with area $s$. Then the maximum area of a rectangle inside $A B C$ is

(A) $s / 4$;
(B) $s / 3$;
(C) $s / 2$;
(D) $s$.

## Some Useful Links:

Here are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2007.

Problem 1 :

The number of ways of going up $7$ steps if we take one or two steps at a time is

(A) $19$ ;
(B) $20$;
(C) $21$ ;
(D) $22$ .

Problem 2 :

Consider the surface defined by $x^{2}+2 y^{2}-5 z^{2}=0$. If we cut the surface by the plane given by the equation $x=z,$ then we obtain a

(A) hyperbola;
(B) circle;
(C) parabola;
(D) pair of straight lines.

Problem 3 :

Let $a, b$ be real numbers. The number of real solutions of the system of equations $x+y=a$ and $x y=b$ is

(A) at most $1$ ;
(B) at most $2$;
(C) at least $1$;
(D) at least $2$.

Problem 4 :

If a fair coin is tossed $100$ times, then the probability of getting at least one head is

(A) $\frac{100}{2^{100}}$;
(B) $\frac{99}{100}$;
(C) $1-\frac{1}{100 !}$;
(D) $1-\frac{1}{2^{100}}$.

Problem 5 :

Let $f(x)$ be a degree five polynomial with real coefficients. Then the number of real roots of $f$ must be

(A) $1$;
(B) $2$ or $4$;
(C) $1$ or $3$ or $5$ ;
(D) none of the above.

Problem 6 :

The number of ways in which $3$ girls and $2$ boys can sit on a bench so that no two girls are adjacent is

(A) $6$ ;
(B) $12$ ;
(C) $32$ ;
(D) $120$ .

Problem 7 :

Let $R_{n}=2+\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}\left(n\right.$ square root signs). Then $\lim _{n \rightarrow \infty} R_{n}$ equals

(A) $4$;
(B) $8$;
(C) $16$ ;
(D) $e^{2}$.

Problem 8 :

Let $a_{n}$ be the sequence whose $n$ th term is the sum of the digits of the natural number $9 n$. For example, $a_{1}=9, a_{11}=18$ etc. The minimum $m$ such that $a_{m}=81$ is

(A) $110111112$ ;
(B) $119111113$ ;
(C) $111111111$;
(D) none of the above.

Problem 9 :

$\lim _{n \rightarrow \infty}\left[2 \log (3 n)-\log \left(n^{2}+1\right)\right]$

(A) is $0$ ;
(B) is $2 \log 3$;
(C) is $4 \log 6$;
(D) does not exist.

Problem 10 :

Let $S=\{x \in \mathbb{R}|1 \leq| x \mid \leq 100\}$ be a subset of the real line. Let $M$ be a non-empty subset of $\bar{S}$ such that for all $x, y$ in $M,$ their product $x y$ is also in $M .$ Then $M$ can have

(A) only one element;
(B) at most $2$ elements;
(C) more than $2$ but only finitely many elements;
(D) infinitely many elements.

Problem 11:

An astronaut lands on a planet and meets a native of the planet. She asks the native "How many days do you have in your year?" He answers "It is the sum of the souares of three consecutive natural numbers but it is also the sum of the sauares of the next two numbers". The answer to the astronaut's question is

(A) $365$;
(B) $1095$ ;
(C) $30000$ ;
(D) $10^{10}$.

Problem 12 :

Let $a_{1}=2$ and for all natural number $n$, define $a_{n+1}=a_{n}\left(a_{n}+1\right) .$ Then, as $n \rightarrow \infty$, the number of prime factors of $a_{n}$

(A) goes to infinity;
(B) goes to a finite limit;
(C) oscillates boundedly;
(D) oscillates unboundedly.

Problem 13 :

Suppose that the equation $a x^{2}+b x+c=0$ has a rational solution. If $a, b, c$ are integers then

(A) at least one of $a, b, c$ is even;
(B) all of $a, b, c$ are even;
(C) at most one of $a, b, c$ is odd;
(D) all of $a, b, c$ are odd.

Problem 14 :

Let $S={1,2,3,4}$. The number of functions $f: S \rightarrow S$ such that $f(i) \leq 2 i$ for all $i \in S$ is

(A) $32$ ;
(B) $64$;
(C) $128$;
(D) $256$ .

Problem 15 :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=x^{2}-\frac{x^{2}}{1+x^{2}}$. Then

(A) $f$ is one-one but not onto;
(B) $f$ is onto but not one-one;
(C) $f$ is both one-one and onto;
(D) $f$ is neither one-one nor onto.

Problem 16 :

Define the sequence $\{a_{n}\}$ by $a_{1}=1, a_{2}=\frac{e}{2}, a_{3}=\frac{e^{2}}{4}, a_{4}=\frac{e^{3}}{8}, \ldots$ Then
$\lim_ {n \rightarrow \infty} a_{n}$ is

(A) $0$;
(B) $1$;
(C) $e^{e}$;
(D) infinite.

Problem 17 :

Let $C$ be the circle of radius 1 around 0 in the complex plane and $z_{0}$ be a fixed point on $C$. Then the number of ordered pairs $\left(z_{1}, z_{2}\right)$ of points on $C$ such that $z_{0}+z_{1}+z_{2}=0$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 18 :

The number of real solutions of $e^{x}+x^{2}=\sin x$ is

(A) $0$;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 19 :

The set of complex numbers $z$ such that $|z+1| \leq|z-1|$ is the half plane

(A) of complex numbers that lie above the real axis;
(B) of complex numbers that lie below the real axis;
(C) of complex numbers that lie left of the imaginary axis;
(D) of complex numbers that lie right of the imaginary axis.

Problem 20 :

$\lim _{x \rightarrow 0} \cos (\sin x)$ is

(A) $-1$ ;
(B) $0$;
(C) $1 / 2$;
(D) $1$ .

Problem 21 :

The number of rational roots of the polynomial $x^{3}-3 x-1$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $3$.

Problem 22 :

$\sum_{n=1}^{\infty} \frac{n^{2}}{n !}$ equals

(A) $e$;
(B) $2 e$;
(C) $e^{2}$;
(D) $\infty$.

Problem 23 :

Let $n>1$ be a natural number and let $A=\left(\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right) .$ Then

(A) $A^{n}=I d$;
(B) $A^{n^{2}+1}=I d$;
(C) $A^{n^{4}+1}=I d$;
(D) none of these numbers.

Problem 24 :

The number of ways of breaking a stick of length $n>1$ into $n$ pieces of unit length (at each step break one of the pieces with length $>1$ into two pieces of integer lengths) is

(A) $(n-1) !$;
(B) $n !-1$;
(C) $2^{n-2}$;
(D) $2^{n-1}-1$.

Problem 25 :

Let $A B C$ be a right angled triangle in the plane with area $s$. Then the maximum area of a rectangle inside $A B C$ is

(A) $s / 4$;
(B) $s / 3$;
(C) $s / 2$;
(D) $s$.

## Some Useful Links:

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### Knowledge Partner

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# Discovery of nuclear fission

The nuclear reaction theorised by Meitner and Frisch.

Nuclear fission was discovered in December 1938 by physicists Lise Meitner and Otto Robert Frisch and chemists Otto Hahn and Fritz Strassmann. Fission is a nuclear reaction or radioactive decay process in which the nucleus of an atom splits into two or more smaller, lighter nuclei. The fission process often produces gamma rays and releases a very large amount of energy, even by the energetic standards of radioactive decay. Scientists already knew about alpha decay and beta decay, but fission assumed great importance because the discovery that a nuclear chain reaction was possible led to the development of nuclear power and nuclear weapons.

Hahn and Strassmann at the Kaiser Wilhelm Institute for Chemistry in Berlin bombarded uranium with slow neutrons, and discovered that barium had been produced. They reported their findings by mail to Meitner in Sweden, who a few months earlier had fled Nazi Germany. Meitner and her nephew Frisch theorised, and then proved, that the uranium nucleus had been split, and published their findings in Nature. Meitner calculated that the energy released by each disintegration was approximately 200 megaelectronvolts, and Frisch observed this. By analogy with the division of biological cells, he named the process "fission". Hahn was awarded the 1944 Nobel Prize in Chemistry for the discovery.

The discovery came after forty years of investigation into the nature and properties of radioactivity and radioactive substances. The discovery of the neutron by James Chadwick in 1932 created a new means of nuclear transmutation. Enrico Fermi and his colleagues in Rome studied the results of bombarding uranium with neutrons, and Fermi concluded that his experiments had created new elements with 93 and 94 protons, which his group dubbed ausonium and hesperium. Fermi won the 1938 Nobel Prize in Physics for his "demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons".[1] However, not everyone was convinced by Fermi's analysis of his results. Ida Noddack suggested that instead of creating a new, heavier element 93, it was conceivable that the nucleus had broken up into large fragments, and Aristid von Grosse suggested that what Fermi's group had found was an isotope of protactinium.

This spurred Hahn and Meitner, the discoverers of the most stable isotope of protactinium, to conduct a four-year-long investigation into the process with their colleague Strassmann. After much hard work and many discoveries, they determined that what they were observing was fission, and that the new elements that Fermi had found were fission products. Their work overturned long-held beliefs in physics and paved the way for the discovery of the real elements 93 (neptunium) and 94 (plutonium), for the discovery of fission in other elements, and for the determination of the role of the uranium-235 isotope in that of uranium. Niels Bohr and John Wheeler reworked the liquid drop model to explain the mechanism of fission.

### Background

In the last years of the 19th century, scientists frequently experimented with the cathode-ray tube, which by then had become a standard piece of laboratory equipment. A common practice was to aim the cathode rays at various substances and to see what happened. Wilhelm Röntgen had a screen coated with barium platinocyanide that would fluoresce when exposed to cathode rays. On 8 November 1895, he noticed that even though his cathode-ray tube was not pointed at his screen, which was covered in black cardboard, the screen still fluoresced. He soon became convinced that he had discovered a new type of rays, which are today called X-rays. The following year Henri Becquerel was experimenting with fluorescent uranium salts, and wondered if they too might produce X-rays.[2] On 1 March 1896 he discovered that they did indeed produce rays, but of a different kind, and even when the uranium salt was kept in a dark drawer, it still made an intense image on an X-ray plate, indicating that the rays came from within, and did not require an external energy source.[3]

The periodic table circa 1930

Unlike Röntgen's discovery, which was the object of widespread curiosity from scientists and lay people alike for the ability of X-rays to make visible the bones within the human body, Becquerel's discovery made little impact at the time, and Becquerel himself soon moved on to other research,[4] Marie Curie tested samples of as many elements and minerals as she could find for signs of Becquerel rays, and in April 1898 also found them in thorium. She gave the phenomenon the name "radioactivity".[5] Along with Pierre Curie and Gustave Bémont, she began investigating pitchblende, a uranium-bearing ore, which was found to be more radioactive than the uranium it contained. This indicated the existence of additional radioactive elements. One was chemically akin to bismuth, but strongly radioactive, and in July 1898 they published a paper in which they concluded that it was a new element, which they named "polonium". The other was chemically like barium, and in a December 1898 paper they announced the discovery of a second hitherto unknown element, which they called "radium". Convincing the scientific community was another matter. Separating radium from the barium in the ore proved very difficult. It took three years for them to produce a tenth of a gram of radium chloride, and they never did manage to isolate polonium.[6]

#### Protactinium

The decay chain of actinium. Alpha decay shifts two elements down; beta decay shifts one element up.

Soddy and Kasimir Fajans independently observed in 1913 that alpha decay caused atoms to shift down two places in the periodic table, while the loss of two beta particles restored it to its original position. In the resulting reorganisation of the periodic table, radium was placed in group II, actinium in group III, thorium in group IV and uranium in group VI. This left a gap between thorium and uranium. Soddy predicted that this unknown element, which he referred to (after Dmitri Mendeleev) as "ekatantalium", would be an alpha emitter with chemical properties similar to tantalium (now known as tantalum).[17][18][19] It was not long before Fajans and Oswald Helmuth Göhring discovered it as a decay product of a beta-emitting product of thorium. Based on the radioactive displacement law of Fajans and Soddy, this was an isotope of the missing element, which they named "brevium" after its short half-life. However, it was a beta emitter, and therefore could not be the mother isotope of actinium. This had to be another isotope.[17]

Otto Hahn and Lise Meitner in 1912

The work was interrupted by the outbreak of the First World War in 1914. Hahn was called up into the German Army, and Meitner became a volunteer radiographer in Austrian Army hospitals.[21] She returned to the Kaiser Wilhelm Institute in October 1916, when not only Hahn, but most of the students, laboratory assistants and technicians had been called up. Meitner therefore had to do everything herself, aided only briefly by Hahn when he came home on leave. By December 1917 she was able to isolate the substance, and after further work was able to prove that it was indeed the missing isotope. She submitted her findings for publication in March 1918.[17]

Although Fajans and Göhring had been the first to discover the element, custom required that an element was represented by its longest-lived and most abundant isotope, and brevium did not seem appropriate. Fajans agreed to Meitner naming the element protactinium, and assigning it the chemical symbol Pa. In June 1918, Soddy and John Cranston announced that they had extracted a sample of the isotope, but unlike Meitner were unable to describe its characteristics. They acknowledged Meitner's priority, and agreed to the name. The connection to uranium remained a mystery, as neither of the known isotopes of uranium decayed into protactinium. It remained unsolved until uranium-235 was discovered in 1929.[17][22]

### Transmutation

Irène Curie and Frédéric Joliot in their Paris laboratory in 1935.

Patrick Blackett was able to accomplish nuclear transmutation of nitrogen into oxygen in 1925, using alpha particles directed at nitrogen. In modern notation for the atomic nuclei, the reaction was:

14
7
N
+ 4
2
He
17
8
O
+ p

This was the first observation of a nuclear reaction, that is, a reaction in which particles from one decay are used to transform another atomic nucleus.[23] A fully artificial nuclear reaction and nuclear transmutation was achieved in April 1932 by Ernest Walton and John Cockcroft, who used artificially accelerated protons against lithium, to break this nucleus into two alpha particles. The feat was popularly known as "splitting the atom", but was not nuclear fission;[24][25] as it was not the result of initiating an internal radioactive decay process.[26] Just a few weeks before Cockcroft and Walton's feat, another scientist at the Cavendish Laboratory, James Chadwick, discovered the neutron, using an ingenious device made with sealing wax, through the reaction of beryllium with alpha particles:[27][28]

11
5
Be
+ 4
2
He
14
7
N
+ n

Irène Curie and Frédéric Joliot irradiated aluminium foil with alpha particles, they found that this results in a short-lived radioactive isotope of phosphorus with a half-life of around three minutes:

27
13
Al
+ 4
2
He
30
15
P
+ n

which then decays to a stable isotope of silicon

30
15
P
30
14
Si
+ e +

They noted that radioactivity continued after the neutron emissions ceased. Not only had they discovered a new form of radioactive decay in the form of positron emission, they had transmuted an element into a hitherto unknown radioactive isotope of another, thereby inducing radioactivity where there had been none before. Radiochemistry was now no longer confined to certain heavy elements, but extended to the entire periodic table.[29][30][31]

Enrico Fermi and his research group (the Via Panisperna boys), circa 1934. Left to right: Oscar D'Agostino, Emilio Segrè, Edoardo Amaldi, Franco Rasetti and Fermi.

Working in assembly-line fashion, they started by irradiating water, and then progressed up the periodic table through lithium, beryllium, boron and carbon, without inducing any radioactivity. When they got to aluminium and then fluorine, they had their first successes. Induced radioactivity was ultimately found through the neutron bombardment of 22 different elements.[35][36] Meitner was one of the select group of physicists to whom Fermi mailed advance copies of his papers, and she was able to report that she had verified his findings with respect to aluminium, silicon, phosphorus, copper and zinc.[34] When a new copy of La Ricerca Scientifica arrived at the Niels Bohr's Institute for Theoretical Physics at the University of Copenhagen, her nephew, Otto Frisch, as the only physicist there who could read Italian, found himself in demand from colleagues wanting a translation. The Rome group had no samples of the rare earth metals, but at Bohr's institute George de Hevesy had a complete set of their oxides that had been given to him by Auergesellschaft, so de Hevesy and Hilde Levi carried out the process with them.[37]

When the Rome group reached uranium, they had a problem: the radioactivity of natural uranium was almost as great as that of their neutron source.[38] What they observed was a complex mixture of half-lives. Following the displacement law, they checked for the presence of lead, bismuth, radium, actinium, thorium and protactinium (skipping the elements whose chemical properties were unknown), and (correctly) found no indication of any of them.[38] Fermi noted three types of reactions were caused by neutron irradiation: emission of an alpha particle (n, α); proton emission (n, p); and gamma emission (n, γ). Invariably, the new isotopes decayed by beta emission, which caused elements to move up the periodic table.[39]

Based on the periodic table of the time, Fermi believed that element 93 was ekarhenium—the element below rhenium—with characteristics similar to manganese and rhenium. Such an element was found, and Fermi tentatively concluded that his experiments had created new elements with 93 and 94 protons,[40] which he dubbed ausonium and hesperium.[41][42] The results were published in Nature in June 1934.[40] However, in this paper Fermi cautioned that "a careful search for such heavy particles has not yet been carried out, as they require for their observation that the active product should be in the form of a very thin layer. It seems therefore at present premature to form any definite hypothesis on the chain of disintegrations involved."[40] In retrospect, what they had detected was indeed an unknown rhenium-like element, technetium, which lies between manganese and rhenium on the periodic table.[38]

Leo Szilard and Thomas A. Chalmers reported that neutrons generated by gamma rays acting on beryllium were captured by iodine, a reaction that Fermi had also noted. When Meitner repeated their experiment, she found that neutrons from the gamma-beryllium sources were captured by heavy elements like iodine, silver and gold, but not by lighter ones like sodium, aluminium and silicon. She concluded that slow neutrons were more likely to be captured than fast ones, a finding she reported to Naturwissenschaften in October 1934.[43][44] Everyone had been thinking that energetic neutrons were required, as was the case with alpha particles and protons, but that was required to overcome the Coulomb barrier; the neutrally charged neutrons were more likely to be captured by the nucleus if they spent more time in its vicinity. A few days later, Fermi considered a curiosity that his group had noted: uranium seemed to react differently in different parts of the laboratory; neutron irradiation conducted on a wooden table induced more radioactivity than on a marble table in the same room. Fermi thought about this and tried placing a piece of paraffin wax between the neutron source and the uranium. This resulted in a dramatic increase in activity. He reasoned that the neutrons had been slowed by collisions with hydrogen atoms in the paraffin and wood.[45] The departure of D'Agostino meant that the Rome group no longer had a chemist, and the subsequent loss of Rasetti and Segrè reduced the group to just Fermi and Amaldi, who abandoned the research into transmutation to concentrate on exploring the physics of slow neutrons.[38]

The current model of the nucleus in 1934 was the liquid drop model first proposed by George Gamow in 1930.[46] His simple and elegant model was refined and developed by Carl Friedrich von Weizsäcker and, after the discovery of the neutron, by Werner Heisenberg in 1935 and Niels Bohr in 1936, it agreed closely with observations. In the model, the nucleons were held together in the smallest possible volume (a sphere) by the strong nuclear force, which was capable of overcoming the longer ranged Coulomb electrical repulsion between the protons. The model remained in use for certain applications in the 21st century, when it attracted the attention of mathematicians interested in its properties,[47][48][49] but in its 1934 form it confirmed what physicists thought they already knew: that nuclei were static, and that the odds of a collision chipping off more than an alpha particle were practically zero.[50]

### Discovery

#### Objections

Fermi won the 1938 Nobel Prize in Physics for his "demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons".[1] However, not everyone was convinced by Fermi's analysis of his results. Ida Noddack suggested in September 1934 that instead of creating a new, heavier element 93, that:

One could assume equally well that when neutrons are used to produce nuclear disintegrations, some distinctly new nuclear reactions take place which have not been observed previously with proton or alpha-particle bombardment of atomic nuclei. In the past one has found that transmutations of nuclei only take place with the emission of electrons, protons, or helium nuclei, so that the heavy elements change their mass only a small amount to produce near neighbouring elements. When heavy nuclei are bombarded by neutrons, it is conceivable that the nucleus breaks up into several large fragments, which would of course be isotopes of known elements but would not be neighbours of the irradiated element.[51]

Noddack's article was read by Fermi's team in Rome, Curie and Joliot in Paris, and Meitner and Hahn in Berlin.[38] However, the quoted objection comes some distance down, and is but one of several gaps she noted in Fermi's claim.[52] Bohr's liquid drop model had not yet been formulated, so there was no theoretical way to calculate whether it was physically possible for the uranium atoms to break into large pieces.[53] Noddack and her husband, Walter Noddack, were renowned chemists who had been nominated for the Nobel Prize in Chemistry for the discovery of rhenium, although at the time they were also embroiled in a controversy over the discovery of element 43, which they called "masurium". The discovery of technetium by Emilio Segrè and Carlo Perrier put an end to their claim, but did not occur until 1937. It is unlikely that Meitner or Curie had any prejudice against Noddack because of her sex,[54] but Meitner was not afraid to tell Hahn Hahnchen, von Physik verstehst Du Nichts ("Hahn dear, of physics you understand nothing").[55] The same attitude carried over to Noddack, who did not propose an alternative nuclear model, nor conduct experiments to support her claim. Although Noddack was a renowned analytical chemist, she lacked the background in physics to appreciate the enormity of what she was proposing.[52]

Former Kaiser Wilhelm Institute for Chemistry building in Berlin. After the Second World War it became part of the Free University of Berlin. It was renamed the Otto Hahn Building in 1956, and the Hahn-Meitner Building in 2010. [56] [57]

Noddack was not the only critic of Fermi's claim. Aristid von Grosse suggested that what Fermi had found was an isotope of protactinium.[58][59] Meitner was eager to investigate Fermi's results, but she recognised that a highly skilled chemist was required, and she wanted the best one she knew: Hahn, although they had not collaborated for many years. Initially, Hahn was not interested, but von Grosse's mention of protactinium changed his mind.[60] "The only question", Hahn later wrote, "seemed to be whether Fermi had found isotopes of transuranian elements, or isotopes of the next-lower element, protactinium. At that time Lise Meitner and I decided to repeat Fermi's experiments in order to find out whether the 13-minute isotope was a protactinium isotope or not. It was a logical decision, having been the discoverers of protactinium."[61]

Hahn and Meitner were joined by Fritz Strassmann. Strassmann had received his doctorate in analytical chemistry from the Technical University of Hannover in 1929,[62] and had come to the Kaiser Wilhelm Institute for Chemistry to study under Hahn, believing that this would improve his employment prospects. He enjoyed the work and the people so much that he stayed on after his stipend expired in 1932. After the Nazi Party came to power in Germany in 1933, he declined a lucrative offer of employment because it required political training and Nazi Party membership, and he resigned from the Society of German Chemists when it became part of the Nazi German Labour Front. As a result, he could neither work in the chemical industry nor receive his habilitation, which was required to become an independent researcher in Germany. Meitner persuaded Hahn to hire Strassmann using money from the director's special circumstances fund. In 1935, Strassmann became an assistant on half pay. Soon he would be credited as a collaborator on the papers they produced.[63]

The 1933 Law for the Restoration of the Professional Civil Service removed Jewish people from the civil service, which included academia. Meitner never tried to conceal her Jewish descent, but initially was exempt from its impact on multiple grounds: she had been employed before 1914, had served in the military during the World War, was an Austrian rather than a German citizen, and the Kaiser Wilhelm Institute was a government-industry partnership.[64] However, she was dismissed from her adjunct professorship at the University of Berlin on the grounds that her World War I service was not at the front, and she had not completed her habilitation until 1922.[65] Carl Bosch, the director of IG Farben, a major sponsor of the Kaiser Wilhelm Institute for Chemistry, assured Meitner that her position there was safe, and she agreed to stay.[64] Meitner, Hahn and Strassmann drew closer together personally as their anti-Nazi politics increasingly alienated them from the rest of the organisation, but it gave them more time for research, as administration was devolved to Hahn's and Meitner's assistants.[63]

#### Research

The nuclear fission display at the Deutsches Museum in Munich. This was touted for many years as the table and experimental apparatus with which Otto Hahn discovered nuclear fission in 1938. The table and instruments are representative of the ones used, but not necessarily the originals, and would not have been together on the one table in the same room. Pressure from historians, scientists and feminists caused the museum to alter the display in 1988 to acknowledge Lise Meitner, Otto Frisch and Fritz Strassmann. [66]

The Berlin group started by irradiating uranium salt with neutrons from a radon-beryllium source similar to the one that Fermi had used. They dissolved it and added potassium perrhenate, platinum chloride and sodium hydroxide. What remained was then acidified with hydrogen sulphide, resulting in platinum sulphide and rhenium sulphide precipitation. Fermi had noted four radioactive isotopes with the longest-lived having 13- and 90-minute half-lives, and these were detected in the precipitate. The Berlin group then tested for protactinium by adding protactinium-234 to the solution. When this was precipitated, it was found to be separated from the 13- and 90-minute half-life isotopes, demonstrating that von Grosse was incorrect, and they were not isotopes of protactinium. Moreover, the chemical reactions involved ruled out all elements from mercury and above on the periodic table.[67] They were able to precipitate the 90-minute activity with osmium sulphide and the 13-minute one with rhenium sulphide, which ruled out them being isotopes of the same element. All this provided strong evidence that they were indeed transuranium elements, with chemical properties similar to osmium and rhenium.[68][69]

Fermi had also reported that fast and slow neutrons had produced different activities. This indicated that more than one reaction was taking place. When the Berlin group could not replicate the Rome group's findings, they commenced their own research into the effects of fast and slow neutrons. To minimise radioactive contamination if there were an accident, different phases were carried out in different rooms, all in Meitner's section on the ground floor of the Kaiser Wilhelm Institute. Neutron irradiation was carried out in one laboratory, chemical separation in another, and measurements were conducted in a third. The equipment they used was simple and mostly hand made.[70]

By March 1936, they had identified ten different half-lives, with varying degrees of certainty. To account for them, Meitner had to hypothesise a new (n, 2n) class of reaction and the alpha decay of uranium, neither of which had ever been reported before, and for which physical evidence was lacking. So while Hahn and Strassmann refined their chemical procedures, Meitner devised new experiments to shine more light on the reaction processes. In May 1937, they issued parallel reports, one in Zeitschrift für Physik with Meitner as the principal author, and one in Chemische Berichte with Hahn as the principal author.[70][71][72] Hahn concluded his by stating emphatically: Vor allem steht ihre chemische Verschiedenheit von allen bisher bekannten Elementen außerhalb jeder Diskussion ("Above all, their chemical distinction from all previously known elements needs no further discussion."[72]) Meitner was increasingly uncertain. They had now constructed three (n, γ) reactions:

1. 238
92
U
+ n → 239
92
U
(10 seconds) → 239
93
ekaRe
(2.2 minutes) → 239
94
ekaOs
(59 minutes) → 239
95
ekaIr
(66 hours) → 239
96
ekaPt
(2.5 hours) → 239
97
ekAu
(?)
2. 238
92
U
+ n → 239
92
U
(40 seconds) → 239
93
ekaRe
(16 minutes) → 239
94
ekaOs
(5.7 hours) → 239
95
ekaIr
(?)
3. 238
92
U
+ n → 239
92
U
(23 minutes) → 239
93
ekaRe

Meitner was certain that these had to be (n, γ) reactions, as slow neutrons lacked the energy to chip off protons or alpha particles. She considered the possibility that the reactions were from different isotopes of uranium; three were known: uranium-238, uranium-235 and uranium-234. However, when she calculated the neutron cross section it was too large to be anything other than the most abundant isotope, uranium-238. She concluded that it must be a case of nuclear isomerism, which had been discovered in protactinium by Hahn in 1922. Nuclear isomerism had been given a physical explanation by von Weizsäcker, who had been Meitner's assistant in 1936, but had since taken a position at the Kaiser Wilhelm Institute for Physics. Different nuclear isomers of protactinium had different half-lives, and this could be the case for uranium too, but if so it was somehow being inherited by the daughter and granddaughter products, which seemed to be stretching the argument to breaking point. Then there was the third reaction, an (n, γ) one, which occurred only with slow neutrons.[73] Meitner therefore ended her report on a very different note to Hahn, reporting that: "The process must be neutron capture by uranium-238, which leads to three isomeric nuclei of uranium-239. This result is very difficult to reconcile with current concepts of the nucleus."[71][74]

Exhibition to mark the 75th anniversary of the discovery of nuclear fission, at the Vienna International Centre in 2013. The table (on loan from the Deutsches Museum Munich) is now described as a replica and images of Meitner and Strassmann are prominently displayed.

After this, the Berlin group moved on to working with thorium, as Strassmann put it, "to recover from, the horror of the work with uranium".[75] However, thorium was not easier to work with than uranium. For a start, it had a decay product, radiothorium (228
90
Th
) that overwhelmed weaker neutron-induced activity. But Hahn and Meitner had a sample from which they had regularly removed its mother isotope, mesothorium (228
88
Ra
), over a period of several years, allowing the radiothorium to decay away. Even then, it was still more difficult to work with because its induced decay products from neutron irradiation were isotopes of the same elements produced by thorium's own radioactive decay. What they found was three different decay series, all alpha emitters—a form of decay not found in any other heavy element, and for which Meitner once again had to postulate multiple isomers. They did find an interesting result: these (n, α) decay series occurred simultaneously when the energy of the incident neutrons was less than 2.5 MeV; when they had more, an (n, γ) reaction that formed 233
90
Th
was favoured.[76]

In Paris, Irene Curie and Pavel Savitch had also set out to replicate Fermi's findings. In collaboration with Hans von Halban and Peter Preiswerk, they irradiated thorium and produced the isotope with a 22-minute half-life that Fermi had noted. In all, Curie's group detected eight different half-lives in their irradiated thorium. Curie and Savitch detected a radioactive substance with a 3.5-hour half-life.[38][32][77] The Paris group proposed that it might be an isotope of thorium. Meitner asked Strassmann, who was now doing most of the chemistry work, to check. He detected no sign of thorium. Meitner wrote to Curie with their results, and suggested a quiet retraction.[78] Nonetheless, Curie persisted. They investigated the chemistry, and found that the 3.5-hour activity was coming from something that seemed to be chemically similar to lanthanum (which in fact it was), which they attempted unsuccessfully to isolate with a fractional crystallization process. (It is possible that their precipitate was contaminated with yttrium, which is chemically similar.) By using Geiger counters and skipping the chemical precipitation, Curie and Savitch detected the 3.5-hour half-life in irradiated uranium.[79]

With the Anschluss, Germany's unification with Austria on 12 March 1938, Meitner lost her Austrian citizenship.[80] James Franck offered to sponsor her immigration to the United States, and Bohr offered a temporary place at his institute, but when she went to the Danish embassy for a visa, she was told that Denmark no longer recognised her Austrian passport as valid.[81] On 13 July 1938, Meitner departed for the Netherlands with Dutch physicist Dirk Coster. Before she left, Otto Hahn gave her a diamond ring he had inherited from his mother to sell if necessary. She reached safety, but with only her summer clothes. Meitner later said that she left Germany forever with 10 marks in her purse. With the help of Coster and Adriaan Fokker, she flew to Copenhagen, where she was greeted by Frisch, and stayed with Niels and Margrethe Bohr at their holiday house in Tisvilde. On 1 August she took the train to Stockholm, where she was met by Eva von Bahr.[82]

#### Eureka!

The Paris group published their results in September 1938.[79] Hahn dismissed the isotope with the 3.5-hour half-life as contamination, but after looking at the details of the Paris group's experiments and the decay curves, Strassmann was worried. He decided to repeat the experiment, using his more efficient method of separating radium. This time, they found what they thought was radium, which Hahn suggested resulted from two alpha decays:

238
92
U
+ n → α + 235
90
Th
→ α + 235
88
Ra

Meitner found this very hard to believe.[83][84]

The mechanism of fission. A neutron caused the nucleus to wobble, elongate, and split.

In November, Hahn travelled to Copenhagen, where he met with Bohr and Meitner. They told him that they were very unhappy about the proposed radium isomers. On Meitner's instructions, Hahn and Strassmann began to redo the experiments, even as Fermi was collecting his Nobel Prize in Stockholm.[85] Assisted by Clara Lieber and Irmgard Bohne, they isolated the three radium isotopes (verified by their half-lives) and used fractional crystallisation to separate them from the barium carrier by adding barium bromide crystals in four steps. Since radium precipitates preferentially in a solution of barium bromide, at each step the fraction drawn off would contain less radium than the one before. However, they found no difference between each of the fractions. In case their process was faulty in some way, they verified it with known isotopes of radium; the process was fine. On 19 December, Hahn wrote to Meitner, informing her that the radium isotopes behaved chemically like barium. Anxious to finish up before the Christmas break, Hahn and Strassmann submitted their findings to Naturwissenschaften on 22 December without waiting for Meitner to reply.[86] Hahn concluded with: "As chemists... we should substitute the symbols Ba, La, Ce for Ra, Ac, Th. As 'nuclear chemists' fairly close to physics we cannot yet bring ourselves to take this step which contradicts all previous experience in physics."[87]

Frisch normally celebrated Christmas with Meitner in Berlin, but in 1938 she accepted an invitation from Eva von Bahr to spend it with her family at Kungälv, and Meitner asked Frisch to join her there. Meitner received the letter from Hahn describing his chemical proof that some of the product of the bombardment of uranium with neutrons was barium. Barium had an atomic mass 40% less than uranium, and no previously known methods of radioactive decay could account for such a large difference in the mass of the nucleus.[88][89] Nonetheless, she had immediately written back to Hahn to say: "At the moment the assumption of such a thoroughgoing breakup seems very difficult to me, but in nuclear physics we have experienced so many surprises, that one cannot unconditionally say: 'It is impossible.'"[90] Meitner felt that Hahn was too careful a chemist to make an elementary blunder, but found the results difficult to explain. All the nuclear reactions that had been documented involved chipping protons or alpha particles from the nucleus. Breaking it up seemed far more difficult. However the liquid drop model that Gamow had postulated suggested the possibility that an atomic nucleus could become elongated and overcome the surface tension that held it together.[91]

According to Frisch:

At that point we both sat down on a tree trunk (all that discussion had taken place while we walked through the wood in the snow, I with my skis on, Lise Meitner making good her claim that she could walk just as fast without), and started to calculate on scraps of paper. The charge of a uranium nucleus, we found, was indeed large enough to overcome the effect of the surface tension almost completely; so the uranium nucleus might indeed resemble a very wobbly unstable drop, ready to divide itself at the slightest provocation, such as the impact of a single neutron.

But there was another problem. After separation, the two drops would be driven apart by their mutual electric repulsion and would acquire high speed and hence a very large energy, about 200 MeV in all; where could that energy come from? Fortunately Lise Meitner remembered the empirical formula for computing the masses of nuclei and worked out that the two nuclei formed by the division of a uranium nucleus together would be lighter than the original uranium nucleus by about one-fifth the mass of a proton. Now whenever mass disappears energy is created, according to Einstein's formula ${\displaystyle E=m\,c^{2}}$, and one-fifth of a proton mass was just equivalent to 200 MeV. So here was the source for that energy; it all fitted![91]

Meitner and Frisch had correctly interpreted Hahn's results to mean that the nucleus of uranium had split roughly in half. The first two reactions that the Berlin group had observed were light elements created by the breakup of uranium nuclei; the third, the 23-minute one, was a decay into the real element 93.[92] On returning to Copenhagen, Frisch informed Bohr, who slapped his forehead and exclaimed "What idiots we have been!"[93] Bohr promised not to say anything until they had a paper ready for publication. To speed the process, they decided to submit a one-page note to Nature. At this point, the only evidence that they had was the barium. Logically, if barium was formed, the other element must be krypton,[94] although Hahn mistakenly believed that the atomic masses had to add up to 239 rather than the atomic numbers adding up to 92, and thought it was masurium (technetium), and so did not check for it:[95]

235
92
U
+ n →
56
Ba
+
36
Kr
+ some n

Over a series of long-distance phone calls, Meitner and Frisch came up with a simple experiment to bolster their claim: to measure the recoil of the fission fragments, using a Geiger counter with the threshold set above that of the alpha particles. Frisch conducted the experiment on 13 February 1939, and found the pulses caused by the reaction just as they had predicted.[94] He decided he needed a name for the newly discovered nuclear process. He spoke to William A. Arnold, an American biologist working with de Hevesy and asked him what biologists called the process by which living cells divided into two cells. Arnold told him that biologists called it fission. Frisch then applied that name to the nuclear process in his paper.[96] Frisch mailed both the jointly-authored note on fission and his paper on the recoil experiment to Nature on 16 January 1939; the former appeared in print on 11 February and the latter on 18 February.[97][98]

### Reception

#### Bohr brings the news to the United States

Before departing for the United States on 7 January 1939 with his son Erik to attend the Fifth Washington Conference on Theoretical Physics, Bohr promised Frisch that he would not mention fission until the papers appeared in print, but during the Atlantic crossing on the SS Drottningholm, Bohr discussed the mechanism of fission with Leon Rosenfeld, and failed to inform him that the information was confidential. On arrival in New York City on 16 January, they were met by Fermi and his wife Laura Capon, and by John Wheeler, who had been a fellow at Bohr's institute in 1934–1935. As it happened, there was a meeting of Princeton University's Physics Journal Club that evening, and when Wheeler asked Rosenfeld if he had any news to report, Rosenfeld told them.[99] An embarrassed Bohr fired off a note to Nature defending Meitner and Frisch's claim to the priority of the discovery.[100] Hahn was annoyed that while Bohr mentioned his and Strassmann's work in the note, he cited only Meitner and Frisch.[101]

News spread quickly of the new discovery, which was correctly seen as an entirely novel physical effect with great scientific—and potentially practical—possibilities. Isidor Isaac Rabi and Willis Lamb, two Columbia University physicists working at Princeton, heard the news and carried it back to Columbia. Rabi said he told Fermi; Fermi gave credit to Lamb. For Fermi, the news came as a profound embarrassment, as the transuranic elements that he had partly been awarded the Nobel Prize for discovering had not been transuranic elements at all, but fission products. He added a footnote to this effect to his Nobel Prize acceptance speech. Bohr soon thereafter went from Princeton to Columbia to see Fermi. Not finding Fermi in his office, Bohr went down to the cyclotron area and found Herbert L. Anderson. Bohr grabbed him by the shoulder and said: "Young man, let me explain to you about something new and exciting in physics."[102]

#### Further research

It was clear to many scientists at Columbia that they should try to detect the energy released in the nuclear fission of uranium from neutron bombardment. On 25 January 1939, a Columbia University group conducted the first nuclear fission experiment in the United States,[103] which was done in the basement of Pupin Hall. The experiment involved placing uranium oxide inside of an ionization chamber and irradiating it with neutrons, and measuring the energy thus released. The next day, the Fifth Washington Conference on Theoretical Physics began in Washington, D.C., under the joint auspices of The George Washington University and the Carnegie Institution of Washington. From there, the news on nuclear fission spread even further, which fostered many more experimental demonstrations.[104]

Bohr and Wheeler overhauled the liquid drop model to explain the mechanism of nuclear fission, with conspicuous success.[105] Their paper appeared in Physical Review on 1 September 1939, the day Germany invaded Poland, starting World War II in Europe.[106] As the experimental physicists studied fission, they uncovered more puzzling results. George Placzek (who had measured the slow neutron absorption of gold in 1934 using Bohr's Nobel Prize medal[99]) asked Bohr why uranium fissioned with both very fast and very slow neutrons. Walking to a meeting with Wheeler, Bohr had an insight that the fission at low energies was due to the uranium-235 isotope, while at high energies it was mainly due to the far more abundant uranium-238 isotope.[107] This was based on Meitner's 1937 measurements of the neutron capture cross-sections.[108] This would be experimentally verified in February 1940, after Alfred Nier was able to produce sufficient pure uranium-235 for John R. Dunning, Aristid von Grosse and Eugene T. Booth to test.[100]

Other scientists resumed the search for the elusive element 93, which seemed to be straightforward, as they now knew it resulted from the 23-minute half-life. At the Radiation Laboratory in Berkeley, California, Emilio Segrè and Edwin McMillan used the cyclotron to create the isotope. They then detected a beta activity with a 2-day half-life, but it had rare-earth element chemical characteristics, and element 93 was supposed to have chemistry akin to rhenium. It was therefore overlooked as just another fission product. Another year passed before McMillan and Philip Abelson determined that the 2-day half-life element was that of the elusive element 93, which they named "neptunium". They paved the way for the discovery by Glenn Seaborg, Emilio Segrè and Joseph W. Kennedy of element 94, which they named "plutonium" in 1941.[109][110]

Another avenue of research, spearheaded by Meitner, was to determine if other elements could fission after being irradiated with neutrons. It was soon determined that thorium and protactinium could. Measurements were also made of the amount of energy released.[20] Hans von Halban, Frédéric Joliot-Curie and Lew Kowarski demonstrated that uranium bombarded by neutrons emitted more neutrons than it absorbed, suggesting the possibility of a nuclear chain reaction.[111] Fermi and Anderson did so too a few weeks later.[112][113] It was apparent to many scientists that, in theory at least, an extremely powerful energy source could be created, although most still considered an atomic bomb an impossibility.[114]

#### Nobel Prize

German stamp honouring Otto Hahn and his discovery of nuclear fission (1979).

Both Hahn and Meitner had been nominated for the chemistry and the physics Nobel Prizes many times even before the discovery of nuclear fission for their work on radioactive isotopes and protactinium. Several more nominations followed for the discovery of fission between 1940 and 1943.[115][116] Nobel Prize nominations were vetted by committees of five, one for each award. Although both Hahn and Meitner received nominations for physics, radioactivity and radioactive elements had traditionally been seen as the domain of chemistry, and so the Nobel Committee for Chemistry evaluated the nominations in 1944.[117]

The committee received reports from Theodor Svedberg in 1941 and Arne Westgren [sv] in 1942. These chemists were impressed by Hahn's work, but felt that the experimental work of Meitner and Frisch was not extraordinary. They did not understand why the physics community regarded their work as seminal. As for Strassmann, although his name was on the papers, there was a long-standing policy of conferring awards on the most senior scientist in a collaboration. In 1944 the Nobel Committee for Chemistry voted to recommend that Hahn alone be given the Nobel Prize in Chemistry for 1944.[117] However, Germans had been forbidden to accept Nobel Prizes after the Nobel Peace Prize had been awarded to Carl von Ossietzky in 1936.[118] The committee's recommendation was rejected by the Royal Swedish Academy of Sciences, which decided to defer the award for one year.[117]

The war was over when the Academy reconsidered the award in September 1945. The Nobel Committee for Chemistry had now become more cautious, as it was apparent that much research had been undertaken by the Manhattan Project in the United States in secret, and it suggested deferring the 1944 Nobel Prize in Chemistry for another year. The Academy was swayed by Göran Liljestrand, who argued that it was important for the Academy to assert its independence from the Allies of World War II, and award the Nobel Prize in Chemistry to a German,[119] as it had done after World War I when it had awarded it to Fritz Haber. Hahn therefore became the sole recipient of the 1944 Nobel Prize in Chemistry "for his discovery of the fission of heavy nuclei".[120]

Meitner wrote in a letter to her friend Birgit Broomé-Aminoff on 20 November 1945:

Surely Hahn fully deserved the Nobel Prize in chemistry. There is really no doubt about it. But I believe that Otto Robert Frisch and I contributed something not insignificant to the clarification of the process of uranium fission – how it originates and that it produces so much energy, and that was something very remote from Hahn. For this reason I found it a bit unjust that in the newspapers I was called a Mitarbeiterin [subordinate] of Hahn's in the same sense that Strassmann was.[121]

In 1946, the Nobel Committee for Physics considered nominations for Meitner and Frisch received from Max von Laue, Niels Bohr, Oskar Klein, Egil Hylleraas and James Franck. Reports were written for the committee by Erik Hulthén, who held the chair of experimental physics at Stockholm University, in 1945 and 1946. Hulthén argued that theoretical physics should be considered award-worthy only if it inspired great experiments. The role of Meitner and Frisch in being the first to understand and explain fission was not understood. There may also have been personal factors: the chairman of the committee, Manne Siegbahn, disliked Meitner, and had a professional rivalry with Klein.[117][122] Meitner and Frisch would continue to be nominated regularly for many years, but would never be awarded a Nobel Prize.[116][117][123]

### In history and memory

At the end of the war in Europe, Hahn was taken into custody and incarcerated at Farm Hall with nine other senior scientists, all of whom except Max von Laue had been involved with the German nuclear weapons program, and all except Hahn and Paul Harteck were physicists. It was here that they heard the news of the atomic bombings of Hiroshima and Nagasaki. Unwilling to accept that they were years behind the Americans, and unaware that their conversations were being recorded, they concocted a story that they had never wanted their nuclear weapons program to succeed in the first place on moral grounds. Hahn was still there when his Nobel Prize was announced in November 1945. The Farm Hall scientists would spend the rest of their lives attempting to rehabilitate the image of German science that had been tarnished by the Nazi period.[124][125] Inconvenient details like the thousands of female slave labourers from Sachsenhausen concentration camp who mined uranium ore for their experiments were swept under the rug.[126]

Lise Meitner in 1946 with physicist Arthur H. Compton and actress Katharine Cornell.

For Hahn, this necessarily involved asserting his claim of the discovery of fission for himself, for chemistry, and for Germany. He used his Nobel Prize acceptance speech to assert this narrative.[124][125] Hahn's message resonated strongly in Germany, where he was revered as the proverbial good German, a decent man who had been a staunch opponent of the Nazi regime, but had remained in Germany where he had pursued pure science. As president of the Max Planck Society from 1946 to 1960, he projected an image of German science as undiminished in brilliance and untainted by Nazism to an audience that wanted to believe it.[66]

In contrast, in the immediate aftermath of the war Meitner and Frisch were hailed as the discoverers of fission in English-speaking countries. Japan was seen as a puppet state of Germany and the destruction of Hiroshima and Nagasaki as poetic justice for the persecution of the Jewish people.[127][128] In January 1946, Meitner toured the United States, where she gave lectures and received honorary degrees. She attended a cocktail party for Lieutenant General Leslie Groves, the director of the Manhattan Project (who gave her sole credit for the discovery of fission in his 1962 memoirs), and was named Woman of the Year by the Women's National Press Club. At the reception for this award, she sat next to the President of the United States, Harry S. Truman. But Meitner did not enjoy public speaking, especially in English, nor did she relish the role of a celebrity, and she declined the offer of a visiting professorship at Wellesley College.[129][130]

In 1966, the United States Atomic Energy Commission jointly awarded the Enrico Fermi Prize to Hahn, Strassmann and Meitner for their discovery of fission. The ceremony was held in the Hofburg palace in Vienna.[131] It was the first time that the Enrico Fermi Prize had been awarded to non-Americans, and the first time it was presented to a woman.[132] Meitner's diploma bore the words: "For pioneering research in the naturally occurring radioactivities and extensive experimental studies leading to the discovery of fission".[133] Hahn's diploma was slightly different: "For pioneering research in the naturally occurring radioactivities and extensive experimental studies culminating in the discovery of fission."[134] Hahn and Strassmann were present, but Meitner was too ill to attend, so Frisch accepted the award on her behalf.[135]

During combined celebrations in Germany of the 100th birthdays of Einstein, Hahn, Meitner and von Laue in 1978, Hahn's narrative of the discovery of fission began to crumble. Hahn and Meitner had died in 1968, but Strassmann was still alive, and he asserted the importance of his analytical chemistry and Meitner's physics in the discovery, and their role as more than just assistants. A detailed biography of Strassmann appeared in 1981, a year after his death, and a prize-winning one of Meitner for young adults in 1986. Scientists questioned the focus on chemistry, historians challenged the accepted narrative of the Nazi period, and feminists saw Meitner as yet another example of the Matilda effect, where a woman had been airbrushed from the pages of history. By 1990, Meitner had been restored to the narrative, although her role remained contested.[66]


text:
## Home

### Note

If com.unraveldata.login.mode is set to LDAP or SAML, you can only toggle the RBAC Access status.

The tags used for RBAC end-users must also be loaded as Application and Workflows tags. If they are not, the pages are filtered only on com.unraveldata.rbac.default. (See RBAC Configuration.)

1. Go to Manage | Role Manager to access the Role Manager.

The RBAC default is set via com.unraveldata.rbac.enabled. You can toggle the status of RBAC; however, when the Unravel daemon is restarted RBAC resets.

2. If you are not using LDAP/SAML login mode, you can add filters for specific end-users.

Any end-user roles you have previously set are displayed. If the Unravel daemon was restarted after you added end-user roles the entries are lost. You can add end-users one at a time via Add New Role. To add multiple users at a time create and upload a csv file.

You limit end-user access through tags. In the example below only two tags are available, project and tenant. If a 3rd tag, department, had been defined it would be available. The end-user filters between the red brackets were loaded using a .csv file

Clicking on Add New Role adds a row to the Roles table containing text boxes (1). You must define the User and at least one tag restriction. To add multiple tag names under a tag type separate the tag names with commas, be sure the string contains no spaces or special characters. To save the entry click . Click to delete your entry click without saving it.

Adding one or more roles via a role file

Click on Select role file to choose the .csv file. The format of a .csv is:

first row is a header row defining the columns tags: user,tagKey[,tagKey]*

• tagKey: is a valid tag key, i.e., department, tenant.

one or more rows defining user and tag values: user,tagValue[,tagValue]*

• tagValue: is empty, a valid tag value for tagKey, or tagString,

• tagString: is a series of tagValues separated by commas and enclosed in quotes, and

• *: means zero or more

### Note:

The file must define at least one tagKey, one user and one tagValue for the user.

After you add your last tagValue you can leave the rest of the row blank. See the userNew filter in the CSV file below for an example.

tagValues must be ordered as defined in the header row.

No special characters or spaces are allowed in file.

The CSV file below was used to load filters within the red brackets.

user,project,tenant
user72,"group1,group2",mm
user25,,"3n,3m" userNew,groupNew
user33,"group3,group2","3m,mm"

4. Editing or deleting roles.

To edit a role, click the edit glyph (). You can add or delete tags, but not edit the end-user's name.

To delete a role, click the delete glyph.

###### Effect of RBAC control
End-user's access with RBAC turned off

The user has access to all the Unravel UI features and all applications.

End-user's access with RBAC turned on.

The user only has access to their applications or those matching their tags.


text:
# Paretheses should match same height on single line in multiple arrays

Given is following formula:

$a^{(k)}=\left(\begin{array}{c} a_1^{(k)}\\ \vdots\\ a_n^{(k)} \end{array}\right)\to\left(\begin{array}{c} a_1\\ \vdots\\ a_n \end{array}\right)=a$

The first vector is a little bit more height than the second one. Is it possible to have the same height for all four parentheses?

-

\documentclass{article}
$a^{(k)}=\left(\begin{array}{c} a_1^{(k)}\\ \vdots\\ a_n^{(k)} \end{array}\right)\to\left(\begin{array}{c} a_1^{\vphantom{(k)}}\\ \vdots\\ a_n^{\vphantom{(k)}} \end{array}\right)=a$

One should also consider the pmatrix environment provided by amsmath, which gives better spacing for the big parentheses. – egreg May 4 '12 at 12:27


text:
Hey there,

I have a small service server node that also happens to have another std::thread in it (i.e. serial communication). The cb_function of the service is using a member-function through std::bind, as suggested by this answere here.

The first thing I do in the constructor of the node-class is to setup the service by this->create_service(...). When I perform a rclcpp::spin_some(this->get_node_base_interface()) afterwards I see the service via ros2 service list for a very short while (just one spin, so very expected).

I end my constructor by this:

RCLCPP_WARN(this->get_logger(), "starting Thread");
RCLCPP_WARN(this->get_logger(), "Thread is running -> should spin now");

As the node only has one service, I cannot further veryfy if it really runs. I see parameter-services from my node, but not the actual service.

When I include this at the very end of my constructor, it operates as expected:

while (rclcpp::ok())
{
rclcpp::spin_some(this->get_node_base_interface());
}

The main from which my node is constructed looks like this:

int main(int argc, char ** argv)
{
rclcpp::init(argc, argv);
rclcpp::NodeOptions options;
auto node = std::make_shared<myPackage::MyClass>(options);
rclcpp::spin(node->get_node_base_interface());
rclcpp::shutdown();

return 0;
}

Is there any obviouse I'm doing wrong? I'm thinking about using a MultiThreadedExecutor, but I only thought that I would use it for responding to multiple Action/Service Requests from the same node.

Kind regards :)

Edit: I use galactic on ROS2 inside an ARM docker-container

edit retag close merge delete


text: [10] Esters are responsible for the aroma of many fruits, including apples, durians, pears, bananas, pineapples, and strawberries. Leen C. Verhagen, in Comprehensive Natural Products II, 2010. Ester names are derived from the parent alcohol and the parent acid, where the latter may be organic or inorganic. Esters are formed by reaction of acids with alcohols: Esters are generally safe to use, with low toxicity (see exception of methyl salicylate and sabinyl acetate found in Spanish sage). Found in essential oils of bergamot, lavender, lavandin, spike lavender, Clary sage, neroli and petitgrain. CAUTION: Methyl salicylate makes up 90% of essential oils of wintergreen and birch. It is well known that acetate esters are synthesized by an enzyme alcohol-acyl-transferase (AAT) which catalyzes the esterification of volatile alcohols with acetyl CoA molecules to produce volatile esters and free CoA-SH (Salas, 2004). Answer. Poly(ethylene terephthalate) is produced by the transesterification of dimethyl terephthalate and ethylene glycol:[12], A subset of transesterification is the alcoholysis of diketene. Have questions or comments? The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The carbonylation of methanol yields methyl formate, which is the main commercial source of formic acid. Esters form the most widespread group of compounds in plant essences and fragrances. Qian et al. (2013) prepared phage-displayed TGN peptide-decorated polymeric micelle-like polyplexes based on PEG-PDMAEMA for efficient brain-targeted gene delivery. Mobile phase A consists of 15 mM phosphate potassium buffer, pH 3, and mobile phase B consists of methanol. Alkaline hydrolysis is rapid but requires care in application as the MCPD formed is rapidly dechlorinated in alkaline conditions. The C–H bonds adjacent to the carbonyl are weakly acidic but undergo deprotonation with strong bases. acidic hydrolysis: carboxylic acid + alcohol; basic hydrolysis: carboxylate salt + alcohol, basic hydrolysis: completion; acidic hydrolysis: incomplete reaction, $$\mathrm{(CH_3)_2CHCOOCH_2CH_3 + H_2O \overset{H^+}{\rightleftharpoons}}$$. Floral, fruity, sweet and herbaceous odour. [20], Direct reduction to give the corresponding ether is difficult as the intermediate hemiacetal tends to decompose to give an alcohol and an aldehyde (which is rapidly reduced to give a second alcohol). Asked by Wiki User. Because of their lack of hydrogen-bond-donating ability, esters do not self-associate. By continuing you agree to the use of cookies. D) 2nd order reaction with molecularity 1. Consequently, esters are more volatile than carboxylic acids of similar molecular weight.[5]. Found in geranium oil. Another method for the dehydration of mixtures of alcohols and carboxylic acids is the Mitsunobu reaction: Carboxylic acids can be esterified using diazomethane: Using this diazomethane, mixtures of carboxylic acids can be converted to their methyl esters in near quantitative yields, e.g., for analysis by gas chromatography. Alcohols react with acyl chlorides and acid anhydrides to give esters: The reactions are irreversible simplifying work-up. Esters of propionic acid are produced commercially by this method: A preparaton of methyl propionate is one illustrative example. Esterification is a reversible reaction. This process is the one that usually initiates condensation reactions. Joseph P.Y. In the first, the esters are transesterified or hydrolyzed to release free MCPD which is determined by any of the methods described above, and in the second method, the intact esters are determined by high performance liquid chromatography with MS (LCMS). Esters are generally identified by gas chromatography, taking advantage of their volatility. Vol. Acid hydrolysis is relatively slow and has concerns regarding the possible formation of MCPD during the reaction. (Benzyl acetate is not a terpene derivative but is found in many essential oils.). The alcohol portion of the ester ends up as the free alcohol. Uses and Application. Authentic 3-nitro-BTBE and 3,3′-di-BTBE are used as standards. Chromatographic conditions are as follows: flow, 1 ml/min; 75% mobile phase B for 25 min, followed by a linear increase to 100% mobile phase B for 10 min, which is essential for column reconstitution and elution of higher oxidation states of BTBE polymerization products and phospholipids. Transesterification, which involves changing one ester into another one, is widely practiced: Like the hydrolysation, transesterification is catalysed by acids and bases. [15] In difficult cases, the silver carboxylate may be used, since the silver ion coordinates to the halide aiding its departure and improving the reaction rate. DCC (dicyclohexylcarbodiimide) is used to activate the carboxylic acid to further reaction. The reaction is widely used for degrading triglycerides, e.g. In a saponification reaction, the base is a reactant, not simply a catalyst. From: Clinical Aromatherapy (Third Edition), 2015, C.W. Direct determination is made difficult by the fact that oils are not easily soluble in the relatively polar solvents required for LC, and the fact that the esters do not ionize easily. Y. Li, D. Ju, in Neurotoxicity of Nanomaterials and Nanomedicine, 2017. The analogous acylations of amines to give amides are less sensitive because amines are stronger nucleophiles and react more rapidly than does water. Volatile esters are important for beer flavor, in particular ethyl acetate and 2- and 3-methylbutyl acetate (often called iso-amyl acetate) because these esters are present in concentration well above their flavor threshold value. Alkaline hydrolysis is rapid but requires care in application as the MCPD formed is rapidly dechlorinated in alkaline conditions. Benzaldehyde reacts with sodium benzyloxide (generated from sodium and benzyl alcohol) to generate benzyl benzoate. The H of HOH joins to the oxygen atom in the OR part of the original ester, and the OH of HOH joins to the carbonyl carbon atom: The products are butyric acid (butanoic acid) and ethanol. Since esterification is highly reversible, the yield of the ester can be improved using Le Chatelier's principle: Reagents are known that drive the dehydration of mixtures of alcohols and carboxylic acids.


text:
# 【题目】Average and Median

AtCoder Beginner Contest 236

E - Average and Median

Time Limit: 2 sec / Memory Limit: 1024 MB

Score : $500$ points

### Problem Statement

We have $N$ cards. The $i$-th card $(1 \leq i \leq N)$ has an integer $A_i$ written on it.

Takahashi will choose any number of cards from these. However, for each $i$ $(1 \leq i \leq N - 1)$, at least one of the $i$-th and $(i+1)$-th cards must be chosen.

Find the following values.

• The maximum possible average of the integers written on the chosen cards
• The maximum possible median of the integers written on the chosen cards

Here, the median of the $n$ integers is defined to be the $\lceil \frac{n}{2} \rceil$-th smallest of them, where $\lceil x \rceil$ is the smallest integer not less than $x$.

### Constraints

• $2 \leq N \leq 10^5$
• $1 \leq A_i \leq 10^{9}$
• All values in input are integers.

### Input

Input is given from Standard Input in the following format:

$N$
$A_1$ $\ldots$ $A_N$

### Output

Print two lines. The first and second lines should contain the maximum possible average and maximum possible median of the integers written on the chosen cards, respectively. For the average, your output will be considered correct when its relative or absolute error from the exact answer is at most $10^{-3}$.

6
2 1 2 1 1 10

### Sample Output 1

4
2

Choosing the $2$-nd, $4$-th, and $6$-th cards makes the average of the written integers $\frac{12}{3} = 4$, which is the maximum possible.

Choosing the $1$-st, $3$-rd, $5$-th, and $6$-th cards makes the median of the written integers $2$, which is the maximum possible.

7
3 1 4 1 5 9 2

### Sample Output 2

5.250000000
4

For the average, your output may contain some degree of error: for example, the output $5.2491$ is still considered correct. For the median, however, the exact value must be printed.

### 我的笔记

#### 用动态规划选择卡片

$f[i]$ 表示在 $1\sim i$ 范围选择卡片，且选择第 $i$ 张卡片的和。
$g[i]$ 表示在 $1\sim i$ 范围选择卡片，且不选第 $i$ 张卡片的和。

$f[i]=\max(f[i-1],g[i-1])+B_i$
$g[i]=f[i-1]$

### 源码

#include <bits/stdc++.h>
#define SIZE 100010
#define esp 1e-4

using namespace std;

int N;
int A[SIZE];

int main(void)
{
ios::sync_with_stdio(false);
cin >> N;
for (int i = 1; i <= N; i++)
{
cin >> A[i];
}
double l1 = 1, r1 = 1e9;
while (r1 - l1 > esp)
{
double mid = (l1 + r1) / 2;
double f[SIZE], g[SIZE];
f[0] = g[0] = 0;
for (int i = 1; i <= N; i++)
{
f[i] = max(f[i - 1], g[i - 1]) + A[i] - mid;
g[i] = f[i - 1];
}
if (max(f[N], g[N]) >= 0)
{
l1 = mid;
}
else
{
r1 = mid;
}
}
cout << l1 << endl;
int l2 = 1, r2 = 1e9;
while (l2 < r2)
{
int mid = (l2 + r2 + 1) / 2;
int f[SIZE], g[SIZE];
f[0] = g[0] = 0;
for (int i = 1; i <= N; i++)
{
f[i] = max(f[i - 1], g[i - 1]) + ((A[i] >= mid) ? 1 : -1);
g[i] = f[i - 1];
}
if (max(f[N], g[N]) > 0)
{
l2 = mid;
}
else
{
r2 = mid - 1;
}
}
cout << l2 << endl;
return 0;
}


text:
Popular

# Magnetism in solids by D. H. Martin

Written in English

## Subjects:

• Solids -- Magnetic properties

Edition Notes

Bibliography: p. 437-443.

## Book details

Classifications The Physical Object Statement [by] D. H. Martin. LC Classifications QC753 .M32 1967a Pagination 452 p. Number of Pages 452 Open Library OL5996774M LC Control Number 66029138

This book presents a phenomenological approach to the field of solid state magnetism. After introducing the basic concepts from statistical thermodynamics and electronic structure theory, the first part discusses the standard models for localized moments (Weiss, Heisenberg) and Cited by: Magnetism in Solids [D.

Martin] on *FREE* shipping on qualifying by: Supplemented with numerous Magnetism in solids book, solutions, and tables as well as many suggestions for further reading, Magnetism and Magnetic Resonance in Solids is an excellent text for senior-undergraduate and graduate courses in physics, materials science, and engineering.

Get Textbooks on Google Play. Rent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone. This book presents a phenomenological approach to the field of solid state magnetism.

After introducing the basic concepts from statistical thermodynamics and electronic structure theory, the first part discusses the standard models for localized moments (Weiss, Heisenberg) and Brand: Springer-Verlag Berlin Heidelberg.

Cite this chapter as: Viola Kusminskiy S. () Magnetism in Solids. In: Quantum Magnetism, Spin Waves, and Optical Cavities. SpringerBriefs in : Silvia Viola Kusminskiy. The Theory of Magnetism is an important subfield of the theory of solids, and has guided a long history of research into the phenomenon of book provides the foundation for further development in this field.

Its contents is divided into four parts. In Part I, Magnetic Ions in Crystals, the electron states of magnetic ions in crystals and the exchange interactions between them are Brand: Springer-Verlag Berlin Heidelberg.

Instructors may use magnetism as a case study in statistical mechanics, but it really depends on the instructor and the textbook adopted. The one course that is most likely to talk about magnetism in some depth is solid state physics.

But from my experience, one semester of solid state physics typically does not cover by:   This book presents a phenomenological approach to the field of solid state magnetism. It surveys the various theories and discusses their applicability in different types of materials.

The text will be valuable as a text for graduate courses in magnetism and magnetic materials. In a metal, as we increase $$\mathbf B$$, the Landau level at $$k_z=0$$ passes through the Fermi energy $$\zeta$$ and the internal energy abruptly decreases.

Many physically observable properties of the system are periodic functions of the magnetic field. The periodic oscillation of the diamagnetic susceptibility of a metal at low temperatures is known as the de Haas–van Alphen : John J.

Quinn, Kyung-Soo Yi. Magnetism, Volume I: Magnetic Ions in Insulators: Their Interactions, Resonances, and Optical Properties summarizes the understanding of magnetically ordered materials.

This book contains 12 chapters that specifically tackle the concepts of ferromagnetism, ferrimagnetism, and antiferromagnetism.

Magnetism in Solids - Free download as Powerpoint Presentation .ppt), PDF File .pdf), Text File .txt) or view presentation slides online. magnetism. The invisible forces of magnetism are at work all around us, even as you read this sentence.

To understand this concept, and how magnets work, can be difficult for students. These books about magnets will help make this concept more concrete. In addition to the Science Stations on Magnetism. Get this from a library. Magnetism in solids. [D H Martin] COVID Resources.

Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist.

Mattis [3],[4] Mattis [3] is a two-volume book; the ﬁrst Magnetism in solids book deals mainly with models of magnetism, and the second with ﬁnite temper-atures. It contains an illuminating and entertaining discussion of the history of magnetism, as well as a careful discussion of conditions un-der which ferromagnetism can be possible.

Mattis [4] is an updated,File Size: KB. COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.

Magnetism of Solids. Authors; Authors and affiliations; Hanmin Jin; Terunobu Miyazaki; Chapter. First Online: 23 August k Downloads; Part of the Springer Series in Materials Science book series (SSMATERIALS, volume ) : Hanmin Jin, Terunobu Miyazaki.

The Theory of Magnetism is an important subfield of the theory of solids, and has guided a long history of research into the phenomenon of magnetism. This book provides the foundation for further Reviews: 1.

The aim of this book is to provide a simple, up-to-date introduction to the study of solid state magnetism, both intrinsic and technical. It is designed to meet the needs and interests of advanced undergraduate students reading physics; of postgraduates in physical and materials sciences and in engineering; and also those of the practising Cited by: Theory of Magnetism: Magnetism, the power of attracting iron by a material, is known to mankind for centuries before Christ.

The oldest magnetic material or simply magnet, so called magnetite (Fe 3O 4) is a mineral was initially found in the district of Magnesia of the modern Turkey. The word magnet is a Greek word and known from the name of. White and Geballe wrote Long Range Order in Solids, a book that treats magnetism and superconductivity as well as other cooperative phenomena from a general point of view.

He spent sabbaticals at the Cavendish Laboratory with Sir Neville Mott, the Ecole Polytechnique, and the Max Planck Institute in : Springer-Verlag Berlin Heidelberg.

Magnetism in Crystalline Materials covers the applications of the theory of groups of cambiant symmetry and the physical properties of magnetically ordered crystalline solids.

This book is divided into eight chapters; the first of which discusses the macroscopic properties of magnetic Edition: 1. This, coupled with Magnetic moments 1 the diversity of types of magnetic interactions that can be found, leads to Classical mechanics and a surprisingly rich variety of magnetic properties in real systems.

The plan magnetic moments 6 of this book. Magnetic phenomena are universal in nature. Vast, distant galaxies, the tiny invisible atoms, men and beasts all are permeated through and through with a host of magnetic fields from a variety of sources.

The earth magnetism predates human Size: KB. Thus magnetism at rst referred to the long-range interaction between ferromagnetic bodies. Indeed, the present course will mainly address magnetic order in solids, of which ferromagnetism is the most straight-forward case.

This begs the question of what it is that is ordering in a ferromagnet. Oersted () found that a compass needle is deFile Size: KB.

This book presents a phenomenological approach to the field of solid state magnetism. After introducing the basic concepts from statistical thermodynamics and electronic structure theory, the first part discusses the standard models for localized moments (Weiss, Heisenberg) and.

Science 01 Dec Vol. Issuepp. DOI: /scienceaAuthor: Stanley Engelsberg. •Solid State Physics, 2nd ed by J. Hook and H. Hall, Wiley This is frequently the book that students like the most. It is a ﬁrst introduction to the subject and is much more introductory than Ashcroft and Mermin.

•The Solid State, by H M Rosenberg, OUP This slightly more advanced book was written a few decades ago to cover what was. Vettier, in Encyclopedia of Materials: Science and Technology, Magnetism in solids covers a large variety of properties ranging from bulk characteristics of permanent magnets or recording and storage devices to fundamental microscopic phenomena in quantum systems.

The measurement of bulk magnetic properties (such as susceptibility, magnetization, and anisotropy) represents the first. THE PHYSICS OF MAGNETISM BACKGROUND: Read chapters on magnetism from your favorite college physics book for review.

Paleomagnetism is the study of the magnetic properties of rocks. It is one of the most broadly applicable disciplines in geophysics, having uses in diverse ﬁelds such as geo. The ground state of an atomor ion with an incomplete shell is determined by Hund’s rules: The ground state has the maximum S consistent with the Pauli exclusion principle.

It has the maximum L consistent with the maximum spin multiplicity 2S + 1 of Rule 1. The J-value is given by L − S when a shell is less than half filled and by L + S when more than half : John J.

Quinn, Kyung-Soo Yi. This book presents a phenomenological approach to the field of solid state magnetism. It surveys the various theories and discusses their applicability in different types of materials. Chapter 12 Thermodynamics and Magnetism Magnetism in Solids Lodestones { fragments of magnetic1 FeO+Fe 2 O 3 (Fe 3 O 4) { although known to the ancients were, according to Pliny the Elder, rst formally described in Greek 6th century B.C.E.

writings. By that time they were already the stu of myth. BSc Sem V & VI Recommended Text-Books. Quantum Mechanics, Special Theory of Relativity & Electromagnetic waves comes under the first book. On the other hand, Atomic/Molecular Physics & Subatomic Physics covers the second book of BSc 3rd year.

Last but not least the third book covers Basic Electronic & Analog/Digital of the best books for each of the three. Electricity and Magnetism Notes.

This lecture note covers the following topics: Coulomb's law, superposition, energy of a system of charges, Basic field concept, flux, Gauss's law, Fields and potentials around conductors, the electrostatic uniqueness theorem,RC circuits, Thevenin equivalence, Forces and fields in special relativity.

Solid: Substances which have definite shape and volume, high density and are not easily compressible are defined as inter- molecular attraction (forces that operate within the particles such as atoms, ions etc) in solid is very strong and it holds the particles of the substance close to each other.

Band Theory of Magnetism in Metals. Application of band theory to magnetic materials was first performed by Stoner, Mott & Slater () • Free atoms vs solids • Density of States • Ferromagnetism Electronic structure of free atoms vs solids 1 File Size: KB. This book presents both experimental and theoretical aspects of topology in magnetism.

It first discusses how the topology in real space is relevant for a variety of magnetic spin structures, including domain walls, vortices, skyrmions, and dynamic excitations, and then focuses on the phenomena that are driven by distinct topology in reciprocal momentum space, such as anomalous and spin Hall.

Magnetism Magnetic materials play critical roles in our daily lives. Magnetism was first discovered by the ancient Greeks and used by the Chinese to create a “south pointing” compass (Mattis, ).

Since the invention of the compass, the number of devices that File Size: 1MB. Solid state magnetism is important and attempts to understand magnetic properties have led to an increasingly deep insight into the fundamental make up of solids.

Both experimental and theoretical research into magnetism continue to be very active, yet there is still much ground to cover before there can be a full understanding. Magnetism in Crystalline Materials covers the applications of the theory of groups of cambiant symmetry and the physical properties of magnetically ordered crystalline solids.

This book is divided into eight chapters; the first of which discusses the macroscopic properties of magnetic crystals.The superb book describes the modern theory of the magnetic properties of solids.

Starting from fundamental principles, this copiously illustrated volume outlines the theory of magnetic behaviour, describes experimental techniques, and discusses current research topics.4/5(18).Cengage Electricity and Magnetism ; Cengage Optics and Modern Physics; This book covers all the topics in electricity and magnetism for JEE Mains and Advanced.

Electricity and Magnetism are the topics which can be understood easily and is actually one of the scoring topics in JEE Mains and Advanced.

This book has a total of pages.

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# Probability Problem choosing $6$

If you have $10$ red marbles and $15$ green marbles in a bag, and you randomly choose $6$, what is the probability that $5$ red marbles will be selected?

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## 2 Answers

$$p(A)=\frac{\binom{10}5.\binom{15}1}{\binom{25}6}$$

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HINT: How many ways are there to choose exactly $5$ red marbles? You may choose any of the $\binom{10}5$ sets of $5$ red marbles and any of the $15$ green marbles to fill out the half dozen, so there are $15\binom{10}5$ ways to get exactly $5$ red marbles. This is the number of succesful outcomes.

• How many (equally likely) possible outcomes are there altogether when you choose $6$ marbles from a set of $25$?

• How do you combine these two numbers to get the probability of a successful outcome?

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# A median of an AVL. How to take advantage of the AVL?

Here is the source of my question.

Given a self-balancing tree (AVL), code a method that returns the median.

(Median: the numerical value separating the higher half of a data sample from the lower half. Example: if the series is

2, 7, 4, 9, 1, 5, 8, 3, 6

then the median is 5.)

I can offer the following solution:

1. Traverse the given tree, return the number of elements.
2. Traverse n / 2 + 1 (if n is odd) the tree again applying an in-order tree walk. The value of the n / 2 + 1th element is the median.

But I can do it with a binary search tree, can't I? Is there a better algorithm for an AVL?

• Usually, search algorithms for a Binary Search Tree work with an AVL tree, but with an AVL you get the extra guarantee that your tree's height is logarithmic in the number of nodes. Apr 17 '15 at 9:29

If you modify the AVL tree by storing the size of the subtree at each node rather than just its height, then you can find the median in time $O(\log n)$ using the fact that the tree is balanced. To accomplish this, you write a more general procedure Select which accepts a node $v$ and a number $k$, and finds the $k$th smallest node at the subtree rooted at $v$.

Suppose that the left subtree (if any) has $L$ nodes. If $k \leq L$ then we recurse to the left subtree. If $k = L+1$ then we return $v$. Otherwise we recurse to the right subtree, reducing $k$ by $L+1$.

The running time of this algorithm is linear in the height of the tree, which is $O(\log n)$.

• Could you please give me an example? Apr 18 '15 at 10:05
• No, you'll have to construct one yourself. Try to understand what my algorithm is trying to accomplish and how. Apr 18 '15 at 14:40
• My solution is on codereview.stackexchange.com/q/104525/23821 Sep 12 '15 at 20:49
• Can you explain the logic behind this algorithm , I checked this method works but not able to figure out logic behind this . Specifically this part "we recurse to the right subtree, reducing k by L+1." Why aren't we reducing k when recursing left subtree . While the first two cases k ≤ L and k = L+1 are quite clear . Apr 12 at 2:02
• If $L=5$ then the seventh smallest element is the smallest element in the right subtree, with the third smallest element is the third smallest element in the left subtree. Can you see why? Apr 12 at 6:04

AVL is binary search tree with some special property: it is self-balancing tree. It's height is always logarithmic. Ordinary binary tree in some worst scenario can be linked list (if you add sorted data) so it's height is n. AVL tree in worst scenario is fibonacci tree.